Split (2)

train · ~13.5M rows (showing the first 104k)

text stringlengths 4 2.78M
--- abstract: 'We investigate the effect of the bulk contents in the DGP braneworld on the evolution of the universe. We find that although the pure DGP model cannot accommodate the transition of the effective equation of state of dark energy, once the bulk matter $T^5_5$ is considered, the modified model can realize the $w_{eff}$ crossing $-1$. However this transition of the equation of state cannot be realized by just considering bulk-brane energy exchange or the GB effect while the bulk matter contribution is not included. $T^5_5$ plays the major role in the modified DGP model to have the $w_{eff}$ crossing $-1$ behavior. We show that our model can describe the super-acceleration of our universe with the equation of state of the effective dark energy and the Hubble parameter in agreement with observations.' author: - 'Shaoyu Yin, Bin Wang' - Elcio Abdalla - 'Chi-Yong Lin' title: The transition of equation of state of effective dark energy in the DGP model with bulk contents --- Introduction ============ The accelerated expansion of our universe is one of the most important discovery in the last decade [@Riess:1998; @Perlmutter:1999; @Riess:2004], having triggered plenty of efforts to understand and explain it. This phenomenon is in conflict with our common sense about attractive gravity. Within the framework of general relativity, the acceleration is attributed to the mysterious “dark energy" existing in our universe. The theoretical nature and origin of this dark energy are a source of much debate. Candidates suggested for this dark energy can be classified according to the behavior of their respective equation of state $w=P/\rho$. The cosmological constant, with $w =-1$, is located at a central position among dark energy models both in theoretical investigation and in data analysis [@Weinberg:1989]. In quintessence [@Caldwell:1998; @et; @al], Chaplygin gas [@Kamenshchik:2001] and holographic dark energy models [@Li:2004], $w$ always remains bigger than $-1$. The phantom models of dark energy have $w <-1$ [@Caldwell; @et; @al]. Recent more accurate data analysis tells us a dramatic result, namely that the time varying dark energy gives a better fit than a cosmological constant and in particular, $w$ can cross $-1$ around $z = 0.2$ from above to below [@Alam; @et; @al]. Theoretical attempts towards the understanding of the $w$ crossing $-1$ phenomenon have been suggested, including the model containing a negative kinetic scalar field and a normal scalar field [@Feng; @et; @al], a single scalar field model [@Li:2005], interacting holographic dark energy models [@Wang; @et; @al] and others [@Nojiri; @et; @al]. An alternative approach which does not need dark energy to explain the late-time acceleration is motivated by string theory via the brane-world scenarios. In this scenario our universe is a 3-d brane embedded in a space-time with extra dimensions. The cosmological evolution on the brane is described by an effective Friedmann equation incorporating non-trivially with the effects of the bulk onto the brane. The presence of the 5-d matter can interact with the matter contents on the brane and alter the cosmic expansion leading to a behavior resembling the dark energy. The cosmic evolution of the Randall-Sundrum(RS) braneworld [@Randall:1999] with energy exchange between brane and bulk has been studied [@Kirstsis; @et; @al; @Cai:2006; @Apostolopoulos:2006; @Bogdanos:2006two; @Sheykhi:2007]. In these models, due to the energy exchange between the bulk and the brane, the usual energy conservation law on the brane is broken and consequently it was found that the equation of state of the effective dark energy can experience the transition behavior [@Cai:2006; @Apostolopoulos:2006; @Bogdanos:2006two; @Sheykhi:2007]. In string theory, in addition to the Einstein action, some higher derivative curvature terms have been included to derive gravity. The combination of the Einstein-Hilbert and Gauss-Bonnet(GB) term constitutes, for 5D spacetimes, the most general Lagrangian to produce second-order field equations . The GB correction changes the bulk field equations and modifies the braneworld Friedmann equation. It influences the evolution of the universe in our brane. Effects of the GB correction on the RS braneworld have been studied in [@Kofinas:2003; @Sheykhi:2007]. In this paper we are going to concentrate on another braneworld model introduced by Dvali, Gabadadze and Porrati (DGP) [@Dvali:2000], where the braneworld is embedded in the flat bulk with infinite extra dimensions. Considering that the graviton propagates into the extra dimension, and at large scale, gravity can become weaker due to its leakage, the DGP model can realize the accelerated expansion naturally. However for the pure DGP model, its effective equation of state never goes down to the phantom phase. Our main motivation here is to investigate the effects of the bulk contents in the DGP braneworld on the evolution of the universe and explore the possibility of the transition of equation of state if there are contributions from the bulk-related energy-momentum tensor components which has been observed in RS model[@Kirstsis; @et; @al; @Cai:2006; @Apostolopoulos:2006; @Bogdanos:2006two; @Sheykhi:2007]. The DGP model only with $T^0_5$ has been investigated in [@Kofinas:2006]. We are going to present a systematic and complete examination of the bulk effects including $T^0_5$ and $T^5_5$ terms on DGP model. Besides we will also study the modification on the brane evolution due to the GB correction together with bulk related energy-momentum tensor components. Influences of the GB correction on the pure DGP braneworld have been studied in [@Brown:2006; @Cai:2005]. Although the effects of the GB correction term on the late time universe is small, we will see that it still plays an important role in the early time cosmic evolution. We will show that in the DGP model the bulk matter contribution $T^5_5$ plays a major role in accommodating the transition of equation of state, while the $T^0_5$ and GB correction alone cannot present a profile of the $w_{eff}$ crossing $-1$ phenomenon found by observations. The organization of the paper is the following: in section II we will give out the basic equation sets for the DGP model by considering different correction terms respectively. The bulk effects due to $T^5_5$ term will be shown in detail in section III. In section IV, we will consider the influence of the energy flow $T^0_5$ on the brane universe evolution. Conclusions and discussions will be presented in the last section. General equations for DGP model with GB correction ================================================== The DGP brane model with GB correction starts from the action $$S=-\frac{1}{2\kappa^2}\int d^5X\sqrt{-g}(R-2\Lambda_5+\alpha L_{GB}) -\frac{1}{2\mu^2}\int d^4x\sqrt{-\widetilde{g}}(\widetilde{R}-2\Lambda_4)+\int d^5\sqrt{-g}L_{EM},$$ where $\kappa$ and $\mu$ are related to the gravitational constants and the Planck masses for the bulk and brane as [@Deffayet:2001]: $$\kappa^2=8\pi G_{(5)}=M_5^{-3};\qquad\mu^2=8\pi G_{(4)}=M_4^{-2},$$ respectively, $\Lambda_5$ and $\Lambda_4$ are cosmological constants for the bulk and brane. $L_{EM}$ is the energy-momentum tensor and $L_{GB}$ is the GB correction term in the form $$L_{GB}=R^2-4R^{AB}R_{AB}+R^{ABCD}R_{ABCD}.$$ $\alpha$ is the coefficient of the GB term, which is positive, as required by string theory and is generally considered to be very small. If we take $\alpha=0$, Eq.(1) reproduces the pure DGP model [@Deffayet:2001]. Throughout the paper the capital letter are used to present the 5-d indices, while the Greek alphabet is used for 4-d brane case. From the action one can obtain the field equation $$G_{AB}+\Lambda_5g_{AB}+2\alpha H_{AB}=\kappa^2\{T_{AB}-[\frac{1}{\mu^2}(\widetilde{G}_{\mu\nu}+ \Lambda_4\widetilde{g}_{\mu\nu})+\widetilde{T}_{\mu\nu}]\delta(y_b)\delta_A^{\mu}\delta_B^{\nu}\},$$ where $H_{AB}=RR_{AB}-2R^C_AR_{BC}-2R^{CD}R_{ACBD}+R^{CDE}_AR_{BCDE}-\frac{1}{4}g_{AB}L_{GB}$ is the second-order Lovelock tensor [@Lovelock:1971], $\delta(y_b)$ comes from the difference between the integration with 4-d metric and 5-d metric. The energy-momentum tensor on the brane is assumed to be that of a perfect fluid, $$\widetilde{T}_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+p\tilde{g}_{\mu\nu},$$ where $u_{\mu}$, $\rho$ and $p$ are the fluid velocity, energy density and pressure, respectively ($c=1$ is used). The non-zero components related to the fifth dimension in the bulk energy-momentum tensor are supposed to be $T^0_5$ and $T^5_5$, whose role in the accelerated expansion will be studied in detail. Generally, the metric in 5-d brane cosmology is written as $$ds^2=-n^2(t,y)dt^2+a^2(t,y)\gamma^{ij}dx_idx_j+b^2(t,y)dy^2,$$ where $y$ stands for the extra dimension orthogonal to the brane, and $\gamma^{ij}$ is the maximally symmetric 3-d tensor. Then $\sqrt{-g}=b\sqrt{-\widetilde{g}}$, thus $\delta(y_b)=\frac{\delta(y)}{b}$. From this metric the Einstein equation can be obtained directly. According to Eq.(4) one can obtain the Einstein tensor components as [@Barcelo:2003] $$\begin{aligned} G_{tt}&=&3[n^2\Phi+\frac{\dot{a}}{a}\frac{\dot{b}}{b}-\frac{n^2}{b^2}(\frac{a''}{a}-\frac{a'}{a}\frac{b'}{b})],\nonumber\\ G_{ty}&=&3(\frac{\dot{a}}{a}\frac{n'}{n}+\frac{a'}{a}\frac{\dot{b}}{b}-\frac{\dot{a}'}{a}),\nonumber\\ G_{ij}&=&\frac{a^2}{b^2}\gamma_{ij}[\frac{a'}{a}(\frac{a'}{a}+2\frac{n'}{n}) -\frac{b'}{b}(\frac{n'}{n}+2\frac{a'}{a})+2\frac{a''}{a}+\frac{n''}{n}]\nonumber\\ &&-\frac{a^2}{n^2}\gamma_{ij}[\frac{\dot{a}}{a}(\frac{\dot{a}}{a}-2\frac{\dot{n}}{n}) -\frac{\dot{b}}{b}(\frac{\dot{n}}{n}-2\frac{\dot{a}}{a})+2\frac{\ddot{a}}{a}+\frac{\ddot{b}}{b}]-k\gamma_{ij},\nonumber\\ G_{yy}&=&3[-b^2\Phi+\frac{a'}{a}\frac{n'}{n}-\frac{b^2}{n^2}(\frac{\ddot{a}}{a}-\frac{\dot{a}}{a}\frac{\dot{n}}{n})],\nonumber\\ H_{tt}&=&6\Phi[\frac{\dot{a}}{a}\frac{\dot{n}}{n}+\frac{n^2}{b^2}(\frac{a'}{a}\frac{b'}{b}-\frac{a''}{a})],\nonumber\\ H_{ty}&=&6\Phi(\frac{\dot{a}}{a}\frac{n'}{n}+\frac{a'}{a}\frac{\dot{b}}{b}-\frac{\dot{a}'}{a}),\nonumber\\ H_{ij}&=&2a^2\gamma_{ij}\{\Phi[\frac{1}{n^2}(\frac{\dot{n}}{n}\frac{\dot{b}}{b}-\frac{\ddot{b}}{b}) -\frac{1}{b^2}(\frac{n'}{n}\frac{b'}{b}-\frac{n''}{n})]\nonumber\\ &+&\frac{2}{a^2bn}[\frac{\dot{a}^2\dot{b}\dot{n}}{n^4} +\frac{a'^2b'n'}{b^4}+\frac{\dot{a}a'}{b^2n^2}(b'\dot{n}-\dot{b}n')]\nonumber\\ &-&2[\frac{1}{n^2}\frac{\ddot{a}}{a}(\frac{1}{n^2}\frac{\dot{a}}{a}\frac{\dot{b}}{b}+\frac{1}{b^2}\frac{a'}{a}\frac{b'}{b}) +\frac{1}{b^2}\frac{a''}{a}(\frac{1}{n^2}\frac{\dot{a}}{a}\frac{\dot{n}}{n}+\frac{1}{b^2}\frac{a'}{a}\frac{n'}{n})]\nonumber\\ &+&\frac{2}{b^2n^2}[\frac{\ddot{a}}{a}\frac{a''}{a}-\frac{\dot{a}^2}{a^2}\frac{n'^2}{n^2}- \frac{a'^2}{a^2}\frac{\dot{b}^2}{b^2}-\frac{\dot{a}'}{a}(\frac{\dot{a}'}{a}-2\frac{\dot{a}}{a}\frac{n'}{n}- 2\frac{a'}{a}\frac{\dot{b}}{b})]\},\nonumber\\ H_{yy}&=&6\Phi[\frac{a'}{a}\frac{n'}{n}+\frac{b^2}{n^2}(\frac{\dot{a}}{a}\frac{\dot{n}}{n}-\frac{\ddot{a}}{a})],\end{aligned}$$ where $$\Phi=\frac{1}{n^2}\frac{\dot{a}^2}{a^2}-\frac{1}{b^2}\frac{a'^2}{a^2}+\frac{k}{a^2}.$$ The dot denotes a derivative with respect to $t$, and the prime the derivative with respect to $y$. Without loosing generality, in the brane world scenario, one usually chooses the metric function $b(t,y)=1$ and $n(t,0)=1$ to simplify the calculation. We choose the brane to be located at $y=0$, and suppose the metric functions to be continuous at this point, but their first derivatives are discontinuous due to the energy-momentum tensor distribution on the brane. Furthermore, the geometry is supposed to display a $Z_2$-symmetry around $y=0$, thus $a'(0_+)=-a'(0_-)$ and $n'(0_+)=-n'(0_-)$. If one integrates the field equation for the $tt$ and $ij$ components at the infinitesimal region near $y=0$, only those terms in the metric with $a''$ or $n''$ and the energy-momentum distribution on the brane can remain. Then the differences of $a'$ and $n'$ on both sides of the brane, say, $a'(0_+)-a'(0_-)\equiv2a'(0_+)$ and $n'(0_+)-n'(0_-)\equiv2n'(0_+)$ can be obtained. For simplicity, throughout the paper we use $a'$ and $n'$ to stand for $a'(0_+)$ and $n'(0_+)$, and in the equation on the brane all bulk terms are taken with their values at $y=0_+$. From $G_{tt}$, $G_{ij}$, $H_{tt}$, $H_{ij}$ in Eq.(7), we find that $a'$ and $n'$ satisfy the following equations: $$\frac{a'}{a}\{-3+4\alpha[\frac{a'^2}{a^2}-3(\frac{k}{a^2}+\frac{\dot{a}^2}{a^2})]\} =\frac{\kappa^2}{2\mu^2}[\mu^2\rho-3(\frac{k}{a^2}+\frac{\dot{a}^2}{a^2})];$$ $$n'+\frac{2a'}{a}+ 4\alpha[n'(\frac{k}{a^2}-\frac{a'^2}{a^2}+\frac{\dot{a}^2}{a^2}) +\frac{2a'}{a}(\frac{\ddot{a}}{a}-\frac{\dot{a}\dot{n}}{a})] =\frac{\kappa^2}{2\mu^2}(\mu^2p+\frac{k}{a^2}+\frac{\dot{a}^2}{a^2} -\frac{2\dot{a}\dot{n}}{a}+\frac{2\ddot{a}}{a}).$$ Generally, one can solve these equations and substitute the results of $a'$ and $n'$ into the field equation for the $ty$ and $yy$ components to obtain the continuity equation and the effective Friedmann equation. But before that, it is helpful to notice that the function $\Phi$ we introduced in Eq.(8) satisfies $$\begin{aligned} \widetilde{\Phi}'&=&\frac{2\kappa^2}{3}(\Lambda_5-T^0_0)a^3a'-\frac{2\kappa^2}{3}T^0_5a^3\dot{a},\\ \dot{\widetilde{\Phi}}&=&\frac{2\kappa^2}{3}(\Lambda_5-T^5_5)a^3\dot{a}-\frac{2\kappa^2}{3}\frac{n^2}{b^2}T^0_5a^3a',\end{aligned}$$ where $\widetilde{\Phi}\equiv(\Phi+2\alpha\Phi^2)a^4$. From (12), if $T^0_5=0$ and $T^5_5$ has a proper ansatz, such as $T^5_5=\frac{F}{\kappa^2}a^{\nu}$, $\widetilde{\Phi}$ can be obtained analytically by an integration with respect to $t$: $$\widetilde{\Phi}=\frac{\kappa^2}{6}\Lambda_5a^4-\frac{2Fa^{\nu+4}}{3(4+\nu)}+C,$$ where $C$ is an integration constant. For the case without GB term, the solution of $\Phi$ is simply $$\Phi=\frac{\kappa^2}{6}\Lambda_5-\frac{2Fa^{\nu}}{3(4+\nu)}+\frac{C}{a^4},$$ where the term $\frac{C}{a^4}$ is usually referred to the dark radiation [@Maartens:2000]. For the case with GB correction, solutions of $\Phi$ are, $$\Phi=\frac{-(4+\nu)\pm\sqrt{(4+\nu)^2}\sqrt{1+8\alpha(\frac{\kappa^2\Lambda_5}{6}+\frac{C}{a^4} -\frac{2Fa^{\nu}}{3(4+\nu)})}}{4\alpha(4+\nu)},$$ while only one solution with finite $\alpha\rightarrow0$ limit can be taken. For $\nu>-4$ the solution reads $$\Phi=\frac{-1+\sqrt{1+8\alpha(\frac{\kappa^2\Lambda_5}{6}+\frac{C}{a^4} -\frac{2Fa^{\nu}}{3(4+\nu)})}}{4\alpha}.$$ When $\alpha\rightarrow0$, Eq.(16) goes back to Eq.(14). From the definition of $\Phi$ in Eq.(8), we see that $a'$ can be expressed in terms of $\dot{a}$ once $\Phi$ is obtained. Integrating the equation of $tt$ component around $y=0$ and substituting $a'$ in terms of $\Phi$ and $\dot{a}$, we can finally arrive at the equation for the Hubble parameter $H(t)\equiv\frac{\dot{a}}{a}$, $$(H^2+\frac{k}{a^2}-\Phi)[1+\frac{8\alpha}{3}(H^2+\frac{k}{a^2}+\frac{\Phi}{2})]^2= \frac{r^2}{4}[H^2+\frac{k}{a^2}-\frac{\mu^2}{3}(\rho+\Lambda_4)]^2,$$ where $r=\kappa^2/\mu^2$ is the DGP crossover radius. For the sake of simplicity and clarity in the following discussion, we give the simplifications we are going to use. We will neglect the cosmological constant, $\Lambda_5=\Lambda_4=0$, since the effect of the cosmological constant can be included in $\rho$ and $p$. We will apply our discussion to the flat universe with $k=0$. Besides, we will employ dimensionless notations in the following calculation by defining $$\begin{aligned} x&=&\frac{H^2}{H_0^2},\nonumber\\ z&=&\frac{a_0}{a}-1,\nonumber\\ u&=&\frac{\mu^2\rho}{3H_0^2},\nonumber\\ y&=&\frac{\Phi}{2H_0^2},\nonumber\\ n&=&\frac{1}{H_0^2r^2},\nonumber\\ m&=&\frac{8H_0^2}{3}\alpha,\nonumber\\ X&=&\frac{a_0^2}{H_0^2(4+\nu)}F,\nonumber\\ \widetilde{X}&=&\frac{a_0^2}{H_0^2}F,\nonumber\\ M&=&\frac{3}{2H_0^2a_0^4}C,\end{aligned}$$ where $a_0$ and $H_0$ are the present values of the scale factor and the Hubble parameter, $z$ is the redshift and $u/x=\frac{\mu^2\rho}{3H^2}$ is the proportion of matter in the total effective energy density. Using dimensionless notations, the expression for the solution of $\Phi$ becomes $$\Phi=2H_0\frac{-X(1+z)^{-\nu}+M(1+z)^4}{3},$$ for the case without the GB correction; if the GB term is included, it reads $$\Phi=2H_0\frac{-1+\sqrt{1+2m(-X(1+z)^{-\nu}+M(1+z)^4)}}{3m}.$$ The equation for $H^2$, Eq.(17), turns into an equation for $x$ [@Cai:2005] $$4n(x-2y)[1+m(x+y)]^2=(x-u)^2.$$ If $T^0_5$ is nonzero, $\Phi$ cannot be solved analytically and we can not take advantage of the simplicity discussed above. To obtain the equation for $H^2$, we need to substitute the solutions of $a'$ and $n'$ into the equation of $yy$ component, and obtain $H(t)$ through onerous calculations. The acceleration of the scale factor can be written as $\ddot{a}\equiv a(H^2+\dot{H})$. By choosing the proper ansatz of $T^0_5$ and expressing the result of $\rho(t)$ as a function of $a(t)$, we can generally obtain the equation as a nonlinear ordinary differential equation of $H(t)$, combining with the unknown function $a(t)$. To solve such a problem in RS model [@Cai:2006; @Bogdanos:2006two] the authors introduced new effective fields related to $H(t)^2$ and separated the equation into two equations, both of which are solvable separately. But for the DGP model, due to the extra 4-d intrinsic curvature terms, the highest order of $H(t)$ is $4$, rather than $2$ in the RS model. With the GB correction, the order goes up to $8$. In Ref.[@Kofinas:2006] when just nonzero $T^0_5$ was included in the pure DGP model, the author solved the problem by introducing the concept of “fix point" and setting $\rho(t)$ and the auxiliary field time-independent. Generally, we do not hope to obtain any analytical solution for such a nonlinear ordinary differential equation. We will count more on the numerical calculations. Considering nonzero $T^0_5$ and $T^5_5$ components, our problem is general and complicated. We will present a general way to solve the problem. We have two time-dependent functions, $H(t)$ and $a(t)$, in the same equation. Considering that in the big-bang cosmology the flat universe is expanding monotonically, $a(t)$ is a monotonic function of $t$, and $H(t)$ can be written as $H(a)$. For the convenience we will use the dimensionless redshift $z$ and write $H(t)$ as $$\dot{H}(t)=-\frac{H_0^2(1+z)}{2}\frac{dx(z)}{dz}.$$ Substituting all the dimensionless notations into the equation of the $yy$ component and expressing the results until the linear order in $\alpha$, the equation for $H(t)$, or equally, $x(z)$, is $$\begin{aligned} 0&=&-mx^4+m[24n+u+(1+z)x']x^3\nonumber\\ &+&[-16n+48mn^2-12mnu+3mu^2-3m(6n+u)(1+z)x']x^2\nonumber\\ &+&\{64n^2+8nu-12mnu^2-5mu^3+[8n(1-3mn)+3mu(8n+u)](1+z)x'\}x\nonumber\\ &+&[8nu^2+2mu^4-\{16n^2+u[8n+mu(6n+u)]\}(1+z)x']\nonumber\\ &+&\frac{32n^2(1+z)^{-\nu}}{3}\widetilde{X},\end{aligned}$$ where the prime here is the derivative with respect to $z$, $x$ and $u$ are functions of $z$, and we have taken $T^5_5=\frac{F}{\kappa^2}a^{\nu}$. It is to be noted that since we don’t need to analytically integrate $T^5_5$ with the term $a^3\dot{a}$ as did in Eq.(12), we can in principle use any form of $T^5_5$ as a function of $a$. Eq.(23) is a nonlinear differential equation of $x(z)$. From the equation of the $ty$ component, assuming $b(t,y)=1$, we get $$3(1+4\alpha\Phi)(\frac{\dot{a}}{a}\frac{n'}{n}-\frac{\dot{a}'}{a})=T_{05}.$$ Taking the value of each term in this equation at $y=0_+$ and substituting the solution of $a'$ and $n'$, we can find that the left hand side of this equation is simply $\frac{1}{2}(\dot{\rho}+3H(\rho+p))$. If $T^0_5=0$, it is the conservation of energy on the brane. If $T^0_5\neq0$, it acts as the energy flow between the brane and the bulk, $$\dot{\rho}+3H(\rho+p)=-2T^0_5.$$ Here $T^0_5$ has a sign difference as compared to $T_{05}$ due to the metric term $g_{tt}\|_{y=0}=-n(t,0)=-1$. Setting the ansatz, $T^0_5=fHa^s$, the equation for $\rho$ can be solved analytically. Expressing $\rho(t)$ as $\rho(a)$, we have $$\frac{d\rho}{da}+\frac{3(1+w)\rho}{a}+2fa^{s-1}=0,$$ with the solution $$\rho=a^{-3-3w}C_1-\frac{2fa^s}{3+3w+s},$$ where $C_1$ is an integration constant. For the cold matter on the brane $w=0$, the first term on the right-hand-side is proportional to $a^{-3}$ and the second term could be attributed to the effective dark energy. Eq.(27) can be expressed by using the dimensionless notation $u(z)$, $$u(z)=P(1+z)^{-s}+\Omega_{m0}(1+z)^3,$$ where $P=-\frac{2\mu^2a_0^s}{3(3+s)H_0^2}f$, and $\Omega_{m0}=\frac{\mu^2C_1}{3H_0^2a_0^3}=\frac{8\pi G}{3H_0^2}\rho_0$ is the present ratio of conservative matter in the total energy density of the universe, $\rho_0$ is the density of the conservative matter today. There is a strong constraint on the value of dimensionless parameter $P$. Since the matter portion of the total energy density should be in the range $[0,1]$, thus $0\leq\frac{u(z)}{x(z)}\leq1$. Here we will use the fitting results on the WMAP data [@Spergel:2003] and take $\Omega_{m0}=0.28$ in our calculation. Of course, that fitting is from a different model, but the generally accepted values of $\Omega_{m0}$ are all close to this value, and the small variation of this value will not change the qualitative conclusion of our calculation. At the present moment $z=0$, we have $-0.28\leq P\leq0.72$. Any solution with $P$ out of this range is physically unreasonable. To describe the effect of the effective dark energy, we can define the effective equation of state [@Linder:2002]: $$w(z)_{eff}\equiv-1+\frac{1}{3}\frac{d\ln\delta H^2}{d\ln(1+z)},$$ where $\delta H^2\equiv H(z)^2-\Omega_{m0}(1+z)^3H_0^2$. $w_{eff}$ can be expressed by using the dimensionless parameters as $$w(z)_{eff}=-1+\frac{(1+z)\frac{dx(z)}{dz}-3\Omega_{m0}(1+z)^3}{3x(z)-3\Omega_{m0}(1+z)^3}.$$ The subscript $eff$ indicates that the effect similar to the dark energy on the brane comes from the bulk contribution. Another important quantity is the deceleration parameter $q$ $$q\equiv-\frac{\ddot{a}a}{\dot{a}^2}=\frac{1}{2}+\frac{3}{2}w(z)_{eff}(1-\frac{\Omega_{m0}(1+z)^3}{x(z)}),$$ which will also be used in the following discussion of the expansion of our universe. Calculation and Discussion without considering the $T^0_5$ term =============================================================== For the case without $T^0_5$ term, we can solve the equation Eq.(21) by substituting $\Phi$ from Eq.(20) and $u$ from Eq.(28) but with $P=0$. We can put the result into Eq.(30) and Eq.(31) to examine the behavior of the effective dark energy. In Eq.(21), if there is no GB correction, the equation for $x$ is quadratic, thus we have two solutions. For the case $\Phi=0$, they recover the two branches of pure DGP model, DGP(+) and DGP(-): $x=2n+(1+z)^3\Omega_{m0}\pm2\sqrt{n^2+n(1+z)^3\Omega_{m0}}$. Since only the DGP(+) solution has late-time self-accelerating behavior [@Brown:2006], we will concentrate our discussion on this solution. When GB correction is added, Eq.(21) becomes a cubic equation and has three roots for $x$, two of which correspond to DGP(+) and DGP(-) in $\alpha\rightarrow0$ limit, and the third one diverges when $\alpha\rightarrow0$. For comparison, we also study the solution with DGP(+) limitation for that case in this work. To compare our model description on the evolution of the universe with the observation, we have several constraints to meet. At present we have $x(z=0)\equiv H(t=0)^2/H_0^2\equiv1$. For the effective equation of state, we require $w_{eff}(z=0.2)=-1$ and $w_{eff}(z=0)=-1.06$ [@Alam; @et; @al]. We will use these constraints to refine our model parameters. Assuming $T^0_5=0$, we have parameters such as $n$ (corresponding to the DGP crossover radius), $m$ (corresponding to the GB correction), $M$ (corresponding to the dark radiation), $X$ and $\nu$ (relating to $T^5_5$ form). We will focus on whether we can accommodate the $w_{eff}$ crossing $-1$ by including bulk related energy-momentum tensor and the GB correction, which cannot be realized in the pure DGP model. Since we have the parameter $\nu$ in the exponential, the equations are not polynomial. We will use FindRoot in our calculation which will raise the problem about the choice of the initial values. To avoid the possibility of failing to find the solution, we will try different initial values in the FindRoot. We find that the solution is not strongly dependent on the initial values so that we are confident that our results are almost the whole collection of all possible solutions to the equations we deal with. To see the consistency of our results with observation, we will plot the Hubble parameter and compare with the observational $H(z)$ data as shown in Table 1. It is interesting to note that all the cases which can accommodate the equation of state transition can fit well the observational $H(z)$ data. Remembering that the $w_{eff}$ crossing $-1$ was observed in the SNIa data fitting containing an integration effect in the luminosity distance, while the Hubble parameter does not suffer from this integrated over effect, the Hubble parameter data can present a complementary and consistent check for our model. z 0.09 0.17 0.27 0.40 0.88 1.30 1.43 1.53 1.75 ----------------------- --------- ---------- --------- ----------- ----------- ----------- ----------- --------- ----------- H(z) (km/s/Mpc) 69 83 70 87 117 168 177 140 202 1$\sigma$ uncertainty $\pm12$ $\pm8.3$ $\pm14$ $\pm17.4$ $\pm23.4$ $\pm13.4$ $\pm14.2$ $\pm14$ $\pm40.4$ : The observational $H(z)$ data [@Jimenez:2003; @Simon:2005]. \[table1\] Now we list out our results step by step. In the following results, all the numbers obtained in numerical calculation are expressed only with 3 digits after decimal unless for the cases where more digits are necessary to be shown. 1\. DGP+M In this step we consider the DGP model with the dark radiation, where $M$ denotes the dark radiation. Now we have two free parameters $n$ and $M$ and we will use two constraints: $x(0)=1$ and $w_{eff}(0.2)=-1$ to see whether the dark radiation can help to realize the $w_{eff}$ crossing $-1$. Actually $n$ and $M$ can be solved by using these constraints as $n=0.157$ and $M=0.263$. But using these values of $n$ and $M$, the $w_{eff}$ behavior is not good. $w_{eff}$ crosses $-1$ at $z=0.2$ from below to up as shown in Fig.1, and the present value is $w_{eff}=-0.950$. This is not in consistent with the observation, especially the transition behavior. ![The $w_{eff}$ curve as function of $z$ in the case DGP+M. The behavior of $w_{eff}$ is bad since it crossed $-1$ at $z=0.2$ from below to above. []{data-label="fig1"}](M.eps){width="10cm"} 2\. DGP+GB+M Including the GB correction, we have three free parameters now, such as $n$, $m$ and $M$. If we apply all three constraints ($x(0)=1$, $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$), the solution is complex ($n=-0.030+0.093i$, $m=-0.002-0.007i$ and $M=6.340+6.244i$). If we apply only two constraints ($x(0)=1$ and $w_{eff}(0.2)=-1$) and search $n$ in a big range $0.001\leq n\leq5$, the similar result to that in case 1 appears: $w_{eff}$ crosses $-1$ from below to above and $m$ is negative. One solution is shown in Fig.2a. We note that there is a singularity about $z=1.246$ in the curve of $w_{eff}$. This singularity comes from the definition in Eq.(29), we see that when $\delta H^2\equiv H(z)^2-\Omega_{m0}(1+z)^3H_0^2\leq0$, the logarithm is not well defined, but one can still calculate the $w_{eff}$ through the simplified expression in Eq.(30). In Fig.2b we show the relation between $H^2/H_0^2$ and the matter component $\Omega_{m0}(1+z)^3$. We see that beyond the redshift $z=1.246$ the matter component is overweight, so $\delta H^2<0$, which means the effective dark energy component is negative. This is obviously unreasonable, and at least it shows that the model fails in explaining the universe before that redshift. In this work we concentrate on those solutions with $w_{eff}(z)$ free of singularity. ![In the case DGP+GB+M, $w_{eff}$ and $q$ as functions of $z$ are shown in Fig.2a. The behavior is not favored as in Fig.1. Singularity is observed at $z=1.246$. In Fig.2b, relation between $H^2/H_0^2$ and the matter component $\Omega_{m0}(1+z)^3$ is shown. When $z>1.246$, $\delta H^2\equiv H(z)^2-\Omega_{m0}(1+z)^3H_0^2$ becomes negative, which breaks the definition of $w_{eff}$ in Eq.(29).[]{data-label="fig2"}](GB+M.eps "fig:"){width="8cm"} ![In the case DGP+GB+M, $w_{eff}$ and $q$ as functions of $z$ are shown in Fig.2a. The behavior is not favored as in Fig.1. Singularity is observed at $z=1.246$. In Fig.2b, relation between $H^2/H_0^2$ and the matter component $\Omega_{m0}(1+z)^3$ is shown. When $z>1.246$, $\delta H^2\equiv H(z)^2-\Omega_{m0}(1+z)^3H_0^2$ becomes negative, which breaks the definition of $w_{eff}$ in Eq.(29).[]{data-label="fig2"}](GB+M_H.eps "fig:"){width="8cm"} 3\. DGP+$T^5_5$ Now we include the bulk related energy-momentum tensor $T^5_5$. In this case we have three free parameters ($n$, $X$ and $\nu$) and we are going to employ all three constraints ($x(0)=1$, $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$). We can find the solution $n=0.046$, $X=2.729$ and $\nu=0.948$. The curves of $w_{eff}$, $q$ and $H$ v.s. redshift $z$ are shown in Fig.3a and Fig.3b respectively. In plotting Fig.3b, we have used $H_0=72km/s/Mpc$ [@Freemann:2000]. It is interesting to find that parameters adjusted to meet the requirement of $w_{eff}$ crossing $-1$ and its value at the present moment automatically fit well to the $H(z)$ data. Recalling that the transition behavior of $w_{eff}$ results from the SN data analysis containing integration in the luminosity distance, while the Hubble parameter is not integrated over, which persists fine structure highly degenerated in the luminosity distance, the simultaneous satisfaction of the $w_{eff}$ behavior and the $H(z)$ data gives complementary and consistent check of the viability of our model. ![$w_{eff}$ and $q$ v.s. $z$ (Fig.3a) and $H(z)$ curve (Fig.3b) in DGP+$T^5_5$ case. We see that $H(z)$ curve fits the data quite well.[]{data-label="fig3"}](T55.eps "fig:"){width="8cm"} ![$w_{eff}$ and $q$ v.s. $z$ (Fig.3a) and $H(z)$ curve (Fig.3b) in DGP+$T^5_5$ case. We see that $H(z)$ curve fits the data quite well.[]{data-label="fig3"}](T55_H.eps "fig:"){width="8cm"} 4\. DGP+GB+$T^5_5$ Here we include the GB correction based on the case discussed above. We now have four free parameters $n$, $m$, $X$ and $\nu$. Employing constraints ($x(0)=1$, $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$) and searching through $0.001\leq n\leq5$, we see that $m$, $X$ and $\nu$ can either be negative or positive, but the latter two are always with the same sign. $m$ decreases with the increase of $n$ and can be positive only when $n<0.05$. The $w_{eff}$ curve has no singularity for positive $m$, but contains singularity when $m<0$ (few solutions without singularity have been found with negative $m$, but then $\nu$ becomes smaller than $-4$, which conflicts with our simplification assumption discussed above). We are interested in the positive $m$, since GB correction coefficient $\alpha$ is always positive, the singularity-free curves of $w_{eff}(z)$ and $q(z)$ with positive solution $m=0.025$ are shown in Fig.4a. When the GB correction is considered, $w_{eff}$ appears more different at larger $z$ if compared to the result without the GB correction. Since the GB effect is only important in the early universe, its stronger modification to the $w_{eff}$ at bigger redshift is natural. In Fig.4b, we plotted the $H(z)$ curve by using the same adjusted parameter from the constraints on the equation of state, and we see again that in the model when the $w_{eff}$ requirement is met, the $H$ parameter automatically fits the data, which gives the consistent check of the model. ![$w_{eff}(z)$ and $q(z)$ curves (Fig.4a) and $H(z)$ curve (Fig.4b) in DGP+GB+$T^5_5$ case. Comparisons with the results without GB corrections have been shown. Differences from the case without the GB correction become bigger at higher redshift.[]{data-label="fig4"}](GB+T55mposi.eps "fig:"){width="8cm"} ![$w_{eff}(z)$ and $q(z)$ curves (Fig.4a) and $H(z)$ curve (Fig.4b) in DGP+GB+$T^5_5$ case. Comparisons with the results without GB corrections have been shown. Differences from the case without the GB correction become bigger at higher redshift.[]{data-label="fig4"}](GB+T55mposi_H.eps "fig:"){width="8cm"} 5.DGP+$T^5_5$+M Based on case 3, we include the dark radiation contribution. We now have four parameters, namely $n$, $X$, $\nu$ and $M$. Employing three constraints ($x(0)=1$, $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$), and searching $n$ in the range $0.001\leq n\leq5$, we find that the solutions exist only when $n<0.36$. We see that $M$ can either be positive or negative: for the negative $M$, $w_{eff}$ never decreases with the increase of z within $z<5$; while for the positive $M$, $w_{eff}$ drops at large $z$, and singularities of $w_{eff}(z)$ and $q(z)$ appear within $z<5$. Pictures of these two cases are shown in Fig.5a and Fig.5b. It is also observed that with the increase of $n$, the positive value of $M$ becomes bigger and the singularity appears at smaller $z$. ![We show in Fig.5a the $w_{eff}(z)$ and $q(z)$ curves in DGP+$T^5_5$+M case, where M is negative and the curves are singularity free; in Fig.5b, M is positive and the curves have a singularity at $z=0.936$.[]{data-label="fig5"}](T55+Mnosing.eps "fig:"){width="8cm"} ![We show in Fig.5a the $w_{eff}(z)$ and $q(z)$ curves in DGP+$T^5_5$+M case, where M is negative and the curves are singularity free; in Fig.5b, M is positive and the curves have a singularity at $z=0.936$.[]{data-label="fig5"}](T55+Msing.eps "fig:"){width="8cm"} 6\. DGP+GB+$T^5_5$+M Now we have all five parameters: $n$, $m$, $X$, $\nu$ and $M$ and three constraints to be used ($x(0)=1$, $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$). First, we try to set $m$ small, e.g. $10^{-9}$, and choose one solution obtained in case 5, but the FindRoot command cannot help to get a solution to meet all three constraints, even though the initial parameters are set closely to those in case 5. This means that the solution is sensitive to $\alpha$, though $\alpha$ (or $m$) is small. Next we search the nonsingular solutions within the parameters’ ranges $0.001\leq m\leq0.02$, $0.01\leq n\leq0.1$, the curves are not quite different from what we have seen in case 3 and case 5. We show one of the solutions in Fig.6a and Fig.6b. ![$w_{eff}(z)$ and $q(z)$ curves (Fig.6a) and $H(z)$ curve (Fig.6b) in DGP+GB+$T^5_5$+M case.[]{data-label="fig6"}](GB+T55+M.eps "fig:"){width="8cm"} ![$w_{eff}(z)$ and $q(z)$ curves (Fig.6a) and $H(z)$ curve (Fig.6b) in DGP+GB+$T^5_5$+M case.[]{data-label="fig6"}](GB+T55+M_H.eps "fig:"){width="8cm"} Calculation and Analysis considering the $T^0_5$ term ===================================================== The study with nonzero $T^0_5$ becomes more complicated. In order to solve the effective Friedmann equation, or equivalently, to obtain $x(z)$, we have to use the differential equation (23) with the full expression of $u$ in Eq.(28). The free parameters we have now are $n$, $m$, $\widetilde{X}$, $\nu$, $P$ and $s$. Here we do not have the explicit dark radiation term $M$, since we do not make the integration to get $\Phi$. In the calculation, we have to solve the differential equation, where the boundary condition $x(z=0)=1$ is needed. The effect of the dark radiation is reflected in the boundary condition of $x$. Solving equations when $T^0_5=0$, the result is the same as the case when dark radiation term $M$ appeared in the last section. We will still use two constraints on $w_{eff}$, which are $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$. Eq.(23) is a differential equation of $x(z)$, where free parameters are involved. More efforts are needed to solve the equation numerically. Here we try to employ the self-consistent method. First we get $x(z)$ solved with the chosen initial values of parameters, then we substitute the numerical result into the expression of $w_{eff}(z)$. The term $\frac{dx(z)}{dz}$ in the expression of $w_{eff}$ can be replaced by the function of $x(z)$ using Eq.(23), thus we can write $w_{eff}$ into a function of $x(z)$. Substituting the solution of $x(z)$, $w_{eff}$ becomes a function of $z$ and we can solve the parameters with the constraints $w_{eff}(0.2)=-1$ and $w_{eff}(0)=-1.06$. If the result is not consistent, we substitute the results back to $x(z)$ as new initial values until convergence is finally arrived. A proper choice of the initial parameters is crucial to obtain a convergent result, we usually do the iteration with many different choices within quite a large reasonable parameter space. Now we show the results we have obtained. First, to show the consistent and efficient of our numerical calculation, we turn off the contribution of $T^0_5$ to recover corresponding cases in section III. 7\. DGP+$T^5_5$ (employing Eq.(23) to solve the problem numerically) We have three parameters $n$, $\widetilde{X}$ and $\nu$ in this case. Searching within ranges $-100\leq\widetilde{X}\leq100$ and $-4\leq\nu\leq100$ by setting $n=0.01$, we can find the solution ($\widetilde{X}=79.824$, $\nu=0.719$), which is singularity free at least for $z<5$. This result can be compared with that in case 5 (DGP+$T^5_5$+M). Here the dimensionless parameter for $T^5_5$ term is $\widetilde{X}$, which differs from $X$ employed in case 5 with a factor $4+\nu$ as shown in Eq.(18). Taking this into account, the solution for $X$ is $16.917$, which is just the result in case 5 with $n=0.01$, where the solution in case 5 reads $X=16.917$, $\nu=0.719$ and with the additional parameter $M=-1.023$, see Fig.5a. They coincide, although they are obtained by completely different methods. This shows the correctness of the self-consistent method we used and also demonstrates the equivalence of the boundary condition in differential equation and the extra freedom of the integration constant. 8\. DGP+GB+$T^5_5$ (employing Eq.(23) to solve the problem numerically) We have now four parameters $n$, $m$, $\widetilde{X}$ and $\nu$. Within ranges $-100\leq\widetilde{X}\leq100$, $0\leq\nu\leq5$ by setting $n=0.05$ and $m=0.007$ (which are the values used in case 6 in Fig.6a and Fig.6b), we can find the solution $\widetilde{X}=11.93120$ and $\nu=0.98898$. Considering the difference between the dimensionless notations, this corresponds to $X=2.39151$ and $\nu=0.98898$, which is a bit different from those directly obtained in case 6 as $X=2.39146$, $\nu=0.98906$. This small difference is due to the approximation we have taken in Eq.(23), where the expansion on $\alpha$ is kept only to the linear order. So when $\alpha\neq0$, the equation systems in case 6 and case 8 are not exactly the same. We can see that the difference between the curves in Fig.7a, 7b and those in Fig.6a, 6b lies in large $z$ region, which shows that the effect of GB correction is important in the large redshift era. ![$w_{eff}(z)$ and $q(z)$ curves (Fig.7a) and $H(z)$ curve (Fig.7b) in DGP+GB+$T^5_5$ case. This result is calculated from the equation considering the $T^0_5$ term, which is not quite different from Fig.6 of case 6, except the curves lie a little higher than the curves in Fig.6a at large $z$. []{data-label="fig7"}](GB+T55_T05.eps "fig:"){width="8cm"} ![$w_{eff}(z)$ and $q(z)$ curves (Fig.7a) and $H(z)$ curve (Fig.7b) in DGP+GB+$T^5_5$ case. This result is calculated from the equation considering the $T^0_5$ term, which is not quite different from Fig.6 of case 6, except the curves lie a little higher than the curves in Fig.6a at large $z$. []{data-label="fig7"}](GB+T55_T05_H.eps "fig:"){width="8cm"} To demonstrate more explicitly the effect of GB term, we calculate in this case with different fixed values of $m$. We shut down the freedom of $\nu$ by setting $\nu=1$ in order to show the effect of GB term more clearly. The results are shown in Fig.8, where we see clearly that the GB term only changes the property in the early universe. ![$w_{eff}(z)$ curve in DGP+GB+$T^5_5$ case. These curves correspond to $m=0.01$, $m=0.001$, $m=0.0001$ and $m=0.00001$ respectively. []{data-label="fig8"}](GBeffect.eps){width="10cm"} 9\. DGP+$T^0_5$ We turn on the $T^0_5$ effect to consider the energy exchange between the bulk and the brane. In this case free parameters are $n$, $P$ and $s$. Searching in the range $0.01\leq n\leq0.2$ with $-0.28\leq P\leq0.72$ and $-100\leq s\leq100$ as initial tries, we find that for $n\leq0.15$, solutions can be found, but all require $P<-0.28$, which is forbidden as we discussed. This tells us that $T^0_5$ alone with the simple form $T^0_5=fHa^s$ cannot lead to the expected behavior of the effective equation of state. 10\. DGP+GB+$T^0_5$ We have four parameters $n$, $m$, $P$ and $s$, one more than those in the previous case. But after searching in ranges $0.001\leq n\leq0.1$, $0.01\leq m\leq0.9$, with $-0.28\leq P\leq0.72$ and $-100\leq s\leq100$ as initial values, all solutions which can be found needs $P<-0.28$. The value of $P$ increases with the increase of $m$ and the decrease of $n$, but it can only go up to $-0.425$ when $n=0.001$ and $m=0.9$, which is almost the most favored parameter set within acceptable ranges for $m$ and $n$. Actually $m$ is related to the GB correction which should be small. Thus introducing one more free parameter, the GB correction, cannot change the unfavored result in case 9. 11\. DGP+$T^5_5$+$T^0_5$ We contain now five parameters in total, e.g., $n$, $\widetilde{X}$, $\nu$, $P$ and $s$, but for simplicity, we keep only three parameters ($n$, $P$ and $\widetilde{X}$) free, by setting $\nu=0.7$, $s=1$. The reason of choosing $\nu=0.7$ is because in case 7 the solutions give the value of $\nu$ around $0.5\sim1$. We find that the value of $n$ cannot be too big, otherwise the curve will have singularity at very small $z$. For small $n$, we can find solutions without singularity, e.g., $n=0.01$, $P=0.020$ and $\widetilde{X}=78.535$. We show the proportion of different components as functions of $z$ in Fig.9, in which $\Omega_{matter}+\Omega_{dark energy}\equiv1$, $\Omega_{matter}$ is the total matter as obtained from the differential equation (26); while $\Omega_{effective}$ is the effective dark energy proportion including the energy exchange effect between the bulk and brane. Since the energy exchange effect is very small ($P$ is small), the difference between $\Omega_{effective}$ and $\Omega_{dark energy}$ can basically be neglected. The behavior of $w_{eff}(z)$, $q(z)$ and $H(z)$ are shown in Fig.10a and Fig.10b. Different from the case 9, we see that when the bulk matter $T^5_5$ is considered, the modified DGP model allows the $w_{eff}$ crossing $-1$ and is consistent with $H(z)$ data. ![Different components as functions of $z$ in DGP+$T^5_5$+$T^0_5$ case. The solid line and the long dashed line, representing the “real" matter component and dark energy component respectively. The short dashed line shows the remainder of the total energy density after subtracting the conserved matter $\Omega_{m0}(1+z)^3H_0^2/H(z)^2$, which acts as the effective dark energy where the energy exchange was considered. Since $P$ is quite small, the effect of energy exchange is negligible, curves of $\Omega_{effective}$ and $\Omega_{dark energy}$ lie almost together. []{data-label="fig9"}](T55+T05_component.eps){width="10cm"} ![$w_{eff}(z)$ and $q(z)$ curves (Fig.10a) and $H(z)$ curve (Fig.10b) in DGP+$T^5_5$+$T^0_5$ case.[]{data-label="fig10"}](T55+T05.eps "fig:"){width="8cm"} ![$w_{eff}(z)$ and $q(z)$ curves (Fig.10a) and $H(z)$ curve (Fig.10b) in DGP+$T^5_5$+$T^0_5$ case.[]{data-label="fig10"}](T55+T05_H.eps "fig:"){width="8cm"} 12\. DGP+GB+$T^5_5$+$T^0_5$ This is the most general case in our discussion, where we have all parameters of our model: $n$, $m$, $\widetilde{X}$, $\nu$, $P$ and $s$. To simplify the calculation, we fix $\nu=0.5$ and $s=1$. Searching in parameters’ ranges $0.01\leq n<10$, $0.001\leq m<1$, $-0.28\leq P\leq0.72$ and $-1000\leq\widetilde{X}\leq1000$, we found that when $n$ becomes larger, the solution becomes worse, either the curves have very bad shapes or the value of $P$ lies far away from the acceptable range. It is found that generally $n$ should not be larger than 0.1. For small $n$, we can find the solution such as $n=0.001$, $m=0.01$, $P=0.166$, $\widetilde{X}=895.044$, whose corresponding curves of $w_{eff}(z)$, $q(z)$ and $H(z)$ are shown in Fig.11a and Fig.11b respectively. ![$w_{eff}(z)$ and $q(z)$ curves (Fig.11a) and $H(z)$ curve (Fig.11b) in DGP+GB+$T^5_5$+$T^0_5$ case.[]{data-label="fig11"}](GB+T55+T05.eps "fig:"){width="8cm"} ![$w_{eff}(z)$ and $q(z)$ curves (Fig.11a) and $H(z)$ curve (Fig.11b) in DGP+GB+$T^5_5$+$T^0_5$ case.[]{data-label="fig11"}](GB+T55+T05_H.eps "fig:"){width="8cm"} Conclusions and Discussions =========================== In this work we have generalized the DGP braneworld by including bulk matter content, bulk-brane energy exchange and adding the GB curvature correction term in the bulk action. We have investigated the effects of the bulk contents and the GB correction on the evolution of the universe. We have found that although the pure DGP model cannot accommodate the transition of the equation of state as indicated by recent observation, once the bulk matter $T^5_5$ is considered, the modified model can accommodate the $w_{eff}$ crossing $-1$. However this transition of the equation of state cannot be realized by just considering bulk-brane energy exchange or the GB effect but without the bulk matter contribution. Thus $T^5_5$ plays the major role in the modified DGP model to have the $w_{eff}$ crossing $-1$ behavior. The GB term can have little influence on the late time behavior of the universe, it gives modification to the equation of state at big redshift. This is because of the fact that the GB correction arises from the high energy theory, being negligible in our present cold universe. Besides the $w_{eff}$ crossing behavior, our model can describe the Hubble parameter consistently with observation. In our parameter space there is a generally favored range $n<0.1$, which is crucial to have singularity free behavior in the equation of state. From the definition $n=\frac{1}{r^2H_0^2}$, this range of $n$ requires that the crossover factor obeys $r>3.16 H^{-1}_0$, which is bigger than the lower bound just due to the GB correction[@Brown:2006; @He:2007]. Since $P$ and $m$ are related, proper $P$ requires a bit bigger value of $m$. The permitted value $P$ is small. Its sign corresponds to the direction of the energy flow. The results we show previously in case 11 and case 12 have positive $P$ standing for the influx of energy, which is considered reasonable as the explanation of the accelerating expansion of the universe for the cosmology without extra dimension. But in the brane cosmology, since we have shown with our results that the $T^5_5$ term dominates the effective equation of state behavior, there is no big difference whether the energy flows into or out of the brane, and in fact the solutions with negative $P$ have also been obtained, which have similar behavior to those shown here. From our result we see that the $T^0_5$ term plays little effect in the transition of equation of state, this could be due to the choice of the ansatz. A more general form of $T^0_5$ can make the calculation more difficult, since the numerical solution rather than the analytical solution of $\rho$ from Eq. (25) will bring more difficulties in the following calculations. The solution of $\rho(a)$ should be substituted into the function of $H(z)$ after it is expressed in dimensionless notation. But in principle this is not impossible. Our method to numerically solve the nonlinear differential equation of $H(z)^2$ supplies a general way to deal with such problem, and it can relax the assumption form for $T^0_5$ and $T^5_5$. We expect to see the influence of a more general form of $T^0_5$ on the behavior of the equation of state of effective dark energy. Acknowledgements {#acknowledgements .unnumbered} ================ This work is partially supported by CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico) and FAPESP (Fundacao de Ampara a Pesquisa do Estado de Sao Paulo). The work of B. Wang was partially supported by NNSF of China and Shanghai Education Commission. The work of C.-Y. L. was supported in part by the National Science Council under Grant No. NSC- 93-2112-M-259-011. [99]{} A. G. Riess [et. al.]{}, Astronphys. J. 116, 1009 (1998). S. Perlmutter [et. al.]{}, Astrophys. J. 517, 565 (1999). A. G. Riess [et. al.]{}, Astrophys. J. 560, 49 (2001). S. Weinberg, Rev. Mod. Phys. 61, 1 (1989); N. Straumann, astro-ph/0203330; T. Padmanabhan, hep-th/0406060. R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998); P. J. E. Peebles and A. Vilenkin, Phys. Rev. D 59, 063505 (1999); P. J. Steinhardt, L. M. Wang, and I. Zlatev, Phys. Rev. D 59, 123504 (1999); M. Doran and J. Jaeckel, Phys. Rev. D 66, 043519 (2002); A. R. Liddle, P. Parson, and J. D. Barrow, Phys. Rev. D 50, 7222 (1994). A. Y. Kamenshchik, U. Moschella, and V. Pasquier, Phys. Lett. B 511, 256 (2001). M. Li, Phys. Lett. B 603, 1 (2004); Q. G. Huang and M. Li, JCAP 0408, 013 (2004). R. R. Caldwell, Phys. Lett. B 545, 23 (2002); R. R. Caldwell, M. Kamionkowski, and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003); J. M. Cline, S. Y. Jeon, and G. D. Moore, Phys. Rev. D 70, 043543 (2004). U. Alam, V. Sahni, and A. Starobinsky, JCAP 0406, 008 (2004); Y. G. Gong, Class. Quant. Grav. 22, 2121 (2005); Y. Wang and M. Tegmark, Phys. Rev. D 71, 103513 (2005); Y. Wang and P. Mukherjee, Astrophys. J. 606, 654 (2004); R. Daly and S. Djorgovski, Astrophys. J. 612, 652 (2004); U. Alam, V. Sahni, T. Saini, and A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004); T. Choudhury and T. Padmanabhan, Astron. Astrophys. 429, 807 (2005). B. Feng, X. L.Wang, and X. M. Zhang, Phys. Lett. B 607, 35 (2005); W. Hu, Phys. Rev. D 71, 047301 (2005); Z. K. Guo, Y. S. Piao, X. M. Zhang, and Y. Z. Zhang, Phys. Lett. B 608, 177 (2005); X. F. Zhang, H. Li, Y. S. Piao, and X. Zhang, astro-ph/0501652. M. Z. Li, B. Feng, and X. M. Zhang, hep-ph/0503268. B. Wang, Y. Gong, and E. Abdalla, Phys. Lett. B 624, 141 (2006); B. Wang, C. Y. Lin, and E. Abdalla, Phys. Lett. B 637, 357 (2006); B. Wang, J. D. Zang, C. Y. Lin, E. Abdalla and S. Micheletti, Nucl.Phys.B778, 69,2007, astro-ph/0607126. S. Nojiri and S. D. Odintsov, Phys. Lett. B 562, 147 (2003); S. Nojiri and S. D. Odintsov, Phys. Rev. D 70, 103522 (2004); A. Vikman, Phys. Rev. D 71, 023515 (2005); A. Anisimov, E. Babichev, and A. Vikman, JCAP 0506, 006 (2005); S. Nojiri, S. D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005); P. Singh, gr-qc/0502086; H. Stefancic, astro-ph/0504518; E. O. Kahya and V. K. Onemli, gr-qc/0612026; T. Koivisto and D. F. Mota, Phys. Lett. B 644, 104 (2007), astro-ph/0606078; T. Koivisto and D. F. Mota, Phys. Rev. D 75, 023518 (2007), hep-th/0609155; V.K. Onemli, R.P. Woodard, Phys.Rev.D 70, 107301 (2004). G. R. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83, 4690 (1999). E. Kiritsis, G. Kofinas, N. Tetradis, T. N. Tomaras, and V. Zarikas, JHEP 0302, (2003) 035; E. Kiritsis, N. Tetradis, and T. N. Tomaras, JHEP 0203, (2002) 019; P. S. Apostolopoulos and N. Tetradis, Phys. Rev. D 71, 043506 (2005); P. S. Apostolopoulos and N. Tetradis, Phys. Lett. B 633, 409 (2006); E. Kiritsis, JCAP 0510, 014 (2005); K. I. Umezu, K. Ichiki, T. Kajino, G. J. Mathews, R. Nakamura, and M. Yahiro, Phys. Rev. D 73, 063527 (2006); M.R. Setare, Phys.Lett.B642,421,2006. R. G. Cai, Y. Gong, and B. Wang, JCAP 0603, 006 (2006). P. S. Apostolopoulos and N. Tetradis, hep-th/0604014; P. S. Apostolopoulos, N. Brouzakis, N. Tetradis, E. Tzavara, arXiv:0708.0469. C. Bogdanos and K. Tamvakis, hep-th/0609100; C. Bogdanos, A. Dimitridis, and K. Tamvakis, hep-th/0611094. A. Sheykhi, B. Wang, and N. Riazi, Phys. Rev. D 75, 123513 (2007). B. Zwiebach, Phys. Lett. B 156, 315 (1985); D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985). D. Lovelock, J. Math. Phys. 12, 498 (1971). G. Kofinas, R. Maartens, and E. Papantonopoulos, JHEP 0310, 066 (2003). G. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B. 485, 208 (2000). G. Kofinas, G. Panotopoulos, and N. Tomaras, hep-th/0510207. R. A. Brown, R. Maartens, E. Papantonopoulos, and V. Zamarias, gr-qc/0508116. R. G. Cai, H. S. Zhang, and A. Wang, hep-th/0505186. C. Deffayet, Phys. Lett. B. 502, 199 (2001). C. Barceló, C. Germani, and C. F. Sopuerta, Phys. Rev. D 68, 104007 (2003). R. Maartens, Phys. Rev. D 62, 084023 (2000). D. N. Spergel et al. \[WMAP Collaboration\], Astrophys. J. Suppl. 148, 175 (2003). E. V. Linder and A. Jenkins, Mon. Not. Roy. Astron. Soc. 346, 573 (2003). R. Jimenez, L. Verde, T. Treu, and D. Stern, Astrophys. J. 593, 622 (2003). J. Simon, L. Verde, and R. Jimenez, Phys. Rev. D 71, 123001 (2005). W. L. Freeman, et al., astro-ph/0012376. J. H. He, B. Wang, and E. Papantonopoulos, arXiv:0707.1180, Phys. Lett. B (in press).
--- abstract: 'We propose a rate equation approach to compute two vertex correlations in scale-free growing network models based in the preferential attachment mechanism. The formalism, based on previous work of Szabó et al. \[Phys. Rev. E 67 056102 (2002)\] for the clustering spectrum, measuring three vertex correlations, is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree $k$, with an appropriate boundary condition. We study the properties of both two and three vertex correlations for linear preferential attachment models, and also for a model yielding a large clustering coefficient. The expressions obtained are checked by means of extensive numerical simulations. The rate equation proposed can be generalized to more sophisticated growing network models, and also extended to deal with related correlation measures. As an example, we consider the case of a recently proposed model of weighted networks, for which we are able to compute a weighted two vertex correlation function, taking into account the strength of the interactions between connected vertices.' author: - Alain Barrat - 'Romualdo Pastor-Satorras' title: Rate equation approach for correlations in growing network models --- Introduction ============ Many natural and man-made complex systems can be fruitfully represented and studied in terms of networks or graphs [@bollobas98], in which the vertices stand for the elementary units that compose the system, while the edges picture the interactions or relations between pairs of units. This topological representation has found many applications in fields as diverse as the Internet [@romuvespibook], the World-Wide Web [@www99], biological interacting networks [@wagner01; @jeong01; @maslov02] or social systems [@wass94], leading to the development of a new branch of statistical mechanics, the modern theory of complex networks [@barabasi02; @mendesbook]. The empirical study of real complex networks, promoted by the recent accessibility to computers powerful enough to deal with very large databases, has uncovered the presence of some typical characteristics. The three most relevant of these are: (i) The small-world property [@watts98], defined by an average shortest path length—average distance between any pair of vertices—increasing very slowly (logarithmically or slower [@havlin03]) with the network size $N$. (ii) The presence of a large transitivity [@wass94], which implies that two neighbors of a given vertex are also connected to each other with large probability. Transitivity can be quantitatively measured by means of the clustering coefficient $c_i$ of vertex $i$ [@watts98], defined as the ratio between the number of edges $m_i$ existing betwen the $k_i$ neighbors of $i$, and its maximum possible value, i.e. $c_i = 2 m_i / (k_i(k_i-1))$. The average clustering coefficient, defined as $C = \sum_i c_i /N$, usually takes quite large values in real complex networks. (iii) A scale-free behaviour for the degree distribution $P(k)$ [@barabasi02; @mendesbook], defined as the probability that a vertex is connected to $k$ other vertices (has degree $k$), that shows a power-law behavior $$P(k) \sim k^{-\gamma},$$ where $\gamma$ is a characteristic degree exponent, usually in the range $2< \gamma < 3$. A major role is especially played by the scale-free nature of many real complex networks, which implies a large connectivity heterogeneity, at the basis of the peculiar behavior shown by dynamical processes taking place on top of these networks, such as the resilience to random damage [@barabasi00; @newman00; @havlin00], the spreading of infectious agents [@pv01a; @pv01b; @lloydsir; @sievolution], or diffusion-annihilation processes [@originalA+A; @michelediffusion]. >From a theoretical point of view, the empirical research has inspired the proposal of new network models, aimed at reproducing and explaining the properties exhibited by complex networks. In this respect, many efforts have been devoted to develop models capable to account for a scale-free degree distribution. Classical network modeling was previously based on the Erdös-Renyi random graph model [@erdos59; @bollobas], which is a static model (i.e. defined for a fixed number of vertices $N$) yielding small-world networks with a Poisson degree distribution. A change of perspective in nework modeling took place after the introduction of the preferential attachment paradigm first proposed by Barabási and Albert (BA) [@barab99]. The insight behind this concept is the realization of two facts: Firstly, most complex networks are the result of a growth process, in which new vertices are added in time to the system. Secondly, new edges are not placed at random, but tend to connect to vertices which already have a large degree. It turns out that these two ingredients are able to reproduce scale-free degree distributions with a tunable degree exponent [@barab99; @mendes99]. Moreover, it has been shown that not all sorts of preferential attachment are able to generate a power law degree distribution, but only those in which new edges attach to vertices with a probability strictly linear in their degree [@krap00], and that some alternative mechanisms, such as the copying model [@kumar00] implicitly define a linear preferential attachment dynamics. While a proper characterization and understanding of the origin of the scale-free degree distribution displayed by most real complex networks is a fundamental task, it has been recently realized that this property does not provide a suffient characterization of natural networks. In fact, these systems seem to exhibit also ubiquitous degree correlations, which translate in the fact that the degrees of the vertices at the end points of any given edge are not independent [@alexei; @alexei02; @assortative; @newmanmixing]. Two vertex degree correlation can be conveniently measured by means of the conditional probability $P(k'\|k)$ that a vertex of degree $k$ is connected to a vertex of degree $k'$. For uncorrelated networks, in which this conditional probability is independent of $k$, it can be estimated as the probability that any edge end points to a vertex of degree $k'$, which is simply given by $P_\mathrm{n.c.}(k'\|k) = k' P(k') / {\langle k \rangle}$ [@hiddenvars]. The empirical evaluation of $P(k'\|k)$ in real networks is usually a cumbersome task, restricted by finite size data yielding noisy results. For this reason, it is more practical to analyze instead the average degree of the nearest neighbors of the vertices of degree $k$, which is formally defined as [@alexei] $${\bar{k}_{nn}}(k) = \sum_{k'} k' P(k'\|k). \label{eq:8}$$ For uncorrelated networks, in which $P(k'\|k)$ does not depend on $k$, we have $${\bar{k}_{nn}}^\mathrm{n.c.}(k) = \frac{{\langle k^2 \rangle}}{{\langle k \rangle}}, \label{eq:13}$$ independent of $k$. Thus, a ${\bar{k}_{nn}}(k)$ function with an explicit dependence on the degree signals the presence of two vertex degree correlations in the network. When ${\bar{k}_{nn}}(k)$ is an increasing function of $k$, the network shows assortative mixing [@assortative] (vertices of large degree connected more preferably with vertices of large degree, and vice-versa). Negative correlations (low degree vertices connected preferably with large degree vertices), on the other hand, give rise to dissasortative mixing, detected by a decreasing ${\bar{k}_{nn}}(k)$ function. Analogously to two vertex correlations, correlations implying three vertices can be mesured by means of the probability $P(k', k''\|k)$ that a vertex of degree $k$ is simultaneously connected to vertices of degree $k'$ and $k''$. Again, the difficulties in directly estimating this conditional probability can be overcome by analyzing the clustering coefficient. The average clustering coefficient of the vertices of degree $k$ (the clustering spectrum), ${\bar{c}}(k)$ [@alexei02; @ravasz02], can be formally computed as the probability that a vertex of degree $k$ is connected to vertices of degree $k'$ and $k''$, and that those two vertices are at the same time joined by an edge, averaged over all the possible values of $k'$ and $k''$ [@hiddenvars]. Thus, we can write ${\bar{c}}(k)$ as a function of the three vertex correlations as $${\bar{c}}(k) = \sum_{k', k''} P(k', k''\|k) p^k_{k', k''},$$ where $p^k_{k', k''}$ is the probability that vertices $k'$ and $k''$ are connected, provided that they have a common neighbor $k$ [^1]. From this expression, the average clustering coefficient can be computed as $$C = \sum_k P(k) {\bar{c}}(k).$$ For uncorrelated networks, we have that $P_\mathrm{n.c.}(k', k''\|k)=P_\mathrm{n.c.}(k'\|k) P_\mathrm{n.c.}(k''\|k)$ [@hiddenvars], and $p^k_{k', k''} = (k'-1)(k''-1)/ {\langle k \rangle}N$ [@newmanrev]. Therefore we obtain $${\bar{c}}_\mathrm{n.c.}(k) = \frac{({\langle k^2 \rangle}- {\langle k \rangle})^2}{ {\langle k \rangle}^3 N}.$$ That is, ${\bar{c}}_\mathrm{n.c.}(k)$ is independent of $k$ and equal to the average clustering coefficient $C$ [@newmanrev]. A functional dependence of ${\bar{c}}(k)$ on the degree can thus be attributed to the presence of a structure in the three vertex correlations. In particular, for scale-free networks it has been observed that in many instances, the clustering spectrum exhibits also a power-law behavior, ${\bar{c}}(k) \sim k^{-\alpha}$. A value of $\alpha$ close to $1$ has been empirically observed in several natural networks, and analytically found in some growing network models [@ravasz02; @szabo; @structurednets]. These findings have led to propose the clustering spectrum ${\bar{c}}(k)$ as a tool to measure hierarchical organization and modularity in complex networks [@ravasz02]. The presence of correlations are thus a very relevant issue in order to understand and classify complex networks, especially in view of the important consequences that they can have on dynamical processes taking place on the topology defined by the networks [@marian1; @morenostructured; @morenopercolation]. While there are quite a few empirical results for real networks, the situation is not the same for network models, and therefore there is no consensus regarding the origin of assortative and dissasortative mixing, and its relation with the power law behavior of the clustering spectrum ${\bar{c}}(k)$. In fact, most works devoted to analytical calculations of correlations in complex network models have been performed only for particular cases [@ravasz02; @szabo; @structurednets; @hiddenvars]. In this respect, it is noteworthy the rate equation formalism proposed by Szabó et al. in Ref. [@szabo] (see also [@szaboproc]) to compute ${\bar{c}}(k)$ in growing network models with preferential attachment. However, and up to our knowledge, no such formalism has been developed to deal with two vertex correlations, as given by the ${\bar{k}_{nn}}(k)$ function. In this paper we revise the formalism proposed in Ref. [@szabo] for computing the clustering spectrum in growing networks models with preferential attachment. Reconsidering the mean field rate equation in the continuous degree approximation for the ${\bar{c}}(k)$ presented in [@szabo], we are able to provide a general expression for the boundary condition of this rate equation, which was neglected in the original treatment and which can have as a matter of fact relevant effects in the final solution, as we will show below. Inspired by this result we also propose a new rate equation for two vertex correlations, as measured by the ${\bar{k}_{nn}}(k)$ function, and work out the correponding boundary condition. We remark that both equations are valid in general for the so-called “citation networks” [@mendesbook], in which neither edge or vertex removal nor edge rewiring are allowed. The general formalism is presented in section \[sec:rate-equat-corr\]. The rate equations obtained can be easily solved for growing networks with linear preferential attachement (LPA) [@mendes99], as shown in section \[sec:lpa\]. In particular, we are able to obtain expressions for the dependence of the correlations on the degree $k$ and the system size $N$, for both the dissasortative and assortative regimes of the model, which are in very good agreement with numerical simulations and previous scaling arguments [@dorogorev]. LPA models generate networks with a vanishing average clustering coefficient $C$. In order to asses the effects of a nonzero clustering, we study in section \[sec:gener-dorog-mend\] a growing model presenting a large final clustering coefficient [@dms], which we are able to compute with very good accuracy. The results obtained are qualitatively similar to those shown by the Holme-Kim model [@szabo; @holme02c]. The rate equation proposed for two vertex correlations can be easily generalized to deal with more involved situations. As an example of its flexibility, we examine in section \[sec:weight-grow-netw\] a recently proposed model for the evolution of weighted complex networks [@barrat04:_weigh]. In this case, we extend our formalism to compute a function estimating weighted two vertex correlations, in which the actual strength of the interactions between neighboring vertices is taken into account. Our results allow us to discuss the scaling form of two and three vertex correlation functions, and signal the possible relations that can be established between them. Rate equations for correlations in growing networks {#sec:rate-equat-corr} =================================================== Let us consider the class of growing network models in which, at each time step, a new vertex with $m$ edges is added to the network. For the vertex introduced at time $t$, each of its emanating edges is connected to an existing vertex introduced at time $s$ ($s< t)$ with a connection probability $\Pi_s(\{k\}, t)$, which is assumed to depend only on the degrees of the existing vertices at time $t$, $\{k\}= \{k_{1}(t), \ldots k_{t-1}(t)\}$. Time runs from $1$ to $N$ (the final network size), and since for each new vertex $m$ edges are added, the average degree is fixed and given by ${\langle k \rangle}=2m$. In the continuous $k$ and $t$ approximation [@dorogorev], the average degree that the vertex $s$ (i.e. the vertex introduced at time $s$) has at time $t$ ($t>s)$ can be computed from the rate equation $$\frac{d k_s(t)}{d t} = m \Pi_s(\{k\}, t), \label{eq:7}$$ with the boundary condition $k_s(s)=m$ (initially all vertices have $m$ connections). From $k_s(t)$, the degree distribution can be obtained as $$P(k, t) = - \frac{1}{t} \left.\left(\frac{\partial k_s(t)}{\partial s} \right)^{-1}\right\|_{s=s(k,t)} ,$$ where $s(k,t)$ is the solution of the implicit equation $k=k_s(t)$. For this class of networks it is possible to obtain a rate equation for the clustering spectrum. Following Ref. [@szabo], we recall that the clustering coefficient $c_s(t)$ of vertex $s$ at time $t$ is defined as the ratio between the number of edges between the neighbors of $s$ and its maximum possible value. Then, if $M_s(t)$ is the number of connections between the neighbors of $s$ at time $t$, we have that $$c_s(t) = \frac{2 M_s(t)}{k_s(t) [ k_s(t)-1]}. \label{eq:17}$$ During the growth of the network, $M_s(t)$ can only increase by the simultaneous addition of an edge to $s$ and one of its neighbors. Therefore, in the continuous $k$ approximation, we can write down the following rate equation [@szabo]: $$\frac{d M_s(t)}{d t} = m(m-1) \Pi_s(\{k\}, t) \sum_{j \in {\cal V}(s)} \Pi_j(\{k\}, t) \ , \label{eq:3}$$ where $\mathcal{V}(s)$ is the set of neighbours of vertex $s$. In order to solve this equation we must provide additionally a boundary condition. To do so, we observe that $M_s(s)$ is the number of triangles created by the introduction of vertex $s$. Therefore $$M_s(s) = \frac{m(m-1)}{2} \sum_{j, n=1}^s \Pi_j(\{k\}, s) \Pi_n(\{k\}, s) \Pi_{j,n}, \label{eq:4}$$ that is, it is proportional to the probability that $s$ is connected to vertices $j$ and $n$, times the probability $\Pi_{j,n}$ that $j$ and $n$ are linked, averaged over all vertices $j$ and $n$ existing in the network at time $s$. The probability $\Pi_{j,n}$ is given by $$\Pi_{j,n} = \Theta(j-n) m \Pi_n(\{k\}, j) + \Theta(n-j) m \Pi_j(\{k\}, n), \label{eq:5}$$ where $\Theta(x)$ is the Heaviside step function. Solving the equation for $M_s(t)$ with the boundary condition Eq. (\[eq:4\]), we can obtain the clustering $c_s(t)$ from Eq. (\[eq:17\]). Then, since in growing network models in the continuous $k$ approximation the degree at time $t$ is uniquely determined by the introduction time $s$, from $c_s(t)$ we can directly obtain the clustering spectrum ${\bar{c}}(k)$ as a function of $k$ and the largest time $t=N$. In the case of the two vertex correlation function ${\bar{k}_{nn}}(k)$, we can proceed along similar lines. Let us define $R_s(t)$ as the sum of the degrees of the neighbors of vertex $s$, evaluated at time $t$. That is, $$R_s(t) = \sum_{j \in {\cal V}(s)} k_j(t). \label{eq:9}$$ The average degree of the neighbors of vertex $s$, ${\bar{k}_{nn}}(s)$ is then given by ${\bar{k}_{nn}}(s) = R_s(t) / k_s(t)$. During the growth of the network, $R_s(t)$ can only increase by the adjunction of a new vertex connected either directly to $s$, or to a neighbour of $s$. In the first case $R_s(t)$ increases by an amount $m$ (the degree of the newly linked vertex), while in the second case it increases by one unit. Therefore, in the continuous $k$ approximation, we can write down the following rate equation: $$\frac{dR_s(t)}{dt} = m [m \Pi_s(\{k\}, t)] + m \sum_{j \in {\cal V}(s)} \Pi_j(\{k\}, t). \label{eq:1}$$ In order to obtain the boundary condition for this equation, we observe that, at time s, the new vertex $s$ connects to an old vertex of degree $k_j(s)$ with probability $\Pi_j(\{k\}, s)$, and that this vertex gains a new connection in the process. Therefore, $$R_s (s) = m \sum_{j=1}^{s} \Pi_j(\{k\}, s) [k_j(s)+1]. \label{eq:2}$$ >From the solution of this rate equation, we can obtain ${\bar{k}_{nn}}(s)$ and from it the two vertex correlation function by the functional dependence of $s$ on $k$ and $t=N$. We must note that Eqs. (\[eq:3\]), (\[eq:4\]), (\[eq:5\]), (\[eq:1\]), and (\[eq:2\]) are only valid for the so-called “citation networks”, in which neither edge removal nor rewiring [@albert00] are allowed, since these two processes can induce nonlocal variations in the conectivity of the nearest neighbors. Linear preferential attachement models {#sec:lpa} ====================================== As an example of the application of the rate equations presented in the previous Section, we consider the general LPA model proposed in Ref. [@mendes99], for which the rate equations for $R_s(t)$ and $M_s(t)$ can be closed and solved analytically. For general LPA, the connection probability takes the form $$\Pi_s(\{k\}, t) = \frac{b_1 k_s(t)+b_2 }{\sum_j [b_1 k_j(t)+b_2]}, \label{eq:6}$$ where $b_i$ are real constants. Since, for each new vertex, $m$ edges are added to the network, the normalization constant in Eq. (\[eq:6\]) takes the form $\sum_j [b_1 k_j(t)+b_2] = (2mb_1+b_2)t$. Thus, the model depends only on the tuning parameter $a=b_2/b_1$, taking values in the interval $a \in ]-m, \infty[$ (since the minimum degree is $m$, $a$ cannot be lower than $-m$ in order for $\Pi_s(\{k\}, t)$ to remain positive). Solving the rate equation for the degrees Eq. (\[eq:7\]), we obtain $$k_s(t) = (m+a) \left(\frac{t}{s} \right)^\beta -a, \ \ \beta=\frac{m}{2m+a}. \label{eq:ks}$$ Therefore, this model yields networks with a power law degree distribution of the form $$P(k) \sim k^{-\gamma}, \ \ \gamma=3+a/m.$$ For $a>0$, we obtain a degree exponent $\gamma>3$, which corresponds to finite degree fluctuations in the thermodynamic limit. The case $a=0$ recovers the original BA model with $\gamma=3$ [@barab99]. Finaly, values $-m < a< 0$ yield scale free networks with a tunable degree exponent, in the range $\gamma\in ]2,3[$. Two vertex degree correlations ------------------------------ The rate equation for $R_s(t)$ takes in this case the form $$\begin{aligned} \nonumber \frac{dR_s(t)}{dt} &=& m^2 \frac{k_s(t)+a}{(2m+a)t} +\sum_{j \in {\cal V}(s)} m \frac{k_j(t)+a}{(2m+a)t} \\ &=& \beta \frac{(m+a)k_s(t)+am}{t} + \beta \frac{R_s(t)}{t}, \label{eq:10}\end{aligned}$$ where we have used the definition of $R_s(t)$, Eq. (\[eq:9\]). The general solution of the previous equation is $$R_s(t)= A_0(s) t^\beta + \beta (m+a)^2 \left( \frac{t}{s} \right)^\beta \ln t +a^2 \label{eq:12}$$ where $A_0(s)$ is given by the boundary condition $R_s(s)$. From Eq. (\[eq:2\]), we have that $$\begin{aligned} R_s (s) &=& m \sum_{j=1}^{s} \frac{ a+ (a+1) k_j(s) + k_j^2(s)}{(2m+a)s} \nonumber \\ &=& \beta a + 2 m \beta (a+1) + \frac{\beta}{s} \sum_{j=1}^{s} k_j^2(s) \ .\end{aligned}$$ Plugging in $k_j (s) = (m+a) (s/j)^{\beta} -a$ into $R_s(s)$ results in $$R_s (s) = m(1-a) + \beta (m+a)^2 s^{2 \beta-1} \sum_{j=1}^{s} j^{-2 \beta}. \label{eq:11}$$ In order to estimate the behaviour of the previous expression, we have to distinguish the different cases corresponding to the possible values of $\beta$ (namely $a$). (i) $-m < a < 0$ (i.e. $\beta > 1/2$, $\gamma<3$). In this case, for large $s$, $ \sum_{j=1}^{s} j^{-2 \beta} \simeq \zeta(2\beta)$, where $\zeta(x)$ is the Riemann Zeta function, and thus, at leading order, $$R_s (s) \simeq \beta \zeta(2\beta) (m+a)^2 s^{2\beta-1}. \label{eq:23}$$ The determination of the integration constant $A_0(s)$ from Eq. (\[eq:12\]) yields then $$R_s(t) \simeq \beta \zeta(2\beta) (m+a)^2 t^\beta s^{\beta-1} + \beta (m+a)^2 \left( \frac{t}{s} \right)^\beta \ln \left( \frac{t}{s} \right) ,$$ where terms independent of $t$ and $s$ and terms going to zero in the large $t$ or $s$ limit have been neglected. From the definition of ${\bar{k}_{nn}}(s)$, and substituting $s$ as a function of $k$ and $t=N$ (the network final size) in the limit of large $k$ and $N$ we obtain the following expression for the average degree of the neighbors of the vertices of degree $k$: $$\begin{aligned} {\bar{k}_{nn}}(k,N) &\simeq& \beta \zeta(2\beta) (m+a)^{3-1/\beta} N^{2\beta-1}k^{-2+1/\beta} \nonumber \\ &+&(m+a)\ln \left( \frac{k}{m+a}\right). \label{eq:15}\end{aligned}$$ >From this expression, we conclude that the LPA with $a<0$ yields in the large $N$ limit networks with dissasortative two vertex correlations, characterized by a power-law decay ${\bar{k}_{nn}}(k,N) \sim N^{2\beta-1} k^{-2+1/\beta}$. This exponent was previously obtained by scaling arguments in Ref. [@dorogorev]. The dependence of the prefactor on $N$ implies that ${\bar{k}_{nn}}(k,N)$ diverges in the thermodynamic limit $N\to\infty$, in agreement with the theoretical arguments provided in Ref. [@marian3]. For finite $N$, however, the logarithmic term with constant prefactor can induce corrections to the power-law scaling. Since $2\beta -1$ is at most $1$, the growth of ${\bar{k}_{nn}}(k,N)$ is not very steep with $N$ and these corrections are observable in numerical simulations, as we will see below in this Section. (ii) $a=0$ (i.e. $\beta = 1/2$, $\gamma=3$). For this value of $\beta$ Eq. (\[eq:11\]) is dominated by a logarithmic divergence, $\sum_{j=1}^{s} j^{-1} \simeq \ln s$, yielding $$R_s(s) \simeq \frac{m^2}{2}\ln s.$$ >From here, we obtain $$R_s(t) \simeq \frac{m^2}{2} \sqrt{\frac{t}{s}} \ln t,$$ and finally $${\bar{k}_{nn}}(k,N) \simeq \frac{m}{2} \ln N. \label{eq:knnBA}$$ That is, two vertex correlations are independent of the degree and grow with the system size as $\ln N$, in agreement with the behavior expected for an uncorrelated scale-free network with degree exponent $\gamma=3$, Eq. (\[eq:13\]). Numerical simulation of the BA model [@alexei02] show actually a very weak dependence on $k$ in the ${\bar{k}_{nn}}(k,N)$ function, compatible nevertheless with the behavior given by our rate equation approach in the large $k$ limit. This $k$ dependence, evidentiated at small values of the degree, cannot be detected within our approach, since it has been formulated in the continuous $k$ approximation. (iii) $a>0$ (i.e. $\beta < 1/2$, $\gamma>3$). In this situation, the summation in Eq. (\[eq:11\]), $\sum_{j=1}^{s} j^{-2\beta} \simeq s^{1-2 \beta}/(1-2\beta)$, is divergent, and therefore $R_s(s)$ becomes independent of $s$. This leads to $$\begin{aligned} R_s(t) &\simeq& \beta (m+a)^2 \left( \frac{t}{s} \right)^\beta \ln \left( \frac{t}{s} \right) \nonumber \\ &+& \left[ m(1-a)+ \frac{\beta(m+a)^2}{1-2\beta} -a^2 \right] \left( \frac{t}{s} \right)^\beta,\end{aligned}$$ and finally the dominant behaviour for the correlation function is $${\bar{k}_{nn}}(k,N) \simeq (m+a) \ln \left(\frac{k}{m+a} \right)$$ In this case, ${\bar{k}_{nn}}(k,N)$ is independent of the network size, and increases logarithmically with $k$: For $\gamma>3$, LPA yields networks with weak assortative mixing. Three vertex correlations ------------------------- In order to estimate three vertex degree correlations by means of the clustering spectrum ${\bar{c}}(k)$, we start from the rate equation Eq. (\[eq:3\]), which for the LPA model takes the form $$\begin{aligned} \nonumber \lefteqn{\frac{d M_s(t)}{d t} = m(m-1) \frac{k_s(t)+a}{(2m+a)t} \sum_{j \in {\cal V}(s)} \frac{k_j(t)+a}{(2m+a)t}} \\ &&= m(m-1) \frac{k_s(t)+a}{(2m+a)^2 t^2} [R_s(t) + a k_s(t)]. \label{eq:14}\end{aligned}$$ The boundary condition $M_s(s)$ can be written as $$\begin{aligned} M_s(s) &=& \frac{m(m-1)}{2} \sum_{j, n} \Pi_j(\{k\}, s) \Pi_n(\{k\}, s) \Pi_{j, n} \nonumber \\ &=& \frac{\beta^2 (m-1) (m+a)^3}{2(2m+a)} s^{2 \beta-2} \left\{ \sum_{n=1}^s n^{-2 \beta} \sum_{j=n+1}^s j^{-1} + \sum_{j=1}^s j^{-2 \beta} \sum_{n=j+1}^s n^{-1} \right\} \nonumber \\ &=& \frac{\beta^2 (m-1) (m+a)^3}{2(2m+a)} s^{2 \beta-2} \times 2 \times \sum_{n=1}^s n^{-2 \beta} \sum_{j=n+1}^s j^{-1} \ . \label{eq:mss}\end{aligned}$$ In order to solve Eq. (\[eq:14\]), we approximate $k_s(t)$ and $R_s(t)$ by their dominant terms for large $t$ and $s$, as computed above for the different possible values of $a$. (i) $-m < a < 0$. In this case we have $$k_s(t) \simeq (m+a) \left(\frac{t}{s} \right)^\beta, \ \ \ \ R_s(t) \simeq \beta \zeta(2\beta) (m+a)^2 t^\beta s^{\beta-1}.$$ Introducing this expression into Eq. (\[eq:14\]), we obtain at leading order $$M_s(t) \simeq \beta^2 \frac{(m-1)(m+a)^3 \zeta(2\beta)}{(2 \beta-1)(2m+a)} (t^{2\beta-1} - s^{2\beta-1})s^{-1} + M_s(s) . \label{eq:18}$$ In order to compute $M_s(s)$, we observe that the double summation in Eq. (\[eq:mss\]) takes the form at large $s$ $$\mathcal{S} = \sum_{n=1}^s n^{-2 \beta} \sum_{j=n+1}^s j^{-1} \simeq \sum_{n=1}^s n^{-2 \beta} (\ln s - \ln n) \simeq \zeta(2\beta) \ln s, \label{eq:19}$$ since $\sum_{n=1}^\infty n^{-2 \beta } \ln n$ is convergent for $\beta> 1/2$. Thus we obtain $$M_s(t) \simeq \beta^2 \frac{(m-1)(m+a)^3}{(2 \beta-1)(2m+a)} (t^{2\beta-1} - s^{2\beta-1}) s^{-1} +\beta^2 \frac{(m-1)(m+a)^3}{2m+a} \zeta(2\beta) s^{2\beta-2} \ln s,$$ and from here the expression for the three vertex correlation function follows: $$\begin{aligned} {\bar{c}}(k,N) &\simeq& \frac{2 \beta^2 (m-1) (m+a)^{3-1/ \beta}}{(2 \beta-1)(2m+a)} N^{2\beta -2} k^{-2 +1 / \beta} \nonumber \\ &+& \frac{2 \beta^2 \zeta(2 \beta)(m-1) (m+a)^{5-2/ \beta}}{2m+a} \ln N N^{2\beta -2} k^{-4 +2 \beta}. \label{eq:ckaneg}\end{aligned}$$ To understand the asymptotic behavior of ${\bar{c}}(k,N)$, two limits have to be taken, corresponding to large $N$ and large $k$: - At fixed and large $N$, the leading behavior at large $k$ is ${\bar{c}}(k,N) \sim N^{2\beta -2} k^{-2 +1 / \beta}$. - At fixed $k \lesssim (\ln N)^{\beta/(2\beta-1)}$ and large $N$, the leading behavior is instead ${\bar{c}}(k,N) \sim N^{2\beta -2} \ln N k^{-4 +2 / \beta}$. Therefore, in the numerical simulations we should expect to observe a crossover between these two scaling regimes. (ii) $a=0$. We now have $$k_s(t) \simeq m \sqrt{\frac{t}{s}}, \ \ \ \ R_s(t) \simeq \frac{m^2}{2} \sqrt{\frac{t}{s}} \ln t,$$ which yields $$M_s(t)\simeq \frac{m^2(m-1)}{16 s} ( \ln^2 t - \ln^2 s ) + M_s(s).$$ Since $\beta=1/2$, $M_s(s)$, as given by Eq. (\[eq:mss\]), can be easily shown to be $$M_s(s) = \frac{m^2(m-1)}{16 s} \ln^2 s \ , \label{eq:bcBA}$$ and we obtain $$M_s(t)\simeq \frac{m^2(m-1)}{16 s} \ln^2 t ,$$ which results in a clustering coefficient at large $N$ $${\bar{c}}(k,N)\simeq \frac{m-1}{8} \frac{\ln^2 N}{N}.$$ We recover the well-known result for the BA model that the clustering spectrum is constant, and scaling as $\ln^2 N/N$, as observed in Ref. [@szabo]. It is worth noting that the computation of the boundary condition (\[eq:bcBA\]) is essential in recovering this result. (iii) $a>0$. For this range of values of $a$ we have $$k_s(t) \simeq (m+a) \left(\frac{t}{s} \right)^\beta, \ \ \ \ R_s(t) \simeq \beta (m+a)^2 \left( \frac{t}{s} \right)^\beta \ln \left( \frac{t}{s} \right),$$ yielding $$M_s(t) \simeq \beta^2 \frac{(m-1)(m+a)^3}{(2m+a)(1-2\beta)} s^{-2 \beta} \left\{ -t^{2\beta-1} \ln \left(\frac{t}{s}\right) + \frac{ s^{2\beta-1} - t^{2\beta-1}}{1-2 \beta} \right\}+ M_s(s).$$ For the evaluation of $M_s(s)$, we observe that the double summation $\mathcal{S}$ defined in Eq. (\[eq:19\]) shows now a power-law divergence, $\mathcal{S} \simeq s^{1-2\beta} /(1-2\beta)^2$. Thus we have $$M_s(t) \simeq \beta^2 \frac{(m-1)(m+a)^3}{(2m+a)(1-2\beta)} s^{-2\beta} \left\{ -t^{2\beta-1} \ln \left(\frac{t}{s}\right) + \frac{2 s^{2\beta-1} - t^{2\beta-1}}{1-2\beta} \right\},$$ yielding a three vertex correlation function $${\bar{c}}(k,N) \simeq \frac{4 \beta^2(m-1) (m+a)^{3-1/ \beta}}{(2m+a)(1-2\beta)^2} N^{-1} k^{-2+ 1/ \beta}. \label{eq:20}$$ Therefore, for $a>0$ (i.e. $\beta < 1/2$), we obtain that the average clustering of the vertices of degree $k$ is a growing function of $k$, scaling as ${\bar{c}}(k,N) \sim N^{-1} k^{-2+ 1/ \beta}$. Since by definition the clustering must be smaller than $1$, this growing behavior must be restricted to degree values $k \lesssim N^{\beta/(1-2\beta)}$. Computer simulations -------------------- In order to check the analytical results obtained in this Section, we have performed extensive numerical simulations of the LPA model. Simulations were performed for system sizes ranging from $N=10^3$ to $N=10^6$, averaging over $100$ network samples for each value of $N$ and $a$. We focus in particular in the ranges $a<0$ and $a>0$, which have not been previously explored (for numerical data corresponding to $a=0$, the BA model, see Refs. [@alexei02; @klemm02b]). In Figs. \[fig:knnlin\] and \[fig:ckaneglin\] we explore the behavior of networks generated for $a<0$. We consider first the average degree of the nearest neighbors ${\bar{k}_{nn}}(k,N)$. Fig. \[fig:knnlin\](a) corresponds to $m=4$, $a=-2$, values that yield $\beta=2/3$ and $\gamma=2.5$, while Fig. \[fig:knnlin\](b) plots data for $m=4$, $a=-3$, corresponding to $\beta=4/5$ and $\gamma=2.25$. The dashed lines represent the power law behavior $k^{-2+1/ \beta }$ expected analytically. We observe that, as the size of the network increases, the data follow the predicted scaling ${\bar{k}_{nn}}(k,N) \sim N^{2\beta-1} k^{-2+1/ \beta}$ on larger and larger ranges. Nevertheless, the logarithmic corrections present in Eq. (\[eq:15\]) are clearly visible from the large $k$ deviations shown by the data (middle plots in Fig. \[fig:knnlin\]). The logarithmic correction can, in fact, be taken into account if one rescales ${\bar{k}_{nn}}(k,N)$ appropriately. Namely, if we define $${\bar{k}_{nn}}^\mathrm{resc}(k,N) = {\bar{k}_{nn}}(k,N) - (m+a)\ln \left( \frac{k}{m+a}\right),$$ then, from see Eq. (\[eq:15\]), we expect $${\bar{k}_{nn}}^\mathrm{resc}(k,N) N^{1-2\beta} \sim k^{-2+1/\beta}.$$ In the bottom plots of Fig. \[fig:knnlin\] we draw the rescaled average degree of the nearest neighbors with logarithmic corrections. The collapse of the data is indeed surprisingly good, given the numerous approximations and leading order cancellations made in our calculations. The remaining discrepancy at very large $k$ is presumably due to the subdominant terms we have neglected. In Fig. \[fig:ckaneglin\] we represent the clustering spectrum ${\bar{c}}(k,N)$ for the same parameters as before, i.e. $m=4$, $a=-2$ (a) and $m=4$, $a=-3$ (b). The top plots represent the corresponding nonrescaled raw data. According to the solution provided in Eq. (\[eq:ckaneg\]), for small values of $k$ it is expected an asymptotic scaling of the form ${\bar{c}}(k,N) \sim N^{2\beta-2} \ln N k^{-4+2/ \beta}$. This behavior is well recovered in the bottom plots in Fig. \[fig:ckaneglin\] for both values of $a$, where we can see that the first points in the graphics for different values of $N$ collapse onto the same curve, with the predicted $k$ dependence. For large values of $k$, on the other hand, we expect instead a scaling ${\bar{c}}(k,N) \sim N^{2\beta-2} k^{-2+1/ \beta}$, which is again recovered in the middle plots of this Figure, showing a better collapse in the intermediate range of $k$ values. At very large $k$ values, finally, the neglected logarithmic terms come into play, affecting the scaling of the data. It is important to notice the important role played by the boundary condition Eq. (\[eq:4\]), which is responsible for the second term in Eq. (\[eq:ckaneg\]), giving the correct scaling behavior for small $k$. In Fig. \[fig:positivelin\] we finally explore the average degree of the nearest neighbors (a) and the clustering spectrum (b) for the LPA model with $a>0$. We focus in particular on the values $m=4$ and $a=2$ (top plots), yielding $\beta=2/5$, $\gamma=3.5$; $a=5$ (middle plots), with $\beta=4/13$, $\gamma=4.25$; and $a=10$ (bottom plots), that corresponds to $\beta=2/9$, $\gamma=5.5$. For the ${\bar{k}_{nn}}(k,N)$ function our theoretical analysis predicts a function independent of the network size, and slowly (logarithmically) growing with the degree. These predictions are confirmed in Fig. \[fig:positivelin\](a). It is noteworthy that the theoretical prediction becomes more accurate for large $a$: While the collapse is quite good for $a\geq5$, the graphs are a bit scattered for the smallest value of $a$ considered. This fact is due to the slow convergence (as $N$ grows) to the theoretical asymptotic form for small $a$. Analogously, the clustering spectrum shows the predicted scaling ${\bar{c}}(k,N) \sim N^{-1} k^{-2+1/ \beta}$, Fig. \[fig:positivelin\](b). The dependence on system size is correctly captured by our analysis for larger values of $a$. In this range, however, the power-law dependence on $k$ seems to depart from the theoretical exponent $-2+1/ \beta$. This apparent departure can be due to the limited range of degrees for such large values of the degree exponent (the degree range decreases for increasing $a$), as well as to the subdominant terms neglected in the asymptotic expression Eq. (\[eq:20\]). Growing networks with large clustering {#sec:gener-dorog-mend} ====================================== As we have seen in the previous Section, the LPA model yields a clustering spectrum ${\bar{c}}(k)$ that, even if presenting a non-trivial scaling, vanishes in the thermodynamic limit, i.e., $\lim_{N\to\infty} {\bar{c}}(k,N) = 0$. However, for many complex networks, such as the Internet [@romuvespibook], we observe a function ${\bar{c}}(k)$ scaling with $k$, together with a finite clustering. Several models have been proposed which reproduce this feature. In particular, Dorogovtsev, Mendes and Samukhin (DMS) introduced in Ref. [@dms] a scale-free growing network with large clustering coefficient $C$. The model is defined as follows: At each time-step, a vertex is added and connected to [the two extremities of a randomly chosen edge]{}, thus forming a triangle. The resulting network has a power-law degree distribution $P(k) \sim k^{-3}$, with ${\langle k \rangle}=4$, and since each new vertex induces the creation of at least one triangle, we expect this model to generate networks with finite clustering coefficient. We consider here a generalization of the DMS model, in which every new node is connected to the extremities of $m/2$ randomly chosen [edges]{}, where $m$ is an even number. The original model corresponds thus to $m=2$, and this generalization allows to tune the average degree, setting it to ${\langle k \rangle}=2m$. It is important to notice that this model actually contains the LPA mechanism in a disguised form. Indeed, the probability to choose a vertex $s$ is clearly proportional to the number of edges arriving to $s$, i.e. to its degree $k_s$. At time $t$ there are $mt$ edges so that $\sum_s k_s=2mt$ and the probability to choose $s$ when choosing one edge is $k_s/(mt)$ ($\sum_s k_s/(mt)=2$ since one chooses indeed $2$ vertices). This process is repeated $m/2$ times and thus, at each time step the probability to choose $s$ is $k_s/(2t)$. This shows that another way of formulating the random choice of an edge is in fact the following: a vertex $s$ is chosen with the usual preferential attachment probability $k_s/(2mt)$, and then one of its neighbors is chosen at random, i.e. with probability $1/k_s$. It is then clear that the rate equation for the degree is given by $$\frac{d k_s(t)}{d t} = \frac{k_s(t)}{2t},$$ leading to $k_s(t) = m (t/s)^{1/2}$ and to a scale free degree distribution of the form $P(k) \approx 2 m^2 k^{-3}$. We are now in position to write down the rate equations for the network correlations, taking into account that, each time a vertex is chosen, so is one of its neighbours. Two vertex degree correlations ------------------------------ At each time step, $R_s(t)$ can increase either if the vertex $s$ is chosen (and then $R_s$ increases by $m+1$ because a neighbour of $s$ is also chosen), or if a neighbour $j$ is chosen together with a neighbour $l$ of $j$, with $l \neq s$ (and then $R_s$ increases by $1$). Therefore, we have that $$\begin{aligned} \frac{dR_s(t)}{dt} &=& (m+1) \frac{k_s(t)}{2t} + \sum_{j \in {\cal V}(s)} \frac{k_j(t)}{2t} \left( 1 - \frac{1}{k_j(t)} \right)\nonumber \\ &=&\frac{m k_s(t)}{2t} + \frac{R_s(t)}{2t} \ .\end{aligned}$$ This is exactly the same equation than for the LPA with $a=0$, i.e. the BA model. Moreover, the boundary condition for $R_s(s)$ can be written as $$R_s(s) = \frac{m}{2} \sum_{j=1}^s \frac{k_j(s)}{2ms} \left\{ k_j(s) + 1 + \sum_{l \in {\cal V}(j)} \frac{1}{k_j(s)} [k_l(s) +1] \right\} \ ,$$ where, for each one of the $m/2$ edges chosen by $s$, the first term corresponds to the contribution of $j$ (chosen with probability $k_j/(2ms)$, and the second term to the contribution of a neighbour $l$ of $j$, which is chosen with probability $1/k_j$. This expression is easily reduced to $$R_s(s) = \frac{1}{2} \sum_{j=1}^s \frac{k_j(s)[k_j(s)+1]}{s} \simeq \frac{m^2}{2} \ln s.$$ Once again we obtain the same result than for the LPA with $a=0$. The conclusion is that the ${\bar{k}_{nn}}(k,N)$ function for the generalized DMS model is given by Eq. (\[eq:knnBA\]): the two vertex correlations are independent of the degree and growing with the network size as $\ln N$, in the same fashion as the BA model. Three vertex correlations ------------------------- In order to write down the rate equation for $M_s(t)$, we have to take into account that, at each time step, $m/2$ triangles are formed by the choice of $m/2$ edges, and that, moreover, additional triangles may be formed for $m>2$ by choosing two different edges with a common vertex. At each time step, the increase in $M_s(t)$ is thus given by two terms: - The first one comes from choosing the vertex $s$ with probability $k_s(t)/(2t)$. In this case, $M_s$ increases by $1$, since one of the neighbours of $s$ is also chosen. - The second contribution comes from the following situation: one of the edges chosen is $s-l$, and another one is $j-l'$, with $j \in {\cal V}(s)$, $j \neq l$ and $l' \neq s$. The resulting rate equation reads: $$\begin{aligned} \frac{dM_s(t)}{dt} &=& \frac{k_s(t)}{2t} + \frac{m}{2} \left( \frac{m}{2} -1 \right) \frac{k_s(t)}{mt} \nonumber \\ &\times& \sum_{j \in {\cal V}(s)} \frac{k_j(t)}{mt}\left(1-\frac{1}{k_s(t)}\right) \nonumber \\ &=& \frac{k_s(t)}{2t} + \frac{m-2}{4mt^2} [k_s(t) - 1] R_s(t).\end{aligned}$$ We use $k_s \simeq m \sqrt{t/s}$ and $R_s \simeq m^2 \ln t \sqrt{t/s}/2$ to obtain $$\frac{dM_s(t)}{dt} \simeq \frac{m}{2\sqrt{t s}} + \frac{m^2(m-2)\ln t}{8 s t},$$ whose solution reads $$M_s(t) \simeq m\left(\sqrt{\frac{t}{s}} -1\right) + \frac{m^2(m-2)}{16 s} (\ln^2 t - \ln^2 s) +M_s(s).$$ The boundary condition is again given by two contributions: First, $m/2$ triangles are created by attaching $s$ to $m/2$ edges. The second contribution is given by $$\frac{m}{2} \left( \frac{m}{2} -1 \right) \sum_{j=1}^s \sum_{l \in {\cal V}(j)} \frac{k_j(s)}{ms} \left( 1 - \frac{1}{k_j(s)} \right) \frac{k_{l}(s)}{ms}$$ i.e. the sum over all vertices $j$ of the probability that, among the $m/2$ edges chosen by the new node $s$, one has $j$ for extremity, and another one has a neighbour $l$ of $j$ for extremity. This yields $$\begin{aligned} M_s(s) &=& \frac{m}{2} + \frac{m-2}{4 m s^2} \sum_{j=1}^s R_j(s) [k_j(s) -1] \nonumber \\ &\simeq& \frac{m}{2} + \frac{m^2(m-2)\ln^2 s}{8s} \end{aligned}$$ and finally $$M_s(t) \simeq m \sqrt{\frac{t}{s}} - \frac{m}{2} + \frac{m^2(m-2)}{16 s} (\ln^2 t + \ln^2 s) \ .$$ The clustering spectrum can therefore be written as $${\bar{c}}(k,N) \simeq \frac{2k-m}{k(k-1)} +\frac{m-2}{8N} \left( \ln^2 N + \ln^2 \left(\frac{m^2 N}{k^2}\right) \right) \ . \label{eq:ckdms}$$ The clustering spectrum is now finite in the infinite size limit, $${\bar{c}}(k) = \lim_{N\to\infty} {\bar{c}}(k,N) \simeq \frac{2k-m}{k(k-1)}.$$ It is interesting to see that, for the original model with $m=2$, the finite-size corrections actually vanish and we obtain the result ${\bar{c}}(k,N) = 2/k$, independent of $N$. This scaling is also similar to that obtained for the Holme-Kim model in [@szabo]. The knowledge of the exact form of the degree distribution for $m=2$, $P(k)=12/(k(k+1)(k+2))$ [@dms], allows us to obtain the average clustering coefficient $C(m=2)=2\pi^2 -19 \approx 0.739$. More generally, for large $m$, approximating $P(k)$ by $2m^2/k^3$, and sums by integrals, yields $$\begin{aligned} \lefteqn{C(m) = \int_m^\infty P(k) {\bar{c}}(k) \; dk} \nonumber \\ &&\simeq 2m^2 -3m -4/3 + 2m^2(2-m)\ln\left(\frac{m}{m-1}\right). \quad \quad \label{eq:cdms}\end{aligned}$$ Computer simulations -------------------- We have performed extensive numerical simulations of the generalized DMS model studied in this Section. We focus on the clustering spectrum ${\bar{c}}(k,N)$ since the results for ${\bar{k}_{nn}}(k)$ are expected to be equal to the case of the BA model. Fig. \[fig:ckdms\](a) shows the excellent agreement between the predicted behaviour, Eq. (\[eq:ckdms\]), and the numerical data for various values of $m$ and sizes ranging from $N=10^4$ to $N=10^6$. As expected, no finite size corrections are present for $m=2$, while they are correctly described by the analytical approach for larger $m$. Moreover, the prediction for the average clustering coefficient $C(m)$, Eq. (\[eq:cdms\]), is also shown to be in excellent agreement with numerical data, Fig. \[fig:ckdms\](b). Weighted growing networks {#sec:weight-grow-netw} ========================= In the previous Sections we have applied the rate equation formalism to analyze two and three vertex correlations in standard models with either vanishing or constant clustering coefficient. The formalism for the two vertex correlations, however, is not limited to these particular cases, and can be easily extended to analyze more complex growing network models. As an example, in this Section we will consider a recently proposed growing weighted network model [@barrat04:_weigh]. Weighted networks [@wiegtedgeneral] are a natural generalization of graphs in which a real quantity is assigned to each edge, representing the importance or weight $w_{ij}$ of the interaction between the vertices $i$ and $j$. Recently [@barratairport], it has been pointed out that real weighted networks present a complex architecture, characterized by broad distributions of weights, as well as nontrivial correlations between the values of the weights and the topological structure of the network. Motivated by these findings, Ref. [@barrat04:_weigh] proposed a dynamic growing weighted network model, in which new edges are attached to old vertices with a connection probability depending on the strength, or total weight, of the vertex. In order to define the model, let us consider a weighted network characterized by the elements $w_{ij}$ defining the weight assigned to the edge connecting vertices $i$ and $j$. We assume the elements $w_{ij}$ to be symmetric, that is, $w_{ij}=w_{ji}$. Each vertex $i$ is characterized by both its degree $k_i$ and its strength $\sigma_i$, defined as $$\sigma_i=\sum_{j \in \mathcal{V}(i)}w_{ij}.$$ For non-weighted networks, in which $w_{ij} = \delta_{ij}$, we obviously recover $\sigma_i = k_i$. The model proposed in Ref. [@barrat04:_weigh] considers a growing network in which at each time step, a new vertex is added to the system and connected with $m$ edges to older vertices. The probability that the new vertex $t$ is connected to $s$ ($s<t$) is given by the connection probability $$\Pi_s(\{s\}, t) = \frac{\sigma_s(t)}{\sum_j \sigma_j(t)}, \label{eq:16}$$ that is, linearly proportional to the strength of the old vertex $s$. Each new edge carries an initial weight $w_0=1$. Additionally, there is a dynamic rearrangement of the weights belonging to the edges of the receiving vertex: When the vertex $s$ receives a new connection, the weight of its edges is increased by an amount $$w_{s j}\to w_{s j} + \delta\frac{w_{s j}}{\sigma_s}, \quad j \in \mathcal{V}(s). \label{rule}$$ This rule implies that, for each new vertex added, the total strength of the network is increased by an amount $2 m + 2 m\delta$, therefore the normalization constant in Eq. (\[eq:16\]) is $\sum_j \sigma_j(t) = 2m(1+\delta)t$. It can be shown, within the continuous $k$ approximation [@barrat04:_weigh], that this model generates scale-free networks, characterized by the quantities $$\sigma_s(t)=m ~\left(\frac{t}{s}\right)^\beta, \quad k_s(t)=\frac{\sigma_s(t) + 2m\delta}{2\delta+1}, \quad P(k) \sim k^{-\gamma}, \label{eq_ss.vs.t}$$ with exponents $$\beta=\frac{2\delta+1}{2\delta+2}, \quad \gamma=\frac{4\delta+3}{2\delta+1}.$$ Therefore, for $\delta>0$, this model yields power-law degree distributions with degree exponent $\gamma \in ]2, 3[$ and $\beta>1/2$. The case $\delta=0$ recovers the BA model. It is easy to see that, at the level of the mean field rate equations in the continuous $k$ approximation, the weighted growing network model described above can be mapped into a growing network with LPA and negative parameter $a$ given by $$a= - \frac{2 m \delta}{2 \delta+1}. \label{eq:24}$$ Therefore, we expect to observe the two and three vertex correlation functions $$\begin{aligned} {\bar{k}_{nn}}(k,N) &\simeq& \frac{m \zeta(2\beta)}{2(1+\delta)} \left( \frac{m}{2\delta+1} \right)^{2-1/ \beta } N^{2 \beta - 1} k^{-2 +1/ \beta } +\frac{m}{2\delta+1}\ln \left(\frac{2 \delta+1}{m} k \right), \label{eq:21}\\ \nonumber {\bar{c}}(k,N) &\simeq & \frac{(m-1)(2\delta+1)^2}{4\delta(\delta+1)^2} \left(\frac{m}{2\delta+1}\right)^{2-1/ \beta} N^{2 \beta - 2} k^{-2 +1/ \beta } \\ &+& \zeta(2\beta)\frac{(m-1)(2\delta+1)^2}{4(\delta+1)^3} \left( \frac{m}{2\delta+1} \right)^{4-2/ \beta } \ln N N^{2 \beta - 2} k^{-4 +1/ \beta}. \label{eq:22}\end{aligned}$$ Weighted two vertex correlations -------------------------------- The definition of the ${\bar{k}_{nn}}(k)$ function we have computed above completely neglects the effect of the weights. Therefore, it provides a biased view of real correlations in the system (for example, two neighbors with the same degree but widely different weights give the same contribution). In order to take into account the effect of the weights associated to the edges, it has been proposed a new correlation measure, the weighted average degree of the nearest neighbors, ${\bar{k}_{nn}}^w(k)$ [@barratairport], defined as follows: $${\bar{k}_{nn}}^w(s) = \frac{1}{\sigma_s(t)} \sum_{j \in \mathcal{V}(s)} w_{s j}(t) k_j(t).$$ This definition implies that ${\bar{k}_{nn}}^w(s) > {\bar{k}_{nn}}(s)$ if the edges with largest weight point to the neighbors with largest degree, while ${\bar{k}_{nn}}^w(s) < {\bar{k}_{nn}}(s)$ in the opposite case. Therefore, ${\bar{k}_{nn}}^w(s)$ measures the effective affinity to connect with large or small degee neighbors, according to the magnitude of the interaction weight. The weighted average degree of the nearest neighbors ${\bar{k}_{nn}}^w(k)$, is defined as the average of ${\bar{k}_{nn}}^w(s)$ for all the vertices with the same degree $k$. We can study analytically the weighted two vertex correlations by seeking a rate equation for the quantity $$Q_s(t) = \sum_{j \in {\cal V}(s)} w_{s j}(t) k_j(t)$$ According to the rules defining the model, at each time step $Q_s(t)$ can increase its value by two mechanisms: - If a new vertex is directly attached to $s$, $Q_s(t)$ increases by an amount $m+\delta Q_s/\sigma_s$ - If a new vertex is attached to a neighbour $j$ of $s$, then $Q_s(t)$ increases by $w_{s j}+\delta w_{s j}/\sigma_j + \delta w_{s j} k_j/\sigma_j$ Therefore, the rate equation fulfilled by $Q_s(t)$ is $$\begin{aligned} \lefteqn{\frac{dQ_s(t)}{d t} = m \Pi_s(\{\sigma\}, t) \left( m+ \frac{\delta}{\sigma_s(t)}Q_s(t) \right)} \nonumber \\ &+& \sum_{j \in {\cal V}(s)} m \Pi_j(\{\sigma\}, t) \left( w_{s j}+\delta \frac{w_{s j}}{\sigma_j(t)} + \delta w_{s j} \frac{k_j(t)}{\sigma_j(t)} \right) \quad \quad\end{aligned}$$ which, in terms of $\sigma_s(t)$ and $Q_s(t)$, yields $$\frac{dQ_s(t)}{dt} = \left(\beta + \frac{\delta}{1+\delta}\right) \frac{Q_s(t)}{t} +\frac{m+\delta-2m\delta}{2(1+\delta)}\frac{\sigma_s(t)}{t}.$$ Inserting the value of $\sigma_s(t)$ given by Eq. (\[eq\_ss.vs.t\]), the general solution of this equation is $$Q_s(t)= A_0(s) t^{\beta+\delta/(1+\delta)} - \frac{m}{2\delta} (m+\delta-2m\delta) \left(\frac{t}{s}\right)^\beta.$$ Since all new edges have an initial weight $w_0=1$, the initial condition for $Q_s(t)$ coincides with that of $R_s(t)$. Solving for $A_0(s)$ from Eq. (\[eq:23\]), substituting for the corresponding value of $a$ given by Eq. (\[eq:24\]), we finally obtain in the large $k$ and $N$ limit $${\bar{k}_{nn}}^w(k,N) \simeq \frac{m\zeta(2\beta)}{2(1+\delta)(2 \delta +1)} N^{2 \beta -1}. \label{eq:25}$$ That is, in this model the weighted average degree of the nearest neighbours is independent of $k$, signaling the absence of two vertex weighted correlations, as indeed found numerically in Ref. [@longweighted]. There is, however, a scaling with the system size, given by the factor $N^{2 \beta -1}$, which is the same as that found for the nonweighted correlations for the same value of $\gamma$. Computer simulations -------------------- We have performed numerical simulations of the weighted growing network model described in Ref. [@barrat04:_weigh], for sizes ranging from $N=10^3$ to $N=10^5$, focusing on the behavior of the average degree of the nearest neighbors, for both its non-weighted and weighted versions. In Fig. \[fig:knnm2d2\] we plot the average degree of the nearest neighbors ${\bar{k}_{nn}}(k,N)$ for $m=2$ and $\delta=2$ (a) which corresponds to a network with $\beta=5/6$, $\gamma=2.20$, and $\delta=5$ (b), that yields $\beta=11/12$, $\gamma= 2.09$. As expected from the analytical analysis performed above, the obtained scaling is analogous to the LPA model: the numerical data follows the predicted form ${\bar{k}_{nn}}(k,N) \sim N^{2\beta-1} k^{-2+1/ \beta}$. The bottom plots highlight the presence of the logarithmic correction of Eq.(\[eq:21\]), by plotting the rescaled function $${\bar{k}_{nn}}^{\mathrm{resc}}(k,N) = {\bar{k}_{nn}}(k,N) - \frac{m}{2 \delta+1} \ln \left(\frac{2 \delta+1}{m}\right).$$ In this case, it is noticeable that the rescaled ${\bar{k}_{nn}}^\mathrm{resc}(k,N)$ function with logarithmic corrections yields a better data collapse than that shown by the LPA model. Even though both models are identical at the mean field level, the existing microscopic differences seem to yield smaller subleading corrections for the weighted growing network model. For this same set of parameters, we have also evaluated the weighted average degree of the nearest neighbors, ${\bar{k}_{nn}}^w(k,N)$, shown in the middle plot in Fig. \[fig:knnm2d2\](a) (filled symbols). We observe that the ${\bar{k}_{nn}}^w(k,N)$ is indeed, as expected, independent of $k$, and scales with the system size with the predicted factor $N^{2 \beta -1}$. Conclusions {#sec:conclusions} =========== A complete theoretical characterization of a growing network model should imply not only the estimation of the corresponding degree distribution, but also an analytical study of the functional form of the correlations between the degrees of neighboring vertices. Capitalizing on the work of Szabó et al. [@szabo; @szaboproc], in this paper we have provided a formalism to compute two vertex correlations, expressed by means of the average degree of the nearest neighbors of the vertices of degree $k$, ${\bar{k}_{nn}}(k)$, valid for growing network models generated by means of the preferential attachment mechanism and belonging to the class of the so-called “citation networks”. The formalism is based on a rate equation in the continuous $k$ approximation, together with the appropriate boundary condition, that can be easily solved in the case in which the preferential attachment is linear in the degree. Additionally, we have presented a more complete description of the rate equation determining the clustering spectrum ${\bar{c}}(k)$, by discussing the effects of boundary conditions. Applying this framework to several growing network models, we have obtained asymptotic expressions for the functions ${\bar{k}_{nn}}(k,N)$ and ${\bar{c}}(k,N)$, evidentiating both the degree dependence and the scaling with the system size, due to finite size effects. As a general result, we conclude that networks generated by LPA with degree exponent $\gamma<3$, exhibit the scaling behavior $${\bar{k}_{nn}}(k,N) \sim N^{2\beta-1}k^{-2+1/\beta},$$ previously obtained by means of scaling arguments [@dorogorev], which is the signature of disassortative (negative) two vertex correlations. We have also been able to identify the presence of logarithmic corrections in models with LPA, which clearly appear in computer simulations of the model. For this LPA model, we also observe the presence of small assortative correlations for degree exponents $\gamma>3$, characterized by a logarithmic growth of the ${\bar{k}_{nn}}(k,N)$ function, which is otherwise independent of the network size. The situation is more complex in what concerns the clustering spectrum ${\bar{c}}(k,N)$. For $\gamma>3$, we observe the presence of a crossover between two power-law decays in the degree, ${\bar{c}}(k)\sim k^{-\alpha}$, with $\alpha = -4 + 2 / \beta$ for $k \lesssim (\ln N)^{\beta/(2\beta-1)}$, and $\alpha = -2+1/\beta$ in the asymptotic limit, while for $\gamma>3$ we obtain an increasing ${\bar{c}}(k,N)$ function, limited by an upper degree cutoff. >From this results we can conclude that the value $\alpha \simeq 1$ observed in the literature [@ravasz02; @szabo] is not a generic feature of all scale-free networks [@szaboproc]. However, we notice that LPA yields networks with a vanishing clustering coefficient. In order to assess the possible effects of this factor, we have considered the DMS model [@dms], that generates networks with a large value of $C$, as observed in real networks. In this case, we obtain a lack of two vertex correlations, while the clustering spectrum scales as ${\bar{c}}(k)\sim k^{-1}$. An analogous result is obtained for the similar Holme-Kim model [@szabo; @holme02c]. From this result we could be tempted to conclude that a clustering spectrum scaling with exponent $\alpha=1$ is related to the lack of correlations. However, this explanation would be in conflict with the empirical observation of this exponent in real networks with clear dissasortative mixing. More work is therefore needed in order to elucidate the relations between the scaling exponents of ${\bar{k}_{nn}}(k)$ and ${\bar{c}}(k)$ in general complex networks. As a final point, we have shown the flexibility of the rate equation approach to compute two vertex correlations by applying it to a recently proposed weighted growing network model, in which edges are further characterized by a distribution of weights that is dynamically coupled to the evolving topology of the network. For this model, we are able to extend our formalism to deal with weighted two vertex correlations, which measure the effect of the strength of the interactions between neighboring vertices. The very good agreement shown between our analytical estimates and numerical simulations suggest that the method proposed in this paper to compute two vertex correlations is in general valid to characterize growing citation network models. An obvious improvement would be to extend it to deal with models in which vertex and edge removal, and edge rewiring, are allowed. This inclusion, however, would probably lead to quite complex non-local rate equation, whose solution would be much harder to tackle. We thank M. Alava, M. Boguñá, and A. Vespignani for helpful comments and discussions. This work has been partially supported by EC-FET Open Project No. IST-2001-33555 and contract 001907 (DELIS). R.P.-S. acknowledges financial support from the Ministerio de Ciencia y Tecnología (Spain), and from the Departament d’Universitats, Recerca i Societat de la Informació, Generalitat de Catalunya (Spain). [10]{} B. Bollob[á]{}s, [Modern Graph Theory]{} (Springer-Verlag, New York, 1998). R. Pastor-Satorras and A. Vespignani, [Evolution and structure of the Internet: A statistical physics approach]{} (Cambridge University Press, Cambridge, 2004). R. Albert, H. Jeong, and A.-L. Barab[á]{}si, Nature [401]{}, 130 (1999). A. Wagner, Mol. Biol. Evol. [18]{}, 1283 (2001). H. Jeong, S. S. Mason, A. L. Barab[á]{}si, and Z. N. Oltvai, Nature [411]{}, 41 (2001). S. Maslov and K. Sneppen, Science [296]{}, 910 (2002). S. Wasserman and K. Faust, [Social Network Analysis: Methods and Applications]{} (Cambridge University Press, Cambridge, 1994). R. Albert and A.-L. Barab[á]{}si, Rev. Mod. Phys. [74]{}, 47 (2002). S. N. Dorogovtsev and J. F. F. Mendes, [Evolution of networks: From biological nets to the [I]{}nternet and [WWW]{}]{} (Oxford University Press, Oxford, 2003). D. J. Watts and S. H. Strogatz, Nature [393]{}, 440 (1998). R. Cohen and S. Havlin, Phys. Rev. Lett. [90]{}, 058701 (2003). R. A. Albert, H. Jeong, and A.-L. Barab[á]{}si, Nature [406]{}, 378 (2000). D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. Lett. [85]{}, 5468 (2000). R. Cohen, K. Erez, D. ben Avraham, and S. Havlin, Phys. Rev. Lett. [85]{}, 4626 (2000). R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. [86]{}, 3200 (2001). R. Pastor-Satorras and A. Vespignani, Phys. Rev. E [63]{}, 066117 (2001). R. M. May and A. L. Lloyd, Phys. Rev. E [64]{}, 066112 (2001). M. Barth[é]{}lemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, Phys. Rev. Lett. [92]{}, 178701 (2004). L. K. Gallos and P. Argyrakis, Phys. Rev. Lett. [92]{}, 138301 (2004). M. Catanzaro, M. Bogu[ñ]{}[á]{}, and R. Pastor-Satorras, Diffusion-annihilation processes in complex networks, e-print cond-mat/0407447. P. Erd[ö]{}s and P. R[é]{}nyi, Publicationes Mathematicae [6]{}, 290 (1959). B. Bollob[á]{}s, [Random graphs]{} (Academic Press, London, 1985). A.-L. Barab[á]{}si and R. Albert, Science [286]{}, 509 (1999). S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Phys. Rev. Lett. [ 85]{}, 4633 (2000). P. L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett. [85]{}, 4629 (2000). R. Kumar [et al.]{}, in [Proceedings of the 41th [IEEE]{} Symposium on Foundations of Computer Science ([FOCS]{})]{} (IEEE, New York, 2000), pp. 57–65. R. Pastor-Satorras, A. V[á]{}zquez, and A. Vespignani, Phys. Rev. Lett. [ 87]{}, 258701 (2001). A. V[á]{}zquez, R. Pastor-Satorras, and A. Vespignani, Phys. Rev. E [65]{}, 066130 (2002). M. E. J. Newman, Phys. Rev. Lett. [89]{}, 208701 (2002). M. E. J. Newman, Phys. Rev. E [67]{}, 026126 (2003). M. Bogu[ñ]{}[á]{} and R. Pastor-Satorras, Phys. Rev. E [68]{}, 036112 (2003). E. Ravasz and A.-L. Barab[á]{}si, Phys. Rev. E [67]{}, 026112 (2003). M. E. J. Newman, in [Handbook of Graphs and Networks: From the Genome to the [I]{}nternet]{}, edited by S. Bornholdt and H. G. Schuster (Wiley-VCH, Berlin, 2003), pp. 35–68. G. Szab[ó]{}, M. Alava, and J. Kert[é]{}sz, Phys. Rev. E [67]{}, 056102 (2003). A. V[á]{}zquez [et al.]{}, Phys. Rev. E [67]{}, 046111 (2003). M. Bogu[ñ]{}[á]{} and R. Pastor-Satorras, Phys. Rev. E [66]{}, 047104 (2002). Y. Moreno and A. V[á]{}zquez, Eur. Phys. J. B [31]{}, 265 (2003). A. V[á]{}zquez and Y. Moreno, Phys. Rev. E [67]{}, 015101 (2003). G. Szab[ó]{}, M. Alava, and J. Kert[é]{}sz, in [Complex Networks]{}, Vol. 650 of [Lecture Notes in Physics]{}, edited by E. Ben-Naim, H. Frauenfelder, and Z. Toroczkai (Springer Verlag, Berlin, 2004). S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. [51]{}, 1079 (2002). S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Phys. Rev. E [63]{}, 062101 (2001). P. Holme and B. J. Kim, Phys. Rev. E [65]{}, 026107 (2002). A. Barrat, M. Barth[é]{}lemy, and A. Vespignani, Phys. Rev. Lett. [92]{}, 228701 (2004). R. Albert and A.-L. Barab[á]{}si, Phys. Rev. Lett. [85]{}, 5234 (2000). M. Bogu[ñ]{}[á]{}, R. Pastor-Satorras, and A. Vespignani, Phys. Rev. Lett. [ 90]{}, 028701 (2003). K. Klemm and V. M. Egu[í]{}luz, Phys. Rev. E [65]{}, 057102 (2002). J. Clark and D. A. Holton, [A first look at graph theory]{} (World Scientific, Singapore, 1990). A. Barrat, M. Barth[é]{}lemy, R. Pastor-Satorras, and A. Vespignani, Proc. Natl. Acad. Sci. USA [101]{}, 3747 (2004). A. Barrat, M. Barth[é]{}lemy, and A. Vespignani, Modeling the evolution of weighted networks, e-print cond-mat/0406238. [^1]: Note that the probability $p^k_{k', k''}$ can depend on the degree $k$ of the common vertex
--- abstract: \| We present a reactive beta model that includes the leverage effect to allow hedge fund managers to target a near-zero beta for market neutral strategies. For this purpose, we derive a metric of correlation with leverage effect to identify the relation between the market beta and volatility changes. An empirical test based on the most popular market neutral strategies is run from 2000 to 2015 with exhaustive data sets including 600 US stocks and 600 European stocks. Our findings confirm the ability of the reactive beta model to withdraw an important part of the bias from the beta estimation and from most popular market neutral strategies.\ Key words: Beta, Correlation, Volatility, Portfolio Management, Market Neutral Strategies.\ JEL classification: C5, G01, G11, G12, G32.\ author: - 'Sebastien Valeyre [^1]' - 'Denis Grebenkov [^2]' - 'Sofiane Aboura [^3]' title: The Reactive Beta Model --- Introduction ============ The correct measurement of market betas is paramount for market neutral hedge fund managers who target a near-zero beta. Contrary to common belief, perfect beta neutral strategies are difficult to achieve in practice, as the mortgage crisis in 2008 exemplified, when most market neutral funds remained correlated with stock markets and experienced considerable unexpected losses. This exposure to the stock index [@Banz81; @Fama92; @Fama93; @Carhart97; @Ang06] is even stronger during down market conditions [@Mitchell01; @Agarwal04; @Bussiere15]. In such a period of market stress, hedge funds may even add no value [@Asness01]. In this paper, we test the quality of hedging for four popular strategies that have often been used by hedge funds. The first and most important strategy captures the low beta anomaly [@Black72; @Black72b; @Haugen75; @Haugen91; @Ang06; @Baker13; @Frazzini14; @Hong16] that defies the conventional wisdom on the risk and reward trade-off predicted by the CAPM [@Sharpe64]. According to this anomaly, high beta stocks underperform low beta stocks. Similarly, stocks with high idiosyncratic volatility earn lower returns than stocks with low idiosyncratic volatility [@Malkiel97; @Goyal03; @Ang06; @Ang09]. The related strategy consists in shorting high beta stocks and buying low beta stocks. The second important strategy captures the size effect [@Banz81; @Reinganum81; @Fama92], in which stocks of small firms tend to earn higher returns, on average, than stocks of larger firms. The related strategy consists in buying stocks with small market capitalization and shorting those with high market capitalization. The third strategy captures the momentum effect [@Jegadeesh93; @Carhart97; @Grinblatt04; @Fama12], where past winners tend to continue to show high performance. This strategy consists in buying the past year’s winning stocks and shorting the past year’s losing ones. The forth strategy captures the short-term reversal effect [@Jegadeesh90], where past winners on the last month tend to show low performance. This strategy consists in buying the past month’s losing stocks and shorting the past month’s winner ones and would be highly profitable if there were no transaction cost and no market impact. Testing the quality of the hedge of the strategies is equivalent to assess the quality of the beta measurements that is difficult to realize directly as the true beta is not known. The implementation of all these strategies requires a reliable estimation of the betas to maintain the hedge. The Ordinary Least Squares (OLS) estimation remains the most frequently employed method, even though it is impaired in the presence of outliers, notably from small companies [@Fama08], illiquid companies [@Amihud02; @Acharyaa05; @Ang13], and business cycles [@Ferson99]. In these circumstances, the OLS beta estimator might be inconsistent. To overcome these limitations, our approach consists in renormalizing the returns to make them closer to Gaussian and thus to make the OLS estimator more consistent. In addition, many papers report that betas are time varying [@Blume71; @Fabozzi78; @Jagannathan96; @Fama97; @Bollerslev88; @Lettau01; @Lewellen06; @Ang07; @Engle16]. This can lead to measurement errors that could create serious bias in the cross-sectional asset pricing test [@Shanken92; @Chan92; @Meng11; @Bali17]. In fact, firms’ stock betas do change over time for several reasons. The firm’s assets tend to vary over time via acquiring or replacing new businesses that makes them more diversified. The betas also change for firms that change in dimension to be safer or riskier. For instance, financial leverage may increase when firms become larger, as they can issue more debt. Moreover, firms with higher leverage are exposed to a more unstable beta [@Galai76; @DeJong85]. One way to account for the time dependence of beta is to consider regime changes when the return history used in beta estimation is long enough. Surprisingly, only one paper [@Chen05] suggests a solution to capture the time dependence and discusses regime changes for the beta using a multiple structural change methodology. The study shows that the risk related to beta regime changes is rewarded by higher returns. Another way is to examine the correlation dynamics. @Francis79 finds that “the correlation with the market is the primary cause of changing betas ... the standard deviations of individual assets are fairly stable”. This finding calls for special attention to the correlation dynamics addressed in our paper but apparently insufficiently investigated in other works. Despite the extended literature on this issue, little attention has been paid to the link between the leverage effect[^4] and the beta. The leverage effect is defined as the negative correlation between the securities’ returns and their volatility changes. This correlation induces residual correlations between the stock overperformances and beta changes. In fact, earlier studies have heavily focused on the role of the leverage effect on volatility [@Black76; @Christie82; @Campbell92; @Bekaert00; @Bouchaud01; @Valeyre13]. Surprisingly, despite its theoretical and empirical underpinnings, the leverage effect has not been considered so far in beta modeling, while it is a measure of risk. We aim to close this gap. Our paper starts by investigating the role of the leverage effect in the correlation measure by extending the reactive volatility model [@Valeyre13], which efficiently tracks the implied volatility by capturing both the retarded effect induced by the specific risk and the panic effect, which occurs whenever the systematic risk becomes the dominant factor. This allows us to set up a reactive beta model incorporating three independent components, all of them contributing to the reduction of the bias of the hedging. First, we take into account the leverage effect on beta, where the beta of underperforming stocks tends to increase. Second, we consider a leverage effect on correlation, in which a stock index decline induces an increase in correlations. Third, we model the relation between the relative volatility (defined as the ratio of the stock’s volatility to the index’s volatility) and the beta. When the relative volatility increases, the beta increases as well. All three independent components contribute to the reduction of biases in the naive regression estimation of the beta and therefore considerably improve hedging strategies. The main contribution of this paper is the formulation of a reactive beta model with leverage effect. The economic intuition behind the reactive beta model is the derivation of a suitable beta measure allowing the implementation of the popular market neutral hedging strategies with reduced bias and smaller standard deviation. In contrast, portfolio managers who use naive beta measures remain exposed to systematic risk factors that create biases in their market neutral strategies. An empirical test is performed based on an exhaustive dataset that includes 600 American stocks and 600 European stocks from the S&P 500, Nasdaq 100, and Euro Stoxx 600 over the period from 2000 to 2015, which includes several business cycles. This test validates the superiority of the reactive beta model over conventional methods. The article is organized as follows. Section \[sec:model\] outlines the methodology employed for the reactive beta model. Section \[sec:empirical\] describes the data and empirical findings. Section \[sec:robustness\] provides several robustness checks to assess the quality of the reactive beta model against alternative methods. Section \[sec:application\] expands the discussion beyond the field of portfolio management, while Section \[sec:conclusion\] concludes. The reactive beta model {#sec:model} ======================= In this section, we present the reactive beta model with three independent components. First, we take into account the specific leverage effect on beta. Second, we consider the systematic leverage effect on correlation. Third, we model the relation between the relative volatility and the beta via the nonlinear beta elasticity. The leverage effect on beta {#sec:RVM} --------------------------- We first account for relations between returns, volatilities, and beta, which are characterized by the so-called leverage effect. This component takes into account the phenomenon when a beta increases as soon as a stock underperforms the index. Such a phenomenon can be fairly well described by the leverage effect captured in the reactive volatility model. We call the specific leverage effect the negative relation between specific returns and the risk (here, the beta), where the specific return is the non-systematic part of the returns (a stock’s overperformance). The specific leverage effect on beta follows the same dynamics as the specific leverage effect introduced in the reactive volatility model. ### The reactive volatility model {#sec:stock_index} This section aims at capturing the dependence of betas on stock overperformance (when a stock is overperforming, its beta tends to decrease). For this purpose, we rely on the methodology of the reactive volatility model [@Valeyre13] to derive a stable measure of beta by using the renormalization factor that depends on the stock’s overperformance. The model describes the systematic and specific leverage effects. The systematic leverage due to the panic effect and the specific leverage due to a retarded effect have very different time scales and intensity. These two different effects were investigated by @Bouchaud01 [@Valeyre13]. We start by recalling the construction of the reactive volatility model, which explicitly accounts for the leverage effect on volatility. Let $I(t)$ be a stock index at day $t$. It is well known that arithmetic returns, $r_I(t) = \delta I(t)/I(t-1)$, are heteroscedastic, partly due to price-volatility correlations. Throughout the text, $\delta$ refers to a difference between successive values, e.g., $\delta I(t) = I(t)-I(t-1)$. The reactive volatility model aims at constructing an appropriate “level” of the stock index, $L(t)$, to substitute the original returns $\delta I(t)/I(t-1)$ by less-heteroscedastic returns $\delta I(t)/L(t-1)$. For this purpose, we first introduce two “levels” of the stock index as exponential moving averages (EMAs) with two time scales: a slow level $L_s(t)$ and a fast level $L_f(t)$. In addition, we denote by $L_{is}(t)$ the EMA (with the slow time scale) of the price $S_i(t)$ of the stock $i$ at time $t$. These EMAs can be computed using standard linear relations: $$\begin{aligned} \label{eq:Lslow} L_s(t) &=& (1-\lambda_s) L_s(t-1) + \lambda_s I(t) , \\ \label{eq:Lf} L_f(t) &=& (1-\lambda_f) L_f(t-1) + \lambda_f I(t) , \\ L_{is}(t) &=& (1-\lambda_s) L_{is}(t-1) + \lambda_s S_{i}(t) ,\end{aligned}$$ where $\lambda_s$ and $\lambda_f$ are the weighting parameters of the EMAs that we set to $\lambda_s = 0.0241$ and $\lambda_f = 0.1484$, relying on the estimates by @Bouchaud01. The slow parameter corresponds to the relaxation time of the retarded effect for the specific risk, whereas the fast one corresponds to the relaxation time of the panic effect for the systematic risk. These two relaxation times are found to be rather universal, as they are stable over the years and do not change among different mature stock markets. The appropriate levels $L(t)$ and $L_i(t)$, accounting for the leverage effect on the volatility, were introduced for the stock index and individual stocks, respectively [^5] $$\begin{aligned} \label{eq:L_Taylor} L(t) & = & I(t) \left(1 + \frac{L_s(t)-I(t)}{I(t)}\right) \left(1 + \ell ~ \frac{L_f(t) - I(t)}{L_f(t)}\right) , \\ \label{eq:Li_Taylor} L_{i}(t) & = & S_i(t) \underbrace{\left(1 + \frac{L_{is}(t)-S_i(t)}{S_i(t)}\right)}_{\textrm{specific risk}} \underbrace{\left(1 + \ell_i ~ \frac{L_f(t) - I(t)}{L_f(t)}\right)}_{\textrm{systematic risk}},\end{aligned}$$ with the parameters $\ell$ and $\ell_i$ quantifying the leverage. The parameter $\ell$ was defined by @Valeyre13 and deduced to be around 8 from another parameter estimated by @Bouchaud01 on 7 major stock indexes. If $\ell = \ell_i$, the correlation between the stock index and the individual stock $i$ is not impacted by the leverage effect. In turn, if $\ell > \ell_i$, the correlation increases when the stock index decreases. Although $\ell_i$ can generally be specific to the considered $i$-th stock, we ignore its possible dependence on $i$ and set $\ell_i = \ell'$. Using the levels $L(t)$ and $L_i(t)$, we introduce the normalized returns: $$\label{eq:tilder} \tilde r_I = \tilde r_I(t) = \frac{\delta I(t)}{L(t-1)}, \qquad \tilde r_i = \tilde r_i(t) = \frac{\delta S_i(t)}{L_i(t-1)}$$ and compute the renormalized variances $\tilde{\sigma}_I^2$ and $\tilde{\sigma}_i^2$ through the EMAs as: $$\begin{aligned} \label{eq:sigmaIt0} \tilde{\sigma}_I^2(t) &=& (1 - \lambda_\sigma) \tilde{\sigma}_I^2(t-1) + \lambda_\sigma \tilde r_I^2(t) , \\ \tilde{\sigma}_i^2(t) &=& (1 - \lambda_\sigma) \tilde{\sigma}_i^2(t-1) + \lambda_\sigma \tilde r_i^2(t) ,\end{aligned}$$ where $\lambda_\sigma$ is a weighting parameter that has to be chosen as a compromise between the accuracy of the estimated renormalized volatility and the reactivity of that estimation. Indeed, the renormalized returns are constructed to be homoscedastic only at short times because the renormalization based on the leverage effect with short relaxation times ($\lambda_s$, $\lambda_f$) cannot account for long periods of changing volatility related to economic cycles. Since economic uncertainty does not change significantly in a period of two months (40 trading days), we set $\lambda_\sigma$ to $1/40 = 0.025$. This sample length leads to a statistical uncertainty of approximately $\sqrt{1/40} \approx 16\%$. Finally, these renormalized variances can be converted into the reactive volatility $\sigma_I(t)$ of the stock index quantifying the systematic risk governed by the panic effect, and the reactive volatility $\sigma_i(t)$ of each individual stock quantifying the specific risk governed by the leverage effect: $$\begin{aligned} \label{eq:sigmaInew} \sigma_{I}(t) &=& \tilde{\sigma}_{I}(t) \frac{L(t)}{I(t)} , \\ \label{eq:sigmainew} \sigma_{i}(t) &=& \tilde{\sigma}_{i}(t) \frac{L_i(t)}{S_i(t)} .\end{aligned}$$ This reactive volatility captures a large part of the heteroscedascticity, i.e., a large part of the volatility variation is completely explained by the leverage effect. For instance, if the stock index loses 1%, $\frac{L(t)}{I(t)}$ increases by $\ell \times 1\%=8\%$, and stock index volatility increases by 8%. That is enough to capture the large part of the VIX variation, with $R^2 = 0.46$, see Fig. 4 by @Valeyre13. In turn, if the stock underperforms the stock index by 1%, $\frac{L_i(t)}{S_i(t)}$ increases by 1%, and the single stock volatility increases by 1%. ### The specific leverage effect on beta {#sec:beta_specific} The volatility estimation procedure naturally impacts the estimation of beta. Many financial instruments rely on the estimated beta, $\beta_i$, which corresponds to the slope of a linear regression of stocks’ arithmetic returns $r_i$ on the index arithmetic return $r_I$: $$\label{eq:simplemarketmodel} r_i = \beta_i r_I + \epsilon_i, \qquad \textrm{with} \quad r_i = \frac{\delta S_i(t)}{S_i(t-1)}, \quad r_I = \frac{\delta I(t)}{I(t-1)} ,$$ where $\epsilon_i$ is the residual random component specific to stock $i$. We consider another beta estimate, $\tilde \beta_i$, based on the reactive volatility model, in which the renormalized stock returns $\tilde r_i$ are regressed on the renormalized stock index returns $\tilde r_I$: $$\label{eq:simplereactivemodel} \tilde r_{i} = \tilde\beta_i \, \tilde r_{I} + \tilde \epsilon_{i}, \qquad \textrm{with} \quad \tilde r_{i} = \frac{\delta S_{i}(t)}{L_{i}(t-1)}, \quad \tilde r_{I} = \frac{\delta I(t)}{L(t-1)}.$$ We then obtain a reactive beta measure: $$\label{eq:reactivevolatility} \beta_{i}(t) = \tilde \beta_{i}(t) \frac{\sigma_{i}(t) \, \tilde \sigma_{I}(t)}{\sigma_{I}(t)\, \tilde \sigma_{i}(t)} =\tilde \beta_{i}\frac{L_{is}(t) I(t)}{L_{s}(t) S_{i}(t)} ,$$ which includes two improvements: - $\tilde \beta_{i}$, which becomes less sensitive to price changes by accounting for the specific leverage effect; - $\sigma_{i} \tilde \sigma_{I}/(\sigma_{I} \tilde \sigma_{i})$, which changes instantaneously with price changes. When taking into account the short-term leverage effect in correlations, the reactive term is reduced to $\frac{L_{is}(t) I(t)}{L_{s}(t) S_{i}(t)}$. This term has a significant impact, as the beta of underperforming stocks should increase. The systematic leverage effect on correlation {#sec:RCM} --------------------------------------------- ### The empirical estimation of $\ell'$ We coin by systematic leverage effect the negative relation between systematic returns and the risk (here, the correlation), where the systematic returns are the non-specific part of the returns (stock index performance). The systematic leverage effect on correlation follows the same dynamics as the systematic leverage effect introduced in the reactive volatility model (the phenomenon’s duration is approximately 7 days for $\lambda_f=0.1484$). All correlations are impacted together in the same way by the systematic leverage effect, and single stocks and their stock indexes should also shift in the same direction. This explains why the stock’s beta will not change with respect to the index. The implication is that betas are not very sensitive to the systematic leverage effect, in contrast to the specific leverage effect. We consider the impact of the short-term systematic leverage effect on correlation. Assuming that the correlation between each individual stock and the stock index is the same for all stocks, one can define the implied correlation as: [^6] $$\label{eq:rho_Taylor} \rho(t) = \frac{\sigma^{2}_{I}(t) - \sum\limits_{i}{w_{i}^{2}\sigma_{i}^{2}(t)}}{\sum\limits_{i\neq j} w_{i}w_{j}\sigma_{i}(t)\sigma_{j}(t)} ,$$ where $w_i$ represents the weight of stock $i$ in the index. Denoting $$\label{eq:e_Taylor} e_{I}(t) = \frac{\hat{L}_s(t)}{I(t)} - 1 , \qquad e_{i}(t) = \frac{\hat{L}_{is}(t)}{S_i(t)} - 1 ,$$ we use Eqs. (\[eq:sigmaInew\], \[eq:sigmainew\]) to obtain: $$\label{eq:rho_Taylor2} \rho = \frac{\tilde{\sigma}^2_I (1+e_{I})^2 \left(1+ \ell\frac{L_f - I}{L_f} \right)^2 - \left(1+ \ell^{\prime}\frac{L_f - I}{L_f} \right)^2 \sum\limits_{i} w_i^2 (1+e_i)^2 \sigma_i^2 } {\left(1+ \ell^{\prime} \frac{L_f - I}{L_f} \right)^2 \sum\limits_{i\neq j} w_i w_j \tilde{\sigma}_i\tilde{\sigma}_j(1+e_i)(1+e_j)} .$$ If the weights $w_i$ are small, we can ignore the second term; in addition, if $e_i$ are small, then $$\sum\limits_{i\neq j} w_i w_j \tilde{\sigma}_i \tilde{\sigma}_j (1+e_i)(1+e_j) \approx (1+e_I)^2 \tilde\sigma_0^2 ,$$ where $\tilde\sigma_0^2$ is an average of $\tilde\sigma_i^2$. Keeping only the leading terms of the expansion in terms of the small parameter $(L_f-I)/L_f$, one thus obtains $$\label{eq:modelcorrel} \rho \approx \frac{\tilde{\sigma_{I}}^2}{\tilde{\sigma_{0}}^2}\left( 1+2(\ell-\ell^{\prime})\frac{L_{f} - I}{L_{f}}\right) .$$ This relation shows the dynamics of the implied correlation $\rho$ induced by the leverage effect (accounted through the factor $(L_f -I)/L_f$). We assume that the same dynamics are applicable to correlations between individual stocks, i.e., $$\label{eq:correl} \rho_{i,j} = \tilde \rho_{i,j}\left( 1+2(\ell-\ell^{\prime})\frac{L_{f} - I}{L_{f}}\right),$$ where $\tilde{\rho}_{i,j}$ are the parameters specific to each pair of stocks $i$ and $j$. From this relation, we derive a measure of correlation accounting for the leverage effect between the single stock $i$ and the stock index: $$\label{eq:correli} \rho_i = \tilde \rho_i\left( 1+(\ell-\ell^{\prime})\frac{L_{f} - I}{L_{f}}\right) ,$$ where $\tilde{\rho}_i$ are the parameters specific to each stock $i$. Note that there is no factor 2 in front of $(\ell-\ell^\prime)$ in Eq. (\[eq:correli\]) because we have a one-factor model here. We use Eq. (\[eq:correli\]) in the reactive beta model (see Eqs. (\[eq:eq1\], \[eq:beta\_final\]) below) to take into account the varying nature of the correlation in the regression. We rescale the measurement by the normalization factor $(1+(\ell -\ell^\prime) (L_f-I)/L_f)$ and then recover the variation of the correlation through the denormalization factor $1/(1+(\ell-\ell^\prime)(L_f-I)/L_f)$. We emphasize that the parameter $\ell$ in Eq. (\[eq:L\_Taylor\]) that quantifies the systematic leverage for the stock index is slightly different from the parameter $\ell'$ in Eq. (\[eq:Li\_Taylor\]) that quantifies the systematic leverage for single stocks. According to Eq. (\[eq:correl\]), when the market decreases, correlations between stocks increase as $\ell > \ell^\prime$, and therefore, the stock index volatility increases more than the single stocks volatility: $\delta (\sigma_i/\sigma_I)<0$. Once again, the beta is, in contrast to the correlation, weakly impacted by the systematic leverage effect, as all correlations increase in the same way. More precisely, it means that the impact of the increase of correlation in the beta measurement is compensated by a decrease of the relative volatility: $\delta (\sigma_i/\sigma_I)<0$, i.e., the single stock volatility increase is lower than that of the stock index volatility. For this reason, the reactive beta model in Eqs. (\[eq:eq1\], \[eq:beta\_final\]) is not very sensitive to the choice of $\ell'$. Nevertheless, we explain in this section how $\ell'$ is calibrated using the implied volatility index. We measure the level of the systematic leverage effect $\ell^\prime$ for a single stock by regressing Eq. (\[eq:modelcorrel\]) with data from the market-implied correlation S&P 500 index. Figure \[fig:RegVarRho\] illustrates the slope of this regression. By regressing $\frac{L_f-I}{L_f}$ against $\frac{\rho}{\tilde{\rho}_0}$, where $\tilde{\rho}_0$ is the average of $\rho$, we deduce that empirically we can set: $$\label{eq:ell_diff} \ell-\ell^{\prime} = 0.91\pm 0.08 ,$$ with a t-statistics of $11.4$. Since $\ell-\ell^{\prime} \ll \ell (= 8)$, we deduce an important result, namely, that the systematic leverage impact on the correlation is more than 8 times smaller than the systematic leverage impact on volatility. The main consequence is that although statistically significant, the leverage effect is not a major component of the correlation. ![ Daily variations of the CBOE S&P 500 Implied Correlation Indices (ICI) since their inception, divided by their mean, versus daily variations of the leverage factor $(L_f-I)/L_f$. A linear regression (solid line) yields the coefficient $1.82 \pm 0.16$ (i.e., $2(\ell-\ell^\prime) = 1.82$), with $R^2 = 0.13$ and t-statistics of $11.4$. Period: 2007-2015.[]{data-label="fig:RegVarRho"}](figure1.eps){width="120mm"} ### The systematic leverage effect component in the reactive model As just discussed, the correlation increases when stock index price decrease. This effect could generate a bias in the beta measurement as stock index prices could fluctuate in a sample used to measure the slope. Our solution is to adjust the beta between renormalized returns through the correction factor $\L(t)$ defined as $$\label{eq:L} \L(t) = 1+(\ell-\ell^{\prime}) \left(\frac{L_f(t-1)-I(t-1)}{L_f(t-1)}\right),$$ The correction factor $\L(t)$ should be used to estimate the slope between the stock index and single stock returns and then to denormalize the slopefor getting the reactive beta that depends directly on $\L(t)$. The relation between the relative volatility and beta {#sec:beta_relation} ----------------------------------------------------- ### The empirical estimation of beta elasticity In this part, we identify correlations between the relative volatility and beta changes. We choose the relative volatility defined as the ratio $\tilde\sigma_i/\tilde\sigma_I$ as an explanatory variable of $\tilde\beta_i$, because $\tilde\beta_i$ is expected to be constant if the ratio $\tilde\sigma_i/\tilde\sigma_I$ is constant. However, empirically, the ratio $\tilde\sigma_i/\tilde\sigma_I$ can change dramatically between periods of high dispersion (i.e., when stocks are, on average, weakly correlated) and low systematic risk (i.e., when stock indexes are not stressed), and periods of low dispersion and high systematic risk. Figure \[fig:dsigmaidsigmaI\] illustrates, for both European and US markets, that the dispersion among stocks decreases, on average, when markets become volatile. A linear regression of rescaled daily variations of $\tilde\sigma_i$ yields: $$\label{eq:delta_sigma_auxil1} \frac{\delta \tilde\sigma_i(t)}{\tilde\sigma_i(t-1)} \approx 0.4 \, \frac{\delta \tilde\sigma_I(t)}{\tilde\sigma_I(t-1)} + \epsilon_i ,$$ where $\epsilon_i$ is the residual (specific) noise. Using the standard rules for infinitesimal increments, we find from this regression: $$\label{eq:delta_sigma_auxil2} \delta \left(\frac{\tilde\sigma_i}{\tilde\sigma_I}\right) \simeq \frac{\delta \tilde\sigma_i}{\tilde\sigma_I} - \frac{\tilde\sigma_i \, \delta \tilde\sigma_I}{\tilde\sigma_I^2} = \frac{\tilde\sigma_i}{\tilde\sigma_I} \left(\frac{\delta \tilde\sigma_i}{\tilde\sigma_i} - \frac{\delta \tilde\sigma_I}{\tilde\sigma_I} \right) \simeq -0.6 \frac{\tilde\sigma_i}{\tilde\sigma_I} \, \frac{\delta \tilde\sigma_I}{\tilde\sigma_I} ,$$ i.e., the relative volatility $\tilde\sigma_i/\tilde\sigma_I$ is relatively stable but its small variations can still impact the beta estimation. This empirical relation shows that when there is a volatility shock in the market, the stock index volatility increases much faster than the average single stock volatility. ![ Normalized daily variations of $\tilde\sigma_{i}$, $\delta \tilde \sigma_i/\tilde \sigma_i = \frac{\tilde \sigma_i(t)-\tilde\sigma_i(t-1)}{\tilde \sigma_i(t-1)}$, versus normalized daily variations of $\tilde\sigma_{I}$, $\delta \tilde \sigma_I/\tilde \sigma_I = \frac{\tilde \sigma_I(t)-\tilde\sigma_I(t-1)}{\tilde \sigma_I(t-1)}$, for the European market (blue crosses) and the US market (red pluses). The two gray lines show the linear regression of both datasets, with regression coefficients of $0.40$ ($R^2=0.60$) and $0.42$ (with $R^2=0.59$) for the European and US markets, respectively. The time frame includes observations from the technology bubble burst, the U.S. subprime, and Euro debt crises. Period: 1998-2015.[]{data-label="fig:dsigmaidsigmaI"}](figure2.eps){width="120mm"} Because we want to take into account the impact of the relative volatility change on the beta measurement, we introduce the derivative of the beta with respect to the logarithm of the squared relative volatility: $$\label{eq:f} f(\tilde \beta_i) = \frac{d\tilde\beta_i}{d\ln(\tilde\sigma_i^2/\tilde\sigma_I^2)} = \frac{d\tilde\beta_i}{d(\tilde\sigma_i/\tilde\sigma_I)} \, \frac{\tilde \sigma_i}{2\tilde\sigma_I} .$$ We expect that $f(\tilde\beta_i)$ is positive and increasing with $\tilde\beta_i$. Indeed, we expect that a stock with a low beta should have a stable beta (less sensitive to its relative volatility increase), as the increase in this case is most likely due to a specific risk increase. In such a case, the sensitivity of beta to the relative volatility is weak. In the opposite case of a high beta, a stock that is highly sensitive to the stock index will face a beta decline as soon as its relative volatility decreases. Consequently, when there is a volatility shock in the market, $\delta(\frac{\tilde\sigma_i}{\tilde\sigma_I})$ is negative, and therefore, the beta of stocks with high beta and high $f$ is reduced. In turn, the stocks with low beta are less impacted because $f$ is smaller and $\delta(\tilde\sigma_i/\tilde\sigma_I)$ is expected to be less negative. When the correlation of the stock with the stock index is constant, we can use a linear model: $f(\tilde \beta_i) = \tilde\beta_i/2$. In fact, using the relation $\tilde \beta_i = \tilde \rho_i \frac{\tilde \sigma_i}{\tilde \sigma_I}$ and the assumption that $\tilde\rho_i$ is constant (i.e., it does not depend on $ \frac{\tilde \sigma_i}{\tilde \sigma_I}$), one obtains from Eq. (\[eq:f\]) $f = \tilde\rho_i \frac{\tilde \sigma_i}{2\tilde \sigma_I} = \tilde\beta_i/2$. In general, however, the correlation can depend on the relative volatility, and thus, the function $f$ may be more complicated. To estimate $f$, one needs the renormalized beta and the relative volatility. For a better estimation, we aim at reducing even further the heteroscedasticity by using an exponential moving regression of the returns $\tilde r_i$ and $\tilde r_I$ that are renormalized by the estimated normalized index volatility $\tilde \sigma_I$. We denote these renormalized returns as: $$\label{eq:hatr} \hat r_i(t) = \frac{\tilde r_i(t)}{\tilde \sigma_I(t-1)}, \qquad \hat r_I(t) = \frac{\tilde r_I(t)}{\tilde \sigma_I(t-1)} .$$ Computing the EMAs, $$\begin{aligned} \label{eq:SigmahatI} \hat{\phi}_{i}(t) &=& (1-\lambda_{\beta})\hat{\phi}_{i}(t-1) + \lambda_{\beta} \hat r_i(t) \, \hat r_I(t) , \\ \label{eq:hat_sigmaI} \hat{\sigma}_{I}^2(t) &=& (1-\lambda_{\beta})\hat{\sigma}_{I}^2(t-1) + \lambda_{\beta}\, \bigl[\hat r_I(t)\bigr]^2 ,\end{aligned}$$ with $\lambda_\beta = 1/90$, we estimate the beta as: $$\label{eq:beta} \hat \beta_i(t) = \frac{\hat{\phi}_{i}(t)}{\hat{\sigma}_{I}^2(t)} .$$ Here, $\hat \phi_i$ is an estimation of the covariance between stock index returns and single stock returns that includes two normalizations: the levels $L_i$ and $L$ from the reactive volatility model, and $\tilde\sigma_I$ to further reduce heteroscedasticity. We write $\hat \beta_i$ instead of $\tilde \beta_i$ to stress this particular way of estimating the beta. Similarly, the hat symbol in Eq. (\[eq:hat\_sigmaI\]) is used to distinguish $\hat{\sigma}_{I}(t)$, computed with renormalized index returns, from $\tilde{\sigma}_I(t)$. In principle, the above estimate $\tilde\beta$ could be directly regressed to the ratio of earlier estimates of $\tilde\sigma_i$ and $\tilde \sigma_I$ from Eqs. (\[eq:sigmaIt0\]). However, to use the normalization by $\tilde{\sigma}_I$ consistently, we consider the ratio of these volatilities obtained in the renormalized form, i.e., $\hat\sigma_{i}(t)/\hat\sigma_I(t)$, where $\hat{\sigma}_{I}(t)$ is given in Eq. (\[eq:hat\_sigmaI\]), and $$\label{eq:hat_sigmai} \hat{\sigma}^2_{i}(t) = (1-\lambda_{\beta})\hat{\sigma}^2_{i}(t-1) + \lambda_{\beta} \bigl[\hat r_i(t)\bigr]^2 .$$ Figure \[fig:beta\_log\] illustrates the sensitivity of beta to relative volatilities by plotting $\hat{\beta}_{i}(t)$ from Eq. (\[eq:beta\]) versus $\ln(\hat{\sigma}_i(t)/\hat{\sigma}_I(t))$ for all stocks $i$ and times $t$ from 2000 to 2015, although we only display the time frame of 2014-2015 for clarity of illustration. On both axes, we subtract the mean values $\langle\hat{\beta}_{i}\rangle$ and $\ln(\langle \hat{\sigma}_i/\hat{\sigma}_I\rangle)$ averaged over all times in the whole sample. This plot enables us to measure the average of the $f(\tilde\beta_i)$ in Eq. (\[eq:f\]), which is close to $0.76/2 = 0.38$. ![ Relation between the beta $\hat{\beta}_{i}$ and the doubled logarithm of the relative volatility $\ln(\hat{\sigma}_i/\hat{\sigma}_I)$, from which the mean values $\langle\hat{\beta}_{i}\rangle$ and $\ln(\langle \hat{\sigma}_i/\hat{\sigma}_I\rangle)$ were subtracted (the mean is obtained by averaging over time for each $i$). A linear regression is shown by the solid line: $\hat{\beta}_{i} - \langle \hat{\beta}_{i}\rangle = 0.76\bigl[\ln (\hat{\sigma}_i/\hat{\sigma}_I) - \ln(\langle \hat{\sigma}_i/\hat{\sigma}_I\rangle)\bigr]$, with $R^2 = 0.14$. For better visualization, only 10,000 randomly selected points are shown (by circles) among 271,958 points from the European dataset. Period: 2014-2015.[]{data-label="fig:beta_log"}](figure3.eps){width="120mm"} To obtain the dependence of $f$ on beta, we estimate the slope between $\hat{\beta}_i(t)- \langle \hat{\beta}_i\rangle$ from Eq. (\[eq:beta\]) and $2\ln(\hat{\sigma}_i(t)/\hat{\sigma}_I(t)) - 2\ln(\langle\hat{\sigma}_i/\hat{\sigma}_I\rangle)$ [locally]{} around each value of $\hat \beta_i$. For this purpose, we sort all collected values of $\hat\beta_i$ and group them into successive subsets, each with 10,000 points. In each subset, we estimate the slope between $\hat{\beta}_i(t)- \langle \hat{\beta}_i\rangle$ from Eq. (\[eq:beta\]) and $2\ln(\hat{\sigma}_i(t)/\hat{\sigma}_I(t)) - 2\ln(\langle\hat{\sigma}_i/\hat{\sigma}_I\rangle)$ by a standard linear regression over 10,000 points. This regression yields the value of $f$ of that subset that corresponds to some average value of $\hat\beta_i$. Repeating this procedure over all subsets, we obtain the dependence of $f$ on $\hat\beta_i$, which is plotted in Figure \[fig:fdependingonbeta\]. We show that $f$ increases with beta. For both European and US markets, we propose the following approximation of the function $f$ with three different regimes: $$\label{eq:ffit} f(\tilde \beta_i) = \begin{cases} 0, \hskip 20mm \tilde\beta_{i}<0.5, \cr 0.6 (\tilde\beta_{i}-0.5) , \quad 0.5<\tilde\beta_{i}<1.6, \cr 0.6 \hskip 20mm \tilde\beta_{i}>1.6 .\end{cases}$$ In the first regime, for low beta stocks (mostly, quality stocks), the beta elasticity is zero that is equivalent to the constant beta case. For the intermediate regime, the elasticity increases linearly with $\tilde \beta_i$ and is close to the constant correlation case with $f(\tilde \beta_i) = \tilde \beta_i/2$. In the third regime for high beta stocks (speculative and growth stocks), the elasticity is constant. The shape of the beta elasticity is similar for the European market and the US market. ![ The function $f$ from Eq. (\[eq:f\]) versus beta for the European market (blue crosses) and the US market (red pluses). This function is estimated locally for 4 different time periods. The black solid line shows the approximation (\[eq:ffit\]). Period: 2000-2015.[]{data-label="fig:fdependingonbeta"}](figure4new1.eps){width="120mm"} ### The component of the nonlinear beta elasticity According to Eq. (\[eq:ffit\]), the sensitivity of the normalized beta to changes in the relative volatility is nonlinear. This elasticity could generate bias in the beta estimation if the relative volatility changes in a sample used to measure the slope. Our solution is to adjust the beta between normalized returns through the correction factor $\F(t)$ defined as: $$\label{eq:F} \F(t) = 1+ \frac{2f(\tilde\beta_i(t))}{\tilde\beta_i(t)} \, \Delta \left(\frac{\tilde\sigma_i}{\tilde\sigma_I}\right) .$$ The function $f$ is approximated by Eq. (\[eq:ffit\]), $\ell-\ell^\prime$ is given by Eq. (\[eq:ell\_diff\]), and $$\Delta \left(\frac{\tilde\sigma_i}{\tilde\sigma_I}\right) = \frac{\tilde\sigma_i(t-1)/\tilde\sigma_I(t-1) - \sqrt{\kappa_i(t-1)}}{\sqrt{\kappa_i(t-1)}}$$ with $$\kappa_i(t) = (1-\lambda_\beta) \kappa_i(t-1) + \lambda_\beta \biggl(\frac{\tilde\sigma_i(t)}{\tilde\sigma_I(t)} \biggr)^2$$ being the EMA of the squared relative volatility $(\tilde \sigma_i/\tilde \sigma_I)^2$. The $\Delta (\tilde\sigma_i/\tilde\sigma_I)$ quantifies deviations of the relative volatility from its average over the sample that will be used to estimate beta. The correction factor $\F(t)$ should be used to estimate the slope between stock index and single stock returns and then to denormalize the slope for getting the reactive beta that depends directly on $\F(t)$. Summary of the reactive beta model ---------------------------------- In this section, we recapitulate the reactive beta model that combines the three independent components that we described in the previous sections: the specific leverage effect on beta, the systematic leverage effect on correlation, and the relation between the relative volatility and the beta. Starting with the time series $I(t)$ and $S_i(t)$ for the stock index and individual stocks, one computes the levels $L_f(t)$, $L(t)$, and $L_i(t)$ from Eqs. (\[eq:Lf\], \[eq:L\_Taylor\], \[eq:Li\_Taylor\]), the normalized stock index and individual stocks returns $\tilde r_I(t)$ and $\tilde r_i(t)$ from Eqs. (\[eq:tilder\]), the normalized stock index volatility $\tilde \sigma_I(t)$ from Eq. (\[eq:sigmaIt0\]), the renormalized stock index and individual stocks returns $\hat r_I(t)$ and $\hat r_i(t)$ from Eq. (\[eq:hatr\]), the associated volatilities $\hat \sigma_I(t)$ and $\hat \sigma_i(t)$ from Eqs. (\[eq:hat\_sigmaI\], \[eq:hat\_sigmai\]), and the renormalized beta $\hat \beta_i(t)$ from Eq. (\[eq:beta\]). From these quantities, one re-evaluates the covariance between $\hat r_i$ and $\hat r_I$ by accounting for the leverage effects and excluding the other effects. In fact, we compute $\hat{\Phi}_{i}(t)$ as an EMA of the normalized covariance of the normalized daily returns: $$\label{eq:eq1} \hat{\Phi}_{i}(t) = (1-\lambda_{\beta}) \hat{\Phi}_{i}(t-1) + \lambda_{\beta} \, \frac{ \hat r_i(t) \, \hat r_I(t) } {\L(t) \, \F(t)} ,$$ where $\L(t)$ and $\F(t)$ are two corrections factors defined in Eq. (\[eq:L\]) and Eq. (\[eq:F\]), used to withdraw bias from the systematic leverage and the beta elasticity. The parameter $\lambda_{\beta}$ describes the look-back used to estimate the slope and is set to $1/90$ as 90 days of look-back appears to us as a good compromise. In fact, for a longer look-back, variations in beta, correlation and volatilities are expected to happen due to changes of market stress and business cycle and are not taken into account properly by our reactive renormalization. In turn, for a shorter look-back, the statistical noise of the slope would be too high. Finally, the stable estimate of the normalized beta is $$\tilde\beta_i(t) = \frac{\hat{\Phi}_i(t)}{\hat\sigma_I^2(t)} ,$$ with $\hat\sigma_I^2(t)$ defined in Eq. (\[eq:hat\_sigmaI\]) from which the estimated reactive beta of stock $i$ is deduced as $$\label{eq:beta_final} \beta_i (t) = \tilde \beta_i(t) \left(\frac{L_i(t) \, I(t)}{S_i(t) \, L(t)} \right) \, \L(t) \, \F(t) .$$ This estimation is close to the slope estimated by an OLS but with exponentially decaying weights to accentuate recent returns and with normalized returns to withdraw different biases. In fact, the normalized stable beta $\tilde \beta_i(t)$ is “denormalized” by the factor that combines the three main components: the specific leverage effect on beta, $(L_i/S_i)(I/L)$, the systematic leverage effect, $\L(t)$, and nonlinear beta elasticity, $\F(t)$. Every term impacts the hedging of a certain strategy: - the term with $\L(t)$ does not have significant impact on beta, as it is compensated in $L_i/L$, which models the short-term systematic leverage effect on correlation in Eqs. (\[eq:eq1\], \[eq:beta\_final\]) (introduced in Sec. \[sec:RCM\]), whereas the levels $L_i$ and $L$ were introduced in the reactive volatility model. However, it could impact the correlation by $+10\%$ if the market decreases by $10\%$. - the term with $L_i I/(L S_i)$ that models the specific leverage effect on volatilities (introduced in Sec. \[sec:beta\_specific\]) could impact beta by $10\%$ if the stocks underperform by $10\%$. This term impacts the hedging of the short-term reversal strategy. - the term with $\F(t)$ that models the nonlinear beta elasticity which is the sensitivity of beta to the relative volatility (introduced in Sec. \[sec:beta\_relation\]) could impact the beta by $10\%$ if the relative volatility increases by $10\%$. This term impacts the hedging of the low volatility strategy. The simple version of the reactive beta model, when only the leverage effect is introduced without beta elasticity and stochastic normalized volatilities, defines an interesting class of stochastic processes that appears to be a mean reverting with a standard deviation linked to $\tilde{\sigma_i} \sqrt{ 1/\lambda_s}$ and a relaxation time linked to $1/\lambda_s$. The reactive beta model is based on the fit of several well identified effects. Implied parameters work universally for all stock markets ($\ell-\ell^{\prime}$ is the only one that was fitted only on the US market as the implied correlations for other countries are not traded). Here we summarize the different parameters used in the reactive beta model: - $\lambda_f=0.1484$ that describes the relaxation time of 7 days for the panic effect; - $\lambda_s = 0.0241$ that describes the relaxation time of 40 days for the retarded effect; - $l= 8$ that describes the leverage intensity of the panic effect; - $\ell-\ell^{\prime}\approx 0.91$ based on implied correlations on the US stock market; - the different thresholds in the function $f(\tilde \beta_i)$ from Eq. (\[eq:ffit\]) that describes the nonlinear beta elasticity. Empirical findings {#sec:empirical} ================== Data description {#sec:data} ---------------- For the empirical calibration of $\ell-\ell^\prime$, we chose the CBOE S&P 500 Implied Correlation Index (ICI), which is the first widely disseminated market-based estimate of implied average correlation of the stocks that comprise the S&P 500 Index (SPX). This index begins in July 2009, with historical data back to 2007. We take the front-month correlation index data from 2007 and roll it to the next contract until the previous one expires. We also use the daily S&P 500 stock index. For the empirical calibration of the other parameters of the reactive beta model, we use the daily S&P 500 stock index and 600 largest US stocks from January 1, 2000, to May 31, 2015. For the European market, we consider the EuroStoxx50 index and the 600 largest European stocks over the same period. The same data are used for both calibration parameters and empirical tests. To be precise we kept the parameters of the reactive volatility models, that describes the intensity, the relaxation time of the specific and systematic leverage effect that appear the most important, identical to those that were calibrated in a period prior to 2000 by [@Bouchaud01]. Empirical results {#sec:results} ----------------- In this section, we show that exposure to the common risk factors can sometimes lead to a high exposure of market neutral funds to the stock market index if the betas are not correctly assessed. Indeed, although market neutral funds should be orthogonal to traditional asset classes, such is not always the case during extreme moves [@Fung97]. For instance, @Patton09 tests the zero correlation against non-zero correlation and finds that approximately 25% of the market neutral funds exhibit some significant non-neutrality, concluding that “many market neutral hedge funds are in fact not market neutral, but overall they are, at least, more market neutral than other categories of hedge funds.” The reactive beta model can help hedge funds be more market neutral than others. To demonstrate this, we empirically test the efficiency of our methodology in estimating the reactive beta model using the most popular market neutral strategies (low volatility, momentum and size): - low volatility (beta) strategy: buying the stocks with the highest $30\%$ beta and shorting those with the lowest $30\%$ beta (estimated by the standard methodology); - short-term reversal strategy: shorting the stocks with the highest $15\%$ one-month returns and buying those with the lowest $15\%$ one-month returns; - momentum strategy: buying the stocks with the highest $15\%$ two-year returns and shorting those with the lowest $15\%$ two-year returns; - size strategy: buying the stocks with the highest $30\%$ capitalization and shorting those with the lowest $30\%$ capitalization. The construction of the four most popular strategies is explained in Appendix \[sec:Afactors\]. For each strategy, we compare two different methods to estimate the beta that use only the past information to avoid look-ahead bias: the ordinary least square (OLS) (that is equivalent to our model with $L_i = S_i$, $L = I$, $\ell = \ell' = 0$, and $f = 0$, with the same exponential weighting scheme) and our reactive method. We analyze two statistics: - Statistics 1: the CorSTD, that describes the unrobustness of the hedge and in consequence the inefficiency of the beta measurement. The CorSTD is defined as the standard deviation of the 90-day correlation of the strategy with the stock index returns. The more robust the strategy is, the lower is the CorSTD statistics. If the strategy was well hedged, the correlation would fluctuate by approximately $0$ within the theoretical $10\%$ standard deviation and CorSTD will be of $10\%$ ($10\%$ is obtained with uncorrelated Gaussian variables for 90-day correlations). - Statistics 2: the Bias, that describes the bias in the hedge of the strategy and of the beta measurement, that is defined as the correlation of the strategy with the stock index returns on the whole period. This statistics are a proxy for assessing the quality of the beta measurement that is very difficult to realize directly as true beta are not known. Table \[tab:schemes\] summarizes the statistics of the four strategies for the US and Europe markets. We see the highest bias for the low volatility strategy when hedged with the standard approach ($-25.5\%$ for USA and $-22.4\%$ for Europe). The CorSTD is approximately $20\%$, i.e., twice as high as expected if the volatility were stable, which means that the efficiency of the hedge is time-varying. This could represent an important risk for fund of funds managers, where hidden risk could accumulate and arise especially when the market is stressed. Indeed, the bias seems to be higher by approximately $-60\%$ for both the USA and Europe when the market was stressed in 2008. We see that the use of the reactive beta model reduces the bias in the low volatility factor, and that the residual bias comes from the selection bias (see Appendix \[sec:selection\]). When using the OLS, the possible loss in 2008 would have been $-9.6\%$ ($= -60\% \times 40\% \times 8\%/20\%$) for a $40\%$ stock decline with a fund invested entirely on a low volatility anomaly with a bias of $-60\%$ and a target annualized volatility $8\%$ for the fund and $20\%$ for the index. We also see a significant bias for the short-term reversal strategy when hedged with the standard approach (approximately $13.1\%$ in the USA and in Europe). The CorSTD is approximately $18\%$. The efficiency of the hedge depends on the recent past performance of the strategy. As soon as the strategy starts to lose, the efficiency will decline and risk will arise, as in 2009. Again, we see that the reactive beta model reduces the bias in the short-term reversal factor. The biases and CorSTD are lower for the momentum strategy ($-6.3\%$ in the USA, with a CorSTD of $18.3\%$) and are of same magnitude for the size strategy ($-7.6\%$ in the USA with a CorSTD of $17.0\%$). The reactive beta model further reduces the bias and the CorSTD. This is also valid for the European market. We conclude that the reactive beta model reduces the bias of the low volatility factor when it is stressed by the market. The remaining residual is most likely explained by the selection bias (see Appendix \[sec:selection\] for a formal proof). The improvement is more significant for the momentum factors and for the size factor in the U.S. only. We also illustrate these findings by presenting the correlation between the stock index and the low volatility strategy (Figure \[fig:betafactor\]) and the short-term reversal strategy (Figure \[fig:momentum\]), which are the strategies with the highest bias. A period surrounding the financial crisis was chosen (2007-2010). One can see that the beta, computed by the OLS, is highly positively exposed to the stock index in 2008. In turn, the exposure is reduced within the reactive model. The improvement becomes even more impressive in extreme cases when the strategies are stressed by the market. We see that in some extreme cases (stress period with extreme strategies), the common approach could generate high biases ($-50\%$ for the short-term reversal strategies in 2008-2009 and $-71\%$ for the beta strategy in 2008). In each case, our methodology allows one to significantly reduce the bias. [\|c\|c\|c\|c\|c\|c\|]{} & Strategy $\backslash$ Method & &\ \[-1mm\] [90]{}US & Statistics : & Bias & CorSTD & Bias & CorSTD\ & low volatility & -25.54% & 21.73% & -16.79% &21.43%\ & short-term reversal & 13.09% &18.96% & -6.06% &18.50%\ & momentum & -6.27% &18.28% & -2.95% &16.54%\ & size & -7.56% &17.00% & -1.84%&17.26%\ \[4mm\] [90]{}Europe & low volatility & -22.39% &19.97% & -14.68% &20.94%\ & short-term reversal & 13.05% &17.51% & 0.64% &14.52%\ & momentum & -4.42% &18.03% & -1.55% &17.23%\ & size & 3.12% &17.15% & 3.79% &15.63%\ ![ Ninety-day correlation of the low volatility factor with the stock index in the European market [(a)]{} and in the USA market [(b)]{}. Solid and dashed lines present the proposed the reactive beta model and the OLS methodology, respectively. The dotted horizontal line shows the selection bias of $-19.10\%$, as shown in Appendix \[sec:selection\]. A time frame surrounding the financial crisis is chosen. Period: 2007-2010.[]{data-label="fig:betafactor"}](figure5a.eps "fig:"){width="120mm"} ![ Ninety-day correlation of the low volatility factor with the stock index in the European market [(a)]{} and in the USA market [(b)]{}. Solid and dashed lines present the proposed the reactive beta model and the OLS methodology, respectively. The dotted horizontal line shows the selection bias of $-19.10\%$, as shown in Appendix \[sec:selection\]. A time frame surrounding the financial crisis is chosen. Period: 2007-2010.[]{data-label="fig:betafactor"}](figure5b.eps "fig:"){width="120mm"} ![ Ninety-day correlation of the short-term reversal factor with the stock index in the European market [(a)]{} and in the USA market [(b)]{}. Solid and dashed lines present the proposed Reactive beta model and the OLS methodology, respectively. A time frame surrounding the financial crisis is chosen. Period: 2007-2010.[]{data-label="fig:momentum"}](figure6a.eps "fig:"){width="120mm"} ![ Ninety-day correlation of the short-term reversal factor with the stock index in the European market [(a)]{} and in the USA market [(b)]{}. Solid and dashed lines present the proposed Reactive beta model and the OLS methodology, respectively. A time frame surrounding the financial crisis is chosen. Period: 2007-2010.[]{data-label="fig:momentum"}](figure6b.eps "fig:"){width="120mm"} Robustness Checks {#sec:robustness} ================= This section presents robustness check analysis by comparing the quality of several methods for beta measurements against the reactive beta model. We build the comparative analysis based on two important articles in order to explore two aspects of the beta estimation. @Chan92 enable to assess robustness statistics of some alternatives methods to classical ordinary least square (OLS) when assuming implicitly that betas are static and returns are homoscedastic. This section extends their work by including alternative dynamics beta estimators to be coherent with our reactive model and with the work by @Engle16 that demonstrates that the betas are significantly time-varying using dynamic conditional betas. The presentation of the models and methods are located in the Appendix \[sec:methods\]. Monte Carlo simulations ----------------------- In financial research, one often resorts to simulated data to estimate the error of measurements. For instance, [@Chan92] built their main results on numerical simulation while applying real data for simple comparison between betas estimated with OLS and quantile regression (QR). The comparative analysis is based on a two-step procedure. The first step simulates returns using different models that capture some markets patterns and the second step estimates the beta from simulated returns by using our reactive method and alternative methods. We tested the same estimators as used by [@Chan92] that includes the OLS, the minimum absolute deviation (ABSD), and the Trimean quantile regression (TRM). We also added two variations of the dynamical conditional correlation (DCC) which has become a mainstream model to measure conditional beta when beta is stochastic [@Bollerslev88; @Bollerslev90; @Engle02; @Cappiello06]. We analyze the error of measurements that we defined as the difference between the measured beta and the true beta of simulated data. ### The first step: simulation The first step simulates 30,000 paths of $T$=1,000 consecutive returns for both the stock index and the single stock. It allows also to generate 1,000 conditional “true” expected beta per path (Fig. \[fig:MC\]). To that end, following [@Chan92], normally distributed residuals and Student-t distributed residuals are considered to take into consideration robustness of different methods to outliers. In our setting, we implemented seven Monte Carlo simulations for the returns $r_i$ and $r_I$. We targeted in simulations the realistic case of an unconditional single stock annualized volatility of 40%, an unconditional stock index volatility of 15% and an unconditional beta of 1. That is important to target the realistic correlation between the index and the stock of 0.4. Indeed the relative precision of the beta measurement is inversely proportional to the square root of the number of returns when correlation is close to zero. First, we consider the naive version of the market model, based on Eq. (\[eq:simplemarketmodel\]), that we call “the basic market model” For the case of constant beta, as in paper by [@Chan92], the simulated data are based on the hypothesis of a null intercept and beta is equal to $1$ to characterize the ideal case with a Gaussian (MC1) or a t-student distribution (MC2) for residuals. In the most simple reactive version of the market model that we call “the reactive market model”, normalized returns $\tilde r_i$ and $\tilde r_I$ are first generated randomly through Eq. (\[eq:simplereactivemodel\]) with a normalized beta set to 1. Then, based on the level $L_{s}$, $L_{is}$ that are respectively the slow moving averages of the stock index and the stock prices defined in Eq. (\[eq:Lslow\]), we generate $\delta I$ and $\delta S$ defined in Eq. (\[eq:tilder\]), then $r_i$ and $r_I$, and finally update $L_{s}$ and $L_{is}$ . That model is sufficient to capture the leverage effect on beta with increasing beta as soon as single stock underperforms the stock index. Even if the normalized beta is set to unity (MC3 and MC4), the denormalized beta in Eq. (\[eq:reactivevolatility\]) becomes time dependent (Fig. \[fig:MC\]). As previously, MC3 and MC4 differ by the distribution of residuals, Gaussian (MC3) versus Student-t (MC4). For the case of time-varying beta (MC 3 to 5), we used two versions of the reactive market model in Eq. (\[eq:simplereactivemodel\]): the reduced version with only the leverage effect components that is enough to generate stochastic beta in Eq. (\[eq:reactivevolatility\]), and the full version with all components including the nonlinear beta elasticity. For the full version (MC5), we generated stochastic $\tilde{\sigma}_i$ and $\tilde{\sigma}_I$ that generate $\tilde r_i$ and $\tilde r_I$ from Eq. (\[eq:simplereactivemodel\]) using the normalized beta fixed to $\mathcal{F}(t)\mathcal{L}(t)$ (see definitions in Eqs. (\[eq:F\]) and (\[eq:L\])). That allows to generate returns that capture the leverage effect pattern and the empirical non-linear beta elasticity (Fig. \[fig:beta\_log\] and Fig. \[fig:fdependingonbeta\]). For the case of time-varying beta (MC 6 to 7), we used another way to generate random returns that capture a time-varying beta through the implementation of the dynamic conditional correlation (DCC) model [@Engle02], which generalizes the GARCH(1,1) process to two dimensions. This is a mainstream model which has two variations: symmetric and asymmetric, the latter capturing the leverage effect. Symmetric and asymmetric versions of DCC model are denoted as MC6 and MC7. To summarize, seven Monte Carlo simulations: - MC 1: The basic market model in Eq. (\[eq:simplereactivemodel\]) where residuals ($\epsilon_i$) are normally distributed and constant beta is set to 1. - MC 2: The basic market model in Eq. (\[eq:simplereactivemodel\]) where residuals ($\epsilon_i$) follow a Student-t distribution (with three degrees of freedom) and constant beta is set to 1. - MC 3: The reduced reactive market model in Eq. (\[eq:simplereactivemodel\]) where residuals ($\tilde \epsilon_i$) are normally distributed with constant volatilities ($\tilde \sigma_i$, $\tilde \sigma_I$) and constant renormalized beta ($\tilde \beta$) set to 1 but the denormalized beta is now depending on time (Fig \[fig:MC\]). The conditional beta ($\beta$) is now a mean reversion process with a relaxation time $1/\lambda_s=50$ days. MC3 uses only the leverage effect component but not the nonlinear beta elasticity. - MC 4: The reduced reactive market model in Eq. (\[eq:simplereactivemodel\]) where residuals ($\tilde \epsilon_i$) follow a Student-t distribution (with three degrees of freedom) with constant relative volatility and constant renormalized beta set to 1. - MC 5: The full reactive market model in Eq. (\[eq:simplereactivemodel\]) where residuals ($\tilde \epsilon_i$) follow a Student-t distribution (with three degrees of freedom) whose standard deviation ($s_i$) is stochastic and where the normalized stock index return ($\tilde r_I$) is a Gaussian whose standard deviation ($s_I$) is also stochastic. We suppose that $\log(s_I)$ and $\log(s_i)-\log(S_I)$ follow two independent Ornstein-Uhlenbeck processes (with the relaxation time of 100 days and the volatility of volatility of 0.04). In that way the stock index annualized volatility could jump up to 40%. Normalized beta, that was set to 1 in MC4, is now set to $\mathcal{F}(t)\mathcal{L}(t)$ to take into account the nonlinear beta elasticity (see definitions in Eqs. (\[eq:F\]) and (\[eq:L\])). Both leverage effect and stochastic normalized volatilities make the beta defined in Eq. (\[eq:beta\_final\]) )and volatilities time-depended (Fig. \[fig:MC\]). - MC 6: The symmetric DCC model in two dimensions, which generates volatilities of volatilities and correlation of similar amplitude as MC5 (Fig. \[fig:MC\]). - MC 7: The ADCC model in two dimensions, which generates volatilities of volatilities and correlation of similar amplitude as MC5 (Fig. \[fig:MC\]). In Fig. \[fig:MC\], we plot a Monte Carlo path generated for true beta for MC 3 to 7 (MC1 and MC2 are excluded as they generate true beta of 1). We also plot the conditional correlation and volatilities that are highly volatile and make the estimation of the conditional beta complicated. ### The second step: measurements The second step is devoted to the analysis of the error measurement of the beta estimations defined as the difference between the measured beta and the true beta of simulated data. In our setting, we test 5 alternative beta estimations that should replicate as close as possible the true beta. Notice that in all five configurations, we use an exponentially weighted scheme to give more weight to recent observations to be in line with the reactive market model ($1/\lambda_\beta=90$). As a consequence, in a path of $T$=1,000 generated returns, only the 90 last returns really matters (note that [@Chan92] based their statistics on 35 returns with an equal weight scheme). The first alternative method is the Ordinary Least Square (OLS) of the returns which was also implemented in the empirical test based on real data. Note that the OLS would give the same measurement than our reactive method if parameters were set differently ($\lambda_s=1$, $\lambda_f=1$, $l=l'=0$, $f=0$). The square errors in the OLS are weighted by $(1-\lambda_\beta)^{T-t}$. The second method estimates the beta by using the Minimum Absolute Deviation (MAD) that is supposed to be less sensitive to outliers as absolute errors instead of square errors are minimized. The absolute errors are weighted by $(1-\lambda_\beta)^{T-t}$. The third alternative is the beta computed from the Trimean quantile regression (TRM) that is reputed to be more robust to outliers according to [@Chan92]. The absolute errors are also weighted by $(1-\lambda_\beta)^{T-t}$. The fourth and fifth methods are the conditional beta computed from the DCC model. The DCC method was calibrated using the same exponential $(1-\lambda_\beta)^{T-t}$ weights introduced in the log-likelihood function to extract the optimal unconditional volatilities and correlations, while other parameters such as the relaxation time and volatilities of volatilities and volatilities of correlations were set to the values that were used for Monte Carlo simulation. We summarize the reactive method and the five alternative methods that were implemented to estimate the beta: - $\beta_{OLS}$: beta estimated by the Ordinary Least Square method; - $\beta_{MAD}$: beta estimated by the Minimum Absolute Deviation method; - $\beta_{TRM}$: beta estimated by the Trimean Quantile regression; - $\beta_{DCC}$: $T^{\rm th}$ conditional beta estimated by using the DCC model; - $\beta_{ADCC}$: $T^{\rm th}$ conditional beta estimated by using the ADCC model; - $\beta_{R}$: beta estimated by the reactive method in Eq. (\[eq:beta\_final\]). ### The statistics We analyze for every path the error of measurement defined as the difference between the measured beta based on different methods applied to $T$ returns and the true value of beta at time $T$. To assess the quality of different methods, we use three statistics following [@Chan92]. The first statistics is the bias and gives the average error of measurement. Yielding the bias is more informative than simply yielding an estimated average estimation of beta as in our case the theoretical expected simulated conditional beta is not always 1 but fluctuates around 1 for time-varying models from MC3 to MC6. As we focused on capturing the leverage effect in the beta measurement we also define winner (loser) stocks that are the stocks that have outperformed (underperformed) the stock index during the last month. Due to the leverage effect, the OLS method is expected to underestimate beta for loser stocks and to overestimate beta for winner stocks. It would be interesting to see how robust is the improvement of the reactive beta. We therefore measure the average error among the loser and winner stocks. The loser and winner biases are related to the bias in hedging of the short term reversal strategy measured on real data and could confirm the robustness of the empirical measurements. We also define the low (high) beta stocks that are the stocks whose conditional true beta is lower (higher) than 1. We measure the average error among low and high beta stocks that are related to the bias in hedging of the low beta strategy measured from real data and could confirm the robustness of the beta measurement when adding the component describing the nonlinear beta elasticity. The second statistics is the ABSolute Deviation (ABSD) of measurement. It reflects the average absolute errors such that the positive and negative sign errors cannot be mutually compensated. It is a measurement of the robustness. The third statistics, that is equivalent to ABSD, is the inverse of the variance of the errors of measurement ($\frac{V_{OLS}}{V_{m}}$) to characterize the relative robustness of the alternative beta estimation. The alternative beta method (with subscript $m$) that brings the highest improvement is the one with the highest ratio. The three statistics that were implemented are summarized: - Statistics 1: the bias, the winner bias and the loser bias, the low bias and the high bias; - Statistics 2: the absolute deviation of measurement (ABSD); - Statistics 3: the relative variance statistics $\frac{V_{OLS}}{V_{m}}$. Empirical tests --------------- We summarize statistics in Table \[tab:MC\]. We see that all methods are unbiased on average in most Monte Carlo simulations. But this is misleading as biases from one group of stocks can be significant and offset others. ### Winner and loser bias The estimated $\beta_{DCC}$ and $\beta_{ADD}$ appear to be biased as soon as fat tails are included (MC2). The reactive beta is the only one to be unbiased for winner and loser stocks when the leverage effect is introduced in Monte Carlo (MC 3, 4, 5). The biases for winner stocks and loser stocks are significant for all methods except for the reactive beta. The biases are amplified when a fat tail of residuals distribution is introduced (MC 4). Winner/loser biases can reach 14%. That is in line with the empirical test implemented on real data where we see that the reactive method reduces the bias of hedging of the short-term reversal strategy (Tab. \[tab:schemes\]). When all components that deviate from the Gaussian market model are mixed in MC5 (fat tails, nonlinear beta elasticity, stochastic volatilities, leverage effect) we see a kind of cocktail effect as bias is generated for most methods on average and not only in some groups of stocks. The reactive method provides the best results and is the only method that has no bias. $\beta_{MAD}$ and $\beta_{TRM}$ that were supposed to be robust appear to perform very badly with high bias (average, loser or winner) as soon as stochastic volatility is added that is confirmed with MC6 and MC7. We also see that the reactive model looks to be incompatible with the DCC or ADCC model. Indeed MC5 generates high bias for $\beta_{DCC}$ and $\beta_{ADD}$ in the winner and loser stocks even if the leverage effect and the dynamic beta are implemented in the ADCC. In the same way MC 6 generates bias for the reactive method that are even amplified when leverage effect is generated through MC7. We can wonder which model is the most realist. Both ADCC and the reactive model capture the volatility clustering and leverage effect patterns but their dynamics is in reality very different. In the reactive model, volatility increases as soon as price decreases, and decreases as soon as price increases whereas ADCC needs to see its volatility increase a negative return, higher than expected ($\gamma \left( \sigma_i^{2} [\xi_i^-(t)]^2 -\tilde\sigma_i^{2}\right)>0$, see Eqs. (\[eq:beta\_aux1\], \[eq:beta\_aux2\])). The reactive beta model has its three components that were fitted to three well identified effect (the specific leverage through the retarded effect, the systematic leverage through the panic effect and the non linear beta elasticity) whose main parameters appears to be stable and universal for all markets. @Bouchaud01 measured most of the parameters for 7 main stock indexes. Relaxation time is around 1 week for the panic effect ($\lambda_s=0.1484$), relaxation is 40 days for the retarded effect ($\lambda_s = 0.0241$), the leverage parameter for the panic effect is $l= 8$. The systematic leverage parameter $\ell-\ell^{\prime} = 0.91$ was the only one to have been measured through the implied correlation only from the US market. The parameters of the beta elasticity were measured similar for both the European and the US market. The different thresholds are $0.5$ and $1.6$ in beta of the non linear beta elasticity separating low beta stocks from speculative stocks). Parameters $a$, $b$, $\gamma$, $a_{\rho}$, $b_{\rho}$, $\gamma_{\rho}$ of the DCC and ADCC were based on the work by [@Sheppard17] but $b$ and $b_{\rho}$ which are the “decay coefficients”, describe relaxation times of 10 days and 13 days that are different from those used in the reactive volatility model. ### High and low beta bias The reactive beta is the only one that reduces the bias for low and high beta stock when stochastic volatility is introduced and when the empirical nonlinear beta elasticity is implemented (MC 5). That is in line with the empirical test implemented on real data where we see that the reactive method reduces the bias of hedging of the low volatility strategy (Tab. \[tab:schemes\]). ### ABSD and $V_{OLS}/V_{m}$ The $\beta_{OLS}$, that is the theoretical optimal estimation for Monte Carlo simulated returns with the Gaussian market model (MC1), gives similar statistics to that of the reactive beta for the MC3. In this case (MC3), the reactive method outperforms the other considered methods. The ABSD of 0.17 is entirely explained by irreducible statistical noise that is intrinsic to any regression based on approximately 90 points with a weak correlation. When a fat tail is incorporated to the residual (MC4), the ABSD of the reactive beta is increased and becomes intermediate between the ABSD of $\beta_{OLS}$, $\beta_{MAD}$ and $\beta_{TRM}$. $\beta_{MAD}$ and $\beta_{TRM}$ are more robust in presence of fat tails. The reactive beta is expected to be as sensitive as the OLS would be due to the outliers. The reactive method could be still improved if a TRM regression was implemented instead of the classical OLS to measure the normalized beta between normalized returns. When stochastic volatility and correlation are introduced (MC5, MC6 and MC7), the reactive beta becomes as robust as $\beta_{MAD}$ and $\beta_{TRM}$ based on ABSD. [ \|c\|c\|c\|c\|c\|c\|c\|c\| ]{} Method & Bias & Winner Bias& Loser Bias& Low Bias& High Bias & ABSD & Vols/Vm\ \ $\beta_{OLS}$&-0.00&-0.00&-0.00&&&0.16&1.00\ $\beta$ Reactive&0.00&-0.05\&0.05\&&&0.18&0.79\ $\beta_{DCC}$&0.04\&0.05\&0.03\&&&0.23&0.51\ $\beta_{ADCC}$&0.09\&0.01&0.17\&&&0.25&0.44\ $\beta_{MAD}$&-0.00&0.00&-0.01&&&0.20&0.65\ $\beta_{TRM}$&-0.00&0.00&-0.01&&&0.20&0.68\ \ $\beta_{OLS}$&-0.00&0.01&-0.01&&&0.28&1.00\ $\beta$ Reactive&0.01&-0.06\&0.08\&&&0.31&0.82\ $\beta_{DCC}$&0.13\&0.14\&0.12\&&&0.39&0.67\ $\beta_{ADCC}$&0.25\&0.15\&0.35\&&&0.46&0.57\ $\beta_{MAD}$&-0.00&-0.00&-0.00&&&0.22&2.18\ $\beta_{TRM}$&-0.00&-0.00&-0.00&&&0.22&2.24\ \ $\beta_{OLS}$&-0.00&0.07\&-0.07\&0.07\&-0.07\&0.19&1.00\ $\beta$ Reactive&-0.00&0.02\&-0.02\&0.02\&-0.02\&0.17&1.27\ $\beta_{DCC}$&0.04\&0.10\&-0.02&0.11\&-0.02&0.24&0.62\ $\beta_{ADCC}$&0.09\&0.06\&0.12\&0.07\&0.11\&0.24&0.66\ $\beta_{MAD}$&-0.01&0.06\&-0.08\&0.06\&-0.08\&0.22&0.73\ $\beta_{TRM}$&-0.01&0.06\&-0.08\&0.06\&-0.08\&0.22&0.75\ \ $\beta_{OLS}$&0.01&0.13\&-0.11\&0.12\&-0.10\&0.35&1.00\ $\beta$ Reactive&-0.01&0.02&-0.04\&0.03&-0.05\&0.31&1.30\ $\beta_{DCC}$&0.12\&0.22\&0.02&0.27\&-0.01&0.47&0.84\ $\beta_{ADCC}$&0.26\&0.24\&0.28\&0.30\&0.21\&0.52&0.83\ $\beta_{MAD}$&-0.03\&0.09\&-0.14\&0.10\&-0.14\&0.27&2.68\ $\beta_{TRM}$&-0.03\&0.09\&-0.14\&0.10\&-0.14\&0.27&2.76\ \ $\beta_{OLS}$&-0.01&0.13\&-0.14\&0.14\&-0.22\&0.50&1.00\ $\beta$ Reactive&-0.04\&-0.00&-0.07\&0.05\&-0.17\&0.41&1.42\ $\beta_{DCC}$&-0.01&0.10\&-0.12\&0.20\&-0.32\&0.52&1.31\ $\beta_{ADCC}$&0.10\&0.10\&0.11\&0.29\&-0.17\&0.54&1.32\ $\beta_{MAD}$&-0.09\&0.04\&-0.22\&0.09\&-0.37\&0.38&2.43\ $\beta_{TRM}$&-0.09\&0.04\&-0.22\&0.09\&-0.36\&0.37&2.46\ \ $\beta_{OLS}$&-0.11\&-0.10\&-0.11\&0.06\&-0.27\&0.32&1.00\ $\beta$ Reactive&-0.07\&-0.11\&-0.02&0.09\&-0.23\&0.33&0.93\ $\beta_{DCC}$&-0.01&-0.00&-0.02\&-0.01&-0.01&0.16&4.09\ $\beta_{ADCC}$&0.02\&-0.08\&0.12\&0.05\&-0.01&0.22&2.06\ $\beta_{MAD}$&-0.14\&-0.13\&-0.15\&0.04\&-0.32\&0.34&0.89\ $\beta_{TRM}$&-0.14\&-0.13\&-0.15\&0.04\&-0.32\&0.34&0.90\ \ $\beta_{OLS}$&-0.09\&0.03&-0.24\&0.09\&-0.25\&0.30&1.00\ $\beta$ Reactive&-0.07\&0.02&-0.17\&0.10\&-0.21\&0.27&1.21\ $\beta_{DCC}$&-0.04\&0.04\&-0.15\&-0.00&-0.08\&0.21&2.08\ $\beta_{ADCC}$&-0.01&-0.01&-0.01&-0.00&-0.01&0.15&3.74\ $\beta_{MAD}$&-0.13\&-0.02&-0.28\&0.06\&-0.29\&0.32&0.92\ $\beta_{TRM}$&-0.13\&-0.01&-0.28\&0.06\&-0.29\&0.32&0.92\ ![ Simulated paths for the models MC4 – MC7. True conditional beta (top), true conditional correlation (middle left), true conditional stock index volatility (middle right), true conditional single stock volatility (bottom left), true conditional relative volatility (bottom right) are plotted. Paths limited to 500 days, that are independent from model to model, capture the same order of magnitude of variation in volatilities, beta and correlation. []{data-label="fig:MC"}](figure7a.eps "fig:"){width="120mm"} ![ Simulated paths for the models MC4 – MC7. True conditional beta (top), true conditional correlation (middle left), true conditional stock index volatility (middle right), true conditional single stock volatility (bottom left), true conditional relative volatility (bottom right) are plotted. Paths limited to 500 days, that are independent from model to model, capture the same order of magnitude of variation in volatilities, beta and correlation. []{data-label="fig:MC"}](figure7b.eps "fig:"){width="60mm"} ![ Simulated paths for the models MC4 – MC7. True conditional beta (top), true conditional correlation (middle left), true conditional stock index volatility (middle right), true conditional single stock volatility (bottom left), true conditional relative volatility (bottom right) are plotted. Paths limited to 500 days, that are independent from model to model, capture the same order of magnitude of variation in volatilities, beta and correlation. []{data-label="fig:MC"}](figure7c.eps "fig:"){width="60mm"} ![ Simulated paths for the models MC4 – MC7. True conditional beta (top), true conditional correlation (middle left), true conditional stock index volatility (middle right), true conditional single stock volatility (bottom left), true conditional relative volatility (bottom right) are plotted. Paths limited to 500 days, that are independent from model to model, capture the same order of magnitude of variation in volatilities, beta and correlation. []{data-label="fig:MC"}](figure7d.eps "fig:"){width="60mm"} ![ Simulated paths for the models MC4 – MC7. True conditional beta (top), true conditional correlation (middle left), true conditional stock index volatility (middle right), true conditional single stock volatility (bottom left), true conditional relative volatility (bottom right) are plotted. Paths limited to 500 days, that are independent from model to model, capture the same order of magnitude of variation in volatilities, beta and correlation. []{data-label="fig:MC"}](figure7e.eps "fig:"){width="60mm"} Open problems in other fields {#sec:application} ============================= The estimated beta is used in a wide range of financial applications, which includes security valuation, asset pricing, portfolio management and risk management. This extends also to corporate finance in many applications such like financing decisions to quantify risk associated with debt, equity and asset and for firm valuation when discounting cash-flows using the weighted average cost of capital. The most likely reason is that the beta describes systematic risk that could not be diversified and that should should be remunerated. However as explained, the OLS estimator of beta is subject to measurement errors, which include the presence of outliers, time dependence, the leverage effect, and the departure from normality. Asset Pricing ------------- [@Bali17] apply the DCC model by [@Engle16] to assess the cross-sectional variation in expected stock returns. They estimate the conditional beta for the S&P 500 using daily data for each year from 1963 to 2009. They test if the betas have predictive power for the cross-section of individual stocks returns over the next on to five days. They show that there is no link between the unconditional beta and the cross-section of expected returns. Most remarkably, they also show that the time-varying conditional beta is priced in the cross-section of daily returns. At the portfolio level, they indicate that a long-short trading strategy buying the highest conditional beta stocks and selling the lowest conditional beta stocks yields average returns of 8% per year. So conditional CAPM is empirically valid whereas unconditional CAPM is empirically not valid. Moreover they showed that conditional beta when comparing to unconditional beta would not have significant pricing effect on major anomalies (size, book, momentum,...). So one can see that DCC greatly improves the empirical validation of the CAPM but does not change pricing of anomalies. We expect that the reactive method can bring further improvements. According to our robustness tests in Sec. \[sec:robustness\], the leverage effect and the nonlinear beta elasticity could also generate bias in the DCC estimation. As our reactive method was designed to correct for these biases, its use can help to reveal pricing effects of the dynamic beta on major anomalies. This point is an interesting perspective for future research. Corporate Finance ----------------- To determine a fair discount rate for valuing cash-flows, the firm’s manager must select the appropriate beta of the project given that the discount rate remains constant over time while the project may exhibit significant variation over time and leverage effect due to the debt-to-equity ratio. As such, [@Ang04] discuss how to discount cash-flows with time-varying expected returns in traditional set-up. For instance, the traditional dividend discount model assumes that the expected return along with the expected rate of cash-flow growth are set constant while they are time-varying and correlated. In practice, in the first step, the manager computes the expected future cash-flows from financial forecasts and then in a second step, the manager uses a constant discount rate, usually relying on the CAPM to discounting factor. In contrast, [@Ang04] derive a valuation formula that incorporates correlation between stochastic cash-flows, betas and risk premiums. They show that the greater the magnitude of the difference between the true discount rates and the constant discount rate, the greater the project’s misvaluation. They even show that when computing perpetuity values from the discounting model, the potential mispricing can even get worse. They conclude that accounting for time-varying expected returns can lead to different prices from using a constant discount rate from the traditional unconditional CAPM. The impact of the leverage effect and of the non-linear elasticity of beta on potential mispricing should be investigated. Conclusion {#sec:conclusion} ========== We propose a reactive beta model with three components that account for the specific leverage effect (when a stock underperforms, its beta increases), the systematic leverage effect (when a stock index declines, correlations increase), and beta elasticity (when relative volatility increases, the beta increases). The three components were fitted and incorporated through elaborate statistical measurements. An empirical test is run from 2000 to 2015 with exhaustive data sets including both American and European securities. We compute the bias in hedging the most popular market neutral strategies (low volatility, momentum and capitalization) using the standard approach of the beta measurement and the reactive beta model. Our main findings emphasize the ability of the reactive beta model to significantly reduce the biases of these strategies, particularly during stress periods. Robustness check confirms that the reactive beta is not biased when the leverage effect and beta elasticity are introduced and appear to be robust when volatility of volatility and volatility of correlation are introduced. [99]{} Acharyaa, V., and L. H., Pedersen. “Asset Pricing with Liquidity Risk.” Journal of Financial Economics, 77 (2005), pp. 375-410. Agarwal, V., and N. Y. Naik. “Risks and Portfolio Decisions Involving Hedge Funds.” Review of Financial Studies, 17 (2004), pp. 63-98. Amihud, Y. “Illiquidity and Stock Returns: Cross-section and Time-series Effects.” Journal of Financial Markets, 5 (2002), pp. 31-56. Ang, A., and J. Liu. “How to Discount Cashflows with Time-Varying Expected Returns.” Journal of Finance, 59, 6 (2004), pp. 2745-2783. Ang, A., and J. Chen. “CAPM over the Long Run: 1926-2001.” Journal of Empirical Finance, 14 (2007), pp. 1-40. Ang, A., R. Hodrick, Y. Xing, and X. Zhang. “The Cross-Section of Volatility and Expected Returns.” Journal of Finance, 61 (2006), pp. 259-299. Ang, A., R., Hodrick, Y. Xing, and X. Zhang. “High Idiosyncratic Volatility and Low Returns: International and Further U.S. Evidence.” Journal of Financial Economics, 91 (2009), pp. 1-23. Ang, A., A. Shtauber, and P. C. Tetlock. “Asset Pricing in the Dark: The Cross-Section of OTC Stocks.” Review of Financial Studies, 26 (2013), pp. 2985-3028. Asness, C., R. Krail, and J. Liew. “Do Hedge Funds Hedge?” Journal of Portfolio Management, 28 (2001), pp. 6-19. Baker, M., B. Bradley, and R. Taliaferro. “The Low Beta Anomaly: A Decomposition into Micro and Macro Effects.” Working paper, Harvard Business School, 2013. Bali, T. G. and R. F. Engle, and Y. Tang. “Dynamic Conditional Beta Is Alive and Well in the Cross Section of Daily Stock Returns.” Management Science, 63, 11 (2017), pp. 3760-3779. Banz, R. W. “The Relationship Between Return and Market Value of Common Stocks.” Journal of Financial Economics, 9 (1981), pp. 3-18. Bekaert, G., and G. Wu. “Asymmetric Volatility and Risk in Equity Markets.” Review of Financial Studies, 13 (2000), pp. 1-42. Black, F. “Capital Market Equilibrium with Restricted Borrowing.” Journal of Business, 4 (1972), pp. 444-455. Black, F., M. Jensen, and M. Scholes. “The Capital Asset Pricing Model: Some Empirical Tests.” In Studies in the Theory of Capital Markets, edited by Michael Jensen. New York: Praeger, 1972. Black, F. “Studies in Stock Price Volatility Changes.” American Statistical Association, Proceedings of the Business and Economic Statistics Section, 177-181, 1976. Blume, M. E. “On the Assessment of Risk.” Journal of Finance, 26 (1971), pp. 1-10. Bollerslev, T., R. Engle, and J. Wooldridge. “A Capital Asset Pricing Model with Time-Varying Covariances.” Journal of Political Economy, 96 (1988), pp. 116-131. Bollerslev, T. “Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model.” Review of Economics and Statistics, 72 (1990), pp. 498-505. Bouchaud, J.-P., A. Matacz, and M. Potters. “Leverage Effect in Financial Markets: The Retarded Volatility Model.” Physical Review Letters, 87 (2001), pp. 1-4. Bussière, M., M. Hoerova, and B. Klaus. “Commonality in Hedge Fund Returns: Driving Factors and Implications.” Journal of Banking & Finance, 54 (2015), pp. 266-280. Campbell, J. Y., and L. Hentchel, “No News is Good News: An Asymmetric Model of Changing Volatility in Stock Returns.” Journal of Financial Economics, 31 (1992), pp. 281-318. Cappiello, L., R. Engle, and K. Sheppard. “Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns.” Journal of Financial Econometrics, 4 (2006), pp. 537-572. Carhart, M. “On Persistence in Mutual Fund Performance.” Journal of Finance, 52 (1997), pp. 57-82. Chan, L., and J. Lakonishok. “Robust Measurement of Beta Risk.” Journal of Financial and Quantitative Analysis, 27 (1992), pp. 265-282. Chen, A-S., T.-W. Zhang, and H. Wu. “The Beta Regime Change Risk Premium.” Working paper, National Chung Cheng University, 2005. Christie, A. “The Stochastic Behavior of Common Stock Variances - Value, Leverage, and Interest Rate Effects.” Journal of Financial Economic Theory, 10 (1982), pp. 407-432. DeJong, D., and D. W. Collins. “Explanations for the Instability of Equity Beta: Risk-Free Rate Changes and Leverage Effects.” Journal of Financial and Quantitative Analysis, 20 (1985), 73-94. Fabozzi, J. F., and C. J. Francis. “Beta Random Coefficient.” Journal of Financial and Quantitative Analysis, 13 (1978), pp. 101-116. Engle, R. F. ”Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business and Economic Statistics, 20 (2002), pp. 339-350. Engle, R. F. “Dynamic Conditional Beta.” Journal of Financial Econometrics, 14, Issue 4 (2016), pp. 643-667. Fama, E. F., and K. R. French. “The cross-section of expected returns.” Journal of Finance, 47 (1992), pp. 427-465. Fama, E. F., and K. R. French. “Common risk factors in the returns on stocks and bonds.” Journal of Financial Economics, 33 (1993), pp. 3-56. Fama, E. F., and K. R. French. “Industry Costs of Equity.”, Journal of Financial Economics, 43 (1997), pp. 153-193. Fama, E. F., and K. R. French. “Dissecting Anomalies.” Journal of Finance, 63 (2008), pp. 1653-1678. Fama, E. F., and K. R. French. “Size, Value, and Momentum in International Stock Returns.” Journal of Financial Economics, 105 (2012), pp. 457-472. Ferson, W. E., and C. R. Harvey. “Conditioning Variables and the Cross-Section of Stock Returns.” Journal of Finance, 54 (1999), pp. 1325-1360. Francis, J. C. “Statistical Analysis of Risk Surrogates for NYSE Stocks.” Journal of Financial and Quantitative Analysis, 14 (1979), pp. 981-997. Frazzini, A., and L. Pedersen. “Betting Against Beta.” Journal of Financial Economics, 111 (2014), pp. 1-25. Fung, W., and D. A. Hsieh. “Empirical Characteristics of Dynamic Trading Strategies: The Case of Hedge Funds.” The Review of Financial Studies, 10 (1997), pp. 275-302. Galai, D. and R. W. Masulis. “The Option Pricing Model and the Risk Factor of Stock.” Journal of Financial Economics, 3 (1976), 53-81. Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On The Relation between The Expected Value and The Volatility of Nominal Excess Return on stocks.” Journal of Finance, 48 (1993), pp. 1779-1801. Goyal, A., and P. Santa-Clara, “Idiosyncratic Risk Matters!” Journal of Finance, 58 (2003), pp. 975-1007. Grinblatt, M., and T., Moskowitz. “Predicting Stock Price Movements from Past Returns: The Role of Consistency and Tax-Loss Selling.” Journal of Financial Economics, 71 (2004), pp. 541-579. Haugen, R. A., and A. Heins. “Risk and the Rate of Return on Financial Assets: Some Old Wine in New Bottles.” Journal of Financial and Quantitative Analysis, 10 (1975), pp. 775-784. Haugen, R. A., and N. L. Baker. “The Efficient Market Inefficiency of Capitalization-Weighted Stock Portfolios.” Journal of Portfolio Management, 17 (1991), pp. 35-40. Hong, H., and D. Sraer. “Speculative Betas.” Forthcoming, Journal of Finance, (2016). Jagannathan, R., and Z. Wang, “The Conditional CAPM and the Cross-Section of Expected Returns.” Journal of Finance, 51 (1996), pp. 3-53. Jegadeesh, N. “Evidence of Predictable Behavior of Securities Returns.” Journal of Finance, 45 (1990), pp. 881-898. Jegadeesh, N., and S. Titman. “Returns to buying winners and selling losers: Implications for stock market efficiency.” Journal of Finance, 48 (1993), pp. 65-91. Koenker, R., and G. Bassett. “Regression Quantiles.” Econometrica, 46 (1978), pp. 33-50. Koenker, R. “Robust Methods in Econometrics.” Econometric Reviews, 1 (1982), pp. 213-255. Lettau, M., and S. Ludvigson. “Resurrecting the (C)CAPM: a Cross-Sectional Test when Risk Premia are Time-Varying.” Journal of Political Economy, 109 (2001), pp. 1238-1287. Lewellen, J., and S. Nagel. “The Conditional CAPM does not Explain Asset Pricing Anomalies.” Journal of Financial Economics, 82 (2006), pp. 289-314. Malkiel, B. G., and Y. Xu. “Risk and Return Revisited.” Journal of Portfolio Management, 23 (1997), pp. 9-14. Meng, J. G., G. Hu, and J. A. Bai. “A Simple Method for Estimating Betas When Factors Are Measured with Error.” Journal of Financial Research, 34 (2011), pp. 27-60. Mitchell, M., and T. Pulvino. “Characteristics of Risk and Return in Risk Arbitrage.” Journal of Finance, 56 (2001), pp. 2135-2175. Patton, A. J. “Are ‘Market Neutral’ Hedge Funds Really Market Neutral?” Review of Financial Studies, 22 (2009), pp. 2295-2330. Reinganum, R. “Misspecification of Capital Asset Pricing: Empirical Anomalies based on Earnings Yields and Market Values.” Journal of Financial Economics, 9 (1981), pp. 19-46. Roll, R. “A Critique of the Asset Pricing Theory’s Tests Part I: On Past and Potential Testability of the Theory.” Journal of Financial Economics, 4 (1977), pp. 129-176. Sharpe, W. F. “Capital Asset Prices: A Theory of Market Equilibrium under Risk.” Journal of Finance, 19 (1964), pp. 425-442. Shanken, J. “On the Estimation of Beta-Pricing Models.” Review of Financial Studies, 5 (1992), 1-33. Sheppard, K. “Univariate volatility modeling.” Notes, Chapter 7, University of Oxford (2017). Valeyre, S., D. S. Grebenkov, S. Aboura, and Q. Liu. “The Reactive Volatility Model.” Quantitative Finance, 13, (2013), pp. 1697-1706. Selection bias {#sec:selection} ============== Here, we provide some evidence that the bias in beta of the low volatility factor comes from the selection bias: selection of the bottom beta stocks yields the stocks whose beta is underestimated. The measured beta $\beta_{im}$ of stock $i$ is obtained by a standard linear regression of the $i$-th stock returns, $r_i$, to the stock index returns, $r_I$, $$\label{eq:ri_epsiloni} r_{i}=\beta_{im} r_I+ \epsilon_i ,$$ where $\epsilon_i$ is the residual return. We suppose that the measured beta of the stock $i$, $\beta_{im}$, is affected by noise, $$\label{eq:beta_im} \beta_{im}=\beta_{iT}+\eta_i ,$$ where $\beta_{iT}$ is the true beta (which is unknown), and $\eta_i$ is the error of the measurement inherent to the linear regression. The standard deviation of $\eta_i$, $\sigma_{\eta_i}$, depends on the average correlation between the single stock $i$ and the stock index and on the number $n$ of independent points used for the regression (which we set at $n = \frac{1}{\lambda_{\beta}}=90$): $$\sigma_{\eta_i}=\frac{\sigma_{\epsilon i}}{\sigma_I} \, \frac{1}{\sqrt{n}} ,$$ where $\sigma_{\epsilon i}$ is the standard deviation of the residual returns $\epsilon_{i}$. Averaging the above relation over all stocks, we obtain $$\sigma_{\eta} = \frac{\langle\sigma_{\epsilon i} \rangle}{\sigma_I}\sqrt{\lambda_\beta} ,$$ where $\langle \sigma_{\epsilon i}\rangle$ denotes the average. According to Eq. (\[eq:ri\_epsiloni\]), the standard deviation of the stock returns, $\sigma_i$, is $$\sigma_i = \sqrt{\beta_{im}^2 \sigma_I^2 + \sigma_{\epsilon i}^2} \approx \sigma_{\epsilon i} ,$$ because $(\beta_{im} \sigma_I /\sigma_i)^2 \ll 1$ (stocks are much more volatile than the index). We thus obtain $$\label{eq:sigma_eta} \sigma_{\eta} \approx \frac{\langle\sigma_i \rangle}{\sigma_I}\sqrt{\lambda_\beta}.$$ The low volatility factor is 50% long of the 30% top $\beta_{im}$ stocks and 50% short of the 30% bottom $\beta_{im}$ stocks (here, we consider only one sector for simplicity). We adjust the most volatile leg to target a beta neutral factor if we suppose that $\eta_i$ are null. In reality, when taking into account the difference between the measured and the true beta, we obtain the beta of the low volatility factor as: $$\label{eq:beta_low} \beta_{\textrm{low factor}}= -50\%\langle\beta_{iT}\vert i\in \textrm{Bottom}\rangle + 50\% \frac{\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle } {\langle\beta_{im}\vert i\in \textrm{Top}\rangle}\langle\beta_{iT}\vert i\in \textrm{Top}\rangle .$$ This is essentially the beta neutral condition that we impose when constructing the factor (see Appendix \[sec:Afactors\]). Here, $\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle$ is the average of the measured beta over the stocks $i$ in the 30% bottom in the [measured]{} beta values $\beta_{im}$ (similar for other averages). Defining $\Delta \beta_B$ and $\Delta \beta_T$ as $$\langle\beta_{iT}\vert i\in \textrm{Bottom}\rangle=\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle+\Delta \beta_B ,$$ $$\langle\beta_{iT}\vert i\in \textrm{Top}\rangle=\langle\beta_{im}\vert i\in \textrm{Top}\rangle+\Delta \beta_T ,$$ we rewrite Eq. (\[eq:beta\_low\]) as $$\begin{aligned} \nonumber \beta_{\textrm{low factor}} &=& -50\%\left(\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle+\Delta\beta_B\right) + 50\% \frac{\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle }{\langle\beta_{im}\vert i\in \textrm{Top}\rangle} \left(\langle\beta_{im}\vert i\in \textrm{Top}\rangle+\Delta\beta_T\right) \\ &=& -50\% \Delta\beta_B + 50\% \frac{\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle }{\langle\beta_{im}\vert i\in \textrm{Top}\rangle} \Delta\beta_T .\end{aligned}$$ Given that $\langle\beta_{im}\vert i\in \textrm{Bottom}\rangle \ll \langle\beta_{im}\vert i\in \textrm{Top}\rangle$ (as the $\beta_{im}$ in the top quantile are higher than the $\beta_{im}$ in the bottom quantile), we obtain the following approximation $$\beta_{\textrm{low factor}}\approx-50\%\Delta\beta_B .$$ If one knew the true $\beta_{iT}$ values and used them for constructing the low volatility factor, the excess $\Delta \beta_B$ would be zero. However, the true values are unknown, and one uses the measured beta $\beta_{im}$ that creates a selection bias and the nonzero $\Delta \beta_B$, as shown below. To estimate $\Delta\beta_B$, we consider the true beta $\beta_{iT}$ and the measurement error $\eta_i$ as independent random variables and replace the average over stocks by the following conditional expectation $$\label{eq:Delta_beta} \Delta \beta_B = \langle\beta_{iT} - \beta_{im} \vert i\in \textrm{Bottom}\rangle \approx\E \{ \beta_{iT} - \beta_{im} \| i \in \textrm{Bottom} \} = B.$$ We have, then, $$\begin{aligned} \nonumber - B &=& \E \{ \eta_i \| i \in \textrm{Bottom} \} = \int\limits_{-\infty}^\infty \eta \, \P \{ \eta_i \in (\eta,\eta+d\eta) \| i \in \textrm{Bottom} \} \\ &=& \int\limits_{-\infty}^\infty \eta \, \frac{\P \{ \eta_i \in (\eta,\eta+d\eta), ~ i \in \textrm{Bottom} \}} {\P \{ i \in \textrm{Bottom} \}} , \end{aligned}$$ where we wrote explicitly the conditional probability. The denominator is precisely the threshold determining the bottom quantile, $\P \{ i \in \textrm{Bottom} \} = p$, which we set to $30\%$. We thus obtain $$- B = \frac{1}{p} \int\limits_{-\infty}^\infty \eta \, \P \{ \eta_i \in (\eta,\eta+d\eta) , ~ \beta_{im} - \beta_0 < Q \},$$ where the event $i \in \textrm{Bottom}$ is equivalently written as $\beta_{im} < \beta_0 + Q$, where $Q$ is the value of the measured beta that corresponds to the quantile $p$, and $\beta_0$ is the mean of $\beta_{im}$. Using Eq. (\[eq:beta\_im\]) and the assumption that $\beta_{iT}$ and $\eta_i$ are independent, one obtains $$\begin{aligned} \nonumber - B &=& \frac{1}{p} \int\limits_{-\infty}^\infty \eta \, \P \{ \eta_i \in (\eta,\eta+d\eta) , ~ \beta_{iT} - \beta_0 < Q - \eta \} \\ &=& \frac{1}{p} \int\limits_{-\infty}^\infty \eta \, \P \{ \eta_i \in (\eta,\eta+d\eta)\}\, \P\{ \beta_{iT} - \beta_0 < Q - \eta \} .\end{aligned}$$ To obtain some quantitative estimates, we make a strong assumption that both $\beta_{iT}$ and $\eta_i$ are Gaussian variables, with means $\beta_0$ and $0$ and standard deviations $\sigma_\beta$ and $\sigma_\eta$, respectively. We then obtain $$- B = \frac{1}{p} \int\limits_{-\infty}^\infty d\eta \, \eta \, \frac{\exp(-\eta^2/(2\sigma_\eta^2))}{\sqrt{2\pi} \, \sigma_\eta} \Phi\bigl((Q - \eta)/\sigma_\beta\bigr) ,$$ where $$\Phi(x) = \int\limits_{-\infty}^x dy\, \frac{e^{-y^2/2}}{\sqrt{2\pi}}$$ is the cumulative Gaussian distribution. Changing the integration variable, one obtains $$- B = \frac{\sqrt{2} \sigma_\eta}{p \sqrt{\pi}} \int\limits_{-\infty}^\infty dx \, x \, \exp(-x^2) \Phi\bigl((Q - x \sqrt{2}\sigma_\eta)/\sigma_\beta\bigr) .$$ Integrating by parts and omitting technical computations, we obtain $$B = \frac{\sqrt{2} \sigma_\eta}{p \sqrt{\pi}} \, \frac{\sigma_\eta}{2\sigma_\beta \sqrt{1+b^2}} \exp\biggl(-\frac{a^2}{1+b^2}\biggr),$$ where $a = Q/(\sqrt{2} \sigma_\beta)$ and $b = \sigma_\eta/\sigma_\beta$. Setting $$Q = \sigma_\beta \sqrt{2} \, q, \qquad q = \erf^{-1}(2p-1),$$ we obtain $$\label{eq:B} B = \frac{\sigma_\eta}{p \sqrt{2\pi}} \, \frac{1}{\sqrt{1 + (\sigma_\beta/\sigma_\eta)^2}} \exp\biggl(-\frac{q^2}{1+(\sigma_\eta/\sigma_\beta)^2}\biggr),$$ from which $$\label{eq:beta_low_final} \beta_{\textrm{low factor}}\approx -50\% \frac{\sigma_\eta}{p \sqrt{2\pi}} \, \frac{1}{\sqrt{1 + (\sigma_\beta/\sigma_\eta)^2}} \exp\biggl(-\frac{q^2}{1+(\sigma_\eta/\sigma_\beta)^2}\biggr).$$ From the data for the USA, we estimate the standard deviation of the measured beta ($\sigma_{\beta}=0.43$), the volatility of the stock index ($\sigma_I = 19.77\%$), the volatility of the low volatility factor ($3.46\%$), and $\langle \sigma_i\rangle/\sigma_I = 1.53$. Setting $\lambda_\beta = 1/90$, we obtain from Eq. (\[eq:sigma\_eta\]) $\sigma_\eta = 1.53 \, \sqrt{1/90} = 0.1613$. For $p = 0.3$ (bottom $30\%$), we obtain $q = -0.3708$ and, thus, $\beta_{\textrm{low factor}} \approx 0.0334$ from Eq. (\[eq:beta\_low\_final\]). Finally, we conclude that $\rho_{\textrm{low factor}}=3.34\%\frac{19.77\%}{3.46\%}=19.1\%$. Construction of the beta-neutral factors {#sec:Afactors} ======================================== We implement the four most popular strategies as four beta-neutral factors that are constructed as follows. First, we split all stocks into six supersectors of similar sizes to minimize sectorial correlations. For each trading day, the stocks of the chosen supersector are sorted according to the indicator (e.g., the capitalization) available the day before (we use the publication date and not the valuation date). The related indicator-based factor is formed by buying the first $pN$ stocks in the sorted list and shorting the last $pN$ stocks, where $N$ is the number of stocks in the considered supersector, and $p$ is a chosen quantile level. As described in Sec. \[sec:results\], we use $p = 0.15$ for short-term reversal and long-term momentum factors and $p = 0.30$ for the capitalization and low volatility factors. The other stocks (with intermediate indicator values) are not included (weighted by $0$). To reduce the specific risk, the weights of the selected stocks are set inversely proportional to the stock’s volatility $\sigma_i$, whereas the weights of the remaining stocks are $0$. Moreover, the inverse stock volatility is limited to reduce the impact of extreme specific risk. Each trading day, we recompute the weight $w_{i}$ as follows $$\label{eq:weights} w_{i} = \left\{ \begin{array}{c l} + \mu_+ \min\{1, \sigma_{\rm mean}/\sigma_i \}, & \qquad \textrm{if $i$ belongs to the first $pN$ stocks in the sorted list}, \\ - \mu_- \min\{1, \sigma_{\rm mean}/\sigma_i \}, & \qquad \textrm{if $i$ belongs to the last $pN$ stocks in the sorted list},\\ 0, & \qquad \textrm{otherwise,} \\ \end{array} \right.$$ where $\sigma_{\rm mean} = \frac{1}{N}(\sigma_1 + \ldots + \sigma_N)$ is the mean estimated volatility over the cluster of sectors. In this manner, the weights of low-volatility stocks are reduced to avoid strongly unbalanced portfolios concentrated in such stocks. The two common multipliers, $\mu_\pm$, are used to ensure the beta market neutral condition: $$\label{eq:beta_neutral} \sum\limits_{i=1}^N \beta_{i} w_{i} = 0 ,$$ where $\beta_{i}$ is the sensitivity of stock $i$ to the market obtained either by an OLS or by our reactive method. In every case, the method to estimate beta uses the rolling daily returns and only past information to avoid the look-ahead bias. If the aggregated sensitivity of the long part of the portfolio to the market is higher than that of the short part of the portfolio, its weight is reduced by the common multiplier $\mu_+ < \frac{1}{2p N}$, which is obtained from Eq. (\[eq:beta\_neutral\]) by setting $\mu_- = \frac{1}{2p N}$ (which implies that the sum of absolute weights $\|w_i\|$ does not exceed $1$). In the opposite situation (when the short part of the portfolio has a higher aggregated beta), one sets $\mu_+ = \frac{1}{2p N}$ and determines the reducing multiplier $\mu_- < \frac{1}{2p N}$ from Eq. (\[eq:beta\_neutral\]). The resulting factor is obtained by aggregating the weights constructed for each supersector. We emphasize that the factors are constructed on a daily basis, i.e., the weights are re-evaluated daily based on updated indicators. However, most indicators do not change frequently, so the transaction costs related to changing the factors are not significant. Appendix C: Description of alternative methods {#sec:methods} ---------------------------------------------- ### Unconditional beta #### The theory. [@Chan92] produce an empirical analysis that describes various robust methods for estimating constant beta as they provide an alternative to Ordinary Least Squares (OLS). Their method is built the work [@Koenker78] that provides robust alternatives to the sample mean using more complex linear combination of order statistics in order to face the case of non-Gaussian errors, which are the source of outliers. Instead of minimizing the sum of squared residuals, they consider an estimator that is based on minimizing the criterion including a penalty function $\varrho$ on the residuals $\epsilon$: $$\label{min} \sum_{t=1}^{T} \varrho_{\theta} (\epsilon_t)$$ for $\varrho_{\theta} (\epsilon_t) = \theta \left\| \epsilon_t \right\|$ if $\epsilon_t \geq 0,$ or $(1 - \theta) \left\| \epsilon_t \right\|$ if $\epsilon_t < 0$, where $0< \theta <1$. [@Chan92] minimize the sum of absolute deviations of the residuals $\epsilon_{it}$ from the market model, instead of the sum of squared deviations. The resulting minimum absolute deviations (MAD) estimator of the regression parameters corresponds to the special case of $\theta=1/2$ where half of the observations lie above the line, while half lie below. More generally, large or small values of the weight $\theta$ attach a penalty to observations with large positive or negative residuals. Varying $\theta$ between 0 and 1 yields a set of regression quantile estimates $\hat{\beta(\theta)}$ that is analogous to the quantiles of any sample of data. However, they recognize that MAD does not prove itself to be a clearly superior method and suggest that it may be improved via linear combinations of sample quantiles such like trimmed means. For that reason, [@Chan92] test different combinations of regressions quantiles serving as the basis for the robust estimators. They discuss the general case of trimmed regression quantile (TRQ) given as a weighted average of the regression quantile statistics: $$\label{trq} \hat{\beta}_{\alpha} = (1-2\alpha)^{-1} \int_{\alpha}^{1-\alpha} \hat{\beta}(\theta) d\theta$$ where $0<\alpha<1/2$ and $0<\theta<1$. More specifically, [@Chan92] suggest a more straightforward and equivalent method that considers estimators that are finite linear combinations of regression quantiles (QR) and computationally simpler: $$\label{regq} \beta_{\omega} = \sum_{i=1}^{N} \omega_i \hat{\beta}(\theta_i)$$ where weights $0<\omega_i<1$, $i=1,...,N$ and $\sum_{i=1}^{N} \omega_i=1$. The specific case of weighted average is given by the Tukey’s trimean (TRM) estimator: $$\begin{aligned} \hat{\beta}_{TRM} = 0.25 \hat{\beta}(1/4) + 0.5 \hat{\beta}(1/2) + 0.25 \hat{\beta}(3/4) \end{aligned}$$ #### The application. Their analysis is based mainly on simulated returns data although they add some tests with actual returns data. The main advantages of a simulation are that the true values of the underlying parameters are known, and that the extent of departures from normality can be controlled. They begin with a baseline simulation with 25,000 replications using data generated from a normal distribution and they also consider the case where the residual term is drawn from a Student-distribution with three degrees of freedom in order to explain the observed leptokurtosis in daily returns data. We follow the same methodology to assess the quality of the OLS, the MAD and the TRM beta estimators using Gaussian and t-Student residuals in the seven types of Monte Carlo simulations (MC1,...,MC7). To replicate the exponential weight scheme of the reactive model ($\lambda_\beta=1/90$), Eq. (\[min\]) is replaced by $$\label{min2} \sum_{t=1}^{T} \left(1-\lambda_\beta\right)^{T-t} \varrho_{\theta} (\epsilon_t)$$ ### Conditional Beta #### The theory. The first application of time-varying beta was proposed in [@Bollerslev88] since the beta was computed as the ratio of the conditional covariance to the conditional variance. [@Engle02] generalizes [@Bollerslev90] constant correlation model by making the conditional correlation matrix time-dependent with the Dynamic Conditional Correlation (DCC) model that constrains the time-varying conditional correlation matrix to be positive definite and the number of parameters to grow linearly by a two step procedure. The first step requires the GARCH variances to be estimated univariately. Their parameter estimates remains constant for the next step. The second stage is estimated conditioning on the parameters estimated in the first stage. Hereafter, we extend the modeling of the DCC beta for inclusion of an asymmetric term in the conditional variance equation. In the case of asymmetry in the conditional variance, we select the GJR-GARCH(1,1) specification by [@Glosten93], which assumes a specific parametric form with leverage effect in the conditional variance (DCC-GJR beta). The basic idea is that negative shocks at period $(t-1)$ have a stronger impact in the conditional variance at period $t$ than positive shocks. Notice that even though the conditional distribution is Gaussian, the corresponding unconditional distribution can still present excess kurtosis. We select the ADCC model by [@Cappiello06] to incorporate asymmetry in correlation. The case mixing asymetry in both located in the variance equation (GJR-GARCH) and in the correlation equation (ADCC) is examined (ADCC-GJR GARCH). In our paper the symmetric GARCH DCC will be called simply DCC and the asymmetric ADCC-GJR will be called simply ADCC Let us consider $r_i$ and $r_I$ as the returns of a single stock and the stock index, respectively. We assume that their respective conditional variances follow a (GJR-)GARCH(1,1) specification. The stock return $r_i$ is defined by its conditional volatility, $\sigma_i$, and a zero-mean white noise $\xi_i(t)$: $$r_i(t)=\sigma_i(t-1) \xi_i(t)$$ The conditional variation specification of the stock return is the following: $$\label{eq:beta_aux1} \sigma_i^{2}(t)= (1-a-b-\gamma/2) \tilde\sigma_i^{2} + a \sigma_i^{2}(t-1) [\xi_i(t)]^2+b \sigma_i^{2}(t-1)+\gamma \sigma_i^{2} [\xi_i^-(t)]^2$$ where $\tilde\sigma_i$ is the unconditional volatility, and $a$, $b$, and $\gamma$ are parameters reflecting respectively, the ARCH, GARCH and asymmetry effects. When $\gamma=0$, the specification collapses to a GARCH model, otherwise, it stands for the GJR-GARCH model, where the asymmetric term is defined such as $\xi_i^-(t)=\xi_i(t)$ if $\xi_i(t)>0$, otherwise $\xi_i^-(t)=0$. The stock index return $r_I$ is defined by its conditional volatility, $\sigma_I$, and a zero-mean white noise $\xi_I(t)$ that is correlated to $\xi_i(t)$: $$r_I(t)=\sigma_I(t-1) \xi_I(t)$$ The conditional variance specification of the stock index return is the following: $$\label{eq:beta_aux2} \sigma_I^{2}(t)= (1-a-b-\gamma/2) \tilde\sigma_I^{2} + a \sigma_I^{2}(t-1) [\xi_I(t)]^2+b \sigma_I^{2}(t-1) +\gamma \sigma_I^{2} [\xi_I^-(t)]^2$$ We define the normalized conditional variance diagonal terms such as: $$q_{ii}(t)=(1-a_\rho-b_\rho- \gamma_\rho/2) + a_\rho \xi_i(t-1) \xi_i(t-1) + b_\rho q_{ii}(t-1) +\gamma_\rho \xi_i^-(t-1) \xi_i^-(t-1)$$ $$q_{II}(t)=(1-a_\rho-b_\rho- \gamma_\rho/2) + a_\rho \xi_I(t-1) \xi_I(t-1) + b_\rho q_{II}(t-1) +\gamma_\rho \xi_I^-(t-1) \xi_I^-(t-1)$$ The normalized conditional covariance term $q_{iI}(t)$ is given by: $$q_{iI}(t)=(1-a_\rho-b_\rho- \gamma_\rho/4) \tilde \rho + a_\rho \xi_i(t-1) \xi_I(t-1) + b_\rho q_{iI}(t-1) +\gamma_\rho \xi_i^-(t-1) \xi_I^-(t-1)$$ When $\gamma_\rho=0$, the specification collapses to a DCC model, otherwise it stands for the ADCC model, where the asymmetric term is defined such as $\xi_i^-(t)=\xi_i(t)$ if $\xi_i(t)>0$, otherwise $\xi_i^-(t)=0$. The conditional correlation between $\xi_I(t+1)$ and $\xi_i(t+1)$ is then updated by: $$\rho_{iI}(t)=q_{iI}(t)/\sqrt{ q_{II}(t) q_{ii}(t)}$$ The beta DCC and beta ADCC estimation are defined in the same way: $$\beta_{DCC}(t)=\rho_{iI}(t) \sigma_i(t)/\sigma_I(t)$$ The log-likelihood function is optimized to calibrate the parameters $\tilde \rho$, $\tilde\sigma_I$ and $\tilde\sigma_i$ for estimation: $$\label{eq:Lbis} L_{DCC} = \frac{1}{2}\sum_{t}^T \left( L_{V}(t)+L_{C}(t)\right)$$ $$L_{V}(t)=- 2 \log(2\pi) -\xi_I(t)^2-\xi_i(t)^2- 2\log(\sigma_I(t))-2\log(\sigma_i(t))$$ $$L_{C}(t)= -\log(det(R(t)))- U'(t)R(t)^{-1}U(t)- U'(t)U(t)$$ with $det$ as the determinant of a matrix, and $$R(t)=\begin{bmatrix} 1 & \rho_{iI}(t) \\ \rho_{iI}(t) & 1 \end{bmatrix}, \qquad U(t)=\begin{bmatrix} \xi_i(t) \\ \xi_I(t) \end{bmatrix}$$ #### The application. For Monte Carlo simulation purposes: - $\xi_i(t)$ is either generated randomly in MC6 and MC7 according to a standard Gaussian or measured through returns $r_i(t)$ and $\sigma_i(t-1)$ for beta DCC estimation. - $\gamma=0$ for MC6 and beta DCC estimation but $\gamma>0$ for MC7 and beta ADCC that captures the asymmetry term of the GJR-GARCH. - $\xi_I(t)$ is either generated randomly in MC6 and MC7 according to a standard Gaussian random variable that is correlated to the random variable $\xi_i(t)$ (correlation between $\xi_i(t)$ and $\xi_I(t)$ is $\rho_{iI}(t-1)$) or measured through returns $r_I(t)$ and $\sigma_I(t-1)$ for beta DCC estimation. - $\gamma_\rho=0$ for MC6 and beta DCC but $\gamma_\rho>0$ for MC7 and beta ADCC that captures the asymmetry term of the ADCC. The fixed parameters that are supposed to be known when testing the beta DCC are set to US market estimates by from [@Sheppard17]: - fixed parameters for univariate symmetric GARCH(1,1) process (MC6, i.e. DCC): $b=0.89$, $b$ is the decay coefficient and $1/(1-b)$ is related to the number of days the process needs to mean revert; $a=0.099$ would describe the level of the volatility of the volatility. - fixed parameters for univariate asymmetric GJR-GARCH(1,1,1) process (MC7, i.e., ADCC): $b=0.901$, $b$ is the decay coefficient and $1/(1-b)$ is related to the number of days the process needs to mean revert; $a=0.0$, $a+\gamma/2$ describe the level of the volatility of the volatility; $\gamma=0.171$, $\gamma$ would describe the asymmetry. The fixed parameters that are supposed to be known when testing the beta DCC and betas ADCC are set to US market estimates from [@Cappiello06]: - fixed parameters for the symmetric cross term process (MC6, i.e., DCC): $b_\rho=0.9261$, $b_\rho$ is the decay coefficient and is linked to the relaxation time; $a_\rho=0.0079$ would describe the level of the volatility. - fixed parameters for the asymmetric cross term process (MC7, i.e., ADCC): $b_\rho=0.9512$, $b_\rho$ is the decay coefficient and is linked to the relaxation time; $a_\rho=0.0020$, $a_\rho+\gamma_\rho/4$ would describe the level of the volatility of the correlation; $\gamma_\rho=0.0040$, $\gamma_\rho$ would describe the asymmetry. The fixed parameters that are not known when testing the beta DCC and estimated through optimization of log-likelihood are set by MC simulation to: - $\tilde \rho=0.15/0.4$, unconditional correlation; - $\tilde\sigma_I=0.15/\sqrt{255}$, $\tilde\sigma_i=0.4/\sqrt{255}$ unconditional stock index volatility; - $\tilde\sigma_i=0.4/\sqrt{255}$ unconditional single stock volatility. To replicate the exponential weight scheme in the reactive model ($\lambda_\beta=1/90$), Eq. (\[eq:Lbis\]) is replaced by $$L_{DCC} = \frac{1}{2}\sum_{t}^T \left(1-\lambda_\beta\right)^{T-t} \left( L_{V}(t)+L_{C}(t)\right)$$ [^1]: John Locke Investments, 38 Avenue Franklin Roosevelt, 77210 Fontainebleau-Avon, France, and Université de Paris XIII, Sorbonne Paris Cité, 93430 Villetaneuse, France [^2]: Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS – Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France [^3]: Université de Paris XIII, Sorbonne Paris Cité, 93430 Villetaneuse, France [^4]: Note that we are not dealing with the restricted definition of the “leveraged beta” that comes from the degree of leverage in the firm’s capital structure. [^5]: In practice, a filtering function is introduced to attenuate the contribution from eventual extreme events or wrong data (outliers). The filter was applied to $z=\frac{L_s(t)-I(t)}{I(t)}$ and $z=\frac{L_{is}(t)-S_i(t)}{S_i(t)}$ in Eqs. (\[eq:L\_Taylor\], \[eq:Li\_Taylor\]) and was defined as $F_\phi(z) = \tanh(\phi z)/\phi$ with $\phi = 3.3$ (in the limit $\phi = 0$, there is no filter: $F_0(z) = z$). [^6]: See <http://www.cboe.com/micro/impliedcorrelation/impliedcorrelationindicator.pdf>
--- abstract: 'We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with $\ell_1$ regularization. The minimization problem can be efficiently solved by a split Bergman iteration type algorithm. We further prove that from any initial condition, the algorithm converges to a minimizer of the variational principle.' address: - 'Department of Mathematics, University of California, Irvine' - 'Department of Mathematics, Physics, and Chemistry, Duke University' - 'Department of Mathematics and Institute for Pure and Applied Mathematics, University of California, Los Angeles' author: - Rongjie Lai - Jianfeng Lu - Stanley Osher title: 'Density matrix minimization with $\ell_1$ regularization' --- [^1] Introduction ============ The low-lying eigenspace of operators has many important applications, including those in quantum chemistry, numerical PDEs, and statistics. Given a $n \times n$ symmetric matrix $H$, and denote its eigenvectors as $\{\Phi_i\}, i = 1, \ldots, n$. The low-lying eigenspace is given by the span of the first $N$ (usually $N \ll n$) eigenvectors. In many scenario, the real interest is the subspace itself, but not a particular set of basis functions. In particular, we are interested in a sparse representation of the eigenspace. The eigenvectors form a natural basis set, but for oftentimes they are not sparse or localized (consider for example the eigenfunctions of the free Laplacian operator $-\Delta$ on a periodic box). This suggests asking for an alternative sparse representation of the eigenspace. In quantum chemistry, the low-lying eigenspace for a Hamiltonian operator corresponds to the physically occupied space of electrons. In this context, a localized class of basis functions of the low-lying eigenspaces is called Wannier functions [@Wannier:1937; @Kohn:1959Wannier]. These functions provide transparent interpretation and understandings of covalent bonds, polarizations, etc. of the electronic structure. These localized representations are also the starting point and the essence for many efficient algorithms for electronic structure calculations (see e.g. the review article [@Goedecker:99]). Our contribution ---------------- In this work, we propose a convex minimization principle for finding a sparse representation of the low-lying eigenspace. $$\label{eqn:l1DensityMatrix} \begin{aligned} & \min_{P \in {\mathbb{R}}^{n\times n}} \operatorname{tr}(H P) + \frac{1}{\mu} {\lVertP\rVert}_1\\ & \text{s.t.} \ P = P^{{\mathrm{T}}},\, \operatorname{tr}P = N,\, 0 \preceq P \preceq I, \end{aligned}$$ where ${\lVert\cdot\rVert}_1$ is the entrywise $\ell_1$ matrix norm, $A \preceq B$ denotes that $B - A$ is a positive semi-definite matrix, and $\mu$ is a penalty parameter for entrywise sparsity. Here $H$ is an $n\times n$ symmetric matrix, which is the (discrete) Hamiltonian in the electronic structure context. The variational principle gives $P$ as a sparse representation of the projection operator onto the low-lying eigenspace. The key observation here is to use the matrix $P$ instead of the wave functions $\Psi$. This leads to a convex variational principle. Physically, this corresponds to looking for a sparse representation of the density matrix. We also noted that in cases where we expect degeneracy or near-degeneracy of eigenvalues of the matrix $H$, the formulation in terms of the density matrix $P$ is more natural, as it allows fractional occupation of states. This is a further advantage besides the convexity. Moreover, we design an efficient minimization algorithm based on split Bregman iteration to solve the above variational problem. Starting from any initial condition, the algorithm always converges to a minimizer. Previous works -------------- There is an enormous literature on numerical algorithms for Wannier functions and more generally sparse representation of low-lying eigenspace. The influential work [@Marzari:1997] proposed a minimization strategy within the occupied space to find spatially localized Wannier functions (coined as “maximally localized Wannier functions”). In [@E:2010PNAS], the second author with his collaborators developed a localized subspace iteration (LSI) algorithm to find Wannier functions. The idea behind the LSI algorithm is to combine the localization step with the subspace iteration method as an iterative algorithm to find Wannier functions of an operator. The method has been applied to electronic structure calculation in [@Garcia-CerveraLuXuanE:09]. As [@Garcia-CerveraLuXuanE:09] shows, due to the truncation step involved, the LSI algorithm does not in general guarantee convergence. As a more recent work in [@OzolinsLaiCaflischOsher:13], $L_1$ regularization is proposed to be used in the variational formulation of the Schrödinger equation of quantum mechanics for creating compressed modes, a set of spatially localized functions $\{\psi_i\}_{i=1}^N$ in $\mathbb{R}^d$ with compact support. $$\label{model:CMs} E = \min_{\Psi_N} \sum_{j=1}^N \left( \frac{1}{\mu}\left\| \psi_j \right\|_1 + \langle \psi_j , \hat{H} \psi_j \rangle \right) \quad \mbox{\text{s.t.}} \quad \langle \psi_j, \psi_k \rangle = \delta_{jk},$$ where $\hat{H} = -\frac{1}{2}\Delta + V({\mathrm{x}})$ is the Hamilton operator corresponding to potential $V({\mathrm{x}})$, and the $L_1$ norm is defined as $\left\| \psi_j \right\|_1 = \int \| \psi_j \| d{\mathrm{x}}$. This $L_1$ regularized variational approach describes a general formalism for obtaining localized (in fact, compactly supported) solutions to a class of mathematical physics PDEs, which can be recast as variational optimization problems. Although an efficient algorithm based on a method of splitting orthogonality constraints (SOC) [@Lai:2014splitting] is designed to solve the above non-convex problem, it is still a big challenge to theoretically analyze the convergence of the proposed the algorithm. The key idea in the proposed convex formulation of the variational principle is the use of the density matrix $P$. The density matrix is widely used in electronic structure calculations, for example the density matrix minimization algorithm [@LiNunesVanderbilt:93]. In this type of algorithm, sparsity of density matrix is specified explicitly by restricting the matrix to be a banded matrix. The resulting minimization problem is then non-convex and found to suffer from many local minimizers. Other electronic structure algorithms that use density matrix include density matrix purification [@McWeeny:60], Fermi operator expansion algorithm [@BaroniGiannozzi:92], just to name a few. From a mathematical point of view, the use of density matrix can be viewed as similar to the idea of lifting, which has been recently used in recovery problems [@CandesStrohmerVoroninski:13]. While a nuclear norm is used in PhaseLift method [@CandesStrohmerVoroninski:13] to enhance sparsity in terms of matrix rank; we will use an entrywise $\ell_1$ norm to favor sparsity in matrix entries. The rest of the paper is organized as follows. We formulate and explain the convex variational principle for finding localized representations of the low-lying eigenspace in Section \[sec:formulation\]. An efficient algorithm is proposed in Section \[sec:Algorithm\] to solve the variational principle, with numerical examples presented in Section \[sec:numerics\]. The convergence proof of the algorithm is given in Section \[sec:proof\]. Formulation {#sec:formulation} =========== Let us denote by $H$ a symmetric matrix [^2] coming from, for example, the discretization of an effective Hamiltonian operator in electronic structure theory. We are interested in a sparse representation of the eigenspace corresponding to its low-lying eigenvalues. In physical applications, this corresponds to the occupied space of a Hamiltonian; in data analysis, this corresponds to the principal components (for which we take the negative of the matrix so that the largest eigenvalue becomes the smallest). We are mainly interested in physics application here, and henceforth, we will mainly interpret the formulation and algorithms from a physical view point. The Wannier functions, originally defined for periodic Schrödinger operators, are spatially localized basis functions of the occupied space. In [@OzolinsLaiCaflischOsher:13], it was proposed to find the spatially localized functions by minimizing the variational problem $$\label{eq:Psi} \min_{\Psi \in {\mathbb{R}}^{n \times N},\, \Psi^{{\mathrm{T}}} \Psi = I} \operatorname{tr}(\Psi^{{\mathrm{T}}} H \Psi) + \frac{1}{\mu} {\lVert\Psi\rVert}_{1}$$ where ${\lVert\Psi\rVert}_{1}$ denotes the entrywise $\ell_1$ norm of $\Psi$. Here $N$ is the number of Wannier functions and $n$ is the number of spatial degree of freedom (e.g. number of spatial grid points or basis functions). The idea of the above minimization can be easily understood by looking at each term in the energy functional. The $\operatorname{tr}(\Psi^{{\mathrm{T}}} H \Psi)$ is the sum of the Ritz value in the space spanned by the columns of $\Psi$. Hence, without the $\ell_1$ penalty term, the minimization $$\min_{\Psi \in {\mathbb{R}}^{n \times N},\, \Psi^{{\mathrm{T}}} \Psi = I} \operatorname{tr}(\Psi^{{\mathrm{T}}} H \Psi)$$ gives the eigenspace corresponds to the first $N$ eigenvalues (here and below, we assume the non-degeneracy that the $N$-th and $(N+1)$-th eigenvalues of $H$ are different). While the $\ell_1$ penalty prefers $\Psi$ to be a set of sparse vectors. The competition of the two terms gives a sparse representation of a subspace that is close to the eigenspace. Due to the orthonormality constraint $\Psi^{{\mathrm{T}}} \Psi = I$, the minimization problem is not convex, which may result in troubles in finding the minimizer of the above minimization problem and also makes the proof of convergence difficult. Here we take an alternative viewpoint, which gives a convex optimization problem. The key idea is instead of $\Psi$, we consider $P = \Psi \Psi^{{\mathrm{T}}} \in {\mathbb{R}}^{n\times n}$. Since the columns of $\Psi$ form an orthonormal set of vectors, $P$ is the projection operator onto the space spanned by $\Psi$. In physical terms, if $\Psi$ are the eigenfunctions of $H$, $P$ is then the density matrix which corresponds to the Hamiltonian operator. For insulating systems, it is known that the off-diagonal terms in the density matrix decay exponentially fast [@Kohn:59; @Panati:07; @Cloizeaux:64a; @Cloizeaux:64b; @Nenciu:83; @Kivelson:82; @NenciuNenciu:98; @ELu:CPAM; @ELu:13]. We propose to look for a sparse approximation of the exact density matrix by solving the minimization problem proposed in . The variational problem is a convex relaxation of the non-convex variational problem $$\label{eq:Pnonconvex} \begin{aligned} & \min_{P \in {\mathbb{R}}^{n\times n}} \operatorname{tr}(H P) + \frac{1}{\mu} {\lVertP\rVert}_1 \\ & \text{s.t.} \ P = P^{{\mathrm{T}}},\, \operatorname{tr}P = N,\, P = P^2, \end{aligned}$$ where the constraint $0 \preceq P \preceq I$ is replaced by the idempotency constraint of $P$: $P = P^2$. The variational principle can be understood as a reformulation of using the density matrix as variable. The idempotency condition $P = P^2$ is indeed the analog of the orthogonality constraint $\Psi^{{\mathrm{T}}} \Psi = I$. Note that $0 \preceq P \preceq I$ requires that the eigenvalues of $P$ (the occupation number in physical terms) are between $0$ and $1$, while $P = P^2$ requires the eigenvalues are either $0$ or $1$. Hence, the set $$\mathcal{C} = \{ P: P = P^{{\mathrm{T}}},\, \operatorname{tr}P = N,\, 0 \preceq P \preceq I \}$$ is the convex hull of the set $$\mathcal{D} = \{ P: P = P^{{\mathrm{T}}},\, \operatorname{tr}P = N,\, P = P^2\}.$$ Therefore is indeed a convex relaxation of . Without the $\ell_1$ regularization, the variational problems and become $$\label{eq:P'} \begin{aligned} & \min_{P \in {\mathbb{R}}^{n\times n}} \operatorname{tr}(H P) \\ & \text{s.t.} \ P = P^{{\mathrm{T}}},\, \operatorname{tr}P = N,\, 0 \preceq P \preceq I, \end{aligned}$$ and $$\label{eq:Pnonconvex'} \begin{aligned} & \min_{P \in {\mathbb{R}}^{n\times n}} \operatorname{tr}(H P) \\ & \text{s.t.} \ P = P^{{\mathrm{T}}},\, \operatorname{tr}P = N,\, P = P^2. \end{aligned}$$ These two minimizations actually lead to the same result in the non-degenerate case. \[prop:equiv\] Let $H$ be a symmetric $n \times n$ matrix. Assume that the $N$-th and $(N+1)$-th eigenvalues of $H$ are distinct, the minimizers of and are the same. This is perhaps a folklore result in linear algebra, nevertheless we include the short proof here for completeness. It is clear that the unique minimizer of is given by the projection matrix on the first $N$ eigenvectors of $H$, given by $$P_N = \sum_{i=1}^N v_i v_i^{{\mathrm{T}}}$$ where $\{v_i\}, i = 1, \ldots, n$ are the eigenvectors of $H$, ordered according to their associated eigenvalues. Let us prove that is minimized by the same solution. Assume $P$ is a minimizer of , we calculate $$\label{eq:trhp} \operatorname{tr}(HP) = \sum_{i=1}^n v_i^{{\mathrm{T}}} H P v_i = \sum_{i=1}^n \lambda_i v_i^{{\mathrm{T}}} P v_i = \sum_{i=1}^n \lambda_i \theta_i(P),$$ where $\theta_i(P) = v_i^{{\mathrm{T}}} P v_i$. On the other hand, we have $$\operatorname{tr}(P) = \sum_{i=1}^n v_i^{{\mathrm{T}}} P v_i = \sum_{i=1}^n \theta_i(P) = N,$$ and $0 \leq \theta_i(P) \leq 1$ since $0 \preceq P \preceq I$. Therefore, if we view as a variational problem with respect to $\{\theta_i\}$, it is clear that the unique minimum is achieved when $$\theta_i(P) = \begin{cases} 1, & i \leq N; \\ 0, & \text{otherwise}. \end{cases}$$ We conclude the proof by noticing that the above holds if and only if $P = P_N$. This result states that we can convexify the set of admissible matrices. We remark that, somewhat surprisingly, this result also holds for the Hartree-Fock theory [@Lieb:77] which can be vaguely understood as a nonlinear eigenvalue problem. However the resulting variational problem is still non-convex for the Hartree-Fock theory. Proposition \[prop:equiv\] implies that the variational principle can be understood as an $\ell_1$ regularized version of the variational problem . The equivalence no longer holds for and with the $\ell_1$ regularization. The advantage of over is that the former is a convex problem while the latter is not. Coming back to the properties of the variational problem . We note that while the objective function of is convex, it is not strictly convex as the $\ell_1$-norm is not strictly convex and the trace term is linear. Therefore, in general, the minimizer of is not unique. Let $\mu \in {\mathbb{R}}_+$, $N = 1$ and $$H = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},$$ The non-uniqueness comes from the degeneracy of the Hamiltonian eigenvalues. Any diagonal matrix $P$ with trace $1$ and non-negative diagonal entries is a minimizer. Let $\mu = 1$, $N = 1$ and $$H = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 2 & 2 \end{pmatrix}$$ The non-uniqueness comes from the competition between the trace term and the $\ell_1$ regularization. The eigenvalues of $H$ are $0, 1$ and $4$. Straightforward calculation shows that $$P_0 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1/2 & -1/2 \\ 0 & -1/2 & 1/2 \end{pmatrix}$$ which corresponds to the eigenvector $(0, \sqrt{2}/2, -\sqrt{2}/2)^{{\mathrm{T}}}$ associated with eigenvalue $0$ and $$P_1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ which corresponds to the eigenvector $(1, 0, 0)^{{\mathrm{T}}}$ associated with eigenvalue $1$ are both minimizers of the objective function $\operatorname{tr}(HP) + {\lVertP\rVert}_{1}$. Actually, due to convexity, any convex combination of $P_0$ and $P_1$ is a minimizer too. It is an open problem under what assumptions that the uniqueness is guaranteed. Algorithm {#sec:Algorithm} ========= To solve the proposed minimization problem , we design a fast algorithm based on split Bregman iteration [@Goldstein:2009split], which comes from the ideas of variables splitting and Bregman iteration [@Osher:2005]. Bregman iteration has attained intensive attention due to its efficiency in many $\ell_1$ related constrained optimization problems [@Yin:2008bregman; @yin2013error]. With the help of auxiliary variables, split Bregman iteration iteratively approaches the original optimization problem by computation of several easy-to-solve subproblems. This algorithm popularizes the idea of using operator/variable splitting to solve optimization problems arising from information science. The equivalence of the split Bregman iteration to the alternating direction method of multipliers (ADMM), Douglas-Rachford splitting and augmented Lagrangian method can be found in [@Esser:2009CAM; @Setzer:2009SSVMCV; @Wu:2010SIAM]. By introducing auxiliary variables $Q = P$ and $R=P$, the optimization problem is equivalent to $$\label{eq:P_split} \begin{aligned} & \min_{P, Q, R \in {\mathbb{R}}^{n\times n}} \frac{1}{\mu} {\lVertQ\rVert}_1 + \operatorname{tr}(H P) \\ & \text{s.t.} \ Q = P,\, R = P,\, \operatorname{tr}P = N,\,R = R^{{\mathrm{T}}},\, 0 \preceq R \preceq I, \end{aligned}$$ which can be iteratively solved by: $$\begin{aligned} (P^k,Q^k,R^k) &= \arg\min_{P, Q, R \in {\mathbb{R}}^{n\times n}} \frac{1}{\mu} {\lVertQ\rVert}_1 + \operatorname{tr}(H P) + \frac{\lambda}{2} \\|P - Q + b\\|_F^2 + \frac{r}{2}\\|P - R + d\\|^2_F \\ & \qquad \text{s.t.} \qquad \operatorname{tr}P = N,\,R = R^{{\mathrm{T}}},\, 0 \preceq R \preceq I, \nonumber \label{eqn:PQR}\\ b^k &= b^{k-1} + P^k - Q^k \\ d^k &= d^{k-1} + P^k - R^k\end{aligned}$$ where variables $b, d$ are essentially Lagrangian multipliers and parameters $r, \lambda$ control the penalty terms. Solving $P^k, Q^k, R^k$ in alternatively, we have the following algorithm. \[alg:CM\_P\] Initialize $Q^0 = R^0 = P^0 \in\mathcal{C} , b^0 = d^0 = 0$ Note that the minimization problem in the steps of Algorithm \[alg:CM\_P\] can be solved explicitly, as follows: $$\begin{aligned} P^k &= B^k - \frac{\operatorname{tr}(B^k) - N}{n}, \\ & \quad \text{where } \quad B^k = \frac{\lambda}{\lambda+r}(Q^{k-1} - b^{k-1}) + \frac{r}{\lambda+r}(R^{k-1} - d^{k-1}) - \frac{1}{\lambda +r} H \notag \\ Q^k &= \operatorname{Shrink}\left(P^k +b^{k-1},\frac{1}{\lambda\mu} \right) = \operatorname{sign}(P^k + b^{k-1})\max\left\{\|P^{k} +b^{k-1}\| - \frac{1}{\lambda\mu},0\right\} \\ R^k &= V\min\{\max\{D,0\},1\}V^T , \text{ where } [V,~ D] = \operatorname{eig}(P^k + d^{k-1}).\end{aligned}$$ Starting form any initial guess, the following theorem guarantees that the algorithm converges to one of the minimizers of the variational problem . \[thm:conv\] The sequence $\big\{(P^k, Q^k, R^k)\big\}_k$ generated by Algorithm \[alg:CM\_P\] from any starting point converges to a minimum of the variational problem . We will prove a slightly more general version of the above (Theorem \[thm:three\]). The idea of the proof follows from the general framework of analyzing split Bregman iteration, i.e. alternating direction method of multipliers (ADDM), see for example [@GlowinskiLeTallec:89]. The standard proof needs to be generalized to cover the current case of “two level splitting” and the non-strictly convexity of the functionals. We defer the detailed proof to Section \[sec:proof\]. Numerical results {#sec:numerics} ================= In this section, numerical experiments are presented to demonstrate the proposed model for density matrix computation using algorithm \[alg:CM\_P\]. We illustrate our numerical results in three representative cases, free electron model, Hamiltonian with energy band gap and a non-uniqueness example of the proposed optimization problem. All numerical experiments are implemented by in a PC with a 16G RAM and a 2.7 GHz CPU. 1D Laplacian ------------ In the first example, we consider the proposed model for the free electron case, in other words, we consider the potential free Schrödinger operator $-1/2\Delta$ defined on 1D domain $\Omega = [0,~ 100]$ with periodic boundary condition. This model approximates the behavior of valence electrons in a metallic solid with weak atomic pseudopotentials. In this case, the matrix $H$ is a central difference discretization of $-1/2\Delta$ on $[0, ~100]$ with equally spaced $256$ points, and we take $N = 10$. Figure \[fig:DensityMatrix\_Lap\](a) illustrates the true density matrix $\sum_{i=1}^{10} \|\phi_i\rangle \langle \phi_i \|$ obtained by the first $10$ eigenfunctions of $H$. As the free Laplacian does not have a spectral gap, the density matrix decays slowly in the off-diagonal direction. Figure \[fig:DensityMatrix\_Lap\](b) and (c) plot the density matrices obtained from the proposed model with parameter $\mu = 10$ and $100$. Note that they are much localized than the original density matrix. As $\mu$ gets larger, the variational problem imposes a smaller penalty on the sparsity, and hence the solution for $\mu = 100$ has a wider spread than that for $\mu = 10$. ![(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively. \[fig:DensityMatrix\_Lap\]](Lap_DensityM_true.eps "fig:"){width=".5\linewidth"}\ ![(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively. \[fig:DensityMatrix\_Lap\]](Lap_DensityM_10.eps "fig:"){width="1\linewidth"}\ ![(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively. \[fig:DensityMatrix\_Lap\]](Lap_DensityM_100.eps){width="1\linewidth"} After we obtain the sparse representation of the density matrix $P$, we can find localized Wannier functions as its action on the delta functions, as plotted in Figure \[fig:Projection\_Lap\] upper and lower pictures for $\mu = 10$ and $100$ respectively. ![Projection of Delta function $\delta(x - x_i)$ using density matrices with $\mu = 10$ (upper) and $\mu = 100$ (lower) respectively.[]{data-label="fig:Projection_Lap"}](Lap_Projection_10.eps "fig:"){width=".8\linewidth"}\ ![Projection of Delta function $\delta(x - x_i)$ using density matrices with $\mu = 10$ (upper) and $\mu = 100$ (lower) respectively.[]{data-label="fig:Projection_Lap"}](Lap_Projection_100.eps "fig:"){width=".8\linewidth"} To indicate the approximation behavior of the proposed model, we consider the energy function approximation of $ \frac{1}{\mu} {\lVertP\rVert}_1 + \operatorname{tr}(H P) $ to $\sum_{i=1}^{10} \langle \phi_i \| H \| \phi_i \rangle$ with different values of $\mu$. In addition, we define $\sum_{i=1}^{10} \\| \phi_i - P \phi_i\\|^2$ as a measurement for the space approximation of the density matrix $P$ to the lower eigen-space $Span\{\phi_i\}_{i=1}^{10}$. Figure \[fig:DensityFunApprox\_Lap\] reports the energy approximation and the space approximation with different values of $\mu$. Both numerical results suggest that the proposed model will converge to the energy states of the Schrödinger operator. We also remark that even though the exact density matrix is not sparse, a sparse approximation gives fairly good results in terms of energy and space approximations. ![Upper: Energy approximation as a function of $\mu$. Lower: Space approximation as a function of $\mu$.[]{data-label="fig:DensityFunApprox_Lap"}](Lap_EnergyFunApprox.eps "fig:"){width=".8\linewidth"}\ ![Upper: Energy approximation as a function of $\mu$. Lower: Space approximation as a function of $\mu$.[]{data-label="fig:DensityFunApprox_Lap"}](Lap_SpaceApprox.eps "fig:"){width=".8\linewidth"}\ 1D Hamiltonian operator with a band gap --------------------------------------- We then consider a modified Kronig–Penney (KP) model [@Kronig:1931quantum] for a one-dimensional insulator. The original KP model describes the states of independent electrons in a one-dimensional crystal, where the potential function $V(x)$ consists of a periodic array of rectangular potential wells. We replace the rectangular wells with inverted Gaussians so that the potential is given by $$V(x) = -V_0\sum_{j=1}^{N_{\text{at}}} \exp\left[ -\frac{(x - x_j)^2}{\delta^2} \right],$$ where $N_{\text{at}}$ gives the number of potential wells. In our numerical experiments, we choose $N_{\text{at}} = 10$ and $x_j = 100 j / 11$ for $j = 1, \ldots, N_{\text{at}}$, and the domain is $[0, 100]$ with periodic boundary condition. The potential is plotted in Figure \[fig:V\_KP\](a). For this given potential, the Hamiltonian operator $H = - \tfrac{1}{2} \Delta + V(x)$ exhibits two low-energy bands separated by finite gaps from the rest of the eigenvalue spectrum (See Figure \[fig:V\_KP\](b)). Here a centered difference is used to discretize the Hamiltonian operator. ![(a): The potential function in the modified Kronig-Penney model. (b): The spectrum of the (discretized) Hamiltonian operator.[]{data-label="fig:V_KP"}](KP_PotentialFun.eps "fig:"){width="1\linewidth"}\ ![(a): The potential function in the modified Kronig-Penney model. (b): The spectrum of the (discretized) Hamiltonian operator.[]{data-label="fig:V_KP"}](KP_LapEigs.eps "fig:"){width=".9\linewidth"}\ \ (a) (b) We consider three choices of $N$ for this model: $N = 10$, $N = 15$ and $N = 20$. They correspond to three interesting physical situations of the model, as explained below. For $N = 10$, the first band of the Hamiltonian is occupied, and hence the system has a spectral gap between the occupied and unoccupied states. As a result, the associated density matrix is exponentially localized, as shown in Figure \[fig:DensityFunApprox\_KP\](a). The resulting sparse representation from the convex optimization is shown in Figure \[fig:DensityFunApprox\_KP\](b) and (c) for $\mu = 10$ and $100$ respectively. We see that the sparse representation agrees well with the exact density matrix, as the latter is very localized. The Wannier functions obtained by projection of delta functions are shown in Figure \[fig:Projection\_KP\]. As the system is an insulator, we see that the localized representation converges quickly to the exact answer when $\mu$ increases. This is further confirmed in Figure \[fig:DensityFunApprox\_KP\_energy\] where the energy corresponding to the approximated density matrix and space approximation measurement $\sum_{i=1}^{10} \\| \phi_i - P \phi_i\\|^2$ are plotted as functions of $\mu$. ![(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively.[]{data-label="fig:DensityFunApprox_KP"}](KP_DensityM_true.eps "fig:"){width=".5\linewidth"}\ ![(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively.[]{data-label="fig:DensityFunApprox_KP"}](KP_DensityM_10.eps "fig:"){width="1\linewidth"}\ ![(a): The true density matrix obtained by the first 10 eigenfunctions of $H$. (b), (c): solutions of the density matrices with $\mu = 10, 100$ respectively.[]{data-label="fig:DensityFunApprox_KP"}](KP_DensityM_100.eps "fig:"){width="1\linewidth"}\ ![Projection of Delta function $\delta(x - x_i)$ using density matrices with $\mu = 10$ (upper) and $\mu = 100$ (lower) respectively.[]{data-label="fig:Projection_KP"}](KP_Projection_10.eps "fig:"){width=".7\linewidth"}\ ![Projection of Delta function $\delta(x - x_i)$ using density matrices with $\mu = 10$ (upper) and $\mu = 100$ (lower) respectively.[]{data-label="fig:Projection_KP"}](KP_Projection_100.eps "fig:"){width=".7\linewidth"}\ ![(a): Energy approximation as a function of $\mu$. (b): Space approximation as a function of $\mu$.[]{data-label="fig:DensityFunApprox_KP_energy"}](KP_EnergyFunApprox.eps "fig:"){width=".8\linewidth"}\ (a)\ ![(a): Energy approximation as a function of $\mu$. (b): Space approximation as a function of $\mu$.[]{data-label="fig:DensityFunApprox_KP_energy"}](KP_SpaceApprox.eps "fig:"){width=".8\linewidth"}\ (b) Next we consider the case $N = 15$. The first band of $10$ eigenstates of $H$ is occupied and the second band of $H$ is “half-filled”. That is we have only $5$ electrons occupying the $10$ eigenstates of comparable eigenvalue of $H$. Hence, the system does not have a gap, which is indicated by the slow decay of the density matrix shown in Figure \[fig:KP\_N15\](a). Nevertheless, the algorithm with $\mu = 100$ gives a sparse representation of the density matrix, which captures the feature of the density matrix near the diagonal, as shown in Figure \[fig:KP\_N15\](b). To understand better the resulting sparse representation, we diagonal the matrix $P$: $$P = \sum_{i} f_i \varphi_i \varphi_i^{{\mathrm{T}}}.$$ The eigenvalues $f_i$, known as the occupation number in the physics literature, are sorted in the decreasing order. The first $40$ occupation numbers are shown in Figure \[fig:KP\_N15\](c). We have $\sum_i f_i = \operatorname{tr}P = 15$, and we see that $\{f_i\}$ exhibits two groups. The first $10$ occupation numbers are equal to $1$, corresponding to the fact that the lowest $10$ eigenstates of the Hamiltonian operator is occupied. Indeed, if we compare the eigenvalues of the operator $PH$ with the eigenvalues of $H$, as in Figure \[fig:KP\_N15\](d), we see that the first $10$ low-lying states are well represented in $P$. This is further confirmed by the filtered density matrix $M_1$ given by the first $10$ eigenstates of $P$ as $$M_1 = \sum_{i=1}^{10} f_i \varphi_i \varphi_i^{{\mathrm{T}}},$$ plotted in Figure \[fig:KP\_N15\](e). It is clear that it is very close to the exact density matrix corresponding to the first $10$ eigenfunctions of $H$, as plotted in Figure \[fig:DensityFunApprox\_KP\](a). The next group of occupation numbers in Figure \[fig:KP\_N15\](c) gets value close to $0.5$. This indicates that those states are “half-occupied”, matches very well with the physical intuition. This is also confirmed by the state energy shown in Figure \[fig:KP\_N15\](d). Note that due to the fact these states are half filled, the perturbation in the eigenvalue by the localization is much stronger. The corresponding filtered density matrix $$M_2 = \sum_{i=11}^{20} f_i \varphi_i \varphi_i^{{\mathrm{T}}},$$ is shown in Figure \[fig:KP\_N15\](f). For this example, we compare with the results obtained using the variational principle as in [@OzolinsLaiCaflischOsher:13] shown in Figure \[fig:CMs\_KP\_N15\]. As the variational principle is formulated with orbital functions $\Psi$, it does not allow fractional occupations, in contrast with the one in terms of the density matrix. Hence, the occupation number is either $1$ or $0$, which is equivalent to the idempotency condition, as shown in Figure \[fig:CMs\_KP\_N15\](b). As a result, even though the states in the second band have very similar energy, the resulting $\Psi$ are forced to choose five states over the ten, as can be seen from the Ritz value plotted in Figure \[fig:CMs\_KP\_N15\](c). The solution is quite degenerate in this case. Physically, what happens is that the five electrons choose $5$ wells out of the ten to sit in (on top of the state corresponding to the first band already in the well), as shown from the corresponding density matrix in Figure \[fig:CMs\_KP\_N15\](a), or more clearly by the filtered density matrix in Figure \[fig:CMs\_KP\_N15\](d) for the five higher energy states. ![(a): The true density matrix corresponds to the first $15$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e): The filtered density matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f) The filtered density matrix $M_2$ corresponds to the next $10$ eigenstates of $P$.\[fig:KP\_N15\]](KP_ExactP_N15.eps "fig:"){width="1\linewidth"}\ ![(a): The true density matrix corresponds to the first $15$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e): The filtered density matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f) The filtered density matrix $M_2$ corresponds to the next $10$ eigenstates of $P$.\[fig:KP\_N15\]](KP_PMatrix_N15.eps "fig:"){width="1\linewidth"}\ \ ![(a): The true density matrix corresponds to the first $15$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e): The filtered density matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f) The filtered density matrix $M_2$ corresponds to the next $10$ eigenstates of $P$.\[fig:KP\_N15\]](KP_P_EigenValue_N15.eps "fig:"){width=".9\linewidth"}\ ![(a): The true density matrix corresponds to the first $15$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e): The filtered density matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f) The filtered density matrix $M_2$ corresponds to the next $10$ eigenstates of $P$.\[fig:KP\_N15\]](KP_StateEnergy_N15.eps "fig:"){width=".9\linewidth"}\ \ ![(a): The true density matrix corresponds to the first $15$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e): The filtered density matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f) The filtered density matrix $M_2$ corresponds to the next $10$ eigenstates of $P$.\[fig:KP\_N15\]](KP_DensityEigM1_N15.eps "fig:"){width="1\linewidth"}\ ![(a): The true density matrix corresponds to the first $15$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e): The filtered density matrix $M_1$ corresponds to the first $10$ eigenstates of $P$. (f) The filtered density matrix $M_2$ corresponds to the next $10$ eigenstates of $P$.\[fig:KP\_N15\]](KP_DensityEigM2_N15.eps "fig:"){width="1\linewidth"}\ ![Results obtained by the first 15 Compressed modes $\Psi = \{\psi_i\}_{i=1}^{15}$ for $\mu = 100$. (a): The density representation $P$ given by $P = \Psi^T\Psi $. (b): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $\Psi^T H \Psi$ compared with the eigenvalues of $H$. (d) The filtered density matrix $M_2$ corresponds to the $5$ states in the second band. \[fig:CMs\_KP\_N15\]](CMs_KP_PMatrix_N15.eps "fig:"){width="1\linewidth"}\ ![Results obtained by the first 15 Compressed modes $\Psi = \{\psi_i\}_{i=1}^{15}$ for $\mu = 100$. (a): The density representation $P$ given by $P = \Psi^T\Psi $. (b): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $\Psi^T H \Psi$ compared with the eigenvalues of $H$. (d) The filtered density matrix $M_2$ corresponds to the $5$ states in the second band. \[fig:CMs\_KP\_N15\]](CMs_KP_P_EigenValue_N15.eps "fig:"){width=".9\linewidth"}\ \ ![Results obtained by the first 15 Compressed modes $\Psi = \{\psi_i\}_{i=1}^{15}$ for $\mu = 100$. (a): The density representation $P$ given by $P = \Psi^T\Psi $. (b): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $\Psi^T H \Psi$ compared with the eigenvalues of $H$. (d) The filtered density matrix $M_2$ corresponds to the $5$ states in the second band. \[fig:CMs\_KP\_N15\]](CMs_KP_StateEnergy_N15.eps "fig:"){width=".9\linewidth"}\ ![Results obtained by the first 15 Compressed modes $\Psi = \{\psi_i\}_{i=1}^{15}$ for $\mu = 100$. (a): The density representation $P$ given by $P = \Psi^T\Psi $. (b): The occupation number (eigenvalues) of $P$. (d) The first $15$ eigenvalues of $\Psi^T H \Psi$ compared with the eigenvalues of $H$. (d) The filtered density matrix $M_2$ corresponds to the $5$ states in the second band. \[fig:CMs\_KP\_N15\]](CMs_KP_DensityEigM2_N15.eps "fig:"){width="1\linewidth"}\ Finally, the $N = 20$ case corresponds to the physical situation that the first two bands are all occupied. Note that as the band gap between the second band from the rest of the spectrum is smaller than the gap between the first two bands, the density matrix, while still exponentially localized, has a slower off diagonal decay rate. The exact density matrix corresponds to the first $20$ eigenfunctions of $H$ is shown in Figure \[fig:KP\_N20\](a), and the localized representation with $\mu = 100$ is given in Figure \[fig:KP\_N20\](b). The occupation number is plotted in Figure \[fig:KP\_N20\](c), indicates that the first $20$ states are fully occupied, while the rest of the states are empty. This is further confirmed by comparison of the eigenvalues given by $HP$ and $H$, shown in Figure \[fig:KP\_N20\](d). In this case, we see that physically, each well contains two states. Hence, if we look at the electron density, which is diagonal of the density matrix, we see a double peak in each well. Using the projection of delta functions, we see that the sparse representation of the density matrix $P$ automatically locate the two localized orbitals centered at the two peaks, as shown in Figure \[fig:KP\_N20\](e). ![(a): The true density matrix corresponds to the first $20$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $20$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e) Projection of Delta function $\delta(x - x_i)$.[]{data-label="fig:KP_N20"}](KP_ExactP_N20.eps "fig:"){width="1\linewidth"}\ ![(a): The true density matrix corresponds to the first $20$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $20$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e) Projection of Delta function $\delta(x - x_i)$.[]{data-label="fig:KP_N20"}](KP_PMatrix_N20.eps "fig:"){width="1\linewidth"}\ \ ![(a): The true density matrix corresponds to the first $20$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $20$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e) Projection of Delta function $\delta(x - x_i)$.[]{data-label="fig:KP_N20"}](KP_P_EigenValue_N20.eps "fig:"){width=".9\linewidth"}\ ![(a): The true density matrix corresponds to the first $20$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $20$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e) Projection of Delta function $\delta(x - x_i)$.[]{data-label="fig:KP_N20"}](KP_StateEnergy_N20.eps "fig:"){width=".9\linewidth"}\ \ ![(a): The true density matrix corresponds to the first $20$ eigenfunctions of $H$. (b): The sparse representation $P$ of the density matrix for $\mu = 100$. (c): The occupation number (eigenvalues) of $P$. (d) The first $20$ eigenvalues of $PH$ compared with the eigenvalues of $H$. (e) Projection of Delta function $\delta(x - x_i)$.[]{data-label="fig:KP_N20"}](KP_Projection_N20.eps "fig:"){width="1\linewidth"}\ An example of non-unique minimizers ----------------------------------- Let us revisit the Example $2$ in Section \[sec:formulation\] for which the minimizers to the variational problem is non-unique. Theorem \[thm:conv\] guarantees that the algorithm will converge to some minimizer starting from any initial condition. It is easy to check that in this case $$P^{\ast} = Q^{\ast} = R^{\ast} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad b^{\ast} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{pmatrix}, \quad d^{\ast} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & -1 & -1 \\ 0 & -1 & -1 \end{pmatrix}$$ is a fixed point of the algorithm. In Figure \[fig:DecayDist\], we plot the sequence $\Big\{ \lambda \\|b^{k} - b^{\ast}\\|^2 + r \\|d^{k} - d^{\ast}\\|^2 + \lambda \\| Q^{k} - Q^{\ast} \\|^2 + r \\| R^{k} - R^{\ast} \\|^2 \Big\}_k$ for a randomly chosen initial data. We see that the distance does not converge to $0$ as the algorithm converges to another minimizer of the variational problem. Nonetheless, as will be shown in the proof of Theorem \[thm:conv\] in Section \[sec:proof\], the sequence is monotonically non-increasing. ![$\lambda \\|b^{k} - b^{\ast}\\|^2 + r \\|d^{k} - d^{\ast}\\|^2 + \lambda \\| Q^{k} - Q^{\ast} \\|^2 + r \\| R^{k} - R^{\ast} \\|^2$ as a function of $k$ for Algorithm \[alg:CM\_P\].[]{data-label="fig:DecayDist"}](DecayDist.eps){width=".8\linewidth"} Convergence of Algorithm \[alg:CM\_P\] {#sec:proof} ====================================== For ease of notation, we will prove the convergence of the algorithm for the following slightly generalized variational problem. $$\label{eq:three} \begin{aligned} & \min_{P,Q,R} f(P) + g(Q) + h(R) \\ & \text{s.t.} \; P = Q, P = R \end{aligned}$$ where $f$, $g$, and $h$ are proper convex functionals, but not necessarily strictly convex. In particular, we will get if we set $$\begin{aligned} & f(P) = \begin{cases} \operatorname{tr}(H P), & \text{if } \operatorname{tr}P = N; \\ + \infty, & \text{otherwise}, \end{cases} \\ & g(Q) = {\lVertQ\rVert}_1 / \mu \\ & h(R) = \begin{cases} 0, & \text{if } R = R^{{\mathrm{T}}}, \text{and } 0 \preceq R \preceq I; \\ + \infty, & \text{otherwise}. \end{cases}\end{aligned}$$ The corresponding algorithm for is given by \[alg:split2\] Initialize $P^0 = Q^0 = R^0, b^0 = d^0 = 0$ We define an augmented Lagrangian $$\begin{gathered} \label{eq:Lag} {\mathcal{L}}(P,Q,R;b,d) = f(P) + g(Q) + h(R) + \frac{\lambda}{2}\\|P - Q\\|^2 + \lambda\langle P - Q, b\rangle \\ + \frac{r}{2}\\|P - R\\|^2 + r\langle P - R, d\rangle\end{gathered}$$ We call $(P^{\ast}, Q^{\ast}, R^{\ast}; b^{\ast}, d^{\ast})$ a saddle point of the Lagrangian , if $$\label{eq:saddle} {\mathcal{L}}(P^{\ast}, Q^{\ast}, R^{\ast}; b, d) \leq {\mathcal{L}}(P^{\ast}, Q^{\ast}, R^{\ast}; b^{\ast}, d^{\ast}) \leq {\mathcal{L}}(P, Q, R; b^{\ast}, d^{\ast})$$ for any $(P, Q, R; b, d) \in {\mathbb{R}}^{n\times n} \times {\mathbb{R}}^{n\times n} \times {\mathbb{R}}^{n\times n}\times {\mathbb{R}}^{n\times n}\times {\mathbb{R}}^{n\times n}$. $(\Ps, \Qs, \Rs)$ is a solution of the optimization problem if any only if there exist $\bs, \ds \in {\mathbb{R}}^{n\times n}$ such that $(\Ps, \Qs, \Rs;\bs, \ds)$ is a saddle point satisfying Given a saddle point $(\Ps, \Qs, \Rs;\bs, \ds)$ satisfying , it is clear that the first inequality in implies $\Ps = \Qs = \Rs$. Substitute $P = Q = R$ in the second inequality of , we can immediately have $(\Ps, \Qs, \Rs)$ is a minimizer . On the other hand, suppose $(\Ps, \Qs, \Rs)$ is a solution of . The first inequality in holds since $\Ps = \Qs = \Rs$. Moreover, there exist $\bs, \ds$ such that $$-\lambda \bs - r \ds \in \partial f(\Ps), \qquad \lambda \bs \in \partial g(\Qs), \qquad r \ds \in \partial h(\Rs)$$ which suggests, for any $P, Q, R \in {\mathbb{R}}^{n\times n}$ $$\begin{aligned} f(\Ps) &\leq f(P) + \lambda \langle \bs, P - \Ps \rangle + r \langle \ds, P - \Ps \rangle \nonumber \\ g(\Qs) &\leq g(Q) - \lambda \langle \bs, Q - \Qs \rangle \nonumber \\ h(\Rs) &\leq h(R) - r \langle \ds, R - \Rs \rangle \nonumber \end{aligned}$$ The summation of the above three inequalities yield the second inequality in . \[thm:three\] The sequence $\Big\{(P^k, Q^k, R^k)\Big\}_k$ generated by Algorithm \[alg:split2\] from any starting point converges to a minimum of the variational problem . We remind the readers that the minimizers of the variational principle might not be unique. In the non-unique case, the above theorem states that any initial condition will converge to some minimizer, while different initial condition might give different minimizers. Let $(P^{\ast}, Q^{\ast}, R^{\ast})$ be an optimal solution of . We introduce the short hand notations $$\begin{split} & {\widebar{P}}^k = P^k - \Ps, \quad {\widebar{Q}}^k = Q^k - \Qs, \quad \text{and} \quad {\widebar{R}}^k = R^k - \Rs. \\ & {\widebar{b}}^k = b^k - \bs, \quad {\widebar{d}}^k = d^k - \ds. \end{split}$$ From Step $4$ and $5$ in the algorithm, we get $${\widebar{b}}^k = {\widebar{b}}^{k-1} + {\widebar{P}}^{k} - {\widebar{Q}}^{k}, \quad \text{and} \quad {\widebar{d}}^k = {\widebar{d}}^{k-1} + {\widebar{P}}^{k} - {\widebar{R}}^{k},$$ and hence $$\label{eqn:bderror} \begin{split} & \\|{\widebar{b}}^{k-1}\\|^2 - \\|{\widebar{b}}^{k}\\|^2 = -2\langle {\widebar{b}}^{k-1}, {\widebar{P}}^{k} - {\widebar{Q}}^{k} \rangle - \\|{\widebar{P}}^{k} - {\widebar{Q}}^{k}\\|^2 \\ & {\lVert{\widebar{d}}^{k-1}\rVert}^2 - \\|{\widebar{d}}^{k}\\|^2 = - 2\langle {\widebar{d}}^{k-1}, {\widebar{P}}^{k} - {\widebar{R}}^{k} \rangle - \\|{\widebar{P}}^{k} - {\widebar{R}}^{k}\\|^2 \end{split}$$ Note that by optimality $$\begin{aligned} \Ps = \arg\min_{P\in \mathcal{C}_P} \mathcal{L}(P,\Qs,\Rs;\bs,\ds) \\ \Qs = \arg\min_{Q\in \mathcal{C}_Q} \mathcal{L}(\Ps,Q,\Rs;\bs,\ds) \\ \Rs = \arg\min_{R\in \mathcal{C}_R} \mathcal{L}(\Ps,\Qs,R;\bs,\ds) \end{aligned}$$ Hence, for any $P, Q, R\in {\mathbb{R}}^{n\times n}$, we have $$\begin{aligned} & f(P) - f(\Ps) + \lambda\langle \Ps - \Qs + \bs, P - \Ps \rangle + r \langle \Ps - \Rs + \ds, P - \Ps \rangle \geq 0 \label{eqn:fPs}\\ & g(Q) - g(\Qs) + \lambda\langle \Qs - \Ps - \bs, Q - \Qs \rangle \geq 0 \label{eqn:gQs} \\ & h(R) - h(\Rs) + r \langle \Rs - \Ps - \ds, R - \Rs \rangle \geq 0 \label{eqn:hRs}\end{aligned}$$ According to the construction of $\{P^k, Q^k, R^k\}$, for any $P, Q, R \in {\mathbb{R}}^{n\times n}$, we have $$\begin{aligned} & \begin{aligned} f(P) - f(P^{k}) & + \lambda\langle P^{k} - Q^{k-1} + b^{k-1}, P - P^{k} \rangle \\ & + r \langle P^{k} - R^{k} + d^{k-1}, P - P^{k} \rangle \geq 0 \end{aligned} \label{eqn:fPk}\\ & g(Q) - g(Q^{k}) + \lambda\langle Q^{k}- P^{k} - b^{k-1}, Q - Q^{k} \rangle \geq 0 \label{eqn:gQk} \\ & h(R) - h(R^{k}) + r \langle R^{k} - P^{k} - d^{k-1}, R - R^{k} \rangle \geq 0 \label{eqn:hRk}\end{aligned}$$ Let $P = P^{k}$ in and $P = \Ps$ in , their summation yields $$\label{eqn:fPsk} \lambda\langle -{\widebar{P}}^{k} + {\widebar{Q}}^{k-1} - {\widebar{b}}^{k-1}, {\widebar{P}}^{k} \rangle + r \langle -{\widebar{P}}^{k} + {\widebar{R}}^{k-1} - {\widebar{d}}^{k-1}, {\widebar{P}}^{k} \rangle \geq 0$$ Similarly, let $Q = Q^{k}$ in and $Q = \Qs$ in , and let $R = R^{k}$ in and $R = \Rs$ in , we obtain $$\begin{aligned} & \lambda\langle -{\widebar{Q}}^{k} + {\widebar{P}}^{k} + {\widebar{b}}^{k-1}, {\widebar{Q}}^{k} \rangle \geq 0 \label{eqn:gQsk} \\ & r \langle -{\widebar{R}}^{k} + {\widebar{P}}^{k} + {\widebar{d}}^{k-1}, {\widebar{R}}^{k} \rangle \geq 0 \label{eqn:hRsk}\end{aligned}$$ The summation of , , and yields $$\begin{gathered} \lambda\langle - {\widebar{b}}^{k-1}, {\widebar{P}}^{k} \rangle + \lambda\langle {\widebar{Q}}^{k-1} -{\widebar{P}}^{k} , {\widebar{P}}^{k} \rangle + r \langle - {\widebar{d}}^{k-1}, {\widebar{P}}^{k} \rangle + r \langle {\widebar{R}}^{k-1} -{\widebar{P}}^{k}, {\widebar{P}}^{k} \rangle \\ + \lambda \langle {\widebar{b}}^{k-1}, {\widebar{Q}}^{k} \rangle + \lambda\langle {\widebar{P}}^{k} - {\widebar{Q}}^{k}, {\widebar{Q}}^{k} \rangle + r \langle {\widebar{d}}^{k-1}, {\widebar{R}}^{k} \rangle + r \langle {\widebar{P}}^{k} - {\widebar{R}}^{k}, {\widebar{R}}^{k} \rangle \geq 0.\end{gathered}$$ This gives us, after organizing terms $$\begin{gathered} -\lambda\langle {\widebar{b}}^{k-1}, {\widebar{P}}^{k} -{\widebar{Q}}^{k} \rangle - \lambda \\|{\widebar{Q}}^{k} - {\widebar{P}}^{k}\\|^2 - \lambda\langle {\widebar{P}}^{k} , {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1} \rangle \\ -r\langle {\widebar{d}}^{k-1}, {\widebar{P}}^{k} -{\widebar{R}}^{k} \rangle - r \\|{\widebar{R}}^{k} - {\widebar{P}}^{k}\\|^2 - r \langle {\widebar{P}}^{k}, {\widebar{R}}^{k} -{\widebar{R}}^{k-1} \rangle \geq 0\end{gathered}$$ Combining the above inequality with , we have $$\label{eqn:Monotone1} \begin{aligned} \bigl(\lambda \\|{\widebar{b}}^{k-1}\\|^2 + r \\|{\widebar{d}}^{k-1}\\|^2 \bigr) & - \bigl(\lambda \\|{\widebar{b}}^{k}\\|^2 + r \\|{\widebar{d}}^{k}\\|^2 \bigr) \\ & = \lambda \bigl(-2\langle {\widebar{b}}^{k-1}, {\widebar{P}}^{k} - {\widebar{Q}}^{k} \rangle - \\|{\widebar{P}}^{k} - {\widebar{Q}}^{k}\\|^2\bigr) \\ & \qquad + r \bigl(-2\langle {\widebar{d}}^{k-1}, {\widebar{P}}^{k} - {\widebar{R}}^{k} \rangle - \\|{\widebar{P}}^{k} - {\widebar{R}}^{k}\\|^2 \bigr) \\ & \geq \lambda \\|{\widebar{Q}}^{k} - {\widebar{P}}^{k}\\|^2 + 2 \lambda\langle {\widebar{P}}^{k}, {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1}\rangle \\ & \qquad + r \\|{\widebar{R}}^{k} - {\widebar{P}}^{k}\\|^2 + 2 r \langle {\widebar{P}}^{k}, {\widebar{R}}^{k} -{\widebar{R}}^{k-1} \rangle \end{aligned}$$ Now, we calculate $ \langle {\widebar{P}}^{k} , {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1}\rangle$. It is clear that $$\begin{gathered} \label{eqn:ErrorPQ1} \langle {\widebar{P}}^{k} , {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1}\rangle = \langle {\widebar{P}}^{k} - {\widebar{P}}^{k-1} , {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1}\rangle + \langle {\widebar{P}}^{k-1} - {\widebar{Q}}^{k-1} , {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1}\rangle \\ + \langle {\widebar{Q}}^{k-1} , {\widebar{Q}}^{k} -{\widebar{Q}}^{k-1}\rangle\end{gathered}$$ Note that $\displaystyle Q^{k-1} = \arg\min_{Q} g(Q) + \frac{\lambda}{2} \\|Q - P^{k-1} - b^{k-2} \\|^2 $. Thus, for any $Q\in {\mathbb{R}}^{n\times n}$, we have $$\begin{aligned} g(Q) - g(Q^{k-1}) + \lambda\langle Q^{k-1} - P^{k-1} - b^{k-2}, Q - Q^{k-1} \rangle \geq 0 \end{aligned}$$ In particular, let $Q = Q^{k}$, we have $$\begin{aligned} \label{eqn:gQkk1} g(Q^{k}) - g(Q^{k-1}) + \lambda\langle Q^{k-1} - P^{k-1} - b^{k-2}, Q^{k} - Q^{k-1} \rangle \geq 0 \end{aligned}$$ On the other hand, set $Q = Q^{k-1}$ in , we get $$\begin{aligned} \label{eqn:gQkk2} g(Q^{k-1}) - g(Q^{k}) + \lambda\langle Q^{k}- P^{k} - b^{k-1}, Q^{k-1} - Q^{k} \rangle \geq 0 \end{aligned}$$ The summation of and yields $$\langle b^{k-1} - b^{k-2}, Q^{k} - Q^{k-1} \rangle + \langle Q^{k-1} - Q^{k} + P^{k} - P^{k-1} , Q^{k} - Q^{k-1} \rangle \geq 0$$ Note that $P^{k} - P^{k-1} = {\widebar{P}}^{k} - {\widebar{P}}^{k-1}, Q^{k} - Q^{k-1} = {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1}, b^{k-1} - b^{k-2} = {\widebar{P}}^{k-1} - {\widebar{Q}}^{k-1}$, thus we have $$\langle {\widebar{P}}^{k-1} - {\widebar{Q}}^{k-1}, {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \rangle + \langle {\widebar{P}}^{k} - {\widebar{P}}^{k-1} , {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \rangle \geq \\|{\widebar{Q}}^{k} - {\widebar{Q}}^{k-1}\\|^2 \label{eqn:ErrorPQ2}$$ Combine with , we have $$\langle {\widebar{P}}^{k}, {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \rangle \geq \\| {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \\|^2 + \langle {\widebar{Q}}^{k-1}, {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \rangle \label{eqn:ErrorPQ3}$$ Similarly, we have $$\langle {\widebar{P}}^{k}, {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \rangle \geq \\| {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \\|^2 + \langle {\widebar{R}}^{k-1}, {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \rangle \label{eqn:ErrorPR3}$$ Substitute and into , we have $$\label{eqn:Monotone2} \begin{aligned} \bigl(\lambda \\|{\widebar{b}}^{k-1}\\|^2 + r \\|{\widebar{d}}^{k-1}\\|^2 ) & - (\lambda \\|{\widebar{b}}^{k}\\|^2 + r \\|{\widebar{d}}^{k}\\|^2 ) \\ & \geq \lambda \\|{\widebar{Q}}^{k} - {\widebar{P}}^{k}\\|^2 + 2 \lambda\langle {\widebar{P}}^{k}, {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1}\rangle \\ & \qquad + r \\|{\widebar{R}}^{k} - {\widebar{P}}^{k}\\|^2 + 2 r \langle {\widebar{P}}^{k} , {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \rangle \\ & \geq \lambda \\|{\widebar{Q}}^{k} - {\widebar{P}}^{k}\\|^2 + 2 \lambda \bigl(\\| {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \\|^2 + \langle {\widebar{Q}}^{k-1}, {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \rangle \bigr) \\ & \qquad + r \\|{\widebar{R}}^{k} - {\widebar{P}}^{k}\\|^2 + 2 r \bigl(\\| {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \\|^2 + \langle {\widebar{R}}^{k-1}, {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \rangle\bigr) \\ &= \lambda \\|{\widebar{Q}}^{k} - {\widebar{P}}^{k}\\|^2 + \lambda \bigl(\\| {\widebar{Q}}^{k} \\|^2 - \\| {\widebar{Q}}^{k-1} \\|^2 + \\| {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \\|^2 \bigr) \\ & \qquad + r \\|{\widebar{R}}^{k} - {\widebar{P}}^{k}\\|^2 + r \bigl(\\| {\widebar{R}}^{k} \\|^2 - \\| {\widebar{R}}^{k-1} \\|^2 + \\| {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \\|^2 \bigr) \end{aligned}$$ which yields $$\begin{gathered} \label{eqn:Monotone3} \bigl(\lambda \\|{\widebar{b}}^{k-1}\\|^2 + r \\|{\widebar{d}}^{k-1}\\|^2 + \lambda \\| {\widebar{Q}}^{k-1} \\|^2 + r \\| {\widebar{R}}^{k-1} \\|^2 \bigr) \\ - \bigl(\lambda \\|{\widebar{b}}^{k}\\|^2 + r \\|{\widebar{d}}^{k}\\|^2 + \lambda \\| {\widebar{Q}}^{k} \\|^2 + r \\| {\widebar{R}}^{k} \\|^2 \bigr) \\ \geq \lambda \\|{\widebar{Q}}^{k} - {\widebar{P}}^{k}\\|^2 + \lambda \\| {\widebar{Q}}^{k} - {\widebar{Q}}^{k-1} \\|^2 + r \\|{\widebar{R}}^{k} - {\widebar{P}}^{k}\\|^2 + r \\| {\widebar{R}}^{k} - {\widebar{R}}^{k-1} \\|^2\end{gathered}$$ This concludes that the sequence $\Big\{ \lambda \\|{\widebar{b}}^{k}\\|^2 + r \\|{\widebar{d}}^{k}\\|^2 + \lambda \\| {\widebar{Q}}^{k} \\|^2 + r \\| {\widebar{R}}^{k} \\|^2 \Big\}_k$ is non-increasing and hence convergent. This further implies, 1. $\{P^k\}_k, \{Q^k\}_k, \{R^k\}_k, \{b^k\}_k, \{d^k\}_k$ are all bounded sequences, and hence the sequences has limit points. 2. $\lim_{k\rightarrow\infty} \\| Q^k - P^k\\| = 0$ and $\lim_{k\rightarrow\infty} \\| R^k - P^k\\| = 0$. Therefore, the sequences have limit points. Let us denote $({\widetilde{P}}, {\widetilde{Q}}, {\widetilde{R}}; {\widetilde{b}}, {\widetilde{d}})$ as a limit point, that is, a subsequence converges $$\lim_{j\to \infty} (P^{k_j}, Q^{k_j}, R^{k_j}; b^{k_j}, d^{k_j}) = ({\widetilde{P}}, {\widetilde{Q}}, {\widetilde{R}}; {\widetilde{b}}, {\widetilde{d}}).$$ We now prove that $({\widetilde{P}}, {\widetilde{Q}}, {\widetilde{R}})$ is a minimum of the variational problem , i.e. $$\label{eq:limitPQR} f({\widetilde{P}}) + g({\widetilde{Q}}) + h({\widetilde{R}}) = \lim_{j\rightarrow \infty} f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) = f(\Ps) + g(\Qs) + h(\Rs)$$ First note that since $(\Ps, \Qs, \Rs; \bs, \ds)$ is a saddle point, we have $$\begin{gathered} f(\Ps) + g(\Qs) + h(\Rs) \leq f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) + \frac{\lambda}{2}\\|P^{k_j} - Q^{k_j}\\|^2 \\ + \lambda\langle P^{k_j} - Q^{k_j}, \bs \rangle + \frac{r}{2}\\|P^{k_j} - R^{k_j}\\|^2 + r\langle P^{k_j} - R^{k_j}, \ds \rangle\end{gathered}$$ Taking the limit $j \to \infty$, we get $$f(\Ps) + g(\Qs) + h(\Rs) \leq f({\widetilde{P}}) + g({\widetilde{Q}}) + h({\widetilde{R}}).$$ On the other hand, taking $P = \Ps$, $Q = \Qs$, and $R = \Rs$ in –, we get $$\begin{aligned} f(\Ps) & + g(\Qs) + h(\Rs) \\ & \geq f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) - \lambda\langle P^{k_j} - Q^{k_j-1} + b^{k_j-1}, \Ps - P^{k_j} \rangle \\ & \qquad - r \langle P^{k_j} - R^{k_j} + d^{k_j-1}, \Ps - P^{k_j} \rangle - \lambda\langle Q^{k_j}- P^{k_j} - b^{k_j-1}, \Qs - Q^{k_j} \rangle \\ & \qquad - r \langle R^{k_j} - P^{k_j} - d^{k_j-1}, \Rs - R^{k_j} \rangle \\ &= f(P^{k_j}) + g(Q^{k_j}) + h(R^{k_j}) \\ & \qquad - \lambda\langle b^{k_j-1}, Q^{k_j} - P^{k_j} \rangle - \lambda\langle P^{k_j} - Q^{k_j-1} , \Ps - P^{k_j} \rangle - \lambda\langle Q^{k_j}- P^{k_j} , \Qs - Q^{k_j} \rangle \\ & \qquad - r \langle d^{k_j-1}, R^{k_j} - P^{k_j} \rangle - r \langle P^{k_j} - R^{k_j} , \Ps - P^{k_j} \rangle - r \langle R^{k_j} - P^{k_j} , \Rs - R^{k_j} \rangle\end{aligned}$$ From , we have $\{P^{k_j}\}, \{Q^{k_j}\}, \{R^{k_j}\}, \{b^{k_j}\}, \{d^{k_j}\}$ are all bounded sequences, and furthermore, $$\begin{aligned} \lim_{j\rightarrow\infty} \\| Q^{k_j }- P^{k_j}\\| = 0, \qquad \lim_{j\rightarrow\infty} \\| Q^{k_j }- Q^{k_j - 1}\\| = 0.\nonumber \\ \lim_{j\rightarrow\infty} \\| R^{k_j} - P^{k_j}\\| = 0, \qquad \lim_{j\rightarrow\infty} \\| R^{k_j }- R^{k_j - 1}\\| = 0. \nonumber\end{aligned}$$ Taking the limit $j \to \infty$, we then get $$f(\Ps) + g(\Qs) + h(\Rs) \geq f({\widetilde{P}}) + g({\widetilde{Q}}) + h({\widetilde{R}}).$$ Hence, the limit point is a minimizer of the variational principle. Finally, repeating the derivation of by replacing $(\Ps, \Qs, \Rs)$ by $({\widetilde{P}}, {\widetilde{Q}}, {\widetilde{R}})$, we get convergence of the whole sequence due to the monotonicity. [^1]: The research of J.L. was supported in part by the Alfred P. Sloan Foundation and the National Science Foundation under award DMS-1312659. The research of S.O. was supported by the Office of Naval Research Grant N00014-11-1-719. [^2]: With obvious changes, our results generalize to the Hermitian case
--- abstract: 'In this article, we derive bounds for values of the global geometry of locally tessellating planar graphs, namely, the Cheeger constant and exponential growth, in terms of combinatorial curvatures. We also discuss spectral implications for the Laplacians.' author: - \| Matthias Keller [^1]\ TU Chemnitz\ Fakultät für Mathematik\ D-09107 Chemnitz, Germany - \| Norbert Peyerimhoff [^2]\ Department of Math. Sciences\ University of Durham\ Durham DH1 2LE, UK title: Geometric and spectral properties of locally tessellating planar graphs --- Introduction ============ A locally tessellating planar graph ${{\cal G}}$ is a tiling of the plane with all faces to be polygons with finitely or infinitely many boundary edges (see Subsection \[loctess\] for precise definitions). The edges of ${{\cal G}}$ are continuous rectifiable curves without self-intersections. Faces with infinitely many boundary edges are called infinigons and occur, e.g., in the case of planar trees. The sets of vertices, edges and faces of ${{\cal G}}$ are denoted by ${{\cal V}}, {{\cal E}}$ and ${{\cal F}}$. $d(v,w)$ denotes the combinatorial distance between two vertices $v,w \in {{\cal V}}$, where each edge is assumed to have combinatorial length one. Useful [local]{} concepts of the graph ${{\cal G}}$ are combinatorial curvature notions. The finest curvature notion is defined on the corners of ${{\cal G}}$. A corner is a pair $(v,f) \in {{\cal V}}\times {{\cal F}}$, where $v$ is a vertex of the face $f$. The set of all corners is denoted by ${{\cal C}}$. The [corner curvature]{} $\kappa_C$ is then defined as $$\kappa_C(v,f) = \frac{1}{\|v\|} + \frac{1}{\|f\|} - \frac{1}{2},$$ where $\|v\|$ and $\|f\|$ denote the degree of the vertex $v$ and the face $f$. If $f$ is an infinigon, we set $\|f\| = \infty$ and $1/\|f\| = 0$. The [curvature at a vertex]{} $v \in {{\cal V}}$ is given by the sum $$\kappa(v) = \sum_{(v,f) \in {{\cal C}}}\kappa_C(v,f) = 1 - \frac{\|v\|}{2} + \sum_{f: v \in f} \frac{1}{\|f\|}.$$ For a finite set $W \subset {{\cal V}}$ we define ${{\kappa}}(W) = \sum_{v \in W} {{\kappa}}(v)$. These combinatorial curvature definitions arise naturally from considerations of the Euler characteristic and tessellations of closed surfaces, and they allow to prove a combinatorial Gau[ß]{}-Bonnet formula (see [@BP1 Thm 1.4]). Similar combinatorial curvature notions have been introduced by many other authors, e.g., [@Gro; @Hi; @St; @Woe]. The aim of this paper is to establish connections between local curvature conditions and characteristic values of the [global geometry]{} of the graph ${{\cal G}}$, in particular the exponential growth and Cheeger constants. For a finite subset $W \subset {{\cal V}}$, let ${{\rm vol}}(W) = \sum_{v \in W} \|v\|$. The exponential growth is defined as follows (note that the value $\mu({{\cal G}})$ does not depend on the choice of center $v \in {{\cal V}}$): The [exponential growth]{} $\mu({{\cal G}})$ is given by $$\mu({{\cal G}}) = \limsup_{n \to \infty} \frac{\log {{\rm vol}}(B_n(v))}{n},$$ where $B_n(v) = \{ w \in {{\cal V}}\mid d(v,w) \le n \}$ denotes the (combinatorial) ball of radius $n$ about $v$. We also consider the following two types of Cheeger constants. \[d:cheeg\] Let $${{\alpha}}({{\cal G}})=\inf_{\scriptsize\begin{array}{c} W\subseteq {{\cal V}}, \\ \|W\|<\infty \end{array}} \frac{\|{{\partial}}_E W\|}{\|W\|}\quad\mathrm{and}\quad {\widetilde{{{\alpha}}}}({{\cal G}})=\inf_{\scriptsize\begin{array}{c} W\subseteq {{\cal V}}, \\ \|W\|<\infty \end{array}} \frac{\|{{\partial}}_E W\|}{{{\rm vol}}(W)}$$ where ${{\partial}}_E W$ is the set of all edges $e \in {{\cal E}}$ connecting a vertex in $W$ with a vertex in ${{\cal V}}\backslash W$. ${{\alpha}}({{\cal G}})$ is called the physical Cheeger constant and ${\widetilde{{{\alpha}}}}({{\cal G}})$ the combinatorial Cheeger constant of the graph ${{\cal G}}$. The attributes [physical]{} and [combinatorial]{} in the previous definition are motivated by the fact that these Cheeger constants are closely linked to two types of Laplacians: The [physical Laplacian]{} is used frequently in the community of Mathematical Physicists and is defined as follows: $$\label{laplp} ({\Delta}{{\varphi}})(v)=\|v\|{{\varphi}}(v) - \sum_{w\sim v}{{\varphi}}(w).$$ Note that ${\Delta}$ is an unbounded operator if there is no bound on the vertex degree of ${{\cal G}}$. The [combinatorial Laplacian]{} ${\widetilde{\Delta}}$ is a bounded operator and appears in the context of spectral geometry (see, e.g., [@DKa; @DKe; @Woe2]): $$\label{laplc} ({\widetilde{\Delta}}{{\varphi}})(v)={{\varphi}}(v) - \frac{1}{\|v\|}\sum_{u\sim v}{{\varphi}}(u).$$ Both operators are defined in and are self-adjoint with respect to different $l^2$-spaces (see Subsection \[lap\]). In the case of fixed vertex degree, both operators are multiples of each other. Our main geometric results are given in Subsection \[cheegexp\], where we - provide lower bounds for both Cheeger constants in terms of combinatorial curvatures (see Theorem \[t:cheeg\] below), - provide upper bounds for the exponential growth in terms of an upper vertex bound (see Theorem \[t:expgr\] below). Even though Theorem \[t:expgr\](b) is formulated in terms of bounds on vertex and face degrees, it can also be considered as an estimate in terms of combinatorial curvature, as is explained in the remark following the theorem. In fact, the proof is based on the corresponding curvature version. Now we discuss connections to the spectrum. The Cheeger constant and the exponential growth were first introduced in the context of Riemannian manifolds and were useful invariants to estimate the bottom of the (essential) spectrum of the Laplacian (see [@Che] and [@Br]). An analogous inequality between the Cheeger constant and the bottom of the spectrum in the discrete case of graphs was first proved by [@Do] and [@Al]. This inequality is also useful in the study of expander graphs. [@Al] noted also the connection between this inequality and the Max Flow-Min Cut Theorem (see also [@Chu] and [@Gri]). For other connections between isoperimetric inequalities and lower bounds of eigenvalues in both continuous and discrete settings see, e.g., [@CGY]. The best results about the relations between the combinatorial Cheeger constant, the exponential growth, and the bottom ${\widetilde{ {{\lambda}}}}_0({{\cal G}})$ and ${\widetilde{ {{\lambda}}}}_0^{ess}({{\cal G}})$ of the (essential) spectrum of the [combinatorial Laplacians]{} ${\widetilde{\Delta}}$ are due to K. Fujiwara (see [@Fu1] and [@Fu2]): $$\label{fuji} 1-\sqrt{1-\widetilde\alpha^2({{\cal G}})} \le {\widetilde{ \lambda}}_0({{\cal G}}) \le {\widetilde{ \lambda}}_0^{ess}({{\cal G}}) \le 1 - \frac{2 e^{\mu({{\cal G}})/2}}{1+e^{\mu({{\cal G}})}}.$$ These estimates are sharp in the case of regular trees. Using these estimates and Theorems \[t:cheeg\] and \[t:expgr\], we obtain - lower and upper estimates on the bottom of the (essential) spectrum of the combinatorial Laplacian in terms of combinatorial curvature (see Corollaries \[c:McKean\] and \[c:essenspecest\]). Since there are estimates to compare the bottom of the (essential) spectrum of the combinatorial Laplacian with the physical Laplacian (see for instance [@Ke]) these results can be also formulated for the physical Laplacian. A lower estimate for the bottom of the essential spectrum of the combinatorial Laplacian via the combinatorial Cheeger constant at infinity can be found in [@Fu2 Cor. 3]. This yields a discrete analogue for the combinatorial Laplacian of the result in [@DL] about the [emptiness of the essential spectrum]{} for complete simply connected manifolds with curvature converging to minus infinity. Corresponding results about the emptiness of the essential spectrum for the physical Laplacian can be found in [@Ke; @Woj]. Finally, let us discuss two other interesting types of eigenfunctions, namely, [strictly positive eigenfunctions]{} and [finitely supported eigenfunctions]{}, and illustrate all concepts in two examples. For the discrete case of a graph, it was shown in [@DKa Prop. 1.5] that the equation ${\widetilde{\Delta}}f = \lambda f$ has a [positive solution]{} if and only if $\lambda \le {\widetilde{ \lambda}}_0({{\cal G}})$. This characterisation of the bottom of the spectrum was well known before in the context of Riemannian manifolds (see, e.g., [@Sull] and the references therein). In the reverse direction, this characterisation might be used in concrete cases to determine the bottom of the spectrum of an infinite graph. On the other hand, [finitely supported solutions]{} of the equation ${\widetilde{\Delta}}f = \lambda f$ are obviously $l^2$-eigenfunctions and, therefore, they can only exist for eigenvalues $\lambda \ge {\widetilde{ \lambda}}_0({{\cal G}})$. Existence of finitely supported eigenfunctions in Penrose tilings was first observed in [@KS]. Their existence is a purely discrete phenomenon, since in the case of a non-compact, connected Riemannian manifold the eigenvalue equation $\Delta f = \lambda f$ cannot have compactly supported eigenfunctions (a fact which is known as the [unique continuation principle]{}; see [@Ar]). These finitely supported eigenfunctions coincide with the discontinuities of the integrated density of states (or spectral density function). See, e.g., the articles [@KLS; @LV] and the references therein for more details about this connection. \(a) We consider the periodic tessellation ${{\cal G}}= ({{\cal V}},{{\cal E}},{{\cal F}})$ in Figure \[example1\]. We assume that all edges are straight Euclidean segments of length one. ![Plane tessellation with regular triangles and hexagons[]{data-label="example1"}](example1.eps){height="6cm"} We first show that $\mu({{\cal G}}) =0$: Choose a fixed radius $0<r<1/2$. Then all Euclidean balls of radius $r$ centered at all vertices in ${{\cal V}}$ are pairwise disjoint. On the other hand, the vertices in the combinatorial ball $B_n(v)$ are contained in the Euclidean ball of radius $n$, centered at $v$. Both facts together imply that combinatorial balls grow only polynomially and the exponential growth is zero. As a consequence, this graph cannot contain a binary tree as a subgraph. Moreover, using , we conclude that $${\widetilde{ \lambda}}_0({{\cal G}}) = {\widetilde{ \lambda}}_0^{ess}({{\cal G}}) =0\quad \mathrm{and} \quad {\widetilde{{{\alpha}}}}({{\cal G}}) = {{\alpha}}({{\cal G}}) = 0.$$ Finally, ${{\cal G}}$ does admit finitely supported eigenfunctions, namely, choose $p \in {{\mathbb R}}^2$ to be the center of a hexagon and define $f(p + e^{2\pi i/6}) = (-1)^i$ (i.e., choose alternating values $1,-1,1,-1,1,-1$ clockwise around the vertices of the hexagon) and $f(v) = 0$ for all other vertices. Then we have ${\widetilde{\Delta}}f =\frac{3}{2} f$. \(b) Let ${{\cal T}}_p$ denote the $p$-regular tree. In this case, spectrum and essential spectrum of the combinatorial Laplacian coincide and are given by the interval (see, e.g., [@Sun App. 3]) $$\left[1-\frac{2\sqrt{p-1}}{p}, 1+\frac{2\sqrt{p-1}}{p}\right].$$ Consequently, ${\widetilde{\Delta}}f = \lambda f$ admits a positive solution if and only if $\lambda \le 1-2\sqrt{p-1}/p$. Moreover, we have ${\widetilde{{{\alpha}}}}({{\cal T}}_p) = \frac{p-2}{p}$, ${{\alpha}}({{\cal T}}_p) = p-2$ and $\mu({{\cal T}}_p) = \log (p-1)$. Note that a regular tree doesn’t admit $l^2$-eigenfunctions. For otherwise, we could choose a vertex $v$ at which our eigenfunction doesn’t vanish and take its radialisation with respect to this vertex. This radialisation would be again a non-vanishing $l^2$-eigenfunction with the same eigenvalue and, since its values would only depend on the distance to $v$, there would be an easy recursion formula for its values. The precise form of the recursion formula would then contradict to the requirement that the function lies in $l^2$. Acknowledgements. Matthias Keller would like to thank Daniel Lenz who encouraged him to study the connection between curvature and spectral theory. Matthias Keller was supported during this work by the German Business Foundation (sdw). Basic notions and main results ============================== In the first two subsections, we provide the notions which haven’t yet been introduced in full detail in the Introduction. In Subsections \[cheegexp\] and \[specappl\], we state our main results. Locally tessellating planar graphs {#loctess} ---------------------------------- Let ${{\cal G}}=({{\cal V}},{{\cal E}})$ be a planar graph (with ${{\cal V}}$ and ${{\cal E}}$ the set of vertices and edges) embedded in ${{\mathbb R}}^2$. The faces $f$ of ${{\cal G}}$ are the closures of the connected components in ${{\mathbb R}}^2\setminus \bigcup_{e\in E} e$. The set of faces is denoted by ${{\cal F}}$. We further assume that ${{\cal G}}$ has no loops, no multiple edges and no vertices of degree one (terminal vertices). We write $e = vw$, if the edge $e$ connects the vertices $v,w$. Moreover, we assume that every vertex has finite degree and that every bounded open set in ${{\mathbb R}}^2$ meets only finitely many faces of ${{\cal G}}$. We call a planar graph with these properties [simple]{}. The [boundary of a face $f$]{} is the subgraph $\partial f = ({{\cal V}}\cap f,{{\cal E}}\cap f)$. We call a sequence of edges $e_1,\dots,e_n$ a [walk of length $n$]{} if there is a corresponding sequence of vertices $v_1,\dots,v_{n+1}$ such that $e_i = v_iv_{i+1}$. A walk is called a [path]{} if there is no repetition in the corresponding sequence of vertices $v_1,\dots,v_n$. A simple planar graph ${{\cal G}}$ is called a [locally tessellating planar graph]{} if the following additional conditions are satisfied: - Any edge is contained in precisely two different faces. - Any two faces are either disjoint or have precisely a vertex or a path of edges in common. In the case that the length of the path is greater then one, then both faces are unbounded. - Any face is homeomorphic to the closure of an open disc ${{\mathbb D}}\subset {{\mathbb R}}^2$, to ${{\mathbb R}}^2\setminus {{\mathbb D}}$ or to the upper half plane ${{\mathbb R}}\times{{\mathbb R}}_+\subset {{\mathbb R}}^2$ and its boundary is a path. Note that these properties force the graph ${{\cal G}}$ to be connected. Examples are tessellations ${{\mathbb R}}^2$ introduced in [@BP1; @BP2], trees in ${{\mathbb R}}^2$, and particular finite tessellations on the sphere mapped to ${{\mathbb R}}^2$ via stereographic projection. When we consider the vertex degree as a function on ${{\cal V}}$ we write $\deg(v)=\|v\|$ for $v\in {{\cal V}}$. Moreover we define the degree $\|f\|$ of a face $f\in F$ to be the length of the shortest closed walk in the subgraph $\partial f$ meeting all its vertices. If there is no such finite walk we set $\|f\|=\infty$. $v \sim w$ means that $d(v,w)=1$, i.e., $v$ and $w$ are neighbors. A (finite or infinite) path with associated vertex sequence $\dots v_i v_{i+1} v_{i+2} \dots$ is called a [geodesic]{}, if we have $d(v_i,v_j) = \|i-j\|$ for all pairs of vertices in the path. Laplacians {#lap} ---------- Let ${{\cal G}}= ({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph. The operators ${\Delta}$ and ${\widetilde{\Delta}}$ were already introduced in and . They are symmetric operators and initially defined on the space $$c_c({{\cal V}}) := \{{{\varphi}}: {{\cal V}}{{\rightarrow}}{{\mathbb R}}\mid\, \|{{\mathrm{supp}\;}}{{\varphi}}\| <\infty \}$$ of functions with finite support. However, they have unique self-adjoint extensions on different $l^2$-spaces: Let $g: {{\cal V}}\to (0,\infty)$ be a weight function on the vertices of the graph ${{\cal G}}$ and $$l^2({{\cal V}},g) := \{{{\varphi}}:{{\cal V}}{{\rightarrow}}{{\mathbb R}}\mid {\langle {{\varphi}},{{\varphi}}\rangle}_g:=\sum_{v\in V} g(v)\|{{\varphi}}(v)\|^2<\infty\}.$$ For $g=1$ we simply write $\l^2({{\cal V}})$. Then the combinatorial Laplacian can be extended to a bounded self-adjoint operator on all of $l^2({{\cal V}},\deg)$. The physical Laplacian has also a unique self-adjoint extension in the space $l^2({{\cal V}})$ (see [@We] or [@Woj]). Note, however, that the adjacency operator need not be essentially self adjoint (see [@MW Section 3] and the references therein). We denote the self-adjoint extensions of both Laplacians, again, by ${\widetilde{\Delta}}$ and ${\Delta}$. Furthermore, we define the restriction of the combinatorial Laplacian on the complement of a finite set $K$ of vertices. Let $P_K:l^2({{\cal V}},\deg){{\rightarrow}}l^2({{\cal V}}\setminus K,\deg)$ be the canonical projection and $i_K:l^2({{\cal V}}\setminus K,\deg){{\rightarrow}}l^2({{\cal V}},\deg)$ be its dual operator, which is the continuation by $0$ on $K$. We write ${\widetilde{\Delta}}_K=P_K {\widetilde{\Delta}}i_K$. Of particular importance is the bottom of the spectrum ${\widetilde{ \lambda}}_0({{\cal G}})$ and of the essential spectrum ${\widetilde{ \lambda}}_0^{ess}({{\cal G}})$. ${\widetilde{ \lambda}}_0({{\cal G}})$ can be characterised as the infimum of the Rayleight-Ritz quotient over all non-zero functions $f \in l^2({{\cal V}},\deg)$, i.e., $${\widetilde{ \lambda}}_0({{\cal G}}) = \inf \left\{ \frac{\langle {\widetilde{\Delta}}f,f\rangle_{\deg}}{\langle f,f\rangle_{\deg}}: f \neq 0, f \in l^2({{\cal V}},\deg) \right\}.$$ Similarly, ${\widetilde{ \lambda}}_0^{ess}({{\cal G}})$ can be obtained via $$\label{specess} {\widetilde{ \lambda}}_0^{ess}({{\cal G}}) = \lim_{n \to \infty} \inf \left\{ \frac{\langle {\widetilde{\Delta}}_{B_n} f,f\rangle_{\deg}}{\langle f,f\rangle_{\deg}}: f \neq 0, f \in l^2({{\cal V}}\backslash B_n,\deg) \right\},$$ where $B_n$ are balls of radius $n$ around any fixed vertex $v \in {{\cal V}}$. A proof of can be found in [@Ke]. Obviously, we have ${\widetilde{ \lambda}}_0({{\cal G}}) \le {\widetilde{ \lambda}}_0^{ess}({{\cal G}})$. Equality holds in the following case: \[specessspeceq\] Assume that there is a subgroup $\Gamma$ of the automorphism group of ${{\cal G}}$ with $\sup_{\gamma \in \Gamma} d(v,\gamma v) = \infty$ for some vertex $v \in {{\cal V}}$. Then we have $${\widetilde{ \lambda}}_0({{\cal G}}) = {\widetilde{ \lambda}}_0^{ess}({{\cal G}}).$$ For the bottom of the spectrum not to lie in the essential spectrum would mean that it is an isolated eigenvalue of finite multiplicity. But this cannot be the case (see Fact 1 in [@Sun p. 259]). Analogous statements hold for the bottom of the (essential) spectrum of the physical Laplacian. Cheeger constant and exponential growth estimates {#cheegexp} ------------------------------------------------- The physical and combinatorial Cheeger constants were introduced in Definition \[d:cheeg\]. It is easy to see that they are linked to the physical and combinatorial Laplacians via the equations: $${{\alpha}}({{\cal G}})=\inf_{\scriptsize\begin{array}{c} W\subseteq {{\cal V}}, \\ \|W\|<\infty \end{array}} \frac{{\langle {\Delta}\chi_W,\chi_W\rangle}}{{\langle \chi_W,\chi_W\rangle}}\quad\mathrm{and}\quad {\widetilde{{{\alpha}}}}({{\cal G}})=\inf_{\scriptsize\begin{array}{c} W\subseteq {{\cal V}}, \\ \|W\|<\infty \end{array}} \frac{{{\langle {\widetilde{\Delta}}\chi_W,\chi_W\rangle}}_{\deg}}{{{\langle \chi_W,\chi_W\rangle}}_{\deg}},$$ where $\chi_W$ denotes the characteristic function of the set $W\subseteq {{\cal V}}$. Note, in particular, that the combinatorial Cheeger constant is always bounded from above by ${\widetilde{{{\alpha}}}}({{\cal G}}) \le 1$. Next, we state the Cheeger constant estimates: \[t:cheeg\] Let ${{\cal G}}=({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph and $3 \le q \le \infty\}$ such that $\|f\|\leq q$ for all faces $f\in F$. - For some $a > 0$, let ${{\kappa}}(v) \le -a$ for all $v \in {{\cal V}}$. Then we have $${{\alpha}}({{\cal G}}) \geq \frac{2q}{q-2}a.$$ - For some $c > 0$, let $\frac{1}{\|v\|}{{\kappa}}(v) \le -c$ for all $v \in {{\cal V}}$. Then we have $${\widetilde{{{\alpha}}}}({{\cal G}}) \geq \frac{2q}{q-2} c.$$ Moreover, the above estimates are sharp in the case of regular trees. (Note that in the case $q=\infty$ we set $\frac{2q}{q-2} = 2$.) The combinatorial Cheeger constant of all non-positively curved [ regular]{} plane tessellation ${{\cal G}}_{p,q}$ (with all vertices satisfying $\|v\| = p$ and faces satisfying $\|f\| = q$) was explicitly calculated in [@HJL] and [@HiShi] as $${\widetilde{{{\alpha}}}}({{\cal G}}_{p,q}) = \frac{p-2}{p} \sqrt{1 - \frac{4}{(p-2)(q-2)}}.$$ Our estimate gives in this case $${\widetilde{{{\alpha}}}}({{\cal G}}_{p,q}) \ge \frac{(p-2)(q-2)-4}{p(q-2)}.$$ Before considering the exponential growth of a locally tessellating planar graph ${{\cal G}}=({{\cal V}},{{\cal E}},{{\cal F}})$, let us first introduce the [cut locus]{} ${{\rm Cut}}(v)$ of a vertex $v \in {{\cal V}}$. ${{\rm Cut}}(v)$ denotes the set of all vertices $w$, at which $d_v := d(v,\cdot)$ attains a local maximum, i.e., we have $w \in {{\rm Cut}}(v)$ if $d_v(w') \le d_v(w)$ for all $w' \sim w$. ${{\cal G}}$ is [without cut locus]{} if ${{\rm Cut}}(v) = \emptyset$ for all $v \in {{\cal V}}$. Obviously, the cut locus of a finite graph is never empty. It was proved in [@BP2 Thm. 1] that plane tessellations with everywhere non-positive corner curvature are graphs without cut locus. Moreover, let ${{\cal T}}_p$ denote the regular tree with $\|v\| = p$ for all vertices. \[t:expgr\] Let ${{\cal G}}=({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph without cut locus. - If there exists $p \ge 3$ such that $$\label{vertexest} \| v \| \le p \quad \forall\, v \in {{\cal V}},$$ then we have $$\mu({{\cal G}}) \le \mu({{\cal T}}_p) = \log(p-1).$$ - If there exist $p \ge 3$ such that is satisfied and $q \in \{3,4,6\}$ such that $$\| f \| = q \quad \forall\, f \in {{\cal F}},$$ (i.e., ${{\cal G}}$ is face-regular) then we have $$\mu({{\cal G}}) \le \mu({{\cal G}}_{p,q}) = \log \left( \frac{p}{2} - \frac{2}{q-2} + \sqrt{ \left( \frac{p}{2} - \frac{2}{q-2} \right)^2 - 1 } \right).$$ For the reader’s convenience, Theorem \[t:expgr\](b) was stated in “more familiar” terms of vertex and face degrees. However, the statement has an equivalent reformulation in terms of curvature: Let ${{\cal G}}$ be a locally tessellating planar graph without cut locus satisfying $\|f\| = q$ for all faces and $q \in \{3,4,6\}$. For some $b \ge 0$, let $-b \le {{\kappa}}(v)$ for all $v \in {{\cal V}}$. Then we have $$\mu({{\cal G}}) \le \log ( \tau + \sqrt{\tau^2 -1} ),$$ where $\tau = 1 + \frac{q}{q-2} b \ge 1$. The inequality is sharp (with the optimal choice of $b$) in the case of regular graphs ${{\cal G}}_{p,q}$. In fact, the proof will be given for this equivalent reformulation. (Note that the constants $p$ and $b$ in the two formulations are related by $b = \frac{q-2}{q} p - 1$.) Since the regular plane tessellations ${{\cal G}}_{p,q}$ can be considered as combinatorial analogues of [constant curvature space forms]{} in Riemannian geometry, it is natural to conjecture the following discrete version of a [Bishop volume comparison result]{} (see, e.g., [@GaHuLa Theorem 3.101] for the case of a Riemannian manifold). Let $p,q \ge 3$ with $1/p+1/q \le 1/2$ be given. Then we have $$\label{bishop} \mu({{\cal G}}) \le \mu({{\cal G}}_{p,q}),$$ for all locally tessellating planar graphs ${{\cal G}}= ({{\cal V}},{{\cal E}},{{\cal F}})$ without cut locus satisfying $\|v\| \le p$, $\|f\| \le q$. Theorem \[t:expgr\] confirms this conjecture for the cases $q = 3$ and $q = \infty$. However, it seems difficult to prove this seemingly obvious estimate for general face degree bounds $q \ge 3$. Assuming the above conjecture to be true, the comparison of the exponential growth of a locally tessellating planar graph with upper vertex degree bound $p$ and of the regular tree ${{\cal T}}_p$, as given in Theorem \[t:expgr\](a), is quite good if all faces of ${{\cal G}}$ satisfy $\|f\| \ge 6$. For example, we have in the case $(p,q)=(5,6)$: $$1.307\dots = \log(2+\sqrt{3}) = \mu({{\cal G}}_{5,6}) \le \mu({{\cal T}}_5) = \log 4 = 1.381\dots.$$ An direct consequence of [@BP1 Corollary 5.2] is the following lower bound for the exponential growth: \[t:bp\] Let ${{\cal G}}= ({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph without cut locus and $a > 0$ such that ${{\kappa}}(v) \le -a$ for all vertices $v \in {{\cal V}}$. Assume there is $3 \le q \le \infty$ such that we have $\|f\| \le q$ for all faces $f \in {{\cal F}}$. Then we have $$\mu(G) \ge \log\left( 1 + \frac{2q}{q-1} a \right).$$ Moreover, this estimate is sharp in the case of regular trees. (In the case $q = \infty$, we set $\frac{2q}{q-1} = 2$.) We like to finish this subsection by a few additional useful facts: Let $$S_n(v) = \{ w \in {{\cal V}}\mid d(v,w) = n \}$$ be the (combinatorial) sphere of radius $n$ about $v \in {{\cal V}}$. If there is a uniform upper bound on the vertex degree and if $s_n := \| S_n(v) \|$ is a non-decreasing sequence, one easily checks that $$\label{muGalt} \mu({{\cal G}}) = \limsup_{n \to \infty} \frac{\log s_n}{n}.$$ Yet another Cheeger constant $h({{\cal G}})$ was considered in [@BS]: $$h({{\cal G}}) = \inf_{\scriptsize\begin{array}{c} W\subseteq {{\cal V}}, \\ \|W\|<\infty \end{array}} \frac{\|{{\partial}}_V W\|}{\|W\|},$$ where ${{\partial}}_V W$ is the set of all vertices $v \in {{\cal V}}\backslash W$ which are end points of an edge in ${{\partial}}_E W$. In the case that $\mu({{\cal G}})$ is presented by , this Cheeger constant is related to the exponential growth by $$e^{\mu({{\cal G}})} \ge 1+h({{\cal G}}),$$ with equality in the case of regular trees. Spectral applications {#specappl} --------------------- An immediate consequence of Fujiwara’s lower estimate and Theorem \[t:cheeg\] is the following [combinatorial analogue of McKean’s Theorem]{} (see [@McK] for the case of a Riemannian manifold): \[c:McKean\] Let ${{\cal G}}= ({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph and $3 \le q \le \infty$ such that $\|f\|\leq q$ for all faces $f\in F$. For some $c > 0$, let $\frac{1}{\|v\|} {{\kappa}}(v) \le -c$ for all $v \in {{\cal V}}$. Then we have $$1-\sqrt{1-\left( \frac{2q}{q-2} c\right)^2} \le {\widetilde{ \lambda}}_0({{\cal G}}).$$ This estimate is sharp in the case of regular trees. Combining Theorem \[t:expgr\](a), the curvature version of Theorem \[t:expgr\](b) (see the remark of the theorem) and Fujiwara’s upper estimate , we obtain: \[c:essenspecest\] Let ${{\cal G}}=({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph without cut locus. - If there exists $p \ge 3$ such that $$\label{vertexest2} \| v \| \le p \quad \forall\, v \in {{\cal V}},$$ then we have $${\widetilde{ \lambda}}_0^{ess}({{\cal G}}) \le {\widetilde{ \lambda}}_0^{ess}({{\cal T}}_p) = 1 - \frac{2\sqrt{p-1}}{p}.$$ - If there exist $q \in \{3,4,6\}$ with $\|f\| = q$ for all $f \in {{\cal F}}$, and $b > 0$ with $-b \le {{\kappa}}(v)$ for all $v \in {{\cal V}}$, then we have $${\widetilde{ \lambda}}_0^{ess}({{\cal G}}) \le 1 - \frac{2 \sqrt{\tau + \sqrt{ \tau^2 - 1 }} }{1 + \tau + \sqrt{ \tau^2 - 1 } },$$ where $\tau = 1 + \frac{q}{q-2}b$. Next we indicate implications of the above results for the spectrum of the [physical Laplacian]{}. Let ${{\lambda}}_0({{\cal G}})$ and ${{\lambda}}_)^{ess}({{\cal G}})$ denote the bottom of the (essential) spectrum of the physical Laplacian $\Delta$ and, for $n \ge 0$, let $$m_n=\inf_{w\in V\setminus B_{n-1}(v)} \|w\| \quad \mathrm{and} \quad M_n=\sup_{w\in V\setminus B_{n-1}(v)} \|w\|,$$ where $v \in {{\cal V}}$ is an arbitrary vertex and $B_{-1}(v) = \emptyset$. Moreover let $m_\infty=\lim_{n \to \infty} m_n$ and $M_\infty=\lim_{n \to \infty} M_n$. Then we have, by [@Do] $$\label{dodziuk} {{\lambda}}_0({{\cal G}}) \ge \frac{{{\alpha}}({{\cal G}})^2}{2M} \quad \mathrm{and} \quad {{\lambda}}_0^{ess}({{\cal G}}) \ge \frac{{{\alpha}}_\infty({{\cal G}})^2}{2M_\infty},$$ where ${{\alpha}}_\infty({{\cal G}})$ denotes the physical Cheeger constant at infinity, defined in [@Ke]. In general we can also estimate, as demonstrated in [@Ke], $$m_0 {\widetilde{ {{\lambda}}}}_0({{\cal G}}) \leq {{\lambda}}_0({{\cal G}}) \leq M_0 {\widetilde{ {{\lambda}}}}_0({{\cal G}}) \quad \mathrm{and} \quad m_\infty {\widetilde{ {{\lambda}}}}_0^{ess}({{\cal G}}) \leq {{\lambda}}_0^{ess}({{\cal G}}) \leq M_\infty {\widetilde{ {{\lambda}}}}_0^{ess}({{\cal G}}).$$ Via this inequalities we can estimate the bottom of the (essential) spectrum of the physical Laplacian ${\Delta}$ by the estimates of Corollary \[c:McKean\] and \[c:essenspecest\] for the combinatorial Laplacian. Before we look at an explicit example, let us mention the following result about the absence of finitely supported eigenfunctions in the case of non-positive corner curvature: \[t:KLPS\] Let ${{\cal G}}= ({{\cal V}},{{\cal E}},{{\cal F}})$ be a plane tessellation (in the restricted sense of [@BP2]) with non-positive corner curvature in all corners. Then the combinatorial Laplacian does not admit finitely supported eigenfunctions. Note that Theorem \[t:KLPS\] becomes wrong if we replace “non-positive corner curvature” by the weaker assumption “non-positive vertex curvature”, since Example (a) of the Introduction is a graph with vanishing vertex curvature which admits finitely supported eigenfunctions. Let us, finally, apply the above results in an example. We consider the regular tessellation ${{\cal G}}_{6,6}$. Using our geometric results in this article, we obtain $${\widetilde{{{\alpha}}}}({{\cal G}}_{6,6}) \ge \frac{1}{2} \quad \textrm{and} \quad \mu({{\cal G}}_{6,6}) = \log \frac{1+\sqrt{21}}{2} \approx 1.5668.$$ Proposition \[specessspeceq\] tells us that ${\widetilde{ \lambda}}_0({{\cal G}}_{6,6}) = {\widetilde{ \lambda}}_0^{ess}({{\cal G}}_{6,6})$, and with our results in this Subsection we can conclude that $$\begin{aligned} {\widetilde{ \lambda}}_0({{\cal G}}_{6,6}) = {\widetilde{ \lambda}}_0^{ess}({{\cal G}}_{6,6}) &\in& \left[ 1-\frac{\sqrt{3}}{2}, 1- 2 \frac{\sqrt{3}+\sqrt{7}}{7+\sqrt{21}} \right] \\ &\approx& [ 0.1340, 0.2441 ]. \end{aligned}$$ Using the explicit formula for the Cheeger constant in [@HJL] in this particular case, we obtain ${\widetilde{{{\alpha}}}}({{\cal G}}_{6,6}) = \frac{1}{\sqrt{3}} \approx 0.5774$ and we can shrink this interval to $$\begin{aligned} {\widetilde{ \lambda}}_0({{\cal G}}_{6,6}) = {\widetilde{ \lambda}}_0^{ess}({{\cal G}}_{6,6}) &\in& \left[ 1-\sqrt{\frac{2}{3}}, 1- 2 \frac{\sqrt{3}+\sqrt{7}}{7+\sqrt{21}} \right] \\ &\approx& [ 0.1835, 0.2441 ]. \end{aligned}$$ Note that the physical Laplacian is just a multiple of the combinatorial Laplacian (${\Delta}= 6 {\widetilde{\Delta}}$). Finally, Theorem \[t:KLPS\] guarantees that there are no finitely supported eigenfunctions in ${{\cal G}}_{6,6}$. Proof of Theorem 1 ================== The heart of the proof of Theorem \[t:cheeg\] is Proposition \[p:Harm\] below. An earlier version of this proposition in the dual setting (see [@BP1 Prop. 2.1]) was originally obtained by helpful discussions with Harm Derksen. Let us first introduce some important notions related to a locally tessellating planar graph ${{\cal G}}=({{\cal V}},{{\cal E}},{{\cal F}})$. For a finite set $W\subseteq {{\cal V}}$ let ${{\cal G}}_W=(W,{{\cal E}}_W,{{\cal F}}_W)$ be the subgraph of ${{\cal G}}$ induced by $W$, where ${{\cal E}}_W$ are the edges in ${{\cal E}}$ with both end points in $W$ and ${{\cal F}}_W$ are the faces induced by the graph $(W,{{\cal E}}_W)$. Euler’s formula states for a finite and connected subgraph ${{\cal G}}_W$ (observe that ${{\cal F}}_W$ contains also the unbounded face): $$\label{e:Euler} \|W\|-\|{{\cal E}}_W\|+\|{{\cal F}}_W\|=2.$$ By ${{\partial}}_F W$, we denote the set of faces in $F$ which contain an edge of ${{\partial}}_E W$. Moreover, we define the inner degree of a face $f\in{{\partial}}_F W$ by $${\|f\|^i}_W=\|f\cap W\|.$$ In the following, we need the two important formulas which hold for arbitrary finite and connected subgraphs ${{\cal G}}_W=(W,{{\cal E}}_W,{{\cal F}}_W)$. The first formula is easy to see and reads as $$\label{e:E_W} \sum_{v\in W}\|v\|=2\|{{\cal E}}_W\|+\|{{\partial}}_E W\|.$$ Since $W$ is finite, the set ${{\cal F}}_W$ contains at least one face which is not in ${{\cal F}}$, namely the unbounded face surrounding ${{\cal G}}_W$, but there can be more. Define $C(W)=\|{{\cal F}}_W\|-\|{{\cal F}}_W\cap {{\cal F}}\| \ge 1$. Note that $\|{{\cal F}}_W \cap {{\cal F}}\|$ is the number of faces in ${{\cal F}}$ which are entirely enclosed by edges of ${{\cal E}}_W$. Sorting the following sum over vertices according to faces gives the second formula $$\begin{aligned} \sum_{ v\in W}\sum_{f\ni v}\frac{1}{\|f\|} &=& \|{{\cal F}}_W \cap {{\cal F}}\| + \sum_{f\in{{\partial}}_F W}\frac{{\|f\|}^i_W}{\|f\|} \nonumber \\ &=& \|{{\cal F}}_W\| - C(W) + \sum_{f\in{{\partial}}_F W}\frac{{\|f\|}^i_W}{\|f\|}. \label{e:F_W}\end{aligned}$$ \[p:Harm\] Let ${{\cal G}}=({{\cal V}},{{\cal E}},{{\cal F}})$ be a locally tessellating planar graph and $W \subset {{\cal V}}$ be a finite set of vertices such that the induced subgraph ${{\cal G}}_W$ is connected. Then we have $${{\kappa}}(W)=2 -C(W) -\frac{\|{{\partial}}_E W\|}{2}+\sum_{f\in{{\partial}}_F W}\frac{{\|f\|}^i_W}{\|f\|}$$ By the equations , and we conclude $$\begin{aligned} {{\kappa}}(W)&=&\sum_{ v\in W}{\left( 1-\frac{\|v\|}{2}+\sum_{f\ni v}\frac{1}{\|f\|}\right)}\\ &=&\|W\|-{\|{{\cal E}}_W\|}-\frac{\|{{\partial}}_E W\|}{2}+{\|{{\cal F}}_W\|}-C(W)+ \sum_{f\in{{\partial}}_F W}\frac{{\|f\|}^i_W}{\|f\|}\\ &=& 2 -C(W) -\frac{\|{{\partial}}_E W\|}{2}+\sum_{f\in{{\partial}}_F W}\frac{{\|f\|}^i_W}{\|f\|}. \end{aligned}$$ \[p:dd\_E W\] Let $G=(V,E,F)$ be a locally tessellating planar graph and $3 \le q \le \infty$ such that $\|f\|\leq q$ for $f\in F$. Let $W \subset {{\cal V}}$ be a finite set of vertices such that the induced subgraph ${{\cal G}}_W$ is connected. Then we have $$\|{{\partial}}_E W\|\geq\frac{2q}{q-2}(2-C(W)-{{\kappa}}(W)).$$ Since ${{\cal G}}$ is locally tessellating, every edge $e \in {{\partial}}_E W$ separates precisely two different faces. The edge obtains a direction by its start vertex to be in ${{\cal V}}\backslash W$ and its end vertex to be in $W$. Thus it makes sense to refer to the faces at the left and right side of the edge $e$. Thus every edge $e \in {{\partial}}_E W$ determines a unique corner $(v,f) \in W \times {{\partial}}_F W$, where $v \in W$ is the end vertex of $e$ and $f$ is the face at the left side of $e$. The so defined map ${{\partial}}_E W \to W \times {{\partial}}_F W$ is clearly injective, and thus we have $$\sum_{f\in{{\partial}}_F W} {{\|f\|}^i_W} = \|\{(v,f)\in W\times {{\partial}}_F W: v\in f\}\| \ge \|{{\partial}}_E W\|.$$ Using this fact and $\|f\| \leq q$ for all $f\in {{\cal F}}$, we conclude with Proposition \[p:Harm\] $$2 - C(W) - {{\kappa}}(W) = \frac{\|{{\partial}}_E W\|}{2} - \sum_{f\in{{\partial}}_F W}\frac{{\|f\|}^i_W}{\|f\|} \le \|{{\partial}}_E W\| \left( \frac{1}{2} - \frac{1}{q}\right),$$ which proves the inequality in the proposition. Note that the Cheeger constants in Definition \[d:cheeg\] are obtained by taking the infimum of a particular expression over all finite subsets $W \subset {{\cal V}}$. In fact, we can restrict ourselves to consider only finite sets $W$ for which the induces graph ${{\cal G}}_W$ is connected. This follows from the observation that, for a given finite set $W \subset {{\cal V}}$, we can always find a non-empty subset $W_0 \subset W$ such that ${{\cal G}}_{W_0}$ is a connected component of ${{\cal G}}_w$ and that $\|{{\partial}}_E W_0\|/{{\rm vol}}(W_0) \le \|{{\partial}}_e W\|/{{\rm vol}}(W)$ or $\|{{\partial}}_E W_0\|/\|W_0\| \le \|{{\partial}}_e W\|/\|W\|$, respectively. We can reduce the sets under consideration even further. Let $W \subset {{\cal V}}$ be a finite set such that ${{\cal G}}_W$ is connected. Note that ${{\cal G}}_w$ has only one unbounded face. By adding all vertices of ${{\cal V}}$ contained in the union of all bounded faces of ${{\cal G}}_w$, we obtain a bigger finite set $P_W \supset W$ such that $C(P_W) = 1$. (Note that all bounded faces of ${{\cal G}}_{P_W}$ are also faces of the original graph ${{\cal G}}$.) We call a finite set $P \subset {{\cal V}}$ with connected graph ${{\cal G}}_P$ and $C(P) = 1$ a [polygon]{}. Clearly, we have $\|{{\partial}}_E P_W \|/{{\rm vol}}(P_W) \le \|{{\partial}}_e W\|/{{\rm vol}}(W)$ and $\|{{\partial}}_E P_W\|/\|P_W\| \le \|{{\partial}}_e W\|/\|W\|$. Thus it suffices for the definition of the Cheeger constants to take the infimum only over all polygons. With this final observation we can now prove Theorem \[t:cheeg\]. Let $W \subset {{\cal V}}$ be a polygon. Since $C(W) = 1$, we conclude from Proposition \[p:dd\_E W\] that $$\frac{\|{{\partial}}_E W\|}{\|W\|} \ge \frac{2q}{q-2} \frac{-{{\kappa}}(W)}{\|W\|} \ge \frac{2q}{q-2} a.$$ Taking the infimum over all polygons yields part (a) of the theorem. For the proof of part (b), recall that $-{{\kappa}}(v) \ge c \cdot \|v\|$ for all vertices $v \in {{\cal V}}$. This implies that $$\frac{-{{\kappa}}(W)}{{{\rm vol}}(W)} = \frac{- \sum_{v \in W} {{\kappa}}(v)}{\sum_{v \in W} \|v\|} \ge c,$$ and, consequently, for polygons $W \subset {{\cal V}}$, $$\frac{\|{{\partial}}_E W\|}{{{\rm vol}}(W)} \ge \frac{2q}{q-2} \frac{-{{\kappa}}(W)}{{{\rm vol}}(W)} \ge \frac{2q}{q-2} c.$$ The statement follows now again by taking the infimum over all polygons. Proof of Theorem 2 ================== Parts (a) and (b) of Theorem \[t:expgr\] have very different proofs. We present them separately. We choose a vertex $v_0 \in {{\cal V}}$ and introduce the following functions $m, M: {{\cal F}}\to \{ 0,1,2,\dots, \infty \}$: $$\begin{aligned} m(f) &=& \min \{ d(w,v_0) \mid w \in \partial f \}, \\ M(f) &=& \max \{ d(w,v_0) \mid w \in \partial f \}. \end{aligned}$$ Note that the face $f$ “opens up” at distance $m(f)$ and “closes up” at distance $M(f)$ from $v_0$. We call a face $f$ [finite]{}, if $M(f) < \infty$. The idea of the proof is to “open up” successively every finite face $f \in {{\cal F}}$ into an infinigon without violating the vertex bound. In this way, we will build up a comparison tree ${{\cal T}}$ with the same vertex bound $p$ and satisfying $\mu({{\cal G}}) \le \mu({{\cal T}})$. It turns out, however, that finite faces $f$ with more than one vertex in the sphere $S_{M(f)}(v_0)$ cause problems in this “opening up” procedure (since the distance relations to the vertex $v_0$ will be changed). Therefore, we first modify the tessellation ${{\cal G}}$ by removing all edges connecting two vertices $v,w$ at the same distance to $v_0$. The modified planar graph is denoted by ${{\cal G}}_0 = ({{\cal V}}_0,{{\cal E}}_0,{{\cal F}}_0)$. To keep track, we add at each of the vertices $v,w$ a short terminal edge. These terminal edges do not belong “officially” to the graph ${{\cal G}}_0$ and serve merely as reminders that an edge can be added in their place without violating the vertex bound of the graph. Moreover, we can only guarantee $\mu({{\cal G}}_0) \ge \mu({{\cal G}})$, if these inofficial edges are included in ${{\cal G}}_0$. (At the end of the procedure we will replace all “inofficial” terminal edges by infinite trees rooted in $v$ and $w$.) The modification ${{\cal G}}\to {{\cal G}}_0$ is illustrated in Figure \[openup1\]. (For convenience, the vertices belonging to distance spheres $S_n(v_0)$ are arranged to lie on concentric Euclidean circles around $v_0$.) ![Removing edges between vertices on the same spheres and replacing them by “inofficial” terminal edges[]{data-label="openup1"}](openup1.eps){width="12cm"} Note that none of the distance relations of the vertices in ${{\cal G}}_0$ (without the inofficial terminal edges) to the vertex $v_0$ are changed and that we still have ${{\rm Cut}}(v_0) = \emptyset$. Moreover, the modified graph $G_0$ (without the inofficial terminal edges) has a new set of faces ${{\cal F}}_0$. Every finite face $f$ of ${{\cal G}}_0$ has now even degree, since $f$ opens up at a single vertex in the sphere $S_{m(f)}(v_0)$ and $f$ closes up at a single vertex in the sphere $S_{M(f)}(v_0)$. We order all finite faces $f_0, f_1, f_2, \dots$ of ${{\cal G}}_0$ such that we have $$M(f_0) \le M(f_1) \le M(f_2) \le ...$$ Next we explain the first step of our procedure, namely, how to open up $f_0$ into an infinigon $\widetilde f_0$. Let $n = M(f_0) \ge 1$ and $w \in \partial f_0$ such that $d(w,v_0) = n$. Since $C(v_0) = \emptyset$, we can find an infinite geodesic ray $w_0= w, w_1, w_2, \dots \in {{\cal V}}$ such that $d(w_i,v_0) = n+i$. We may think of $v_0$ as being the origin of the plane and of $w_0, w_1, \dots$ as being arranged to lie on the positive vertical coordinate axis at heights $n,n+1, \dots$ with straight edges between them. Now we cut our plane along this geodesic ray, i.e., replace the ray by two parallel copies of the ray and thus preventing the face $f_0$ from closing up at distance $n$. In this way, $f_0$ becomes an infinigon, which we denote by $\widetilde f_0$. (In fact, we rotationally shrink the angle $2\pi$ to $2\pi-\epsilon$ around $v_0$ to open up a conic sector of angle $\epsilon$ containing the infinigon $\widetilde f_0$.) The procedure is illustrated in Figure \[openup2\]. Note that the vertices $w_i$ are replaced by two copies $w_i^{(1)}, w_i^{(2)}$, such that $w_i^{(j)}$ is connected to $w_{i+1}^{(j)}$ for $j=1,2$ and $w_i^{(1)}$ inherits all previous neighbors of $w_i$ at one side of the ray and $w_i^{(2)}$ inherits all previous neighbors of $w_i$ at the other side of the ray (this concerns in particular also the “inofficial” vertices). In this way we obtain a new planar graph ${{\cal G}}_1 = ({{\cal V}}_1,{{\cal E}}_1,{{\cal F}}_1)$. ![Changing the finite face $f_0$ into an infinigon $\widetilde f_0$[]{data-label="openup2"}](openup2.eps){width="12cm"} The graph ${{\cal G}}_1$ is still connected. Note also that we have $$\begin{aligned} \| w_0^{(1)} \| + \| w_0^{(2)} \| &=& \| w_0 \| + 1, \label{w0}\\ \| w_i^{(1)} \| + \| w_i^{(2)} \| &=& \| w_i \| + 2, \quad \forall \, i \ge 1. \label{wi} \end{aligned}$$ After including the inofficial terminal edges in the graph ${{\cal G}}_1$, we still have $$\| v \| \le p \quad \forall\, v \in {{\cal V}}_1,$$ and , imply that $\mu({{\cal G}}_1) \ge \mu({{\cal G}}_0) \ge \mu({{\cal G}})$. In the second step we carry out the same procedure with the face $f_1 \in {{\cal F}}_1$, and obtain a new connected planar graph ${{\cal G}}_2 = ({{\cal V}}_2,{{\cal E}}_2,{{\cal F}}_2)$, with $f_1 \in {{\cal F}}_1$ replaced by the infinigon $\widetilde f_1 \in {{\cal F}}_2$. Again, after including the inofficial terminal edges, the graph ${{\cal G}}_2$ has vertex bound $p$ and satisfies $\mu({{\cal G}}_2) \ge \mu({{\cal G}}_1) \ge \mu({{\cal G}})$. It is now clear how to repeat the procedure. Note that for every radius $n \ge 1$ there is a large enough $j \ge 1$ such that the graphs ${{\cal G}}_j, {{\cal G}}_{j+1}, {{\cal G}}_{j+2}, \dots$ remain unaltered inside the balls $B_n({{\cal G}}_k,v_0)$. This fact guarantees that there is a well-defined limiting graph associated to the sequence ${{\cal G}}_j$. This limit is a connected tree ${{\cal T}}_0$ (since all faces of ${{\cal T}}_0$ are infinigons). In ${{\cal T}}_0$, we replace now finally the inofficial terminal edges by infinite trees, rooted at the corresponding proper vertices of the tree ${{\cal T}}_0$, with branching sequence $1,p-1,p-1,p-1,\dots$. These infinite trees can be nicely fitted into the infinigons to yield an infinite planar tree ${{\cal T}}$ with vertex bound $p$ and satisfying $\mu({{\cal T}}) \ge \mu({{\cal G}})$. Since we obviously have $\mu({{\cal T}}) \le \mu({{\cal T}}_p) = \log (p-1)$, the proof of part (a) of the theorem is finished. We prove the equivalent curvature version of the statement, given in the remark after the theorem. Since $\|f\| = q < \infty$ for all faces $f$, ${{\cal G}}$ is a tessellating plane graph in the sense of [@BP1] and we have $${{\kappa}}(v) = 1 - \frac{q-2}{q} \|v\|.$$ Since $\{ {{\kappa}}(v) \mid v \in {{\cal V}}\}$ is a discrete set and bounded from below by $-b$, we can assume, without loss of generality, that $-b$ is of the form $1 - \frac{q-2}{q} p$, for some integer value $p \ge 3$. (In fact, $p$ is the optimal upper bound on the vertex degree of ${{\cal G}}$.) Let $S_n, B_n$ be the combinatorial spheres and balls in ${{\cal G}}$ with respect to a reference vertex $v_0 \in {{\cal V}}$ and $s_n = \|S_n\|$. Corollary 6.4 of [@BP1] states that we have $$s_{n+1} - s_n = \frac{2q}{q-2} (1-\kappa(B_n)).$$ Applying this equation twice, we derive $$s_{n+2} - 2 s_{n+1} + s_n = - \frac{2q}{q-2} \kappa(S_{n+1}) \le \frac{2q}{q-2} b s_{n+1}.$$ Hence we obtain the following recursion inequality $$s_{n+2} \le 2 \tau s_{n+1} - s_n, \quad s_1 \le p, \ s_0 = 1,$$ with $\tau = 1 + \frac{q}{q-2} b \ge 1$. It is easy to see that the sequence $$\label{alphan} \sigma_{n+2} = 2 \tau \sigma_{n+1} - \sigma_n, \qquad \sigma_1 = p, \ \sigma_0 = 1,$$ is strictly increasing and dominates the sequence $s_n$. Moreover, $\sigma_n$ describes the cardinality of a sphere of radius $n$ in the regular tessellation ${{\cal G}}_{p,q}$. This implies that $\mu({{\cal G}}) \le \mu({{\cal G}}_{p,q})$. Now, we return to the sequence $\sigma_n$, as defined in . We first consider the case $\tau > 1$. The recursion formula implies that $$\sigma_n = u \left( \tau - \sqrt{\tau^2 -1} \right)^n + v \left( \tau + \sqrt{\tau^2 -1} \right)^n,$$ with constants $u,v \in {{\mathbb R}}$ chosen in such a way that the initial conditions are satisfied. Since $$0 < \tau - \sqrt{\tau^2 -1} < 1,$$ we conclude that $v \neq 0$, for otherwise we would have $\sigma_n \to 0$, contradicting to the fact that ${{\cal G}}_{p,q}$ is an infinite graph. Hence, $\sigma_n$ behaves asymptotically like $$\sigma_n \sim v \left( \tau + \sqrt{\tau^2 -1} \right)^n,$$ with a positive constant $v$. This, together with implies that $$\label{mugpq} \mu({{\cal G}}_{p,q}) = \lim_{n \to \infty} \frac{\log \sigma_n}{n} = \log \left( \tau + \sqrt{\tau^2 -1} \right).$$ In the case $\tau = 1$, the sequence is simply given by $\sigma_n = n (p-1) +1$. Linear growth of $\sigma_n$ implies that $\mu({{\cal G}}_{p,q})=0$, which also coincides with . [Tho]{} N. Alon, Eigenvalues and expanders, Theory of computing (Singer Island, Fla., 1984). Combinatorica 6 (1986), no. 2, 83–96. N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. O. Baues, N. Peyerimhoff, [Curvature and geometry of tessellating plane graphs]{}, Discrete Comput. Geom. (2001), no. 1, 141-159 O. Baues, N. Peyerimhoff. Geodesics in Non-Positively Curved Plane Tessellations, Advances of Geometry 6, no. 2, (2006), 243-263. R. Brooks. A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981), 501-508. I. Benjamini, O. Schramm. Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant, Geom. funct. anal. 7, (1997), 403-419. J. Cheeger, [A lower bound for the lowest eigenvalue of the Laplacian]{}, in: Problems in analysis, asymposium in honor of S. Bochner, Princeton Univ. Press, Princeton, (1970), 195-199. F. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. F. Chung, A. Grygor’yan, S.-T. Yau, Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs, Comm. Anal. Geom. 8 (2000), no. 5, 969–1026. J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. J. Dodziuk, L. Karp, Spectral and function theory for combinatorial Laplacians, Geometry of Random Motion, (R. Durrett, M.A. Pinsky ed.) AMS Contemporary Mathematics, Vol 73, (1988), 25-40. J. Dodziuk, W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, From Local Times to Global Geometry, Control and Physics, (K. D. Elworthy ed.) Longman Scientific and Technical, (1986), 68-75. H. Donnelly, P. Li, Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J. 46 (1979), no. 3, 497–503. K. Fujiwara, [Growth and the spectrum of the Laplacian of an infinite graph]{}, T[ô]{}hoku Math. J. [48]{} (1996), 293-302 K. Fujiwara, Laplacians on rapidly branching trees, Duke Math Jour. 83, no 1, (1996), 191-202. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer Verlag (1990). D. Grieser, The first eigenvalue of the Laplacian, isoperimetric constants, and the max flow min cut theorem, Arch. Math. (Basel) 87 (2006), no. 1, 75–85. M. Gromov, Hyperbolic groups, Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987 O. H[ä]{}ggstr[ö]{}m, J. Jonasson, R. Lyons, Explicit isoperimetric constants and phase transitions in the random-cluster model, Ann. Probab. 30 (2002), no. 1, 443-473. Y. Higuchi, Combinatorial curvature for planar graphs, J. Graph Theory 38 (2001), no. 4, 220–229. Y. Higuchi, T. Shirai, Isoperimetric constants of $(d,f)$-regular planar graphs, Interdiscip. Inform. Sci. 9 (2003), no. 2, 221-228. M. Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, arXiv:0712.3816. S. Klassert, D. Lenz, N. Peyerimhoff, P. Stollmann, Elliptic operators on planar graphs: unique continuation or eigenfunctions and nonpositive curvature, Proc. Amer. Math. Soc. 134, (2005), 1549-1559. S. Klassert, D. Lenz, P. Stollmann, Discontinuities of the integrated density of states for random operators on Delone sets, Commun. Math. Phys. 241, (2003), 235-243. M. Kohmoto, B. Sutherland, Electronic states on a Penrose lattice, Phys. Rev. Lett. 56, (1986), 2740-2743. D. Lenz, I. Veseli[ć]{}, Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence, arXiv:0709.2836. H. P. McKean, An upper bound to the spectrum of $\Delta$ on a manifold of negative curvature, J. Diff. Geom. 4, (1970), 359-366. B. Mohar, W. Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21, (1989), 209-234. D. A. Stone, A combinatorial analogue of a theorem of Myers and Correction to my paper: “A combinatorial analogue of a theorem of Myers”, Illinois J. Math. 20 (1976), no. 1, 12–21, and Illinois J. Math. 20 (1976), no. 3, 551–554 D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327–351. T. Sunada, Fundamental groups and Laplacians, Selected papers on number theory, algebraic geometry, and differential geometry, 19–32, Amer. Math. Soc. Transl. Ser. 2, 160, Amer. Math. Soc., Providence, RI, 1994. A. Weber, Analysis of the physical Laplacian and the heat flow on a locally finite graph, arXiv:0801.0812. W. Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 385–393 W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics 138, Cambridge University Press, (2000). R. K. Wojciechowski, Stochastic Completeness of Graphs, arXiv:0712.1570. [^1]: e-mail: matthias.keller@mathematik.tu-chemnitz.de [^2]: e-mail: norbert.peyerimhoff@durham.ac.uk
--- abstract: \| The growing popularity of cloud-based machine learning raises a natural question about the privacy guarantees that can be provided in such a setting. Our work tackles this problem in the context where a client wishes to classify private images using a convolutional neural network (CNN) trained by a server. Our goal is to build efficient protocols whereby the client can acquire the classification result without revealing their input to the server, while guaranteeing the privacy of the server’s neural network. To this end, we design , a scalable and low-latency system for secure neural network inference, using an intricate combination of homomorphic encryption and traditional two-party computation techniques (such as garbled circuits). makes three contributions. First, we design the homomorphic encryption library which provides fast algorithms for basic homomorphic operations such as SIMD (single instruction multiple data) addition, SIMD multiplication and ciphertext permutation. Second, we implement the homomorphic linear algebra kernels which map neural network layers to optimized homomorphic matrix-vector multiplication and convolution routines. Third, we design optimized encryption switching protocols which seamlessly convert between homomorphic and garbled circuit encodings to enable implementation of complete neural network inference. We evaluate our protocols on benchmark neural networks trained on the MNIST and CIFAR-10 datasets and show that outperforms the best existing systems such as MiniONN (ACM CCS 2017) by $20\times$ and Chameleon (Crypto Eprint 2017/1164) by $30\times$ in online runtime. Similarly when compared with fully homomorphic approaches like CryptoNets (ICML 2016) we demonstrate [three orders of magnitude]{} faster online run-time. author: - - - - bibliography: - 'References.bib' title: '[[gazelle]{}: A Low Latency Framework for Secure Neural Network Inference]{}' --- =1 =0 ### acknowledgments {#acknowledgments .unnumbered} We thank Kurt Rohloff, Yuriy Polyakov and the PALISADE team for providing us with access to the PALISADE library. We thank Shafi Goldwasser, Rina Shainski and Alon Kaufman for delightful discussions. We thank our sponsors, the Qualcomm Innovation Fellowship and Delta Electronics for supporting this work.
--- address: 'University of Rochester, Rochester, New York 14628' author: - 'Steven R. Blusk, for the CDF Collaboration' title: Top Physics from Run 1 and Run 2 Prospects at CDF --- Introduction ============ In ${p\bar{p}}$ collisions at the Tevatron ($\sqrt{s}$=1.8 TeV), top quark pairs are produced through the strong interaction with an expected cross section (at NLO) of 5.1 pb [@ttbar-xs-theory]. Single top quarks are also expected to be produced through a t-W-b electroweak vertex with an expected total cross section of $\approx$1/2 that of ${t\bar{t}}$ [@singletop-xs-theory]. Within the SM, the top quark is expected to decay with a lifetime of $\approx~10^{-24}$ seconds into a W boson and a b quark. ${t\bar{t}}$ final states are classified according to the decays modes of the two W bosons. Dilepton final states consist of events where both W bosons decay to an $e$ or $\mu$ (BR=5%). Lepton + jets final states include events where one of the W bosons decays leptonically ($e$ or $\mu$) and the other hadronically (BR=30%). The All-Jets mode includes events in which both W-bosons decay hadronically (BR=44%). ${t\bar{t}}$ Cross Section ========================== Cross section measurements have been made in all three decay channels. In the dilepton channel [@dil-xsec], we observe 9 events with an expected background of 2.5$\pm$0.5 events, which leads to a ${t\bar{t}}$ cross section of $8.2^{+4.4}_{-3.4}$ pb. In the lepton+$\ge$3 jets channel, there are 29 (25) events which are SVX (SLT) tagged with expected backgrounds of 8.1 (13.2) events, leading to a measurement of $5.7^{+1.9}_{-1.5}$ pb [@ljet-xsec]. For the All-Jets mode, we measure $7.6^{+3.5}_{-2.7}$ pb [@allhad-mass-xsec]. Results from all three channels are combined to obtain a ${t\bar{t}}$ cross section of $6.5^{+1.7}_{-1.4}$ pb [@ptohos] which is within one standard deviation from the theoretical prediction. Top Quark Mass ============== The most precise measurements in the top quark sector thus far have been in the mass. In the dilepton channel, we use a weighting technique which compares the observed in each event to the expected value as a function of the assumed top mass. Using a likelihood technique we extract a top mass of =167.4$\pm$11.4 ${ {\rm GeV}/c^{2} }$ [@dil-mass]. In the lepton+$\ge$4 jets events, we perform a 2C fit of the final state particles to the decay chain, which results in a measured top mass of $176.1\pm7.4~{ {\rm GeV}/c^{2} }$ [@ljet-mass]. Full reconstruction of events in the All-Jets mode is also performed from which we measure =186.0$\pm 11.5~{ {\rm GeV}/c^{2} }$ [@allhad-mass-xsec]. The result from combining all three measurements is $176.1\pm 6.6~{ {\rm GeV}/c^{2} }$, roughly 35 times the mass of the next heaviest quark! The ${t\bar{t}}$ Invariant Mass ($M_{{t\bar{t}}}$) ================================================== The $M_{{t\bar{t}}}$ analysis [@mttbar] proceeds in a similar way to the top mass analysis. To improve the resolution on the four momenta of the final state particles (and thus $M_{{t\bar{t}}}$), we constrain the top quark mass to 175 ${ {\rm GeV}/c^{2} }$ in the fit. We also require when we remove this constraint that the fitted top quark mass lie in the range from 150-200 ${ {\rm GeV}/c^{2} }$. The data do not show an excess above the SM prediction, and we therefore present limits on the cross section times branching ratio (see Fig. \[fig:mttbar-limits\]). At the 95% confidence level (CL), the data rule out a topcolor $Z^{\prime}$ with mass less than 480 (780) ${ {\rm GeV}/c^{2} }$ and natural width equal to 0.012 (0.04) $M_{Z^{\prime}}$. Top $\Pt$ ========= Like the $M_{{t\bar{t}}}$ analysis, we use $l$+4 jet data and constrain the top quarks mass to 175 ${ {\rm GeV}/c^{2} }$. Because of the strong correlation between the top and antitop quarks’ $\Pt$, we use only the hadronically decaying top quark. We measure the fraction of top quarks produced in four bins of true $\Pt$: 0-75 ${ {\rm GeV}/c^{2} }$, 75-150 ${ {\rm GeV}/c^{2} }$, 150-225 ${ {\rm GeV}/c^{2} }$, and 225-300 ${ {\rm GeV}/c^{2} }$. First, we determine initial response functions which give the distribution of reconstructed $\Pt$ in each of the four true $\Pt$ bins. The data are then fit to a combination of the four Monte Carlo (MC) reconstructed $\Pt$ distributions using an iterative procedure to minimize the sensitivity of the final result to the initial assumptions of the true top $\Pt$ distribution. Within the limited statistics, the data are consistent with SM expectations. We measure the 95% CL limit for the fraction of top quarks with true $\Pt$ larger than 225 ${ {\rm GeV}/c }$ to be 0.114. W Helicity in Top Decays ======================== The V-A structure of the t-W-b vertex results in a specific prediction for the W polarization in top decays. At tree level, we expect the fraction of longitudinal $W$ bosons, $F_0$, to be 70.1$\pm$1.6%. The $\Pt$ spectrum of the leading lepton is sensitive to the W polarization. Using MC distributions of longitudinal and left-handed W’s, we fit the data to extract the fraction $F_0$. Using both the lepton+jets and dilepton data samples, we measure $F_0=91\pm 37(stat)\pm 13$% [@w-hel]. Rare Decays =========== The FCNC decays $t\to Zq$ and $t\to\gamma q$ are strongly suppressed in the S.M. at the level of $\sim$10$^{-12}$, and therefore an observation of such events is a signature of new physics. We have performed searches for these decays [@rare-decays] and find one event in each channel, consistent with background expectations. We therefore set 95% CL limits of 33% and 3.2% respectively for these two FCNC decays. Single Top Production ===================== We have searched for single top in the lepton+jets data. One analysis searches for events in both the W-gluon fusion and the s-channel $W^$ processes. We select W+1,2,3 jet events which have a SVX b-tag and a top invariant mass, $M_{l\nu b}$ in the range 140 to 210 ${ {\rm GeV}/c^{2} }$. We observe 65 events with an expected background of 62.5$\pm$11.5 events. We expect to 4.3 signal events. Fitting the $H_T=\sum {{\rm E}_{\scriptscriptstyle\rm T}}(lepton, {\mbox{$\protect \raisebox{.3ex}{$\not$}{{\rm E}_{\scriptscriptstyle\rm T}}\ $}}, jets)$ distribution in data to MC signal and background distributions, we extract a cross section limit of 13.5 pb at 95% CL. A second analysis which looks just for the W-gluon fusion process selects W+2 jet events with an SVX tag and the same cut on $M_{l\nu b}$. An interesting and exploitable feature of these events is that, unlike the backgrounds, the product of the leading lepton’s charge ($Q$) and the pseudorapidity of the untagged jet ($\eta$) peaks at positive $Q\times\eta$. We observe 15 events with an expected background of 12.9$\pm$2.1 events (we expect 1.2$\pm$0.3 signal events). From a fit of the $Q\times\eta$ distribution in data to MC signal and background distributions, we extract a 95% CL limit of 15.4 pb. Run 2 Expectations ================== Run 2 will provide $\approx$40-50 times more ${t\bar{t}}$ events than Run 1. In addition to a large reduction in statistical uncertainties, systematic uncertainties such as the jet energy scale and MC modelling will also be reduced. For example, the large sample of $Z\to{b\bar{b}}$ events can be used to check the $b-jet$ energy scale. The invariant mass of the two untagged jets in double SVX tagged W+4 jet events can be used to check the light quark jet energy scale. A comparison of extra jets in a high purity top sample can be used to put constraints on gluon radiation in the MC simulation. Moreover, we expect to undertake new physics analyses in Run 2, such as studying the spin correlations in ${t\bar{t}}$ events. Given the size of the Run 2 data sample, we have made projections for the precision we can expect for a variety of measurements. Some of these projections are given in Table \[tab:run2proj\]. Run 2 and a future Run 3 will clearly provide very rich top samples with which to probe the SM and beyond. Measurement Precision ----------------------------- ------------------ -- -- 1.5% ${t\bar{t}}$ cross section 9% Single top cross section 24% $V_{tb}$ (from Single top) 13% $F_0$ 5.5% $\sigmaBR(X\to{t\bar{t}})$ 0.1 pb at 1 TeV $ BR(t\to \gamma c)$ $<$2.8x10$^{-3}$ $ BR(t\to Zc)$ $<$1.3x10$^{-2}$ $ BR(t\to Hb)$ $<12$% : Projections for the expected precision for measurements with an integrated luminosity of 2 $fb^{-1}$.[]{data-label="tab:run2proj"} Acknowledgments {#acknowledgments .unnumbered} =============== We thank the Fermilab staff and our CDF collaborators for their vital contributions to these physics analyses. [99]{} E. Berger and H. Contopanagos, [Phys. Rev.]{} D [57]{}, 253 (1998), [and references therein]{}. T. Stelzer, Z. Sullivan and S. Willenbrock, [Phys. Rev.]{} D [58]{}, 094021 (1998), [and references therein]{}. F. Abe [et al.,]{} Phys. Rev. Lett. [80]{}, 2779 (1998). F. Abe [et al.,]{} Phys. Rev. Lett. [80]{}, 2773 (1998). F. Abe [et al.,]{} Phys. Rev. Lett. [79]{}, 1992 (1997). F. Ptohos (for the CDF Collaboration), Proceedings of the International Europhysics Conference on High Energy Physics 99, Tampere, Finland, July 17, 1999. F. Abe [et al.,]{} Phys. Rev. Lett. [82]{}, 271 (1999). T. Affolder [et al.,]{} preprint hep-ex/0006028 (submitted to [Phys. Rev.]{} D); F. Abe [et al.,]{} Phys. Rev. Lett. [80]{}, 2767 (1998). T. Affolder [et al.,]{} preprint hep-ex/0003005 (accepted in Phys. Rev. Lett.) T. Affolder [et al.,]{} to be submitted to [Phys. Rev. Lett.]{} T. Affolder [et al.,]{} [Phys. Rev. Lett.]{} [84]{} 216 (2000). T. Affolder [et al.,]{} [Phys. Rev. Lett.]{} [80]{} 2525 (1998).
--- abstract: 'The model of $p$ Ising spins coupled to 2d gravity, in the form of a sum over planar 3 graphs, is studied and in particular the two-point and spin-spin correlation functions are considered. We first solve a toy model in which only a partial summation over spin configurations is performed and, using a modified geodesic distance, various correlation functions are determined. The two-point function has a diverging length scale associated with it. The critical exponents are calculated and it is shown that all the standard scaling relations apply. Next the full model is studied, in which all spin configurations are included. Many of the considerations for the toy model apply for the full model, which also has a diverging geometric correlation length associated with the transition to a branched polymer phase. Using a transfer function we show that the two-point and spin-spin correlation functions decay exponentially with distance. Finally, by assuming various scaling relations, we make a prediction for the critical exponents at the transition between the magnetized and branched polymer phases in the full model.' --- epsf by 8mm 3[$\phi^3$]{} Introduction ============ The work of Knizhnik, Polyakov and Zamolodchikov (KPZ) [@KPZ] as well as of David, Distler and Kawai (DDK) [@DDK] made it possible to understand many aspects two-dimensional quantum gravity coupled to conformal field theories. At the same time, it became clear that the models of dynamical triangulations coupled to matter fields provide us with a statistical mechanical realisation of the models described by KPZ and DDK, in the same way as the two-dimensional Ising model at its critical point can be viewed as a conformal $c=1/2$ field theory. Two aspects of 2d quantum gravity coupled to matter fields remain puzzling: the $c=1$ barrier and the concept of correlation length. The formulae for critical exponents derived by KPZ cease to be valid for $c >1$ and the critical exponents are derived by general scaling arguments applied to globally defined operators. At no point is the concept of a divergent correlation length introduced. For ordinary statistical systems the divergence of a correlation length when a critical temperature is approached is believed to be the underlying reason that general scaling arguments work well. The difficulties of defining a local length scale in quantum gravity are well known and have so far prevented a proper treatment of correlation functions by means of continuum methods. Working entirely in the context of dynamical triangulations the problems mentioned are not seen directly. Statistical models with multiple Ising spins living on dynamically triangulated surfaces are perfectly well defined even if $c >1$ and they have a critical point. At least superficially, the only difference compared to $c <1$ is that we cannot solve the theory. Nevertheless, low temperature expansions and mean field calculations seem reliable for $c \to \infty$, as is confirmed by the agreement between the theoretical calculations and Monte Carlo simulations of the systems with large $c$ [@MC; @Wexlerq]. The picture which emerges for large $c$ from the low temperature expansion [@Wexler] is as follows: for large $\beta$ (low temperature) there is a magnetized phase for which $\gamma_{str}=-1/2$ separated from a branched polymer phase, where $\gamma_{str}=1/2$, by a transition at a critical $\beta^$ where $\gamma^_{str}=1/3$. Likewise it has been possible by the use of dynamical triangulations to address the question of correlation length in two-dimensional quantum gravity. A two-point correlator between “punctures” has been calculated in pure gravity as a function of a “quantum” geodesic distance and it is found that standard scaling relations, known from the theory of critical phenomena, are satisfied, although with unusual critical exponents [@AmbWat95]. So far it has not been possible in 2d quantum gravity to calculate correlation functions of matter fields as a function of the geodesic distance. The attempts to measure the spin-spin correlation functions by Monte Carlo simulations and to define a divergent correlation length as one approaches the critical point, have so far been ambiguous [@CTBJ; @MC]. The question arises as to whether there is a divergent spin-spin correlation length associated with the phase transition between a magnetized and a non-magnetized phase in the two-dimensional Ising model coupled to gravity. Rather surprisingly from a continuum point of view, we can, using dynamical triangulations, begin to answer this question in the limit of large $c$, i.e. in the limit where a large number of Ising models is coupled to quantum gravity. The statistical model we will define and solve in the following sections is a toy model of 2d quantum gravity coupled to Ising spins in the sense that the correct summation over all triangulations is performed, but not all the spin configurations are included. The spin configurations we include are precisely the ones which dominate in the large $c$ limit, at large $\beta$, namely those for which the domains are connected in a tree-like fashion with domain boundaries of minimal size. This model allows us to calculate the two-point functions (using a certain definition of distance, which is based on the geodesic distance) and extract the critical exponents. We will verify that these exponents satisfy standard scaling relations and that the geometrical interpretation of some of the exponents is related to the fractal structure of the underlying “space-time”, in agreement with general arguments; that is, we explicitly verify that the exponent $\nu$ is related to the Hausdorff dimension by $\nu=1/d_H$. The model has a third order transition between a tree-like (i.e. branched polymer) phase and a magnetized phase. It is found that there is a diverging correlation length in the tree phase associated with the geometric two-point function, but no diverging correlation length associated with the spin-spin correlation function. Next we turn our attention to the full model of $p$ independent Ising spins coupled to 2d gravity, which has a central charge of $c=p/2$. Following the analysis of the toy model, we show that various two-point functions decay exponentially with distance in the magnetized phase. Again there is a diverging correlation length associated with the geometric two-point function. By assuming that all the standard scaling relations still hold for this model and making a few further fairly modest assumptions, we show that the critical exponents for the transition between the magnetized and branched polymer phases, in the full model, are the same as those in the toy model. The rest of this article is organized as follows: in section \[sec:toynomag\] we define the toy model, define a variant of geodesic distance in the model and calculate the two-point function. In sections \[sec:toycrit\] and \[sec:dh\] the critical exponents and Hausdorff dimension are calculated for the toy model, whilst section \[sec:spinspin\] discusses the spin-spin correlation function. In section \[sec:toymag\] we verify by direct calculation in an external magnetic field, that the magnetic exponents found by scaling arguments are indeed the correct ones. In section \[sec:full\] we address some of the questions mentioned above for the full model, i.e. in the model where we sum over all the spin configurations. Finally, section \[sec:concl\] contains our conclusions. Toy model, without a magnetic field {#sec:toynomag} =================================== Definition ---------- Before looking at the full model of $p$ spins coupled to 2d gravity (in the form of a sum over planar 3 graphs), we will solve a toy model, which is very similar to the one studied in [@ADJ]. As for that model we will show that there is a magnetized phase for which $\gst=-\half$ and a branched polymer phase with $\gst=\half$. On the boundary $\gst^=\third$, which is the same value as that in [@ADJ; @JonWhe94] and agrees with the result $\gst^=\overline{\gamma}/(\overline{\gamma}-1)$ for $\overline{\gamma}=-\half$, the pure gravity value, which one might expect from the calculation in [@Dur94]. For our model we sum over all possible rooted planar 3 graphs (let us denote this set of graphs by $\gone$), but we only sum over a subset of the possible spin configurations, namely those for which the domains are connected in a tree-like fashion, with at most one link connecting any two domains. In this paper we shall only consider planar graphs, that is, $\chi=2$ throughout. Each vertex will be weighted with a factor of $x$ and has $p$ independent spins on it which can take the values $\pm 1$ (i.e. vertex $i$ has spins $S^\alpha_i$ on it, with $\alpha = 1, \cdots, p$). Links joining vertices with dissimilar spin configurations will be given a factor of $e^{-2 \beta}$ for each spin flavour which differs. Thus the grand canonical partition function, T, is \[eqn:gcpf\] T = \_[G ]{} x\^N \_[{S}’]{} \_[<i j>]{} ( \_[=1]{}\^[p]{} ( {S\_i\^S\_j\^-1 ) ) , where the first summation is over graphs, with $N$ being the number of vertices in the graph. The product is over the nearest neighbour pairs on the graph $G$ (referred to as “links”) and the second summation is over the following set of spin configurations. Take the graph $G$ and decompose it into a set of one-particle irreducible (“1PI”) graphs (which we shall call “blobs”) connected in a tree-like fashion (see fig. \[fig:tree\]); note that these blobs are essentially just minimum necked baby universes (“minbu”s). The blobs are fully magnetized, that is, all the vertices within a 1PI blob have the same set of spins, and we will sum over the $2^p$ possible spin configurations for each blob (except the root blob for which all the spins will be fixed to be $+1$). (100,55)(0,0) In the next sections we will solve this model exactly and also calculate various correlation functions, but in order to do this we need first to define how distances are to be measured on the graphs. One possible measure of the distance between two vertices $A$ and $B$, is the geodesic distance between them. That is, find the shortest path from $A$ to $B$ along links, counting a distance of one unit for every link traversed—this gives the geodesic distance between the two points. Unfortunately, this is quite difficult to deal with analytically, although some results are known for the pure gravity case [@KKMW93]. In this paper we use a slightly different definition of distance, which is much simpler to handle. The distance between two vertices will be defined as the shortest path between them along links, counting a distance of one unit only for links that separate two 1PI blobs. Since we are interested in correlations between the spins in the root blob, which are held fixed and spins further away, distances will be measured from the root blob. Thus all the vertices in the root blob have a distance zero, all the vertices within blobs connected by a single link to the root blob are at distance one and so on (see fig. \[fig:tree\]). It will turn out that the average number of vertices in each blob is very small, so that for many graphs this definition of distance will be quite similar to the geodesic distance, especially in the branched polymer phase for which the partition function is dominated by tree-like graphs. The advantage of measuring distances in this way is that it makes it very easy to define a transfer function, $f(y)$, \[eqn:f\] f(y)= ( x (1- y)\^[-]{} ) + y\^2, where $\lambda = 2 x \left(1+ e^{-2 \beta} \right)^p$ and (x) = \_[G ]{} x\^N . This summation is over the set $\gtwo$ of rooted planar 1PI graphs. This function, $f(y)$, takes a rooted 1PI blob and glues an arbitrary number of trees, each weighted with a factor of $y$, on to the blob. Note that each time we glue a tree on to a link in the blob, we pick up a factor of $x$, for the new vertex that is created, a factor of two because we can hang the tree in one of two directions and a factor of $(1+e^{\-2 \beta})^p$ to take account of the $2^p$ different ways the spins on the blob and those on the tree’s root blob can differ—this accounts for the factor of $\lambda$ multiplying each $y$. When we sum over all possible rooted 1PI blobs and all possible ways of gluing trees to links this gives the first term in (\[eqn:f\]). More specifically, the blob without any trees attached has a weight of $\ztwo(x)$. Consider a rooted graph with $N$ vertices, it has $L =\half (3N-1)$ internal links and we can add an arbitrary number of trees to each link, giving a total contribution of $x^N (1 + \lambda y + (\lambda y)^2 + \cdots )^L$ for the graph. When we sum over all 1PI graphs this gives a contribution of \_[G ]{} x\^N (1- y)\^[-L]{} = ( x (1- y)\^[-]{} ) . The second term in (\[eqn:f\]) comes from the special case in which the root blob only consists of a single vertex, in which case we get a factor of $x$ for the vertex, $(1+ e^{-2 \beta})^{2p}$ for the possible ways the spins may differ and $y^2$ for the two trees. (100,25)(0,0) The partition function, $T$, satisfies $T=f(T)$ and this is represented diagrammatically in fig. \[fig:tft\]; the shaded circle represents the sum over 1PI graphs, $\ztwo(x)$. For fixed $p$ and $\beta$, this equation gives $T$ as a function of $x$. By finding the closest singularity to the origin, which occurs at $x_c(\beta,p)$, say, we can determine the free energy for the model. For what follows it will be convenient to define $x' = x(1- \lambda T)^{-\frac{3}{2}}$. Now, there are essentially two different types of singularity. One occurs when $x'$ equals $\xctwo$, the critical value of $x'$ for $\ztwo(x')$ and this will correspond to the magnetized phase. The other occurs when the graphs become tree-like and we shall come back to this case later. $x_c$ in the magnetized phase ----------------------------- Consider the first case, for which $x'=\xctwo$ and hence $\ztwo(x')=\ztwo_c$, then writing $T(x_c)$ as $T_c$, T\_c = \_c + T\_c\^2 and $\xctwo = x_c (1- \lambda_c T_c)^{-\frac{3}{2}}$, where $\lambda_c = 2 x_c (1+e^{-2 \beta})^p$. Defining, h ()\^= , we get \[eqn:quadhh\] 1- h\^2 = (1 + e\^[-2 ]{})\^p . However, $\ztwo(x)$ is a known function, from Brézin [et al]{} [@BIPZ] one can show that \[eqn:model2z\] (x)= (1-3 ), where $x^2= \tau (1-2 \tau)^2$. This gives $\tau_c =\frac{1}{6}$, $\xctwo=\sqrt{\frac{2}{27}}$ and $\ztwo_c = \frac{1}{4}\sqrt{\frac{3}{2}}$. Hence we can solve (\[eqn:quadhh\]) to get $x_c=\xctwo h^3$ with \[eqn:xcmag\] h\^2 = , where we have defined $d \equiv \left( 1 + e^{-2 \beta} \right)^p$. This gives $x_c$ in the magnetized phase—we shall prove that it is magnetized later. Number of blobs and $\gst$ {#sec:blobgam} -------------------------- Next we will calculate $\nmean_r$, the average number of 1PI blobs at distance $r$. Define $G_r =f^{(r)}(v T)$, where the notation means $f(f(f(...f(v T))))$; this weights each blob at distance $r$ with an extra factor of $v$. Note that $v T$ can be regarded as the standard partition function, but with the root blob weighted by an extra factor of $v$. Applying the function $f$ to this gives $f(v T)$, where now each blob at distance one is weighted with the extra factor of $v$. Each application of $f$ just moves the weights of $v$ down the tree, by one unit of distance. Thus, \_r = \_[v=1]{} = . \_[v=1]{} , where we have used that $G_r(v \! = \! 1)=T$. However, $G_r=f(G_{r-1})$ and thus $$\begin{aligned} \nmean_r &=& \frac{1}{T} \left[ \left. \frac{\partial f}{\partial y} \right\vert_{y=G_{r-1}} \frac{\partial G_{r-1}}{\partial v} \right]_{v=1} \\ &=& \left.\frac{\partial f}{\partial y} \right\vert_{y=T} \nmean_{r-1} \label{eqn:expo} = \left[ \left.\frac{\partial f}{\partial y}\right\vert_{y=T} \right]^r ,\end{aligned}$$ since $\nmean_0=1$ (there is only one root blob). It will be convenient to define, B(x, , p) = . \_[y=T]{}. Using the formula (\[eqn:f\]) for $f(y)$, one can calculate $B$, \[eqn:b\] B=. \_T = ( 3 (x’) -1 ) + 2 x ( 1+e\^[-2 ]{} )\^[2p]{} T, where \[eqn:model2m\] (x) = . Note that $\mtwo(\xctwo)=\frac{5}{3}$. Evaluating at $x_c$ will give $B_c(\beta,p)$. In the magnetized phase this gives us, \[eqn:bc\] B\_c= 1 + ( d -1 ) - . Thus at $x_c$, $\nmean_r=(B_c)^r$ and the average total number of blobs is = . For large enough $\beta$, $B_c$ is less than one and positive, that is, the number of blobs decreases exponentially with distance. As $\beta$ is reduced, $B_c$ increases (since $\lambda$ increases and this encourages the tree to branch), until $B_c=1$ at some critical value, $\beta^$; this corresponds to the boundary of the tree-like region. Using $T=f(T)$, one can show that, = . For $\beta > \beta^$, $0<B_c<1$ and $\left. \frac{\partial T}{\partial x} \right\vert_{x_c}$ is finite, whilst for $\beta \le \beta^$, $\frac{\partial T}{\partial x}$ diverges as $x \to x_c$. That is, $B_c=1$ throughout the entire tree-like region and $\gst>0$ in this region, since $\frac{\partial T}{\partial x} \sim (x_c -x)^{-\gst}$. One can easily calculate $\gst$ in the various regions of the phase diagram. In the tree-like region and on the boundary, $\frac{\partial T}{\partial x} \sim \frac{1}{1-B}$, hence $1-B \sim {(\Delta x)}^{\gst}$, where $\Delta x \equiv x_c -x$. However, we also have an expression (\[eqn:b\]) for $B$ and this gives $1-B \sim {(\Delta x)}^{1-\gst}$ inside the tree phase. Hence, $\gst=\half$ here as expected. On the boundary $1-B \sim {(\Delta x)}^{\half (1-\gst)}$ giving $\gst^ =\third$. In the magnetized phase $\frac{\partial^2 T}{\partial x^2} \sim \frac{\partial^2 \ztwo(x')}{\partial {x'}^2} \sim (\Delta x')^{-\half} \sim (\Delta x)^{-\half}$, giving $\gst=-\half$ as expected. Equation (\[eqn:bc\]) applies in the magnetized phase and also on the boundary between the two phases. At $\beta^$, $B_c=1$ and this gives using (\[eqn:bc\]), \^\= - ( 2\^ -1 ). Note that this gives an estimate for the location, in the full model, of the transition between the tree and magnetized phases; compare this with the estimated location of the transition from the tree-like to the unmagnetized (pure gravity) phase, given in [@paper3]. $x_c$ in the tree phase ----------------------- Next we will calculate $x_c$ in the tree phase. At $x_c$ we have from $T_c=f(T_c)$, \[eqn:a1\] T\_c = (x’) + x\_c (1+ e\^[-2 ]{} )\^[2p]{} T\_c\^2 and from $B_c=1$, using (\[eqn:b\]), \[eqn:b1\] ( 3 (x’) -1 ) + 2 x\_c ( 1+e\^[-2 ]{} )\^[2p]{} T\_c = 1 , where $x'=x_c (1- \lambda_c T_c)^{-\frac{3}{2}}$. Remembering that $d=\left( 1+ e^{-2 \beta} \right)^p$ and defining $X \equiv 1 - \lambda_c T_c$, (\[eqn:b1\]) gives (3-1) x’ X d + 2 d\^2 T\_c x’ X\^ =1 and (\[eqn:a1\]) gives \[eqn:c1\] d\^2 T\_c\^2 x’ X\^ = T\_c - X\^ . Thus, ( 3-1 ) x’ X d +1 - X\^=0 . Moreover, T\_c = = and thus (3 -1 ) X(1-X) x’ d + 1-X - 4 x’ X\^2 d =0. Also (\[eqn:c1\]) gives \[eqn:d1\] 4x’ X\^2 d = (1-X) . Combining these last two equations gives X d + d-1=0. Using (\[eqn:model2z\]) and (\[eqn:model2m\]), X= and substituting into (\[eqn:d1\]) gives = \^[-1]{} and thus $x_c=x' X^{\frac{3}{2}}$ is given by \[eqn:xctree\] x\_c = d\^[-]{} in the tree phase. From equations (\[eqn:xctree\]) and (\[eqn:xcmag\]) one can easily show that there is a third order phase transition with a finite discontinuity, that is, the critical exponent $\alpha=-1$. Number of vertices ------------------ One can also calculate the average number of vertices at distance $r$. Define $G'_r =f^{(r)}\left( f(T)\vert_{xz} \right)$, where the notation $f(y)\vert_{xz}$ means the function $f(y)$, but with $x$ replaced everywhere by $xz$. Thus $f(T) \vert_{xz}$ is just a tree, but with each vertex in the root blob weighted by an extra factor of $z$. Applying $f$, $r$ times, just pushes these weights down the tree to a distance $r$, so that in $G'_r$ all the vertices at distance $r$ have an extra weight of $z$. Thus the average number of vertices at distance $r$ is given by $$\begin{aligned} \Nmean_r &=& \left[ \frac{z}{G'_r} \frac{\partial G'_r}{\partial z} \right]_{z=1} = \frac{1}{T}\left[ \left. \frac{\partial f}{\partial y} \right\vert_{y=G'_{r-1}} \frac{\partial G'_{r-1}}{\partial z} \right]_{z=1} \\ \label{eqn:nmeanr} &=& B^r \Nmean_0 \end{aligned}$$ and this is essentially just the (geometric) two-point function (see section \[sec:dh\]). The average total number of vertices is = , where $\Nmean_0$ is the average number of vertices in the root blob. This is given by \_0 = B + (x’) - x d\^2 T ( (x’) +1 ). In the magnetized phase this gives, at $x_c$, \_0 = + (d-1) + and in the tree phase, \_0 = . Note that $d \equiv (1+e^{-2 \beta})^p$, with $1<d<2$ in the magnetized phase and $d > 2$ in the tree phase. The formulae also show that the average number of vertices in a blob is very small, for any values of $\beta$ and $p$; in fact $\Nmean_0=2$ at $\beta^$. Critical exponents from the scaling relations {#sec:toycrit} ============================================= From equation (\[eqn:expo\]) we see that there is an exponential decay of the number of blobs with distance. One can define a mass, $m$, in the model through, \[eqn:a2\] \_r = B\^r = e\^[-m r]{}. The mass is a function of $\beta$, $p$ and $x$. At $x_c$, as the critical line is approached from the magnetized phase, the mass vanishes (i.e. there is a correlation length equal to $1/m$, which diverges). Note that at $x_c$, $m=0$ throughout the whole tree phase. Let us use this definition of the mass and various scaling relations to calculate the critical exponents. Later we will calculate the magnetic exponents directly from the model, for the case $p=1$, gaining the same results for $\beta_m$ and $\delta_m$; this will show that our definition of the mass is reasonable and that the scaling relations hold for this toy model. First let us consider the geometric exponents; note that we have already calculated $\gst$ in section \[sec:blobgam\]. The exponent $\nu$ is defined by $m \sim (\Delta x)^\nu$ for $\Delta x \to 0$ (where $\Delta x \equiv x_c-x$), but $m=-\ln B$ so that $ m \sim 1- B \sim {(\Delta x)}^{\gst}$ and hence $\nu=\gst$; in the tree phase $\nu=\gst=\half$ and on the boundary $\nu^=\gst^=\third$. Since there are no power law corrections to (\[eqn:a2\]) we have $\eta=1$; note that at small $\Delta x$, for $1 \ll r \ll 1/m$, one might expect [@AmbWat95] that $\nmean_r \sim r^{1-\eta}$. These sets of exponents satisfy Fisher’s scaling relation $\gst = \nu (2- \eta)$. In reference [@AmbWat95] it is shown that if the two-point function has associated with it a vanishing mass, then for a suitable definition of the Hausdorff dimension, $\nu d_H =1$. This gives that $d_H=2$ in the tree phase, as we might expect for branched polymers and $d_H^=3$ on the critical line. Consider now the magnetic exponents, which we will write with a subscript $m$ to avoid confusion. Evaluating at $x_c$ and letting $\beta \to \beta^$ from the magnetized phase, we have, defining $\Delta \beta \equiv \beta - \beta^$, $m \sim \Delta \beta$ from (\[eqn:bc\]) and hence $\nu_m=1$. If we take $\eta_m=1$ (we shall see later that the spin-spin correlation function has no power law corrections either), then using $\gamma_m = \nu_m (2- \eta_m)$ gives $\gamma_m=1$. Applying the other scaling relations $2- \alpha= \nu_m d_H^$, $\beta_m \delta_m = \beta_m + \gamma_m$ and $\alpha +2 \beta_m + \gamma_m =2$, yields $\alpha = -1$, $\beta_m=1$ and $\delta_m=2$. The various exponents are listed in table \[tab:exp\]. Phase $\gst$ $\nu$ $\eta$ $d_H$ ------------ ---------- ---------- -------- ------- Magnetized $-\half$ (1) Critical $\third$ $\third$ 1 3 Tree $\half$ $\half$ 1 2 : Critical exponents[]{data-label="tab:exp"} [c]{}\ $\alpha$ $\beta_m$ $\gamma_m$ $\delta_m$ $\nu_m$ $\eta_m$ ---------- ----------- ------------ ------------ --------- ---------- $-1$ 1 1 2 1 1 : Critical exponents[]{data-label="tab:exp"} Before proceeding to check the magnetic exponents explicitly, we shall define $d_H$ and calculate it directly from the model. Hausdorff dimension {#sec:dh} =================== As before in [@AmbWat95] we define the Hausdorff dimension, $d_H$, in terms of the two-point function T\_2(r) = \_[ G ]{} x\^N W\_G, where $\gmark$ is the set of planar 3 graphs, with two marked points that are separated by a distance $r$. One of the marked points will be taken to be the vertex, in the root blob, which is connected to the external leg. $W_G$ is just the usual weight for the spin configurations, which appears in (\[eqn:gcpf\]). Then $d_H$ is defined, in the continuum limit, by N(r) \~r\^[d\_H]{}, r , m(x) r = [const.]{}, where N(r) \_[G ]{} N x\^N W\_G = x . That is, by tuning $x$ to $x_c$ we are taking a continuum limit; however this definition of $d_H$ only really makes sense in the tree phase and on the boundary, where the mass vanishes at $x_c$. Now $T_2(r)$ is just T\_2(r) = \_[G ]{} x\^N W\_G N\_r = T \_r = T B\^r \_0, where $N_r$ is the number of vertices at distance $r$ for graph $G$; N(r) = x . As $r \to \infty$, N(r) \~r \~r ( - x’ )\^[-]{} (x )\^[-]{}. Since we are taking the continuum limit with $m r$ fixed, $r \sim m^{-1} \sim (\Delta x)^{-\gst}$, N(r) \~r\^2 ( - x’ )\^[-]{}. In the tree phase $N(r) \sim r^2$ giving $d_H=2$ and on the critical line $N(r) \sim r^2 (\Delta x')^{-\half} \sim r^2 (\Delta x)^{- \third} \sim r^3$, so that $d_H^ =3$, as expected. Unfortunately, due to the way distances and the Hausdorff dimension have been defined in this model, $d_H$ is not well-defined in the magnetized phase. In reference [@AmbWat95] it is shown, using the geodesic distance, that $d_H=4$ for pure gravity (also the geometric exponents are $\nu=1/4$ and $\eta=4$) and this may well be true in the whole of the magnetized phase. It is certainly the correct value at $p=0$, where the toy model reduces to a pure gravity model. Thus our definition of distance appears to correctly capture the behaviour of the model in the tree phase and on the critical line, where $\gst>0$, but not within the magnetized phase. Spin-spin correlation function {#sec:spinspin} ============================== In this section we will calculate $\smean_r$ the average total spin at distance $r$; the summation is over all vertices at distance $r$ and all spin flavours on those vertices. Since the spins in the root blob are fixed to be $+1$, this is essentially a spin-spin correlation function, for spins separated by a distance $r$. To calculate this we will add a magnetic field for vertices at distance $r$. First we will solve the model with $p=1$ and then the general case. Let us define f\_+(y) = ( x e\^H ( 1 - y e\^H )\^[-]{} ) + x (1+e\^[-2 ]{})\^2 y\^2 e\^H, with $\lambda = 2 x (1+e^{-2 \beta})$. This is just $f(y)$, but with $x$ replaced by $x e^H$. The function takes a number of trees with weights $y$ and glues them on to a blob, where each vertex of the blob has a spin of $+1$ on it and there is a magnetic field $+H$ applied on these spins. The function $f_-(y)$ is defined in the same way, but with $H$ replaced by $-H$ throughout. Let $T^+_r$ be a tree with positive spins in the root blob and a magnetic field of $+H$ applied at distance $r$ from the root. Similarly $T^-_r$ has spins of $-1$ in the root blob. Then $T^+_0 =f_+(T)$ and $T^-_0=f_-(T)$. Define, F(y)= ( x (1-2xy)\^[-]{} ) + x y\^2 . Then $$\begin{aligned} T^+_r &=& F(T^+_{r-1} + e^{-2 \beta} T^-_{r-1}) \\ T^-_r &=& F(T^-_{r-1} + e^{-2 \beta} T^+_{r-1}) ,\end{aligned}$$ that is, to make a graph with spins of $+1$ on the root blob and a magnetic field at distance $r$, one takes a number of trees with magnetic fields at distance $r-1$ and glues them on to a blob with positive spins, picking up a factor of $e^{-2 \beta}$ if the spins on the root of the tree are negative. Now, \_r\^+ = \_[H=0]{} = . \_[H=0]{} ; the superscript on $\smean_r^+$ denotes the fact that the vertices in the root blob have a single spin each, which is fixed to be $+1$. = . \_[y=T\_[r-1]{}\^+ + e\^[-2 ]{} T\_[r-1]{}\^-]{} \_r\^+ = . \_[y=(1 + e\^[-2 ]{}) T]{} but $\smean_{r-1}^- = - \smean_{r-1}^+$ and $F(y)=f(2xy/\lambda)$ so that . \_[y=(1 + e\^[-2 ]{}) T]{} = . Thus defining $t\equiv \tanh \beta$, \_r\^+ = B t \_[r-1]{}\^+ = (B t)\^r \_0\^+ =(Bt)\^r \_0. Let us consider the general case of $p$ spin flavours for which $\lambda = 2 x \left(1 + e^{-2 \beta} \right)^p$, $$T_r^{++ \cdots +}=F\Bigl(T_{r-1}^{+\cdots+} + e^{-2 \beta} \left( T_{r-1}^{+\cdots+-} + T_{r-1}^{+\cdots+-+} + \cdots \right) +$$ e\^[-4 ]{} ( T\_[r-1]{}\^[++–]{} + ) + + e\^[-2 p]{} T\_[r-1]{}\^[--]{} ) \_r\^[+++]{} = . However one can easily show that, \_r\^ = (p-2n) \_r\^[+++]{}, where there are $p \! - \! n$ plus signs and $n$ minus signs in the superscript on the left-hand side. Thus, $$\begin{aligned} \smean_r^{++\cdots+} &=& \frac{B}{\left(1+e^{-2 \beta} \right)^p} \smean_{r-1}^{+\cdots+} \sum_{n=0}^p e^{-2 \beta n} \frac{1}{p} (p-2n) \left( p \atop n \right) \\ &=& (Bt) \smean_{r-1}^{++\cdots+} = (Bt)^r p \Nmean_0 .\end{aligned}$$ Now defining the magnetization at distance $r$ by \_r , we have that ${\cal M}_r = t^r$. It should be noted that if one uses the exponential decay of $\smean_r^{+ \cdots +}$ or ${\cal M}_r$ to define a spin-spin correlation length, then this quantity will not diverge at the phase transition; this behaviour should be contrasted with that of the geometric correlation length, which does diverge. If we define the total magnetization ${\cal M}$, for the case in which all the spins in the root blob are positive, by \[eqn:mgce\] [M]{} , where $\smean^{+\cdots+}$ is the average of the total spin, which is a sum over all vertices and all spin flavours, then we have, at $x_c$, = . Note that ${\cal M}=1$ for $\beta = \infty$, and that ${\cal M}=0$ throughout the tree phase, showing that this phase is unmagnetized. The other phase has $0<B_c<1$ and hence $0<{\cal M}\le 1$; this is the magnetized phase as claimed earlier. Also at $x_c$, near $\beta^$, $1-B_c \sim \Delta \beta$, where $\Delta \beta \equiv \beta - \beta^$, so that ${\cal M} \sim \Delta \beta$ and thus $\beta_m=1$, as calculated from the scaling relations. In fact for finite $p$, = + O(()\^2), so that the coefficient in front of $\Delta \beta$ is non-zero in general (note that $p=0$ would give $\beta^=-\infty$). Toy model, with a magnetic field {#sec:toymag} ================================ In the previous section we defined the magnetization in the grand canonical ensemble by (\[eqn:mgce\]). A different definition is possible namely, \_[ce]{} = \_[N ]{} \_N, where now we are working in the canonical ensemble, that is, we are using the set, $\gone(N)$ of rooted $N$-vertex graphs. In this section we will add a magnetic field, $H$, to the toy model. However, we shall only consider the case $p=1$, as $p>1$ appears to be more difficult to solve. It will be shown that the magnetic exponents $\beta_m$ and $\delta_m$ are the same for both definitions of the magnetization and agree with the results from the scaling relations. Rather than use the transfer function $f(y)$, which glues trees on to a 1PI blob, we shall use a different transfer function $\fbar(y)$, that glues trees on to a domain; this is more convenient for the $p=1$ model with a magnetic field. For $H=0$, it will give a sum over the same set of graphs that we had previously, with the same weights, but we will no longer be able to keep track of distances within the graphs. Consider first the case $H=0$, the function $\fbar(y)$ is given by \[eqn:fbar\] (y)= ( x (1-2x K )\^[-]{} ) + x K\^2 , where K= ( 1 - ) , with $\mu = 4 x e^{-2\beta}$ and (x)=\_[G ]{} x\^N . The function defined in (\[eqn:fbar\]) takes a rooted 3 graph, which will form the core of a domain, and glues on trees weighted with factors of $K$. As before one picks up a factor of $2x$ for each tree glued on. In this case however we allow trees to be glued on to the root link, which changes the power of the factor in front of $\zone$ compared with that in (\[eqn:f\]). This is necessary to make sure that we correctly sum over all possible domain structures. The factor $K$ is the solution of K= e\^[-2]{} y + x K\^2 . This generates tree graphs whose vertices are weighted with $x$ and which have a factor of $e^{-2\beta} y$ at the end of each branch. The domain corresponds to the core 3 graph plus all the vertices in $K$. The ends of the branches in $K$, which we have weighted with $e^{-2\beta} y$, correspond to the domain boundaries; setting $y=T$ will give us a set of domains glued together in a tree-like fashion. Thus we have (y) = ( x ( 1 - y )\^[- ]{} ) + ( 1 - )\^2 and the grand canonical partition function (\[eqn:gcpf\]) is the solution of $T= \fbar(T)$. This is shown diagrammatically in fig. \[fig:domain\]. (100,55)(0,0) Note that this is just a different way of formulating the same model that we solved previously. One can easily use $T=\fbar(T)$ to calculate $x_c$ in the two phases, gaining the same results as before (to solve the $p$-flavour case one just uses $\mu=4x(d-1)$). The advantage of this new formulation is that as we use $\fbar$ to move down the tree from one domain to the next, the sign of the spins alternates (at least for $p=1$). This simplifies greatly the task of adding a magnetic field. For the original version of the model it is quite difficult to keep track of the spins on the different blobs. Now let us add a magnetic field of $+H$, the transfer function which makes a domain with spins of $+1$ is \[eqn:fbarplus\] \_+(y) = ( x e\^H ( 1 - y e\^H )\^[- ]{} ) + ( 1 - )\^2 . Each spin in the domain is given an extra weight of $e^H$. It will be convenient to define $x_+'=x e^H \left( 1 - \mu y e^H \right)^{- \frac{3}{4}}$. Similarly $\fbar_-(y)$, which makes a domain with negative spins is (\[eqn:fbarplus\]), but with $H$ replaced everywhere by $-H$. Let $T^+$ be the partition function for graphs with positive spins in the root domain and $T^-$ that for those with a negative root domain. Then $T^+=\fbar_+(T^-)$ and $T^-=\fbar_-(T^+)$. These equations determine $T^+(x,\beta,H)$. For $H>0$ in the magnetized phase, the critical value of $x$ is determined by $x_+' = \xcone$, where $\xcone$ is the critical value for $\zone(x_+')$. Defining ( )\^, we have T\^-\_c = ( 1 - \^4 ) and T\^+\_c = \^[-1]{} \_c + ( 1 - \^2 )\^2 . Note that $\xcone=1/(2.3^{\frac{3}{4}})$, $\zone_c=3^\frac{3}{4} \left( 1 - \frac{\sqrt{3}}{2} \right)$ and $\mone_c = 1/(\sqrt{3} -\frac{3}{2}) -1 $ from [@BIPZ]. However we still have $T^-=\fbar_-(T^+)$, and using this gives an equation determining $\hbar$. Putting X 1 - \_c e\^[-H]{} T\^+\_c = 1 - e\^[-2]{} e\^[-2H]{} ( 1 - \^2 + \^4 ) and $x'=x_c e^{-H} X^{-\frac{3}{4}}$, then we have \[eqn:hbar\] e\^[2 ]{} (1- \^4) = e\^[2H]{} . This gives $\hbar(\beta,H)$ and hence $x_c(\beta,H)$. Now, (,H) = - = 1 - . At $H \to 0$ we get by differentiating (\[eqn:hbar\]), (,H 0\^+) = 1 - , with \^2 = ; note that at $H=0$, $\hbar^2 = 2 h^2 / \sqrt{3}$, see equation (\[eqn:xcmag\]). At $\beta$ close to $\beta^=0$, (, H 0\^+ ) = 2 + O(()\^2). Hence, $\beta_m =1$ as previously, but note that $\mce$ is not equal to ${\cal M}$. Now consider $\delta_m$, defined by $\mce(\beta^, \Delta H) \sim (\Delta H)^{\frac{1}{\delta_m}}$. From (\[eqn:hbar\]), one can show that $\mce$ is a certain function of $\hbar$, $e^{-2 \beta}$, $e^{2 H}$, $x'$, $\zone(x')$, $\mone(x')$ and $X$, where (x’) . At $\beta^$ and small $\Delta H$, barring accidental cancellations, then all these functions are a constant plus terms of order $\Delta H$ (or smaller), except for $\mone(x')$, (x’) = \_c + (- x’ )\^+ . Since $x' = \xcone + O(\Delta H)$ then $\mone(x') = \mone_c + O((\Delta H)^\half)$, giving $\delta_m =2$. Note that the value of $\delta_m$ depends crucially on the fact that $\gst=-\half$ in the magnetized phase. Suppose that we next try to calculate $\gamma_m$, defined by (,H=0) \~()\^[-\_m]{}, then we find that $\chi \sim \frac{\partial^2 x_c}{\partial H^2}$ and that this contains terms such as $\frac{\partial^2 \zone(x')}{\partial {x'}^2}$, which diverge as we take $H$ to zero. Thus $\gamma_m$ seems not to be well-defined, even though the scaling relations give $\gamma_m=1$. Consider now the magnetization in the grand canonical ensemble, ${\cal M}$, defined by (\[eqn:mgce\]), $\beta_m$ has already been calculated and this just leaves $\delta_m$. We see that ${\cal M} \to 0$ at $\beta^$, due to the divergence of $\Nmean$, (\^\,H) \~. \_[\^\]{} . For graphs with a positive root domain, at $x_c$, \^+ = . \_[x\_c]{} , where $T^+=\fbar_+(\fbar_-(T^+))$. This gives = . \_[T\^-]{} + . \_[T\^-]{} . \_[T\^+]{} . Thus (\^\, H) \~1- . \_[T\^-\_c]{} . \_[T\^+\_c]{} and again this is a function of various variables all of which are a constant plus $O(\Delta H)$, except that $\mone(x')=\mone_c + O((\Delta H)^\half)$, giving $\delta_m=2$. This completes our analysis of the toy model; we have derived all the critical exponents from the model (except for $\gamma_m$) and shown that all the usual scaling relations hold. Now we shall turn our attention to the full model of $p$-Ising spins coupled to 2d gravity, and will find that many of the considerations in the previous sections apply for the transition between the tree and magnetized phases in this model. Full model {#sec:full} ========== Definition ---------- Consider the full model where we have $p$ independent Ising spins on each vertex and are summing over all rooted planar 3 graphs and all spin configurations. Then the partition function is, with $t=\tanh \beta$, \[eqn:full\] (x,) = \_[G ]{} x\^N \^p , where we have divided out various factors of two and $\cosh \beta$ in order to simplify the formulae later on. $N$ is the number of vertices in graph $G$. Note that there is no vertex and are no spins at the end of the root, so that the product over links does not include the root link. In fact we shall consider a generalization of this model that has $p$ coupling constants. Expanding out the above equation gives, with the extra coupling constants, \[eqn:general\] (x,{}) = \_[G ]{} x\^N \_[{S}]{} \_[<ij>]{} ,$$ where $S_i^\alpha$ is the flavour $\alpha$ spin on vertex $i$, the second summation is over all spin configurations and $\{\lambda\}$ is the set of coupling constants $\{\lambda_1,\lambda_2,\cdots,\lambda_p\}$, with $ 0 \le \lambda_j \le 1$ for all $j$. Of course by setting $\lambda_j =t^j$ one recovers the ordinary model (\[eqn:full\]). Alternatively by setting $\lambda_j=0$ for $j>1$ we recover the O(n) models studied in [@onmodels; @charlotte] with $n=p$. To save writing we define $T \equiv \zone(x,\{\lambda\})$. We will show that $T$ satisfies $T=f(T)$ where \[eqn:fullf\] f(y)= (x’,{’}) + x y\^2 , with \[eqn:fullmap\] x’=x(1-2xy)\^[-]{} , \_j’ =\_j ; $\ztwo(x',\{\lambda'\})$ is defined as in (\[eqn:general\]), but using the set of rooted 1PI graphs, $\gtwo$. In a similar fashion to (\[eqn:f\]) the function $f(y)$ takes a 1PI graph and glues trees on to it, but this time as well as $x$ being changed to $x'$, the coupling constants are also renormalized. Note that a similar renormalization is studied in [@Dur94]. To justify the above equations let us first consider the case $p=1$. For each link we have a factor such as $(1+\lambda_1 S_i S_j)$ and when we multiply these factors together, the sum over spins will cause any terms containing odd powers of a given spin to vanish. The only non-vanishing terms correspond to sets of closed loops; we shall refer to such a non-vanishing term as a loop configuration. Thus for each graph $G$, we sum over all ways of drawing sets of closed non-intersecting, non-backtracking loops on the graph (call this set of loop configurations $\setloop$). So, (x,\_1) = \_[G ]{} x\^N \_ \_1\^l , where $N$ is the number of vertices in graph $G$, and $l$ is the number of links making up the loops in the loop configuration $\lcal$. As before, we will make the graphs $G$ (where $G \in \gone$) by gluing together 1PI blobs (fig. \[fig:tree\]). Note that because of the tree-like structure, any given closed loop is wholly contained within a single blob. This means that the partition function for a given graph $G$, factorizes into a product of contributions from the individual blobs. To calculate the transfer function, $f(y)$, we take a rooted 1PI blob, summing over all graphs and loop configurations, as well as over all ways of attaching trees to the blob. If no trees were attached, one would just have $\ztwo(x,\lambda_1)$, which is (x,\_1) = \_[G ]{} x\^N \_[{S}]{} \_[<ij>]{} = \_[G ]{} x\^N \_ \_1\^l 1\^[L-l]{}, where $L$ is the number of internal links in graph $G$ (i.e. excluding the root link); that is, $L=\half (3N-1)$. Attaching a tree weighted with $y$, gives a factor of $2xy$ (there is an extra vertex and there are two ways of hanging the tree off the link). For a given $G$ and $\lcal$, each link either contributes $\lambda_1$ if it is part of a closed loop or $1$ if it is not. Attaching an arbitrary number of trees to an internal link that contributed $1$ changes this contribution to $(1-2xy)^{-1}$; however for those that contributed $\lambda_1$ we get $\lambda_1 (1-2xy \lambda_1)^{-1}$, since adding a vertex on a link which was part of a closed loop, increases the length of that loop by one and hence gives an extra factor of $\lambda_1$. Thus summing over all ways of attaching trees causes the change: (1+\_1 S\_i S\_j) ( 1 + S\_i S\_j ) and hence (x,\_1) (x’,\_1’). The extra term in (\[eqn:fullf\]) comes from the case in which the blob is just a single vertex. The general case for $p>1$ follows with only minor modifications to the argument. Each closed loop is labelled with a spin flavour (an integer $\alpha$, with $1\le \alpha \le p$) and loops with different spin flavours can intersect. Thus for a given graph $G$ and loop configuration $\lcal$, a given link will have $j$ spin flavours running through it ($0 \le j \le p$), giving a corresponding factor of $\lambda_j$. Adding an arbitrary number of trees to this link changes the contribution to $\lambda_j (1-2xy \lambda_j)^{-1}$. Hence, we get (\[eqn:fullf\]) and (\[eqn:fullmap\]) for the general case of a rooted 1PI blob with trees, each weighted by $y$, hanging off it. Setting $y=T$, we recover the partition function for graphs with $G \in \gone$, that is, $T=f(T)$; $f(y)$ is the transfer function for the full model. Exponential decay of blobs -------------------------- As before, we define distance to be the geodesic distance, but count only the links connecting 1PI blobs. Again one has $G_r = f^{(r)}(vT)$, giving the partition function for a model where all blobs at distance $r$ are weighted by $v$. The average number of 1PI blobs at distance $r$ is \_r = \_[v=1]{} = . \_[v=1]{} . Since $G_r=f(G_{r-1})$, \_r = . \_[y=T]{} .\_[v=1]{} = . \_T \_[r-1]{}, so that $\nmean_r=B^r$ with $B=\left. \frac{\partial f}{\partial y} \right\vert_T$ . Thus we still have an exponential decay of the average number of blobs with distance and \[eqn:fullB\] B= \_[y=T]{}, where $\ztwo=\ztwo(x',\{\lambda'\})$. One can also show that $$\begin{aligned} \left( \frac{\partial T}{\partial x} \right)_{\! \{\lambda\}} &=& \frac{1}{(1-B)} \left. \left( \frac{\partial f(y)}{\partial x} \right)_{\! \{\lambda\}} \right\vert_{y=T} \\ &=& \frac{T}{x(1-B)} \left[B-x T + (1-xT) \mtwo(x',\{\lambda'\}) \right] \label{eqn:dtdx} \\ & \sim & (x_c -x)^{-\gst},\end{aligned}$$ where (x’,{’}) ( )\_[ {’}]{}. Thus as in the previous case, at $x_c$, $B_c=1$ in the tree phase and on its boundary, where $\gst \ge 0$, and we have $0<B_c<1$ in the magnetized phase. Using equations (\[eqn:fullB\]) and (\[eqn:dtdx\]) one can show, barring accidental cancellations, that $\gst=\half$ in the tree phase and that on the boundary, $\gst^=\gstwo /(\gstwo -1 )$, which is just Durhuus’ formula [@Dur94]; where $\gstwo(\{\lambda'\})$, which is negative, is the value of the string susceptibility for $\ztwo(x',\{\lambda'\})$, that is, ( )\_[{’}]{} \~( x\_c’({’}) - x’ )\^[-(1 + ({’}))]{} . The spin-spin correlation function {#sec:fullspin} ---------------------------------- Next we consider the spin-spin correlation function. Rather than fixing all the spins in the root blob, as we did for the toy model, it is more convenient to add a set of $p$ external spins on the root, which are all fixed to be $+1$. The partition function is as in (\[eqn:general\]), but with an extra factor for the link connecting the external spins to the root blob, which we shall denote $W_{ext}$ (we will not add an extra factor of $x$). Thus, (x,{}) = \_[G ]{} x\^N \_[{S}]{} , where the weight $W_{ij}$ for link $< \! ij \! >$ is given by the square bracket in equation (\[eqn:general\]). Note that when the sum over spins is performed, the factor $W_{ext}$ just gives a contribution of one, so that the function $\zone$ is unchanged by adding the external spins. The average total spin at distance $r$ is \_r\^+ = \_[G ]{} x\^N \_[{S}]{} , where the last summation is over all spins, $k$, at distance $r$ and all spin flavours, $\alpha$. Consider $\zone \smean_r^+$, the only non-zero contributions come from the configurations in which there is a flavour $\alpha$ line from the external spins to vertex $k$, in addition to the usual sets of closed loops. Define $\gover_r$ by \_r\^+ = p \_[G ]{} x\^N \_[{S}]{} = p \_r , Then $\gover_r$ is a sum over all rooted graphs and all loop configurations that have a flavour $1$ line from the external spins to a vertex $k$ at distance $r$; the location of the vertex $k$ is also summed over. However, we can write $\gover_r$ in terms of $\gover_{r-1}$. This is shown diagrammatically in fig. \[fig:gr\], where links included in the flavour $1$ line are drawn thicker than those which are not included. The symbol on the left-hand side is used to represent $\gover_r$, it has a flavour $1$ line of length $r$, note that we are still measuring distances from the root blob (not from the external spins). (100,20)(0,0) That is, \[eqn:fullD\] \_r= \_[r-1]{}, where $ \ztwo_2(x,\{\lambda\})$, which is drawn as a shaded circle with a thick line passing through it, is the partition function for 1PI graphs with two distinct legs and a flavour $1$ line running through (it does not include factors of $\lambda_1$ on the two legs); $x'$ and $\{\lambda'\}$ are given by (\[eqn:fullmap\]) evaluated at $y=T$. Define $D$ to be equal to the square bracket above, then $\gover_r=D^r \gover_0$. So that we have an exponential decay with distance, \[eqn:fullmagdecay\] \_r\^+ = D\^r \_0 = D\^r \_0\^+. The total average spin is \^+=\^+\_0 . The magnetization in the grand canonical ensemble is given by \[eqn:fullmag\] [M]{} = = ( ) = [M]{}\_0 ( ), where ${\cal M}_0$ is the magnetization of the root blob ($0 \le {\cal M}_0 \le 1$). We have used the fact that the average number of vertices at distance $r$ is $\Nmean_r = B^r \Nmean_0$, so that $\Nmean = \Nmean_0/(1-B)$. This is easily shown by defining $G_r'=f^{(r)}(f(T)\vert_{xz})$ and following the derivation of (\[eqn:nmeanr\]). Note that, at $x_c$, in the tree phase $B_c=1$, so that ${\cal M}=0$ throughout this phase as one might expect. In the next section it is assumed that $D<1$, that is, that the spin-spin correlation length does not diverge; the easiest way to justify this would be to show that $D<B$, as we already know that $B\le 1$. The following calculation will make this assumption plausible. From (\[eqn:fullmagdecay\]) one clearly has $D \le B$, (at least provided that $\smean_0^+ \neq 0$), since $\smean_r^+ \le p \Nmean_r$. To improve this inequality we need an equation for $B$ which is simpler than (\[eqn:fullB\]). Define $G_r''$ to be the partition function for rooted graphs, with a marked vertex at distance $r$ — it is a sum over all graphs, $G \in \gone$, all loop configurations and all ways of marking a vertex at distance $r$; that is, $G_r''=T\Nmean_r$. Then following the derivation of (\[eqn:fullD\]) we have G\_r”=G\_[r-1]{}”=B G\_[r-1]{}” , where $\ztwo_t$ is the partition function for 1PI graphs, with two distinct legs — it is a sum over all such graphs and all loop configurations. Note, $\ztwo_t$ differs from $\ztwo_2$ in that there is no flavour 1 line running through it. Thus, $$\begin{aligned} B &=& \phantom{\lambda_1} \ \ 2xT + \ztwo_t\xld (1-2xT) \\ D &=& \lambda_1 \left[2xT + \ztwo_2\xld (1-2xT) \right] .\end{aligned}$$ Now, if one could show that $\ztwo_t\xld \ge \ztwo_2\xld$, then this would imply that $D \le \lambda_1 B$. Consider $\ztwo_{+-}\xld$, the partition function for 1PI graphs with two distinct legs, in which the flavour 1 spins on the two vertices connected to the legs are held fixed at $+1$ and $-1$ respectively; all other spins are summed over. Then one can easily see that $\ztwo_{+-}\xld = \ztwo_t\xld - \ztwo_2\xld$, since the only non-vanishing terms correspond either to sets of closed loops, or configurations in which there is also a flavour 1 line running between the two fixed spins; this last set of terms is multiplied by the product of the fixed spins, namely $-1$. For small enough values of $\{\lambda'\}$, $\ztwo_{+-}\xld$ is manifestly positive; the weights $W_{ij}$ for each link will be positive for any spin configuration (see equation (\[eqn:general\])), and hence $\ztwo_{+-}$ is just the sum of positive terms. At $\lambda_j'=1$ for all $j$, the system is completely magnetized and hence $\ztwo_{+-}=0$. One would expect that $\ztwo_{+-}\xld >0$ for other values of $\{\lambda'\}$. In which case $D \le \lambda_1 B$, with $D<B$ in general and $D=B$ only when $\lambda_j=\lambda_j'=1$ for all $j$. Thus in general $D<1$ and the spin-spin correlation length does not diverge. Note that our definition of distance is only good in the tree-like phase and on its boundary, and hence this result does not say anything about the possible divergence of the proper spin-spin correlation length (defined using the geodesic distance) at the magnetization transition, between the U and M phases (see figure \[fig:phase\]). Figure \[fig:phase\] shows a tentative phase diagram for the standard model defined by (\[eqn:full\]); T is the (unmagnetized) tree-like or branched polymer phase, U the unmagnetized phase and M the magnetized phase. See [@paper3; @paper2] for a discussion of various possible alternative phase diagrams. Each line of this phase diagram for constant $p$ (where $p$ is a non-negative integer) corresponds to a line through the $p$-dimensional phase space of the generalized model. Within the U and M phases the model behaves in a similar fashion to the pure gravity case and it seems probable that $\gst=-\half$ throughout both these phases. The transition between the magnetized (M) and unmagnetized (U) phases is caused by ${\cal M}_0$ vanishing as the coupling constants $\{\lambda\}$ are varied. From the KPZ results, we expect to have $-\half \le \gst^ \le 0$ along the corresponding critical line, at least for $p\le 2$. The tree-like phase has $\gst=\half$ throughout, the generic value for branched polymers. (100,77)(0,0) Critical exponents ------------------ Now consider the transition from the magnetized to the tree phase, which is caused by $B_c \to 1$. The behaviour of the full model at this transition is very similar to that of the toy model. Following the analysis in section \[sec:toycrit\], we define a mass by $m=-\ln B$. As before there is an exponential decay, so we will take $\eta=1$, which gives $\gst=\nu$ provided that the scaling relation $\gst=\nu(2-\eta)$ holds. Consider the magnetic exponents, at $x_c$ and fixed integer $p$ we vary the $\{\lambda\}$ in order to approach the phase transition; the variable $\Delta \lambda$ will be used to parameterize this, with $\Delta \lambda=0$ at the transition. Then by definition $m \sim (\Delta \lambda)^{\nu_m}$, but $m=-\ln B_c$ so that $B_c \sim 1 - (\Delta \lambda)^{\nu_m}$. Now $\beta_m$ is defined by ${\cal M} \sim (\Delta \lambda)^{\beta_m}$, but ${\cal M} \sim 1-B_c$ from (\[eqn:fullmag\]), thus $\beta_m=\nu_m$; note, we have assumed that we are not close to any point at which ${\cal M}_0$ vanishes ($P$ in fig. \[fig:phase\] is such a point). It has also been assumed that $D_c \neq 1$ at the transition, i.e. that the spin-spin correlation length does not diverge; this was justified in the previous section. If we assume that $\gamma_m=\nu_m(2-\eta_m)$ with $\eta_m=1$, then we have $\gamma_m=\nu_m=\beta_m$. Assuming the scaling relations $2-\alpha=\nu_m d_H^$ and $\alpha + 2 \beta_m + \gamma_m=2$, gives $d_H^=3$. The calculation in [@AmbWat95], which gave that $\nu d_H=1$, still applies, so that $\nu^=\third$ and hence $\gst^=\third$. This is encouraging as the analysis in [@Dur94] leads one to suppose that $\gst^=\third$ provided $\gst=-\half$ in the magnetized phase. Also, applying the relation $\beta_m \delta_m = \beta_m + \gamma_m$ gives $\delta_m=2$. Now $B$ is given by (\[eqn:fullB\]) and in the magnetized phase at $x_c$, $x'=\xctwo(\{\lambda'\})$. Provided that we are away from points at which ${\cal M}_0$ vanishes, $\ztwo(x',\{\lambda'\})$ should be analytic and Taylor expandable about $\{\lambda'^\}$ (the value of $\{\lambda'\}$ at the M to T phase transition). Assuming that $x_c(\{\lambda\}) T_c(\{\lambda\})$ is expandable in powers of $\Delta \lambda$ as far as $\Delta \lambda$, then barring accidental cancellations $B_c \sim 1 - \Delta \lambda$ and hence $\nu_m= \beta_m=\gamma_m=1$ and $\alpha=-1$. Thus on the assumption of various scaling relations and that our definitions of $m$ and ${\cal M}$ are adequate, and barring problems such as accidental cancellations, the set of exponents given in table \[tab:exp\] should also apply for the full model for the M to T transition, away from point $P$. Again it is not clear what the value of $d_H$ is, within the M phase; however one might guess that $d_H=4$, the pure gravity value [@AmbWat95; @CTBJ; @AJW], throughout the U and M phases (in which case it is probable that $\nu=1/4$ and $\eta=4$ here). Note that the mechanism by which the U to T transition occurs is essentially the same and so we would expect to have the same set of exponents, although the magnetic exponents are not relevant since ${\cal M}=0$ in both the U and T phases. Perhaps one should note that a low-temperature expansion matrix model calculation in the limit $c \to \infty$ [@Wexler] gives that $\gst^=\third$ at the transition between the tree and magnetized phases, and that the truncated model studied in [@paper3], which is supposed to approximate the U to T transition, has $\gst^=\third$ and $\alpha = -1$, agreeing with our predictions. In addition the q-state Potts model at large $q$ [@Wexlerq] , which is equivalent to the multiple Ising model in the limit $q=p=\infty$ and may be related to it for finite $q$, has a branched polymer region (with $\gst=\half$) separating two pure gravity regions (with $\gst=-\half$) and has $\gst^=\third$ and $\alpha=-1$ at both transitions. At point $P$, the above argument fails since ${\cal M}_0 \to 0$ and hence it is possible to have $\beta_m \neq \nu_m$; some of the other assumptions may break down here too. Thus at this point one might expect to get a different set of exponents. In any case the calculations in this section only apply for non-negative integer $p$ and it is not entirely clear whether $p^$, the value of $p$ at point $P$, is an integer. It is tempting to suppose that $p^=2$, in which case one could understand the breakdown of the KPZ formula in terms of the appearance of the branched polymer phase, however Monte Carlo simulations do not seem to support this hypothesis [@MC]. Conclusion {#sec:concl} ========== In this article we have studied correlation functions in two-dimensional quantum gravity coupled to Ising spins. For the toy model two approximations are used: (i) the spin configurations allowed are the ones dominant in the low temperature expansion of multiple Ising spins on dynamically triangulated surfaces and (ii) the distance between two spins is identified with the number of links separating 1PI subgraphs along a path connecting the spins. In the full model only the second approximation is made. We do not expect this last approximation to be important for values of $\beta$ where $\gamma_{str}(\beta) >0$, i.e. for values of $\beta$ where the average number of vertices of a generic $\phi^3$ graph diverges for $x \mbox{{\scriptsize $\nearrow$}} \,x_c(\beta)$, since the average number of vertices in an 1PI part of a generic $\phi^3$ graph in our ensemble is very small. However, if $\gamma_{str}(\beta) < 0$ the average number of vertices in a generic $\phi^3$ graph is itself small, even at the critical point $x_c(\beta)$ and our definition of distance can not be used to extract in a reliable way the fractal properties of the ensemble of $\phi^3$ graphs. Approximation (i) has been shown to be exact in the $c\to \infty$ limit [@Wexler] and numerical simulations [@MC] suggest that it is an excellent approximation except for the smallest values of $\beta$ if many Ising spins are coupled to two-dimensional gravity. For moderate values of $c$ it is difficult to judge if approximation (i) is reliable all the way down to $\beta^$. For $c=1/2$ and $c=1$ (i) not a good approximation for $\beta \in [0,\beta^]$. The toy model allows us to analyze the two-point function (the puncture-puncture correlator) and the spin-spin correlator as a function of the distance $r$. For $\beta \leq \beta^$ we have $\gamma_{str} > 0 $ and the fractal structure is reliably extracted from the two-point function. We find that $d_H =2$ for $\beta < \beta^$, while it jumps to $d_H^ =3$ at $\beta^$. Whilst we can not calculate $d_H$ for $\beta>\beta^$, one might expect it to equal four, the pure gravity value [@AmbWat95]. The fact that $d_H$ is different from 4 for $\beta \leq \beta^$ is an indication of the very strong interaction between gravity and matter for $c > 1$. A further indication of the strong link between geometry and matter configurations present in the model is found in the scaling relations for the magnetic exponents. They are found from the behaviour of the [two-point function]{} $\langle n \rangle_r$ in the region $\beta \geq \beta^$. The exponential decay of $\langle n \rangle_r$ [at]{} $x_c (\beta)$ as a function of the distance $r$ defines a length scale () \~, which diverges for $\beta \to \beta^$ and allows us to define the magnetic exponents using the standard hyper-scaling relations (note that $\xi(\beta) = \infty$ for $\beta \le \beta^$). The same exponents are found, for the toy model, by a direct calculation in the canonical and grand canonical ensembles. Strictly speaking, it seems more natural to define magnetic scaling properties from one of the spin-spin correlators, i.e. either $\langle \Sigma S\rangle_r^{++\cdots+}$ or ${\cal M}_r$, however the corresponding correlation length does not diverge at $\beta^$. This is an indication of the geometric nature of the transition. For the full model, a summation over all spin configurations is performed and the only approximation made is in the measure of distance that is used. As before there is an exponential decay of both the two-point function and the spin-spin correlation function, and one can define, in the magnetized phase, a geometric correlation length, which diverges as the tree phase is approached. Based on the analysis in section \[sec:fullspin\] we do not expect the spin-spin correlation length, at least as we have defined it, to diverge at the magnetized to tree-like transition. Using the geometric length scale and the scaling relations, which have been shown to hold for the toy model, all the critical exponents are calculated for the branched polymer phase and its boundary, on the basis of a small number of assumptions. The exponents are the same as those for the toy model (see table \[tab:exp\]) and again $d_H=2$ within the tree phase and $d_H$ is equal to three on the critical line, showing the strong interaction between the matter fields and the geometry. Some interesting questions remain to be answered, such as what happens outside the branched polymer phase, in particular what the values of the geometric exponents and $d_H$ are, and whether there is a diverging length associated with the spin-spin correlation function at the magnetization transition (between the U and M phases). Unfortunately our measure of distance is inadequate in this region and it will require further work using probably the full geodesic distance in order to determine this. It would also be useful to locate point $P$ on the phase diagram, as it has been suggested [@AJW; @MC] that $p^>2$, in which case there is an intermediate region, for $2<p<p^*$, between the KPZ regime and the branched polymer phase. Finally, it should perhaps be noted that most of the considerations in this paper also apply to the O(n) models [@onmodels; @charlotte], as these are just a special case of our generalized multiple Ising model. Acknowledgements {#acknowledgements .unnumbered} ---------------- MGH would like to acknowledge the support of the Royal Society through their European Science Exchange Programme. [99]{}
--- abstract: 'This article relates dominant and codominant dimensions of modules to Gorenstein homological properties. We call a module Gorenstein projective-injective in case it is Gorenstein projective and Gorenstein injective. For gendo-symmetric algebras we find a characterisation of Gorenstein projective-injective modules in terms of dominant and codominant dimensions and find representation theoretic and homological properties of the category of Gorenstein projective-injective modules. In particular, we give a new construction of non-selfinjective algebras having Auslander-Reiten components consisting only of Gorenstein projective modules. We introduce the class of nearly Gorenstein algebras, where the Gorenstein projective-injective modules have especially nice properties in case the algebra is gendo-symmetric. We also show that the Nakayama conjecture holds for nearly Gorenstein algebras.' address: 'Institute of algebra and number theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany' author: - René Marczinzik title: 'Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra' --- Introduction ============ We assume always that our algebras are finite dimensional, connected and non-semisimple over a field $K$. While the dominant dimension of an algebra is a well known concept (see for example [@ARS] VI.5.), dominant dimensions of modules are rarely studied. In this paper we relate dominant dimensions of modules with properties from Gorenstein homological algebra. We establish new features of Gorenstein homological algebra for gendo-symmetric algebras, which are defined as endomorphism rings of generators over a symmetric algebra, see [@FanKoe]. This class of algebras includes all symmetric algebras but also algebras like the Schur algebras $S(n,r)$ for $n \geq r$ or blocks of category $\mathcal{O}$. We call a module Gorenstein projective-injective, in case it is Gorenstein projective and Gorenstein injective. Our main result is the following: (see \[mainresult\] and \[CorCM\]) Let $A$ be a gendo-symmetric algebra. Then a module $M$ is Gorenstein projective-injective iff $M$ has infinite dominant and infinite codominant dimension. In case an Auslander-Reiten component of $A$ contains a nonprojective Gorenstein projective-injective module, then this component consists only of Gorenstein projective-injective modules. We define the finitistic dominant dimension of an algebra as the supremum over all dominant dimensions of modules having finite dominant dimension. We call an algebra $A$ nearly Gorenstein in case the set of modules $M$ with $Ext^{i}(M,A)=0$ for all $i>0$ coincides with the set of Gorenstein projective modules and additionally the same condition holds for the opposite algebra of $A$. For example all Gorenstein algebras and all representation-finite algebras are nearly Gorenstein as we will explain in \[gorexamp\] and \[examples\]. A first example of a non-nearly Gorenstein algebra appeared in [@JS], about 40 years after the pioneering work in [@AB]. No example of a gendo-symmetric algebra, which is not nearly Gorenstein seems to be known at the moment, having asked some experts (see the end of the paper in section 5, why this is a hard and important question related to the Nakayama conjecture). With those notions our second main result is as follows: (see \[finitistic\]) Let $A$ be a nonselfinjective gendo-symmetric algebra, which additionally is nearly Gorenstein. Then a module is Gorenstein projective-injective iff it has infinite dominant dimension. In case $A$ is additionally CM-finite and Gorenstein with Gorenstein dimension $g$, the finitistic dominant dimension of $A$ is bounded by $g+1$. We will show that the bound $g+1$ is really obtained and thus optimal. The second section includes the preliminaries. One of our main tools will be Theorem , which is a special case of results in [@APT]. In the third section, we define nearly Gorenstein algebras and show that many classes of algebras are nearly Gorenstein. The next topic in this chapter are the Gorenstein dominant algebras, which always have dominant dimension at least 1 and additionally the non-projective Gorenstein projective modules have dominant dimension at least two. The motivation comes from the fact, that for a Gorenstein dominant algebra $A$ one can try to characterise the Gorenstein projective modules in terms of some special modules in the module category of $eAe$, where $eA$ is a minimal faithful projective-injective module. All algebras with dominant dimension at least 2 are Gorenstein dominant algebras. We collect here some smaller results of section 3 and 4 worth mentioning: - Nearly Gorenstein algebras satisfy the Nakayama conjecture, see \[nakaconj\]. - We give a new formula for the representation dimension, see \[repdim\]. - We give a very short proof of Iyama’s higher Auslander correspondence for finite dimensional algebras, see \[auscor\]. - We show that every Nakayama algebra is Gorenstein dominant, see \[nakgordom\]. - We generalize the classical formula $\Omega^{2}(M) \cong \tau(M)$ for symmetric algebras to the gendo-symmetric situation, see \[tauomega\]. Section 4 contains our main results, where we will always deal with gendo-symmetric algebras and we find a close connection between Gorenstein homological properties and dominant and codominant dimensions. We will see that the Gorenstein projective-injective modules coincide with modules of infinite dominant and codominant dimension. The nonprojective Gorenstein projective-injective modules form components of the Auslander-Reiten quiver of the algebra, and those modules behave much as modules over a symmetric algebra. We also introduce the finitistic dominant dimension of an arbitrary algebra. Surprisingly, the finitistic dominant dimension of a CM-finite gendo-symmetric $g$-Gorenstein algebra is bounded by $g$+1 and this bound is optimal. In forthcoming work [@ChMar2] we show that for gendo-symmetric Nakayama algebras the finitistic dominant dimension always equals the Gorenstein dimension and calculate the finitistic dominant dimension for general representation-finite, gendo-symmetric biserial algebras, which has surprising relations to number theory. I thank Steffen Koenig for useful comments and proof reading and Ragnar-Olaf Buchweitz for telling me about the results in [@JS]. Preliminaries ============= General preliminaries --------------------- We start by fixing some notations and giving definitions. For the standard notations on Auslander-Reiten theory we refer to [@SkoYam]. Let an algebra always be a finite dimensional, connected and non-semisimple algebra over a field $K$ and a module over such an algebra is always a finite dimensional right module, unless otherwise stated. $D=Hom_K(-,K)$ denotes the duality for a given finite dimensional algebra $A$. We define the dominant dimension domdim($M$) of a module $M$ with a minimal injective resolution $(I_i): 0 \rightarrow M \rightarrow I_0 \rightarrow I_1 \rightarrow ...$ as: domdim($M$):=$\sup \{ n \| I_i $ is projective for $i=0,1,...,n \}$+1, if $I_0$ is projective, and domdim($M$):=0, if $I_0$ is not projective. The codominant dimension of a module $M$ is defined as the dominant dimension of the dual module $D(M)$. The dominant dimension of a finite dimensional algebra is defined as the dominant dimension of the regular module $A_A$ and the codominant dimension is the codominant dimension of the module $D(_AA)$. In the following we always assume that $A$ is a finite dimensional algebra with dominant dimension larger than or equal to 1. It is well known that $A$ has dominant dimension larger than or equal to 1 iff there exists an idempotent such that $eA$ is a minimal faithful projective-injective right module iff there exists and idempotent $f$ such that $Af$ is a minimal faithful projective-injective left module, see [@Yam]. Note that the dominant dimension of $A_A$ is always equal to the dominant dimension of $_AA$ and thus the dominant dimension always equals the codominant dimension (see [@Yam]). Parts of the theory could also be interesting for algebras with dominant dimension 0, but we stick to dominant dimension larger than or equal to 1, since we know no interesting applications for algebras with dominant dimension 0. $eA$ always denotes the minimal faithful injective-projective right $A$-module and $Af$ denotes the minimal faithful injective-projective left $A$-module for some idempotents $e$ and $f$. For $1 \leq i \leq \infty$ $Dom_i(A)$ (resp. $Codom_i(A)$) denotes the full subcategory of mod-$A$ constisting of modules with dominant dimension (resp. codominant dimension) larger than or equal to $i$. We often use the notation $Dom(A):=Dom_{\infty}(A)$ and $Codom(A):=Codom_{\infty}(A)$. Let $proj(A)$ denote the full subcategory of finitely generated projective modules and $inj(A)$ the subcategory of finitely generated injective modules. We often just write $proj,Dom,...$ for $proj(A),Dom(A),...$ if it is clear over which algebra we work. An algebra $A$ is called $g$-Gorenstein for a natural number $g$, in case the left and right injective dimensions of $A$ coincide and are equal to $g$. We call $A$ Gorenstein, if $A$ is $g$-Gorenstein for some $g$. In this case $Gordim(A):=injdim(A)$ is called the Gorenstein dimension of $A$. $\nu_A=\nu=DHom_A(-,A)$ denotes the Nakayama functor and $\nu^{-1}=Hom(D(-),A)$ its inverse. For section 2 and 3 of this article, fix the notations $I:=eA$, $P:=\nu_A^{-1}(eA)=Hom_A(D(eA),A)$, $B_1:=End_A(I)$ and $B_2:=End_A(P)$. We note that $B_1$ and $B_2$ are isomorphic, in case the dominant dimension of $A$ is at least two, see for example [@Yam], after theorem 3.4.1. there. As an $A$-right module $P$ is isomorphic to $fA$. We call a module $W$ $l$-periodic for a natural number $l \geq 1$, in case $\Omega^{l}(W) \cong W$. For $i \geq 1$, define the set of $i$-torsionless modules as $\{ X \| Ext^{l}(Tr(X),A)=0$ for $l=1,...,i \}$. Recall that $2$-torsionless modules are exactly the reflexive modules. For $n \in \mathbb{Z}$, denote by $\Omega^{n}(mod-A)$ the full subcategory of all modules, which are $n-$th syzygy modules including all projective modules. Recall that $A$ is called torsionless-finite, if $\Omega^{1}(mod-A)$ is a representation-finite subcategory of $mod-A$. For an $A$-module $M$, add($M$) denotes the full subcategory of mod-$A$ consisting of all direct summands of a finite direct sum of $M$. Then a map $f:M_0 \rightarrow X$, with $M_0 \in$ add($M$), is called a right add($M$)-approximation of $X$ iff the induced map $Hom(N,M_0) \rightarrow Hom(N,X)$ is surjective for every $N \in$ add($M$). Note that in case $M$ is a generator, such an $f$ must be surjective. When $f$ is also a right minimal homomorphism, we call it a minimal right add($M$)-approximation (or add($M$)-Cover). Note that minimal right add($M$)-approximations always exist for finite dimensional algebras. The kernel of such a minimal right add($M$)-approximation $f$ is denoted by $\Omega_M(X)$. Inductively one defines $\Omega_M^{0}(X):=X$ and $\Omega_M^{n}(X):=\Omega_M(\Omega_M^{n-1}(X))$. The add($M$)-resolution dimension of a module $X$ is defined as: $M\text{-resdim}(X):=\inf \{ n \geq 0 \| \Omega_M^{n}(X) \in \text{add}(M) \}.$ Left approximations and coresolution dimensions are defined dually. If $C$ is a subcategory of mod-$A$, we denote by $C/[proj]$ the subcategory modulo projectives, called stable category, and $C/[inj]$ the subcategory modulo injectives, called costable category. $A$ is called a Morita algebra iff it has dominant dimension larger than or equal to 2 and $D(Ae) \cong I$ as $A$-right modules (recall that $I=eA$). This is equivalent to $A$ being isomorphic to $End_B(M)$, where $B$ is a selfinjective algebra and $M$ a generator of mod-$B$ (see [@KerYam]). $A$ is called a gendo-symmetric algebra iff it has dominant dimension larger than or equal to 2 and $D(Ae) \cong I$ as $(eAe,A)-$bimodules iff it has dominant dimension larger than or equal to 2 and $D(eA) \cong Ae$ as $(A,eAe)$-bimodules. This is equivalent to $A$ being isomorphic to $End_B(M)$, where $B$ is a symmetric algebra and $M$ a generator of mod-$B$ (see [@FanKoe]). For other characterisations of gendo-symmetric algebras we refer to [@FanKoe] and [@Mar2]. Note that in case the algebra is gendo-symmetric, $I=P$ and $B_1=B_2$. For a subcategory $\mathcal{X}$ of mod-$A$ we define for $1 \leq n \leq \infty$: $\mathcal{X}^{\perp n}:= \{ M \in mod-A \| Ext^{i}(Y,M)=0 $ for $1 \leq i \leq n $ and all $Y \in \mathcal{X} \}$ and $^{\perp n}\mathcal{X}:= \{ M \in mod-A \| Ext^{i}(M,Y)=0 $ for $1 \leq i \leq n $ and all $Y \in \mathcal{X} \}$. We often use the shorter notation $\mathcal{X}^{\perp} := \mathcal{X}^{\perp \infty}$ and $^{\perp }\mathcal{X} :=^{\perp \infty}\mathcal{X}$. In case $\mathcal{X}=add(X)$ for a module $X$ we write $X^{\perp n}$ instead of $\mathcal{X}^{\perp n}$ and similar for the other notations involving $\mathcal{X}$. Recall (see [@Iya3]) that a subcategory of the form $add(M)$ (or the module $M$) is called maximal $(n-1)$ orthogonal, in case $add(M)=M^{\perp (n-1)}= ^{\perp (n-1)}M$ holds. Note that in case $M$ is a generator-cogenerator this is equivalent to the single conditions $add(M)=M^{\perp (n-1)}$ or $add(M)=^{\perp (n-1)}M$. The following theorem collects results from [@APT] in a special case. \[ARSmaintheorem\] 1. The functors $F_1:=Hom_A(I,-) : Codom_2 \rightarrow mod-B_1$ and $F_2:=Hom_A(P,-) : Dom_2 \rightarrow mod-B_2$ are equivalences of categories. $F_1$ restricts to an equivalence between add($I$) and the category of projective $B_1$-modules and $F_2$ restricts to an equivalence between add($I$) and the category of injective $B_2$-modules. 2. There are natural isomorphisms of functors: $Hom_A(P,-) \cong (-) \otimes_A (D(eA)) \cong DHom_A(-,D(eA))$. In case $A$ is gendo-symmetric the following holds true: $F_2 \cong (-)e $ and $(-)e \cong F_1$. 3. The Functor $G_1:= (-) \otimes_{B_1} I : mod-B_1 \rightarrow Codom_2$ is inverse to $F_1$ and the functor $G_2:=Hom_{B_2}(P,-) : mod-B_2 \rightarrow Dom_2$ is inverse to $F_2$. 4. For $i \geq 3$, $F_1$ restricts to an equivalence $F_1: Codom_i \rightarrow ^{\perp i-2}(Ae)$ and $F_2$ restricts to an equivalence $F_2: Dom_i \rightarrow (Af)^{\perp i-2}$. <!-- --> 1. This is a special case of Lemma 3.1. of [@APT]. 2. By [@SkoYam] Chapter III. Lemma 6.1., there is the following natural isomorphism of functors: $Hom_A(Hom_A(D(eA),A),-) \cong (-) \otimes_A (D(eA))$. Now there is another natural isomorphism $(-) \otimes_A (D(eA)) \cong DHom_A(-,D(eA))$ by [@ASS] Appendix 5, Propostion 4.11. When $A$ is gendo-symmetric $F_2(-) \cong (-) \otimes_A (D(eA)) \cong (-) \otimes_A Ae \cong (-)e$, is clear since $D(eA) \cong Ae$ as $(A,eAe)$-bimodules. 3. This follows from [@APT], in the passages before Proposition 3.9. and before Proposition 3.10. 4. This follows from Proposition 3.7. in [@APT]. One can use (4) of the previous theorem to calculate dominant dimensions via Ext. This was first noticed by Mueller, see [@Mue]. We sometimes refer to this as Mueller’s theorem. The finitistic dominant dimension of an algebra $A$ is $fdomdim(A):= \sup \{ domdim(M) \| M \in mod-A$ and $domdim(M) < \infty \}$. In [@Mar], we proved that the finitistic dominant dimension of a Nakayama algebra with $n$ simple modules is bounded by $2n-2$. \[torsionless\] Assume $A$ has dominant dimension $d \geq 1$. 1. $Dom_i = \Omega^{i}(A-mod)$ for all $1 \leq i \leq d$ and $Codom_i = \Omega^{-i}(A-mod)$ for all $1 \leq i \leq d$. 2. $Dom_i=\{ X \| Ext^{l}(Tr(X),A)=0$ for $l=1,...,i \}$, for $i=1,...,d$. <!-- --> 1. See [@MarVil] proposition 4. 2. Combine proposition 1.6 of [@AR] and part (1) of this lemma. The following Lemma is proven in Direction 1 of [@Rin2]. \[direction\] Let $M$ be a generator and cogenerator of $B$. Let $A:= End_B(M)$. Then the basic versions of $Af$ and $M$ are isomorphic, where $Af$ is the minimal faithful projective-injective left $A$-module. The following theorem is proven in [@APT] Theorem 3.2. (a). \[auslanderext\] Let $A$ be an algebra and let $X,Y$ be $A$-modules with dominant dimension at least 1. For all $n$ with $0 \leq n \leq domdim(X)+domdim(Y)-2$ there is an isomorphism $Ext_A^{n}(X,Y) \cong Ext_B^{n}(Hom_A(P,X),Hom_A(P,Y))$. The following result is proposition 3.11. from [@CheKoe]: \[theoremchekoe\] Let $A$ be a finite dimensional algebra and $M$ a nonprojective generator and cogenerator of mod-$A$ and define $B:=End_A(M)$. Let $B$ have dominant dimension $z+2$, with $z \geq 0$. Then, for the right injective dimension of $B$ the following holds: $$\text{injdim}(B_B)=z+2\ +\ M\text{-resdim}(\tau_{z+1}(M)\oplus D(A)).$$ For the left injective dimension of $B$ the following holds: $$\text{injdim}(_BB)=z+2\ +\ M\text{-coresdim}(\tau_{-(z+1)}(M)\oplus A).$$ Here we use the notations $\tau_{z+1}=\tau\Omega^{z}$ and $\tau_{-(z+1)}=\tau^{-1}\Omega^{-z}$, introduced by Iyama (see [@Iya]). \[extrechnen\] Let $B$ be a symmetic algebra and $M,N$ two $B$-modules. 1. For the Auslander-Reiten translate $\tau$, the following holds: $\tau(M) \cong \Omega^{2}(M)$, for every indecomposable nonprojective module $M$. 2. $Ext^{i}(M,N) \cong \underline{Hom}(\Omega^{i}(M),N) \cong \underline{Hom}(M,\Omega^{-i}(N))$. 3. $Ext^{1}(M,N) \cong \underline{Hom}(N,\Omega^{2}(M)) \cong \underline{Hom}(\Omega^{-2}(N),(M))$ 4. $\underline{Hom}(M,N) \cong \underline{Hom}(N,\Omega^{1}(M))$. <!-- --> 1. see [@SkoYam], Chapter IV Corollary 8.6. 2. see [@SkoYam], Chapter IV Theorem 9.9. 3. Those are the Auslander-Reiten formulas in the special case of a symmetric algebra, see [@SkoYam] Chapter III. Theorem 6.3. 4. This follows from the previous part using that $\Omega$ is an equivalence of the stable module category of $B$ with inverse $\Omega^{-1}$: $\underline{Hom}(N,\Omega^{1}(M)) \cong \underline{Hom}(\Omega^{1}(N),\Omega^{2}(M)) \cong Ext^{1}(M,\Omega^{1}(N)) \newline \cong \underline{Hom}(\Omega^{1}(M),\Omega^{1}(N)) \cong \underline{Hom}(M,N).$ We also need the following well known lemma, which can be found in [@Ben] as Corollary 2.5.4.: \[benson\] Let $A$ be a finite dimensional algebra and $N$ be an indecomposable $A$-module and $S$ a simple $A$-module. Let $(P_i)$ be the terms of a minimal projective resolution of $N$ and $(I_i)$ the terms of a minimal injective resolution of $N$. 1. For $l \geq 0$, $Ext^{l}(N,S) \neq 0$ iff $S$ is a quotient of $P_l$. 2. For $l \geq 0$, $Ext^{l}(S,N) \neq 0$ iff $S$ is a submodule of $I_l$. Gorenstein-projective and Gorenstein-injective modules ------------------------------------------------------ For the neccessary background on notions from Gorenstein homological algebra, we refer to [@Che] and [@AB]. A module $M$ is called Gorenstein projective, if $Ext^{i}(M,A)=0=Ext^{i}(Tr(M),A)$ for every $i \geq 1$. $M$ is called Gorenstein injective in case $D(M)$ is Gorenstein projective. If $M$ is Gorenstein projective and Gorenstein injective, we call $M$ Gorenstein projective-injective. We refer to [@Che] and [@AB] for many other characterisations and properties of Gorenstein projective and Gorenstein injective modules. For a finite dimensional algebra $A$, we denote by $Gp(A)$ the category of finite dimensional Gorenstein projective modules and we denote by $RGp(A)$ the category of finite dimensional Gorenstein projective modules without a projective summand. We denote by $Gi(A)$ the category of finite dimensional Gorenstein injective modules and $RGi(A)$ denotes the category of finite dimensional Gorenstein injective modules without an injective summand. $Gpi(A)$ is defined as the subcategory of finite dimensional Gorenstein projective-injective modules. $A$ is called CM-finite in case $Gp(A)$ is representation-finite and $A$ is called CM-free in case $Gp(A)=proj$. Recall that a Gorenstein algebra is CM-free iff it has finite global dimension. Let $A$ be a $d$-Gorenstein algebra and $B:=End_A(M)$, where $M$ is Gorenstein projective and a generator of $mod-A$, then $B$ is called a gendo-$d$-Gorenstein algebra, see [@GaKo]. Examples where the Gorenstein projective modules are easy to describe are CNakayama algebras. These are the Nakayama algebras with a cyclic quiver (Nakayama algebras with a noncyclic quiver have finite global dimension and therefore have no nontrivial Gorenstein-projective modules). We collect here some needed results on Nakayama algebras and refer to [@ASS] Chapter 5 for the representation theory of Nakayama algebras. Let $A$ be a basic and elementary (we always assume those two conditions in the following for CNakayama algebras) CNakayama algebra with $n$ simple modules and with Kupisch series $(c_0,c_1,...,c_{n-1})$. We denote by $l_l(i)$ the dimension of the $i$-th injective indecomposable $A$-module $D(Ae_i)$. We will always calculate modulo $n$ for the indices of the quiver and number those indices from $0$ to $n-1$ (corresponding to the simple, indecomposable projective and indecomposable injective modules). The quiver of a CNakayama algebra looks as follows: $$Q=\begin{xymatrix}{ & n-1 \ar[r] & 0 \ar[dr] & \\ n-2 \ar[ur] & & & 1 \ar[d] \\ n-3 \ar[u] & & & 2 \ar[dl] \\ & 4 \ar @{} [ul] \|{\ddots} & 3 \ar[l] & }\end{xymatrix}$$ Recall that every Nakayama algebra has dominant dimension at least 1, see [@Abr]. Denote by $M=e_iA/e_iJ^k$ an arbitrary indecomposable module in $A$. We refer to [@Mar] for the calculation of minimal projective or injective resolutions in Nakayama algebras. We quickly repeat the basics in the following. We get a minimal injective resolution of $M$ as follows (with $k=c_i$, if $M$ is projective): We have soc($M$)=$S_{i+k-1}$ (the simple module corresponding to the point $i+k-1$). Therefore, the injective hull of $M$ is $D(Ae_{i+k-1})$ and thus $\Omega^{-1}(M)=D(J^k e_{i+k-1})$ and $\Omega^{-1}(D(J^k e_{i+k-1}))=D(J^{l_l(i+k-1)-k} e_{i-1})$, by comparing dimensions and using that submodules form a chain. Defining $g:\mathbb{Z}/n \rightarrow \mathbb{Z}/n$ as $g(x):=x-l_l(x)$, the minimal injective resolution of $M$ looks like this by repeating the above process: $$0 \rightarrow M \rightarrow D(Ae_{j-1}) \rightarrow D(Ae_{i-1}) \rightarrow D(Ae_{g(j-1)}) \rightarrow D(Ae_{g(i-1)}) \rightarrow D(Ae_{g^2(j-1)})$$ $$\rightarrow D(Ae_{g^2(i-1)}) \rightarrow \cdots \rightarrow D(Ae_{g^e(j-1)}) \rightarrow D(Ae_{g^e(i-1)}) \rightarrow \cdots .$$ We denote $D(J^y e_x)$ for short by $[x,y] \in \mathbb{Z}/n \times \mathbb{N}$ and then one has that $\Omega^{-1}(D(J^y e_x))=[x-y,d_x-y]$. Like this it is easy to calculate the cosyzygies successively. We start to recall the necessary definitions from [@Rin]. Let $A$ be a basic nonselfinjective CNakayama algebra. The resolution quiver of $A$ is defined as the directed graph having as vertices the simple $A$-modules and an arrow from $S_1$ to $S_2$ iff $S_2=\tau(soc(P(S_1)))$, where $P(S)$ denotes the projective cover of a simple module $S$. A vertex is called black, in case the corresponding simple module $S$ has projective dimension at least two. A vertex is called cyclically black, in case $S$ lies on a cycle in the resolution quiver and all the vertices in that cycle are black. The Gorenstein core is defined as the full subcategory consisting of all nonprojective Gorenstein-projective modules and their projective covers. By definition, an indecomposable projective module or simple module has a property like being cyclically black in case the corresponding vertex has that property. The following is one of the main results of [@Rin] (see proposition 3 there): \[ringeltheorem\] Let $A$ be a basic nonselfinjective CNakayama algebra, then a nonprojective indecomposable $A$-module $M$ is Gorenstein-projective iff top($M$) and top($\Omega^{1}(M)$) are cyclically black iff in the projective presentation $P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$, the modules $P_1$ and $P_0$ are cyclically black. The following two propositions can be found in [@Che] as proposition 2.2.3. and theorem 2.3.3.: \[taugorenstein\] 1. A module $M$ is Gorenstein-injective iff $\nu^{-1}(M)$ is Gorenstein-projective and the natural morphism $\nu\nu^{-1}(M) \rightarrow M$ is an isomorphism. 2. There is an equivalence of categories: $\nu: Gp(A) \rightarrow Gi(A)$ with quasi-inverse $\nu^{-1}$. \[gorensteinkrit\] The following are equivalent: 1. $A$ is Gorenstein of Gorenstein dimension $g$. 2. $Gp(A)= \Omega^{g}(A-mod)$. 3. $Gi(A)= \Omega^{-g}(A-mod)$. In case $A$ is Gorenstein, the Gorenstein projective modules coincide with the modules $M$ with $Ext^{i}(M,A)=0$ for all $i \geq 1$ and the Gorenstein injective modules coincide with the modules $M$ with $Ext^{i}(D(A),M)=0$ for all $i \geq 1$. \[gorensteingrund\] 1. If a module satisfies $M \cong \Omega^{n}(M)$ for a natural number $n\geq 1$ in the stable category and $M \in ^{\perp}\!A$, then $M$ is Gorenstein projective. 2. A module $M$ is Gorenstein projective iff $\tau(M)$ is Gorenstein injective and a module $N$ is Gorenstein injective iff $\tau^{-1}(N)$ is Gorenstein projective. 3. The functor $\Omega^{1}: ^{\perp}\!A/[proj] \rightarrow ^{\perp}\!A/[proj]$ is fully faithful 4. If $X \in ^{\perp}\!A$ is indecomposable then also $\Omega^{1}(X)$ is indecomposable in the stable category. <!-- --> 1. This is Proposition 2.2.17. in [@Che]. 2. This is Proposition 2.2.13 in [@Che]. 3. This is well-known, see for example [@AB]. 4. Assume that $\Omega^{1}(X) \cong Y_1 \oplus Y_2$ in the stable category with two nonprojective modules $Y_1$ and $Y_2$. This contradicts the fact that $\underline{End_A(X)}$ is isomorphic to $\underline{End_A(\Omega^{1}(X))}$ (because $\Omega^{1}$ is fully faithful). To see this note that $\underline{End_A(X)}$ is local as a quotient of a local ring, but $\underline{End_A(Y_1 \oplus Y_2)}$ is not. Nearly Gorenstein algebras and Gorenstein dominant algebras =========================================================== Basics and examples for nearly Gorenstein algebras -------------------------------------------------- We call an algebra $A$ right nearly Gorenstein, if the subcategory of Gorenstein-injective modules coincides with $D(A)^{\perp}$. We call an algebra $A$ left nearly Gorenstein, if the subcategory of Gorenstein-projective modules coincides with $^{\perp}\!A$. We call an algebra $A$ nearly Gorenstein, if $A$ is left and right nearly Gorenstein. We remark that left nearly Gorenstein algebras are examined also in [@Bel] (without naming them); there is given a characterisation in theorem 5.2. under the assumption that $Gp(A)$ is contravariantly finite. While some of our results can also be deduced with methods from [@Bel], our proofs are more elementary. \[gorexamp\] Every Gorenstein algebra is nearly Gorenstein by \[gorensteinkrit\], hence the name. By [@Mar] Lemma 1.2.3. and its dual, every Nakayama algebra is nearly Gorenstein. But not every Nakayama algebra is Gorenstein, take for example the CNakayama algebra with Kupisch series $[5,6]$. Here is a natural characterisation of nearly Gorenstein algebras that does not mention Gorenstein projective modules: An algebra $A$ is nearly Gorenstein iff there is an equivalence of categories induced by the Auslander-Reiten translate: $\tau: ^{\perp}\!A/[proj] \rightarrow D(A)^{\perp}/[inj]$ with inverse $\tau^{-1}: D(A)^{\perp}/[inj] \rightarrow ^{\perp}\!A/[proj]$. Recall that $\tau: $ mod-$A/[proj] \rightarrow $mod$-A/[inj]$ is an equivalence with inverse $\tau^{-1}$. Assume first that $A$ is nearly Gorenstein. Then $^{\perp}\!A=Gp(A)$ and $D(A)^{\perp}=Gi(A)$ and the result follows since $\tau(X)$ is Gorenstein injective, if $X$ is Gorenstein projective and vice versa by \[gorensteingrund\] . Thus $\tau$ restricts to an equivalence: $^{\perp}\!A/[proj] \rightarrow D(A)^{\perp}/[inj]$. Now assume that $\tau: ^{\perp}\!A/[proj] \rightarrow D(A)^{\perp}/[inj]$ is an equivalence with inverse $\tau^{-1}: D(A)^{\perp}/[inj] \rightarrow ^{\perp}\!A/[proj]$. We will show $^{\perp}\!A=Gp(A)$. Assume that $X$ is a nonprojective indecomposable $A$-module and $X \in ^{\perp}\!A$. Then $\tau(X) \in D(A)^{\perp}$. Thus $Ext^{i}(D(A),\tau(X))=0$ for all $i \geq 1$, which is equivalent to $Ext^{i}(Tr(X),A)=0$ for all $i \geq 1$. Thus, by definition, $X$ is Gorenstein projective. $D(A)^{\perp}=Gi(A)$ follows dually, and thus $A$ is nearly Gorenstein. Let $A$ be an algebra of dominant dimension $d \geq 1$. Then the following holds: $Gp(A) \subseteq ^{\perp}\!A \subseteq Tr((Dom_d)^{op}) \cup proj$, where $(Dom_d)^{op}$ denotes the subcategory of mod-$A^{op}$ of all left modules of dominant dimension at least $d$. $Gp(A) \subseteq ^{\perp}\!A$ holds by definition of Gorenstein projective modules. Now $^{\perp}\!A = \{X \in mod-A\| Ext_{A}^{i}(X,A)=0$ for all $i\geq 1 \} =Tr( \{ Y \in mod-A^{op} \| Ext_{A^{op}}^{i}(Tr(Y),A)=0$ for all $i \geq 1 \} \cup proj \subseteq Tr( \{ Y \in mod-A^{op} \| Ext_{A^{op}}^{i}(Tr(Y),A)=0$ for all $1 \leq i \leq d \} \cup proj =Tr((Dom_d)^{op}) \cup proj$, where the last inclusion holds because the $i$-torsionless modules coincide with the modules of dominant dimension at least $i$ for $i\leq d$ by . The following theorem provides alot of examples of nearly Gorenstein algebras that are in general not Gorenstein: \[examples\] 1. If $^{\perp}\!A$ and $D(A)^{\perp}$ are representation-finite, then $A$ is nearly Gorenstein. In particular, all representation-finite algebras are nearly Gorenstein. 2. Assume $A$ has dominant dimension $d \geq 2$, with minimal faithful injective-projective module $eA$ such that $B_2$ ($B_2$ as in \[ARSmaintheorem\]) is representation-finite. Then $A$ is CM-finite and nearly Gorenstein. 3. If $A$ is torsionless-finite, then $A$ is nearly Gorenstein. 4. Let $A$ be an algebra with $rad^2(A)=0$, then $A$ is nearly Gorenstein. In case $A$ additionally is not selfinjective, then $^{\perp}\!A$ coincides with the category of finitely generated projective modules and thus $A$ is not Gorenstein or has finite global dimension. <!-- --> 1. We show that $Gp(A)=^{\perp}\!A$ in case $^{\perp}\!A$ is representation-finite: First recall that there is a fully faithful functor $\Omega^{1}: ^{\perp}\!A/[proj] \rightarrow ^{\perp}\!A/[proj]$ induced by the syzygy functor. Assume $X \in ^{\perp}\!A$ is nonprojective and indecomposable. That $\Omega^{1}(X)$ must be nonprojective is clear, since the functor is fully faithful. Then also $\Omega^{1}(X)$ is indecomposable and nonprojective. Since $^{\perp}\!A$ is representation-finite, there must be an $i \geq 1$ with $\Omega^{i}(X) \cong X$ in the stable category. Thus $X$ is periodic in the stable category and $X \in ^{\perp}\!A$. By \[gorensteingrund\], $X$ must be Gorenstein-projective then. The dual argument works in case $D(A)^{\perp}$ is representation-finite. Thus $A$ is nearly Gorenstein. 2. We write $B=B_2$ in the following. By the previous lemma, $^{\perp}\!A \subseteq Tr((Dom_d)^{op}) \cup proj$. By \[ARSmaintheorem\], there is an equivalence of categories $Dom_2^{op} \cong mod-B$ and thus the subcategory $Dom_2^{op}$ is representation-finite because $mod-B^{op}$ is representation-finite. Because of $Dom_d^{op} \subseteq Dom_2^{op}$, also $^{\perp}\!A$ is representation-finite. The proof that $D(A)^{\perp}$ is representation-finite is dual. 3. The argument is similar to the argument in (1). Recall that $A$ is torsionless-finite iff $A^{op}$ is torsionless finite (see [@Rin3]). Thus, by using duality, it suffices to show $^{\perp}\!A=Gp(A)$. Let $M \in ^{\perp}\!A$. Then also $\Omega^{i}(M) \in ^{\perp}\!A$, for every $i \geq 1$. But the subcategory $\Omega^{1}(A-mod)$ is representation-finite and in the stable category $\Omega^{i}(M)$ is always indecomposable for every $i \geq 1$. Thus there have to be some indices $p>q$ with $ \Omega^{p}(M) \cong \Omega^{q}(M)$ in the stable category. But since $\Omega$ is fully faithful in $^{\perp}\!A$: $\Omega^{p-q}(M) \cong M$ in the stable category. Thus by , $M$ is Gorenstein projective. 4. Note that $rad^2(A)=0$ implies that $A$ is torsionless-finite and thus nearly Gorenstein. Assume now that $A$ is not selfinjective and has infinite global dimension (the result is clear in case of finite global dimension). By [@Che] Theorem 2.3.9., the algebra is CM-free and thus not Gorenstein. Because it is nearly Gorenstein, this forces that $^{\perp}\!A=Gp(A)$ coincides with the category of all finitely generated projective modules. Since our main concern will be Gorenstein homological algebra of gendo-symmetric algebras in the next section, we give here a concrete example of a gendo-symmetric nearly Gorenstein algebra that is not Gorenstein. Here we use . We choose $A$ to be the symmetric CNakayama algebra with Loewy length $7$ and $3$ simple modules and we set $M=e_0 J^2$. The algebra $B:=End_A(W)$, with $W:= A \oplus M$, is nearly Gorenstein by (2) of the previous theorem. It is easy to see that $Ext^{1}(M,M)=0$, but $Ext^{2}(M,M)\neq 0$ and so domdim($B$)$=3$. $\tau_2(M)=\tau(\Omega^{1}(M))= \tau(e_2 J^{5})=e_0 J^{5}$ and thus we have to calculate the right $W$-resolution dimension of $ e_0 J^{5}$. First one calculates the start of this $W$-resolution and the first kernel. Note first that because of $Ext_A^{1}(e_0 J^2,e_0 J^2)=0$, the subcategory $\mathbb{X}:=add(A \oplus e_0 J^2)$ is extension-closed. Then by Wakamatsus lemma (see [@EJ] Chapter 7.2), a map $f:A \rightarrow B$ is an $\mathbb{X}-$approximation iff $A \in \mathbb{X}$ and $Ext^{1}(L,ker(f))=0$ for every $L \in \mathbb{X}$. The minimal $add(W)$-cover $\pi$ of $e_0 J^5$ looks as follows: $$0 \rightarrow e_0 J^4 \rightarrow e_0 J^{2} \stackrel{\pi}\rightarrow e_0 J^5 \rightarrow 0.$$ $e_0J^2 \cong e_0 A / e_0 J^5$ and $e_0 J^5 \cong e_2 A/ e_2 J^2$ and $\pi$ is the surjective map which is left multiplication by the unique arrow of length $1$ from $e_2$ to $e_0$. Now we calculate a minimal $W$-Cover $\pi_2$ of the kernel $e_0 J^4$. We have $$0 \rightarrow e_0J^4 \oplus e_1 J \rightarrow e_1 A \oplus e_0 J^2 \stackrel{\pi_2}\rightarrow e_0 J^4 \rightarrow 0.$$ Note that $e_0 J^4 \cong e_1 A/e_1 J^3$ and $\pi_2=(f,g)$, where, $f: e_1 A \rightarrow e_0J^4$ is the projective Cover of $e_0 J^4$ and $g: e_0 A /e_0J^4 \rightarrow e_0 J^4$ is left multiplication by the unique arrow of length $2$ from $e_1$ to $e_0$. Now it is clear that the minimal $W$-resolution dimension of $e_0 J^5$ has to be infinite, since the second kernel of the resolution has the first kernel as a direct summand. By , $B$ is not Gorenstein. In [@Mar], we proved that all gendo-symmetric Nakayama algebras are Gorenstein. Therefore, there are probably no much easier examples of non-Gorenstein, gendo-symmetric algebras than the previous one. Homological conjectures for nearly Gorenstein algebras ------------------------------------------------------ Recall the following famous homological conjectures for a finite dimensional algebra $A$ (see for example [@Yam] for a discussion of some of the conjectures): 1. Strong Nakayama conjecture: For every non-zero module $M$ there is an $i \geq 0$ with $Ext_A^{i}(M,A) \neq 0$. 2. The generalized Nakayama conjecture: For every simple module $S$ there is an $i \geq 0$ with $Ext_A^{i}(S,A) \neq 0$. 3. The Nakayama conjecture: Every nonselfinjective algebra has finite dominant dimension. 4. The Gorenstein symmetry conjecture: The right injective dimension equals the left injective dimension of an algebra. It is well-known and easy to see that $(1) \Rightarrow (2) \Rightarrow (3)$. \[nakaconj\] 1. The strong Nakayama is true for every left nearly Gorenstein algebra and thus every nonselfinjective left nearly Gorenstein algebra has finite dominant dimension. 2. The Gorenstein symmetry conjecture is true for left nearly Gorenstein algebras. <!-- --> 1. Let $M$ be a module with $Ext^{i}(M,A)=0$ for every $i \geq 1$. Then $M$ is Gorenstein projective, since $A$ is left nearly Gorenstein. But then there is an embedding $M \rightarrow A^n$, for some $n \geq 1$. Thus $Hom(M,A) \neq 0$. 2. If the right injective dimension of $A_A$ is zero, then $A$ is selfinjective and being selfinjective is left-right symmetric. So assume that the right injective dimension of $A_A$ is $n \geq 1$. Then for every module $X: \newline Ext^{n+i}(X,A)=0$ for every $i \geq 1$. Now for every $i \geq 1: Ext^{n+i}(X,A) \cong Ext^{i}(\Omega^{n}(X),A)=0$ and thus $\Omega^{n}(X) \in ^{\perp}\!A=Gp(A)$. This gives us $\Omega^{n}(A-mod) \subseteq Gp(A)$. But by definition of Gorenstein projective $Gp(A) \subseteq \Omega^{j}(A-mod)$ for every $j \geq 1$ and thus $\Omega^{n}(A-mod) = Gp(A)$, which by means that $A$ is Gorenstein with Gorenstein dimension $n$ and thus also has left injective dimension $n$. Gorenstein dominant algebras ---------------------------- We call a finite dimensional algebra $A$, a Gorenstein dominant algebra if it has the following properties: - it has dominant dimension larger than or equal to 1. - all modules in $RGp(A)$ have dominant dimension at least 2. First we show that interesting classes of finite dimensional algebras are Gorenstein dominant algebras. \[intclass\] Let $A$ be an algebra with dominant dimension $d\geq 1.$ Every Gorenstein projective module has dominant dimension at least $d$. Especially: A finite dimensional nonselfinjective algebra $A$ with domdim($A)=d \geq 2$ is a Gorenstein dominant algebra and dually every Gorenstein injective module has codominant dimension at least $d$. Every Gorenstein projective module is contained in $\Omega^{i}(A-mod)$ for an arbitrary $i\geq 1$. Using $\Omega^{d}(A-mod) = Dom_d$, we see: $Gp(A) \subseteq \bigcap\limits_{k=1}^{\infty}{\Omega^{k}(A-mod)} \subseteq \Omega^{d}(A-mod) = Dom_d$. The dual result follows similarly, since the codominant dimension of $A$ equals the dominant dimension of $A$. A Gorenstein algebra $A$ of finite Gorenstein dimension $g$ and dominant dimension $d$ has the following properties: 1. $Dom_g \subseteq Gp(A)$. 2. $A$ has finite finitistic dominant dimension, in case $A$ is also CM-finite. <!-- --> 1. Assume $X \in Dom_g$. Then there is the following beginning of a minimal injective resolution of $X$: $$0 \rightarrow X \rightarrow I_0 \rightarrow I_1 \rightarrow \cdots \rightarrow I_{g-1} \rightarrow \Omega^{-g}(X) \rightarrow 0.$$ This exact sequence is also a minimal projective resolution of $\Omega^{-g}(X)$, since $X$ has dominant dimension at least $g$. Thus $\Omega^{g}(\Omega^{-g}(X))=X$ and so $X \in \Omega^{g}(A-mod)$ is Gorenstein projective, by . 2. Since $A$ is CM-finite, $Dom_g$ is representation-finite by 1., since it is a subcategory of the representation finite category $Gp(A)$. Thus there are only finitely many indecomposable modules having dominant dimension larger than or equal to $g$ and so the finitistic dominant dimension must be finite. We note that in the next section, we give an optimal bound for the finitistic dominant dimension in case of a CM-finite gendo-symmetric Gorenstein algebra. We give two applications of the previous lemma to gendo-$d$-Gorenstein algebras and the representation dimension of algebras. Let $A$ have dominant dimension at least 2 and minimal faithful injective-projective module $eA$ and let $B=B_2$ as in the preliminaries. Define $C:=End_A(A \oplus G)$, where $G$ is a Gorenstein projective module. Let $M:=F_2(A \oplus G)$, with $F_2$ as in \[ARSmaintheorem\]. Then $C$ has dominant dimension equal to $\inf\{ i \geq 1 \| Ext_B^{i}(M,M) \neq 0 \}$+1 and $C \cong End_B(M)$. There is the equivalence $F_2:Dom_2 \rightarrow mod-B$, which restricts to an equivalence $F_2:Gp(A)\rightarrow X$, for a subcategory $X$ of $mod-B$. $F_2$ induces an isomorphism of the algebras $C$ and $End_B(M)$ and then one can apply Mueller’s theorem to calculate the dominant dimension of $C$. Note that the previous proposition applies to gendo-$d$-Gorenstein algebras. We give an interesting special case as a corollary, which allows one to find endomorphism algebras of a generator-cogenerator of finite global dimension for a given algebra, when one is able to find endomorphism rings of a generator-cogenerator which are CM-finite and Gorenstein. Let $A=End_B(N)$ have dominant dimension at least 2 and Gorenstein dimension $d \geq 2$ and minimal faithful injective-projective module $eA$. Assume also that $A$ is CM-finite. Let $C:=End_A(G)$ be the gendo-$d$-Gorenstein algebra such that $G$ is the direct sum of all Gorenstein projective modules ($C$ is then the so called Cohen-Macauley Auslander algebra, see for examples). Then $C$ has finite global dimension equal to the Gorenstein dimension of $A$ and is isomorphic to an algebra of the form $End_{B}(M)$, where $M$ is a generator-cogenerator. The only missing part is that the global dimension of $C$ is equal to the Gorenstein dimension of $A$. This is proven in Corollary 6.8. of [@Bel]. For the next proposition we recall that the representation dimension of an algebra $A$ is defined as $Repdim(A):=\inf \{ gldim(End_A(M) \| M $ is a generator-cogenerator of $mod-A \}$. Iyama proved in [@Iya2] that the representation dimension is always finite. \[repdim\] $Repdim(A)= \inf \{ Gordim(End_A(M)) \| M$ is a generator-cogenerator and $End_A(M)$ is CM-finite $\}$. First note that $\inf \{ Gordim(End_A(M)) \| M$ is a generator-cogenerator and $End_A(M)$ is CM-finite$\}$ $\leq$ $Repdim(A)$, since the set $\{M \| M$ is a generator-cogenerator and $End_A(M)$ is CM-finite $\}$ contains the set $\{M \| M $ is a generator-cogenerator of $mod-A$ such that $End_A(M)$ has finite global dimension $\}$. Now assume that the infimum in $\inf \{ Gordim(End_A(M)) \| M$ is a generator-cogenerator and $End_A(M)$ is CM-finite $\}$ is attained at the module $N$. Let $T:=End_A(N)$ and $C:=End_T(T\oplus G)$, where $G$ is the direct sum of all nonprojective Gorenstein projective modules of $mod-T$. By the previous corollary the Gorenstein dimension of $T$ coincides with the global dimension of $C$ and $C$ is isomorphic to $End_A(L)$ for some generator-cogenerator $L$ of $mod-A$. Thus the result follows. The following proposition characterises when the category of Gorenstein projective modules coincides with the category of modules of dominant dimension $d$, for an algebra with dominant dimension $d \geq1$. \[domdimgordim\] Let $A$ be an algebra with dominant dimension $d \geq 1$. Then the following two statements are equivalent: 1. $Gp(A)=Dom_d$. 2. $A$ is Gorenstein with Gorenstein dimension $d$. First assume that $Gp(A)=Dom_d$. Note that by $Dom_d=\Omega^{d}(A-mod)$ and thus $Gp(A)=\Omega^{d}(A-mod)$ and so $A$ has Gorenstein dimension $d$ by . Now assume that $A$ has Gorenstein dimension $d$. Then $Gp(A)=\Omega^{d}(A-mod)=Dom_d$, by and . We note that in [@Mar] the gendo-symmetric Nakayama algebras with dominant dimension equal to the Gorenstein dimension are classified. With the previous proposition, we are also able to give a very short proof of the higher Auslander correspondence (see [@Iya3] Theorem 2.6) and add another equivalent condition. \[auscor\] Let $M$ be a generator-cogenerator of $B$ and $A:=End_B(M)$ and $d \geq 2$. Then the following are equivalent: 1. $proj=Dom_d$. 2. $A$ has dominant and global dimension equal to $d$ 3. $M$ is maximal $(d-2)$-orthogonal. $1. \implies 2.$: We have that $Gp(A) \subseteq Dom_d =proj$ and thus $Dom_d=Gp(A)=proj$. By the previous proposition, this means that $A$ has Gorenstein dimension $d$. But $Gp(A)=proj$ implies that $A$ is CM-free and thus has global dimension equal to the Gorenstein dimension $d$. $2. \implies 1.$: The category of Gorenstein projective modules of an algebra with finite global dimension coincides with $proj$. Thus by the previous proposition $proj=Dom_d$ holds. $1. \Leftrightarrow 3.$: Using the functor $F_2$, the condition $proj=Dom_d$ translates into $add(M)=M^{\perp (d-2)}$ and this implies that $M$ is maximal $d-2$ orthogonal, since $M$ is assumed to be a generator-cogenerator. Surprisingly there are also many examples of algebras that are Gorenstein dominant algebras and with dominant dimension 1. In fact every Nakayama algebra is a Gorenstein dominant algebra, despite the fact that many Nakayama algebras have dominant dimension equal to one. Note that Nakayama algebras with a line as a quiver are trivially Gorenstein dominant algebras, since $RGp(A)$ consists only of the zero module. \[nakgordom\] Let $A$ be a nonselfinjective CNakayama algebra. 1. If a vertex $i$ is cyclically black, then $D(Ae_{i-1})$ is a projective module. 2. The Gorenstein core of $A$ is a subcategory of $Dom_2$ and thus $A$ is a Gorenstein dominant algebra. <!-- --> 1. Since every Nakayama algebra has dominant dimension at least 1, the minimal injective hull of a projective module $e_i A$ in the Gorenstein core looks as follows: $0 \rightarrow e_i A \rightarrow D(Ae_{i+c_i-1})$, where $D(Ae_{i+c_i-1})$ is projective and injective. By the definition of cyclically black vertices, with the vertex $i$ also the vertex $l:=i+c_i$ is cyclically black and every cyclically black vertex can be written in the form $j+c_j$, where $j$ is another cyclically black vertex. Thus for every cyclically black vertex $i$, the module $D(Ae_{i-1})$ is projective. 2. A minimal injective corepresentation of an arbitrary indecomposable module $M=e_iA/e_iJ^k$ in the Gorenstein core looks as follows: $$0 \rightarrow M \rightarrow D(Ae_{i+k-1}) \rightarrow D(A_{i-1}).$$ Note that a minimal projective presentation of $M$ is given by: $$e_{i+k}A \rightarrow e_iA \rightarrow M \rightarrow 0.$$ But by $i$ and $i+k$ are cyclically black vertices and thus by the previous part $D(Ae_{i+k-1})$ and $D(A_{i-1})$ are projective. Since the dominant dimension of a finite direct sum of modules is the minimal dominant dimension of the direct summands, we see now that every module in the Gorenstein core has dominant dimension at least 2. Gorenstein projective-injective modules in gendo-symmetric algebras =================================================================== The next propostion will be one of our main tools in this section. We recall the following from [@FanKoe2], which gives a first clue for a connection between dominant dimensions and Gorenstein homological properties in gendo-symmetric algebras: \[domdimform\] Let $A$ be a nonselfinjective gendo-symmetric algebra and $M$ an $A$-module. 1. $M$ has dominant dimension larger than or equal to 2 iff $\nu^{-1}(M) \cong M$ and in this case the dominant dimension of $M$ is equal to $\inf \{ i \geq 1 \| Ext^{i}(D(A),M) \neq 0 \}$+1. 2. $M$ has codominant dimension larger than or equal to 2 iff $\nu(M) \cong M$ and in this case the codominant dimension of $M$ is equal to $\inf \{ i \geq 1 \| Ext^{i}(M,A) \neq 0 \}$+1. <!-- --> 1. This is proven in [@FanKoe2] in Proposition 3.3. 2. This is dual to (1). Let $A$ be a finite dimensional algebra and $M$ an $A$-module. The following holds: 1. Let $P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$ be a minimal projective presentation of $M$. Then there exists an exact sequence in mod$-A$ of the form: $0 \rightarrow \tau(M) \rightarrow \nu(P_1) \rightarrow \nu(P_0) \rightarrow \nu(M) \rightarrow 0.$ 2. Let $0 \rightarrow M \rightarrow I_0 \rightarrow I_1$ be a minimal injective corepresentation of $M$. Then there exists an exact sequence in mod$-A$ of the form: $0 \rightarrow \nu^{-1}(M) \rightarrow \nu^{-1}(I_0) \rightarrow \nu^{-1}(I_1) \rightarrow \tau^{-1}(M) \rightarrow 0.$ This follows from the definitions of Auslander-Reiten translates, see [@SkoYam] Chapter III., Proposition 5.3. The following proposition is inspired by [@FanKoe3] Lemma 3.4.: \[tauomega\] Let $A$ be a gendo-symmetric algebra and $M$ an $A$-module. 1. If $M$ has codominant dimension larger than or equal to 2 iff $\tau(M) \cong \Omega^{2}(M)$. 2. If $M$ has dominant dimension larger than or equal to 2 iff $\tau^{-1}(M) \cong \Omega^{-2}(M)$. We prove only (1), since the proof of (2) is dual. Assume that $M$ has codominant dimension larger than or equal to 2. Assume that $P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$ is the minimal projective presentation of $M$. Then by the previous proposition, there is the following exact sequence: $0 \rightarrow \tau(M) \rightarrow \nu(P_1) \rightarrow \nu(P_0) \rightarrow \nu(M) \rightarrow 0.$ But since $P_1,P_0$ are also injective and thus have codominant dimension larger than or equal to 2: $\nu(P_1) \cong P_1$ and $\nu(P_0) \cong P_0$. As $M$ also has codominant dimension larger than or equal to 2: $\nu(M) \cong M$ and the exact sequence looks like the beginning of a minimal projective resolution of $M$: $0 \rightarrow \tau(M) \rightarrow P_1 \rightarrow P_0 \rightarrow M \rightarrow 0.$ Thus $\Omega^{2}(M) \cong \tau(M)$. Assume now that $\Omega^{2}(M) \cong \tau(M)$. Then look at the minimal projective presentation $P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$ of $M$ and by the previous proposition we get a minimal injective coresolution of $\tau(M)$ as follows. $0 \rightarrow \tau(M) \rightarrow \nu(P_1) \rightarrow \nu(P_0) \rightarrow \nu(M) \rightarrow 0$. But now since $\Omega^{2}(mod-A)=Dom_2$, $\tau(M) \cong \Omega^{2}(M)$ has dominant dimension at least two and thus in the above minimal injective coresolution $\nu(P_1)$ and $\nu(P_0)$ are projective-injective, which implies that $P_0$ and $P_1$ are also projective-injective, since in a gendo-symmetric algebra a module $P$ is projective-injective iff $P \in add(eA)$. Thus $M$ has codominant dimension at least two. The following gives the first link between dominant dimensions of modules and Gorenstein homological algebra: \[mainresult\] Let $A$ be a gendo-symmetric algebra and $M$ an $A$-module. Then the following are equivalent: 1. $M$ is Gorenstein projective-injective. 2. $M$ has infinite dominant dimension and infinite codominant dimension. $(1) \rightarrow (2)$: A projective module is projective-injective iff it is a Gorenstein projective-injective module. Assume now that $M$ is Gorenstein projective-injective and not projective. We conclude using that $M$ has dominant dimension and codominant dimension larger than or equal to 2, since $A$ has dominant dimension at least 2 and thus is a Gorenstein dominant algebra. Since $M$ is Gorenstein injective, $Ext^{i}(D(A),M)=0$ for all $i \geq 1$ and thus $M$ has infinite dominant dimension by \[domdimform\]. Since $M$ is Gorenstein projective $Ext^{i}(M,A)=0$ for all $i \geq 1$ and thus $M$ has infinite codominant dimension again by \[domdimform\]. $(2) \rightarrow (1)$: Assume now that $M$ has infinite dominant and infinite codominant dimension. Clearly then also all the modules $\Omega^{i}(M)$ have infinite dominant and codominant dimensions for every $i \in \mathbb{Z}$. Then $\nu^{-1}(M) \cong M$, since the dominant dimension of $M$ is larger than or equal 2. And dually $\nu(M) \cong M$, since the codominant dimension of $M$ is larger than or equal to 2. Since $M$ has infinite codominant dimension, we get $Ext^{i}(M,A)=0$ for all $ i \geq 1$ and with the previous lemma the following holds: $\tau(M) \cong \Omega^{2}(M)$. Now $M$ is Gorenstein projective iff additionally $Ext^{i}(Tr(M),A)=0$ for all $i \geq 1$. But $Ext^{i}(Tr(M),A)= Ext^{i}(D(A),\tau(M))=Ext^{i}(D(A),\Omega^{2}(M)))=0$ for all $i \geq 1$, since with $M$ also $\Omega^{2}(M)$ has infinite dominant dimension. Thus $M$ is Gorenstein projective. Since $M$ has infinite dominant dimension, $Ext^{i}(D(A),M)=0$ for all $ i \geq 1$ and by the previous lemma the following holds: $\tau^{-1}(M) \cong \Omega^{-2}(M)$. Now $M$ is Gorenstein injective iff additionally $Ext^{i}(\tau^{-1}(M),A)=0$ for all $i \geq 1$. But $Ext^{i}(\tau^{-1}(M),A)=Ext^{i}(\Omega^{-2}(M),A)=0$ for all $i \geq 1$, since with $M$ also $\Omega^{-2}(M)$ has infinite codominant dimension. \[CorCM\] Let $A$ be a gendo-symmetric nonselfinjective algebra. 1. If an Auslander-Reiten component contains a nonprojective Gorenstein projective-injective module then this component consists only of Gorenstein projective-injective modules. Thus the indecomposable nonprojective Gorenstein projective-injective modules form unions of stable Auslander-Reiten components of the algebra. 2. If $A$ is CM-finite, then $A$ contains no nonprojective Gorenstein projective-injective module. <!-- --> 1. Assume that $M$ is a nonprojective Gorenstein projective-injective module. With $M$ being Gorenstein projective-injective also $\tau(M) \cong \Omega^{2}(M)$ and $\tau^{-1}(M) \cong \Omega^{-2}(M)$ are Gorenstein projective-injective, since being Gorenstein projective-injective is equivalent to having infinite dominant and infinite codominant dimension. In a Auslander-Reiten sequence:$ 0 \rightarrow \tau(M) \rightarrow S \rightarrow M \rightarrow 0$, also $S$ is Gorenstein projective-injective, since with $M$ and $\tau(M) \cong \Omega^{2}(M)$ also the middle term $S$ has infinite dominant and codominant dimension by the Horseshoe lemma. Thus every module in the Auslander-Reiten component containing $M$ is a Gorenstein projective-injective module. 2. Assume that $A$ is CM-finite and contains a nonprojective module $M$ that is Gorenstein projective-injective. Then $A$ contains a whole Auslander-Reiten component of modules that are Gorenstein projective-injective by 1. Then there are 2 possible cases: 1. The Auslander-Reiten component is infinite. But then $A$ is not CM-finite. This is a contradiction. 2. The Auslander-Reiten component is finite. Then $A$ is representation finite and thus the Nakayama conjecture holds true for $A$. But by 1., every module in the Auslander-Reiten component has infinite dominant and codominant dimension despite that fact that the dominant dimension of $A$ is finite. This is a contradiction. We have the following diagram for gendo-symmetric algebras, where we use the notations as in and $\mathcal{W}:= \{ X \in ^{\perp}{Ae} \cap {Ae}^{\perp}\| G_1(X) \cong G_2(X) \}$ (we will see later that $\mathcal{W}$ can be desribed much better in case the algebra is nearly Gorenstein): $$\xymatrix@1{ & \text{Gpi}(A)=\text{Dom}\cap \text{Codom} \mystrut\ar@{^{(}->}[dr] \ar@{_{(}->}[dl] \ar[dddd]_{(-)e} & \\ \text{Dom} \ar@/^/[dd]^{F_2=(-)e} & & \text{Codom} \ar@/^/[dd]^{F_1=(-)e} \\ & & \\ {(Ae)}^{\perp} \ar@/^/[uu]^{G_2=\text{Hom}_{eAe}(eA,-)}& & {}^{\perp}(Ae) \ar@/^/[uu]^{G_1=(-)\otimes_{eAe} eA} \\ & \mathcal{W} \mystrut\ar@{_{(}->}[ru]\ar@{^{(}->}[lu] & }$$ Using this diagram, we can prove the next theorem: \[diagram\] Let $A$ be a gendo-symmetric nonselfinjective algebra with minimal faithful projective-injective module $eA$. Then $(-)e: Gpi(A) \rightarrow \mathcal{W}$ is an equivalence. Recall that $F_i$ is an equivalence with quasi-inverse $G_i$ as explained in \[ARSmaintheorem\] for $i=1,2$. This makes it clear that $(-)e$ really maps $Gpi(A)$ to $\mathcal{W}$. Since $(-)e$ is fully faithful on $Dom$, also the restiction to $Gpi(A)$ is fully faithful. Now let $X \in \mathcal{W}$, then just note that $(G_i(X))e \cong F_i(G_i(X)) \cong X$ for $i=1,2$ and thus $(-)e$ is dense. \[finitistic\] Let $A$ be a nonselfinjective gendo-symmetric algebra, which additionally is a nearly Gorenstein algebra. 1. The following are equivalent for a noninjective and nonprojective indecomposable module $M$: 1. $M$ has infinite dominant dimension. 2. $M$ is Gorenstein projective-injective and $\nu^{-1}(M) \cong M$. 3. $M$ is Gorenstein projective-injective. 4. $M$ has infinite codominant dimension. 2. If $A$ is CM-finite and Gorenstein with Gorenstein dimension $g$, then every noninjective module has finite dominant dimension and $fdomdim(A) \leq g+1$. <!-- --> 1. $i) \rightarrow ii)$: Assume $M$ has infinite dominant dimension. Since $A$ is a gendo-symmetric algebra, this is equivalent to the two conditions $\nu^{-1}(M)=Hom(D(A),M) \cong M$ and $Ext^{i}(D(A),M)=0$ for all $i\geq 1$. Since we assume that the subcategory of Gorenstein-injective modules coincides with $D(A)^{\perp}$, $M$ is Gorenstein-injective. Now by \[taugorenstein\] (1), if a module $X$ is Gorenstein-injective, then $Hom_A(D(A),X)$ is Gorenstein-projective. Using this with $X=M$, we get that $M \cong Hom_A(D(A),M)$ is Gorenstein-projective. $ii) \rightarrow i)$: Since $\nu^{-1}(M)=M$, $M$ has dominant dimension larger than or equal to 2. Because $M$ is Gorenstein-injective, $Ext^{i}(D(A),M)=0$ for all $i\geq 1$. Then domdim($M$)=$\infty$ is clear. $ii) \leftrightarrow iii)$: Since $A$ is gendo-symmetric and nearly Gorenstein, $A$ is also Gorenstein dominant and thus all Gorenstein projective modules have dominant dimension at least 2. But in a gendo-symmetric algebra this is equivalent to $\nu^{-1}(M) \cong M$. $iv) \leftrightarrow iii)$: Now codomdim($M$)=$\infty$ is equivalent to domdim($D(M)$)=$\infty$ and using the equivalence of $i)$ and $ii)$, this is equivalent to $\nu^{-1}(D(M)) \cong D(M)$ and $D(M)$ is Gorenstein-injective and Gorenstein-projective. But, since $\nu^{-1}=Hom_A(-,A) \circ D$, this is equivalent to $Hom_A(M,A) \cong D(M)$ and that $M$ is Gorenstein-projective and Gorenstein-injective. Now $\nu^{-1}(M)=DD(M) \cong M$ and the equivalence of $iii)$ and $iv)$ is clear. 2. By , $A$ does not contain a nonprojective Gorenstein projective-injective module and thus no noninjective module of infinite dominant dimension. Assume now $M$ is a module with $\infty>i=domdim(M) > g+1$. Then $Ext^{i-1}(D(A),M) \neq 0$ by \[domdimform\], contradicting the fact that $D(A)$ has projective dimension equal to $g$. The next example shows that the bound for the finitistic dominant dimension in the previous theorem is optimal: Let $A$ be the so called penny-farthing algebra (see [@GR]) with two simple modules given by quiver and relations as follows: $$\begin{xy} \xymatrix{ & {\bullet}^2 \ar@/ _1pc/[r]_{\beta_2} & {\bullet}^1 \ar@(ur,dr)[]^{\alpha} \ar@/ _1pc/[l]_{\beta_1}} \end{xy}$$ The relations are: $I=<\alpha^2-\beta_1 \beta_2, \beta_2 \beta_1>.$ Thus $A=kQ/I$ and $A$ is a symmetric algebra. Let $S_i$($e_i$) be the simple module (primitive idempotent) corresponding to the vertex $i$ and $J$ the radical of $A$. We show that the algebra $B:=End_A(A \oplus S_2)$ has dominant dimension and Gorenstein dimension equal to 3 and finitistic dominant dimension equal to 4. Note that $B$ is a CM-finite nearly Gorenstein algebra using (2), since the penny-farthing algebra is representation-finite. By explicitly giving the relevant minimal projective resolutions (see below) and using \[benson\], we will show that $Ext^{1}(S_2,S_2)=0$ but $Ext^{2}(S_2,S_2) \neq 0$ and $Ext^{i}(S_2,e_2J^2)=0$ for $i=1$ and $i=2$, but $Ext^{3}(S_2,e_2J^2) \neq 0$. Thus the gendo-symmetric algebra $B$ has dominant dimension 3 and the module $Hom_A(A \oplus S_2, e_2 J^{2})$ has dominant dimension 4 by and . Because of $\tau(\Omega^{1}(S_2))=\Omega^3(S_2)=S_2$, one also conlcudes that $B$ has Gorenstein dimension 3 using . The relevant minimal projetive resolutions of $S_2$ and $e_2J^{2}$ needed to proof the above statements are (note that both modules have period 3 and thus one can read off minimal projective and minimal injective resolutions): $$\begin{xy} \xymatrix@C-1pc@R-1pc{ & e_2A\ar[rd]^{} & & e_2A \ar[rd]^{} & & e_1A \ar[rd]^{} & & e_2A\ar[r] & S_2\ar[r]^{} & 0 \\ \cdots\ar[ru]^{} & & S_2 \ar[ru]^{} & & \beta_1 A \ar[ru]^{} & & e_2J^{1}\ar[ru]^{} & & & } \end{xy}$$ $$\begin{xy} \xymatrix@C-1pc@R-1pc{ & e_1A \ar[rd]^{} & & e_2A \ar[rd]^{} & & e_1A \ar[rd]^{} & & e_1A \ar[r] & e_2J^{2} \ar[r]^{} & 0 \\ \cdots\ar[ru]^{} & & e_2J^{2} \ar[ru]^{} & & \alpha \beta_1A \ar[ru]^{} & & \alpha A \ar[ru]^{} & & & } \end{xy}$$ \[perpcor\] Let $A$ be a gendo-symmetric and nearly Gorenstein algebra with minimal faithful injective-projective module $eA$. Then the functor $(-)e : Gpi(A) \rightarrow eA^{\perp}$ is an equivalence of categories and $eA^{\perp} = ^{\perp}{Ae} = ^{\perp}{Ae} \cap {Ae}^{\perp}$. This follows from and . Assume that $A$ is a symmetric algebra and $M$ a generator of $A$-mod and let $B:=End_A(M)$. Assume that there is an equivalence $H:^{\perp}M \cong M^{\perp}$, then there is an equivalence of categories $G_1HF_2: Dom \rightarrow Codom$ with inverse $G_2HF_1: Codom \rightarrow Dom$. This follows with the diagram 1, using $M \cong D(eA) \cong Ae$ as $eAe$-modules. We will see in forthcoming work (see [@ChMar2]) that the finitistic dominant dimension of representation-finite, gendo-symmetric and special biserial algebras (which are Gorenstein and generalize the class of Brauer tree algebras, see [@ChMar1]) always equals $g$ or $g+1$, when $g$ is the Gorenstein dimension of such an algebra. In particular, there we show that gendo-symmetric Nakayama algebras always have finitistic dominant dimension equal to their Gorenstein dimension. The following proposition allows us to check whether a module is Gorenstein projective-injective by calculating only finitely many terms in a projective or injective minmal resolution of this module. Let $A$ be a gendo-symmetric nonselfinjective algebra of Gorenstein dimension $g$. Then for a module $M$ the following are equivalent: 1. $M$ is Gorenstein projective-injective. 2. $domdim(M)+codomdim(M) \geq 2g$. 3. $domdim(M) \geq 2g$. 4. $codomdim(M) \geq 2g$. $1. \Rightarrow 2.:$ We saw in that a module $M \in Gpi(A)$ has infinite dominant and codominant dimension, thus $domdim(M)+codomdim(M) \geq 2g$. $2. \Rightarrow 1.:$ Assume $domdim(M)=i$ and $codomdim(M)=j$, and $i+j \geq 2g$. Without loss of generality, assume that $i \geq j$. Then the module $W=\Omega^{-i+g}(M)$ can be written $W=\Omega^{g}(\Omega^{-i}(M))$ and $W=\Omega^{-g}(\Omega^{s}(M))$ for some $s \leq j$. Thus by the characterisation of Gorenstein algebras , $W$ is Gorenstein projective-injective and so is every syzygy of $W$ and thus also $M=\Omega^{-k}(W)$. $1. \Rightarrow 3.:$ This is clear since a Gorenstein projective-injective module has infinite dominant dimension in a gendo-symmetric algebra by \[mainresult\]. $3. \Rightarrow 2.:$ This is clear. $4. \Leftrightarrow 1. $ is dual to $3. \Leftrightarrow 1.$ We now give a class of examples of nonselfinjective gendo-symmetric algebras containing Auslander-Reiten compontent of Gorenstein projective-injective modules. Let $B$ be a symmetric algebra and $M:=B \oplus W$, where $W$ is a nonzero 2-periodic module. 1. The gendo-symmetric algebra $A:=End_B(M)$ has dominant dimension 2 and Gorenstein dimension 2. 2. $^{\perp}M= M^{\perp}$ 3. $M^{\perp}=\{X \in $mod$-B \| Ext^{i}(M,X)=0$, for $i=1,2 \}$. 4. If $W$ is even 1-periodic, then $M^{\perp}=\{X \in $mod$-B \| Ext^{1}(M,X)=0 \} \}$. <!-- --> 1. We use :$Ext^{1}(M,M) \cong \underline{Hom}(M,\Omega^{2}(M)) \cong \underline{Hom}(M,M) \neq 0$, since the identity does not factor over the projectives. Thus $A$ has dominant dimension 2. Now we use to calculate the Gorenstein dimension. But the right $M$-approximitation of $\tau(M) \cong \Omega^{2}(M) \cong M$ is an isomorphism $f:\Omega^{2}(M) \rightarrow M$, and thus the length of a resolution is 0 and so the right Gorenstein dimension is also equal to two. Similar the left Gorenstein dimension is calculated to be 2. 2. This follows by . 3. We can write every natural number $i$ as $i=2r+p,$ for $r \geq 0$ and $p \in \{1,2\}$. Then $Ext^{i}(M,X) \cong \underline{Hom}(\Omega^{i}(M),X) \cong \underline{Hom}(\Omega^{p}(\Omega^{2r}(M),X) \cong Ext^{p}(M,X)$, since $M$ is 2-periodic. 4. In this case $Ext^{i}(M,X) \cong \underline{Hom}(\Omega^{i}(M),X) \cong \underline{Hom}(\Omega^{1}(M),X) \cong Ext^{1}(M,X) $ for every $i \geq 1.$ In a tame algebra, all but finitely many indecomposable modules $M$ of a given dimension have the property that $\tau(M) \cong M$, by a famous result of Crawley-Boevey, see [@Cra]. Thus with the previous proposition one might construct alot of examples, since in a tame symmetric algebra $\tau(M) \cong \Omega^{2}(M) \cong M$ then holds for all but finitely many indecomposable modules of a given dimension. We noted that for every 2-periodic module $M$ in a symmetric algeba the following holds: $^{\perp}M= M^{\perp}$. We give an alternative direct proof of this fact using the formulas in : First note that $^{\perp}M = \{ X \| Ext^{i}(X,M)=0 $ for $i=1,2 \}$ with a similar argument as in the previous proposition: We can write every natural number $i$ as $i=2r+p,$ for $r \geq 0$ and $p \in \{0,1\}$. Then $Ext^{i}(X,M) \cong \underline{Hom}(\Omega^{i}(X),M) \cong \underline{Hom}(M,\Omega^{i+1}(X)) \cong \underline{Hom}(\Omega^{-(i+1)}(M),X) \cong \underline{Hom}(\Omega^{-p+1}(\Omega^{2r}M),X) \cong Ext^{1+p}(M,X)$. Then note that $Ext^{1}(X,M) \cong \underline{Hom}(X,\Omega^{-1}(M)) \cong \underline{Hom}(\Omega^{-1}(M),\Omega^{1}(X)) \newline \cong \underline{Hom}(\Omega^{-2}(M),X) \cong \underline{Hom}(M,X) \cong \underline{Hom}(\Omega^{-1}(M),\Omega^{-1}(X)) \newline \cong \underline{Hom}(\Omega^{1}(M),\Omega^{-1}(X)) \cong Ext^{1}(M,\Omega^{-1}(X)) \cong Ext^{2}(M,X)$. Furthermore: $Ext^{2}(X,M) \cong \underline{Hom}(\Omega^{2}(X),X) \cong \underline{Hom}(M,\Omega^{3}(X)) \newline \cong \underline{Hom}(\Omega^{-3}(M),X) \cong \underline{Hom}(\Omega^{1}(M),(X)) \cong Ext^{1}(M,X)$. Assume $k$ is an infinite field of characteristic 2. Let $A$ be the symmetric algebra $k[x,y]/(x^2,y^2,xy-yx)$. We use the notation and results as in Example 10.7 in page 417 of [@SkoYam]. There $M(a,b)$ is defined as the module $A/(ax+by)A$, for $a,b$ nonzero elements of $k$. Note that $M(a,b)$ is isomorphic to $M(c,d)$ for nonzero $c,d$ iff there is an $l \in k$ with $c=l \cdot a$ and $d=l \cdot b$. Then $\Omega^{1}(M(a,b))=(ax+by)A \cong M(-a,b)$ (note this holds despite the fact that they excluded $\lambda=1$ in [@SkoYam]). But $M(-a,b)=M(a,b)$ since we assume that the field has characteristic 2 and thus $M(a,b)$ is 1 periodic and we can apply the previous proposition. We now want to calculate $Ext^{1}(M(a,b),M(c,d))$, for nonzero $c$ and $d$ such that $M(c,d)$ is not isomorphic to $M(a,b)$. For this we use the short exact sequence $0 \rightarrow M(a,b) \rightarrow A \rightarrow M(a,b) \rightarrow 0$ and apply the functor $Hom_A(-,M(c,d))$ to get the long exact sequence: $0 \rightarrow Hom(M(a,b),M(c,d)) \rightarrow Hom(A,M(c,d)) \rightarrow Hom(M(a,b),M(c,d)) \newline \rightarrow Ext^{1}(M(a,b),M(c,d)) \rightarrow \cdots $. Now we see that $Ext^{1}(M(a,b),M(c,d))=0$ iff $0 \rightarrow Hom(M(a,b),M(c,d)) \rightarrow Hom(A,M(c,d)) \rightarrow Hom(M(a,b),M(c,d)) \rightarrow 0$ is a short exact sequence, which is the case iff $2=dim(M(c,d))=dim(Hom(A,M(c,d)))=2dim(Hom(M(a,b),M(c,d))).$ But the last equation is true since $Hom(M(a,b),M(c,d))=1$, since up to multiplication by a scalar the only homomorphism is the projection from the top into the socle(note that both module have dimension $2$). Thus $Ext^{1}(M(a,b),M(c,d))=0$ and $Ext^{k}(M(a,b),M(c,d))=0$ for all $k \geq 1$ since $Ext^{k}(M(a,b),M(c,d))=Ext^{1}(\Omega^{k-1}(M(a,b)),M(c,d))=Ext^{1}(M(a,b),M(c,d))=0$. Then $B:=End_A(A\oplus M(a,b))$ is a Gorenstein and gendo-symmetric algebra. Thus by the previous proposition, the algebra $B$ has $Hom_A(A \oplus M(a,b),M(c,d))$ as a Gorenstein projective-injective module and by it has a whole Auslander-Reiten component consisting of Gorenstein projective-injective modules. We note that most of the previous results are special to gendo-symmetric algebras as the following example shows: Take the CNakayama algebra $A$ with Kupisch series $(3s+1,3s+2,3s+2), s \geq 1$ ($A$ is not gendo-symmetric). We first calculate the Gorenstein dimension and dominant dimension of $A$ and then the finitistic dominant dimension of $A$. First note that $e_1 A \cong D(A e_2)$ is injective. Also $e_2 A \cong D(A e_0)$ is injective. The only noninjective indecomposable projective module is then $e_0 A$ and the only nonprojective injective indecomposable module is $D(Ae_1)$. We have the following injective resolution: $0 \rightarrow e_0 A \rightarrow D(A e_0) \rightarrow D(A e_2) \rightarrow D(A e_1) \rightarrow 0$. Thus the dominant dimension and the Gorenstein dimension of $A$ are both 2. Now take an indecomposable module $M=e_aA/e_aJ^k$ and calculate the minimal injective presentation of $M$: $0 \rightarrow M \rightarrow D(A e_{a+k-1}) \rightarrow D(A e_{a-1})$. Thus $M$ has dominant dimension larger than or equal to 2 iff $a+k-1 \in \{0,2 \}$ mod $3$ and $a-1 \in \{0,2 \}$ mod $3$ iff ($a=0$ mod $3$ and $k \in \{0,1 \}$ mod $3$) or ($a=1$ mod $3$ and $ k \in \{0,2 \}$ mod $3$). The following table gives the relevant values of the dominant dimensions: [l\[6]{}[c]{}r]{} a= & 0 & 1\ $k \equiv 0$ & 4 & 2\ $k \equiv 1$ & 2 & -\ $k \equiv 2$ & - & 2\ Thus the finitistic dominant dimension equals 4, while the finitistic dimension equals the Gorenstein dimension which is 2 (note that by , this can not happen for gendo-symmetic CM-finite Gorenstein algebras). The resolution quiver looks as follows: $$\begin{xy} \xymatrix{ & {\bullet}^0 \ar@/ _1pc/[r]_{} & {\bullet}^1 \ar@/ _1pc/[l]_{} & {\bullet}^2 \ar@/ _1pc/[l]_{}} \end{xy}$$ Thus the Gorenstein-projective modules have the form $e_iA/e_iJ^{k}$, with $i \in \{0,1\}$ and $i+k \in \{0,1\}$. Note that in general a module $M$ is Gorenstein-injective iff $\tau^{-1}(M)$ is Gorenstein-projective. So $e_jA/e_jJ^{l}$ is Gorenstein-injective iff $e_{j-1}A/e_{j-1}J^{l}$ is Gorenstein-projective iff $j-1 \in \{0,1\}$ and $j-1+l \in \{0,1\}$ iff $j \in \{1,2\}$ and $j+l \in \{1,2\}$. Thus the modules of the form $e_1 A/e_1 J^k$ with $k \equiv 0$ are exactly the Gorenstein projective-injective modules but those modules have dominant dimensions equal to 2. There is also no Auslander-Reiten component consisting only of Gorenstein projective-injective modules. Questions and comments ====================== 1. Is every left nearly Gorenstein algebra also right nearly Gorenstein? 2. Is there a natural construction of (gendo-symmetric) algebras with dominant dimension at least two that are not nearly Gorenstein? Since we proved that the Nakayama conjecture holds for nearly Gorenstein algebras, it is a natural question to search for algebras with dominant dimension at least two and not being nearly Gorenstein. Algebras not being nearly Gorenstein seem to have been found first in [@JS], but those algebras are local algebras having dominant dimension 0. Thus, when one attempts to disprove the Nakayama conjecture (or the Tachikawa conjecture for symmetric algebras), one first has to find (gendo-symmetric) algebras which are not nearly Gorenstein. This questions seems to be highly non-trivial concerning the fact that only after about 40 years after the definition of Gorenstein projective modules, a first counterexample of a non-nearly Gorenstein algebra appeared in [@JS] for finite dimensional algebras. 3. How do the Auslander-Reiten components of Gorenstein projective-injective modules look like? Can they be classified? 4. Is there an example of a gendo-symmetric algebra with infinite finistic dominant dimension? Is the finitistic dominant dimension somehow related to the classical finitistic dimension for general algebras (for example by some inequalities)? Note that in case of gendo-symmetric CM-finite Gorenstein algebras of Gorenstein dimension $g$, there is a positive answer by Theorem B from the introduction since the finitistic dimension coincides with the Gorenstein dimension in case the Gorenstein dimension is finite. [Gus]{} Abrar, Muhammad : [Dominant dimensions of finite dimensional algebras.]{} PhD-thesis Stuttgart (2012) <http://elib.uni-stuttgart.de/opus/volltexte/2012/7548/pdf/Abrar.pdf>. retrieved 2015-06-18. Auslander, Maurice; Platzeck, Maria Ines; Todorov, Gordana: [Homological theory of idempotent Ideals.]{} Transactions of the American Mathematical Society, Volume 332, Number 2 , August (1992). Auslander, Maurice; Reiten, Idun: [Syzygy modules for noetherian rings.]{} Journal of algebra 183, 167-185 (1996). Auslander, Maurice; Bridger, Mark: [Stable Module Theory.]{} Memoirs of the American Mathematical Society, No. 94 American Mathematical Society, Providence, R.I. (1969) 146 pp. Auslander, Maurice; Reiten, Idun; Smalo, Sverre: [Representation Theory of Artin Algebras.]{} Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, (1997). xiv+425 pp. Assem, Ibrahim ; Simson, Daniel; Skowronski, Andrzej: [Elements of the Representation Theory of Associative Algebra 1.]{} London Mathematical Society Student Texts, Volume 65, (2006). Beligiannis, Apostolos: [On algebras of finite Cohen-Macaulay type.]{} Adv. Math. 226 (2011), no. 2, 1973-2019. Benson, David J.: [Representations and cohomology I: Basic representation theory of finite groups and associative algebras.]{} Cambridge Studies in Advanced Mathematics, Volume 30, Cambridge University Press, (1991). Chan, Aaron; Marczinzik, René: [On representation-finite gendo-symmetric biserial algebras.]{} http://arxiv.org/abs/1607.05965. Chan, Aaron; Marczinzik, René: [Gorenstein homological algebra of special Gendo-Brauer tree algebras.]{} preprint. Chen, Xiao-Wu: [Gorenstein Homological Algebra of Artin Algebras.]{} <http://home.ustc.edu.cn/~xwchen/Personal%20Papers/postdoc-Xiao-Wu%20Chen%202010.pdf> retrieved 2015-06-18. Chen, Hongxing; Koenig, Steffen: [Ortho-symmetric modules, Gorenstein algebras and derived equivalences.]{} Int. Math. Res. Not. IMRN (2016) electronically published. doi:10.1093/imrn/rnv368. Crawley-Boevey, William: [On tame algebras and bocses]{} Proc. London Math. Soc.56 (1988),451-483 Enochs, Edgar; Jenda, Overtoun: [Relative homological algebra Vol. 1]{} de Gruyter Expositions in Mathematics, (2011). Fang, Ming; Koenig, Steffen: [Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension.]{} Trans. Amer. Math. Soc. 368 (2016), no. 7, 5037-5055. Fang, Ming, Koenig, Steffen: [Endomorphism algebras of generators over symmetric algebras.]{} J. Algebra 332 (2011), 428-433. Fang, Ming, Koenig, Steffen: [Schur functors and dominant dimension]{} Trans. Amer. Math. Soc. 363 (2011), no. 3, 1555-1576. Gao, Nan; Koenig, Steffen: [Grade, dominant dimension and Gorenstein algebras.]{} J. Algebra 427 (2015), 118-141. Gabriel, Peter; Roiter, Andrei: [Representations of Finite-Dimensional Algebras.]{} Reprint of the 1992 English translation. Springer-Verlag, Berlin, (1997). iv+177 pp. Hu, Wei; Xi, Changchang: [Derived equivalences and stable equivalences of Morita type, I.]{} Nagoya Mathematical Journal, Volume 200, 107-152, (2010). Iyama, Osamu: [Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories.]{} Advances in Mathematics, Volume 210, 22-50, (2007). Iyama, Osamu: [Finiteness of representation dimension.]{} Proc. Amer. Math. Soc. 131 (2003), no. 4, 1011-1014. Iyama, Osamu: [Auslander-Reiten theory revisited]{} Trends in representation theory of algebras and related topics, 349-397, EMS Ser. Congr. Rep., Eur. Math. Soc., Zuerich, (2008). Jorgensen, David ; Sega, Liana: [Independence of the total reflexivity conditions for modules.]{} Algebr. Represent. Theory 9 (2006), no. 2, 217-226. Kerner, Otto; Yamagata, Kunio: [Morita algebras.]{} Journal of Algebra, Volume 382, 185-202, (2013). Marczinzik, René: [Upper bounds for the dominant dimension of Nakayama and related algebras.]{} http://arxiv.org/abs/1605.09634. Marczinzik, René: [A bocs theoretic characterization of gendo-symmetric algebras.]{} http://arxiv.org/abs/1606.07272. Martinez Villa, Roberto: [Modules of dominant and codominant dimension]{} Communications in algebra, 20(12), 3515-3540, (1992). Mueller, Bruno: [The classification of algebras by dominant dimension.]{} Canadian Journal of Mathematics, Volume 20, 398-409, (1968). Ringel, Claus Michael : [The Gorenstein projective modules for Nakayama algebras I.]{} . J. Algebra 385 (2013), 241-261. Ringel, Claus Michael : [Artin algebras of dominant dimension at least 2.]{} <https://www.math.uni-bielefeld.de/~ringel/opus/domdim.pdf>, retrieved 2015-06-18. Ringel, Claus Michael : [On the representation dimension of artin algebras.]{} Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 1, 33-70. Skowronski, Andrzej; Yamagata, Kunio: [Frobenius Algebras I: Basic Representation Theory.]{} EMS Textbooks in Mathematics, (2011). Yamagata, Kunio: [Frobenius Algebras*]{} [Hazewinkel, M. (editor): Handbook of Algebra, North-Holland, Amsterdam, Volume I, 841-887, (1996).]{}
--- abstract: \| The first quantized theory of N=2, D=3 massive superparticles with arbitrary fixed central charge and (half)integer or fractional superspin is constructed. The quantum states are realized on the fields carrying a finite dimensional, or a unitary infinite dimensional representation of the supergroups $\rm OSp(2\|2)$ or $\rm SU(1,1\|2)$. The construction originates from quantization of a classical model of the superparticle we suggest. The physical phase space of the classical superparticle is embedded in a symplectic superspace $T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$, where the inner Kähler supermanifold $\rm{\cal L}^{1\|2}\cong OSp(2\|2)/[U(1)\times U(1)]\cong SU(1,1\|2)/[U(2\|2)\times U(1)]$ provides the particle with superspin degrees of freedom. We find the relationship between Hamiltonian generators of the global Poincaré supersymmetry and the “internal” $\rm SU(1,1\|2)$ one. Quantization of the superparticle combines the Berezin quantization on ${\cal L}^{1\|2}$ and the conventional Dirac quantization with respect to space-time degrees of freedom. Surprisingly, to retain the supersymmetry, quantum corrections are required for the classical N=2 supercharges as compared to the conventional Berezin method. These corrections are derived and the Berezin correspondence principle for ${\cal L}^{1\|2}$ underlying their origin is verified. The model admits a smooth contraction to the N=1 supersymmetry in the BPS limit. --- 6.0in 21.5cm 0.30truein 0.30truein =1.5pc =0 IC/98/138\ hep-th/9809104\ 10 September, 1998 and S.L. Lyakhovich$^{\rm 1,2}$\ \ PACS numbers: 11.30.Pb, 71.10.Pm\ Keywords: quantization, superparticles, anyons, $\rm OSp(2\|2)$. INTRODUCTION {#s0} ================ In this paper we construct N=2, D=1+2 massive spinning superparticle model and study the symplectic supergeometry behind it. This supergeometry is compatible to the Berezin quantization method which is applied to construct the one-particle quantum theory. The main part of our consideration is based on the observation that the N=2 superextension of D=3 spinning particle results in the classical model which possesses simultaneously Poincaré supersymmetry and Lorentz supersymmetry of the superspin degrees of freedom. This “double” supersymmetry can be lifted to the quantum level and we obtain the realization of N=2, D=3 Poincaré supermultiplet on the fields carrying an irreducible representation of the supergroup $\rm SU(1,1\|2)$ (“Lorentz supergroup” whose even part is $\rm SO^\uparrow(1,2)\times U(2)\times central\ charge$). A nonlinear mutual involvement of the Hamiltonian generators of two supersymmetries requires the careful geometric quantization of the superparticle. At first, we try to explain the most important motivations of the problem. In the hierarchy of all known entities, the particles living in three-dimensional space-time stand out mostly due to a possibility of fractional spin and statistics (anyons). Anyon excitations are actually presented in some planar physics phenomena [@Laugh; @Wilczekbook] and the relevant theoretical concept has both topological [@LeiMyr; @Wilc; @Goldin] and group-theoretical [@JackNair; @Forte; @SorVol] grounds. It is well known that in the field theory fractional statistics originates usually from a coupling of the matter fields to the gauge field with the Chern-Simons mass term [@Chern-Sim]. The supersymmetric extension of this approach [@HlousSpect] implies a direct interaction between anyon excitations. The group-theoretical methods may give an alternative way to understand the anyon concept. One can start from the mechanical model of D=3 spinning particle, whose quantization leads to the one-particle quantum mechanics for the fractional spin state [@Balach; @Plyush; @SorVol; @CorPly; @Ghosh95; @GKL1; @Ners]. It is established that D=3 spinning particles possess the following remarkable features: (i) the spinning particle carries as many physical degrees of freedom as a spinless one; (ii) there is the so-called [canonical]{} model [@Balach] of spinning particle, which implies a deformation of the canonical symplectic structure of spinless particle by the use of [the Dirac monopole two-form]{}, without extension of the phase space introducing any “spinning” variables; (iii) it is promising feature of the canonical model to be adapted for the construction of consistent couplings of the particle to external fields [@ChouNairPol; @Chou; @Ghosh95; @GhoshMukh; @GKL1] and self-interaction of anyons [@BenGhosh; @Ghosh97.1; @Ghosh97.2]. In higher dimensions, the interaction problem for spinning particles becomes more involved, although some progress has recently been achieved there as well [@LyakSegShar; @LShSh98]; (iv) the anyon wave equations may be formulated in analogy with the ones for bosons and fermions. An essential difference is that the fractional spin, in contrast to the (half)integer, is naturally described in terms of infinite component fields carrying infinite dimensional representations of the universal covering group $\overline{{\rm SO}^\uparrow(1,2)}\cong\overline{{\rm SU}(1,1)}$; (v) representations of fractional spin are multivalued. There is no consistent quantum field theory of anyons up to now, nevertheless the Chern-Simons and group-theoretical constructions are deemed to lead to a unified consistent theory. In this regard, it would be interesting to understand, how the supersymmetry may be included into a group-theoretical description of anyons in terms of the infinite component fields. Another reason to investigate the D=3 superparticle is the exceptional fact that not only the Poincaré supersymmetry is possible in 1+2 dimensions, but the Lorentz one is too. The Lorentz group $\rm SO^\uparrow(1,2)$ coincides with the D=2 anti-de Sitter group, the latter admits the superextension regardless of specific space-time dimension. Although the Lorentz and the Poincaré supersymmetries are not compatible with each other, surprisingly, we will show that the Lorentz supersymmetry of D=3 spinning superparticle (which is invariant by construction with respect to the global Poincaré SUSY transformations) manifests itself as a hidden supersymmetry of internal degrees of freedom associated to the particle superspin and to the underlying superextended monopole-like symplectic structure. The hidden $\rm OSp(2\|2)$ supersymmetry of N=1 superanyons has been found in Ref. [@GKL2] where the respective model is constructed. The presence of the $\rm OSp(2\|2)$ supersymmetry already in the classical mechanics appears to be crucial for a consistent first quantization of N=1, D=3 superanyon. As a result, one obtains in quantum theory the realization of the N=1 Poincaré supermultiplet on the fields carrying an atypical unitary infinite dimensional representation of the $\rm OSp(2\|2)$ [@GKL2]. It is a direct N=1 superextension of description in terms of infinite dimensional unitary representation of the D=3 Lorentz group [@JackNair; @Forte; @SorVol] or the ones of the deformed Heisenberg algebra [@Ply1]. We argued in this manner the relevance of the group-theoretical approach for N=1 supersymmetric anyons. In this paper we suggest a nontrivial generalization of this construction to the case of D=3, N=2 massive spinning superparticle with arbitrary fixed central charge. We construct a superparticle model, which gives N=2 superextension of the canonical description of the D=3 spinning particle mentioned above. It is essential for our consideration that the Hamiltonian formalism of the canonical model may be built either in terms of the minimal phase space, or in an extended phase space restricted by constraints [@CorPly; @GKL1; @Ners; @GKL2]. In both cases the reduced phase space could be thought of as a space of motion of a Souriau’s “elementary system” [@Sour]. A general concept of elementary physical systems, including spinning particles and superparticles, is based on the so-called Kostant-Souriau-Kirillov construction [@Kost; @Sour; @Kiril]. The idea of the KSK construction is to identify the [physical phase space (= space of motion)]{} of any elementary system with a [coadjoint orbit]{} $\cal O$ of the symmetry group $G$. The symplectic action $G$ on $\cal O$ (classical mechanics) lifts to a representation of the group in a space of functions $\cal H$ on the classical manifold (prequantization). Then the quantization problem reduces to an appropriate choice of polarization, that is a global Lagrangian section in $T({\cal O})$ being invariant under the action of the symmetry group. In the special case of Kähler homogeneous spaces perfect results can be achieved in the framework of the Berezin quantization method [@Berezin1; @Berezin2], which implies one-to-one correspondence between the phase-space functions (covariant Berezin symbols) and linear operators in a Hilbert space. The latter is realized by holomorphic sections, because the Kähler homogeneous manifold admits a natural complex polarization [@Wood]. Moreover, the multiplication of the operators in the Hilbert space induces a noncommutative binary for the covariant Berezin symbols and a correspondence principle can be proved [@Berezin1; @Berezin2]. Physically speaking, it would not always be satisfactory to describe elementary systems in terms of the coadjoint orbits. In particular, the dynamics of relativistic particles and superparticles is usually supposed to evolve in a fibre bundle $\cal M$ over a [space-time manifold]{} that is crucial for the interaction problem. Thus, the coadjoint orbit of the spinning (super)particle arises from embedding into evolution (super)space. The projection $\pi:{\cal M}\to{\cal O}_G\,$, where $G$ is a Poincaré (super)group, generates $G$-invariant constraints and gauge symmetries in $\cal M$. The construction of interactions, being consistent with the gauge symmetries, and the quantization problem for $\pi$ provide a subject of current interest in the problems of spinning particle and superparticle models [@Balach; @LyakSegShar; @LShSh98; @MarnMart; @Frydr; @KuzLyakSeg]. Concerning the D=3 spinning particle, we know [@GKL1; @GKL2] that the quantization problem for the canonical model is naturally solved by means of an embedding of the maximal (four-dimensional) coadjoint orbit of the group ${\rm ISO}^\uparrow (1,2)$ into eight dimensional phase space (that is [extended phase space]{}) ${\cal M}^8\cong T^\ast({\rm R}^{1,2})\times {\cal L}$. Here ${\cal L}\cong\rm SU(1,1)/U(1)$ is a Lobachevsky plane and the character $\cong$ denotes a symplectomorphism. The projection $\pi:{\cal M}^8\to{\cal O}_{m,s}$ onto co-orbit ${\cal O}_{m,s}$ of the particle of mass $m$ and spin $s$ is provided by the constraints. The auxiliary variables parametrizing $\cal L$ are used to describe the particle spin. One can interpret the (holomorphic) automorphisms of the Lobachevskian Kähler metric as a hidden symmetry of the internal particle’s structure, which is related to the spin. We observed in Refs. [@GKL1; @GKL2] that the quantization of anyon could be achieved as a compromise of the conventional Dirac quantization on $T^\ast({\rm R}^{1,2})$ and of the geometric quantization in the Lobachevsky plane. Constraints of the classical mechanics are converted into wave equations of anyon according to the Dirac prescriptions. The starting point of this paper is a mechanical model of N=2 superparticle with arbitrary fixed mass $m>0$, superspin $s\neq0$ and central charge ${\cal Z}\equiv mb$, ${\|b\|\leq1}$ briefly announced before [@GL98]. For this elementary system the maximal coadjoint orbit ${\cal O}_{m,s,b}$ of real dimension $4/4$ is related to the case $\|b\|<1$. In our model, this orbit appears embedded into $8/4$-dimensional extended phase superspace ${\cal M}^{8\|4}$ of a special geometry: ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times {\cal L}^{1\|2}$, where $\rm {\cal L}^{1\|2} =SU(1,1\|2)/[U(2\|2)\times U(1)]\cong OSp(2\|2)/[U(1)\times U(1)]$ is an atypical Kähler coadjoint orbit of the supergroup $\rm SU(1,1\|2)$ and the typical one of $\rm OSp(2\|2)$. The inner supermanifold ${\cal L}^{1\|2}$, providing the particle model with a nonzero superspin, was studied originally in Refs. [@Balant; @Grad] in relation to $\rm OSp(2\|2)$ supercoherent states and called [N=2 superunit disc]{}. The projection of ${\cal M}^{8\|4}$ onto physical subspace follows similarly to the nonsupersymmetric model. In fact, introducing the supersymmetry for D=3 particle, we need to superextend only the inner submanifold $\cal L$ of the extended phase space. The extended phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times {\cal L}^{1\|2}$ carries “double supersymmetry”: one is related to the Poincaré supergroup and acts on the associated co-orbit ${\cal O}_{m,s,b}\subset{\cal M}^{8\|4}$, another one lives in the inner subsupermanifold ${\cal L}^{1\|2}$. Moreover, the model allows an extended hidden N=4 supersymmetry with special values of the central charges saturating the BPS bound. We will quantize the theory similarly to quantization of the canonical model of the particle on ${\cal M}^8$ [@GKL1; @GKL2]. Specifically, we combine the geometric quantization in the inner subsupermanifold ${\cal L}^{1\|2}$ for the internal $\rm SU(1,1\|2)$ supersymmetry and the canonical Dirac quantization in $T^\ast({\rm R}^{1,2})$. This quantization scheme implies from the outset that the mentioned “double supersymmetry” must survive in the quantum theory. The crucial point is to express the Hamiltonian generators of the Poincaré supersymmetry in ${\cal M}^{8\|4}$ in terms of the ones of internal $\rm SU(1,1\|2)$ supersymmetry (as well as of space-time coordinates and momenta). These expressions appear to be [nonlinear]{}. As a consequence, some renormalization of the Poincaré supergenerators should be required for the closure of the Poincaré supersymmetry algebra. Roughly speaking, the corrections to generators could be treated as a manifestation of the ordering ambiguity for operators in quantum theory. We will see, that the origin of the corrections may also be clarified from the viewpoint of the Berezin quantization in ${\cal L}^{1\|2}$ and the underlying correspondence principle. However, the Berezin method itself does not provide a regular technique of deriving the closing corrections which have to recover the representation of the Poincaré superalgebra in quantum theory. Moreover, it is unclear a priori whether the consistent corrections exist at all. Surprisingly, the problem is solved if a simple ansatz is taken for the renormalized Poincaré generators. Then the closing corrections, which appear in the order of ${\cal O}(s^{-2})$, can be [exactly]{} calculated. We arrive eventually to the realization of the unitary representation of N=2, D=3 supermultiplet on the fields carrying atypical irreps of the supergroup $\rm SU(1,1\|2)$ and the typical ones of the subsupergroup $\rm OSp(2\|2)$. These irreps are certainly infinite dimensional for the case of fractional superspin, but for the habitual case of (half)integer superspin they may be chosen to be finite dimensional. The model of N=2 superparticle reduces to the one of N=1 superparticle in the Bogomol’ny-Prassad-Sommerfield (BPS) limit for central charge, when $\|b\|=1$. One can trace the BPS limit both at the classical and quantum levels. Classically, it corresponds to the degenerate coadjoint orbit of D=3, N=2 superparticle of dimension $4/2$. When $\|b\|=1$, the extended phase superspace becomes degenerate and reduces to ${\cal M}^{8\|2}\cong T^\ast({\rm R}^{1,2})\times {\cal L}^{1\|1}$ with inner supermanifold ${\cal L}^{1\|1}\cong\rm OSp(2\|2)/U(1\|1)\cong OSp(1\|2)/U(1)$. ${\cal M}^{8\|2}$ is exactly the extended phase superspace of N=1 superanyon [@GKL2]. In this exceptional case, the generators of N=1 Poincaré supersymmetry and the internal $\rm OSp(2\|2)$ one are [linearly]{} expressible to one another. Thus, the geometric quantization immediately gives the quantum theory of N=1 superparticle, without extra constructions and corrections. In particular, we don’t need the detailed Berezin correspondence principle for ${\cal L}^{1\|1}$. The geometric quantization in the $\rm OSp(2\|2)$ coadjoint orbits was constructed in Refs. [@Balant; @Grad] and we follow these results. At the same time we have to clarify two important points, which have seemingly been unknown. First, we found out that the Kähler geometry of the regular co-orbit ${\cal L}^{1\|2}$ admits the symplectic holomorphic action of the supergroup $\rm SU(1,1\|2)$, which is larger than the supergroup $\rm OSp(2\|2)$ in itself. We construct the geometric quantization on ${\cal L}^{1\|2}$ provided for this extended supersymmetry supergroup. Secondly, we perform Berezin quantization for ${\cal L}^{1\|2}$ to establish a correspondence principle and to explain the origin of quantum corrections to the N=2 Poincaré supercharges in ${\cal M}^{8\|4}$. The paper is organized as follows. In Sec. II we recall briefly the canonical model of D=3 spinning particle in terms of the minimal and extended phase spaces. Specifically, we focus at symplectic structure and symmetries of the minimal and extended spaces. Then we are going to construct the superextension of the canonical model. The classical mechanics of N=2, D=3 massive spinning superparticle with arbitrary central charge is considered in Sec. III. Starting from a first order Lagrangian we study the supergeometry of the phase superspace and identify it with ${\cal M}^{8\|4}=T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$. We construct explicitly the embeddings of the N=2 Poincaré and Lorentz supergroup’s coadjoint orbits into ${\cal M}^{8\|4}$ and find out the Hamiltonian generators of corresponding supersymmetries. The relation, being crucial for quantization, is established between the N=2 Poincaré and $\rm SU(1,1\|2)$ Hamiltonian generators. We also reveal a degenerate N=4 supersymmetry in the model and a special case of degenerate co-orbits, which appear in the BPS limit. Furthermore, a reduction of the model with respect to a part of constraints is shown to lead to a minimal $6/4$-dimensional phase superspace with superextended symplectic monopole-like structure and with the mass-shell condition to be the only constraint. We obtain, in particular, N=2 superextension of the Dirac monopole two-form, which supplies the particle with superspin. In Sec. IV we suggest a quantization procedure for the classical mechanics constructed in Sec. III. At first the Berezin quantization is considered on the regular $\rm OSp(2\|2)$ coadjoint orbit. In particular, we construct the correspondence between symbols and operators on ${\cal L}^{1\|2}$ and prove the underlying correspondence principle. Then these results are applied to the consistent quantization of D=3, N=2 superparticle, which is the final object of construction. The summary and a general outlook are given in Sec. V. Finally, the Appendix contains the calculation of N=2 Poincaré supercharge’s quantum anticommutator. The calculation provides manifest verification of consistency of the renormalization procedure for the Poincaré supercharges. MINIMAL AND EXTENDED PHASE SPACES\ OF A CANONICAL MODEL OF SPINNING PARTICLE {#s1} =========================================== First consider the nonsupersymmetric canonical model of the particle (various formulations see [@Balach; @JackNair; @Forte; @CorPly; @Ners]), which serves as an initial subject for further generalizations. The particle lives originally on six-dimensional phase space ${\cal M}^6$ with a symplectic two-form[^1] $$\Omega_s=-{\rm d}x^a\wedge{\rm d}p_a+\Omega_{\rm m}\qquad \Omega_{\rm m}=\frac{s}2 \frac{\epsilon^{abc}p_a{\rm d}p_b\wedge{\rm d}p_c}{(-p^2)^{3/2}}\qquad (p^2<0)\,, \label{2.1}$$ where $\Omega_{\rm m}$ is known as the Dirac monopole form. The Poincaré transformations are generated by the following functions $${\cal P}_a=p_a\qquad{\cal J}_a =\epsilon_{abc}x^bp^c-s\frac{p_a}{(-p^2)\lefteqn{{}^{1/2}}}\qquad , \label{2.2}$$ which constitute D=3 Poincaré algebra with respect to Poisson brackets (PB’s) $$\{{\cal P}_a\,,\,{\cal P}_b\}=0\qquad \{{\cal J}_a\,,\,{\cal P}_b\}=\epsilon_{abc}{\cal P}^c\qquad \{{\cal J}_a\,,\,{\cal J}_b\}=\epsilon_{abc}{\cal J}^c\,.\label{2.3}$$ The fundamental PB’s read $$\{x^a\,,\,x^b\}=s\frac{\epsilon^{abc}p_c}{(-p^2)^{3/2}}\qquad \{x^a\,,\,p_b\}=\delta^a{}_b\qquad \{p_a\,,\,p_b\}=0\,.\label{2.4}$$ The last two PB’s mean that $x^a$ and $p_a$ transform as coordinates and momenta by Poincaré translations. Moreover, they are Lorentz vectors because of $\{{\cal J}_a\,,\,x_b\}=\epsilon_{abc}x^c$ and $\{{\cal J}_a\,,\,p_b\}=\epsilon_{abc}p^c$. Let us assume that the particle dynamics on ${\cal M}^6$ is governed by the mass shell [constraint]{} $$p^2+m^2=0\,,\label{2.5}$$ whereas the canonical Hamilton function is identically zero. On the mass shell, the Casimir functions of the enveloping Poincaré algebra are identically conserved: ${\cal P}^2=-m^2$, $({\cal P},{\cal J})=ms$. We conclude that D=3 particle of mass $m$, spin $s$ and energy sign $p^0/\|p^0\|$ lives on mass shell. From now on, we take a further restriction $p^0>0$ bearing in mind the supersymmetric theory, when the energy is positive essentially. The mass shell constraint generates the reparametrization (gauge) invariance for every world line of the particle. The set of world lines, being considered modulo to the gauge equivalence, is named the particle history space, the latter is isomorphic to the physical state space ${\cal O}_{m,s}$ of the spinning particle. The reduced symplectic manifold ${\cal O}_{m,s}$ is symplectomorphic to the maximal [coadjoint orbit]{} [@Sour; @Kiril; @Wood] of the D=3 Poincaré group. There is a standard way to extend the canonical model to the Poincaré supersymmetry. One may substitute ${\rm d}x^a\to {\rm d} x^a-i(\gamma^a)_{\alpha \beta}\theta^{\alpha\,I}{\rm d}\theta^{\beta\,I}$ in Eq. (\[2.1\]) introducing real Grassmann variables $\theta^{\alpha\,I},\,I=1,\dots,N$. The resulting symplectic superform appears to be invariant under the $N$-extended Poincaré supergroup without central charges. One may further generate central charges introducing some Wess-Zumino type terms [@AzcLuk; @Frydr] in Eq. (\[2.1\]). Then, imposing the mass shell constraint (\[2.5\]), one may build the classical model of D=3 superparticle of mass $m$, superspin $s$ and arbitrary fixed central charges in the $6/2N$-dimensional phase superspace. However, it is hardly possible to conceive satisfactory quantization of this model. Even for the canonical model without supersymmetry the realization of the coordinate operators ${\widehat x}{}^a$ is a nontrivial problem accounting for the complicated form of the first Poisson bracket in Eqs.(\[2.4\]). A detailed analysis of Ref. [@CorPly] shows that the manifest covariance of the canonical model, being formulated in terms of the “minimal” phase space ${\cal M}^6$, is inevitably lost in quantum theory. The superextension of the canonical model makes the Poisson brackets, being quantized, much more complicated. In fact, the quantization problem in the reduced nonlinear phase superspace is not solved even for spinless D=3 superparticle. Thus we will reformulate from the outset the canonical model in an “extended” phase space, where a hidden symmetry of spinning particle becomes transparent and gives an efficient method for quantization making use of this symmetry. Moreover, the construction will be appropriate for intriguing superextension. An adapted reformulation of the canonical model is suggested in Refs. [@GKL1; @GKL2]. We observe, that the monopole two-form $\Omega_{\rm m}$ in Eq. (\[2.1\]) is nothing else but the Kähler two form on the mass hyperboloid (\[2.5\]), which gives the realization of the Lobachevsky plane ${\cal L}$. It will be convenient to make use of another realization of ${\cal L}\cong\{z\in {\rm C}^1, \|z\|<1\}$ by an open unit disc of complex plane ${\rm C}^1$. We rewrite the symplectic two-form (\[2.1\]) as follows $$\Omega_s=-{\rm d}x^a\wedge{\rm d} p_a+\Omega_{\cal L}\qquad\Omega_{\cal L}= -2is\frac{{\rm d}z\wedge{\rm d}\bar z}{(1-z\bar z)^2}\,, \label{2.6}$$ where (recall that $p^2<0$ and we have taken $p^0>0$) $$p^a=\sqrt{-p^2}n^a\qquad n^a=\left(\frac{1+z\bar z}{1-z\bar z}, -\frac{z+\bar z}{1-z\bar z},i\frac{\bar z-z}{1-z\bar z}\right)\quad n^2\equiv-1\,. \label{2.7}$$ The unit timelike Lorentz vector $n^a$ parametrizes the points of the Lobachevsky plane. Let us look at Eq. (\[2.6\]) from a different viewpoint. Consider a new phase space ${\cal M}^8\cong T^\ast({\rm R}^{1,2})\times{\cal L}$ with a symplectic two-form (\[2.6\]) and an elementary system on ${\cal M}^8$, whose dynamics is subjected by three constraints $$p^a=mn^a \label{2.8}$$ Apparently these constraints project the extended phase space ${\cal M}^8$ into the same coadjoint orbit as the mass shell constraint (\[2.5\]) does for ${\cal M}^6$. Alternatively, one can solve explicitly only two constraints $p^a=\sqrt{-p^2}n^a$ providing the reduction ${\pi_1:{\cal M}^8}\to{\cal M}^6$ of extended phase space to the minimal one. In other words, we have constructed the sequence of embeddings ${\cal O}_{m,s}\subset {\cal M}^6\subset{\cal M}^8$. Hence we get an equivalent description of D=3 spinning particle in terms of the extended phase space $T^\ast({\rm R}^{1,2})\times{\cal L}$. The Hamiltonian generators of the canonical Poincaré transformations in ${\cal M}^8$ read $${\cal P}_a=p_a\qquad {\cal J}_a=\epsilon_{abc}x^bp^c+J_a\,, \label{2.9}$$ where the spin vector $J_a$ is expressed in terms of the ‘inner’ space $\cal L$: $$J_a=-sn_a \, . \label{2.10}$$ The Hamiltonians (\[2.9\]) generate the Poincaré algebra with respect to PB in ${\cal M}^8$, whereas the spin generators (\[2.10\]) span internal Lorentz algebra related to the (holomorphic) automorphism group of the Lobachevsky plane. The latter group can be recognized as a hidden symmetry of the internal structure of spinning particle. Although this concept looks may seem artificial at the moment, below, we will observe essentially nontrivial superextension of the hidden symmetry. The Poincaré Casimir functions are identically conserved owing to constraints (\[2.8\]) $$p^2+m^2=0\qquad (p,J)-ms=0\,.\label{2.11}$$ A crucial detail is that the equations (\[2.11\]) define the same surface in the extended phase space as the constraints (\[2.8\]) do. The quantization of the model in ${\cal M}^8$ is almost transparent. We can combine the canonical Dirac quantization in $T^\ast({\rm R}^{1,2})$ and the Berezin quantization in the Lobachevsky plane [@GKL2]. Constraints (\[2.11\]) will be imposed in Hilbert space to separate the one-particle states. Finally, write down the Lagrangian of the theory. One may choose the action functional as an integrand of the one-form $\Theta$, where ${\rm d}\Theta=\Omega_s+V$ and $V$ vanishes on shell. Let us take $$S=\int\Theta \, , \qquad \Theta=p_a{\rm d}x^a +is\frac{\bar z{\rm d}z - z{\rm d}\bar z}{1-z\bar z}\equiv p_a{\rm d}x^a+\Sigma_{\cal L}\, , \qquad {\rm d}\Sigma_{\cal L}=\Omega_{\cal L}\,. \label{2.12}$$ It is implied here that the virtual paths lay in the constraint surface (\[2.8\]). Excluding the momenta accounting for constraints (\[2.8\]) and making pull back of $\Theta$, one obtains the action functional $$S=\int\limits_{\tau_1}^{\tau_2}L\,{\rm d}\tau\qquad L=m(\dot x,n) +is\frac{\bar z{\dot z} -z\dot{\bar z}}{1-z\bar z} \label{2.13}$$ with the first order Lagrangian being invariant under reparametrizations. Notice that the Lagrangian is also strongly invariant under translations and spatial rotations, whereas the Lorentz boosts change it by a total derivative. CLASSICAL MODEL OF $\normalsize\bf D=3$ SPINNING SUPERPARTICLE {#s3} ================================================================== A first order Lagrangian {#sbs3.1} ----------------------------- Introduce a N=2 superextension of the Lagrangian (\[2.13\]) providing both the superpoincaré invariance of the theory and other hidden supersymmetry as well. Introducing a pair of Majorana anticommuting spinors $\theta^{\alpha\,I}=(\theta^\alpha,\chi^\alpha), I=1,2$ we suggest $$L=m(\Pi,n)+mb(\theta_\alpha\dot\chi^\alpha-\chi_\alpha\dot\theta^\alpha)- mb\theta^\alpha n_{\alpha\gamma}\dot n^{\gamma}{}_\beta\chi^\beta +is\frac{\bar z{\dot z} -z\dot{\bar z}}{1-z\bar z}\,, \label{3.1}$$ where $m,b,s$ are real parameters, $n^a$ is a unit Lorentz vector in the Lobachevsky plane, being defined by Eq. (\[2.7\]) and $$n_{\alpha\beta}\equiv n^a\gamma_{a\,\alpha\beta}\qquad\qquad \Pi^a=\dot x^a-i\gamma^a_{\alpha\beta}(\theta^\alpha\dot\theta^\beta +\chi^\alpha\dot\chi^\beta)\,.$$ The three dimensional Dirac matrices $\gamma_a$ are chosen in the form[^2] $$\begin{array}{c} (\gamma_0)_{\alpha\beta}= \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)\quad (\gamma_1)_{\alpha\beta}= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\quad (\gamma_2)_{\alpha\beta}= \left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array}\right)\\ (\gamma_a)_{\alpha\gamma}(\gamma_b)^\gamma{}_\beta= i\epsilon_{abc}(\gamma^c)_{\alpha\beta}-\eta_{ab}\epsilon_{\alpha\beta}\,. \end{array}$$ The first term in the Lagrangian (\[3.1\]) is a conventional superextension of the respective expression in Eq. (\[2.13\]) and the second addend represents the Wess-Zumino type term generating the central charge for the supersymmetry [@AzcLuk]. At last, the third term accounts for the specific of D=3 spinning superparticle model. Owing to this addend the supertranslations, underlying Poincaré supersymmetry of the Lagrangian, read rather unusual: $$\begin{array}{llll} \displaystyle \delta_\epsilon x^a=i\gamma^a_{\alpha\beta}\epsilon^\alpha\theta^\beta+ ib\epsilon^{abc} n_b \gamma_{c\,\alpha\beta}\epsilon^\alpha\chi^\beta+ bn^a\epsilon^\alpha\chi_\alpha & \displaystyle \delta_\epsilon \theta^\alpha=\epsilon^\alpha & \displaystyle \delta_\epsilon \chi^\alpha=0 & \displaystyle \delta_\epsilon z=0 \\ \displaystyle \delta_\eta x^a=i\gamma^a_{\alpha\beta}\eta^\alpha\chi^\beta- ib\epsilon^{abc} n_b \gamma_{c\,\alpha\beta}\eta^\alpha\theta^\beta- bn^a\eta^\alpha\theta_\alpha & \displaystyle \delta_\eta \theta^\alpha=0 & \displaystyle \delta_\eta \chi^\alpha=\eta^\alpha & \displaystyle \delta_\eta z=0\,. \end{array} \label{3.2}$$ Here $\epsilon^\alpha,\eta^\alpha$ are odd real parameters. For completeness, expose also the even infinitesimal Poincaré transformations and $\rm U(1)$ transformations as well $$\begin{array}{llll} \displaystyle \delta_\omega x^a=\epsilon^{abc}\omega_bx_c& \displaystyle \delta_\omega \theta^\alpha=-\frac{i}2 \omega^a\gamma_a{}^\alpha{}_\beta \theta^\beta& \displaystyle \delta_\omega \chi^\alpha=-\frac{i}2 \omega^a\gamma_a{}^\alpha{}_\beta\chi^\beta& \displaystyle \delta_\omega z=i\omega^a\xi_a \\ \displaystyle \delta_{\rm f} x^a=f^a& \displaystyle \delta_{\rm f} \theta^\alpha=\delta_{\rm f}\chi^\alpha=0 & \displaystyle \delta_{\rm f} z=0& \\ \displaystyle \delta_{\mu} x^a=0& \displaystyle \delta_{\mu} \theta^\alpha=-\mu\theta^\alpha & \displaystyle \delta_{\mu} \chi^\alpha=\mu\theta^\alpha & \displaystyle \delta_{\mu} z=0 \end{array} \label{3.3}$$ with the even real parameters $\omega^a,f^a,\mu$ and the holomorphic object $\xi_a=-1/2(2z, 1+z^2,i(1-z^2))$. The infinitesimal transformations (\[3.3\]), (\[3.2\]) generate N=2 Poincaré superalgebra, which is discussed in Subsection 3. Extended phase superspace ----------------------------- \[sbs3.2\]We show in this subsection that the superparticle being described by the Lagrangian (\[3.1\]) lives in a supersymplectic phase space ${\cal M}^{8\|4}$ of very special supergeometry: ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$. Then we identify ${\cal L}^{1\|2}$ with regular (when $\|b\|<1$) or degenerate (when $\|b\|=1$) coadjoint orbit of the $\rm OSp(2\|2)$ supergroup. Having the goal to quantize the theory in ${\cal M}^{8\|4}$ we will need for detailed information about SUSY’s and quantization in ${\cal L}^{1\|2}$. The supersymplectic geometry of ${\cal L}^{1\|2}$ is considered in Subsec. 4, while the Berezin quantization will be constructed in Subsec. IV.1. The model (\[3.1\]) fits naturally into the formulation in symplectic language. The theory originates from the action functional $$S=\int\Theta^{\rm SUSY} \qquad \Theta^{\rm SUSY}=p_a{\rm d}x^a+ \Sigma_{{\cal L}^{1\|2}} \label{3.4}$$ $$\begin{aligned} &\displaystyle \Sigma_{{\cal L}^{1\|2}}=-imn_{\alpha\beta}\theta^\alpha{\rm d}\theta^\beta -imn_{\alpha\beta}\chi^\alpha{\rm d}\chi^\beta+mb\theta_\alpha{\rm d} \chi^\alpha-mb\chi_\alpha{\rm d}\theta^\alpha &\nonumber\\&\displaystyle\phantom{001} -2mb\frac{z^\alpha z^\beta\theta_\alpha \chi_\beta{\rm d}\bar z- \bar z^\alpha\bar z^\beta\theta_\alpha\chi_\beta {\rm d}z}{(1-z\bar z)^2}+is\frac{\bar z{\rm d}z -z{\rm d}\bar z}{1-z\bar z}\,,& \label{3.5}\end{aligned}$$ where the virtual paths belong the surface $$p^a=mn^a\,, \label{3.6}$$ as follows from the definition $p_a=\partial L/\partial\dot x^a$. Introduce the objects $$z^\alpha\equiv(1,z)\qquad\qquad\bar z^\alpha\equiv(\bar z,1)\, , \quad \alpha =0,1 \, , \label{3.7}$$ that simplifies Eq. (\[3.5\]) and many of the forthcoming formulae. In this section $z^\alpha,\bar z^\alpha$ are used for notation only. Remarkable origin and transformation properties of the objects $z^\alpha,\bar z^\alpha$ will be considered later in Subsec. IV.4. Relation (\[3.4\]) shows that the particle dynamics is embedded in phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$ with some inner superspace of a real dimension $2/4$ denoted by ${\cal L}^{1\|2}$. The symplectic two-superform in ${\cal M}^{8\|4}$ reads $$\Omega_s^{\rm SUSY}={\rm d}\Theta^{\rm SUSY}=-{\rm d}x^a\wedge{\rm d}p_a+ \Omega_{{\cal L}^{1\|2}} \qquad\qquad \Omega_{{\cal L}^{1\|2}} ={\rm d}\Sigma_{{\cal L}^{1\|2}} \label{3.8}$$ The inner superspace is a N=2 superextension of the Lobachevsky plane. We show at first that ${\cal L}^{1\|2}$ coincides with a coadjoint orbit of the $\rm OSp(2\|2)$ supergroup. Let us introduce new complex Grassmann variables ($m\neq 0, {s\neq 0}$) $$\begin{array}{l} \displaystyle \theta=\sqrt\frac{m}{s}(iz^\alpha\chi_\alpha-z^\alpha\theta_\alpha) \left[1+m\frac{1-b}{4s}(\theta^\alpha\theta_\alpha+\chi^\alpha\chi_\alpha) \right]\qquad \bar\theta=\overline{(\theta)}\\[1mm] \displaystyle \chi=\sqrt\frac{m}{s}(iz^\alpha\theta_\alpha-z^\alpha\chi_\alpha) \left[1+m\frac{1+b}{4s}(\theta^\alpha\theta_\alpha+\chi^\alpha\chi_\alpha) \right]\qquad \bar\chi=\overline{(\chi)}\,, \end{array} \label{3.9}$$ which are in one-to-one correspondence with the Majorana spinors $\theta^\alpha,\chi^\alpha$ used before. It is easy to check that the symplectic two-superform $\Omega_{{\cal L}^{1\|2}}$ of inner superspace reads in new variables as $$\begin{aligned} \displaystyle \Omega_{{\cal L}^{1\|2}}&=& \displaystyle -\frac{is}{2}\left(2 -a_+\frac{1+z\bar z}{1-z\bar z}\theta\bar\theta -a_-\frac{1+z\bar z}{1-z\bar z}\chi\bar\chi +a_+a_-\frac{1+2z\bar z}{(1-z\bar z)^2} \theta\bar\theta\chi\bar\chi\right) \nonumber \frac{{\rm d}z\wedge{\rm d}\bar z}{(1-z\bar z)^2} \\&&\displaystyle +\frac{isa_+\bar z\theta}{2}\left(1-\frac{a_-\chi\bar\chi}{1-z\bar z}\right) \frac{{\rm d}z\wedge{\rm d}\bar\theta}{(1-z\bar z)^2} \nonumber +\frac{isa_-\bar z\chi}{2}\left(1-\frac{a_+\theta\bar\theta}{1-z\bar z}\right) \frac{{\rm d}z\wedge{\rm d}\bar\chi}{(1-z\bar z)^2}\\&&\displaystyle -\frac{isa_+z\bar\theta}{2}\left(1-\frac{a_-\chi\bar\chi}{1-z\bar z}\right) \frac{{\rm d}\theta\wedge{\rm d}\bar z}{(1-z\bar z)^2} \nonumber -\frac{isa_-z\bar\chi}{2}\left(1-\frac{a_+\theta\bar\theta}{1-z\bar z}\right) \frac{{\rm d}\chi\wedge{\rm d}\bar z}{(1-z\bar z)^2}\\&&\displaystyle +\frac{isa_+}{2}\left(1-\frac{a_-\chi\bar\chi}{1-z\bar z}\right) \frac{{\rm d}\theta\wedge{\rm d}\bar\theta}{1-z\bar z} \label{3.10} +\frac{isa_-}{2}\left(1-\frac{a_+\theta\bar\theta}{1-z\bar z}\right) \frac{{\rm d}\chi\wedge{\rm d}\bar\chi}{1-z\bar z}\\&&\displaystyle +\frac{isa_+a_-\theta\bar\chi}{4} \nonumber \frac{{\rm d}\chi\wedge{\rm d}\bar\theta}{(1-z\bar z)^2} +\frac{isa_+a_-\chi\bar\theta}{4} \frac{{\rm d}\theta\wedge{\rm d}\bar\chi}{(1-z\bar z)^2}\\ &&\displaystyle a_+= 1+b\qquad\qquad\qquad a_-= 1-b\,. \nonumber\end{aligned}$$ This superform exactly coincides to the one deduced by Gradechi and Nieto [@Grad] in the supercoherent state’s approach[^3] for the $\rm OSp(2\|2)$ coadjoint orbits. $\Omega_{{\cal L}^{1\|2}}$ is nondegenerate iff $\|b\|\neq1$. In the case $\|b\|<1$, the supermanifold ${\cal L}^{1\|2}$ is the regular $\rm OSp(2\|2)$ coadjoint orbit ${\cal L}^{1\|2}\cong\rm OSp(2\|2)/[U(1)\times U(1)]$ and is called N=2 superunit disc. The degenerate orbit $\rm OSp(2\|2)/U(1\|1)$, which is denoted usually by ${\cal L}^{1\|1}$ and called N=1 superunit disc, appears when $\|b\|=1$. The other possibility $\|b\|>1$ has no physical significance: neither the Poincaré supersymmetry, nor the internal $\rm OSp(2\|2)$ one admit unitary representations. It is seen from further consideration that the inequality $\|b\|>1$ contradicts to the BPS bound. Observables and the physical subspace ----------------------------------------- \[sbs3.3\]Consider in detail the realization of the Poincaré supersymmetry in the extended phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$. The Poincaré supergroup is realized by a symplectic action leaving the coadjoint orbit (\[3.6\]) invariant. The vector superfields generating the transformations (\[3.2\]) and (\[3.3\]) are related to the corresponding canonical Hamiltonian generators by $$X_H\pint \Omega_s^{\rm SUSY}=-(-1)^{\epsilon_H}{\rm d}H\,,\label{3.11}$$ where $\epsilon_H$ is the Grassmann parity of the Hamiltonian $H$. Solving these equations one gets the following Hamiltonian generators (we denote the generator of isotopic $\rm U(1)$ rotations by $P_3$) $$\begin{array}{l}\displaystyle {\cal P}_a=p_a\qquad{\cal J}_a=\epsilon_{abc} x^bp^c-sn_a+\frac12mn_a (\theta^\alpha\theta_\alpha+\chi^\alpha\chi_\alpha-2ibn_{\alpha\beta} \theta^\alpha\chi^\beta)\\ \displaystyle {\cal Q}^1_\alpha=ip_{\alpha\beta}(\theta^\beta-ibn^\beta{}_\gamma\chi^\gamma) +m(in_{\alpha\beta}\theta^\beta+b\chi_\alpha)\qquad (p_{\alpha\beta}\equiv p_a\gamma^a_{\alpha\beta}) \\ \displaystyle {\cal Q}^2_\alpha=ip_{\alpha\beta}(\chi^\beta+ibn^\beta{}_\gamma\theta^\gamma) +m(in_{\alpha\beta}\chi^\beta-b\theta_\alpha)\\ \displaystyle P_3=imn_{\alpha\beta}\theta^\alpha\chi^\beta-\frac{mb}2(\theta^\alpha \theta_\alpha+\chi^\alpha\chi_\alpha)\,. \end{array} \label{3.12}$$ With respect to Poisson superbrackets on ${\cal M}^{8\|4}$ they generate the following superalgebra $$\begin{array}{ll}\displaystyle \{{\cal J}_a\;,\;{\cal J}_b\}=\epsilon_{abc}{\cal J}^c &\displaystyle \{{\cal J}_a\;,\;{\cal P}_b\}=\epsilon_{abc}{\cal P}^c \qquad \{{\cal J}_a\;,\;{\cal Q}^I_\alpha\}=-\frac{i}{2}(\gamma_a)_\alpha{}^\beta {\cal Q}^I_\beta\\ \displaystyle \{{\cal Q}^I_\alpha\,,\,P_3\}=-\frac12\epsilon^{IJ}{\cal Q}^J_\alpha& \displaystyle \{{\cal Q}^I_\alpha\;,\;{\cal Q}^J_\beta\}\approx -2i\delta^{IJ}p_{\alpha\beta}-2\epsilon^{IJ} \epsilon_{\alpha\beta}{\cal Z}\qquad {\cal Z}=mb\,, \end{array} \label{3.13}$$ the other brackets being equal to zero and $I,J=1,2$, $\epsilon^{IJ}=-\epsilon^{JI}$, $\epsilon^{01}=1$. We stress that the latter bracket $\{{\cal Q}^I_\alpha,{\cal Q}^J_\beta\}$ is closed only in a weak sense, that is modulo to constraints (\[3.6\]). What we have obtained is N=2, D=3 Poincaré superalgebra with central charge ${\cal Z}=mb$ and isotopic charge $P_3$ acting on the internal indices of supercharges ${\cal Q}^I_\alpha$. One can easily examine that the mass and the spin Casimir functions of the superalgebra (\[3.13\]) read $C_1\equiv{\cal P}^a{\cal P}_a=p^2$ and $C_2\equiv{\cal P}^a{\cal J}_a+\frac{1}{8}{\cal Q}^{I\,\alpha}{\cal Q}^I_\alpha -{\cal Z}P_3=-s(p,n)$. On the constraint surface (\[3.6\]) $$p^2+m^2=0\qquad(p,n)+m=0\label{3.14}$$ the Casimirs are conserved identically. Eqs. (\[3.14\]) and (\[3.6\]) are completely equivalent to each other, in other words, they define one and the same surface in the phase superspace ${\cal M}^{8\|4}$. We conclude that the mechanical model describes N=2, D=3 superparticle of mass $m$, superspin $s$ and central charge $mb$. Regular and degenerate cases are essentially distinguished for the coadjoint orbit, being associated for the superparticle. Since the massless and spinless particles are not covered in our model, the Bogomol’nyi-Prassad-Sommerfield bound of central charge (see, for instance, [@sohn]) assumes the only possibility for the degeneracy. The BPS bound $m\geq\|{\cal Z}\|$ provides, as is known, consistency of the quantum theory; the opposite inequality breaks the unitarity. As we have the goal to construct the quantum theory, we may restrict the consideration to the case of $\|b\|\leq 1$. Furthermore, the limiting point $\|b\|=1$ corresponds to the multiplet-shortening [@sohn]. It is the case $m=\|{\cal Z}\|$ when the massive multiplet contains the same number of particles as a massless one. These massive multiplets are called hypermultiplets. In the case of N=2, D=3 Poincaré superalgebra, a massive supermultiplet of superspin $s$ describes a quartet of particles with spins $s,s+\frac12,s+\frac12,s+1$ for $m>\|{\cal Z}\|$ and a doublet $s,s+\frac12$ for $m=\|{\cal Z}\|$. The shortening of the superparticle multiplet has the respective origin in the classical mechanics: the number of odd physical degrees of freedom of the superparticle halfed in the BPS limit. Let us show that it is the case which is described by our model. Reducing to the constraints (\[3.14\]) (or, equivalently, (\[3.6\])) we come to the smaller $5/4$-dimensional phase space ${\cal M}^{5/4}\subset{\cal M}^{8\|4}$ with a degenerate symplectic two-superform $$\left.\Omega_s^{\rm SUSY}\right\|{}_{p_a=mn_a} \equiv \Omega_s^{\rm red}=-m{\rm d}x^a\wedge{\rm d}n_a+ \Omega_{{\cal L}^{1\|2}} \qquad {\rm d} n_a\equiv\frac{2\xi_a{\rm d}\bar z+2\bar\xi_a{\rm d}z}{(1-z\bar z)^2} \,,\label{3.15}$$ where $\Omega_{{\cal L}^{1\|2}}$ is defined by Eq. (\[3.10\]) and $$\xi_a= -\frac{1}{2}(\gamma_a )_{\alpha\beta} z^\alpha z^{\beta}=-\frac{1}{2}(2z,1+z^2,i(z^2-1))\qquad \bar{\xi}_a=\overline{(\xi_a)}\,. \label{3.16}$$ The kernel of the two-superform (\[3.15\]) contains obviously the even one-dimensional null space ${\rm Ker}_0\Omega_s^{red}$, related to the reparametrization invariance of the world lines. In the coset superspace ${\cal O}_{m,s,b}={\cal M}^{5\|4}/{\rm Ker}_0\Omega_s^{red}$ the induced symplectic two-superform is nondegenerate when $\|b\|<1$, the same is true in ${\cal L}^{1\|2}$ for the respective superform $\Omega_{{\cal L}^{1\|2}}$. Therefore, ${\cal O}_{m,s,b}$, ${\rm dim\,}{\cal O}_{m,s,b}=4/4\,, \|b\|<1$ is isomorphic to a regular coadjoint orbit of N=2, D=3 Poincaré supergroup. We have established both the embedding of the regular orbit into the original phase superspace and the underlying projection [$\pi:{\cal M}^{8\|4}\to{\cal O}_{m,s,b}$]{}, provided by constraints (\[3.14\]). In the BPS limit $\|b\|=1$ the inner two-superform $\Omega_{{\cal L}^{1\|2}}$ generates $0/2$-dimensional null-vector superspace. Thus, the full kernel ${\rm Ker}\,\Omega_s^{red}$ of the symplectic two-superform on ${\cal M}^{5\|4}$ becomes $1/2$-dimensional if $\|b\|=1$. The $4/2$-dimensional coset superspace ${\cal O}_{m,s}={\cal M}^{5\|4}/ {\rm Ker}\,\Omega_s^{red}$ corresponds to a degenerate orbit of the N=2 Poincaré supergroup. Hence, the number of odd physical degrees of freedom of N=2, D=3 superparticle halfed actually in the BPS limit and we observe an evident classical analogue of the multiplet-shortening. Some more peculiarities of the BPS limit for the superparticle model will be discussed in Subsec. 6. We have described the embedding of coadjoint orbits of the N=2 superparticle in the phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$. This description should be treated as a natural superextension of the D=3 spinning particle model with extended phase space ${\cal M}^8\cong T^\ast({\rm R}^{1,2})\times{\cal L}$, where the particle spin is realized in terms of the Lobachevsky plane. One can also construct a different embedding of the superparticle dynamics generalizing the canonical model with the minimal six-dimensional phase space ${\cal M}^6$. This embedding is obtained by reduction $\pi_1: {\cal M}^{8\|4}\to{\cal M}^{6\|4}$ with respect to two second class constraints $p_a=\sqrt{-p^2}n_a$ of (\[3.6\]). If the coordinates in ${\cal M}^{6\|4}$ are chosen to be $(x^a,p_a,\theta^{\alpha\,I})$, then the induced symplectic two-superform reads as $$\begin{aligned} \displaystyle \Omega^{\rm SUSY}_s&=&{\rm d}p_a\wedge{\rm d}x^a+ \left(\frac{s}2+imb\frac{p_{\alpha\beta}\theta^\alpha\chi^\beta}{\sqrt{-p^2}} \right)\frac{\epsilon^{abc}p_a{\rm d}p_b\wedge{\rm d}p_c}{(-p^2)^{3/2}} -2imb\epsilon_{\alpha\beta}{\rm d}\theta^\alpha\wedge {\rm d}\chi^\beta\nonumber \nonumber \\ &&\displaystyle -\frac{im}{\sqrt{-p^2}}(\gamma^a)_{\alpha\beta}\left(\Pi_{ab}\theta^\alpha -b\frac{\epsilon_{abc}p^c}{\sqrt{-p^2}}\chi^\alpha\right){\rm d}p^b\wedge {\rm d}\theta^\beta \nonumber -im\frac{p_{\alpha\beta}}{\sqrt{-p^2}}{\rm d}\theta^\alpha\wedge {\rm d}\theta^\beta \\ &&\displaystyle -\frac{im}{\sqrt{-p^2}}(\gamma^a)_{\alpha\beta}\left(\Pi_{ab}\chi^\alpha +b\frac{\epsilon_{abc}p^c}{\sqrt{-p^2}}\theta^\alpha\right){\rm d}p^b\wedge {\rm d}\chi^\beta -im\frac{p_{\alpha\beta}}{\sqrt{-p^2}}{\rm d}\chi^\alpha\wedge {\rm d}\chi^\beta\,, \nonumber \\ && \label{3.17}\end{aligned}$$ where $\Pi_{ab}=\eta_{ab}-p_ap_b/p^2$. It is nondegenerate again if $\|b\|\neq1$. The superparticle dynamics on ${\cal M}^{6\|4}$ is governed by the mass shell constraint $p^2+m^2=0$ only and provides the straightforward N=2 supergeneralization of the canonical description of spinning particle. Rel. (\[3.17\]) is an N=2 analogue of the monopole Dirac symplectic structure (\[2.1\]). It is likely to be interesting to invert the symplectic two-superform on ${\cal M}^{6\|4}$ and to represent the analogue of the fundamental Poisson brackets (\[2.4\]). The nonvanishing brackets are (we mark the PB’s on ${\cal M}^{6\|4}$ by the star) $$\begin{array}{c} \displaystyle \{x^a\,,\,x^b\}^\ast=s\frac{\epsilon^{abc}p_c}{(-p^2)^{3/2}} \left[1-\frac{m}{2s} (\theta^\alpha\theta_\alpha+\chi^\alpha\chi_\alpha)+\frac{imb}s\frac{ p_{\alpha\beta}\theta^\alpha\chi^\beta}{\sqrt{-p^2}}\right]\\ \displaystyle \{\theta^{\alpha\,I}\,,\,\theta^{\beta\,J}\}^\ast=-\frac{1}{2m(1-b^2)}\left( i\delta^{IJ}\frac{p^{\alpha\beta}}{\sqrt{-p^2}}+b\epsilon^{IJ} \epsilon^{\alpha\beta}\right)\\ \displaystyle \{x^a\,,\,p_b\}^\ast=\delta^a{}_b\qquad \{x^a\,,\, \theta^{\alpha\,I}\}^\ast= -\frac{i}{2p^2}\epsilon^{abc}p_b(\gamma_c)^\alpha{}_\beta\theta^{\beta\,I} \,.\label{3.18} \end{array}$$ These nonlinear brackets defy the usual attempts of operator realization in a Hilbert space. An efficient alternative to the direct realization is in the use of the extended phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$, which allows more supersymmetry, that affects on the quantization procedure drastically. Hidden $\bf su(1,1\|2)$ supersymmetry of the superspin degrees of freedom ---------------------------------------------------------------------------- \[sbs3.4\]We have shown that the superparticle dynamics is embedded in the phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$. One can imply that the inner supermanifold ${\cal L}^{1\|2}$ carries internal (both even and odd) degrees of freedom of D=3 particle. Then the symplectomorphisms of ${\cal L}^{1\|2}$ should be treated as the hidden supersymmetry of the particle internal structure. Consider this supersymmetry in more detail. To be specific, let us assume that $\|b\|<1$. The degenerate case will be discussed separately in Subsec. 6. We have already mentioned that ${\cal L}^{1\|2}$ is a homogeneous $\rm OSp(2\|2)$ superspace. Introducing new odd complex variables (\[3.9\]) we established that the symplectic two-superform (\[3.10\]) reduces to the superform on the regular $\rm OSp(2\|2)$ coadjoint orbit obtained earlier in Refs. [@Balant; @Grad] in the framework of the supercoherent state technique. A crucial point is that ${\cal L}^{1\|2}$ reveals a [Kähler]{} supermanifold structure with the superpotential $$\Phi=-2s\ln(1-z\bar z)-s(1+b)\frac{\theta\bar\theta}{1-z\bar z} -s(1-b)\frac{\chi\bar\chi}{1-z\bar z} +\frac{s(1-b^2)}2\frac{\theta\bar\theta\chi\bar\chi}{(1-z\bar z)^2}\,, \label{3.19}$$ so that $$\displaystyle \Omega_{{\cal L}^{1\|2}}=\frac{i}2 \left({\rm d}\bar z\frac\partial{\partial\bar z} +{\rm d}\bar\theta\frac{\vec\partial}{\partial\bar\theta}+ {\rm d}\bar\chi\frac{\vec\partial}{\partial\bar\chi}\right)\wedge \left({\rm d}z\frac\partial{\partial z} +{\rm d}\theta\frac{\vec\partial}{\partial\theta}+ {\rm d}\chi\frac{\vec\partial}{\partial\chi}\right)\Phi\,,$$ and $\rm OSp(2\|2)$ acts on N=2 superunit disc by the [superholomorphic]{} transformations. Moreover, the supergroup of the superholomorphic symplectomorphisms of ${\cal L}^{1\|2}$ is in fact essentially larger than $\rm OSp(2\|2)$ and it contains at least the supergroup $\rm SU(1,1\|2)$. The corresponding infinitesimal transformations read $$\begin{aligned} &&\displaystyle \delta z=i\omega^a\xi_a-\frac{\sqrt{1+b}}2\epsilon_\alpha z^\alpha\theta -\frac{\sqrt{1-b}}2\eta_\alpha z^\alpha\chi\label{3.20}\\ &&\displaystyle \delta\theta=\frac{i}2\omega^a\partial\xi_a\theta+\frac{i}{2} \sqrt\frac{1-b}{1+b}\mu_1\chi-\frac{i}{2}(\mu_2+\mu_3)\theta- \frac{1}{\sqrt{1+b}}\bar\epsilon_\alpha z^\alpha-\frac{\sqrt{1-b}}2\eta_\alpha\partial z^\alpha\theta\chi\nonumber\\ &&\displaystyle \delta\chi=\frac{i}2\omega^a\partial\xi_a\chi-\frac{i}{2} \sqrt\frac{1+b}{1-b}\bar\mu_1\theta-\frac{i}{2}(\mu_2-\mu_3)\chi +\frac{\sqrt{1+b}}2\epsilon_\alpha\partial z^\alpha\theta\chi -\frac{1}{\sqrt{1-b}}\bar\eta_\alpha z^\alpha\,,\nonumber\end{aligned}$$ where $\partial\equiv\partial/\partial z$, even parameters $\omega^a,\mu_2,\mu_3$ are real, even parameter $\mu_1$ is complex and the odd ones $\epsilon_\alpha,\eta_\alpha$ are complex. Transformations (\[3.20\]) are generated by the following Hamiltonians, which may be obtained straightforwardly solving Eqs.(\[3.11\]). There are seven (real) even Hamiltonians $$\begin{aligned} &&\hspace{-7mm} \displaystyle J_a=-sn_a\left(1-\frac{1+b}{2}\frac{\theta\bar\theta}{1-z\bar{z}} -\frac{1-b}{2}\frac{\chi\bar\chi}{1-z\bar{z}} +\frac{1-b^2}{2}\frac{\theta\bar\theta\chi\bar\chi}{(1-z\bar{z})^2} \right)\nonumber\\&&\hspace{-7mm} \displaystyle P_1=s\frac{\sqrt{1-b^2}}{2}\frac{\theta\bar\chi-\bar\theta\chi}{1-z\bar{z}} \quad\hspace{2mm} P_3=-s\left(\frac{1+b}{2}\frac{\theta\bar\theta}{1-z\bar{z}} -\frac{1-b}{2}\frac{\chi\bar\chi}{1-z\bar{z}}\right)\label{3.21a}\\&& \displaystyle\hspace{-7mm} P_2=is\frac{\sqrt{1-b^2}}{2}\frac{\theta\bar\chi+\bar\theta\chi}{1-z\bar{z}} \quad P_4=-s\left(\frac{1+b}{2}\frac{\theta\bar\theta}{1-z\bar{z}} +\frac{1-b}{2}\frac{\chi\bar\chi}{1-z\bar{z}} -\frac{1-b^2}{2}\frac{\theta\bar\theta\chi\bar\chi}{(1-z\bar{z})^2}\right) \nonumber\end{aligned}$$ and eight odd ones $$\begin{array}{ll} \displaystyle E^\alpha=s\sqrt{1+b} \left(\frac{z^\alpha\bar\theta-\bar z^\alpha\theta}{1-z\bar z}\right) \left(1-\frac{1-b}{2} \frac{\chi\bar\chi}{1-z\bar{z}}\right) & \displaystyle F^\alpha=in^\alpha{}_\beta E^\beta \\ \displaystyle G^\alpha=s\sqrt{1-b} \left(\frac{z^\alpha\bar\chi-\bar z^\alpha\chi}{1-z\bar z}\right) \left(1-\frac{1+b}{2} \frac{\theta\bar\theta}{1-z\bar{z}}\right) & \displaystyle H^\alpha=in^\alpha{}_\beta G^\beta\,. \end{array} \label{3.21b}$$ These Hamiltonians, together with one more even element $Z\equiv s$, generate a closed superalgebra with respect to Poisson superbrackets on ${\cal L}^{1\|2}$ (here $I,J,K=1,2,3$): $$\begin{aligned} &&\displaystyle\hspace{-5mm} \begin{array}{lll} \{J_a,J_b\}=\epsilon_{abc}J^c & \displaystyle \{J_a,E^\alpha\}=\frac{i}{2}(\gamma_a)^\alpha{}_\beta E^\beta & \displaystyle \{J_a,F^\alpha\}=\frac{i}{2}(\gamma_a)^\alpha{}_\beta F^\beta \\[2mm] \{P_I,P_J\}=-\epsilon_{IJK}P_K & \displaystyle \{J_a,G^\alpha\}=\frac{i}{2}(\gamma_a)^\alpha{}_\beta G^\beta & \displaystyle \{J_a,H^\alpha\}=\frac{i}{2}(\gamma_a)^\alpha{}_\beta H^\beta \end{array} \nonumber\\&&\hspace{-5mm} \begin{array}{llll} \displaystyle \{E^\alpha,P_1\}=\frac12 H^\alpha& \displaystyle \{E^\alpha,P_2\}=-\frac12 G^\alpha& \displaystyle \{E^\alpha,P_3\}=-\frac12 F^\alpha& \displaystyle \{E^\alpha,P_4\}=-\frac12 F^\alpha\\[2mm] \displaystyle \{F^\alpha,P_1\}=-\frac12 G^\alpha& \displaystyle \{F^\alpha,P_2\}=-\frac12 H^\alpha& \displaystyle \{F^\alpha,P_3\}=\frac12 E^\alpha& \displaystyle \{F^\alpha,P_4\}=\frac12 E^\alpha\\[2mm] \displaystyle \{G^\alpha,P_1\}=\frac12 F^\alpha& \displaystyle \{G^\alpha,P_2\}=\frac12 E^\alpha& \displaystyle \{G^\alpha,P_3\}=\frac12 H^\alpha& \displaystyle \{G^\alpha,P_4\}=-\frac12 H^\alpha\\[2mm] \displaystyle \{H^\alpha,P_1\}=-\frac12 E^\alpha& \displaystyle \{H^\alpha,P_2\}=\frac12 F^\alpha& \displaystyle \{H^\alpha,P_3\}=-\frac12 G^\alpha& \displaystyle \{H^\alpha,P_4\}=\frac12 G^\alpha\\[3mm] \end{array} \nonumber\\&&\displaystyle\hspace{-5mm} \begin{array}{lll} \displaystyle \{E^\alpha,F^\beta\}=\epsilon^{\alpha\beta}(Z-P_3) & \displaystyle \{E^\alpha,G^\beta\}=-\epsilon^{\alpha\beta}P_2 & \displaystyle \{E^\alpha,H^\beta\}=\epsilon^{\alpha\beta}P_1 \\[2mm] \displaystyle \{G^\alpha,H^\beta\}=\epsilon^{\alpha\beta}(Z+P_3) & \displaystyle \{F^\alpha,H^\beta\}=-\epsilon^{\alpha\beta}P_2 & \displaystyle \{F^\alpha,G^\beta\}=\epsilon^{\alpha\beta}P_1\\[1.5mm] \end{array}\label{3.22} \\&&\hspace{-5mm} \displaystyle \ \, \{E^\alpha,E^\beta\}=\{F^\alpha,F^\beta\}= \{G^\alpha,G^\beta\}=\{H^\alpha,H^\beta\}=i(\gamma_a)^{\alpha\beta}J^a \nonumber \\[1.5mm]&&\hspace{-5mm}\displaystyle\ \, \{J_a,P_I\}=0\quad \{P_I,P_4\}=0\quad \{J_a,P_4\}=0\quad \{Z,{\rm anything}\}=0\,.\nonumber\end{aligned}$$ What we have obtained it is the explicit Poisson realization of the so-called $\rm su(1,1\|2)$ superalgebra [@Dictionary], whose even part is $\rm su(1,1\|2)_0=su(1,1) \bigoplus u(2)\bigoplus R$ and the odd part constitutes an eight dimensional module of the even part; $Z$ presents a central charge. The $\rm osp(2\|2)$ subsuperalgebra found in Refs. [@Balant; @Grad] is spanned by $J_a,B,\sqrt{ms}\,V^\alpha,\sqrt{ms}\,W^\alpha$, where $V^\alpha,W^\alpha$ are defined below by Eqs. (\[3.24\]) and $B=P_3-bZ$. We reveal that $\rm N=2$ superunit disc is not only a typical coadjoint orbit of the $\rm OSp(2\|2)$ supergroup, ${\cal L}^{1\|2}\cong \rm OSp(2\|2)/[U(1)\times U(1)]$, but it can be treated simultaneously as an atypical Kähler orbit of the supergroup $\rm SU(1,1\|2)$: ${\cal L}^{1\|2}\cong \rm SU(1,1\|2)/[U(2\|2)\times U(1)]$. Hidden N=4 Poincaré supersymmetry ------------------------------------- \[sbs3.5\]Subalgebra $\rm u(2)$ of the internal $\rm su(1,1\|2)$ superalgebra acts on the odd variables, as is seen from Eqs.(\[3.20\]). It is exactly the subalgebra of the isotopic symmetry. However, the isotopic $\rm U(2)$ symmetry may now be involved in the Poincaré supersymmetry. The isotopic rotations together with the N=2 Poincaré transformations (\[3.2\]), (\[3.3\]) generate (when $\|b\|\neq1$) more wide D=3, N=4 Poincaré superalgebra. In addition to (\[3.2\]) there are the following supersymmetry transformations: $$\begin{array}{lll} \displaystyle\hspace{-1.45mm} \delta_{\tilde\epsilon}x^a= ib\gamma^a_{\alpha\beta}\tilde\epsilon^\alpha\chi^\beta -i\epsilon^{abc} n_b \gamma_{c\,\alpha\beta}\tilde\epsilon^\alpha\theta^\beta+ n^a\tilde\epsilon^\alpha\theta_\alpha & \displaystyle \delta_{\tilde\epsilon}\theta^\alpha=-in^\alpha{}_\beta\tilde\epsilon^\beta & \displaystyle \delta_{\tilde\epsilon} \chi^\alpha= \delta_{\tilde\epsilon} z=0 \\ \displaystyle \hspace{-1.45mm} \delta_{\tilde\eta}x^a=-ib\gamma^a_{\alpha\beta}\tilde\eta^\alpha\theta^\beta -i\epsilon^{abc}n_b \gamma_{c\,\alpha\beta}\tilde\eta^\alpha\chi^\beta+ n^a\tilde\eta^\alpha\chi_\alpha & \displaystyle \delta_{\tilde\eta}\chi^\alpha=-in^\alpha{}_\beta\tilde\eta^\beta & \displaystyle \delta_{\tilde\eta}\theta^\alpha= \delta_{\tilde\eta} z=0\,, \end{array} \label{3.2a}$$ where $\tilde\epsilon^\alpha,\tilde\eta^\alpha$ are odd infinitesimal parameters. The respective Hamiltonians on ${\cal M}^{8\|4}$ read $$\begin{array}{l}\displaystyle \tilde{\cal Q}^1_\alpha= ip_{\alpha\beta}(in^\beta{}_\gamma\theta^\gamma+b\chi^\beta) -m(\theta_\alpha-ibn_{\alpha\beta}\chi^\beta)\\ \displaystyle \tilde{\cal Q}^2_\alpha= ip_{\alpha\beta}(in^\beta{}_\gamma\chi^\gamma-b\theta^\beta) -m(\chi_\alpha+ibn_{\alpha\beta}\theta^\beta)\,. \end{array} \label{3.12a}$$ New supercharges together with N=2 superpoincaré Hamiltonians (\[3.12\]) and isotopic $\rm U(2)$ Hamiltonians $P_I\,,I=1,2,3,4$ generate closed N=4 Poincaré superalgebra with one central charge. It can be seen by introducing new basis for supercharges $2{\cal R}^I_\alpha=({\cal Q}^1_\alpha+\tilde{\cal Q}^2_\alpha\,,\,{\cal Q}^2_\alpha-\tilde{\cal Q}^1_\alpha)$, $2\tilde{\cal R}^I_\alpha=(\tilde{\cal Q}^1_\alpha+{\cal Q}^2_\alpha\,,\, \tilde{\cal Q}^2_\alpha-{\cal Q}^1_\alpha)\,, I=1,2$. On shell (\[3.6\]) we have $$\begin{array}{l}\displaystyle \{{\cal R}^I_\alpha\;,\;{\cal R}^J_\beta\}\approx(1-b)( -i\delta^{IJ}p_{\alpha\beta}+m\epsilon^{IJ}\epsilon_{\alpha\beta})\\ \displaystyle\{\tilde{\cal R}^I_\alpha\;,\;\tilde{\cal R}^J_\beta\}\approx(1+b) (-i\delta^{IJ}p_{\alpha\beta}+m\epsilon^{IJ}\epsilon_{\alpha\beta})\\ \displaystyle\{{\cal R}^I_\alpha\;,\;\tilde{\cal R}^J_\beta\}\approx0\,. \end{array}$$ The invariance of the original Lagrangian (\[3.1\]) under the transformations (\[3.2a\]) can be examined straightforwardly. Thus, the model, being N=2 superpoincaré invariant by construction, allows the hidden N=4 supersymmetry. The appearance the enhanced supersymmetry is hardly surprising in the model. This N=4 supersymmetry is degenerate in a sense that the corresponding central charges equals to $m$ and, so, they saturate the BPS bound for N=4 Poincaré superalgebra. It reflects the degeneracy of N=4 supersymmetry and the shortening of the N=4 superparticle multiplet to the N=2 supermultiplet in quantum theory. Moreover, it is a general property of extended supersymmetry that some of the degenerate multiplets of a larger SUSY (those which saturate the BPS bound) have the same particle content, as is observed in the respective multiplets of a smaller SUSY. This fact provides a simple reason why some of supersymmetric theories may have the extended supersymmetries. The precedents are known both for D=4,6,10 superparticle models [@Brink] and supersymmetric field theories (for example, the theories with non-trivial topological charge [@SpecHlou]). D=3, N=1 superparticle allows the hidden N=2 SUSY [@GKL2]. The degeneracy of the hidden N=4 supersymmetry can be observed already in the classical model. New supercharges $\tilde{\cal Q}^I_\alpha$ are functionally dependent from the N=2 Hamiltonians. On the constraint surface (\[3.6\]) we have $$\tilde{\cal Q}^I_\alpha=-\frac{i}{m}p_\alpha{}^\beta{\cal Q}^I_\beta\,. \label{relN4}$$ We conclude that the hidden N=4 supersymmetry can be treated as an artifact of the embedding of N=4 Poincaré superalgebra into the universal enveloping algebra of N=2 one. The transformations (\[3.2a\]) are, in fact, special linear combinations of the N=2 transformations (\[3.2\]) with the coefficients depending from the on shell conserved quantities. Bogomol’ny-Prassad-Sommerfield limit ----------------------------------------- \[sbs3.6\]Let us briefly discuss the special case $\|b\|=1$. To make the mentioned degeneracy more evident we introduce for a while new odd variables $$\tilde\theta^\alpha=\theta^\alpha-in^\alpha{}_\beta\chi^\beta\qquad \tilde\chi^\alpha=\chi^\alpha-in^\alpha{}_\beta\theta^\beta$$ instead of $\theta^\alpha,\chi^\alpha$. This change of the odd variables is one-to-one, and the original Lagrangian (\[3.1\]) reads in new variables as $$L=m(\dot{x},n)-im\frac{1+b}{2}n_{\alpha\beta}\tilde\theta^\alpha \dot{\tilde\theta}{}^\beta-im\frac{1-b}{2}n_{\alpha\beta}\tilde\chi^\alpha \dot{\tilde\chi}{}^\beta+is\frac{\bar z{\dot z} -z\dot{\bar z}}{1-z\bar z}\,. \eqno{(\rm 3.1a)}$$ It is seen immediately that half of the odd degrees of freedom of the superparticle drops out from the theory in the case of $\|b\|=1$. Moreover, in the BPS limit expression (3.1a) reduces to the Lagrangian of N=1, D=3 superparticle [@GKL2] and does describe not a superquartet, but a supersymmetric doublet of particles of equal mass $m$ and spins $s$ and $s+\frac12$ only. The inner symplectic two-superform (\[3.10\]) turns out to be a Kähler superform on an atypical $\rm OSp(2\|2)$-coadjoint orbit ${\cal L}^{1\|1}$ of complex dimension $1/1$. Thus the phase superspace ${\cal M}^{8\|4}$ reduces to ${\cal M}^{8\|2}\cong T^\ast({\rm R}^{1,2}) \times{\cal L}^{1\|1}$ with internal $\rm OSp(2\|2)$ supersymmetry. The latter is realized by all superholomorphic transformations of N=1 superunit disc ${\cal L}^{1\|1}$. The respective model of N=1, D=3 superparticle on ${\cal M}^{8\|2}$ was considered in detail in Ref. [@GKL2]. In the present paper we give an appropriate N=2 extension for the N=1 model retaining the hidden supersymmetries. It is worth noting that the hidden N=4 supersymmetry vanishes if $\|b\|=1$, whereas N=2 supersymmetry of the N=1 superparticle could be treated as the hidden one [@GKL2]. Almost all the equations of this Section still remain valid in the BPS limit if one takes formally $\tilde\theta^\alpha= \theta^\alpha\,,\chi^\alpha=0$ and $b=0$. Relationship between Hamiltonian generators of the Poincaré\ and internal supersymmetries --------------------------------------------------------------- \[sbs3.7\]We have observed that the model contains both the global Poincaré SUSY and the hidden $\rm SU(1,1\|2)$, the latter is closely related to the superspin intrinsic structure. Thus, the relevant quantization procedure should make a provision for either symmetries to survive in quantum theory. This quantization can be based on a simple fact that the Hamiltonian generators (\[3.12\]) and (\[3.12a\]) of the Poincaré supersymmetries, being the functions on ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times {\cal L}^{1\|2}$, can be expressed in terms of the Minkowski-space co-ordinates and momenta $(x^a,p_a)$ and of the $\rm su(1,1\|2)$ Hamiltonians $J_a,P_I,E^\alpha,F^\alpha,G^\alpha$ and $H^\alpha$ , which parametrize the coadjoint orbit ${\cal L}^{1\|2}$. We give here the explicit form of these expressions: $$\begin{array}{l}\displaystyle {\cal J}_a=\epsilon_{abc}x^bp^c+J_a\qquad\qquad {\cal P}_a=p_a\qquad\qquad {\cal Z}=mb\\[1mm] \displaystyle {\cal Q}^1_\alpha=(ip_{\alpha\beta}W^\beta+m\tilde{W}_\alpha) [1+{\rm q}^{cl}(bP_3-\sqrt{1-b^2}\,P_2-P_4)] \\[1mm] \displaystyle {\cal Q}^2_\alpha=(ip_{\alpha\beta}V^\beta+m\tilde{V}_\alpha) [1+{\rm q}^{cl}(bP_3+\sqrt{1-b^2}\,P_2-P_4)] \end{array} \label{3.23}$$ $$\begin{array}{l}\displaystyle \displaystyle \tilde{\cal Q}^1_\alpha=(ip_{\alpha\beta}\tilde W^\beta-mW_\alpha) [1+{\rm q}^{cl}(bP_3-\sqrt{1-b^2}\,P_2-P_4)] \\[1mm] \displaystyle \tilde{\cal Q}^2_\alpha=(ip_{\alpha\beta}\tilde V^\beta-mV_\alpha) [1+{\rm q}^{cl}(bP_3+\sqrt{1-b^2}\,P_2-P_4)]\,. \end{array} \label{N4}$$ where $$\begin{array}{ll} \displaystyle W^\alpha=\frac{1}{2\sqrt{ms}}(\sqrt{1+b}\,E^\alpha+\sqrt{1-b}\,H^\alpha) & \displaystyle\tilde W^\alpha= \frac{1}{2\sqrt{ms}}(\sqrt{1+b}\,F^\alpha-\sqrt{1-b}\,G^\alpha)\\ \displaystyle V^\alpha=\frac{1}{2\sqrt{ms}}(\sqrt{1+b}\,F^\alpha+\sqrt{1-b}\,G^\alpha) & \displaystyle\tilde V^\alpha= \frac{1}{2\sqrt{ms}}(\sqrt{1-b}\,H^\alpha-\sqrt{1+b}\,E^\alpha) \end{array} \label{3.24}$$ and constant ${\rm q}^{cl}$ reads as $${\rm q}^{cl}=\frac{1}{4s}\label{3.25}\,.$$ To construct an appropriate operator realization of these expressions we shall quantize $(x^a,p_a)$ canonically and extend simultaneously $\rm su(1,1\|2)$ Hamiltonian vector fields to a representation by Hermitian operators in Hilbert space. Notice at once some important details in relation to the quantization which should be compatible to the full symmetry of the superparticle. First, expressions (\[3.23\]) and (\[N4\]) are essentially [nonlinear]{} in the generators of the inner $\rm su(1,1\|2)$ superalgebra. Thus, even though the operator realization of the Poisson $\rm su(1,1\|2)$ superalgebra (\[3.22\]) is found and the corresponding operators are substituted in Eq. (\[3.23\]), we may not be sure that the representation of the Poincaré superalgebra (neither N=2 nor N=4) is reproduced for certain in quantum theory. Because of the nonlinearity, the superalgebra of operators, corresponding to (\[3.23\]), might be disclosed, and it is the parameter ${\rm q}$, which controls the possible disclosure of the Poincaré superalgebra. We will see that the parameter ${\rm q}^{cl}$ should be renormalized in quantum theory to reproduce a representation of the Poincaré supersymmetry. Second, it is a matter of direct verification that the Hamiltonians $W^\alpha,\tilde W^\alpha$ have vanishing Poisson superbrackets with $bP_3-\sqrt{1-b^2}P_2-P_4$, whereas $V^\alpha,\tilde V^\alpha$ commute to $bP_3+\sqrt{1-b^2}\,P_2-P_4$. This point will be important for Hermitian properties of operators in quantum mechanics. FIRST QUANTIZATION OF THE SUPERPARTICLE {#s2} =========================================== It is a primary objective of previous consideration to present the classical model of N=2, D=3 superparticle in the form, well-adapted for a quantizing procedure. We have obtained an embedding of the (maximal) coadjoint orbit of N=2 Poincaré supergroup in the extended phase superspace ${\cal M}^{8\|4}\cong T^\ast({\rm R}^{1,2})\times {\cal L}^{1\|2}$. Going to quantum theory we will combine the canonical Dirac quantization on $T^\ast({\rm R}^{1,2})$ and the geometric quantization methods on the $\rm SU(1,1\|2)$ co-orbit ${\cal L}^{1\|2}$. In particular, a combination of the standard real polarization in $T^\ast({\rm R}^{1,2})$ and the Kähler one in ${\cal L}^{1\|2}$ will be used to construct the superparticle’s Hilbert space. The quantization scheme implies from the outset that the internal $\rm SU(1,1\|2)$ supersymmetry must survive at the quantum level. Mutual relation between Hamiltonians of $\rm SU(1,1\|2)$ and Poincaré supersymmetries, being expressed by Eq. (\[3.23\]) and (\[N4\]), is crucial in our approach. At first, we construct the operator realization for the Hamiltonians of $\rm su(1,1\|2)$ superalgebra in the framework of Berezin quantization. Then the expressions (\[3.23\]) (possibly, together with (\[N4\])) are used to obtain the realization of a UIR for the N=2 (respectively, enhanced N=4) Poincaré superalgebra. We find that the classical meaning of the parameter $\rm q$ (\[3.25\]) in the relations (\[3.23\]) and (\[N4\]) should be accompanied by certain quantum corrections, referred to as a renormalization, for consistency of the quantum theory. Eventually we obtain the straightforward N=2 supergeneralization of conventional realization of the unitary irreducible representations (UIR’s) of D=3 Poincaré group on the fields carrying representations of $\rm\overline{SO^\uparrow(1,2)}$ [@JackNair]. Two cases should be distinguished among these representations. The fields describing fractional superspin (superanyons) carry an atypical unitary [infinite dimensional]{} UIR’s of $\rm SU(1,1\|2)$, whereas the UIR’s of (half)integer superspin can be realized on the spin-tensor fields carrying atypical [finite dimensional]{} non-unitary representations of $\rm SU(1,1\|2)$. The realization of the superparticle Hilbert space is slightly different in these two cases. Berezin quantization on ${\cal L}^{1\|2}$.\ {#sbs4.1} -------------------------------------------- The Berezin technique [@Berezin1; @Berezin2; @Perel; @BarMar] provides the perfect quantization method for the Kähler homogeneous spaces. We consider here briefly the application of this method to the supermanifold $\rm{\cal L}^{1\|2}\cong OSp(2\|2)/[U(1)\times U(1)]\cong SU(1,1\|2)/[U(2\|2)\times U(1)]$ with the nondegenerate symplectic structure when $\|b\|<1$. The geometric quantization on ${\cal L}^{1\|2}$, being considered as a regular coadjoint orbit, is studied in Refs. [@Balant; @Grad] in detail. However, as we know, ${\cal L}^{1\|2}$ has not been considered as an irregular $\rm SU(1,1\|2)$ co-orbit nor as a detailed Berezin quantization and the underlying correspondence principle is not explicitly established. In the following Subsections we apply the obtained results for quantization of D=3 superparticle. ### Antiholomorphic sections and an inner product Let us consider the space ${\cal O}_{s,b}$ of superantiholomorphic sections of the superholomorphic line bundle over ${\cal L}^{1\|2}$, whose elements are represented by functions $$\begin{aligned} \displaystyle f(\bar{\mit\Gamma})\equiv f(\bar{z},\bar{\theta},\bar{\chi})&\displaystyle =& \displaystyle f_0(\bar{z})+\sqrt{s(1+b)}\,\bar\theta f_1(\bar{z}) +\sqrt{s(1-b)}\,\bar\chi f_2(\bar{z})\label{4.1}\\&&\displaystyle +\sqrt{s(s+1/2)(1-b^2)}\,\bar\theta\bar\chi f_3(\bar{z})\,,\nonumber\end{aligned}$$ where $f_i(\bar z)\,,i=0,1,2,3$ are ordinary antiholomorphic functions on the unit disc of the complex plane. We denote by ${\mit\Gamma}\equiv\{{\mit\Gamma}^A\}=\{z,\theta,\chi\}$ and $\bar{\mit\Gamma}\equiv\{{\mit\Gamma}^{\bar A}\}=\{\bar z,\bar\theta,\bar\chi\}$ the sets of the superholomorphic and superantiholomorphic variables respectively. The space ${\cal O}_{s,b}$ is equipped naturally by an inner product $$\langle f\|g\rangle_{{\cal L}^{1\|2}}=\int_{{\cal L}^{1\|2}}\overline{f(\bar{\mit\Gamma})}g(\bar{\mit\Gamma}) {\rm e}^{-\Phi({\mit\Gamma},\bar{\mit\Gamma})} d\mu({\mit\Gamma},\bar{\mit\Gamma})\,.\label{4.2a}$$ Here $\Phi({\mit\Gamma},\bar{\mit\Gamma})$ is the Kähler superpotential (\[3.19\]) and $d\mu({\mit\Gamma},\bar{\mit\Gamma})$ is an $\rm SU(1,1\|2)$ invariant Liouville supermeasure on ${\cal L}^{1\|2}$. Taking into account the definition of the symplectic two-superform (\[3.10\]) $\Omega_{{\cal L}^{1\|2}}\equiv{\rm d}{\mit\Gamma}^A\Omega_{A\bar{B}}{\rm d} {\mit\Gamma}^{\bar B}$, one can derive the supermeasure explicitly [@Balant; @Grad] $$\hspace{-2.5mm} d\mu({\mit\Gamma},\bar{\mit\Gamma})=-\frac{1}{4\pi}{\rm sdet} \\|\Omega_{A\bar{B}}\\|d{\mit\Gamma}\, d\bar{\mit\Gamma} =\frac{d{\mit\Gamma}\,d\bar{\mit\Gamma}}{i\pi s(1-b^2)} \, , \qquad\! d{\mit\Gamma}\, d\bar{\mit\Gamma}\equiv dz\,d\bar z\,d\theta\, d\bar\theta\,d\chi\,d\bar\chi\,.\!\! \label{4.3}$$ Using Eqs. (\[3.19\]), (\[4.1\]) and (\[4.3\]) we can integrate out the Grassmann variables in Eq. (\[4.2a\]), that reduces the inner product to the following form: $$\langle f\|g\rangle_{{\cal L}^{1\|2}}=\langle f_0\|g_0\rangle_{\cal L}^s +\langle f_1\|g_1\rangle_{\cal L}^{s+1/2} +\langle f_2\|g_2\rangle_{\cal L}^{s+1/2}+\langle f_3\|g_3\rangle_{\cal L}^{s+1} \,,\label{4.2b}$$ where $$\langle\varphi \|\chi\rangle_{{\cal L}}^l=(2l-1) \int\limits_{\|z\|<1}\frac{dzd\bar{z}}{2\pi i}(1-z\bar z)^{2l-2} \overline{\varphi (\bar{z})}\chi (\bar{z}) \label{4.4}$$ is an inner product in the representation space $D^l_+$ of $\rm\overline{SO^\uparrow(1,2)}$ discrete series bounded below, being realized by antiholomorphic functions[^4] in the unit disc $\|z\|<1$. The inner product (4.4) is well defined and positive if $l>1/2$. Moreover, for values $0<l<1/2$ one can still use Eq. (\[4.4\]) if suitable analytic continuations are made. The case of $l=1/2$ should be understood in the sense of the limit. We conclude that the inner product in ${\cal O}_{s,b}$ is well defined if $s>0$ (and, of course, if $\|b\|<1$). In view of the transformation law for Kähler superpotential $\Phi({\mit\Gamma},\bar{\mit\Gamma})$ under the action of $\rm SU(1,1\|2)$ supergroup, the inner product holds to be $\rm SU(1,1\|2)$ invariant, if an appropriate transformation law for $f({\mit\Gamma})\in{\cal O}_{s,b}$ is implemented. In other terms, the Hamiltonian action of $\rm SU(1,1\|2)$ on ${\cal L}^{1\|2}$ can be lifted to a unitary representation in ${\cal O}_{s,b}$. We give below an infinitesimal form of this representation only, that is explicit representation of corresponding superalgebra $\rm su(1,1\|2)$. To obtain it, we first consider a conventional correspondence between linear operators in ${\cal O}_{s,b}$ and Berezin’s symbols. ### Classical observables and operators Let $A({\mit\Gamma},\bar{\mit\Gamma})$ be a “classical observable”, that means it is a real function on ${\cal L}^{1\|2}$ to be continuously differentiable in $z,\bar z$ that the integrals considered below do exist. We associate a linear operator $\widehat A$ in ${\cal O}_{s,b}$ to the classical observable $A({\mit\Gamma},\bar{\mit\Gamma})$ by the rule $$({\widehat A}f)(\bar{\mit\Gamma})=\int_{{\cal L}^{1\|2}} A({\mit\Gamma_1},\bar{\mit\Gamma})f(\bar{\mit\Gamma}_1) L_{s,b}({\mit\Gamma}_1,\bar{\mit\Gamma}) {\rm e}^{-\Phi({\mit\Gamma}_1,\bar{\mit\Gamma}_1)} d\mu({\mit\Gamma}_1,\bar{\mit\Gamma}_1)\,.\label{4.5}$$ where $A({\mit\Gamma_1},\bar{\mit\Gamma})$ serves only as an analytic continuation in ${\cal L}^{1\|2}\times{\cal L}^{1\|2}$ for classical observable $A({\mit\Gamma},\bar{\mit\Gamma})$. The generating kernel $L_{s,b}({\mit\Gamma}_1,\bar{\mit\Gamma})$ can be constructed by the use of an arbitrary complete orthonormal basis $f_k(\bar{\mit\Gamma})$ in ${\cal O}_{s,b}$, and appears to be related immediately to the analytic continuation in ${\cal L}^{1\|2}\times{\cal L}^{1\|2}$ of the Kähler superpotential: $$\begin{aligned} &\displaystyle\hspace{-4mm} L_{s,b}({\mit\Gamma}_1,\bar{\mit\Gamma})& \displaystyle =\sum\limits_{k=1}^\infty f_k(\bar{\mit\Gamma})\overline{f_k(\bar{\mit\Gamma}_1)} =(1-z_{\scriptscriptstyle 1}\bar z)^{-2s}\left[1-s(1+b)\frac{\theta_{\scriptscriptstyle 1}\bar\theta}{1-z_{\scriptscriptstyle 1}\bar z} -s(1-b)\frac{\chi_{\scriptscriptstyle 1}\bar\chi}{1-z_{\scriptscriptstyle 1} \bar z}\right.\nonumber \\ &&\displaystyle\left. +s(s+\frac12)(1-b^2) \frac{\theta_{\scriptscriptstyle 1}\bar\theta\chi_{\scriptscriptstyle 1} \bar\chi}{(1-z_{\scriptscriptstyle 1}\bar z)^2}\right] =\exp[\Phi({\mit\Gamma}_1,\bar{\mit\Gamma})]\,. \label{4.6}\end{aligned}$$ The state, being presented by the function $\Phi_{\bar{\mit\Gamma}} (\bar{\mit\Gamma}_1)=L_{s,b}({\mit\Gamma},\bar{\mit\Gamma}_1)$ with fixed $\bar{\mit\Gamma}\equiv\{\bar z,\bar\theta,\bar\chi\}$ is denoted by $\|\bar z,\bar\theta,\bar\chi\rangle$, is called as an $\rm SU(1,1\|2)$ (or $\rm OSp(2\|2)$) supercoherent state. The analytic continuation in ${\cal L}^{1\|2}\times{\cal L}^{1\|2}$ for any classical observable could be expressed in terms of the supercoherent states as follows $$A({\mit\Gamma_1},\bar{\mit\Gamma_2})=\frac{\langle\Phi_{\bar{\mit\Gamma}_2} \|{\widehat A}\|\Phi_{\bar{\mit\Gamma}_1}\rangle \lefteqn{{}_{{\cal L}^{1\|2}}}} {\langle\Phi_{\bar{\mit\Gamma}_2}\|\Phi_{\bar{\mit\Gamma}_1}\rangle \lefteqn{{}_{{\cal L}^{1\|2}}}} \qquad.\label{4.7}$$ So, the symbol of the unit operator $\hat{\rm I}$ is just $1$. Hence, the one-to-one correspondence between classical observables on ${\cal L}^{1\|2}$ and linear operators in ${\cal O}_{s,b}$ is established. In view of Eq. (\[4.7\]) classical observables are also referred to as (covariant) Berezin symbols. ### Atypical unitary and finite dimensional representations\ of the $\bf su(1,1\|2)$ superalgebra Using equation (\[4.5\]), one can now obtain the operators which correspond to the Hamiltonian generators of holomorphic transformations of N=2 superunit disc. One gets $$\begin{array}{l}\displaystyle {\widehat J}{}_a=-\bar\xi_a\bar\partial- (\bar\partial\bar\xi_a)\left( s+\frac12\bar\theta\frac{\partial}{\partial\bar\theta} +\frac12\bar\chi\frac{\partial}{\partial\bar\chi}\right)\qquad{\widehat Z}= s{\hat{\rm I}} \end{array}$$ $$\begin{array}{ll}\displaystyle {\widehat P}_1=-\frac{1}{\sqrt{1-b^2}}\left(\frac{1-b}{2}\bar\chi \frac\partial{\partial\bar\theta} +\frac{1+b}{2}\bar\theta\frac\partial{\partial\bar\chi}\right) &\displaystyle {\widehat P}_3=\frac12\bar\theta\frac\partial{\partial\bar\theta} -\frac12\bar\chi\frac\partial{\partial\bar\chi} \\ \displaystyle {\widehat P}_2=\frac{i}{\sqrt{1-b^2}}\left(\frac{1-b}{2}\bar\chi \frac\partial{\partial\bar\theta} -\frac{1+b}{2}\bar\theta\frac\partial{\partial\bar\chi}\right) &\displaystyle {\widehat P}_4=\frac12\bar\theta\frac\partial{\partial\bar\theta} +\frac12\bar\chi\frac\partial{\partial\bar\chi} \end{array}\label{4.8}$$ $$\begin{array}{l}\displaystyle {\widehat E}{}^\alpha=\frac{\sqrt{1+b}}{2}\bar\theta\left[\bar{z}^\alpha\bar \partial +(\bar\partial\bar{z}^\alpha)\left(2s+\bar\chi\frac\partial{\partial\bar\chi} \right)\right] -\frac1{\sqrt{1+b}}\bar{z}^\alpha\frac\partial{\partial\bar\theta} \\ \displaystyle {\widehat F}{}^\alpha= -i\frac{\sqrt{1+b}}{2}\bar\theta\left[\bar{z}^\alpha\bar\partial +(\bar\partial\bar{z}^\alpha)\left(2s+\bar\chi\frac\partial{\partial\bar\chi} \right)\right] -i\frac1{\sqrt{1+b}}\bar{z}^\alpha\frac\partial{\partial\bar\theta} \\ \displaystyle {\widehat G}{}^\alpha=\frac{\sqrt{1-b}}{2}\bar\chi\left[\bar{z}^\alpha\bar \partial +(\bar\partial\bar{z}^\alpha)\left(2s+\bar\theta\frac\partial{\partial\bar \theta}\right)\right] -\frac1{\sqrt{1-b}}\bar{z}^\alpha\frac\partial{\partial\bar\chi} \\ \displaystyle {\widehat H}{}^\alpha=-i\frac{\sqrt{1-b}}{2}\bar\chi\left[\bar{z}^\alpha\bar\partial +(\bar\partial\bar{z}^\alpha)\left(2s+\bar\theta\frac\partial{\partial\bar \theta}\right)\right] -i\frac1{\sqrt{1-b}}\bar{z}^\alpha\frac\partial{\partial\bar\chi}\,, \end{array}$$ where $\bar\partial\equiv\partial/\partial\bar{z}$, all the derivatives are left and $\bar\xi_a,\bar z^\alpha$ are defined by Eqs. (\[3.16\]) and (\[3.7\]) respectively. It is readily verified that the derived operators generate an irreducible representation of $\rm su(1\|1\|2)$ superalgebra, and it is the same case for any values $s$ and $b$, not only for $s>0$ and $\|b\|<1$. The anticommutation relations for operators (\[4.8\]) completely correspond to the Poisson superbrackets (\[3.22\]), and it is sufficient to apply the correspondence rules, that is, to replace $\{\,,\,\}\to1/i[\,,\,]_\mp$ (anticommutator for two odd operators and commutator in the rest cases) in Rel. (\[3.22\]). By reduction to the orthosymplectic subsuperalgebra we reproduce just the typical UIR’s of the $\rm osp(2\|2)$ obtained in Refs. [@Balant; @Grad]. The constructed representation is infinite dimensional for $s>0$, $\|b\|<1$ and unitary in the sense that the operators (\[4.8\]) are Hermitian with respect to inner product . It means, in particular, that $\langle f\|{\widehat J}{}_a\|g\rangle_{{\cal L}^{1\|2}}=\overline{\langle g\|{\widehat J}{}_a\|f\rangle}_{{\cal L}^{1\|2}}$, $\langle f\|{\widehat P}{}_I\|g\rangle_{{\cal L}^{1\|2 }}=\overline{\langle g\|{\widehat P}{}_I\|f\rangle}_{{\cal L}^{1\|2}}$ for any ${f,g\in{\cal O}_{s,b}}$. The Hermitian selfconjugation conditions for the odd operators may reveal some subtlety. Any odd classical observable among is the Majorana spinor and we have, for example, $\overline{E^0}=-E^1$, $\overline{E^1}=-E^0$ with respect to the reality condition (\[Major\]). ${\widehat E}{}^\alpha$ (and any odd operator with the spinor index) is Hermitian in the sense that $\langle f\|{\widehat E}{}^0\|g\rangle_{{\cal L}^{1\|2}}= -\overline{\langle g\|{\widehat E}{}^1\|f\rangle}_{{\cal L}^{1\|2}}$. We denote the UIR obtained by ${\bf D}_+^{s,b}$. With respect to the $\rm su(1,1)$ subalgebra, it is decomposed into the direct sum $D^s_+\bigoplus D^{s+1/2}_+\bigoplus D^{s+1/2}_+\bigoplus D^{s+1}_+$ of the unitary representations of discrete series, and the components $f_0,f_{1,2},f_3$ of the state (\[4.1\]) transform by the representations of higher weights $s,s+1/2$ and $s+1$ respectively. The representations being obtained for $s\leq0$ or $\|b\|>1$ are non-unitary. The case of $s+1=-j$, $j$ is non-negative integer or half integer, is special. Then the operators (\[4.8\]) generate a finite dimensional representation ${\bf D}^j$ of dimension $8j+8$. It is a superquartet of finite dimensional representations of $\rm su(1,1)$, ${\bf D}^j=D^{j+1}\bigoplus D^{j+1/2} \bigoplus D^{j+1/2}\bigoplus D^{j}$, and the state’s components $f_0,f_{1,2}, f_3$ transform by the $2j+3, 2j+2$ and $2j+1$ dimensional representations respectively. It should be mentioned that the representations of the $\rm su(1,1\|2)$ being considered here correspond to an [irregular]{} coadjoint orbit ${\cal L}^{1\|2}$ of the supergroup $\rm SU(1,1\|2)$ and, hence, they are [atypical]{} representations. By reduction to the orthosymplectic subsuperalgebra we get just the [typical]{} representations of the $\rm osp(2\|2)$. Keeping in mind the spinning superparticle we remember that the representations $D^s_+$ of the universal covering $\rm\overline {SO^\uparrow(1,2)}$ are commonly used for conventional realizations of the UIR’s of D=3 Poincaré symmetry of fractional spin [@JackNair; @Forte; @CorPly], whereas the finite dimensional irreps $D^j$ serve the ones of integer or of half integer spin. It will be natural to extend these realizations to N=2 Poincaré supersymmetry by means of representations ${\bf D}^{s,b}_+$ and ${\bf D}^j$ of the inner $\rm su(1,1\|2)$ superalgebra. ### The correspondence principle To complete the quantization procedure on ${\cal L}^{1\|2}$ let us return to the relation between observables and linear operators. We have examined for the supersymmetry generators of $\rm su(1,1\|2)$ superalgebra that there is an exact correspondence between supercommutators of the operators in ${\cal O}_{s,b}$ and the Poisson superbrackets of respective classical observables. In this sense we do have “the quantization” of a classical mechanics on the N=2 superunit disc. Consider now the correspondence between the algebras of [arbitrary]{} linear operators in the Hilbert space and their symbols. The problem we concern with is thoroughly studied for Kähler homogeneous manifolds. Berezin proved the general “correspondence principle” [@Berezin1; @Berezin2], which roughly consists in the following. The multiplication of operators induces a binary $\ast$-operation for corresponding symbols; $\ast$-multiplication is noncommutative. Furthermore, the theory contains a “Planck constant” $h$ related to one of the quantum numbers, and in the limit when $h\to0$ $\ast$-algebra transforms to the ordinary commutative algebra of functions on the manifold. Finally, the first order reset with respect to $h$ of the commutator of symbols coincides with their Poisson bracket. The Lobachevsky plane has originally served as a test example for the Berezin technique [@Berezin2]. The parameter $s^{-1}$ plays the role of the Planck constant. Similar principles hold true for N=2 superunit disc being a natural N=2 superextension of the Lobachevsky plane. Let ${\widehat A}{}_1,{\widehat A}{}_2$ be two linear operators in ${\cal O}_{s,b}$ and $A_1({\mit\Gamma},\bar{\mit\Gamma}),A_2({\mit\Gamma},\bar{\mit\Gamma})$ being the respective Berezin covariant symbols. It follows from Eq. (\[4.5\]) that the symbol being corresponded to the product ${\widehat A}{}_2\cdot {\widehat A}{}_1$ (and denoted by $A_2\ast A_1$) reads $$A_2\ast A_1({\mit\Gamma},\bar{\mit\Gamma})=\int_{{\cal L}_{1\|2}} A_2({\mit\Gamma}_1,\bar{\mit\Gamma})A_1({\mit\Gamma},\bar{\mit\Gamma}_1) \frac{L_{s,b}({\mit\Gamma},\bar{\mit\Gamma}_1)L_{s,b}({\mit\Gamma}_1, \bar{\mit\Gamma})}{L_{s,b}({\mit\Gamma}_1,\bar{\mit\Gamma}_1)L_{s,b} ({\mit\Gamma},\bar{\mit\Gamma})}d\mu({\mit\Gamma}_1,\bar{\mit\Gamma}_1)\,. \label{4.9}$$ Hence the multiplication of the operators induces the $\ast$-multiplications of the symbols. [Theorem]{} (the correspondence principle): The following estimations take place: $$\begin{array}{ll}\displaystyle 1)&\displaystyle \lim\limits_{s\to\infty}A_2\ast A_1({\mit\Gamma},\bar{\mit\Gamma})= A_2({\mit\Gamma},\bar{\mit\Gamma})\cdot A_1({\mit\Gamma},\bar{\mit\Gamma})\\ \displaystyle 2)&\displaystyle \lim\limits_{s\to\infty}s\left(A_2\ast A_1({\mit\Gamma},\bar{\mit\Gamma}) -A_1\ast A_2({\mit\Gamma},\bar{\mit\Gamma})\right)=is \{A_2\,,\,A_1\}\,, \end{array}$$ where $\{\,,\,\}$ is the Poisson superbracket on ${\cal L}^{1\|2}$. To examine the correspondence principle we need for the explicit form of the fundamental Poisson superbrackets $\Omega^{A\bar B}=\{\Gamma^A,\Gamma^{\bar B}\}$. They are derived accounting for the condition $\Omega^{A\bar B}\Omega_{\bar{B}C}= \delta^A{}_C$, where $\Omega_{\bar{A}B}$ is the supermatrix of the symplectic two-superform (\[3.10\]), $\Omega_{{\cal L}^{1\|2}}\equiv{\rm d} {\mit\Gamma}^{\bar A}\Omega_{\bar AB}{\rm d}{\mit\Gamma}^B$. A slightly cumbersome calculation leads to $$\begin{aligned} \displaystyle && \{z,\bar z\}=-\frac{i}{2s}(1-z\bar z)^2 \left(1+\frac{1+b}{2}\frac{\theta\bar\theta}{1-z\bar z} +\frac{1-b}{2}\frac{\chi\bar\chi}{1-z\bar z}\right)\nonumber\\ &&\displaystyle \{z,\bar\theta^I\}=\frac{i}{2s}z\bar\theta^I(1-z\bar z) \left(1+\frac{1+b}{2}\frac{\theta\bar\theta}{1-z\bar z} +\frac{1-b}{2}\frac{\chi\bar\chi}{1-z\bar z}\right)\quad \nonumber\\[1mm]&&\displaystyle \{\theta^I,\bar z\}=\frac{i}{2s}\bar z\theta^I(1-z\bar z) \left(1+\frac{1+b}{2}\frac{\theta\bar\theta}{1-z\bar z} +\frac{1-b}{2}\frac{\chi\bar\chi}{1-z\bar z}\right)\label{4.11}\\ &&\displaystyle \{\theta,\bar\theta\}=-\frac{i}{s}(1-z\bar z) \left(\frac1{1+b}+\frac{1-b}{2(1+b)}\frac{\chi\bar\chi}{1-z\bar z} +\frac{z\bar z}{2}\frac{\theta\bar\theta}{1-z\bar z} -\frac{1-b}4\frac{\theta\bar\theta\chi\bar\chi}{1-z\bar z}\right)\nonumber\\ &&\displaystyle\{\chi,\bar\chi\}=-\frac{i}{s}(1-z\bar z) \left(\frac1{1-b}+\frac{1+b}{2(1-b)}\frac{\theta\bar\theta}{1-z\bar z} +\frac{z\bar z}{2}\frac{\chi\bar\chi}{1-z\bar z} -\frac{1+b}4\frac{\theta\bar\theta\chi\bar\chi}{1-z\bar z}\right)\nonumber\\ &&\displaystyle\{\theta,\bar\chi\}=\frac{i}{2s}(1-z\bar z)\theta\bar\chi \qquad\{\chi,\bar\theta\}=\frac{i}{2s}(1-z\bar z)\chi\bar\theta\,,\nonumber\end{aligned}$$ where $\theta^I\equiv(\theta,\chi)$. It is seen, in particular, that the r.h.s. of the fundamental Poisson superbrackets contains the order $s^{-1}$. [Proof]{}: It is based on the asymptotic estimation $$A_2\ast A_1({\mit\Gamma},\bar{\mit\Gamma})= A_2({\mit\Gamma},\bar{\mit\Gamma})\cdot A_1({\mit\Gamma},\bar{\mit\Gamma}) +iA_2({\mit\Gamma},\bar{\mit\Gamma})\frac{\stackrel{\leftarrow} {\partial}\phantom{0}}{\partial{\mit\Gamma}^A}\Omega^{A\bar B} \frac{\stackrel{\rightarrow}{\partial}\phantom{0}} {\partial{\mit\Gamma}^{\bar B}}A_1({\mit\Gamma},\bar{\mit\Gamma}) +{\cal O}(s^{-2})\,,\label{4.12}$$ from which both propositions of the theorem are easily obtained. The validity of the latter relation is sufficient to prove when $z=0$. If it is the case, Eqs. (\[4.12\]) hold true at any $z$ in consequence of the $\rm SU(1,1)$ invariance of the symplectic structure. Taking this fact into account the verification of Eq. (\[4.12\]) is made by means of an ordinary expansion of the symbols in (finite) series in the odd variables and the comparison of l.h.s. and r.h.s. of Eqs. (\[4.12\]) for the respective components. It is a trivial but cumbersome exercise, which may be successfully performed using the known estimation [@Berezin2] $${\widehat T}{}_l[\varphi]\equiv\frac{2l-1}{2\pi i}\int\limits_{\|z\|<1} \varphi(z,\bar z)(1-z\bar z)^{2l-2}dzd\bar z=\varphi(0,0)+\frac{1}{2l}\left. \bigtriangleup\varphi(z,\bar z)\right\|_{\scriptstyle z=\bar z=0} +{\cal O}(l^{-2})\label{4.13}$$ and its consequence $$l({\widehat T}{}_l[\varphi]- {\widehat T}{}_{l+1/2}[\varphi])=\frac1{4l}\left.\bigtriangleup \varphi(z,\bar z)\right\|_{\scriptstyle z=\bar z=0}+{\cal O}(l^{-2})\,.$$ Here $l>1/2$, $\varphi(z,\bar z)$ is an arbitrary function to be continuously differentiable into the unit disc in a complex plane, and $\bigtriangleup= (1-z\bar z)^2\partial\bar\partial$ is an invariant Laplace-Beltrami operator in ${\cal L}$. It is exactly the estimation (\[4.13\]), which has been originally applied by Berezin for the proof of the correspondence principle in the Lobachevsky plane [@Berezin2]. In this sense, we reduce the correspondence principle in ${\cal L}^{1\|2}$ to the one in ${\cal L}$ by means of the expansion in the odd variables. ------------------------------------------------------------------------ Operator realization of the Poincaré superalgebra.\ Renormalization of the supercharges {#sbs4.2} ----------------------------------------------------- Now we are in a position to proceed directly to the quantization of D=3 spinning superparticle. Consider the space ${\cal H}$ of functions of the form $$\begin{aligned} \displaystyle F(p,\bar{\mit\Gamma})\equiv F(p,\bar{z},\bar{\theta},\bar{\chi})&\displaystyle =& \displaystyle F_0(p,\bar{z})+\sqrt{s(1+b)}\,\bar\theta F_1(p,\bar{z}) +\sqrt{s(1-b)}\,\bar\chi F_2(p,\bar{z})\nonumber\\&&\displaystyle +\sqrt{s(s+1/2)(1-b^2)}\,\bar\theta\bar\chi F_3(p,\bar{z})\,,\label{4.14}\end{aligned}$$ where $p\equiv p^a\in{\rm R}^{1,2}$, and $F_p(\bar{\mit\Gamma})\equiv F(p,\bar{\mit\Gamma})\in{\cal O}_{s,b}$ at each fixed $p$. We would like suppose that the Hamiltonians (\[3.23\]) (which are the same as in Rel.(\[3.12\])) present “the classical symbols” of respective operators of the N=2 Poincaré superalgebra acting in ${\cal H}$. We take the following ansatz for these operators $$\begin{array}{l}\displaystyle {\widehat{\cal J}}{}_a=-i\epsilon_{abc}p^b\frac\partial{\partial p_c} +{\widehat J}{}_a\qquad\qquad {\widehat{\cal P}}_a=p_a\qquad\qquad {\widehat{\cal Z}}=mb\\ \displaystyle {\widehat{\cal Q}}{}^1_\alpha=(ip_{\alpha\beta}{\widehat W}^\beta+m{\widehat{\tilde W}}{}_\alpha)[1+{\rm q}(b{\widehat P}{}_3-\sqrt{1-b^2}\,{\widehat P}{}_2-{\widehat P}{}_4)] \\ \displaystyle {\widehat{\cal Q}}{}^2_\alpha =(ip_{\alpha\beta}{\widehat V}^\beta+m{\widehat{\tilde V}}{}_\alpha) [1+{\rm q}(b{\widehat P}{}_3+\sqrt{1-b^2}\,{\widehat P}{}_2-{\widehat P}{}_4)]\,. \end{array} \label{4.15}$$ Here the operators ${\widehat W}^\alpha,{\widehat{\tilde W}}{}^\alpha,{\widehat V}^\alpha, {\widehat{\tilde V}}{}^\alpha$ are expressed as linear combinations of ${\widehat E}{}^\alpha,{\widehat F}{}^\alpha$, ${\widehat G}{}^\alpha$, ${\widehat H}{}^\alpha$ according to relations (\[3.24\]), whereas the latter, together with the operators ${\widehat J}{}_a$ and ${\widehat P}{}_I$, are defined by the expressions (\[4.8\]). Recall that the classical observables (\[3.23\]) or (\[3.12\]) generate the Poincaré superalgebra on shell only, that is modulo to the constraints (\[3.14\]). The operator counterparts of the constraints are now imposed to annihilate the physical states according to Dirac quantization prescriptions. The linear operator in ${\cal O}_{s,b}$, which corresponds to the Berezin covariant symbol $-sn_a$, ($s>0$) reads $$\widehat{-sn_a}={\widehat J}{}_a\left(1-\frac{2}{2s+1}{\widehat P}{}_4+\frac{2}{(2s+1)(2s+2)} \bar\theta\frac{\stackrel\rightarrow\partial}{\partial\bar\theta} \bar\chi\frac{\stackrel\rightarrow\partial}{\partial\bar\chi}\right)= {\widehat J}{}_a\left(1+\frac1s\widehat P_4\right)^{-1}\,.$$ Thus, the wave equations for the superparticle are easily brought to the form[^5] $$\begin{array}{l}\displaystyle (p^2+m^2)F^{\rm phys}(p,\bar z,\bar\theta,\bar\chi)=0\\ \displaystyle [(p,{\widehat J})-m{\widehat P}{}_4-ms]F^{\rm phys}(p,\bar z,\bar\theta,\bar\chi)=0\,. \end{array} \label{4.16}$$ Solutions of the wave equations generate a subspace ${\cal H}_{m,s,b}$ in ${\cal H}$. Furthermore, if and ${\widehat{\cal S}}$ is any one of the operators (\[4.15\]), then ${\widehat{\cal S}}F$ is a physical state again, regardless of the particular value of the parameter $\rm q$. In this sense, the wave equations are superpoincaré invariant. It is crucial now to examine explicitly whether the operators (\[4.15\]) actually generate the N=2 Poincaré superalgebra. One gets in ${\cal H}$ (compare with Eq. (3.13)) $$\begin{array}{lll}\displaystyle [{\widehat{\cal J}}{}_a,{\widehat{\cal J}}{}_b]_-=i\epsilon_{abc}{\widehat{\cal J}}{}^c & \displaystyle [{\widehat{\cal J}}{}_a,{\widehat{\cal P}}{}_b]_-=i\epsilon_{abc} {\widehat{\cal P}}{}^c & \displaystyle [{\widehat{\cal P}}{}_a,{\widehat{\cal P}}{}_b]_-=0\\ \displaystyle [{\widehat{\cal J}}{}_a,{\widehat{\cal Q}}{}^I_\alpha]_- =\frac{1}{2}(\gamma_a)_\alpha{}^\beta {\widehat{\cal Q}}{}^I_\beta &\displaystyle [{\widehat{\cal P}}{}_a, {\widehat{\cal Q}}{}^I_\alpha]_-=0&\displaystyle [{\widehat{\cal Q}}{}^I_\alpha,{\widehat P}_3]_-=-\frac{i}{2}\epsilon^{IJ}{\widehat{\cal Q}}{}^J_\alpha\,. \end{array}\label{4.17}$$ These relations hold true with an arbitrary number taken for [q]{}. However, the anticommutator of the supercharges is strongly dependent on the particular value of ${\rm q}$: $$[{\widehat{\cal Q}}{}^I_\alpha,{\widehat{\cal Q}}{}^J_\beta]_+= 2\delta^{IJ}p_{\alpha\beta}-2imb\epsilon^{IJ}\epsilon_{\alpha\beta} +c^{1\,}{}^{IJ}_{\alpha\beta}(p^2+m^2)+c^{2\,}{}^{IJ}_{\alpha\beta} ((p,{\widehat J})- m{\widehat P}{}_4-ms)+{\cal O}(s^{-2})\,,\label{4.18}$$ where $c^{\,\cdot\,}{}^{IJ}_{\alpha\beta}$ are some functions, and ${\cal O}(s^{-2})$ are the corrections of higher orders in $s^{-1}$, which depend on $\rm q$ and the other parameters like $m$ and $b$. These corrections do not vanish if ${\rm q}={\rm q}^{cl}=1/4s$. Eq. (\[4.18\]) is presented in more detailed notation in the Appendix (Eqs. (\[A5\])) from the view of the closing of the operator superalgebra. Notice that the quantum value of ${\rm q}$ is not uniquely determined. The value is derived from the expressions (\[3.23\]), when the relationship between the superpoincaré and $\rm su(1,1\|2)$ generators is taken into account. However, one can start immediately from the symbols (\[3.12\]) and restore the operators applying the correspondence rule (\[4.5\]). What is remarkable is that one obtains the same operators (\[4.15\]), but the parameter ${\rm q}$ changes, and ${\rm q}^{cl}$ appears to be ${\rm q}^{cl}_1=1/(4s+2)$. But the Poincaré superalgebra is disclosed by ${\rm q}=1/(4s+2)$; the same is true for too. Both the appearance of corrections ${\cal O}(s^{-2})$ in r.h.s. of Eq. (\[4.18\]) and the ambiguity in the definition of ${\rm q}$ have the same origin. That is, a [nonlinearity]{} of the Poincaré supercharge operators (\[4.15\]) in the generators (\[4.8\]) of the inner $\rm su(1,1\|2)$ superalgebra. In consequence of the nonlinearity, different operator factor orderings may lead to the different forms for ${\widehat{\cal Q}}{}^I_\alpha$, and the corrections appear in response to the correspondence principle in ${\cal L}^{1\|2}$. We show that the disclosure of the Poincaré superalgebra at the quantum level has transparent mathematical ground in view of the Berezin correspondence principle. However, this disclosure is quite unsatisfactory from the physical viewpoint for the quantization of the elementary system. The latter is completely characterized by its inherent symmetries (in the present case it is the D=3 Poincaré SUSY). [It is the representation of these symmetries in Hilbert space, which allows to identify the obtained quantum theory with the quantized elementary system.]{} According to these reasons, to quantize D=3 superparticle we now have to provide an [exact]{} realization of the representation of the Poincaré superalgebra in the physical Hilbert space, without any corrections in the parameters of the model. To find the true quantum realization for the representation, we can try, starting from Eqs. (\[4.15\])–(\[4.18\]), to introduce some renormalized terms in the observables (\[4.15\]), which should be sufficient for the closure of the anticommutators (\[4.18\]). Certainly, we don’t have an a priori reason, which may ensure the consistency of the renormalization procedure; a structure of the possible higher order corrections to (\[4.15\]) is unclear in general. Surprisingly, the exact corrections may be obtained from the simplest ansatz (\[4.15\]) for the quantum observables. In other words, a true ordering exists for the $\rm su(1,1\|2)$ superalgebra operators, entering in ${\widehat{\cal Q}}{}^I_\alpha$ in Eq. (\[4.15\]), that [allows to restore a representation of the Poincaré superalgebra by the renormalization of the only parameter ${\rm q}$]{}. It is examined by a direct calculation that the corrections ${\cal O}(s^{-2})$ in r.h.s. of Eqs. (\[4.18\]) vanish and the operators (\[4.15\]) generate the closed Poincaré superalgebra if, and only if $${\rm q}={\rm q}^{quant}_\mp=1\mp\sqrt{1-\frac{1}{2s+1}}\ . \label{4.19}$$ Some details of calculations of the anticommutators (\[4.18\]) are given in the Appendix. The renormalized value ${\rm q}^{quant}_-={\rm q}^{cl}+{\cal O}(s^{-2})$ can be treated as a perturbative correction to the classical symbols of the supercharges. The other possible value ${\rm q}^{quant}_+$ emerges from the hidden N=4 supersymmetry and could be understood from the following reasons. Let ${\rm q=q_-}\,$. The operators of supercharges corresponding to the classical observables (\[N4\]) (see also Eqs. (\[relN4\]), (\[3.12a\])) and providing the hidden N=4 supersymmetry in ${\cal H}_{m,s,b}$ are presented by $${\widehat{\tilde{\cal Q}}}{}^I_\alpha =-\frac{i}{m}p_\alpha{}^\beta\widehat{\cal Q}^I_\beta= \left.i\widehat{K}\cdot\widehat{\cal Q}^I_\alpha\right\|_{{\rm q}\to2-{\rm q}} \,, \label{N4op}$$ where the [parity operator]{} $\widehat{K}$ is introduced. It acts on the components of the wave function (\[4.14\]) by the rule $$\widehat{K}:F=(F_0,F_1,F_2,F_3)\to\widehat{K}F=(F_0,-F_1,-F_2,F_3)\qquad \widehat{K}^2=\hat{\rm I}\,, \label{parityop}$$ and $\left.\widehat{\cal Q}^I_\alpha\right\|_{{\rm q}\to2-{\rm q}}$ denote the supercharges (\[4.15\]) being considered when the constant ${\rm q}$ is substituted for the expression $(2-{\rm q})$. The same critical values (\[4.19\]) evidently provide the closure of the N=4 Poincaré superalgebra. Moreover, the parity operator (\[parityop\]) possesses remarkable features: it commutes with the even generators of the N=4 Poincaré superalgebra and anticommutes with the supercharges: $$\begin{array}{c}\displaystyle [{\widehat{\cal J}}{}_a,\widehat{K}]_-=[\widehat{\cal P}{}_a,\widehat{K}]_-=0 \qquad [\widehat{\cal P}{}_I,\widehat{K}]_-=0\quad (I=1,2,3,4)\\ \displaystyle [{\widehat{\cal Q}}{}^I_\alpha,{\widehat K}]_+= [\widehat{\tilde{\cal Q}}{}^I_\alpha,{\widehat K}]_+=0\quad I=1,2\,. \end{array}\label{paritycomm}$$ Therefore, the operators $\widehat{{\cal Q}}{}^{\prime}{}^I_\alpha= -i\widehat{K}\widehat{{\cal Q}}^I_\alpha= \widehat{\tilde{\cal Q}}{}^I_\alpha\|_{{\rm q}\to2-{\rm q}}$ and $\widehat{\tilde{\cal Q}}{}^{\prime}{}^I_\alpha=-i\widehat{K} \widehat{\tilde{\cal Q}}{}^I_\alpha =\widehat{\cal Q}{}^I_\alpha\|_{{\rm q}\to2-{\rm q}}$ satisfy the (anti)commutation relations being identical with Eqs. (\[4.17\]), (\[4.18\]) for the supercharges $\widehat{\cal Q}^I_\alpha$ and ${\widehat{\tilde{\cal Q}}}{}^I_\alpha$ themselves. This observation clarifies to some extent the origin of the nonperturbative value ${\rm q}^{quant}_+$ for the parameter ${\rm q}$. Notice that two representations of the Poincaré superalgebra corresponding to either possible value $\rm q$ are equivalent to each other. It is seen straightforwardly from relations $$\begin{array}{c} \widehat{\cal Q}{}^{\prime}{}^I_\alpha=\widehat U \widehat{\tilde{\cal Q}}{}^I_\alpha \widehat U\qquad \widehat{\tilde{\cal Q}}{}^{\prime}{}^I_\alpha=\widehat U \widehat{\cal Q}{}^I_\alpha\widehat U \qquad [{\widehat{\cal J}}_a, \widehat U]_-=[{\widehat{\cal P}}_a, \widehat U ]_- =[\widehat P_I, \widehat U ]_-=0\,, \end{array}$$ where the operator $\widehat U$ reads $$\widehat U=1-2\bar\theta\frac{\stackrel\rightarrow\partial}{\partial\bar \theta}\bar\chi\frac{\stackrel\rightarrow\partial}{\partial\bar\chi}\qquad \qquad\widehat U^2=\hat{\rm I}\,.$$ We don’t observe, however, either any classical counterpart for the supercharges $\widehat{\cal Q}{}^{\prime}{}^I_\alpha\,, \widehat{\tilde{\cal Q}}{}^{\prime}{}^I_\alpha$ or any algebraic construction (for instance, superalgebra) involving both sets of the N=4 supercharges $\widehat{\cal Q}^I_\alpha\,, \widehat{\tilde{\cal Q}}{}^I_\alpha$ and $\widehat{\cal Q}{}^{\prime}{}^I_\alpha\,, \widehat{\tilde{\cal Q}}{}^{\prime}{}^I_\alpha$ on equal footing. To summarize briefly, the “double” SUSY of the classical mechanics of D=3 spinning superparticle can be lifted to the operator representation in the quantum theory. The key step of construction is the renormalization (\[4.19\]) for the Poincaré supercharges (\[4.17\]). Eq. (\[4.19\]) displays two exceptional values of the parameter ${\rm q}$ providing the closure for the anticommutator of supercharges (\[4.18\]) and recovering the consistent representation of the Poincaré superalgebra. We will suppose below that the parameter ${\rm q}$ is equal to either of two ${\rm q}^{quant}_\mp$. We will show below, that the representation space ${\cal H}_{m,s,b}$ is endowed by a natural Hilbert space structure, thus the operators (\[4.14\]) and (\[N4op\]) of the Poincaré representation become Hermitian. To be specific, we will consider explicitly the operators (\[4.14\]) only, which give a representation of N=2 Poincaré superalgebra. The appearance of the hidden N=4 supersymmetry, being presented by additional supercharges (\[N4op\]), becomes thereby obvious. Hilbert space for N=2 superanyon {#sbs4.3} -------------------------------- For an arbitrary $s>0$ and $\|b\|<1$ the physical space ${\cal H}_{m,s,b}$ is naturally endowed by an inner product $$(F\|G)={\cal N}\int\frac{d{\bf p}}{p^0}\langle F\|G\rangle_{{\cal L}^{1\|2}} \qquad p^0=\sqrt{{\bf p}^2+m^2}>0\,,\label{4.20}$$ where $\langle F\|G\rangle_{{\cal L}^{1\|2}}$ denotes the inner product in ${\cal O}_{s,b}\, $, $p^a=(p^0,{\bf p})$ and ${\cal N}$ is an arbitrary normalization constant. The operators (\[4.15\]) of the N=2 Poincaré superalgebra are Hermitian with respect to the introduced inner product. This fact follows immediately from that the Hermitian property for the operators of the inner $\rm su(1,1\|2)$ superalgebra with respect to $\langle\cdot\|\cdot\rangle_{{\cal L}^{1\|2}}$ and $$[{\widehat S}^{I\,\alpha}_\pm\,,\,b{\widehat P}{}_3\pm\sqrt{1-b^2}\,{\widehat P}{}_2-{\widehat P}{}_4]_- =0 \qquad{\widehat S}^{I\,\alpha}_+\equiv({\widehat V}{}^\alpha,{\widehat{\tilde V}}{}^\alpha) \qquad {\widehat S}{}^{I\,\alpha}_-\equiv({\widehat W}{}^\alpha,{\widehat{\tilde W}}{}^\alpha)\,$$ (compare with the latter notion in Subsec. III.7). Thus, we have constructed a unitary representation of N=2, D=3 superalgebra in the Hilbert space ${\cal H}_{m,s,b}$. In terms of the expansion (\[4.14\]), the wave equations (\[4.16\]) for components reduce to $$(p^2+m^2)F_\imath^{\rm phys}(p,\bar z)=0\qquad [(p,{\widehat J}{}^l)-ml]F_\imath^{\rm phys}(p,\bar z)=0\,, \label{4.21}$$ where $i=0,1,2,3$; $l=s$ for $i=0$, $l=s+1/2$ for $i=1,2$, $l=s+1$ for $i=3$ and ${\widehat J}{}^{\,l}_a= -\bar\xi_a\bar\partial-l(\bar\partial\bar\xi_a)$; $\bar\xi_a$ is defined by Eq. (\[3.16\]). It is well known [@JackNair; @CorPly; @GKL2] that the solutions of equations (\[4.21\]) generate the physical Hilbert space of D=3 particle of mass $m$, arbitrary fixed fractional spin $l>0$ and positive energy $p^0>0$. In this realization the fields $F_i(p,\bar z)$ carry an infinite dimensional UIR’s $D^l_+$ of the group $\rm\overline{SO^\uparrow(1,2)}$. Now, we may conclude that the supersymmetric theory describes the irreducible superquartet of anyons of mass $m$, superspin $s$ and central charge $\|b\|<1$. The corresponding representation is realized on the fields carrying the atypical UIR ${\bf D}^{s,b}_+$ of the superalgebra $\rm su(1,1\|2)$ (or the typical one of $\rm osp(2\|2)$). It is worth recalling that the Poincaré superalgebra admits unitary representations, when $p^0>0$ and the central charge satisfies the Bogomol’nyi-Prassad-Sommerfield bound $m\geq\|{\cal Z}\|$ (that is $\|b\|\leq1$ in our case) [@sohn]. In our realization, the wave equations (\[4.16\]), (\[4.21\]) admit, in fact, the positive energy solutions only, and the Hermitian conditions for operators (\[4.8\]) and (\[4.15\]) are broken when $\|b\|>1$. In addition we observe a shortening of the massive N=2 supermultiplet to a hypermultiplet in the BPS limit $\|b\|=1$; the latter case will be briefly discussed in the last subsection. Hilbert space for N=2 superparticle\ of (half)integer superspin {#sbs4.4} ------------------------------------ In contrast to the anyon case, the ordinary states carrying (half)integer spin have conventional realization in terms of the [finite]{} component spin-tensor fields in the Minkowski space. We have mentioned above that the finite dimensional representations of the superalgebra $\rm su(1,1\|2)$ may emerge at $s+1=-j\,,2j=0,1,2, \dots$. These representations are non-unitary. In particular, the inner product (and, thus, (\[4.20\])) has the inherent singularity at $\|z\|=1$, and the consideration of the previous subsection becomes inadequate in this case. We construct here a correct realization of the Hilbert space in the case of (half)integer (super)spin, which enables to reproduce the conventional description in terms of the spin-tensor fields. Let us start from the spinning particle without supersymmetry living classically in the phase space ${\cal M}^8=T^\ast({\rm R}^{1,2})\times {\cal L}$, that is, the model of Sec. II. The Hilbert space of the particle of (half)integer spin $j>0$ can be realized in the space ${\cal H}_j$ of the fields on ${\cal M}^8$ of the following form: $$F(p,\bar z)=F_{\alpha_1\alpha_2\dots\alpha_{2j}}(p){\bar z}^{\alpha_1} {\bar z}^{\alpha_2}\dots{\bar z}^{\alpha_{2j}}\qquad \alpha_k=0,1\,,\label{4.22}$$ where the coefficients $F_{\alpha_1\alpha_2\dots\alpha_{2j}}(p)$ are totally symmetric in their indices. $F(p,\bar z)$ appears to be polynomial of the degree $2j$ in the variable $\bar z$. The following consideration is based on remarkable transformation properties of the twistorlike objects $z^\alpha$ and ${\bar z}^\alpha$ defined by Eqs. (\[3.7\]). The Lorentz group $\rm SO^\uparrow(1,2)\cong SU(1,1)/{\bf Z}_2$ acts on $\cal L$ by fractional linear transformations $$N:\;\;z\rightarrow z^{\prime}=\frac{az-b}{\bar a-\bar{b} z} \qquad \qquad \\|N_\alpha{}^\beta\\|= \left(\begin{array}{cc}a&b\\ \bar b&\bar a\end{array}\right)\in SU(1,1)\;, \label{4.23}$$ which may be rewritten identically as $$N:\ z^{\alpha}\to z^{\alpha\,\prime} = \left( \frac{\partial z^\prime}{\partial z}\right )^{1/2} N^{-1}{}_\beta {}^\alpha z^\beta\qquad \bar{z}^\alpha\rightarrow \bar{z}^{\alpha\,\prime}= \left( \frac{\partial \bar{z}^\prime}{\partial \bar{z}}\right )^{1/2} N^{-1}{}_\beta {}^\alpha \bar{z}^\beta\;, \label{4.24}$$ or, in the infinitesimal form, $$\delta z=\frac{i}{2}\omega_{\alpha\beta} z^{\alpha} z^{\beta}\qquad \delta \bar{z}=-\frac{i}{2}\omega_{\alpha\beta} \bar{z}^{\alpha} \bar{z}^{\beta}\,,$$ where $\omega_{\alpha\beta}\equiv(\omega^a\gamma_a )_{\alpha\beta}$ are the parameters of infinitesimal Lorentz transformations. As is seen, each of $z^\alpha$, ${\bar z}^\alpha$ transforms simultaneously as a D=3 Lorentz spinor and as field density in $\cal L$. They are, in fact, the only independent twistorlike fields associated with the homogeneous space structure on the Lobachevsky plane. We have in particular $$n^{\alpha\beta}=n^a\gamma_{a}^{\alpha\beta}=-\frac{z^\alpha{\bar z}^\beta+z^\beta{\bar z}^\alpha}{ (\epsilon_{\gamma\delta}z^\gamma{\bar z}^\delta)}\qquad \Omega_{\cal L}=-2is\frac{{\rm d}z\wedge{\rm d}\bar z}{ (\epsilon_{\gamma\delta}z^\gamma{\bar z}^\delta)^2}\,.\label{4.25}$$ for the unit timelike Lorentz vector $n^a$ (\[2.7\]) and the Kähler two-form (\[2.6\]) in the Lobachevsky plane. Let us suppose the coefficients $F_{\alpha_1\alpha_2\dots\alpha_{2j}}(p)$ in Eq. (\[4.22\]) to be Lorentz spin-tensor field of the type $j$. Then $F(p,\bar z)$ possesses the following transformation law with respect to the action of Lorentz group $$N:F(p,\bar z)\to F^\prime(p,{\bar z}^\prime)=\left(\frac{\partial{\bar z}^\prime}{\partial{\bar z}}\right)^jF(p,{\bar z}) \,.$$ We have a standard realization of the finite dimensional representation $D^j$ [@GelfGrVil; @Perel; @JackNair]. Extending this construction to the representation of the Poincaré group in ${\cal H}_j$ one takes the following transformation law of the fields $$F(p,\bar z)\to F^\prime(p^\prime,{\bar z}^\prime)=\left(\frac{\partial{\bar z}^\prime}{\partial{\bar z}}\right)^jF(p,{\bar z})\,.\label{4.26}$$ The Poincaré generators read $${\widehat{\cal P}}{}_a=p_a \qquad{\widehat{\cal J}}{}_a=-i\epsilon_{abc}p^b \frac{\partial}{\partial p_c}+{\widehat J}{}^j_a\,, \label{4.27}$$ where $${\widehat J}{}^j_a=-\bar\xi_a{\bar\partial}+j({\bar\partial}{\bar\xi}_a)$$ and we recall again that $\bar\xi_a$ is defined by Eq. (\[3.16\]). Now, to separate a subspace of irreducible representation of the Poincaré group from ${\cal H}_j$ it is sufficient to impose the operator counterparts of the constraints (\[2.11\]) to annihilate the physical states: $$(p^2+m^2)F(p,\bar z)=0\qquad [(p,{\widehat J}{}^j)+mj]F(p,\bar z)=0\label{4.28}$$ The space ${\cal H}_{m,j}$ of solutions of the wave equations (\[4.28\]) generates the irreducible one- or two-valued massive representation of $\rm ISO^\uparrow(1,2)$. Moreover, using the identity ${\bar z}^\alpha\bar\partial{\bar z}^\beta-{\bar z}^\beta\bar\partial{\bar z}^\alpha= \epsilon^{\alpha \beta}$ one can verify that the irreducibility conditions are equivalent to the following set of equations for Lorentz spin-tensors: $$p_{\alpha_1}{}^{\beta}F_{\beta\alpha_2\dots\alpha_{2j}}(p)= mF_{\alpha_1\alpha_2\dots\alpha_{2j}}(p)\label{4.29}\,.$$ On-shell, the only independent component survives among $2j+1$ ones (for instance, in the rest system, where $p^a=(m,0,0)$, the only nonvanishing component is $F_{11\dots1}(p)$). It is now easy to write down the well-defined Poincaré invariant inner product in the space ${\cal H}_{m,j}$. For each two fields $F(p,\bar z)\in{\cal H}_{m,j}$, $G(p,\bar z)\in{\cal H}_{m,j}$ it reads $$\langle F\|G\rangle_{m,j}=m^{2\kappa+2j-1}{\cal N}(\kappa,j) \int\frac{d{\bf p}}{p^0}\int\limits_{\|z\|<1} \frac{dzd\bar z}{2\pi i} \frac{(1-z\bar z)\lefteqn{{}^{-2-2j}}}{\|(p,n)\| \lefteqn{{}^{2\kappa+2j}}} \qquad\overline{F(p,\bar z)}G(p,\bar z)\,,\label{4.30}$$ where $\kappa>1/2$ is a real parameter and $$\displaystyle [{\cal N}(\kappa,j)]^{-1}=\int\limits_{\|z\|<1} \frac{dzd\bar z}{2\pi i} \frac{(1-z\bar z)\lefteqn{{}^{-2-2j}}}{(1+z\bar z) \lefteqn{{}^{2\kappa+2j}}}\qquad=\frac{1}{2^{2j+1}}\sum\limits_{k=0}^{2j} \frac{(2j)!}{k!(2j-k)!(k+2\kappa-1)}$$ is a normalization constant. The operators (\[4.27\]) are Hermitian with respect to the inner product. Furthermore, parameter $\kappa$ is, in fact, inessential, because the UIR’s, corresponding different $\kappa$ in Eq. (\[4.30\]) and the same $m,j$, are unitary equivalent to each other. It is explicitly seen, if one takes the integral (\[4.30\]) over ${\cal L}$ with account of the expansion (\[4.22\]) for the wave functions. Then the inner product (\[4.30\]) transforms to the following form $$\langle F_j\|G_j\rangle_{m,j}=\frac{1}{m}\int\frac{d{\bf p}}{p^0}\overline{ F^{\,\alpha_1\alpha_2\dots\alpha_{2j}}(p)} G_{\,\alpha_1\alpha_2\dots\alpha_{2j}}(p) \,,\label{4.31}$$ which is independent from $\kappa$. The superextension appears immediately. We need only to consider the space ${\cal H}_{m,s,b}$ introduced in subsec. 2 and to put $s+1=-j\leq0$ being a (half)integer number. The wave function has the following component expansion $$\begin{aligned} F(p,\bar{\mit\Gamma})&=& F_{0\,\alpha_1\alpha_2\dots\alpha_{2j+2}}(p)\bar z^{\alpha_1} \bar z^{\alpha_2}\dots\bar z^{\alpha_{2j+2}}\nonumber\\&&+ i\sqrt{(j+1)(1+b)}\bar\theta F_{1\,\alpha_1\alpha_2\dots\alpha_{2j+1}}(p)\bar z^{\alpha_1} \bar z^{\alpha_2}\dots\bar z^{\alpha_{2j+1}}\label{4.32}\\&&+ i\sqrt{(j+1)(1-b)}\bar\chi F_{2\,\alpha_1\alpha_2\dots\alpha_{2j+1}}(p)\bar z^{\alpha_1} \bar z^{\alpha_2}\dots\bar z^{\alpha_{2j+1}}\nonumber\\&& +\sqrt{(j+1/2)(j+1)(1-b^2)}\bar\theta\bar\chi F_{3\,\alpha_1\alpha_2\dots \alpha_{2j}}(p)\bar z^{\alpha_1}\bar z^{\alpha_2}\dots\bar z^{\alpha_{2j}} \nonumber\end{aligned}$$ and it is subjected the wave equations (\[4.16\]). The latter reduce to the “Dirac equations” (\[4.29\]) for each of the component $F_{\imath\,\alpha_1 \alpha_2\dots\alpha_{j_i}}\,,i=0,1,2,3$. The Hermitian inner product in ${\cal H}_{m,j,b}\equiv{\cal H}_{m,s,b}$ may be introduced by analogy with Eq. (\[4.2b\]) and for each two $F(p,\bar{\mit\Gamma})\,, G(p,\bar{\mit\Gamma})\in {\cal H}_{m,j,b}$ is expressed in terms of the inner product $$(F\|G)=\langle F_0\|G_0\rangle_{m,j+1}+\langle F_1\|G_1\rangle_{m,j+1/2}+ \langle F_2\|G_2\rangle_{m,j+1/2}+\langle F_3\|G_3\rangle_{m,j}\,.\label{4.33}$$ It is a matter of direct verification to prove that the operators (\[4.15\]), which generate the representation of the N=2 Poincaré superalgebra, are truly Hermitian with respect to this inner product. Moreover, the BPS bound $\|b\|\leq1$ and the reality of the renormalized value of ${\rm q}$ (\[4.19\]) by $s\leq-1$ provides the necessary and sufficient conditions for operators (\[4.15\]) to be Hermitian. Thus, the quantization of the model reproduces the Hilbert space of the (half)integer superspin superparticle. Each component of the wave function (\[4.32\]) describes a particle with fixed spin. N=2 supersymmetry unifies four particles of the equal mass $m$ and of the (half)integer spins $j+1,j+1/2,j+1/2,j,(j\geq0)$ in the irreducible superquartet. N=2 hypermultiplet {#sbs4.5} ------------------ The theory is strongly simplified in the BPS limit when $\|b\|=1$. The classical phase superspace appears to be ${\cal M}^{8\|2}=T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|1}$, where the atypical $\rm OSp(2\|2)$ co-orbit ${\cal L}^{1\|1}$ of complex dimension $1/1$ plays the role of the inner supermanifold associated to the superparticle superspin. The atypical coadjoint orbit of the $\rm OSp(2\|2)$ substitutes the typical one in the BPS limit. The supermanifold ${\cal M}^{8\|2}$ serves the extended phase superspace of the N=1, D=3 superparticle allowing the hidden N=2 supersymmetry. Therefore, the quantization of the N=2 superparticle reduces in the BPS limit to the one of the N=1 superparticle. Following the same method we may combine the canonical Dirac procedure and the geometric quantization. The respective theory of N=1 superanyon has been considered earlier [@GKL2] and it results in the description of N=1, D=3 supersymmetric doublet of anyons in terms of the fields carrying the atypical UIR’s of the $\rm OSp(2\|2)$. Moreover, it is shown in Ref. [@GKL2] that the N=1 superdoublet allows extended N=2 SUSY and it can be treated as the N=2 hypermultiplet of anyons. It is exactly N=2 fractional spin superparticle emerged in the BPS limit of the N=2 model suggested in the present paper. Let us briefly consider some details of this limiting case. One can see, that the consideration of this Section is easily modified to the case, which saturates the BPS bound, when $\|b\|=1$. Consider, for instance, $b=1$. Then the wave function (\[4.14\]) does not depend on $\chi$. It is equivalent to the vanishing of half of the odd variables in the BPS limit mentioned at the classical level. The generators (\[4.15\]) of the N=2 Poincaré superalgebra reduce to $$\begin{array}{l}\displaystyle {\widehat{\cal J}}_a=-i\epsilon_{abc}p^b\frac\partial{\partial p_c} +{\widehat J}{}_a\qquad\qquad {\widehat{\cal P}}_a=p_a\qquad\qquad {\widehat{\cal Z}}=m\\ \displaystyle {\widehat{\cal Q}}{}^1_\alpha=(ip_{\alpha\beta}{\widehat W}{}^\beta+m{\widehat{\tilde W}}{}_\alpha) \qquad {\widehat{\cal Q}}{}^2_\alpha=(ip_{\alpha\beta}{\widehat{\tilde W}}{}^\beta -m {\widehat W}{}_\alpha)\,, \end{array} \label{4.34}$$ where $$\begin{aligned} &\displaystyle {\widehat J}{}_a=-\bar\xi_a\bar\partial-(\bar\partial\bar\xi_a) \left(1+\frac12\bar\theta \frac{\partial}{\partial\bar\theta}\right)& \hspace{-2cm}\label{4.35}\\&\displaystyle\hspace{-6mm} \sqrt{ms}{\widehat W}^\alpha=\bar\theta(\bar z^\alpha\bar\partial+2s(\bar\partial \bar z^\alpha))-\bar z^\alpha\frac\partial{\partial\bar\theta}\qquad \sqrt{ms}{\widehat{\tilde W}}{}^\alpha=-i\bar\theta(\bar z^\alpha\bar\partial+2s(\bar\partial \bar z^\alpha))-i\bar z^\alpha\lefteqn{\frac\partial{\partial\bar\theta}\,.}&\nonumber\end{aligned}$$ The operators (\[4.35\]) together with one more scalar $\rm U(1)$-generator $${\widehat B}\equiv{\widehat P_4}{}-s=\frac12\bar\theta\frac\partial{\partial \bar\theta}-s$$ form an irreducible representation of the $\rm OSp(2\|2)$, which is unitary for $s>0$ (it is an atypical UIR of the $\rm OSp(2\|2)$ mentioned above) and it is finite dimensional for $s=-j$, $j$ being positive (half)integer. The expressions (\[4.35\]) can also be obtained by straightforward geometric quantization on ${\cal L}^{1\|1}$ [@Balant; @Grad]. The operators (\[4.34\]) are [linear]{} in the generators of the inner $\rm osp(2\|2)$ superalgebra. Therefore, they generate N=2 Poincaré superalgebra with central charge ${\cal Z}=m$ immediately, without any corrections in $1/s$, and the renormalization is not required for the case. The wave equations (\[4.16\]) hold their form in the BPS limit and the inner products are given by Eqs. (\[4.2b\]), (\[4.20\]), (\[4.33\]), where one should account for the vanishing of the last two components of the wave functions in the expansion (\[4.14\]). The peculiarities of the model mentioned, mean that we obtain in the BPS limit an adequate description of the N=2 particle hypermultiplet. In particular, we have a natural smooth reduction for both the Poincaré supersymmetry and the internal one. The comparison of the classical mechanics given in subsec. III.6 and of the presented quantum theory demonstrates the direct relationship between the contraction of the classical phase superspace $T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$ to $T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|1}$ and the shortening of the N=2 particle supermultiplet to the N=2 hypermultiplet in the BPS limit. SUMMARY AND DISCUSSION {#s5} ========================== In the present paper we have constructed the consistent first quantized theory of N=2, D=3 superanyon as well as the one of massive superparticle of the habitual (half)integer superspin. The starting point for the quantization is the classical model of the superparticle in the nonlinear phase superspace ${\cal M}^{8\|4}=T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$, that is different from the standard approach. A traditional viewpoint in the construction of the spinning particle models [@Frydr] is to describe the spinning degrees of freedom by some variables being simultaneously translation invariant and Lorentz covariant (as usual, those are Lorentz vectors or spinors). Such variables parametrize some linear space $L$ and then the extended phase space is chosen to be ${\cal M}=T^\ast({\rm R}^{1,D-1})\times L$ or ${\cal M}=T^\ast({\rm R}^{1,D-1}\times L)$. The only difference for superparticles is to replace D-dimensional Minkowski space ${\rm R}^{1,D-1}$ by the respective superspace. The advantage of the covariant (super)space ${\cal M}$ is in the linear (“covariant”) action of the Poincaré supergroup. In this approach, however, an embedding of the (super)particle physical space ${\cal O}$ (that is the underlying coadjoint orbit) in the covariant phase (super)space may be ambiguous. Moreover, it is a common usage in this approach to give little attention to the geometry underlying the embedding ${\cal O}\to {\cal M}$. We have demonstrated that the nonlinear phase superspace ${\cal M}^{8\|4}=T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$ of D=3 spinning superparticle has the following remarkable features: \(i) The embedding of an appropriate coadjoint ${\cal O}_{m,s,b}$ orbit, being associated to the N=2, D=3 superparticle of arbitrary fixed mass $m>0$, superspin $s\neq0$ and central charge $mb$ ( $\|b\|<1$ in ${\cal M}^{8\|4}$), is realized by two constraints, which provide the identical conservation of any Casimir function of the Hamiltonian Poincaré superalgebra. These constraints have transparent geometric origin and, after quantization, they are converted into wave equations of the superparticle in a natural way. \(ii) The ‘inner’ subsupermanifold ${\cal L}^{1\|2}$ of ${\cal M}^{8\|4}$ appears to be in itself the coadjoint orbit for some supergroups. ${\cal L}^{1\|2}$ is shown to be symplectomorphic to the Kähler homogeneous superspace of the supergroup $\rm SU(1,1\|2)$ or its subsupergroup $\rm OSp(2\|2)$. In this sense the model admits the second supersymmetry ($\rm SU(1,1\|2)$ SUSY) along with the original Poincaré one. To describe the superparticle in a standard way, it is convenient, starting from an ordinary particle living in ${\rm R}^{1,D-1}$, to extend the geometry of the Minkowski space to the supergeometry of the respective Minkowski superspace. We have found an alternative way, at least for dimension D=3. The intrinsic structure of D=3 spinning particle may be described in terms of the Lobachevsky geometry. To introduce the supersymmetry we may extend the inner manifold, going to the Lobachevsky supergeometry. The following interpretation is admissible. D=3 particle lives in an ordinary Minkowski space ${\rm R}^{1,2}$. In addition the [super]{}spin degrees of freedom are associated to its internal structure and generate the internal phase superspace ${\cal L}^{1\|2}$ with an inherent ${\rm SU(1,1\|2)}$ supersymmetry, which is different from the Poincaré (super)symmetry. \(iii) The BPS limit of the model looks slightly different if compared to the standard picture for the superparticle [@AzcLuk]. In the standard approach the extended superparticle model with central charge reveals the generation of new gauge degrees of freedom in the BPS limit and it corresponds to an appropriate reduction for the physical phase space. Furthermore, it is usually impossible to impose the gauge in a covariant way and then to forget about the nonphysical variables. In contrast to the standard approach, the phase superspace $T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|2}$ can be explicitly truncated to $T^\ast({\rm R}^{1,2})\times{\cal L}^{1\|1}$ in the BPS limit. In this case the inner N=2 Lobachevsky supergeometry reduces to the N=1 one, while the gauge variables drop out from the theory at all. The reduction of the phase superspace in the classical mechanics is directly related to the shortening of the particle supermultiplet in quantum theory considered in the BPS limit. \(iv) We suggest nontrivial quantization of the superparticle in the extended phase superspace ${\cal M}^{8\|4}$, which combines the canonical quantization in $T^\ast({\rm R}^{1,2})$ and the Berezin quantization in the inner phase superspace ${\cal L}^{1\|2}$. This quantization scheme leads naturally to the fields carrying infinite dimensional or finite dimensional representation of the supergroup $\rm SU(1,1\|2)$ depending on the fractional or habitual (half)integer value of spin. The results are completely consistent with the previous known description of D=3 nonsupersymmetric particles as mechanical systems [@JackNair; @CorPly]. Surprisingly, there are two, unitary equivalent to one another, series of N=2 supercharges in quantum theory, which correspond to different possibilities for the parameter $\rm q$ in Eq. (\[4.19\]). Only one of them, namely $\rm q_-$, is related directly to a conventional classical limit. Another possible value $\rm q_+$ is shown to be related to the special properties (\[paritycomm\]) of the parity operator (\[parityop\]). However, the classical counterpart of this parity structure remains unclear, and the origin of the second possible value for $\rm q$ may seem enigmatic. Notice that the parity operator generates the structure of the deformed Heisenberg algebra in Hilbert space of anyon [@Ply1] or N=1 superanyon [@GKL2]. It would be interesting to understand, what is a geometry behind the parity operator for N=2 superanyon. The significance of this one-particle theory may vary, in particular, depending on the possibility of an efficient second quantization of the model. One of the problems here is to construct a Lagrangian of the theory, which leads to the one-particle wave equation we have deduced from the classical mechanical action. The first step of construction may be to present two independent wave equations of superanyon (like Eqs. (\[4.16\])) in the form of one spinor equation, when the mass and spin shell fixing conditions may emerge as integrability conditions. It is known that the similar construction for anyons gives a simple action functional [@SorVol; @Ply1], which may be relevant for the second quantization of fractional spin particles. An adequate superextension (at least for N=1) may be constructed probably using the representations of the $\rm su(1,1\|2)$ superalgebra in the same way, as the spinor set of the anyon wave equations was constructed in Ref. [@Ply1] using the atypical UIR’s of $\rm osp(2\|2)$. In this connection it should be noted that the exploitation of the atypical UIR’s of the $\rm osp(2\|2)$ superalgebra and of the deformed Heisenberg algebra produces the linear set of spinor wave equations of N=1, D=3 superparticle only for special (half)integer $j=1/2$ and $j=1$ values of the superspin [@Ply1]. And, of course, the consistent interaction of (super)anyons remains an intriguing problem. Even in the first quantized theory the suggested approach to the description of anyon, being attempted for the extension to an interaction with an external field, implies (in the framework of minimal phase space) a perturbative representation for nonlinear commutation relations in terms of a series in powers of the field strengths [@ChouNairPol]. In particular, it is unclear whether any consistent generalization exists for the wave equations of (super)anyons obtained in this paper in the presence of arbitrary external fields. ACKNOWLEDGMENTS {#s6 .unnumbered} =================== S.L.L. is thankful to Professor S. Randjbar-Daemi for the hospitality at the High Energy Section, ICTP, where the manuscript of the paper has been completed. This work is partially supported by Grants of the International Soros Science Education Program a98-166, d-98-932 and by the RBRF Grant 98-02-16261. S.L.L. was supported in part by the Joint INTAS-RBRF Grant 95-829. I.V.G. is supported partially by the INTAS Grant 96-0457 within the research program of the International Center for Fundamental Physics in Moscow. Appendix. Anticommutation relations and renormalization\ for the N=2 Poincaré supercharges {#appendix.-anticommutation-relations-and-renormalization-for-the-n2-poincaré-supercharges .unnumbered} -------------------------------------------------------- We give here the calculation of the anticommutator (\[4.18\]) of N=2 Poincaré supercharges in more detail. The object is to show that the closure of the Poincaré superalgebra requires a renormalization of the parameter ${\rm q}$ entering in the definition of the supercharge’s (\[4.15\]) and the renormalized value is one of given by Eq. (\[4.19\]). Before coming into explicit formulas, it is helpful to introduce a convenient notation, which is slightly different from that used in the paper. First, we redefine the coefficients in the expansion (\[4.14\]) and write down the wave function in the form $$F(p,\bar{\mit\Gamma})=F_0(p,\bar{z})+\bar\theta F_1(p,\bar{z}) +\bar\chi F_2(p,\bar{z})+\bar\theta\bar\chi F_3(p,\bar{z})\,.\label{A1}$$ Hereafter, we shall represent the wave function $F\in{\cal H}$ as a four-column $$F=\left(\begin{array}{c}F_0\\F_1\\F_2\\F_3\end{array}\right)\,,\label{A2}$$ where the components $F_i\,, i=0,1,2,3$ depend on $p$ and $\bar z$. In these terms each linear operator $\widehat A$ in ${\cal H}$ will be presented by a matrix $({\widehat A}{}_i^j)$ of dimension $4\times4$, whose elements are operators acting on the components. The matrix elements $({\widehat A}{}_i^j)$ are defined by the rule $({\widehat A} F)_i=\sum_{j=0}^3{\widehat A}{}_i^jF_j$. The matrix representation gives a convenient tool for the explicit calculations performed below. Second, we take $q^\prime=1-{\rm q}$ to squeeze the notations. Let us note that the Poincaré supercharge’s (\[4.15\]) may be identically presented in the following form $${\widehat{\cal Q}}{}^{1\,\alpha}=ip^\alpha{}_\beta{\widehat W}^\beta_q+ m{\widehat{\tilde W}}{}^\alpha_q \, ,\qquad {\widehat{\cal Q}}{}^{2\,\alpha}=ip^\alpha{}_\beta{\widehat V}^\beta_q+ m{\widehat{\tilde V}}{}^\alpha_q\,, \label{A3}$$ where $$\begin{aligned} \displaystyle {\widehat W}^\alpha_q=\frac{1}{2\sqrt{ms}}\left( \begin{array}{rccc} \displaystyle 0\qquad&\displaystyle-\bar z^\alpha&\displaystyle-i\bar z^\alpha&0\\ \displaystyle\frac{1+b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &iq^\prime\bar z^\alpha\\ \displaystyle -i\frac{1-b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &-q^\prime\bar z^\alpha\\ 0\qquad&\displaystyle iq^\prime\frac{1-b}2{\hat Q}^\alpha_1&\displaystyle q^\prime\frac{1+b}2{\hat Q}^\alpha_1&0 \end{array}\right)&&\nonumber\\[5mm] \displaystyle {\widehat{\tilde W}}{}^\alpha_q=\frac{1}{2\sqrt{ms}}\left( \begin{array}{rccc} \displaystyle 0\qquad&\displaystyle-i\bar z^\alpha&\displaystyle\bar z^\alpha&0\\ \displaystyle-i\frac{1+b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &-q^\prime\bar z^\alpha\\ \displaystyle-\frac{1-b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &-iq^\prime\bar z^\alpha\\ 0\qquad&\displaystyle q^\prime\frac{1-b}2{\hat Q}^\alpha_1&\displaystyle -iq^\prime\frac{1+b}2{\hat Q}^\alpha_1&0 \end{array}\right)&&\nonumber\\&&\label{A4}\\ \displaystyle {\widehat V}^\alpha_q=\frac{1}{2\sqrt{ms}}\left( \begin{array}{rccc} \displaystyle0\qquad&\displaystyle-i\bar z^\alpha&\displaystyle-\bar z^\alpha&0\\ \displaystyle-i\frac{1+b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &q^\prime\bar z^\alpha\\ \displaystyle\frac{1-b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &-iq^\prime\bar z^\alpha\\ 0\qquad&\displaystyle -q^\prime\frac{1-b}2{\hat Q}^\alpha_1&\displaystyle -iq^\prime\frac{1+b}2{\hat Q}^\alpha_1&0 \end{array}\right)&&\nonumber\\[5mm] \displaystyle {\widehat{\tilde V}}{}^\alpha_q=\frac{1}{2\sqrt{ms}}\left( \begin{array}{rccc}\displaystyle 0\qquad&\displaystyle\bar z^\alpha&\displaystyle-i\bar z^\alpha&0\\ \displaystyle-\frac{1+b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &iq^\prime\bar z^\alpha\\ \displaystyle-i\frac{1-b}2{\hat Q}^\alpha_0&\displaystyle 0&0 &q^\prime\bar z^\alpha\\ 0\qquad&\displaystyle iq^\prime\frac{1-b}2{\hat Q}^\alpha_1&\displaystyle -q^\prime\frac{1+b}2{\hat Q}^\alpha_1&0 \end{array}\right)\,,&&\nonumber\end{aligned}$$ and ${\hat Q}^\alpha_\epsilon=\bar z^\alpha\bar\partial+(2s+\epsilon) (\bar\partial\bar z^\alpha)\,;\epsilon=0,1$. When $q^\prime=1$ we have ${\widehat W}^\alpha_0\equiv{\widehat W}^\alpha \,,{\widehat{\tilde W}}{}^\alpha_0\equiv{\widehat{\tilde W}}{}^\alpha$ and similar identities for $\widehat V$’s. The calculations of the anticommutators of N=2 supercharges give $$[{\widehat{\cal Q}}{}^{1\,\alpha}\,,{\widehat{\cal Q}}^{1\,\beta}]_+=2p^{\alpha\beta}- \frac{1}{4ms}{\widehat X}_+{\widehat J}^{\alpha\beta}(p^2+m^2)+ \frac{1}{2ms}{\widehat X}_+p^{\alpha\beta}((p, \widehat J)-m\widehat P_4-ms)+ p^{\alpha\beta}{\widehat O}_+$$ $$[{\widehat{\cal Q}}{}^{2\,\alpha}\,,{\widehat{\cal Q}}{}^{2\,\beta}]_+=2p^{\alpha\beta}- \frac{1}{4ms}{\widehat X}_-{\widehat J}^{\alpha\beta}(p^2+m^2)+ \frac{1}{2ms}{\widehat X}_-p^{\alpha\beta}((p, \widehat J)-m\widehat P_4-ms)+ p^{\alpha\beta}{\widehat O}_- \,,$$ $$\begin{aligned} \displaystyle [{\widehat{\cal Q}}{}^{2\,\alpha}\,,{\widehat{\cal Q}}{}^{1\,\beta}]_+& =&\displaystyle-2imb \epsilon^{\alpha\beta}-\frac{i}{8ms}(\widehat X_1\epsilon^{\alpha\beta} +i\widehat X_0\widehat J^{\alpha\beta})(p^2+m^2)\label{A5}\\&&\displaystyle -\frac{i}{4ms}(m\widehat X_2\epsilon^{\alpha\beta} +i\widehat X_0p^{\alpha\beta}) ((p,\widehat J)-m\widehat P_4-ms)+m\widehat O^{\alpha\beta}\nonumber\,,\end{aligned}$$ where ${\widehat J}^{\alpha\beta}=(\gamma^a)^{\alpha\beta}{\widehat J}_a$, $$\widehat X_\pm=\left( \begin{array}{cccc} 2&0&0&0\\ 0&(1+b)+q^{\prime\,2}(1-b)&\pm i(1+b)(1-q^{\prime\,2})&0\\ 0&\mp i(1-b)(1-q^{\prime\,2})&(1-b)+q^{\prime\,2}(1+b)&0\\ 0&0&0&2q^{\prime2} \end{array}\right)$$ $$\widehat X_0=2(1-q^{\prime\,2})\left( \begin{array}{cccc} 0&0&0&0\\ 0&0&1+b&0\\ 0&1-b&0&0\\ 0&0&0&0 \end{array}\right)\,,$$ $\widehat X_{1,2}$ are diagonal operators $$\begin{aligned} \displaystyle \widehat X_1F&=&\displaystyle 4bsF_0+[(1+b)(2s-1)-q^{\prime\,2}(1-b)(2s+1)]F_1 \nonumber\\&& \displaystyle-[(1-b)(2s-1)-q^{\prime\,2}(1+b)(2s+1)]F_2+4bsq^{\prime\,2}F_3 \nonumber\\ \displaystyle \widehat X_2F&=&\displaystyle 4bF_0+2[(1+b)-q^{\prime\,2}(1-b)]F_1 -2[(1-b)-q^{\prime\,2}(1+b)]F_2+4bq^{\prime\,2}F_3\,, \nonumber\end{aligned}$$ and $$\begin{array}{c}\displaystyle \widehat O_\pm=\left(q^{\prime\,2}\frac{2s+1}{2s}-1\right)\left( \begin{array}{cccc} 0&0&0&0\\ 0&1-b&\mp i(1+b)&0\\ 0&\pm i(1-b)&1+b&0\\ 0&0&0&2 \end{array}\right)\\[10mm] \displaystyle \widehat O^{\alpha\beta}=\left(q^{\prime\,2}\frac{2s+1}{2s}-1\right)\left( \begin{array}{cccc} 0&0&0&0\\ 0&i(1-b)\epsilon^{\alpha\beta}&\displaystyle -\frac{(1+b)}{m}p^{\alpha\beta} &0\\ 0&\displaystyle -\frac{(1-b)}{m}p^{\alpha\beta}&-i(1+b)\epsilon^{\alpha\beta} &0\\0&0&0&-2ib\epsilon^{\alpha\beta} \end{array}\right)\,. \end{array}\label{A6}$$ What we have obtained in Eqs. (\[A5\]) is in fact a detailed notation for Eqs. (4.15). In this notation the structure of the anticommutation relations becomes transparent and a number of helpful features stands out. Let us consider the physical subspace in ${\cal H}$, which is generated by solutions of the wave equations (\[4.16\]). Then the relations (\[A5\]) simplify themselves and we have $$[\widehat{\cal Q}^I_\alpha,\widehat{\cal Q}^J_\beta]_+= 2\delta^{IJ}p_{\alpha\beta}-2imb\epsilon^{IJ}\epsilon_{\alpha\beta} -m\delta^{IJ}\epsilon_{\alpha\beta}\widehat O_\pm-m\epsilon^{IJ}\widehat O_{\alpha\beta}\,.\label{A7}$$ The presence of the operators $\widehat O_\pm$ and $\widehat O_{\alpha\beta}$ breaks off N=2 Poincaré superalgebra in general. In the neighborhood of the value $q^\prime=1-1/4s$, being derived from the combination of the classical mechanics and of the correspondence rules, the operators $\widehat O_\pm,\widehat O_{\alpha\beta}$ should be treated as the corrections of order $s^{-2}$. However, it contains more. The explicit expressions (\[A6\]) show immediately that in the case of $s\neq-1/2$ the renormalization of $q^\prime$ is possible, which provides the identical vanishing of any corrections and the closure of the Poincaré superalgebra on shell. The renormalized values are presented by Eq. (\[4.19\]). The following observation is that the anticommutators obtained are invariant under the substitution $q^\prime\to-q^\prime$ (that is ${\rm q}\to2-q$). As mentioned in Subsec. 4.2, this invariance comes from the degenerate N=4 supersymmetry and from specific properties of the parity operator (\[parityop\]). Finally, the structure of the anticommutators (\[A5\]) changes drastically in the BPS limit $\|b\|=1$. Consider, for instance, $b=1$. In this case, the two latter components of the wave function (\[A1\]) vanish, $F_2=F_3\equiv0$ (see Eq. (\[4.14\])). However, when $b=1$, in the linear subspace $F_2=F_3\equiv0$ the parameter $q$ becomes inessential and the corrections (\[A6\]) vanish identically. Thus, we do not need any renormalization in the BPS limit, which corresponds to the N=1 superparticle, which was considered earlier in Ref. [@GKL2]. [MM]{} R.B. Laughlin, Phys. Rev. Lett. [50]{}, 1395 (1983). F. Wilczek, in [Fractional Statistics and Anyon Superconductivity,]{} (World Scientific, Singapore, 1990), edited by F. Wilczek. J. M. Leinaas and J. Myrheim, Nuovo Cim. [B37]{}, 1 (1977). F. Wilczek, Phys. Rev. Lett. [48]{}, 1144 (1982); [ibid.]{} [49]{}, 957 (1982); G. A. Goldin, R. Menikoff and D. H. Sharp, J. Math. Phys. [21]{}, 650 (1980); [ibid.]{} [22]{}, 1664 (1981). R. Jackiw and V. P. Nair, Phys. Rev. [D43]{}, 1933 (1991). S. Forte and T. Jolicoeur, Nucl. Phys. [B350]{}, 589 (1991). D.P. Sorokin and D.V. Volkov, Nucl. Phys. [B409]{}, 547 (1993). G.W. Semenoff, Phys. Rev. Lett. [61]{}, 517 (1988);\ J. Fröhlich and P.A. Marchetti, Lett. Math. Phys. [16]{}, 347 (1988); Commun. Math. Phys. [121]{}, 177 (1988); Nucl. Phys. [B356]{}, 533 (1991), and references therein. Z. Hlousek and D. Spector, Nucl. Phys. [B344]{}, 763 (1990). A. Balachandran, G. Marmo, B. Skagerstam and A. Stern, [Gauge Symmetries and Fibre Bundles,]{} (Springer, Berlin, 1983). M.S. Plyushchay, Phys. Lett. [B248]{}, 107 (1990); Int. J. Mod. Phys. [A7]{}, 7045 (1992). J. L. Cortés and M. S. Plyushchay, Mod. Phys. Lett. [A10]{}, 409 (1995); Int. J. Mod. Phys. [A11]{}, 3331 (1996). S. Ghosh, Phys. Rev. [D51]{}, 5827 (1995). I. V. Gorbunov, S. M. Kuzenko and S. L. Lyakhovich, Int. J. Mod. Phys. [A12]{}, 4199 (1997). A. Nersesian, Mod. Phys. Lett. [A12]{}, 1783 (1997). C. Chou, V.P. Nair and A.P. Polychronakos, Phys. Lett. [B304]{}, 105 (1993). C. Chou, Phys. Lett. [B323]{}, 147 (1994). S. Ghosh and S. Mukhopadhyay, Phys. Rev. [D51]{}, 6843 (1995). N.Banerjee and S. Ghosh, Phys. Rev. [D52]{}, 6130 (1995). S. Ghosh, J. Phys. [A30]{}, L821 (1997). S. Ghosh, [Planar two-particle Coulomb interaction: classical and quantum aspects,]{} preprint hep-th/9708045. S. L. Lyakhovich, A. Yu. Segal and A. A. Sharapov, Phys. Rev. [D54]{}, 5223 (1996). S. L. Lyakhovich, A. A. Sharapov and K. M. Shekhter. [Massive spinning particle in any dimension. 1. Integer spins,]{} preprint hep-th/9805020, to appear Nucl. Phys. [B]{} I. V. Gorbunov, S. M. Kuzenko and S. L. Lyakhovich, Phys. Rev. [D56]{}, 3744 (1997). M.S. Plyushchay, Ann. Phys. [245]{}, 339 (1996); Mod. Phys. Lett. [A11]{}, 2953 (1996); Mod. Phys. Lett. [A12]{}, 1153 (1997). J. M. Souriau, [Structure des Systémes Dynamique,]{} (Dunod, Paris, 1970). B. Kostant, [Lecture Notes in Mathematics,]{} [170]{} (Springer-Verlag, New York, 1970), p. 87. A. A. Kirillov, [Elements of the Representation Theory,]{} (Mir, Moscow, 1974). F. A. Berezin, Math. USSR - Izvestija [38]{}, 1116, (1974); [ibid.]{} [39]{}, 363 (1975). F. A. Berezin, Commun. Math. Phys. [40]{}, 153, (1975) N. M. J. Woodhouse, [Geometric quantization,]{} (Clarendon Press, Oxford, 1980). R. Marnelius and U. Martensson, Nucl. Phys., [B335]{}, 395 (1991); Int. J. Mod. Phys. [A6]{}, 807 (1991). A. Frydryszak, [Lagrangian models of particles with spin: the first seventy years,]{} preprint hep-th/9601020, and more than 80 references therein. S. M. Kuzenko, S. L. Lyakhovich and A. Yu. Segal, Int. J. Mod. Phys. [A10]{}, 1529 (1995); Phys. Lett. [B348]{}, 421 (1995). A. B. Balantekin, H. A. Schmitt and P. Halse, J. Math. Phys. [30]{}, 274 (1989). I.V.Gorbunov and S.L.Lyakhovich, Phys. Lett. [B422]{}, 293 (1998). El A.M. Gradechi and L. M. Nieto, Commun. Math. Phys. [175]{}, 521 (1996). J. A. de Azcáraga and J. Lukierski, Phys. Lett. [B113]{}, 170 (1982); Phys. Rev. [D28]{}, 1337 (1983). P. Fayet, Nucl. Phys. [B51]{}, 135 (1976); M. F. Sohnius, Nucl. Phys. [B138]{}, 109 (1978); Phys. Rep. [128]{}, 39 (1985). L. Frappat, A. Sciarrino and P. Sorba, [Dictionary on Lie Superalgebras,]{} preprint ENSLAPP-AL-600/96, DSF-T-30/96, hep-th/9607161. L. Brink and M.B. Green, Phys. Lett. [B106]{}, 393 (1981). Z. Hlousek and D. Spector, Nucl. Phys. [B370]{}, 143 (1992); [ibid.]{} [B397]{}, 173 (1993). M. A. Perelomov, [Generalized Coherent States,]{} (Nauka, Moscow, 1987); D. Bar-Moshe and M.S. Marinov, in [Topics in statistical and theoretical physics,]{} ed by R.L. Dobrushin et al.; J. Phys. [A27]{}, 6287 (1994). I. M. Gelfand, M. I. Graiev and N. Ya. Vilenkin, [Generalized Functions,]{} (Academic Press, New York, 1966) Vol. 5. [^1]: We use Latin letters to denote D=3 Lorentz vectors and Greek letters for the $\rm SU(1,1)$ spinors; Minkowski metric is chosen to be $\eta_{ab}={\rm diag}(-1,1,1)$, totally antisymmetric tensor is normalized by condition $\epsilon_{012}=-\epsilon^{012}=1$; the spinor indices are raised and lowered with the use of the spinor metric $\epsilon^{\alpha\beta}=-\epsilon^{\beta\alpha}= -\epsilon_{\alpha\beta}$ $(\alpha, \beta=0, 1)$, $\epsilon^{01}=-1$ by the rule $\psi_{\alpha}=\epsilon_{\alpha\beta}\psi^{\beta}, \psi^{\alpha}= \epsilon^{\alpha\beta}\psi_{\beta}$. [^2]: One may wonder, why the $\gamma$-matrices are not Hermitian. It is instructive to note that the reality condition for $\rm SU(1,1)$ spinor formalism is not trivial, as for isomorphic $\rm SL(2,R)$ ones. For any $g\in\rm SU(1,1)$ the complex conjugation reads $\bar g=cgc$, where $c=c^{-1}=\rm antidiag(-1,-1)$. The matrices $c\gamma^a$ are truly Hermitian. The covariant Majorana (reality) condition looks like $$c\bar\psi=\psi\label{Major}$$ for two-component $\rm SU(1,1)$-spinor $\psi$.\[ftn2\] [^3]: Explicit form of Eq. (\[3.10\]) depends on the grading conventions for the exterior superalgebra. We use ${\bf Z}\times{\bf Z_2}$ grading by analogy with Ref. [@Grad]. The only difference, as compared to Ref. [@Grad], is in convention for complex conjugation of the odd variables. We take $\overline{\theta_1\theta_2}=\bar\theta_2\bar\theta_1$, in particular $\theta\bar\theta$ is a real $c$-number, while in the Ref. [@Grad] it is an imaginary. [^4]: The monomials $\phi_n^l=[\Gamma(2l+n)/ \Gamma(n-1)\Gamma(2l)]^{1/2}\bar z^n$, $n$ is integer non-negative, serve as a standard orthonormal basis in $D^l_+$. [^5]: It is worth noting that the second constraint equation (\[3.14\]) should be written in an equivalent form $(p,J)-mP_4-ms=0$.
--- abstract: \| Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third, fourth, fifth, sixth, seventh - degree algebraic equations exists in gravity theory. This fact, together with the derivation of the algebraic equations for a generally defined contravariant tensor components in this paper, are important in view of finding new solutions of the Einstein’s equations, if they are treated as algebraic ones. Some important properties of the introduced in hep-th/0107231 more general connection have been also proved - it possesses affine transformation properties and it is an equiaffine one. Basic and important knowledge about the affine geometry approach and about gravitational theories with covariant and contravariant connections and metrics is also given with the purpose of demonstrating when and how these theories can be related to the proposed algebraic approach and to the existing theory of gravity and relativistic hydrodynamics. author: - \| Bogdan G. Dimitrov [^1]\ Bogoliubov Laboratory for Theoretical Physics\ Joint Institute for Nuclear Research\ 6 Joliot-Curie str.\ Dubna 141 980, Russia title: 'Elliptic Curves and Algebraic Geometry Approach in Gravity Theory I.The General Approach ' --- INTRODUCTION ============ Inhomogeneous cosmological models have been intensively studied in the past in reference to colliding gravitational waves \[1\] or singularity structure and generalizations of the Bondi - Tolman and Eardley-Liang-Sachs metrics \[2, 3\]. In these models the inhomogeneous metric is assumed to be of the form \[2\] $$ds^{2}=dt^{2}-e^{2\alpha (t,r,y,z)}dr^{2}-e^{2\beta (t,r,y,z)}(dy^{2}+dz^{2}) \tag{1.1}$$(or with $r\rightarrow z$ and $z\rightarrow x$) with an energy-momentum tensor $G_{\mu \nu }=k\rho u_{\mu }u_{\nu }$ for the irrotational dust. The functions $\alpha (t,r,y,z)$ and $\beta (t,r,y,z)$, determined by the Einstein’s equations, are chosen in a special form \[4\], so that the integrations of (some) of the components of the Einstein’s equations is ensured. A nice feature of the approach is that in the limit $t\rightarrow \infty $ \[5\] and under a special choice of the pressure as a definite function of time the metric approaches an isotropic form \[4\]. Other papers, also following the approach of Szafron-Szekerez are \[6,7\]. In \[7\], after an integration of one of the components - $G_{1}^{0}$ of the Einstein’s equations, a solution in terms of an elliptic function is obtained. In different notations, but again in the framework of the Szafron-Szekerez approach the same integrated in \[7\] nonlinear differential equation $$\left( \frac{\partial \Phi }{\partial t}\right) ^{2}=-K(z)+2M(z)\Phi ^{-1}+\frac{1}{3}\Lambda \Phi ^{2} \tag{1.2}$$was obtained in the paper \[8\] of Kraniotis and Whitehouse. They make the useful observation that (1.2) in fact defines a (cubic) algebraic equation for an elliptic curve, which according to the standard algebraic geometry prescribtions (see \[9\] for an elementary, but comprehensive and contemporary introduction) can be parametrized with the elliptic Weierstrass function $$\rho (z)=\frac{1}{z^{2}}+\sum\limits_{\omega }\left[ \frac{1}{(z-\omega )^{2}}-\frac{1}{\omega ^{2}}\right] \tag{1.3}$$and the summation is over the poles in the complex plane. Two important problems immediately arise, which so far have remained without an answer: 1\. The parametrization procedure with the elliptic Weierstrass function in algebraic geometry is adjusted for cubic algebraic equations with number coefficients! Unfortunately, equation (1.2) is not of this type, since it has coefficient functions in front of the variable $\Phi $, which depend on the complex variable $z$. In view of this, it makes no sense to define Weierstrass invariants as $$g_{2}=\frac{K^{2}(z)}{12}\text{ \ \ ; \ \ \ }g_{3}=\frac{1}{216}K^{3}(z)-\frac{1}{12}\Lambda M^{2}(z)\text{ \ \ ,} \tag{1.4}$$since the above functions have to be set up equal to the complex numbers $g_{2}$ and $g_{3}$ (the s. c. Eisenstein series) $$g_{2}=60\sum\limits_{\omega \subset \Gamma }\frac{1}{\omega ^{4}}=\sum\limits_{n,m}\frac{1}{(n+m\tau )^{4}}\text{ \ \ \ ,} \tag{1.5}$$$$g_{3}=140\sum\limits_{\omega \subset \Gamma }\frac{1}{\omega ^{6}}=\sum\limits_{n,m}\frac{1}{(n+m\tau )^{6}}\text{ \ \ \ \ } \tag{1.6}$$and therefore additional equations have to be satisfied in order to ensure the parametrization with the Weierstrass function. 2\. Is the Szekerez - Szafron metric the only case, when the parametrization with the Weierstrass function is possible? Closely related to this problem is the following one - is only one of the components of the Einstein’s equation parametrizable with $\rho (z)$ and its derivative? This series of three papers has the aim to present an adequate mathematical algorithm for finding solutions of the Einstein’s equations in terms of elliptic functions. This approach is based on the clear distinction between covariant and contravariant metric tensor components within the s.c affine geometry approach, which will be clarified further in Section 2. Afterwords, a cubic algebraic equation in terms of the contravariant metric components will be obtained, which according to the general prescription and the algorithm in the previous paper \[10\] can be parametrized with the Weierstrass function and its derivative. Respectively, if the contravariant components are assumed to be known, then a cubic (or a quartic) algebraic equation with respect to the covariant components can be investigated and parametrized again with the Weierstrass function. Thus it will turn out that the parametrization with the Weierstrass function will be possible not only in the Szafron-Szekeres case, but also in the general case due to the “cubic” algebraic structure of the gravitational Lagrangian. This is an important point since valuable cosmological characteristics for observational cosmology such as the Hubble’s constant $H(t)=\frac{\overset{.}{R}(t)}{R(t)}$ and the deceleration parameter $q=-\frac{\overset{..}{R}(t)R(t)}{\overset{.}{R}^{2}(t)}$ may be expressed in terms of the Jacobi’s theta function and of the Weierstrass elliptic function respectively \[8\]. Unfortunately, in the paper \[8\] the Eisenstein series (1.5-1.6) have not been taken into account, due to which the obtained expression for the metric will be another one and will be modified. Instead of searching elliptic solutions of the Einstein’s equations for each separate case of a given metric, as in nearly all of the mentioned papers, in this series of papers another approach will be proposed. First, a cubic algebraic equataion will be parametrized with respect to one of the contravariant components, following the approach in a previous paper \[10\]. In the second part, this parametrization will be extended to more than one variable in the multivariable cubic algebraic equation. This will be a substantial and new development, different from the standard algebraic geometry approach, in which only two-dimensional cubic equations are parametrized with the (elliptic) Weierstrass function and its derivative. Finally, in the third part the dependence of the generalized coordinates $X^{i}=X^{i}(x_{1},x_{2},x_{3},.....,x_{n})$ on the complex variable $z$ will be established from a derived system of first-order nonlinear differential equations. The generalized coordinates can be regarded as $n-$ dimensional hypersurfaces, defining a transition from an initially defined set of coordinates $x_{1},x_{2},x_{3},.....,x_{n}$ on a chosen manifold to another set of the generalized coordinates $X^{1},X^{2},.....,X^{n}$. Since the covariant metric components $g_{ij}$ also depend on these coordinates, this means that their dependence on the complex variable $z$ will also be known. In other words, at the end of the applied approach, each initially given function $g_{ij}(t,\mathbf{x})$ of the time and space coordinates will be expressed also as $g_{ij}(z)$. The algebraic approach will be applied to the s .c. cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian, but further it will be shown that not only the approach will be applicable in the general case of an arbitrary contravariant tensor, but also concrete solutions for the metric $g_{ij}(z)$ will be given in the case of specially chosen simple metrics. The first part of the present paper continues and develops further the approach from the previous paper \[10\], where a definite choice of the contravariant metric tensor was made in the form of the factorized product $\widetilde{g}^{ij}=dX^{i}dX^{j}$. The differentials $dX^{i}$ are assumed to lie in the tangent space $T_{X}$ of the generalized coordinates. In Section 2 of the present paper some basic facts about the affine geometry approach and the s.c. gravitational theory with covariant and contravariant metrics and connections (GTCCMC) will be reminded, but also some new material, related to relativistic hydrodynamics in the context of GTCCMC is added. The basic and very important idea in this section is to show that GTCCMC are already “incorporated” in the theoretical framework of the already known gravitational theories - as an example the known projective formalism is taken, but at the same time in certain theories (such as the Arnowitt-Deser-Misner $3+1$ decomposition), certain assumptions are made so that they do not fall within the class of GTCCMC. This is an interesting observation, since it clearly shows that the limiting assumptions can naturally be removed. In the next Section 3 it will be demonstrated briefly how the cubic algebraic equation with respect to the differentials $dX^{i}$ was derived in \[10\], but in fact the aim will be to show that depending on the choice of variables in the gravitational Lagrangian or in the Einstein’s equations, a wide variety of algebraic equations (of third, fourth, fifth, seventh, tenth- degree) in gravity theory may be treated, if a distinction between the covariant metric tensor components and the contravariant ones is made. This idea, originally set up by Schouten and Schmutzer, was further developed in the papers \[13, 14\] mainly with the purpose of classification of such more general GTCCMC \[13, 14\]. Also, in Section 3 the important and new physical notion of a “tensor length scale” is introduced in a natural way within the GTCCMC, and this notion is a generalization of the metrical (scale) function $l(x)=ds^{2}=g_{ij}dx^{i}dx^{j}$ in usual gravity theory. In Section 4 intersecting algebraic varieties will be proposed as a method for obtaining the known solutions in the standard gravity theory. In Section 5 it will be shown that the previously investigated in \[10\] under some restrictive assumptions cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian and the Einstein’s equations can be written as algebraic ones also in the general case of an arbitrary contravariant tensor $g^{ij}$. The physical idea, which will be exploited in this paper will be: can such a gravitational theory with a more general contravariant tensor be equivalent to the usual and known to us theory with a contravariant metric tensor, which is at the same time the inverse one of the covariant one? By equivalence it is meant that the gravitational Lagrangian in both approaches should be equal, on the base of which the s.c. cubic algebraic equation for reparametrization invariance (of the gravitational Lagrangian) was obtained in \[10\]. The derivation was based also on the construction of another connection $\widetilde{\Gamma }_{kl}^{s}\equiv \frac{1}{2}dX^{i}dX^{s}(g_{ik,l}+g_{il,k}-g_{kl,i})$, which is with contravariant tensor component replaced with the factorized product $dX^{i}dX^{s}$. The connection $\widetilde{\Gamma }_{kl}^{s}$ has two very useful properties: 1. It may have an affine transformation law under a broad variety of coordinate transformations (see Append. A2), which can be found after solving a system of nonlinear differential equations. 2. $\widetilde{\Gamma }_{kl}^{s}$ is an equiaffine connection (see also Appendix A3 for the elementary proof), which is a typical notion, introduced in classical affine geometry \[15, 16\] and meaning that there exists a volume element, which is preserved under a parallel displacement of a basic $n-$dimensional vector $e\equiv e_{i_{1}i_{2}....i_{n}}$. Equivalently defined, $\widetilde{\Gamma }_{kl}^{s}$ is an equiaffine connection \[15, 16\] if it can be represented in the form $\widetilde{\Gamma }_{ks}^{s}=\partial _{k}lge$, where $e$ is a scalar quantity. This notion turns out to be very convenient and important, since for such types of connections we can use the known formulae for the Ricci tensor, but with the connection $\widetilde{\Gamma }_{kl}^{s}$ instead of the usual Christoffell one $\Gamma _{kl}^{s}$. Moreover, the Ricci tensor $\widetilde{R}_{ij}$ will again be a symmetric one, i.e. $\widetilde{R}_{ij}=\widetilde{R}_{ji}=\partial _{k}\widetilde{\Gamma }_{ij}^{k}-\partial _{i}\widetilde{\Gamma }_{kj}^{k}+\widetilde{\Gamma }_{kl}^{k}\widetilde{\Gamma }_{ij}^{l}-\widetilde{\Gamma }_{ki}^{m}\widetilde{\Gamma }_{jm}^{k}$. In usual gravity theory, the contravariant components are at the same time inverse to the covariant ones , and thus the correspondence between covectors (in our terminology - these are the vectors) and the vectors (i.e. the contravariant vectors) is being set up, respectively there is correspondence between covariant and contravariant tensors. By “correspondence” it is meant that both these kinds of tensors satisfy the matrix equation $g_{ij}g^{jk}=\delta _{i}^{k}$. However, within the framework of affine geometry, such a correspondence is not necessarily to be established (see again \[15-18\]) and both tensors have to be treated as different mathematical objects, defined on one and the same manifold. If both components constitute the algebraic variety, satisfying the Einstein’s equations, considered as a set of intersecting multivariable cubic and quartic algebraic surfaces (further instead of cubic surfaces we shall continue to use the terminology cubic curves), then one can speak about separate classes of solutions for the covariant metric tensor components and for the contravariant ones. If one assumes the existence of inverse contravariant metric tensor components $\widetilde{g}^{jk}$ and considers the quadratic system $g_{ij}\widetilde{g}^{jk}=g_{ij}dX^{j}dX^{k}=\delta _{i}^{k}$ as intersecting with the set of cubic and quartic algebraic Einstein’s equations, then it might be expected that the standardly known solutions of the Einstein’s equations should be recovered. However, this is not yet mathematically proved, neither has this been formulated as a problem. General theorems for intersection of algebraic curves of different (arbitrary) degrees are given in \[19, 21, 22\]. AFFINE GEOMETRY APPROACH AND GRAVITATIONAL THEORIES WITH COVARIANT AND CONTRAVARIANT CONNECTIONS AND METRICS ================================================================================================================== This section has the purpose to review some of the basic aspects of gravitational theories with covariant and contravariant metrics and connections (GTCCMC), which would further allow the application of algebraic geometry and of the theory of algebraic equations in gravity theory. The section contains also some new material, concerning the application of GTCCMC in relativistic hydrodynamics. It is known in gravity theory that the knowledge of the metric tensor $g_{ij} $ determines the space - time geometry, which means that the Christoffell connection $$\Gamma _{ik}^{l}\equiv \frac{1}{2}g^{ls}(g_{ks,i}+g_{is,k}-g_{ik,s}) \tag{2.1}$$and the Ricci tensor $$R_{ik}=\frac{\partial \Gamma _{ik}^{l}}{\partial x^{l}}-\frac{\partial \Gamma _{il}^{l}}{\partial x^{k}}+\Gamma _{ik}^{l}\Gamma _{lm}^{m}-\Gamma _{il}^{m}\Gamma _{km}^{l}\text{ \ \ \ } \tag{2.2}$$can be calculated. It is useful to remember also from standard textbooks \[24\] the s. c. Christoffell connection of the first kind: $$\Gamma _{i;kl}\equiv g_{im}\Gamma _{kl}^{m}=\frac{1}{2}(g_{ik,l}+g_{il,k}-g_{kl,i})\text{ \ ,} \tag{2.3}$$obtained from the expression for the zero covariant derivative $0=\nabla _{l}g_{ik}=g_{ik,l}-g_{m(i}\Gamma _{k)l}^{m}$ $.$ By contraction of (2.3) with another contravariant tensor field $\widetilde{g}^{is}$, one might as well define another connection: $$\widetilde{\Gamma }_{kl}^{s}\equiv \widetilde{g}^{is}\Gamma _{i;kl}=\widetilde{g}^{is}g_{im}\Gamma _{kl}^{m}=\frac{1}{2}\widetilde{g}^{is}(g_{ik,l}+g_{il,k}-g_{kl,i})\text{ \ ,} \tag{2.4}$$not consistent with the initial metric $g_{ij}$. Clearly the connection (2.4) is defined under the assumption that the contravariant metric tensor components $\widetilde{g}^{is}$ are not to be considered to be the inverse ones to the covariant components $g_{ij}$ and therefore $\widetilde{g}^{is}g_{im}\equiv f_{m}^{s}(\mathbf{x})$. In fact, the definition $\widetilde{g}^{is}g_{im}\equiv f_{m}^{s}$ turns out to be inherent to gravitational physics. For example, in the projective formalism one decomposes the standardly defined metric tensor (with $g_{ij}g^{jk}=\delta _{i}^{k}$) as $$g_{ij}=p_{ij}+h_{ij}\text{ \ \ ,} \tag{2.5}$$together with the additional assumption that the two subspaces, on which the projective tensor $p_{ij}$ and the tensor $h_{ij}$ are defined, are orthogonal. This means that $$p_{ij}h^{jk}=0\text{ \ \ \ .} \tag{2.6}$$As a consequence $$p_{ij}p^{jk}=\delta _{i}^{k}-h_{ij}h^{jk}\neq \delta _{i}^{k}\text{ \ \ ,} \tag{2.7 }$$ meaning that the contravariant projective metric components $p^{jk}$ are no longer inverse to the covariant ones $p_{ij}$. An example of gravitational theories with more than one connection are the so called theories with affine connections and metrics \[13\], in which there is one connection $\Gamma _{\alpha \beta }^{\gamma }$ for the case of a parallel transport of covariant basic vectors $\nabla _{e_{\beta }}e_{\alpha }=\Gamma _{\alpha \beta }^{\gamma }$ $e_{\gamma }$ and a separate connection $P_{\alpha \beta }^{\gamma }$ for the contravariant basic vector $e^{\gamma }$, the defining equation for which is $\nabla _{e_{\beta }}e^{\alpha }=P_{\gamma \beta }^{\alpha }$ $e^{\gamma }$. In these theories, the contravariant vector and tensor fields are assumed to be not the inverse ones to the covariant vector and tensor fields. This implies that $$e_{\alpha }e^{\beta }\equiv f_{\alpha }^{\beta }(x)\neq \delta _{\alpha }^{\beta } \tag{2.8}$$for such theories and consequently, a distinction is made between covariant and contravariant metric tensors (and vectors too). Clearly, in the above given case (2.7) of projective gravity, this theory should be considered as a GTCCMC. In the same spirit, since the well - known Arnowitt - Deser - Misner (ADM) (3+1) decomposition of spacetime \[44, 45\] is built upon the projective transformation (2.5), it might be thought that it should also be considered as such a theory. But in fact, the ADM (3+1) formalism definitely is not an example for this, because due to the special identification of the vector field’s components \[44, 45\] with certain components of the projective tensor $$g_{00}:=-(N^{2}-N_{i}N^{i})\text{ \ ; \ }g^{00}:=-\frac{1}{N^{2}}\text{ \ ,} \tag{2.9}$$$$g_{ij}:=p_{ij}\text{ \ ; \ }g^{ij}:=p^{ij}-\frac{N^{i}N^{j}}{N^{2}}\text{ \ \ ,} \tag{2.10}$$$$g_{0i}:=N_{i}\text{ \ ; \ \ }g^{0i}:=\frac{N^{i}}{N^{2}}\text{ \ \ ,} \tag{2.11}$$all the contravariant projective tensor components $p^{\alpha \beta }$ ($\alpha ,\beta ,\gamma =0,1,2,3$ ; $\ i,j=1,2,3$) turn out to be the inverse ones to the covariant projective components $p_{\alpha \gamma }$. Indeed, it follows that $$p_{ij}p^{jk}=\delta _{i}^{k}\text{ \ \ \ ; \ \ }N_{i}N^{i}=N^{2}\text{ \ ; \ }N_{i}N^{j}=\delta _{i}^{j}\text{\ \ \ .} \tag{2.12}$$In the case of the ADM (3+1) decomposition, such an identification is indeed possible and justified, since the gravitational field posseses coordinate invariance, allowing to disentangle the dynamical degrees of freedom from the gauge ones. But in the case when the tensors $h_{ij}$ are related with some moving matter (with a prescribed motion) and an observer, “attached” to this matter “measures” all the gravitational phenomena in his reference system by means of the projective metric $p_{ij}$, this will be no longer possible. Then the relation (2.7) will hold, and the resulting theory will be a gravitational theory with covariant and contravariant metrics and connections (GTCCMC). Naturally, if the tensor $h_{ij}$ in (2.5) and (2.7) is taken in the form $h_{ij}=\frac{1}{e}u_{i}u^{j}$ and if the vector field $u$ (tangent at each point of the trajectory of the moving matter) is assumed to be non-normalized (i.e. $e(x)=u_{i}u^{i}\neq 1$), then one would have to work not within the standard relativistic hydrodynamics theory (where $p_{ij}=g_{ij}-u_{i}u_{j}$ and $p_{ij}p^{jk}=\delta _{i}^{k}-u_{i}u^{k}$), but within the formalism of GTCCMC (where $p_{ij}p^{jk}=f_{i}^{k}=\delta _{i}^{k}-\frac{1}{e}u_{i}u^{k}\neq \delta _{i}^{k}$). One may wonder why this should be so, since the last two formulaes for $p_{ij}p^{jk}$ for both cases look very much alike, with the exception of the “normalization” function $\frac{1}{e}$ in the second formulae. But in what follows it shall become clear that in the first case the right-hand side has a tensor transformation property, while in the second case due to the function $\frac{1}{e}$ there would be no such property. And this shall turn out to be crucial. In order to understand this also from another point of view, let us perform a covariant differentiation of both sides of the relation (2.8). Then one can obtain that the two connections are related in the following way \[13\] $$f_{j,k}^{i}=\Gamma _{jk}^{l}\text{ }f_{l}^{i}+P_{lk}^{i}f_{j}^{l}\text{ \ \ \ \ \ ;\ \ \ \ \ \ (}f_{j,k}^{i}=\partial _{k}f_{j}^{i}\text{) \ \ \ \ . } \tag{2.13}$$ Note also the following important moment - $f_{\alpha }^{\beta }(x)$ are considered to be the components of a function. Otherwise, if they are considered to be a (mixed) tensor quantity, the covariant differentiation of the mixed tensor $f_{\alpha }^{\beta }(x)$ in the right-hand side of $e_{\alpha }e^{\beta }\equiv f_{\alpha }^{\beta }(x)$ would give exactly the same expression as in the left-hand side. This would mean that from a mathematical point of view there would be no justification for the introduction of the second covariant connection $P_{lk}^{i}$ - the covariant differentiation would give a quantity on the left-hand side, which would be identically equal to $\nabla _{\gamma }f_{\alpha }^{\beta }(x)$ for every choice of the two connections $\Gamma _{jk}^{l}$ and $P_{lk}^{i}$, including also for the standard case (Einsteinian gravity) $P_{lk}^{i}=-\Gamma _{jk}^{l}$. However, in view of the fact that $f_{\alpha }^{\beta }(x)$ are related with the description of some moving matter in the Universe, then a tensor transformation law should not be prescribed to them. So they should remain components of a function and consequently, the introduction of the second connection $P_{lk}^{i}$ is inevitable. In confirmation of this, it can easily be seen that the quantity $\delta _{i}^{k}-h_{ij}h^{jk}\neq \delta _{i}^{k}$ in (2.7), which is to be set up equal to $f_{i}^{k}(x)$, does not have a tensor transformation property for arbitrarily chosen tensor fields $h_{ij}$. More concretely, it would have such a transformation property if the equality $$\left( \delta _{i}^{j}-\frac{1}{e}h_{ik}h^{kj}\right) ^{^{\prime }}(\mathbf{X})=$$$$=\frac{\partial x^{\alpha }}{\partial X^{i}}\frac{\partial X^{j}}{\partial x^{\beta }}\left( \delta _{\alpha }^{\beta }-\frac{1}{e}h_{\alpha \gamma }h^{\gamma \beta }\right) (\mathbf{x}) \tag{2.14}$$holds. Now since* $h_{ik}h^{kj}$ transforms as a tensor, then the fulfillment of (2.14) would mean that the equality $$\delta _{i}^{j}=\frac{\partial x^{\alpha }}{\partial X^{i}}\frac{\partial X^{j}}{\partial x^{\beta }}\delta _{\alpha }^{\beta } \tag{2.15}$$ should hold for any* derivatives $\frac{\partial x^{\alpha }}{\partial X^{i}}$ and $\frac{\partial X^{j}}{\partial x^{\beta }}$. But if $t_{i}^{\alpha }:=$ $\frac{\partial x^{\alpha }}{\partial X^{i}}$ and $t_{\beta }^{j}:=$ $\frac{\partial X^{j}}{\partial x^{\beta }}$ are the components of some set of tetrad fields, this would imply that this set is orthonormal, i.e. $\delta _{i}^{j}=t_{i}^{\alpha }t_{\beta }^{j}\delta _{\alpha }^{\beta }$ - a property, which now we shall prove to be not consistent with equality (2.7) $p_{ij}p^{jk}=\delta _{i}^{k}-h_{ij}h^{jk}\neq \delta _{i}^{k}$. The reason is that (2.7) already implies the existence of a basis of covariant and contravariant basic vector fields $e_{i}$ and $\widetilde{e}^{j}$, such that $e_{i}\widetilde{e}^{j}=f_{i}^{j}$- in fact, this will be the essence of a proposition, which shall soon be proved. Also, if $e^{j}$ is another system of basic fields for which $e_{i}e^{j}=\delta _{i}^{j}$, then $\widetilde{e}^{j}=f_{k}^{j}e^{k}$ and the orthonormality condition can be written as (for $\alpha =\beta $) $$\delta _{i}^{j}=t_{i}^{\alpha }t_{\alpha }^{j}=\overline{t}_{i}^{\alpha }e_{i}\widetilde{e}^{\alpha }\overline{t}_{\alpha }^{j}e_{\alpha }\widetilde{e}^{j}=\overline{t}_{i}^{\alpha }\overline{t}_{\alpha }^{j}f_{i}^{\alpha }f_{\alpha }^{j}\text{ \ \ .} \tag{2.16}$$But the orthonormality condition is defined and should have one and the same form in all reference frames, including the reference frame $\left( e_{\alpha },\widetilde{e}^{j}\right) $, in which the components of the tetrad field are $\overline{t}_{i}^{\alpha }$. Consequently $f_{i}^{\alpha }f_{\alpha }^{j}=1$, which however is in contradiction with the arbitrariness in determining $f_{i}^{\alpha }$. The contradiction is due to the assumption that the tensor transformation property (2.14) holds, and since the expression in (2.14) equals $f_{i}^{\alpha }$, it should not transform as a tensor (note also that $f_{i}^{\alpha }\neq f_{\alpha }^{i}$), at least for the investigated case of the projective transformation (2.5). Also, the contradiction is that (2.15) is fulfilled for any $t_{i}^{\alpha }:=$ $\frac{\partial x^{\alpha }}{\partial X^{i}}$ and $t_{\beta }^{j}:=$ $\frac{\partial X^{j}}{\partial x^{\beta }}$, or in other words - it should hold in any basis, but at the same time we found a basis, in which (2.16) holds and not (2.15). For the case of standard relativistic hydrodynamics, although $f_{i}^{k}=\delta _{i}^{k}-u_{i}u^{k}\neq \delta _{i}^{k}$, such a problem of course will not appear because of the unit vector normalization $u^{i}u_{i}=1 $ for every vector field, which is imposed apriori. Now it is easy to understand why and in what cases the distinction between covariant and contravariant metric components will lead to an inevitable introduction of two different connections $\Gamma _{ij}^{k}$* and* $P_{ij}^{k}$. For the purpose, let us prove the following statement: If $e_{1},e_{2},...,e_{n}$ is a basis of covariant vector fields and $f_{i}^{\alpha }$ are certain functions or constants, then a basis of contravariant basic fields $\widetilde{e}^{\alpha _{1}},\widetilde{e}^{\alpha _{2}},...,\widetilde{e}^{\alpha _{n}}$ can be found so that for each $i$ and $\alpha _{j}$ one has $e_{i}\widetilde{e}^{\alpha _{j}}=f_{i}^{\alpha }$. This statement in fact is a generalization of the well-known theorem from differential geometry that if a basis of (covariant) vector fields is given then a dual basis of (contravariant) vector fields can be found, so that the contravariant vector fields are the inverse ones to the covariant ones, i.e. $e_{i}\widetilde{e}^{\alpha _{j}}=\delta _{i}^{\alpha }$. The proof is very simple, but essentially based on the relation (2.13). If the covariant basic vector fields are given, then the contravariant connection components $\Gamma _{ij}^{k}$ will be known too. Since $f_{j,k}^{i}$ are derivatives of a function, one may take the expression (2.13) $f_{j,k}^{i}=\Gamma _{jk}^{l}$ $f_{l}^{i}+P_{lk}^{i}f_{j}^{l}$ , which for the moment shall be treated as a system of $n.[\frac{n(n+1)}{2}]$ linear algebraic equations with respect to the (unknown) connection components $P_{lk}^{i}$. A solution of this system can be found for the connection components $P_{lk}^{i}$. Then the condition for the parallel transport of the contravariant basic vector fields $\nabla _{e_{\beta }}\widetilde{e}^{\alpha }=P_{\gamma \beta }^{\alpha }$ $\widetilde{e}^{\gamma }$ can be written as $\partial _{\beta }\widetilde{e}^{\alpha }=P_{\gamma \beta }^{\alpha }$ $\widetilde{e}^{\gamma } $ and considered as a system of $n$* ordinary differential equations with respect to the components* $\widetilde{e}^{\alpha }$. From this system, the unique solution for $\widetilde{e}^{\alpha _{1}},\widetilde{e}^{\alpha _{2}},...,\widetilde{e}^{\alpha _{n}}$ can be found up to integration constants, obtained after the integration of the differential equations. After proving this proposition, the difference between standard relativistic hydrodynamics and “modified” relativistic hydrodynamics with a variable length can be easily understood. In the first case, the right-hand side in $p_{ij}p^{jk}=\delta _{i}^{k}-u_{i}u^{k}=f_{i}^{k}\neq \delta _{i}^{k}$ transforms as a tensor, which is ensured also by normalization property $u_{i}u^{i}=1$.Therefore (2.13) and the proposition will not hold, so the contravariant basic vector fields are determined in the standard way $e_{i}e^{j}=\delta _{i}^{j}$ and more importantly, they cannot be determined in another way, in spite of the fact that again $f_{i}^{k}\neq \delta _{i}^{k}$. In the second case, the situation is just the opposite - the right-hand side of $p_{ij}p^{jk}=\delta _{i}^{k}-\frac{1}{e}u_{i}u^{k}=f_{i}^{k}\neq \delta _{i}^{k}$ transforms not as a tensor because of the “normalization” factor $\frac{1}{e}$, the proposition holds and thus the basic vector fields are determined as $e_{i}\widetilde{e}^{j}=f_{i}^{j}$. Therefore, the treatment of relativistic hydrodynamics with “variable length” should be within the GTCCMC. In the present case, the introduced new connection (2.4) should not be identified with the connection $P_{\alpha \beta }^{\gamma }$, since the connection $\widetilde{\Gamma }_{kl}^{s}\equiv \widetilde{g}^{is}\Gamma _{i;kl}$ is introduced by means of modifying the contravariant tensor and not on the base of any separate parallel transport. Moreover, the connection $\widetilde{\Gamma }_{kl}^{s}$ turns out to be a linear combination of the Christoffell connection components $\Gamma _{\alpha \beta }^{\gamma }$, and the relation between them is not of the type (2.13). In such a way, there will not be a contradiction with the case when the* two connections* $\Gamma _{\alpha \beta }^{\gamma }$* and* $\widetilde{\Gamma }_{kl}^{s}$ are not defined as separate ones, since later on, in deriving the cubic algebraic equation in the general case and for the case $\widetilde{g}^{jk}=dX^{j}dX^{k}$ also, it would be supposed that $\widetilde{g}^{is}$ is a tensor. This would mean (from $\widetilde{g}^{is}g_{im}\equiv f_{m}^{s}(\mathbf{x})$) that $f_{m}^{s}(\mathbf{x})$ will also be a (mixed) tensor quantity, and therefore the covariant differentiation of $e_{\alpha }e^{\beta }\equiv f_{\alpha }^{\beta }(x)$ will not produce any new relation. BASIC ALGEBRAIC EQUATIONS IN GRAVITY THEORY. TENSOR LENGTH SCALE ====================================================================== Now if one applies again the new definition $\widetilde{g}^{ij}\equiv dX^{i}dX^{j}$ of the contravariant tensor with respect to the Ricci tensor, then the following fourth - degree algebraic equation can be obtained $$R_{ik}=dX^{l}\left[ g_{is,l}\frac{\partial (dX^{s})}{\partial x^{k}}-\frac{1}{2}pg_{ik,l}+\frac{1}{2}g_{il,s}\frac{\partial (dX^{s})}{\partial x^{k}}\right] +$$$$+\frac{1}{2}dX^{l}dX^{m}dX^{r}dX^{s}\left[ g_{m[k,t}g_{l]r,i}+g_{i[l,t}g_{mr,k]}+2g_{t[k,i}g_{mr,l]}\right] \text{ \ \ \ \ \ ,} \tag{3.1}$$where $p$ is the scalar quantity $$p\equiv div(dX)\equiv \frac{\partial (dX^{l})}{\partial x^{l}}\text{,} \tag{3.2}$$ which measures the divergency of the vector field $dX$. The algebraic variety of the equation consists of the differentials $dX^{i\text{ }}$ and their derivatives $\frac{\partial (dX^{s})}{\partial x^{k}}$. In the same spirit, one can investigate the problem whether the gravitational Lagrangian in terms of the new contravariant tensor can be equal to the standard representation of the gravitational Lagrangian. This standard (first) representation of the gravitational Lagrangian is based on the standard Christoffell connection $\Gamma _{ij}^{k}$ (given by formulae (2.1)), the Ricci tensor $R_{ik}$ (formulae (2.2)) and the other contravariant tensor $\widetilde{g}^{ij}=dX^{i}dX^{j}$ $$L_{1}=-\sqrt{-g}\widetilde{g}^{ik}R_{ik}=-\sqrt{-g}dX^{i}dX^{k}R_{ik}\text{ \ .} \tag{3.3}$$In the second representation the* Christoffell connection $\widetilde{\Gamma }_{ij}^{k}$ and the Ricci tensor $\widetilde{R}_{ik}$ are “tilda” quantities, meaning that the “tilda” Christoffell connection is determined by formulae (2.4) with the new contravariant tensor $\widetilde{g}^{ij}=dX^{i}dX^{j}$ and the “tilda” Ricci tensor $\widetilde{R}_{ik}$ - by formulae (2.2), but with the “tilda” connection $\widetilde{\Gamma }_{ij}^{k}$ instead of the usual Christoffell connection $\Gamma _{ij}^{k}$. Thus the expression for the second representation* of the gravitational Lagrangian acquires the form $$L_{2}=-\sqrt{-g}\widetilde{g}^{il}\widetilde{R}_{il}=-\sqrt{-g}dX^{i}dX^{l}\{p\Gamma _{il}^{r}g_{kr}dX^{k}-\Gamma _{ik}^{r}g_{lr}d^{2}X^{k}-\Gamma _{l(i}^{r}g_{k)r}d^{2}X^{k}\}\text{ .} \tag{3.4}$$The condition for the equivalence of the two representations (i.e. $L_{1}=L_{2}$) gives a cubic algebraic equation with respect to the algebraic variety of the first differential $dX^{i}$ and the second one $d^{2}X^{i}$ \[10\] $$dX^{i}dX^{l}\left( p\Gamma _{il}^{r}g_{kr}dX^{k}-\Gamma _{ik}^{r}g_{lr}d^{2}X^{k}-\Gamma _{l(i}^{r}g_{k)r}d^{2}X^{k}\right) -dX^{i}dX^{l}R_{il}=0\text{ \ \ \ \ .} \tag{3.5}$$Note the following essential peculiarity of the second representation (3.4) - due to the choice of the “modified” contravariant tensor, the second quadratic term with the “tilda” connection in the expression for $\widetilde{R}_{ij}$ is equal to zero$$-\sqrt{-g}dX^{i}dX^{k}(\widetilde{\Gamma }_{ik}^{l}\widetilde{\Gamma }_{lm}^{m}-\widetilde{\Gamma }_{il}^{m}\widetilde{\Gamma }_{km}^{l})\text{ }=-\frac{1}{2}\sqrt{-g}dX^{i}dX^{k}dX^{l}dX^{m}(-dg_{lm}dX^{s}g_{ks,i}-dg_{ik}dX^{r}g_{mr,l}+$$$$+dg_{il}dX^{r}g_{mr,k}+dg_{km}dX^{s}g_{ls,i})-$$$$-\sqrt{-g}dX^{i}dX^{k}dX^{l}dX^{m}dX^{s}dX^{r}(g_{ks,i}g_{mr,l}-g_{ls,i}g_{mr,k})=0\text{.} \tag{3.6}$$The $\mathit{first\ differential}$ $dg_{ij}$ in (3.6) is represented as $dg_{ij}\equiv \frac{\partial g_{ij}}{\partial x^{s}}dX^{s}\equiv \Gamma _{s(i}^{r}g_{j)r}dX^{s}$. Following the same approach, in \[10\] the Einstein’s equations in vacuum for the general case were derived under the assumption that the contravariant metric tensor components are the “tilda” ones: $$0=\widetilde{R}_{ij}-\frac{1}{2}g_{ij}\widetilde{R}=\widetilde{R}_{ij}-\frac{1}{2}g_{ij}dX^{m}dX^{n}\widetilde{R}_{mn}=$$$$=-\frac{1}{2}pg_{ij}\Gamma _{mn}^{r}g_{kr}dX^{k}dX^{m}dX^{n}+\frac{1}{2}g_{ij}(\Gamma _{km}^{r}g_{nr}+\Gamma _{n(m}^{r}g_{k)r})d^{2}X^{k}dX^{m}dX^{n}+$$$$+p\Gamma _{ij}^{r}g_{kr}dX^{k}-(\Gamma _{ik}^{r}g_{jr}+\Gamma _{j(i}^{r}g_{k)r})d^{2}X^{k}\text{ \ \ \ .} \tag{3.7 }$$ This equation represents again a system of cubic equations. In addition, if the differentials $dX^{i}$ and $d^{2}X^{i}$ are known, but not the covariant tensor $g_{ij}$, the same equation can be considered also as a cubic algebraic equation with respect to the algebraic variety of the metric tensor components $g_{ij}$ and their first derivatives $g_{ij,k}$. It might be thought that the definite choice of the contravariant tensor is a serious restriction, in view of the fact that the second derivatives of the covariant tensor components $g_{ij,kl}$ are not present in the equation. This is indeed so, because the algebraic structure of the equation is simpler to deal with in comparison with the general case, and so it is easier to implement the algorithm for parametrization, developed in \[10\]. But there is one argument in favour of this choice (although the case for an arbitrary contravariant tensor is also more important) - since the metric can be expressed as $ds^{2}=l(x)=g_{ij}dX^{i}dX^{j}$ (consequently $dX^{i}dX^{j}=l(x)g^{ij}$), the obtained cubic algebraic equations (3.5) and (3.7) can be considered with regard also to the length function $l(x)$. Since for Einsteinian gravity $g_{ij}g^{jk}=\delta _{i}^{k}$ (i.e. $g^{jk}=\widetilde{g}^{jk}=dX^{j}dX^{k}$), then for this case the length function is “postulated” to be $l=1$. Note that this choice of the contravariant tensor $\widetilde{g}^{jk}$ in the form of a factorized product is a partial (and not a general) choice, but further it shall be shown that the cubic equation for reparametrization invariance of the gravitational Lagrangian can be written also for a generally chosen tensor $\widetilde{g}^{jk}$. Then from $g_{ij}\widetilde{g}^{jk}=l_{i}^{k}$ and $l_{i}^{k}=l\delta _{i}^{k}$, the length function is again recovered. The important point here is that the length function can also be obtained as a solution of the cubic equation, and thus in more general theories of gravity solutions with $l\neq 1$ may exit. In fact, for a general contravariant tensor $\widetilde{g}^{ij}=l_{k}^{i}g^{kj}\neq dX^{i}dX^{j}$, it would be natural to propose to call $l_{k}^{i}$ a “tensor length scale”, and the previously defined length function $l(x)$ is a partial case of the tensor length scale for $l_{j}^{i}=l\delta _{j}^{i}$. The physical meaning of the notion of tensor length scale is simple - in the different directions (i.e. for different $i$ and $j$) the length scale is different. In particular, some motivation for this comes from Witten’s paper \[46\], where in discussing some aspects of weakly coupled heterotic string theory (when there is just one string couplings ) and the obtained too large bound on the Newton’s constant it was remarked that the problem might be ameliorated by considering an anisotropic Calabi - Yau with a scale $\sqrt{\alpha ^{^{\prime }}}$* in* $d$* directions and* $\frac{1}{M_{GUT}}$* in* $(6-d)$directions. For example, this can be realized if one takes $$l_{i}^{k}=g_{ij}dX^{j}dX^{k}=l_{1}\delta _{i}^{k}\text{ \ for \ }i,j,k=1,....,d\text{ \ \ \ \ ,} \tag{3.8}$$$$l_{a}^{b}=g_{ac}dX^{c}dX^{b}=l_{2}\delta _{a}^{b}\text{ \ for \ }a,b,c=d+1,....,6\text{\ \ \ \ .} \tag{3.9}$$Note also the justification for the name “tensor length scale” - if $l_{k}^{i}$ is a tensor quantity, so will be the “modified” contravariant tensor $\widetilde{g}^{ij}=l_{k}^{i}g^{kj}$, and consequently in accord with section 2 there will be no need for the introduction of a new covariant connection $P_{ij}^{k}$. And this is indeed the case, because the relation between the two connections $\Gamma _{ij}^{k}$ and $\widetilde{\Gamma }_{ij}^{k}$ is given by formulae (2.4) $\widetilde{\Gamma }_{kl}^{s}:=\widetilde{g}^{is}g_{im}\Gamma _{kl}^{m}$. In other words, since these two connections are not considered to be “separately introduced” and so they do not depend on one another by means of the equality (2.13), this particular investigated case does not fall within the classification of spaces with covariant and contravariant metrics and connections (Table I in a previous paper \[47\]). This is an important “terminological” clarification, since it turns out that it is possible to have a theory with (separate) covariant and contravariant metrics, but not (with separate) connections as well. And such a theory is fully consistent from a mathematical point of view, as demonstrated above. However, at this point it is important to clarify what is meant by “a theory with (separate) covariant and contravariant metrics” - it should be understood only with respect to the metrics $g_{\mu \nu }$ and * $\widetilde{g}^{is}$. But if we take the contravariant metric $\widetilde{g}^{is}$ (and ignore for the moment the metric $g_{\mu \nu }$), then from the equality $\overline{g}_{ij}\widetilde{g}^{jk}=\delta _{i}^{k}$ one can determine an inverse to the contravariant metric $\widetilde{g}^{is}$ new covariant metric* $\overline{g}_{ij}$, and consequently, the following new contravariant connection $\overline{\Gamma }_{kl}^{s}$ can also be determined $$\overline{\Gamma }_{kl}^{s}:=\widetilde{g}^{is}\overline{\Gamma }_{i;kl}=\widetilde{g}^{is}\overline{g}_{im}\overline{\Gamma }_{kl}^{m}=\frac{1}{2}\widetilde{g}^{is}(\overline{g}_{ik,l}+\overline{g}_{il,k}-\overline{g}_{kl,i})\text{.} \tag{3.10}$$Evidently, with respect to the metric $\overline{g}_{ij}$ (and its inverse contravariant one $\widetilde{g}^{jk}$),we have the usual gravitational theory with the contravariant $\overline{\Gamma }_{\alpha \beta }^{\gamma }$ and covariant $\overline{P}_{\alpha \beta }^{\gamma }$ connections $$\nabla _{e_{\beta }}e_{\alpha }=\overline{\Gamma }_{\alpha \beta }^{\gamma }e_{\gamma }\text{ ; \ }\nabla _{e_{\beta }}e^{\alpha }=\overline{P}_{\gamma \beta }^{\alpha }e^{\gamma }\text{ ; \ }\overline{P}_{\gamma \beta }^{\alpha }=-\overline{\Gamma }_{\alpha \beta }^{\gamma }\text{ \ \ .} \tag{3.11}$$However, although with respect to the metrics $g_{\mu \nu }$ and * $\widetilde{g}^{is}$ and the connections $\Gamma _{ij}^{k}$ and $\widetilde{\Gamma }_{kl}^{s}:=\widetilde{g}^{is}g_{im}\Gamma _{kl}^{m}$ (3.4) the theory is with covariant and contravariant metrics (only),* connections $\overline{\Gamma }_{ij}^{k}$ and $\widetilde{\Gamma }_{kl}^{s}$ can be determined (by means of the additional metric $\overline{g}_{\mu \nu } $) in the following way $$\nabla _{e_{\beta }}e_{\alpha }=\overline{\Gamma }_{\alpha \beta }^{\gamma }e_{\gamma }\text{ ; }\ \ \ \ \ \ \ \ \text{\ }\nabla _{e_{\beta }}e^{\alpha }=\widetilde{\Gamma }_{\gamma \beta }^{\alpha }e^{\gamma }\text{ , \ \ \ } \tag{3.12}$$so that with respect to these connections the theory can be considered a GTCCMC. This also means that Table I in \[47\] correctly does not account for theories with different covariant and contravariant metrics only, because the different GTCCMC are in principle with different covariant and contravariant metrics and with different connections. The purpose of the present paper further will be: how can one extend the proposed in \[10\] approach for the definitely determined contravariant metric components to the case of a generally defined contravariant tensor $\widetilde{g}^{ij}\neq dX^{i}dX^{j}$? INTERSECTING ALGEBRAIC VARIETIES AND STANDARD (EINSTEINIAN) GRAVITY THEORY =========================================================================== A more general theory with the definition of the contravariant tensor as $\widetilde{g}^{ij}\equiv dX^{i}dX^{j}$ should contain in itself the standard gravitational theory with $g_{ij}g^{jk}=\delta _{i}^{k}$. From a mathematical point of view, this should be performed by considering the intersection \[19\] of the cubic algebraic equations (3.7) with the system of $n^{2}$ quadratic algebraic equations for the algebraic variety of the $n$ variables $$g_{ij}dX^{j}dX^{k}=\delta _{i}^{k}\text{ \ \ .} \tag{4.1}$$In its general form $g_{ij}\widetilde{g}^{jk}=\delta _{i}^{k}$ with an arbitrary contravariant tensor $\widetilde{g}^{jk}$, this system can also be considered together with the Einstein’s “algebraic” system of equations, which in the next section shall be derived for a generally defined contravariant tensor. From an algebro - geometric point of view, this is the problem about the intersection of the Einstein’s algebraic equations with the system of $n^{2}$ (linear) hypersurfaces for the $\left[ \left( \begin{array}{c} n \\ 2\end{array}\right) +n\right] $ contravariant variables, if the covariant tensor components are given. Since the derived Einstein’s algebraic equations are again cubic ones with respect to the contravatiant metric components, this is an analogue to the well - known problem in algebraic geometry about the intersection of a (two-dimensional) cubic curve with a straight line. However, in the present case the straight line and the cubic curve are multi - dimensional ones, which is a substantial difference from the standard case. The standardly known solutions of the Einstein’s equations can be obtained as an intersection variety of the Einstein’s algebraic equations with the system $g_{ij}\widetilde{g}^{jk}=\delta _{i}^{k}$ . However, the strict mathematical proof that such an intersection will give the known solutions is still lacking. It might seem that the system of equations (4.1) does not have solutions with respect to $g_{ij\text{ }}$(and thus no solutions of the Einstein’s equations can be found for the standard case), since the determinant $det\parallel dX^{i}dX^{j}\parallel _{i,j=1,..n}=0$ equals to zero! In another paper it will be proved that such a matrix operator system of equations \[20\] $Y_{ij}g^{jk}=\delta _{i}^{k}$ with unknown variables $Y_{ij}\equiv g_{ij}$ (which is not a system of linear algebraic equations, but instead a system of matrix equations) can be transformed to a system of linear algebraic equations $\widetilde{A}_{ij}\widetilde{Y}^{j}$ $=T_{i}$ ($T_{i}$ is a vector - column). This system always has a solution at least for some of the variables - the others may be determined arbitrarily. Therefore, solutions will exist and will be well-determined even in the case of a zero determinant. ALGEBRAIC EQUATIONS FOR A GENERAL CONTRAVARIANT METRIC TENSOR ===================================================================== Let us write down the algebraic equations for all admissable parametrizations of the gravitational Lagrangian for the generally defined contravariant tensor $\widetilde{g}^{ij}$, following the same prescription as in section 3, where the equality of the two representations of the gravitational Lagrangian has been supposed: $$\widetilde{g}^{i[k}\widetilde{g}_{,l}^{l]s}\Gamma _{ik}^{r}g_{rs}+\widetilde{g}^{i[k}\widetilde{g}^{l]s}\left( \Gamma _{ik}^{r}g_{rs}\right) _{,l}+$$$$+\widetilde{g}^{ik}\widetilde{g}^{ls}\widetilde{g}^{mr}g_{pr}g_{qs}\left( \Gamma _{ik}^{q}\Gamma _{lm}^{p}-\Gamma _{il}^{p}\Gamma _{km}^{q}\right) -R=0\text{ \ \ \ \ .} \tag{5.1}$$This equation is again a cubic algebraic equation with respect to the algebraic variety of the variables $\widetilde{g}^{ij}$ and $\widetilde{g}_{,k}^{ij}$, and the number of variables in the present case is much greater than in the previous case for the contravariant tensor $\widetilde{g}^{ij}\equiv dX^{i}dX^{j}$ . At the same time, this equation is a fourth - degree algebraic equation with respect to the covariant metric tensor $g_{ij}$ and its first and second partial derivatives. With respect to the algebraic variety of all the variables $\widetilde{g}^{ij}$, $\widetilde{g}_{,k}^{ij}$, $g_{ij}$, $g_{ij,k}$, $g_{ij,kl}$, the above algebraic equation is of seventh order and with coefficient functions, due to the presence of the terms with the affine connection $\Gamma _{ik}^{q}$ and its derivatives, which contain the contravariant tensor $g^{ij}$ and $g_{,k}^{ij}$. If the connection is assumed to be the tilda connection $\widetilde{\Gamma }_{kl}^{s}\equiv dX^{i}dX^{s}\Gamma _{i;kl}$, then the same equation can be regarded as a sixth - degree equation with respect to the algebraic variety of $dX^{i}$ and its derivatives. Similarly, the Einstein’s equations can be written as a system of third - degree algebraic equations with respect to the (generally chosen) contravariant variables and their derivatives $$0=\widetilde{R}_{ij}-\frac{1}{2}g_{ij}\widetilde{R}=$$$$=\widetilde{g}^{lr}(\Gamma _{r;i[j}),_{l]}+\widetilde{g}_{,[l}^{lr}\Gamma _{r;ij]}+\widetilde{g}^{lr}\widetilde{g}^{ms}(\Gamma _{r;ij}\Gamma _{s;lm}-\Gamma _{s;il}\Gamma _{r;km})-$$$$-\frac{1}{2}g_{ij}\widetilde{g}^{m[k}\widetilde{g}_{,l}^{l]s}\Gamma _{mk}^{r}g_{rs}-\frac{1}{2}g_{ij}\widetilde{g}^{m[k}\widetilde{g}^{l]s}\left( \Gamma _{mk}^{r}g_{rs}\right) _{,l}-$$$$-\frac{1}{2}g_{ij}\widetilde{g}^{nk}\widetilde{g}^{ls}\widetilde{g}^{mr}g_{pr}g_{qs}\left( \Gamma _{nk}^{q}\Gamma _{lm}^{p}-\Gamma _{nl}^{p}\Gamma _{km}^{q}\right) \text{ \ .} \tag{5.2}$$Interestingly, the same system of equations can be considered as a system of fifth - degree equations with respect to the covariant variables (which is the difference from the previous case). The mathematical treatment of fifth - degree equations is known since the time of Felix Klein’s famous monograph \[25\], published in 1884. A way for resolution of such equations on the base of earlier developed approaches by means of reducing the fifth - degree equations to the so called modular equation has been presented in the more recent monograph of Prasolov and Solov’yev \[9\]. Some other methods for solution of third-, fifth- and higher- order algebraic equations have been given in \[26, 27\]. A complete description of elliptic, theta and modular functions has been given in the old monographs \[28, 29\]. Also, solutions of $n-$ th degree algebraic equations in theta - constants \[30\] and in special functions \[31\] are interesting in view of the not yet proven hypothesis in the paper by Kraniotis and Whitehouse \[8\] that all nonlinear solutions of general relativity are expresed in terms of theta - functions, associated with Riemann - surfaces. The basic knowledge about the parametrization of a cubic algebraic equation with the Weierstrass function and its derivative, which shall be extensively used in the subsequent parts of this paper, is given in almost all basic textbooks on elliptic functions \[9, 11, 32, 33\] and many others. However, the most complete, detailed and exhaustive knowledge about elliptic functions and automorphic forms is contained in the two two - volume books \[34, 35\] of Felix Klein and Robert Fricke, written more than 100 years ago. More specific and advanced topics on elliptic curves from a mathematical point of view such as the group of rational points, cubic curves over finite fields, families of elliptic curves and torsion points and etc. are contained in the monographs \[36, 37\]. A very understandable exposition of the classical topics on cubic algebraic curves and at the same time the most contemporary issues such as the Mordell’s and Dirichlet’s theorems and $L$ functions, modular forms and theories of Eichler - Shimura are given in the book of Knapp \[38\], which can be used for first acquintance in these topics. A consistent, modern and full exposition of elliptic curves in the language of modern mathematics is given in the (two consequent)monographs of Silverman \[39, 40\]. A classical and very understandable exposition of the relation of elliptic curves with modular forms is given in \[41\], also in \[42\]. From a modern standpoint the relation of elliptic curves with number theory and modular forms is given in the review articles of Cohen and Don Zagier in \[43\], also introductory knowledge on hyperelliptic integrals, compact Riemann surfaces and Abelian varieties are presented in the review article by Bost also in \[43\]. Two other important problems can be pointed out with reference to algebraic equations in gravity theory: 1\. One can find solutions of the system of Einstein’s equations not as solutions of a system of nonlinear differential equations, but as elements of an algebraic variety, satisfying the Einstein’s algebraic equations. The important new moment is that this gives an opportunity to find solutions of the Einstein’s equations both for the components of the covariant metric tensor $g_{ij}$ and for the contravariant ones $\widetilde{g}^{jk}$. This means that solutions may exist for which $g_{ij}\widetilde{g}^{jk}\neq \delta _{i}^{k}$. In other words, a classification of the solutions of the Einstein’s equations can be performed in an entirely new and nontrivial manner - under a given contravariant tensor, the covariant tensor and its derivatives have to be found from the algebraic equation, or under a given covariant tensor, the contravariant tensor and its derivatives can be found. 2\. The condition for the zero - covariant derivative of the covariant metric tensor $\nabla _{k}g_{ij}=0$ and of the contravariant metric tensor $\nabla _{k}\widetilde{g}^{ij}=0$ can be written in the form of the following cubic algebraic equations with respect to the variables $g_{ij}$, $g_{ij,k}$ and $\widetilde{g}^{ls}$ : $$\nabla _{k}g_{ij}\equiv g_{ij,k}-\widetilde{\Gamma }_{k(i}^{l}g_{j)l}=g_{ij,k}-\widetilde{g}^{ls}\Gamma _{s;k(i}g_{j)l}=0 \tag{5.3}$$and $$0=\nabla _{k}\widetilde{g}^{ij}=\widetilde{g}_{,k}^{ij}+\widetilde{g}^{r(i}\widetilde{g}^{j)s}\Gamma _{r;sk}\text{ \ \ \ \ .} \tag{5.4}$$The first equation (5.3) is linear with respect to $\widetilde{g}^{ls}$ and quadratic with respect to $g_{ij}$, $g_{ij,k}$, while the second equation (5.3) is linear with respect to $g_{ij}$, $g_{ij,k}$ and quadratic with respect to $\widetilde{g}^{ls}$. Since the treatment of the above cubic algebraic equations is based on singling out one variable, let us rewrite equation (5.1) for the effective parametrization of the gravitational action for the case of diagonal metrics $g_{\beta \beta }$ and $\widetilde{g}^{\alpha \alpha }$, singling out the variable $\widetilde{g}^{44}$: $$A(\widetilde{g}^{44})^{3}+B_{\alpha }(\widetilde{g}^{44})^{2}\widetilde{g}^{\alpha \alpha }+C_{\alpha \alpha }\widetilde{g}^{44}\widetilde{g}^{\alpha \alpha }+(\Gamma _{44}^{\alpha }g_{\alpha \alpha })\widetilde{g}^{44}\widetilde{g}_{,\alpha }^{\alpha \alpha }+$$$$+D_{\alpha \gamma }\widetilde{g}^{44}\widetilde{g}^{\alpha \alpha }\widetilde{g}^{\gamma \gamma }+F_{\alpha \gamma }=0\text{ \ \ \ \ \ ,} \tag{5.5}$$where the coefficient functions $A$, $B_{\alpha }$, $C_{\alpha \alpha }$, $D_{\alpha \gamma }$ and the free term $F_{\alpha \gamma }$ denotes an expression, depending on the covariant metric tensor and the affine connection. In (5.5) the Greek indices run the values $\alpha ,\beta ,\gamma =1,2,3$, while all the other indices run from $1$ to $4$. The representation (5.5) of the cubic equation is the starting point for the parametrization with the Weierstrass function, which will be performed elsewhere, following the algorithm in the paper \[10\]. In the second part of this paper, this would be performed for the case a multivariable cubic algebraic equation (although again within the framework of the factorizing approximation $\widetilde{g}^{ij}\equiv dX^{i}dX^{j}$) and this is entirely different from the standardly known case in algebraic geometry of parametrization of a two-dimensional cubic algebraic equation in its parametrizable form. DISCUSSION ========== In this paper we continued the investigation of cubic algebraic equations in gravity theory, which has been initiated in a previous paper \[10\]. Unlike in the paper \[10\], where the treatment of cubic algebraic equations has been restricted only to the choice of the contravariant tensor $\widetilde{g}^{ij}=dX^{i}dX^{j}$, in the present paper it has been demonstrated that under a more general choice of $\widetilde{g}^{ij}$, there is a wide variety of algebraic equations of various order, among which an important role play the cubic equations. Their derivation is based on two important initial assumptions: 1\. The covariant and contravariant metric components are treated independently, which is a natural approach in the framework of affine geometry \[15 - 18\]. 2\. Under the above assumption, the gravitational Lagrangian (or Ricci tensor) should remain the same as in the standard gravitational theory with inverse contravariant metric tensor components. It will be proved in Appendix A that the new connection $\widetilde{\Gamma }_{ij}^{k}=\frac{1}{2}dX^{k}dX^{s}(g_{js,i}+g_{is,j}-g_{ij,s})$ has again an affine connection transformation property, provided that a complicated system of nonlinear differential equations are satisfied. This system is expected to have a broad class of solutions. The proposed approach allows to treat the Einstein’s equations as algebraic equations, and thus to search for separate classes of solutions for the covariant and contravariant metric tensor components. It can be supposed also that the existence of such separate classes of solutions might have some interesting and unexplored until now physical consequences. Some of them will be demonstrated in reference to theories with extra dimensions, but no doubt the physical applications are much more numerous. Also, it has been shown that the “transition” to the standard Einsteinian theory of gravity can be performed by investigating the intersection with the corresponding algebraic equations. Acknowledgments {#acknowledgments .unnumbered} --------------- The author is grateful to Dr. I. Pestov (BLTP, JINR, Russia), Dr. D. M. Mladenov (Theor.Physics Departm., Fac. of Physics, Sofia Univ., Bulgaria), and especially to Prof. V. V. Nesterenko (BLTP, JINR, Russia), Dr. O. Santillan (IAFE, Buenos Aires, Argentina) and Prof. Sawa Manoff (INRNE, BAS, Bulgaria) for valuable comments, discussions and critical remarks. This paper is written in memory of Prof. S. S. Manoff (1943 - 27.05.2005) - a specialist in classical gravitational theory. The author is grateful also to Dr. G. V. Kraniotis (Max Planck Inst., Munich, Germany) for sending to me his published paper (ref. \[8\]). APPENDIX A: SOME PROPERTIES OF THE NEWLY INTRODUCED CONNECTION $\widetilde{\Gamma }_{ij}^{k}=\frac{1}{2}dX^{k}dX^{l}(g_{jl,i}+g_{il,j}-g_{ij,l})$ ======================================================================================================================================================== A1: A PROOF OF THE AFFINE TRANSFORMATION LAW FOR THE CONNECTION $\widetilde{\Gamma }_{ij}^{k} $ {#a1-a-proof-of-the-affine-transformation-law-for-the-connection-widetildegamma-_ijk .unnumbered} --------------------------------------------------------------------------------------------------------- ### A1.1 THE NECESSARY CONDITION {#a1.1-thenecessary-condition .unnumbered} Next we proceed with the proof that the defined ( in the preceeding paper \[10\] also) connection $$\widetilde{\Gamma }_{ij}^{k}=\frac{1}{2}dX^{k}dX^{l}(g_{jl,i}+g_{il,j}-g_{ij,l})=dX^{k}dX^{r}g_{sr}(X)\Gamma _{ij}^{s}(X) \tag{A1}$$has the transformation property of an affine connection under the coordinate transformations $X^{i}=X^{i}(x^{1},x^{2},...,x^{n})$. From the defining equation (A1) for $\widetilde{\Gamma }_{ij}^{k}$, the tensor transformation property for $g_{ij}^{^{\prime }}(\mathbf{X})$ $$g_{ij}^{^{\prime }}(\mathbf{X})=\frac{\partial x^{k}}{\partial X^{i}}\frac{\partial x^{l}}{\partial X^{j}}g_{kl}(\mathbf{x})\text{ \ \ ,} \tag{A2}$$the affine transformation law for the “usual” connection $\Gamma _{ij}^{k}$ $$\Gamma _{ij}^{k^{\prime }}(\mathbf{X})=\Gamma _{np}^{m}(\mathbf{x})\frac{\partial X^{k}}{\partial x^{m}}\frac{\partial x^{n}}{\partial X^{i}}\frac{\partial x^{p}}{\partial X^{j}}+\frac{\partial ^{2}x^{m}}{\partial X^{i}\partial X^{j}}\frac{\partial X^{k}}{\partial x^{m}} \tag{A3}$$and from the expressions for the differentials $dX^{k}$ and $dX^{r}$ we may write down $$\widetilde{\Gamma }_{ij}^{k^{\prime }}(\mathbf{X})==dX^{k}(\mathbf{X})dX^{r}(\mathbf{X})g_{sr}^{^{\prime }}(\mathbf{X})\Gamma _{ij}^{s^{\prime }}(\mathbf{X})= \tag{A4}$$$$=\Gamma _{np}^{m}(\mathbf{x})\frac{\partial X^{k}}{\partial x^{\alpha }}\frac{\partial x^{n}}{\partial X^{i}}\frac{\partial x^{p}}{\partial X^{j}}g_{m\beta }(\mathbf{x})dx^{\alpha }dx^{\beta }+\frac{\partial ^{2}x^{m}}{\partial X^{i}\partial X^{j}}\frac{\partial X^{k}}{\partial x^{\alpha }}g_{m\beta }(\mathbf{x})dx^{\alpha }dx^{\beta }\text{ .} \tag{A5}$$ On the other hand, if $\widetilde{\Gamma }_{ij}^{k}(\mathbf{X})$ is an affine connection, then it should satisfy the affine connection transformation law (A3) $$\widetilde{\Gamma }_{ij}^{k^{\prime }}(\mathbf{X})=\widetilde{\Gamma }_{np}^{m}(\mathbf{x})\frac{\partial X^{k}}{\partial x^{m}}\frac{\partial x^{n}}{\partial X^{i}}\frac{\partial x^{p}}{\partial X^{j}}+\frac{\partial ^{2}x^{m}}{\partial X^{i}\partial X^{j}}\frac{\partial X^{k}}{\partial x^{m}}\text{ \ \ \ .} \tag{A6}$$Making use of the defining equation (A1) (but in terms of the initial coordinates $x^{1},x^{2},....,x^{n}$), the above expression can be written also as $$\widetilde{\Gamma }_{ij}^{k^{\prime }}(\mathbf{X})=\Gamma _{np}^{m}(\mathbf{x})\frac{\partial X^{k}}{\partial x^{\alpha }}\frac{\partial x^{n}}{\partial X^{i}}\frac{\partial x^{p}}{\partial X^{j}}g_{m\beta }(\mathbf{x})dx^{\alpha }dx^{\beta }+\frac{\partial ^{2}x^{\alpha }}{\partial X^{i}\partial X^{j}}\frac{\partial X^{k}}{\partial x^{\alpha }}\text{ \ \ \ .} \tag{A7}$$Clearly, if $\widetilde{\Gamma }_{ij}^{k^{\prime }}(\mathbf{X})$ is an affine connection, from the R. H. S. of (A5) and (A7) it would follow that the following relation has to be satisfied $$dx^{\alpha }dx^{\beta }g_{m\beta }(\mathbf{x})\frac{\partial ^{2}x^{m}}{\partial X^{i}\partial X^{j}}\frac{\partial X^{k}}{\partial x^{\alpha }}-\frac{\partial ^{2}x^{\alpha }}{\partial X^{i}\partial X^{j}}\frac{\partial X^{k}}{\partial x^{\alpha }}=0\text{ \ \ ,} \tag{A8}$$which in fact represents the necesasy condition for the definition of the connection $\widetilde{\Gamma }_{ij}^{k^{\prime }}(\mathbf{X})$ as an affine connection. It can easily be proved that in case of commuting operators of differentiation $\frac{\partial }{\partial x_{i}}$ and $\frac{\partial }{\partial X_{j}}$ (in the general case, however, they do not commute), equation (A8) is fulfilled. ### A1.2 THE TWO-DIMENSIONAL GENERALIZED CONNECTION $\widetilde{\Gamma }_{ij}^{k^{\prime }}$ IN THE GENERAL CASE {#a1.2-the-two-dimensional-generalized-connection-widetildegamma-_ijkprime-in-the-general-case .unnumbered} Now let us investigate the two-dimensional case, but when the assumption about the commutation of the derivatives is dropped out. Then equation (A8) on the integral curves $dx^{1}=C_{1}$ and $dx^{2}=C_{2}$ can be written as (the relation $\frac{\partial x^{p}}{\partial X^{s}}\frac{\partial X^{s}}{\partial x^{t}}=\delta _{t}^{p}$ is also taken into account) $$\frac{\partial X^{k}}{\partial x^{1}}\{(C_{1}^{2}g_{11}+C_{1}C_{2}g_{12}-1)\overset{(21)}{M^{lij}}-(C_{2}^{2}g_{22}+C_{1}C_{2}g_{12}-1)\overset{(12)}{M^{lij}}+$$$$+(C_{1}^{2}g_{12}+C_{1}C_{2}g_{22})\overset{(22)}{M^{lij}}-(C_{2}^{2}g_{12}+C_{1}C_{2}g_{11})\overset{(11)}{M^{lij}}\}=0\text{ ,} \tag{A9}$$where $\overset{(kk)}{M^{lij}}$ and $\overset{(kn)}{M^{lij}}$ ($k,n=1$ or $2$) are the introduced notations for the expressions $$\overset{(kk)}{M^{lij}}:=\frac{\partial x^{k}}{\partial X^{l}}\frac{\partial ^{2}x^{k}}{\partial X^{i}\partial X^{j}}\text{ \ ; \ \ }\overset{(kn)}{M^{lij}}:=\frac{\partial x^{k}}{\partial X^{l}}\frac{\partial ^{2}x^{n}}{\partial X^{i}\partial X^{j}}\text{\ \ \ .} \tag{A10}$$Now interchanging the functions $x^{1}\leftrightarrow x^{2}$ in (A9) and substracting the derived equation from (A9), one can obtain $$\frac{\partial X^{k}}{\partial x^{1}}\{(C_{1}^{2}g_{11}-C_{2}^{2}g_{22})T^{lij}+$$$$+\left[ g_{12}(C_{1}^{2}+C_{2}^{2})+C_{1}C_{2}(g_{11}+g_{22})\right] (\overset{(22)}{M^{lij}}-\overset{(11)}{M^{lij}})\}=0\text{ \ \ \ ,} \tag{A11}$$where $T^{lij}$ is an introduced notation for $$T^{lij}:=\frac{\partial x^{2}}{\partial X^{l}}\frac{\partial ^{2}x^{1}}{\partial X^{i}\partial X^{j}}-\frac{\partial x^{1}}{\partial X^{l}}\frac{\partial ^{2}x^{2}}{\partial X^{i}\partial X^{j}}\text{ \ .} \tag{A12}$$It can easily be checked that $$T^{[lij]}:=T^{lij}-T^{jil}=\frac{\partial }{\partial X^{i}}\left( \{x^{1},x^{2}\}_{X^{j},X^{l}}\right) \text{ \ \ ,} \tag{A13}$$where $\{x^{1},x^{2}\}_{X^{j},X^{l}}$ is the notation for the s.c. one-dimensional Poisson bracket $$\{x^{1},x^{2}\}_{X^{j},X^{l}}:=\frac{\partial x^{1}}{\partial X^{j}}\frac{\partial x^{2}}{\partial X^{l}}-\frac{\partial x^{1}}{\partial X^{l}}\frac{\partial x^{2}}{\partial X^{j}}\text{ \ \ .} \tag{A14}$$It can be proved that $$\overset{}{\overset{(kk)}{M^{(lij)}}=\frac{\partial }{\partial X^{i}}\left[ \frac{\partial x^{k}}{\partial X^{l}}\frac{\partial x^{k}}{\partial X^{j}}\right] }\text{ ;}\overset{(kk)}{M^{[lij]}=0}\text{ \ .} \tag{A15}$$Since we would like to obtain a relation by combining all the components of eq. (A15) in the two-dimensional case, we can write down equation (A11) with interchanged indices $l\leftrightarrow j$. Substracting the obtained equation from (A11) and taking into account the antisymmetric relations (A13) and (A15), one obtains the simple equation $$\frac{\partial X^{k}}{\partial x^{1}}(C_{1}^{2}g_{11}-C_{2}^{2}g_{22})\frac{\partial }{\partial X^{i}}\{x^{1},x^{2}\}_{X^{j},X^{l}}=0\text{ \ \ \ .} \tag{A16}$$Therefore in the general two-dimensional case of non-commuting operators of differentiation, the modifiedconnection $\widetilde{\Gamma }_{ij}^{k}$ has affine transformation properties in each one of the following cases 1\. If the generalized coordinates $X^{1}$ and $X^{2}$ do not depend on $x^{1} $. 2\. If the Poisson bracket $\{x^{1},x^{2}\}_{X^{j},X^{l}}$ is constant on the integral curves $dx^{1}=C_{1}$ and $dx^{2}=C_{2}.$ 3\. If the following relation is fulfilled for the metric components $g_{11}$ and $g_{22}$ and for the (arbitrary) constants $C_{1}$ and $C_{2}$ $$C_{1}^{2}g_{11}-C_{2}^{2}g_{22}=0\text{ \ \ .} \tag{A17}$$ A2: THE CONNECTION $\widetilde{\Gamma }_{ij}^{k}$ AS AN EQUIAFFINE CONNECTION {#a2-the-connection-widetildegamma-_ijk-as-an-equiaffine-connection .unnumbered} ---------------------------------------------------------------------------------- We have to prove that the connection $\widetilde{\Gamma }_{ij}^{k}$ for $j=k$ can be represented in the form of a gradient of a scalar quantity, i. e. $$\widetilde{\Gamma }_{ij}^{k}=\partial _{i}lne\text{ \ \ .} \tag{A18}$$In the approximation ($dX^{i})_{,k}=0$ one can prove that the connection $\widetilde{\Gamma }_{ij}^{k}$ is indeed an equiaffine one, since one can set up $$lne\equiv \frac{1}{2}dX^{k}dX^{s}g_{ks}\text{ \ \ \ .} \tag{A19}$$ The more complicated and interesting task is to prove that even in the case ($dX^{i})_{,k}\neq 0$, the connection $\widetilde{\Gamma }_{ij}^{k}$ will again be an equiaffine one. For the purpose, note that $$\widetilde{\Gamma }_{ik}^{k}=\frac{1}{2}dX^{s}dX^{k}g_{ks,i}=\frac{1}{2}dX^{k}dX^{s}g_{r(s}\Gamma _{k)i}^{r}=W_{i}\text{ \ } \tag{A20}$$and consequently $\widetilde{\Gamma }_{ik}^{k}$ will be an equiaffine connection if the scalar quantity $e$ can be determined as a solution of the differential equation $$\partial _{i}lne=W_{i}\text{ \ } \tag{A21}$$as $$e=g(X_{1},X_{2},..,X_{i-1},X_{i+1},..,X_{n})e^{\int W_{i}(X_{1},.....,X_{n})dX^{i}}\text{ \ .} \tag{A22}$$Note that the function $g$ depends on all variables $X_{1},X_{2},..,X_{i-1},X_{i+1},..,X_{n}$ with the exception of $X_{i}$, while the function $W_{i}$ depends on all the variables, including also $X_{i}$. Unfortunately, the proof at this stage will be incomplete, since $e$ will depend on the choice of the variable $X_{i}$, which should not happen with a scalar quantity. Consequently, it should be proved that the function $g(X_{1},X_{2},..,X_{i-1},X_{i+1},..,X_{n})$ can be determined in a proper way (so that for every choice of $W_{i})$ the expression (A22) for $e$ would be a scalar quantity. Until we have not proved it, we shall denote the L.H. S. of (A22) with $e^{(i)}$. Let us differentiate both sides of (A22) for $e\equiv e^{(i)}$ and $e\equiv e^{(j)}$ by $X^{j}$ and $X^{i}$ respectively ($i\neq j$). We shall write down only the first equation, since the second one is obtained from the first after a change of the indices $i\Longleftrightarrow $ $j$. $$\frac{\partial e^{(i)}}{\partial X^{j}}=\frac{\partial \ln g(X_{1},X_{2},.,X_{i-1},X_{i+1},.,X_{n})}{\partial X^{j}}e^{(i)}+$$$$+g(X_{1},X_{2},..,X_{i-1},X_{i+1},..,X_{n})e^{\int \frac{\partial W_{i}(X_{1},.....,X_{n})}{\partial X^{j}}dX^{i}}\text{ \ \ \ .} \tag{A23}$$Now differentiate again the derived equation (A23) for $\frac{\partial e^{(i)}}{\partial X^{j}}$ by $X^{i}$ and the other equation for $\frac{\partial e^{(j)}}{\partial X^{i}}$ by $X^{j}$. Taking into account also that $\frac{\partial e^{(i)}}{\partial X^{i}}=e^{(i)}W_{i}$ , applying again (A23) and defining summation over the indices $i$ and $j$, the result will be $$\sum_{i,j}\frac{\partial }{\partial X^{j}}\left( \frac{\partial e^{(i)}}{\partial X^{i}}\right) =grad\left[ \ln g(X_{1},X_{2},.,X_{i-1},X_{i+1},.,X_{n})\right] (\mathbf{e}\text{ }.\mathbf{W})+$$$$+\sum_{i,j}\left[ \frac{\partial W_{i}}{\partial X^{j}}\frac{\partial e^{(i)}}{\partial X^{j}}-\frac{\partial \widetilde{W}_{ij}}{\partial X^{j}}.e^{(i)}\right] +(\mathbf{e}\text{ }.\mathbf{W})\bigtriangleup \ln g(X_{1},X_{2},.,X_{i-1},X_{i+1},.,X_{n})\text{ \ \ \ \ ,} \tag{A24}$$where $(\mathbf{e}$ $.\mathbf{W})$ denotes a scalar product and the following notation has been introduced $$\widetilde{W}_{ij}\equiv W_{i}\frac{\partial \ln g(X_{1},X_{2},.,X_{i-1},X_{i+1},.,X_{n})}{\partial X^{j}}\text{ \ \ \ .} \tag{A25}$$ Again, the second equation will be the same as (A24), but with $i\Leftrightarrow j$. Substracting the two equations and taking into account the formulae for $graddive=\sum_{i,j}\frac{\partial }{\partial X^{i}}\left( \frac{\partial e^{(i)}}{\partial X^{j}}\right) $, one can derive $$\sum_{i,j}\left[ \frac{\partial W_{i}}{\partial X^{j}}\frac{\partial e^{(i)}}{\partial X^{i}}-\frac{\partial W_{j}}{\partial X^{i}}\left( \frac{\partial e^{(j)}}{\partial X^{i}}\right) \right] -$$$$-\sum_{i,j}\left[ \frac{\partial \widetilde{W}_{ij}}{\partial X^{j}}e^{(i)}-\frac{\partial \widetilde{W}_{ji}}{\partial X^{i}}e^{(j)}\right] =0\text{ \ \ \ \ .} \tag{A26}$$But it can be written also $$\sum_{i,j}\left[ \frac{\partial W_{i}}{\partial X^{j}}\frac{\partial e^{(i)}}{\partial X^{i}}-\frac{\partial \widetilde{W}_{ij}}{\partial X^{j}}e^{(i)}\right] =$$$$=\sum_{i,j}\left[ \frac{\partial W_{i}}{\partial X^{j}}\frac{\partial e^{(i)}}{\partial X^{i}}-e^{(i)}\frac{\partial W_{i}}{\partial X^{j}}\frac{\partial \ln g}{\partial X^{j}}-(\mathbf{e}\text{ .}\mathbf{W})\bigtriangleup \ln g\right] \text{ \ \ .} \tag{A27}$$Taking the above expression into account, equation (A26) acquires the form $$\sum_{i,j}\left[ \frac{\partial W_{i}}{\partial X^{j}}\frac{\partial e^{(i)}}{\partial X^{i}}-\frac{\partial W_{i}}{\partial X^{j}}\frac{\partial e^{(i)}}{\partial X^{i}}\right] +$$$$+\sum_{i,j}\left[ e^{(j)}\frac{\partial W_{j}}{\partial X^{i}}\frac{\partial \ln g}{\partial X^{i}}-e^{(i)}\frac{\partial W_{i}}{\partial X^{j}}\frac{\partial \ln g}{\partial X^{j}}\right] =0\text{ \ \ \ .} \tag{A28}$$Further we shall assume that each term in the sum is zero, i.e. the equation is fulfilled for each $i$ and $j$. Substituting expressions (A22) for $e^{(i)\text{ }}$and $e^{(j)}$ and (A23) for $\frac{\partial e^{(i)}}{\partial X^{j}}$ and $\frac{\partial e^{(j)}}{\partial X^{i}}$, differentiating the obtained expression by $X^{i}$ and making use again of (A22) and (A23), the following simple differential equation can be obtained: $$W_{j,i}\frac{\partial \ln g(X_{1},.,X_{j-1,}X_{j+1},..,X_{n})}{\partial X^{i}}+(W_{j,ii}-W_{i,j}W_{j,i}-$$$$-\frac{W_{i,ji}W_{j,i}}{W_{i,j}})+W_{j,i}e^{\int \left[ \frac{\partial ^{2}W_{j}}{\partial X^{i2}}-\frac{\partial W_{j}}{\partial X^{i}}\right] dX^{j}}=0\text{ \ \ .} \tag{A29}$$The first case, when this equation will be satisfied will be $$W_{j,i}=\frac{\partial W_{j}}{\partial X^{i}}=0\text{ \ \ }\Longrightarrow W_{j}=f(X_{1},..,X_{i-1},X_{i+1},..,X_{n})\text{ \ \ .} \tag{A30}$$Since this will be fulfilled for every $i$, then $W_{j}$ should be a constant, which of course is a very rare and special case. The second, more realistic case is when the function $g$ is a solution of the differential equation (A23) for every $i$ and $j$ ($i\neq j$): $$g(X_{1},.,X_{j-1,}X_{j+1},..,X_{n})=F(X_{1},.,X_{i-1,}X_{i+1},..,X_{n})e^{\int \widetilde{Q}(X_{1},...,X_{n})dX^{i}}\text{ \ \ ,} \tag{A31}$$where $$\widetilde{Q}(X_{1},...,X_{n})\equiv \left( W_{i,j}+\frac{W_{i,ji}}{W_{i,j}}-\frac{W_{j,ii}}{W_{j,i}}\right) -e^{\int \left( \frac{\partial ^{2}W_{j}}{\partial X^{i2}}-\frac{\partial W_{j}}{\partial X^{i}}\right) dX^{j}}\text{ .} \tag{A32}$$Since the function $g(X_{1},.,X_{j-1,}X_{j+1},..,X_{n})$ on the L. H. S. of (A31) does not depend on the variable $X_{j}$, then for each $j$ the unknown function $F(X_{1},.,X_{i-1,}X_{i+1},..,X_{n})$ can be obtained after differentiating both sides of (A31) by $X^{j}$. Thus the function $F$ is a solution of the following differential equation $$0=\frac{\partial F(X_{1},.,X_{i-1,}X_{i+1},..,X_{n})}{\partial X^{j}}e^{\int \widetilde{Q}dX^{i}}+F(X_{1},.,X_{i-1,}X_{i+1},..,X_{n})e^{\int \frac{\partial \widetilde{Q}}{\partial X^{j}}dX^{i}}\text{ \ \ .} \tag{A33}$$This precludes the proof that the function $g$ in (A22) can be determined in such a way that $e^{(i)}$ would be indeed a scalar quantity and therefore $e\equiv e^{(i)}$. Throughout the whole proof, we assumed that $W_{i}$, determined by (A20), is a vector. This of course should be proved in the same way, in which it was proved that the connection $\widetilde{\Gamma }_{ij}^{k}$ has affine transformation properties. [99]{} P. Szekerez 1972 J. Math. Phys. 13 (3) 286 - 294 P. Szekerez 1975 Commun. Math. Phys. 41 55 - 64 D. Eardley, E. Liang, and R. Sachs 1972 J. Math. Phys. 13 (1) 99 - 107 D. A. Szafron 1977 J. Math. Phys. 18 (8) 1673 - 1677 D. A. Szafron, and J. Wainwright 1977 J. Math. Phys. 18 (8) 1668 - 1672 S. W. Goode, and J. Wainwright 1982 Phys. Rev. D26 (12) 3315 - 3326 J. D. Barrow, and J. Stein - Schabez 1984 Phys. Lett. 103A (6, 7) 315 -317 G. V. Kraniotis, and S. B. Whitehouse 2002 Class. Quant. Grav. 19 5073 - 5100 (Preprint gr-qc/0105022) V. V. Prasolov, and Y. P. Solov’yev 1997 Elliptic Functions and Elliptic Integrals (AMS Translations of Mathematical Monographs 170) (R.I.: Providence) \[Russian original: V. V. Prasolov, and Y. P. Solov’yev 1997 Elliptic Functions and Algebraic Equations (Moscow: Factorial Publishing House) B. G. Dimitrov 2003 J. Math. Phys. 44 (6) 2542 - 2578 (Preprint hep-th/0107231) A. Hurwitz, and R. Courant 1964 Allgemeine Funktionentheorie und Elliptische Funktionen (Berlin-Heidelberg: Springer-Verlag) A. Hurwitz, and R. Courant 1929 Funktionentheorie (Berlin: Verlag von J. Springer) S. Manoff 1999 Part. Nucl. 30 517 - 549 \[Rus. Edit. 1999 Fiz. Elem. Chast. Atomn.Yadra. 30 (5) 1211 - 1269\] (Preprint gr-qc/0006024) S. Manoff 2004 Phys. Part. Nucl. 35 633 - 674 \[Russian Edit. 2004 Fiz. Elem. Chast. Atom. Yadra 35 1185 - 1258\] (Preprint gr-qc/0309050) A. P. Norden 1950 Spaces of Affine Connection (Moscow: Nauka Publ.House) P. A. Shirokov, and A. P. Shirokov 1959 Affine Differential Geometry (Moscow: Fizmatgiz) J. A. Wolf 1972 Spaces of Constant Curvature (Berkeley: University of California) W. Slebodzinski 1998 Exterior Forms and Their Applications (University of Beijing: College Press) W. V. D. Hodge, and D. Pedoe 1952 Methods of Algebraic Geometry (Cambridge:Cambridge University Press) F. R. Gantmacher 1959 Theory of Matrices (Chelsea) R. J. Walker 1950 Algebraic Curves (New Jersey: Princeton University Press) V. V. Prasolov, and O. V. Schwarzman 1999 The Alphabet of Riemann Surfaces (Moscow: Fazis Publ. House) B. G. Dimitrov 2001, in Perspectives of Complex Analysis, Differential Geometry and Mathematical Physics Proceedings of the 5th International Workshop on Complex Structures and Vector Fields, St. Konstantin, Bulgaria, 3 - 9 September 2000, eds. S. Dimiev, K. Sekigawa (Singapore: World Scientific) p. 171 - 179 (Preprint gr-qc/0107089) L. Landau, and E. Lifschiz 1988 Theoretical Physics, vol. II. Field Theory (Moscow: Nauka) F. Klein 1884 Vorlesungen Uber das Ikosaeder und Auflosung der Gleichungen vom Funften Grade (Leipzig) L. K. Lahtin 1893 Algebraic Equations, Solvable in Hypergeometric Functions (Moscow: University Press) (in Russian) L. K. Lahtin 1897 Differential Resolvents of Higher Order Algebraic Equations (Moscow: University Press) (in Russian) H. Weber 1896 Lehrbuch der Algebra. Band 2. Gruppen. Lineare Gruppen. Anwendungen der Gruppen Theorie. AlgebraischeZahlen (Braunschweig) H. Weber 1908 Lehrbuch der Algebra. Band 3. Elliptische Funktionen und Algebraische Zahlen (Braunschweig) D. Mumford 1893 1894 Tata Lectures on Theta vol. 1,2 (Boston-Basel-Stuttgart: Birkhauser) G. Belardinelli 1960 Fonctions Hypergeometriques De Plusieurs Variables et Resolution Analytique Des Equations Algebriques Generales (Paris: Gauthier - Villars) S. Stoilov 1954 Theory of Functions of Complex Variables (Editura Academiei Romans) N. I. Akhiezer 1979 Elements of the Elliptic Functions Theory (Moscow:Nauka Publish. House) F. Klein, und R. Fricke 1890 Vorlesungen Uber Die Theorie Der Elliptischen Modulfunctionen Band I und II (Leipzig: Druck und Verlag von B.G. Teubner) R. Fricke, und F. Klein 1926 Vorlesungen Uber Die Theorie Der Automorphen Functionen Band I und II (Leipzig- Berlin: Verlag und Druck von B. G. Teubner) J. Silverman, and J. Tate 1992 Rational Points on Elliptic Curves (New York: Springer - Verlag) D. Husemoller 1987 Elliptic Curves (Graduate Texts in Mathematics 111) (New York: Springer-Verlag) A. W. Knapp 1992 Elliptic Curves (Mathematical Notes 40) (Princeton N.J.: Princeton University Press) J. Silverman 1986 The Arithmetic of Elliptic Curves (New York: Springer-Verlag) J. Silverman 1994 Advanced Topics in the Arithmetic of Elliptic Curves (New York: Springer-Verlag) T. M. Apostol 1976 Modular Functions and Dirichlet Series in Number Theory (New York: Springer-Verlag) A. Robert 1973 Elliptic Curves (Lecture Notes in Mathematics 326) (New York: Springer-Verlag) M. Waldschmidt, P. Moussa, J.-M. Luck, and C. Itzykson (eds.) 1992 From Number Theory to Physics (Berlin Heidelberg:Springer-Verlag) R. Arnowitt, S. Deser, and C. Misner 1962 The Dynamics of General Relativity, in “Gravitation: An Introduction to Current Research”, ed. by L. Witten (New York - London: John Wiley & Sons Inc.) (Preprint gr-qc/0405109) R. Arnowitt, S. Deser, and C. Misner 1959 Phys. Rev. 116 1322 E.Witten 1996 Nucl. Phys. B 471 135 S. Manoff, and B. G. Dimitrov 2002 Class. Quant. Grav. 19 4377 - 4397 (Preprint gr-qc/0207060) [^1]: Electronic mail: bogdan@theor.jinr.ru
-25mm A new stationary cylindrically symmetric solution\ of the Einstein’s equations\ admiting Time machine Elena V. Palesheva Department of Mathematics, Omsk State University\ 644077 Omsk-77 RUSSIA\ E-mail: m82palesheva@math.omsu.omskreg.ru\ July 28, 2001\ ABSTRACT In this article a new stationary solution of the Einstein’s equations with cosmological constant and Time machine is given. The garavitational field is created by ideal liquid with three massless scalar fields or by ideal liquid with electric-magnetic field and massless scalar field. Introduction ============ In this article a new stationary solution of the Einstein’s equations with cosmological constant and Time machine is given. There are two interpretations of stress-energy tensor for this spacetime. In the first case cosmological constant can be arbitrary and gravitational field is created by ideal liquid with three massless scalar fields. In second case the cosmological constant is non-negative and matter which creates this gravitational field is an ideal liquid with electric-magnetic field and massless scalar field. In 1937 the first similar solution was found by W.J. van Stockum [@1]. But problem of Time machihe is discoursed from 1949, after that cosmological model admiting smooth closed timelike curves was found by the famous logic K. Gödel [@2]. He was the first who interpreted similar curves as Time machine. Metric and stress-energy tensor =============================== Let us look the metric $$ds^2=\frac{{dx^{\scriptscriptstyle 0}}^2}{2\Omega}+2 (x^{\scriptscriptstyle 2}dx^ {\scriptscriptstyle 1}-x^{\scriptscriptstyle 1}dx^{\scriptscriptstyle 2}) dx^ {\scriptscriptstyle 0}+\Omega (2{x^{\scriptscriptstyle 2}}^2-1) {dx^{\scriptscriptstyle 1}}^2+ \Omega (2{x^{\scriptscriptstyle 1}}^2-1){dx^{\scriptscriptstyle 2}}^2-$$ $$-4\Omega x^ {\scriptscriptstyle 1}x^{\scriptscriptstyle 2}dx^{\scriptscriptstyle 1} dx^{\scriptscriptstyle 2}-{dx^{\scriptscriptstyle 3}}^2,\eqno (1)$$ where $\Omega=const>0$. The nonzero components of the Christoffel’s symbols are $$\Gamma^0_{01}=2x^{\scriptscriptstyle 1},\,\Gamma^0_{02}= 2x^{\scriptscriptstyle 2} ,\,\Gamma^0_{11}=8x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2} \Omega, \,\Gamma^0_{12}=4\Omega ({x^{\scriptscriptstyle 2}}^2- {x^{\scriptscriptstyle 1}}^2), \,\Gamma^0_{22}=-8x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2}\Omega,$$ $$\Gamma^1_{02}=-\frac{1}{\Omega},\,\Gamma^1_{12}=-2x^{\scriptscriptstyle 2},\, \Gamma^1_{22}=4x^{\scriptscriptstyle 1},\,\Gamma^2_{01}=\frac{1}{\Omega} ,\,\Gamma^2_{11}=4x^{\scriptscriptstyle 2},\,\Gamma^2_{12}= -2x^{\scriptscriptstyle 1},$$ and nonzero components of the Ricci tensor are $$R_{00}=\frac{2}{{\Omega}^{\lefteqn{\scriptstyle 2}}},\,R_{01}=\frac{4x^{ \scriptscriptstyle 2}}{\Omega},\,R_{02}= \frac{-4x^{\scriptscriptstyle 1}}{\Omega} ,\,R_{11}=4+8{x^{\scriptscriptstyle 2}}^2,\,R_{12}=-8x^{ \scriptscriptstyle 1}x^{\scriptscriptstyle 2},\,R_{22}= 4+8{x^{\scriptscriptstyle 1}}^ 2.$$ Also was calculated scalar curvature $R=-4/\Omega$. By using the Einstein’s equations[^1] $$R_{ik}-\frac{1}{2} g_{ik}R=\kappa T_{ik}+\Lambda g_{ik},$$ we find the following tensor: $$\kappa T_{ik}+\Lambda g_{ik}=\left[\begin{array}{cccr} {\displaystyle \frac{3}{{\mathstrut\Omega}^{\lefteqn{\scriptstyle2}}}}& {\displaystyle \frac{6x^{\scriptscriptstyle 2}}{\Omega}}& {\displaystyle -\frac{6x^{\scriptscriptstyle 1}}{\Omega}}&0\\ {\displaystyle \frac{{\mathstrut 6x} ^{\scriptscriptstyle 2}}{\mathstrut \Omega}}&12{x^{\scriptscriptstyle 2}}^2+2&-12x^{ \scriptscriptstyle 1}x^{\scriptscriptstyle 2}&0\\ {\displaystyle -\frac{{\mathstrut 6x}^{ \scriptscriptstyle 1}}{\Omega}}&- 12x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2}&12{x^{ \scriptscriptstyle 1}}^2+2&0\\ 0&0&0&{\displaystyle -\frac{2}{\Omega}} \end{array}\right].\eqno (2)$$ Note that if we introduce the cylindrical coordinats $$\left\{\begin{array}{l}x^{\scriptscriptstyle 0}= x^{\scriptscriptstyle 0}\\ x^{ \scriptscriptstyle 1}=r\cos \varphi\\ x^{\scriptscriptstyle 2}= r\sin \varphi\\ x^{\bigskip \scriptscriptstyle 3}=x^{\scriptscriptstyle 3}\end{array}\right.$$ then metric is transformed to the expression $$ds^2=\frac{1}{2\Omega}d{x^{\scriptscriptstyle 0}}^2-2r^2dx^{\scriptscriptstyle 0}d\varphi- \Omega dr^2+\Omega r^2\left(2r^2-1\right)d{\varphi}^2-d{x^{\scriptscriptstyle 3}}^2.$$ The physical interpretations of the stress-energy tensor ======================================================== Interpretation 1 ---------------- In this section we show that geometry of spacetime with metric (1) can be created by ideal liquid, for which we must take $$\left\{\begin{array}{l} T_{{\mathstrut ik}}^{(i.liquid)}=(c^2\rho +p)u_iu_k-p\,g_{ik}\\ g^{{\mathstrut ik}}u_iu_k=1\end{array}\right. ,\eqno (3)$$ and for scalar fields $$T_{ik}^{scalar}=\frac{\partial\varphi}{\partial x^{\scriptscriptstyle i}} \frac{\partial\varphi}{\partial x^{\scriptscriptstyle k}}+\frac{1}{2}g_{ik}(m^2{\varphi}^2- g^{mn}\frac{\partial\varphi}{\partial x^{\scriptscriptstyle m}}\frac{\partial\varphi}{\partial x^ {\scriptscriptstyle n}}),\eqno (4)$$ $$-\frac{1}{\sqrt{-g}}\,\frac{\partial}{\partial x^{\scriptscriptstyle i}} (\sqrt{-g}\,g^{ik}\frac {\partial\varphi}{\partial x^{\scriptscriptstyle k}})-m^2\varphi=0.\eqno (5)$$ Here (5) is the Klein-Fock’s equation. Let there are three real massless scalar fields and ideal liquid. According to these equations gravitational field will be determinated by equality $$\kappa T_{ik}+\Lambda g_{ik}=\kappa (c^2\rho +p)u_iu_k+\kappa\frac{\partial \varphi}{\partial x^{\scriptscriptstyle i}}\frac{\partial\varphi}{\partial x^{\scriptscriptstyle k}}+\kappa\frac{\partial\psi}{\partial x^{\scriptscriptstyle i}}\frac{\partial\psi}{\partial x^{\scriptscriptstyle k}} +\kappa\frac{\partial\theta}{\partial x^{\scriptscriptstyle i}}\frac{\partial \theta}{\partial x^{\scriptscriptstyle k}}+g_{ik}\{\Lambda -\kappa p-$$ $$-\frac{ \kappa}{2}g^{mn}(\frac{\partial\varphi}{\partial x^{\scriptscriptstyle m}} \frac{\partial\varphi}{\partial x^{\scriptscriptstyle n}}+\frac{\partial\psi} {\partial x^{\scriptscriptstyle m}}\frac{\partial\psi}{\partial x^{ \scriptscriptstyle n}}+\frac{\partial\theta}{\partial x^{\scriptscriptstyle m}} \frac{\partial\theta}{\partial x^{\scriptscriptstyle n}})\}.\eqno (6)$$ Now we assume that $\varphi=\varphi(x^{\scriptscriptstyle 1})$, $ \psi= \psi(x^{ \scriptscriptstyle 2})$ and $\theta=\theta(x^{\scriptscriptstyle 3})$, or that only $ \partial\varphi/\partial x^{\scriptscriptstyle 1}, \partial\psi/ \partial x^{\scriptscriptstyle 2}$ and $ \partial\theta/\partial x^{ \scriptscriptstyle 3}$ are not equial to zero. The vector $$u_i=(\pm {\displaystyle \frac{1}{\sqrt{2\Omega}}},\pm \sqrt{2\Omega}x^{ \scriptscriptstyle 2},\mp\sqrt{2\Omega}x^{\scriptscriptstyle 1},0)\eqno (7)$$ satisfies to the restriction (3) for 4-velocity. By using these conjectures and (5), and substituting (2) into (6) we obtain $$\begin{array}{l}{\displaystyle \frac{3}{{\mathstrut\Omega}^{\lefteqn{ \scriptstyle 2}}}=\frac{\kappa (c^2\rho +p)}{2\Omega}+\frac{1}{2\Omega}\left( \Lambda -\kappa p+\frac{\kappa}{2\Omega}{\left(\frac{\partial\varphi}{\partial x^{\scriptscriptstyle 1}}\right)}^2+\frac{\kappa}{2\Omega}{\left(\frac{ \partial\psi}{\partial x^{\scriptscriptstyle 2}}\right)}^2+\right. }\\ \hspace{1.5cm}{\displaystyle \left. +\frac{\kappa}{2 \Omega}{\left(\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}}\right)} ^2\right)}\\ {\displaystyle \frac{{\mathstrut}6x^{\scriptscriptstyle 2}}{\mathstrut \Omega} =\kappa (c^2\rho +p)x^{\scriptscriptstyle 2}+x^{\scriptscriptstyle 2}\left( \Lambda-\kappa p+\frac{\kappa}{2\Omega}{\left(\frac{\partial\varphi}{\partial x^{\scriptscriptstyle 1}}\right)}^2+\frac{\kappa}{2\Omega}{\left(\frac{ \partial\psi}{\partial x^{\scriptscriptstyle 2}}\right)}^2+\right. }\\ \hspace{1.5cm}{\displaystyle \left. +\frac{\kappa}{2 \Omega}{\left(\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}}\right) }^2\right)}\\ {\displaystyle -\frac{{\mathstrut 6x}^{\scriptscriptstyle 1}}{\mathstrut \Omega}=-\kappa (c^2\rho+p)x^{\scriptscriptstyle 1}-x^{\scriptscriptstyle 1} \left(\Lambda -\kappa p+\frac{\kappa}{2\Omega}{\left(\frac{\partial\varphi}{ \partial x^{\scriptscriptstyle 1}}\right)}^2+\frac{\kappa}{2\Omega}{\left( \frac{\partial\psi}{\partial x^{\scriptscriptstyle 2}}\right)}^2+\right. }\\ \hspace{1.5cm}{\displaystyle \left.+\frac{\kappa }{2\Omega}{\left(\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}} \right)}^2\right)}\\ {\displaystyle {\mathstrut -12x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2 }}=-2\Omega\,x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2}\kappa (c^2\rho + p)-2\Omega\,x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2}\left(\Lambda - \kappa p+\frac{\kappa}{2\Omega}{\left(\frac{\partial\varphi}{\partial x^{ \scriptscriptstyle 1}}\right)}^2+\right.}\\ \hspace{1.5cm}{\displaystyle \left.+\frac{\kappa}{2\Omega} {\left(\frac{\partial \psi}{\partial x^{\scriptscriptstyle 2}}\right)}^2 +\frac{\kappa}{2\Omega}{ \left(\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}} \right)}^2\right)}\\ {\displaystyle {\mathstrut 12{x^{\scriptscriptstyle 2}}^2}+2=2\Omega{x^{ \scriptscriptstyle 2}}^2\kappa (c^2\rho +p)+\kappa{\left(\frac{\partial \varphi}{\partial x^{\scriptscriptstyle 1}}\right)}^2+\Omega ({2x^{ \scriptscriptstyle 2}}^2-1)\left(\Lambda -\kappa p+\right.}\\ \hspace{1.5cm}{\displaystyle \left.+\frac{\kappa}{2\Omega}{ \left(\frac{\partial\varphi}{\partial x^{\scriptscriptstyle 1}}\right)}^2+ \frac{\kappa}{2\Omega}{\left(\frac{\partial\psi} {\partial x^{\scriptscriptstyle 2}}\right)}^2+\frac{\kappa}{2\Omega} {\left(\frac{\partial\theta}{\partial x^{ \scriptscriptstyle 3}}\right)}^2\right)}\\ \end{array}$$ $$\begin{array}{l} {\displaystyle {\mathstrut 12{x^{\scriptscriptstyle 1}}^2}+2=2\Omega{x^{ \scriptscriptstyle 1}}^2\kappa (c^2\rho +p)+\kappa{\left(\frac{\partial\psi}{ \partial x^{\scriptscriptstyle 2}}\right)}^2+\Omega ({2x^{\scriptscriptstyle 1}}^2-1)\left(\Lambda -\kappa p+\right.}\\ \hspace{1.5cm}{\displaystyle \left.+\frac{\kappa} {2\Omega}{\left(\frac{\partial \varphi}{\partial x^{\scriptscriptstyle 1}}\right)}^2+\frac{\kappa}{2\Omega} {\left(\frac{\partial\psi}{\partial x^{\scriptscriptstyle 2}} \right)}^2+\frac {\kappa}{2\Omega}{\left(\frac{\partial\theta} {\partial x^{\scriptscriptstyle 3 }}\right)}^2\right)}\\ {\displaystyle -\frac{\mathstrut 2}{\mathstrut \Omega}= \kappa{\left(\frac{ \partial\theta}{\partial x^{\scriptscriptstyle 3}}\right)}^2- \left(\Lambda -\kappa p+\frac{\kappa}{2\Omega}{\left(\frac{\partial\varphi} {\partial x^{\scriptscriptstyle 1}}\right)}^2+\frac{\kappa}{2\Omega}{\left(\frac{\partial\psi}{\partial x^{ \scriptscriptstyle 2}}\right)}^2+\frac{\kappa}{2\Omega}{\left(\frac{\partial \theta}{\partial x^{\scriptscriptstyle 3}}\right)}^2\right)}\\ {\displaystyle \frac{\partial}{\partial x^{\scriptscriptstyle k}} \left(g^{1k}\frac{ \partial\varphi}{\partial x^{\scriptscriptstyle 1}}\right)= \frac{\partial}{\partial x^{\scriptscriptstyle k}}\left(g^{2k}\frac{\partial\psi}{\partial x^{ \scriptscriptstyle 2}}\right)= \frac{\partial}{\partial x^{\scriptscriptstyle k}}\left(g^ {3k}\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}}\right)=0}. \end{array}$$ The next formulas are directly checked: $$\left\{\begin{array}{l} \kappa p=\Lambda+{\displaystyle \frac{2}{\mathstrut\Omega}+\frac{\kappa}{2}{ \left(\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}}\right)}^2}\\ \kappa c^2\rho={\displaystyle \frac{\mathstrut 2}{\mathstrut\Omega}-\Lambda- \frac{3}{2}\kappa{\left(\frac{\partial\theta} {\partial x^{\scriptscriptstyle 3}} \right)}^2}\\ u_i=(\pm {\displaystyle \frac{\mathstrut 1}{\mathstrut\sqrt{2\Omega}}},\pm \sqrt{2\Omega}x^{\scriptscriptstyle 2},\mp \sqrt{2\Omega}x^{\scriptscriptstyle 1},0)\\ {\displaystyle \kappa{\left(\frac{\mathstrut\partial\varphi} {\mathstrut\partial x^{ \scriptscriptstyle 1}}\right)}^2= \kappa{\left(\frac{\partial\psi}{\partial x^{ \scriptscriptstyle 2}}\right)}^2= 4+\kappa\Omega{\left(\frac{\partial\theta}{\partial x^{ \scriptscriptstyle 3}}\right)}^2} \\ {\displaystyle \mathstrut\varphi=A_1x^{\scriptscriptstyle 1}+A_2, \psi= B_1x^{ \scriptscriptstyle 2}+B_2, \theta=C_1x^{\scriptscriptstyle 3}+C_2}\\ A_1,A_2,B_1,B_2,C_1,C_2=const.\end{array}\right.\eqno (8)$$ So cosmological constant must change in the following domain $${\displaystyle -\frac{2}{\Omega}- \frac{\kappa}{2}{\left(\frac{\partial\theta}{ \partial x^{\scriptscriptstyle 3}}\right)}^2\leq\Lambda\leq\frac{2} {\Omega}-\frac{3} {2}\kappa{\left(\frac{\partial\theta}{\partial x^{\scriptscriptstyle 3}} \right)}^2}.$$ Interpretation 2 ---------------- Above one of the some interpretations of stress-energy tensor was considered. Here we shall attempt to demonstrate another variant of matter. We call attention to the well-known fact that electric-magnetic stress-energy tensor $$T_{ik}^{el.mag.}=\frac{1}{4\pi}\left(\frac{1}{4}F_{lm}F^{lm}g_{ik}-F_{il}F^ {\;l}_k\right).\eqno (9)$$ Together with (9) we must take the Maxwell’s equations $$\left\{\begin{array}{l}F_{ik}={\displaystyle \frac{\partial A_k}{\partial x^ {\scriptscriptstyle i}}-\frac{\partial A_i}{\partial x^{\scriptscriptstyle k} }}\\ {\nabla}_kF^{ik}={\displaystyle -\frac{4\pi}{c}j^{\,i}},\end{array} \right.\eqno (10)$$ where $A_k$ is 4-potential of this electric-magnetic field and ${\nabla}_k$ is a covariant derivation. As early we consider ideal liquid and real massless scalar field. Moreover we shall use the electric-magnetic field. And now we show that considered field’s system satisfies to (2). We have $$\kappa T_{ik}+\Lambda g_{ik}=\kappa(c^2\rho+p)u_iu_k+\kappa\frac{\partial \varphi}{\partial x^{\scriptscriptstyle i}}\frac{\partial\varphi}{\partial x^ {\scriptscriptstyle k}}-\frac{\kappa}{4\pi}F_{i\,l}F^{\;l}_k+\left\{ \frac{ \kappa}{16\pi}F_{lm}F^{lm}+\Lambda-\kappa p-\right.$$ $$\left.-\frac{1}{2}\kappa g^{mn}\frac{ \partial\varphi}{\partial x^{\scriptscriptstyle m}}\frac{\partial\varphi}{ \partial x^{\scriptscriptstyle n}}\right\}g_{ik}.\eqno(11)$$ Now we use suggestion (7) and as early consider massless scalar field $\varphi$, so that $$\frac{\partial\varphi}{\partial x^{\scriptscriptstyle 0}}=\frac{\partial \varphi}{\partial x^{\scriptscriptstyle 1}}=\frac{\partial\varphi}{\partial x ^{\scriptscriptstyle 2}}=0.$$ Let only $F_{12}\neq0$. Then $$\frac{\kappa}{16\pi}F_{ lm}F^{lm}=\frac{\kappa} {8\pi{\Omega}^{\lefteqn{\scriptstyle 2}}}\, {(F_{12})}^2,$$ $$-\frac{\kappa}{4\pi}F_{1l}F^{\;l}_1=\frac{\kappa}{4\pi\Omega}\,{(F_{12})}^2,$$ $$-\frac{\kappa}{4\pi}F_{2l}F^{\;l}_2=\frac{\kappa}{4\pi\Omega}\,{(F_{12})}^2.$$ By substituting these formulas and (2) into (11) we obtain $$\begin{array}{l} {\displaystyle u_i=(\pm {\frac{1}{\mathstrut\sqrt{2\Omega}}}, \pm\sqrt{2\Omega}x ^{\scriptscriptstyle 2},\mp \sqrt{2\Omega}x^{\scriptscriptstyle 1},0)}\\ {\displaystyle \frac{\mathstrut 3}{{\mathstrut}{\Omega}^{\lefteqn{ \scriptstyle2}}}=\frac{\kappa}{2\Omega}(c^2\rho+p)+\frac{1}{2\Omega}\left\{ \frac{\kappa}{8\pi{\Omega}^{\lefteqn{\scriptstyle2}}}\,{(F_{12})}^2+\Lambda- \kappa p+\frac{1}{2}\kappa{\left(\frac{\partial\varphi}{\partial x^{ \scriptscriptstyle 3}}\right)}^2\right\}}\\ {\displaystyle \frac{{\mathstrut}6x^{\scriptscriptstyle 2}} {\mathstrut\Omega}= \kappa (c^2\rho+p)x^{\scriptscriptstyle 2}+x^{\scriptscriptstyle 2}\left\{ \frac{\kappa}{8\pi{\Omega}^{\lefteqn{\scriptstyle2}}}\,{(F_{12})}^2+\Lambda- \kappa p+\frac{1}{2}\kappa{\left(\frac{\partial\varphi}{\partial x^{ \scriptscriptstyle 3}}\right)}^2\right\}}\\ {\displaystyle -\frac{{\mathstrut}6x^{\scriptscriptstyle 1}} {\mathstrut\Omega }=-\kappa (c^2\rho+p)x^{\scriptscriptstyle 1}-x^{\scriptscriptstyle 1} \left\{ \frac{\kappa}{8\pi{\Omega}^{\lefteqn{\scriptstyle2}}}\,{(F_{12})}^2+\Lambda- \kappa p+\frac{1}{2}\kappa{\left(\frac{\partial\varphi}{\partial x^{ \scriptscriptstyle 3}}\right)}^2\right\}}\\ {\displaystyle -{\mathstrut}12x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2 }=-2\kappa (c^2\rho+p)\Omega x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2} -2\Omega x^{\scriptscriptstyle 1}x^{\scriptscriptstyle 2}\left\{\frac{ \mathstrut\kappa}{\mathstrut 8\pi{\Omega}^{\lefteqn{\scriptstyle2}}}\, {(F_{12 })}^2+\Lambda-\kappa p+\frac{1}{2}\kappa{\left(\frac{\partial\varphi} {\partial x^{ \scriptscriptstyle 3}}\right)}^2\right\}}\\ \end{array}$$ $$\begin{array}{l} {\displaystyle {\mathstrut}12{x^{\scriptscriptstyle 2}}^2+2= 2\kappa\Omega (c^2 \rho+p){x^{\scriptscriptstyle 2}}^2+\frac{\mathstrut\kappa}{\mathstrut 4\pi \Omega}\,{(F_{12})}^2+\Omega (2{x^{\scriptscriptstyle 2}}^2-1)\left\{\frac{ \kappa}{8\pi{\Omega}^{\lefteqn{\scriptstyle2}}}\, {(F_{12})}^2+\Lambda-\right. }\\ \hspace{1.5cm}{\displaystyle \left.-\kappa p+\frac{1}{2}\kappa{\left(\frac{\partial\varphi} {\partial x^{\scriptscriptstyle 3} }\right)}^2\right\}}\\ {\displaystyle {\mathstrut}12{x^{\scriptscriptstyle 1}}^2+2= 2\kappa\Omega (c^2 \rho+p){x^{\scriptscriptstyle 1}}^2+\frac{\kappa}{4\pi\Omega}\,{(F_{12})}^2+ \Omega (2{x^{\scriptscriptstyle 1}}^2-1)\left\{\frac{\mathstrut\kappa}{ \mathstrut 8\pi{\Omega}^{\lefteqn{\scriptstyle2}}}\, {(F_{12})}^2+\Lambda-\right. }\\ \hspace{1.5cm}{\displaystyle \left.- \kappa p+\frac{1}{2}\kappa{\left(\frac{\partial\varphi}{\partial x^{ \scriptscriptstyle 3}}\right)}^2\right\}}\\ \end{array}$$ $$\begin{array}{l} {\displaystyle -\frac{\mathstrut 2}{\mathstrut\Omega}= \kappa{\left(\frac{\partial \varphi}{\partial x^{\scriptscriptstyle 3}}\right)}^2- \left\{\frac{\kappa}{8\pi{ \Omega}^{\lefteqn{\scriptstyle2}}}\,{(F_{12})}^2+\Lambda-\kappa p+\frac{1}{2} \kappa{\left(\frac{\partial\varphi} {\partial x^{\scriptscriptstyle 3}}\right)}^2\right\} }\\ {\displaystyle \frac{{\partial}^2\varphi}{\partial x^{\scriptscriptstyle 3} \partial x^{\scriptscriptstyle 3}}=0}.\end{array}$$ After nondifficult calculations the solution of the system can be written in the form $$\left\{\begin{array}{l}{\displaystyle \Lambda=\kappa p}\\ {\displaystyle u_i=(\pm {\frac{1}{\sqrt{2\Omega}}},\pm\sqrt{2\Omega}x^{ \scriptscriptstyle 2},\mp\sqrt{2\Omega}x^{\scriptscriptstyle 1},0)}\\ {\displaystyle {(F_{12})}^2= \frac{16\pi\Omega}{\kappa}+4\pi{\Omega}^2{\left(\frac{ \partial\varphi}{\mathstrut\partial x^{\scriptscriptstyle 3}}\right)}^2}\\ {\displaystyle \kappa c^2\rho= \frac{4}{\Omega}-\Lambda-\kappa{\left(\frac{ \mathstrut\partial\varphi}{\mathstrut \partial x^{\scriptscriptstyle 3}}\right)}^2}\\ {\displaystyle \frac{\mathstrut\partial\varphi} {\mathstrut\partial x^{\scriptscriptstyle 3}} =const, \frac{\mathstrut\partial\varphi} {\mathstrut\partial x^{\scriptscriptstyle 0}}= \frac{\mathstrut\partial\varphi} {\mathstrut\partial x^{\scriptscriptstyle 1}}= \frac{\mathstrut\partial\varphi}{\mathstrut\partial x^{\scriptscriptstyle 2 }}=0}\\ {\displaystyle 0\leq\Lambda\leq\frac{4}{\Omega}- \kappa{\left(\frac{\mathstrut\partial \varphi}{\partial x^{\scriptscriptstyle 3}}\right)}^2}. \end{array}\right.\eqno (12)$$ By using (10) we obtain for 4-vector of current $$j^{\,i}=(\frac{c}{4\pi\Omega}F_{12},0,0,0).$$ Strength and induction of electric and magnetic fields are [@3 p.331] $$E_\alpha=0,\;D^\alpha=\left(\frac{2x^{\scriptscriptstyle 1}} {\sqrt{2{\Omega}^{ \lefteqn{\scriptstyle 3}}}}F_{12},\frac{2x^{\scriptscriptstyle 2}} {\sqrt{2{ \Omega}^{\lefteqn{\scriptstyle 3}}}}F_{12},0\right),$$ $$H_\alpha=\left(0,0,-\frac{1}{\sqrt{2{\Omega}^{\lefteqn{\scriptstyle 3}}}}\ \ F_{12}\right),\ \ B^\alpha=\left(0,0,-\frac{1}{\Omega}F_{12}\right).$$ Time machine ============ The metric (1) admits the closed smooth timelike curves. For example we consider the following smooth closed curve $$L= \{x^{\scriptscriptstyle 0}=const, x^{\scriptscriptstyle 1}=a\sin t, x^{\scriptscriptstyle 2}=a\cos t,x^{\scriptscriptstyle 3}=const\}$$ $$a=const>\frac{1}{\sqrt{2}}.$$ It is timelike in metric (1), that is $g_{ik}d{x}^id{x}^k>0$: $$g_{ik}d{x}^id{x}^k=a^2 \left\{g_{11}{\cos}^2t+g_{22}{\sin}^2t-2g_{12}\sin t\cos t\right\}= a^2\Omega (2a^2-1)>0,$$ as $a>1/\sqrt{2}$. As it known the distance which Time traveler must go, and his chronometric invariant time are calculated with the help of following formulas [@3]: $$\tau(L)=\frac{1}{c}\oint\limits_{L}\frac{g_{0i}dx^{ \scriptscriptstyle i}}{\sqrt{g_{00}}}=\frac{2\pi a^2\sqrt{2\Omega}}{c},$$ $$l(L)=\oint \limits_{L}\sqrt{\left(-g_{\alpha\beta} +\frac{g_{0\alpha}g_{0\beta}} {g_{00}}\right) \frac{dx^{\scriptscriptstyle\alpha}}{dt} \frac{dx^{\scriptscriptstyle \beta}}{dt}}dt=2\pi a\sqrt{\Omega}.\eqno (13)$$ So “diameter” of domain which contains a Time machine $L$, has order $l(L)\sim a\sqrt{\Omega}$. Proper time and time $\tau(l)$ are connected by relation $$s(L)=\frac{1}{c}\oint\limits_{L}\sqrt{g_{ik} \frac{dx^{\scriptscriptstyle i}}{dt}\frac{dx^{ \scriptscriptstyle k}}{dt}}dt= \frac{1}{c}\oint\limits_{L}\sqrt{1-{\left(\frac{dl}{d\tau}\right)}^2}d\tau.$$ The proper time goes to zero when the velocity of Time machine goes to the velocity of light. Conclusion ========== In the previous sections we showed that gravitational field, for which geometry of spacetime is discribed by metric (1), can be created by real phisical matter. But it is intresting how the our results and experimental data are in agreement? At the begining we consider the cosmological solution. As it known in this case density of matter of the Universe is equal to $3\cdot 10^{-31}$[g/cm]{}$^3$. If we wish no large distance for Time expedition than $\tau(L)$ and $l(L)$ must be sufficiently small. By using (13) we conclude that under considered assumption we must take small $\Omega$. In result, as our solution discribe the Universe, using (8) and (12), we obtain that scalar field which depend only on third coordinate must be sufficiently large in both interpretations. (But only if in the first interpretation the pressure of liquid is large). And also we denote that in case of cosmological solution with small $\Omega$ in second interpritation the current and the strength and induction of electric and magnetic fields are large. If we take a solution with large density then considered in previous paragraph scalar field $\theta=\theta(x^{\scriptscriptstyle 3})$ must be small. Also we notice that in first interpretation we can remove the scalar field $\theta$ or ideal liquid, and in second interpretation we can remove the considering scalar field. The investigations of the results without analogous scalar field were discribed in our paper [@4]. The evaluations of the nessesary distance and chronometric invariant time for Time travel in model without ideal liquid agree with the evaluations for first interpretation of the stress-energy tensor in [@4]. Moreover we denote that there exists interpretation of matter of gravitational field (1) as electric-magnetic field and three real massless scalar field in ideal liquid. The variants of matter which were considered by this article, are particular cases of such interpretation. Author is grateful to A.K.Guts for recommended topic and consulting during the work. [99]{} Van Stockum W.J. [Gravitational field of a distribution of particles rotating about an axis of symmetry]{}//Roc. R. Soc. Edin. 1937. V.57. P.135-154. Gödel K. [An example of a new type of cosmological solution of Einstien’s field equation of gravitation]{} //Phys. Rev. mod. 1949. V.21. P.447-450. Landau L., Lifshits E. [Theory of Field]{}. Moscow: Nauka, 1967. Palesheva E.V. [A new solution of the Einstein’s equations admiting Time machine]{} // Mathematical Structures ans Modeling / Ed. A.K.Guts. 2000. No.6. P.128-135 (Omsk, Russian).\ – ftp://cmm.univer.omsk.su/pub/sbornik6/palesh.zip [^1]: The Greek indexes are 1,2,3, and Latin indexes are 0,1,2,3.
--- abstract: 'It is shown that, for a small quantaloid ${\mathcal{Q}}$, the category of small ${\mathcal{Q}}$-categories and ${\mathcal{Q}}$-functors is total and cototal, and so is the category of ${\mathcal{Q}}$-distributors and ${\mathcal{Q}}$-Chu transforms.' address: - \| Lili Shen\ Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3 - \| Walter Tholen\ Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3 author: - Lili Shen - Walter Tholen title: 'Limits and colimits of quantaloid-enriched categories and their distributors' --- [^1] Introduction ============ The importance of (small) categories enriched in a (unital) quantale rather than in an arbitrary monoidal category was discovered by Lawvere [@Lawvere1973] who enabled us to look at individual mathematical objects, such as metric spaces, as small categories. Through the study of lax algebras [@Hofmann2014], quantale-enriched categories have become the backbone of a larger array of objects that may be viewed as individual generalized categories. Prior to this development, Walters [@Walters1981] had extended Lawvere’s viewpoint in a different manner, replacing the quantale at work by a quantaloid (a term proposed later by Rosenthal [@Rosenthal1996]), thus by a bicategory with the particular property that its hom-objects are given by complete lattices such that composition from either side preserves suprema; quantales are thus simply one-object quantaloids. Based on the theory of quantaloid-enriched categories developed by Stubbe [@Stubbe2005; @Stubbe2006], recent works [@Hohle2011; @Pu2012; @Stubbe2014; @Tao2014] have considered in particular the case when the quantaloid in question arises from a given quantaloid by a “diagonal construction” whose roots go far beyond its use in this paper; see [@Grandis2000; @Grandis2002]. Specifically for the one-object quantaloids (i.e., quantales) whose enriched categories give (pre)ordered sets and (generalized) metric spaces, the corresponding small quantaloids of diagonals lead to truly partial structures, in the sense that the full structure is available only on a subset of the ambient underlying set of objects. In the first instance then, this paper aims at exploring the categorical properties of the category ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ of small ${\mathcal{Q}}$-enriched categories and their ${\mathcal{Q}}$-functors for a small quantaloid ${\mathcal{Q}}$. By showing that ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is topological [@Adamek1990] over the comma category ${{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$ (Proposition \[QCat\_topological\]) one easily describes small limits and colimits in this category, and beyond. In fact, one concludes that categories of this type are total [@Street1978] and cototal, hence possess even those limits and colimits of large diagrams whose existence is not made impossible by the size of the small hom-sets of ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ (see [@Borger1990]). Our greater interest, however, is in the category ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ whose objects are often called ${\mathcal{Q}}$-Chu spaces, the prototypes of which go back to [@Barr1991; @Pratt1995] and many others (see [@Barwise1997; @Ganter2007]). Its objects are ${\mathcal{Q}}$-distributors of ${\mathcal{Q}}$-categories (also called ${\mathcal{Q}}$-(bi)modules or ${\mathcal{Q}}$-profunctors), hence they are compatible ${\mathcal{Q}}$-relations (or ${\mathcal{Q}}$-matrices) that have been investigated intensively ever since B[é]{}nabou [@Benabou1973] introduced them (see [@Benabou2000; @Borceux1994a]). While when taken as the morphisms of the category whose objects are ${\mathcal{Q}}$-categories, they make for a in many ways poorly performing category (as already the case ${\mathcal{Q}}={\bf 2}$ shows), when taken as objects of ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ with morphisms given by so-called ${\mathcal{Q}}$-Chu transforms, i.e., by pairs of ${\mathcal{Q}}$-functors that behave like adjoint operators, we obtain a category that in terms of the existence of limits and colimits behaves as strongly as ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ itself. In analogy to the property shown in [@Giuli2007] in a different categorical context, we first prove that the domain functor ${{\mathcal{Q}}\text{-}{{\bf Chu}}}\to{{\mathcal{Q}}\text{-}{{\bf Cat}}}$ allows for initial liftings [@Adamek1990] of structured cones over small diagrams (Theorem \[dom\_initial\_lifting\]), which then allows for an explicit description of all limits and colimits in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ over small diagrams. But although the domain functor fails to be topological, just as for ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ we are able to show totality (and, consequently, cototality) of ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$. A key ingredient for this result is the existence proof of a generating set in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$, and therefore also of a cogenerating set (Theorem \[QChu\_generator\]). Limits and colimits in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ =========================================================== Throughout, let ${\mathcal{Q}}$ be a small quantaloid, i.e., a small category enriched in the category ${{\bf Sup}}$ of complete lattices and sup-preserving maps. A small ${\mathcal{Q}}$-category is given by a set $X$ (its set of objects) and a lax functor $$a:X\to{\mathcal{Q}},$$ where the set $X$ is regarded as a quantaloid carrying the chaotic structure, so that for all $x,y\in X$ there is precisely one arrow $x\to y$, called $(x,y)$. Explicitly then, the ${\mathcal{Q}}$-category structure on $X$ is given by - a family of objects $\|x\|_X:=ax$ in ${\mathcal{Q}}$ $(x\in X)$, - a family of morphisms $a(x,y):\|x\|\to\|y\|$ in ${\mathcal{Q}}$ $(x,y\in X)$, subject to $$1_{\|x\|}\leq a(x,x)\quad\text{and}\quad a(y,z)\circ a(x,y)\leq a(x,z)$$ $(x,y,z\in X)$. When one calls $\|x\|=\|x\|_X$ the extent (or type) of $x\in X$, a ${\mathcal{Q}}$-functor $f:(X,a)\to(Y,b)$ of ${\mathcal{Q}}$-categories $(X,a)$, $(Y,b)$ is an extent-preserving map $f:X\to Y$ such that there is a lax natural transformation $a\to bf$ given by identity morphisms in ${\mathcal{Q}}$; explicitly, $$\|x\|_X=\|f(x)\|_Y\quad\text{and}\quad a(x,y)\leq b(f(x),f(y))$$ for all $x,y\in X$. Denoting the resulting ordinary category by ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$, we have a forgetful functor $$\bfig \morphism<1200,0>[{{\mathcal{Q}}\text{-}{{\bf Cat}}}`{{\bf Set}}/\operatorname{ob}{\mathcal{Q}};\operatorname{ob}] \Vtriangle(-300,-500)<300,300>[X`Y`{\mathcal{Q}};f`a`b] \Vtriangle(900,-500)<300,300>[X`Y`\operatorname{ob}{\mathcal{Q}};f`\|\text{-}\|`\|\text{-}\|] \place(600,-350)[\mapsto] \place(0,-320)[\leq] \efig$$ \[QCat\_exmp\] - If ${\mathcal{Q}}$ is a quantale, i.e., a one-object quantaloid, then the extent functions of ${\mathcal{Q}}$-categories are trivial and ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ assumes its classical meaning (as in [@Kelly1982], where ${\mathcal{Q}}$ is considered as a monoidal (closed) category). Prominent examples are ${\mathcal{Q}}={\bf 2}=\{0<1\}$ and ${\mathcal{Q}}=([0,\infty],\geq,+)$, where then ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is the category ${{\bf Ord}}$ (sets carrying a reflexive and transitive relation, called order here but commonly known as preorder, with monotone maps) and, respectively, the category ${{\bf Met}}$ (sets $X$ carrying a distance function $a:X\times X\to[0,\infty]$ required to satisfy only $a(x,x)=0$ and $a(x,z)\leq a(x,y)+a(y,z)$ but, in accordance with the terminology introduced by Lawvere [@Lawvere1973] and used in [@Hofmann2014], nevertheless called metric here, with non-expanding maps $f:X\to Y$, so that $b(f(x),f(y))\leq a(x,y)$ for all $x,y\in X$). - (Stubbe [@Stubbe2014]) Every quantaloid ${\mathcal{Q}}$ gives rise to a new quantaloid ${{{\sf D}}{\mathcal{Q}}}$ whose objects are the morphisms of ${\mathcal{Q}}$, and for morphisms $u,v$ in ${\mathcal{Q}}$, a morphism $(u,d,v):u\to/~>/v$ in ${{{\sf D}}{\mathcal{Q}}}$, normally written just as $d$, is a ${\mathcal{Q}}$-morphism $d:\operatorname{dom}u\to\operatorname{cod}v$ satisfying $$(d{\swarrow}u)\circ u=d=v\circ(v{\searrow}d),$$ also called a diagonal from $u$ to $v$: $$\bfig \square[\bullet`\bullet`\bullet`\bullet;v{\searrow}d`u`v`d{\swarrow}u] \morphism(0,500)<500,-500>[\bullet`\bullet;d] \efig$$ (Here $d{\swarrow}u$, $v{\searrow}d$ denote the internal homs of ${\mathcal{Q}}$, determined by $$z\leq d{\swarrow}u\iff z\circ u\leq d,\quad t\leq v{\searrow}d\iff v\circ t\leq d$$ for all $z:\operatorname{cod}u\to\operatorname{cod}d$, $t:\operatorname{dom}d\to\operatorname{dom}v$.) With the composition of $d:u\to/~>/v$ with $e:v\to/~>/w$ in ${{{\sf D}}{\mathcal{Q}}}$ defined by $$e\diamond d=(e{\swarrow}v)\circ d=e\circ(v{\searrow}d),$$ and with identity morphisms $u:u\to/~>/u$, ${{{\sf D}}{\mathcal{Q}}}$ becomes a quantaloid whose local order is inherited from ${\mathcal{Q}}$. In fact, there is a full embedding $${\mathcal{Q}}\to{{{\sf D}}{\mathcal{Q}}},\quad(u:t\to s)\mapsto(u:1_t\to/~>/1_s)$$ of quantaloids. We remark that the construction of ${{\sf D}}$ works for ordinary categories; indeed it is part of the proper factorization monad on ${{\bf CAT}}$ [@Grandis2002]. - For ${\mathcal{Q}}={\bf 2}$, the quantaloid ${{{\sf D}}{\mathcal{Q}}}$ has object set $\{0,1\}$. There are exactly two ${{{\sf D}}{\mathcal{Q}}}$-arrows $1\to/~>/1$, given by $0,1$, and $0$ is the only arrow in every other hom-set of ${{{\sf D}}{\mathcal{Q}}}$; composition is given by infimum. A ${{{\sf D}}{\mathcal{Q}}}$-category is given by a set $X$, a distinguished subset $A\subseteq X$ (those elements of $X$ with extent $1$) and a (pre)order on $A$. Hence, a ${{{\sf D}}{\mathcal{Q}}}$-category structure on $X$ is a (truly!) partial order on $X$. With morphisms $f:(X,A)\to(Y,B)$ given by maps $f:X\to Y$ monotone on $A=f^{-1}B$ we obtain the category $${{\bf ParOrd}}={{{{\sf D}}{\mathcal{Q}}}\text{-}{{\bf Cat}}},$$ which contains ${{\bf Ord}}$ as a full coreflective subcategory. - For ${\mathcal{Q}}=([0,\infty],\geq,+)$, the hom-sets of ${{{\sf D}}{\mathcal{Q}}}$ are easily described by $${{{\sf D}}{\mathcal{Q}}}(u,v)=\{s\in[0,\infty]\mid u,v\leq s\},$$ with composition given by $t\diamond s=s-v+t$ (for $t:v\to/~>/w$). A ${{{\sf D}}{\mathcal{Q}}}$-category structure on a set $X$ consists of functions $\|\text{-}\|:X\to[0,\infty]$, $a:X\times X\to[0,\infty]$ satisfying $$\|x\|,\|y\|\leq a(x,y),\quad a(x,x)\leq\|x\|,\quad a(x,z)\leq a(x,y)-\|y\|+a(y,z)$$ $(x,y,z\in X)$. Obviously, since necessarily $\|x\|=a(x,x)$, these conditions simplify to $$a(x,x)\leq a(x,y),\quad a(x,z)\leq a(x,y)-a(y,y)+a(y,z)$$ $(x,y,z\in X)$, describing $a$ as a partial metric on $X$ (see [@Hohle2011; @Matthews1994; @Pu2012][^2]). With non-expanding maps $f:(X,a)\to(Y,b)$ satisfying $a(x,x)=b(f(x),f(x))$ for all $x\in X$ one obtains the category $${{\bf ParMet}}={{{{\sf D}}{\mathcal{Q}}}\text{-}{{\bf Cat}}},$$ which contains ${{\bf Met}}$ as a full coreflective subcategory: the coreflector restricts the partial metric $a$ on $X$ to those elements $x\in X$ with $a(x,x)=0$. To see how limits and colimits in the (ordinary) category ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ are to be formed, it is best to first prove its topologicity over ${{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$. Recall that, for any functor $U:{\mathcal{A}}\to{\mathcal{X}}$, a $U$-structured cone over a diagram $D:{\mathcal{J}}\to{\mathcal{A}}$ is given by an object $X\in{\mathcal{X}}$ and a natural transformation $\xi:{\Delta}X\to UD$. A lifting of $(X,\xi)$ is given by an object $A$ in ${\mathcal{A}}$ and a cone ${\alpha}:{\Delta}A\to D$ over $D$ with $UA=X$, $U{\alpha}=\xi$. Such lifting $(A,{\alpha})$ is $U$-initial if, for all cones ${\beta}:{\Delta}B\to D$ over $D$ and morphisms $t:UB\to UA$ in ${\mathcal{X}}$, there is exactly one morphism $h:B\to A$ in ${\mathcal{A}}$ with $Uh=t$ and ${\alpha}\cdot{\Delta}h={\beta}$. We call $U$ small-topological [@Giuli2007] if all $U$-structured cones over small diagrams admit $U$-initial liftings, and $U$ is topological when this condition holds without the size restriction on diagrams. Recall also the following well-known facts: - Topological functors are necessarily faithful [@Borger1978], and for faithful functors it suffices to consider discrete cones to guarantee topologicity. - $U:{\mathcal{A}}\to{\mathcal{X}}$ is topological if, and only if, $U^{{{\rm op}}}:{\mathcal{A}}^{{{\rm op}}}\to{\mathcal{X}}^{{{\rm op}}}$ is topological. - The two properties above generally fail to hold for small-topological functors. However, for any functor $U$, a $U$-initial lifting of a $U$-structured cone that is a limit cone in ${\mathcal{X}}$ gives also a limit cone in ${\mathcal{A}}$. - Every small-topological functor is a fibration (consider singleton diagrams) and has a fully faithful right adjoint (consider the empty diagram). \[QCat\_topological\] For every (small) quantaloid ${\mathcal{Q}}$, the “object functor” ${{\mathcal{Q}}\text{-}{{\bf Cat}}}\to{{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$ is topological. Given a (possibly large) family $f_i:(X,\|\text{-}\|)\to(Y_i,\|\text{-}\|_i)$ $(i\in I)$ of maps over $\operatorname{ob}{\mathcal{Q}}$, where every $Y_i$ carries a ${\mathcal{Q}}$-category structure $b_i$ with extent function $\|\text{-}\|_i$, we must find a ${\mathcal{Q}}$-category structure $a$ on $X$ with extent function $\|\text{-}\|$ such that (1) every $f_i:(X,a)\to(Y,b)$ is a ${\mathcal{Q}}$-functor, and (2) for every ${\mathcal{Q}}$-category $(Z,c)$, any extent preserving map $g:Z\to X$ becomes a ${\mathcal{Q}}$-functor $(Z,c)\to(X,a)$ whenever all maps $f_i g$ are ${\mathcal{Q}}$-functors $(Z,c)\to(Y_i,b_i)$ $(i\in I)$. But this is easy: simply define $$a(x,y):={\bigwedge}_{i\in I}b_i(f_i(x),f_i(y))$$ for all $x,y\in X$. Hence, $a$ is the $\operatorname{ob}$-initial structure on $X$ with respect to the structured source $(f_i:X\to(Y_i,b_i))_{i\in I}$. \[QCat\_complete\] ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is complete and cocomplete, and the object functor has both a fully faithful left adjoint and a fully faithful right adjoint. \[QCat\_limit\] - The set of objects of the product $(X,a)$ of a small family of ${\mathcal{Q}}$-categories $(X_i,a_i)$ $(i\in I)$ is given by the fibred product of $(X_i,\|\text{-}\|_i)$ $(i\in I)$, i.e., $$X=\{((x_i)_{i\in I},q)\mid q\in\operatorname{ob}{\mathcal{Q}},\ \forall i\in I(x_i\in X_i,\|x_i\|=q)\},$$ and (when writing $(x_i)_{i\in I}$ instead of $((x_i)_{i\in I},q)$ and putting $\|(x_i)_{i\in I}\|=q$) we have $$a((x_i)_{i\in I},(y_i)_{i\in I})={\bigwedge}_{i\in I}a_i(x_i,y_i):\|(x_i)_{i\in I}\|\to\|(y_i)_{i\in I}\|$$ for its hom-arrows. In particular, $(\operatorname{ob}{\mathcal{Q}},\top)$ with $$\top(q,r)=\top:q\to r$$ the top element in ${\mathcal{Q}}(q,r)$ (for all $q,r\in\operatorname{ob}{\mathcal{Q}}$), is the terminal object in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. - The coproduct $(X,a)$ of ${\mathcal{Q}}$-categories $(X_i,a_i)$ $(i\in I)$ is simply formed by the coproduct in ${{\bf Set}}$, with all structure to be obtained by restriction: $$X=\coprod_{i\in I}X_i,\quad \|x\|_X=\|x\|_{X_i}\ \text{if}\ x\in X_i,\quad a(x,y)=\begin{cases} a_i(x,y) & \text{if}\ x,y\in X_i,\\ \bot:\|x\|\to\|y\| & \text{else}. \end{cases}$$ In particular, $\varnothing$ with its unique ${\mathcal{Q}}$-category structure is an initial object in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. - The equalizer of ${\mathcal{Q}}$-functors $f,g:(X,a)\to(Y,b)$ is formed as in ${{\bf Set}}$, by restriction of the structure of $(X,a)$. The object set of their coequalizer $(Z,c)$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is also formed as in ${{\bf Set}}$, so that $Z=Y/\sim$, with $\sim$ the least equivalence relation on $Y$ with $f(x)\sim g(x)$, $x\in X$. With $\pi:Y\to Z$ the projection, necessarily $\|\pi(y)\|_Z=\|y\|_Y$, and $c(\pi(y),\pi(y'))$ is the join of all $$b(y_n,y'_n)\circ b(y_{n-1},y'_{n-1})\circ\dots\circ b(y_2,y'_2)\circ b(y_1,y'_1),$$ where $\|y\|=\|y_1\|,\|y'_1\|=\|y_2\|,\dots,\|y'_{n-1}\|=\|y_n\|,\|y'_n\|=\|y'\|$ $(y_i,y'_i\in Y, n\geq 1)$. - The fully faithful left adjoint of ${{\mathcal{Q}}\text{-}{{\bf Cat}}}\to{{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$ provides a set $(X,\|\text{-}\|)$ over $\operatorname{ob}{\mathcal{Q}}$ with the discrete ${\mathcal{Q}}$-structure, given by $$a(x,y)=\begin{cases} 1_{\|x\|} & \text{if}\ x=y,\\ \bot:\|x\|\to\|y\| & \text{else}; \end{cases}$$ while the fully faithful right adjoint always takes $\top:\|x\|\to\|y\|$ as the hom-arrow, i.e., it chooses the indiscrete ${\mathcal{Q}}$-structure. The product of partial metric spaces $(X_i,a_i)$ $(i\in I)$ provides its carrier set $$X=\{((x_i)_{i\in I},s)\mid s\in[0,\infty],\ \forall i\in I(x_i\in X_i,\|x_i\|=s)\}$$ with the “sup metric”: $$a((x_i)_{i\in I},(y_i)_{i\in I})=\sup_{i\in I}a_i(x_i,y_i).$$ $[0,\infty]$ is terminal in ${{\bf ParMet}}$ when provided with the chaotic metric that makes all distances $0$, and it is a generator when provided with the discrete metric $d$: $$d(s,t)=\begin{cases} 0 & \text{if}\ s=t,\\ \infty & \text{else}. \end{cases}$$ Beyond small limits and colimits, ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ actually has all large-indexed limits and colimits that one can reasonably expect to exist. More precisely, recall that an ordinary category ${\mathcal{C}}$ with small hom-sets is (see [@Borger1990]) - hypercomplete if a diagram $D:{\mathcal{J}}\to{\mathcal{C}}$ has a limit in ${\mathcal{C}}$ whenever the limit of ${\mathcal{C}}(A,D-)$ exists in ${{\bf Set}}$ for all $A\in\operatorname{ob}{\mathcal{C}}$; equivalently: whenever, for every $A\in\operatorname{ob}{\mathcal{C}}$, the cones ${\Delta}A\to D$ in ${\mathcal{C}}$ may be labeled by a set; - totally cocomplete if a diagram $D:{\mathcal{J}}\to{\mathcal{C}}$ has a colimit in ${\mathcal{C}}$ whenever the colimit of ${\mathcal{C}}(A,D-)$ exists in ${{\bf Set}}$ for all $A\in\operatorname{ob}{\mathcal{C}}$; equivalently: whenever, for every $A\in\operatorname{ob}{\mathcal{C}}$, the connected components of $(A{\downarrow}D)$ may be labelled by a set. The dual notions are hypercocomplete and totally complete. It is well known (see [@Borger1990]) that - ${\mathcal{C}}$ is totally cocomplete if, and only if, ${\mathcal{C}}$ is total, i.e., if the Yoneda embedding ${\mathcal{C}}\to{{\bf Set}}^{{\mathcal{C}}^{{{\rm op}}}}$ has a left adjoint; - total cocompleteness implies hypercompleteness but not vice versa (with Ad[á]{}mek’s monadic category over graphs [@Adamek1977] providing a counterexample); - for a solid (=semi-topological [@Tholen1979]) functor ${\mathcal{A}}\to{\mathcal{X}}$, if ${\mathcal{X}}$ is hypercomplete or totally cocomplete, ${\mathcal{A}}$ has the corresponding property [@Tholen1980]; - in particular, every topological functor, every monadic functor over ${{\bf Set}}$, and every full reflective embedding is solid. It is also useful for us to recall [@Borger1990 Corollary 3.5]: \[total\_condition\] A cocomplete and cowellpowered category with small hom-sets and a generating set of objects is total. Since ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is topological over ${{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$ which, as a complete, cocomplete, wellpowered and cowellpowered category with a generating and a cogenerating set, is totally complete and totally cocomplete, we conclude: \[QCat\_total\] ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is totally complete and totally cocomplete and, in particular, hypercocomplete and hypercomplete. \[QCat\_generator\] Of course, we may also apply Proposition \[total\_condition\] directly to obtain Theorem \[QCat\_total\] since the left adjoint of ${{\mathcal{Q}}\text{-}{{\bf Cat}}}\to{{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$ sends a generating set of ${{\bf Set}}/\operatorname{ob}{\mathcal{Q}}$ to a generating set of ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$, and the right adjoint has the dual property. Explicitly then, denoting for every $s\in\operatorname{ob}{\mathcal{Q}}$ by $\{s\}$ the discrete ${\mathcal{Q}}$-category whose only object has extent $s$, we obtain the generating set $\{\{s\}\mid s\in\operatorname{ob}{\mathcal{Q}}\}$ for ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. Similarly, providing the disjoint unions $D_s=\{s\}+\operatorname{ob}{\mathcal{Q}}$ $(s\in\operatorname{ob}{\mathcal{Q}})$ with the identical extent functions and the indiscrete ${\mathcal{Q}}$-category structures, one obtains a cogenerating set in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. Limits and colimits in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ =========================================================== For ${\mathcal{Q}}$-categories $X=(X,a)$, $Y=(Y,b)$, a ${\mathcal{Q}}$-distributor [@Benabou1973] ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ (also called ${\mathcal{Q}}$-(bi)module [@Lawvere1973], ${\mathcal{Q}}$-profunctor) is a family of arrows ${\varphi}(x,y):\|x\|\to\|y\|$ $(x\in X,y\in Y)$ in ${\mathcal{Q}}$ such that $$b(y,y')\circ{\varphi}(x,y)\circ a(x',x)\leq{\varphi}(x',y')$$ for all $x,x'\in X$, $y,y'\in Y$. Its composite with $\psi:Y{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Z$ is given by $$(\psi\circ{\varphi})(x,z)={\bigvee}_{y\in Y}\psi(y,z)\circ{\varphi}(x,y).$$ Since the structure $a$ of a ${\mathcal{Q}}$-category $(X,a)$ is neutral with respect to this composition, we obtain the category $${{\mathcal{Q}}\text{-}{{\bf Dis}}}$$ of ${\mathcal{Q}}$-categories and their ${\mathcal{Q}}$-distributors which, with the local pointwise order $${\varphi}\leq{\varphi}'\iff\forall x,y:\ {\varphi}(x,y)\leq{\varphi}'(x,y),$$ is actually a quantaloid. Every ${\mathcal{Q}}$-functor $f:X\to Y$ gives rise to the ${\mathcal{Q}}$-distributors $f_{{\natural}}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ and $f^{{\natural}}:Y{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}X$ with $$f_{{\natural}}(x,y)=b(f(x),y)\quad\text{and}\quad f^{{\natural}}(y,x)=b(y,f(x))$$ $(x\in X,y\in Y)$. One has $f_{{\natural}}{\dashv}f^{{\natural}}$ in the 2-category ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$, and if one lets ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ inherit the order of ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$ via $$f\leq g\iff f^{{\natural}}\leq g^{{\natural}}\iff g_{{\natural}}\leq f_{{\natural}}\iff 1_{\|x\|}\leq b(f(x),g(x))\quad (x\in X),$$ then one obtains 2-functors $$(-)_{{\natural}}:({{\mathcal{Q}}\text{-}{{\bf Cat}}})^{{{\rm co}}}\to{{\mathcal{Q}}\text{-}{{\bf Dis}}},\quad (-)^{{\natural}}:({{\mathcal{Q}}\text{-}{{\bf Cat}}})^{{{\rm op}}}\to{{\mathcal{Q}}\text{-}{{\bf Dis}}}$$ which map objects identically; here “${{\rm op}}$” refers to the dualization of 1-cells and “${{\rm co}}$” to the dualization of 2-cells. - A ${\bf 2}$-distributor is an order ideal relation; that is, a relation ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ of ordered sets that behaves like a two-sided ideal w.r.t. the order: $$x'\leq x\ \&\ x{\varphi}y\ \&\ y\leq y'{}{\Longrightarrow}{}x'{\varphi}y'.$$ A $[0,\infty]$-distributor ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ introduces a distance function between metric spaces $(X,a)$, $(Y,b)$ that must satisfy $${\varphi}(x',y')\leq a(x',x)+{\varphi}(x,y)+a(y,y')$$ for all $x,x'\in X$, $y,y'\in Y$. - A ${{\sf D}}{\bf 2}$-distributor ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ is given by a ${\bf 2}$-distributor $A{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}B$ where $A=\{x\in X\mid x\leq x\}$, $B=\{y\in Y\mid y\leq y\}$ are the coreflections of $X$, $Y$, respectively. Likewise, a ${{\sf D}}[0,\infty]$-distributor ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ is given by a distributor of the metric coreflections of the partial metric spaces $X$ and $Y$. In our context ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$ plays only an auxiliary role for us in setting up the category $${{\mathcal{Q}}\text{-}{{\bf Chu}}}$$ whose objects are ${\mathcal{Q}}$-distributors and whose morphisms $(f,g):{\varphi}\to\psi$ are given by ${\mathcal{Q}}$-functors $f:(X,a)\to(Y,b)$, $g:(Z,c)\to(W,d)$ such that the diagram $$\label{Chu_transform_def_diagram} \bfig \square<700,500>[X`Y`W`Z;f_{{\natural}}`{\varphi}`\psi`g^{{\natural}}] \place(350,0)[\circ] \place(350,500)[\circ] \place(0,250)[\circ] \place(700,250)[\circ] \efig$$ commutes in ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$: $$\label{Chu_transform_def} \psi(f(x),z)={\varphi}(x,g(z))$$ for all $x\in X$, $z\in Z$. In particular, with ${\varphi}=a$, $\psi=b$ one obtains that the morphisms $(f,g):1_{(X,a)}\to 1_{(Y,b)}$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ are precisely the adjunctions $f{\dashv}g:(Y,b)\to(X,a)$ in the 2-category ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. With the order inherited from ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$, ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ is in fact a 2-category, and one has 2-functors $$\begin{array}{lll} \operatorname{dom}: & {{\mathcal{Q}}\text{-}{{\bf Chu}}}\to{{\mathcal{Q}}\text{-}{{\bf Cat}}}, & (f,g)\mapsto f,\\ \operatorname{cod}: & {{\mathcal{Q}}\text{-}{{\bf Chu}}}\to({{\mathcal{Q}}\text{-}{{\bf Cat}}})^{{{\rm op}}}, & (f,g)\mapsto g. \end{array}$$ In order for us to exhibit properties of ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$, it is convenient to describe ${\mathcal{Q}}$-Chu transforms, i.e., morphisms in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$, alternatively, with the help of presheaves, as follows. For every $s\in\operatorname{ob}{\mathcal{Q}}$, let $\{s\}$ denote the discrete ${\mathcal{Q}}$-category whose only object has extent $s$. For a ${\mathcal{Q}}$-category $X=(X,a)$, a ${\mathcal{Q}}$-presheaf ${\varphi}$ on $X$ of extent $\|{\varphi}\|=s$ is a ${\mathcal{Q}}$-distributor ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}\{s\}$. Hence, ${\varphi}$ is given by a family of ${\mathcal{Q}}$-morphisms ${\varphi}_x:\|x\|\to\|{\varphi}\|$ $(x\in X)$ with ${\varphi}_y\circ a(x,y)\leq{\varphi}_x$ $(x,y\in X)$. With $$[{\varphi},\psi]={\bigwedge}_{x\in X}\psi_x{\swarrow}{\varphi}_x,$$ ${{{\sf P}}X}$ becomes a ${\mathcal{Q}}$-category, and one has the Yoneda ${\mathcal{Q}}$-functor $${{\sf y}}_X={{\sf y}}:X\to{{{\sf P}}X},\quad x\mapsto(a(-,x):X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}\{\|x\|\}).$$ ${{\sf y}}$ is fully faithful, i.e., $[{{\sf y}}(x),{{\sf y}}(y)]=a(x,y)$ $(x,y\in X)$. The point of the formation of ${{{\sf P}}X}$ for us is as follows (see [@Heymans2010; @Shen2015]): \[cograph\_Kan\_adjunction\] The 2-functor $(-)^{{\natural}}:({{\mathcal{Q}}\text{-}{{\bf Cat}}})^{{{\rm op}}}\to{{\mathcal{Q}}\text{-}{{\bf Dis}}}$ has a left adjoint ${{\sf P}}$ which maps a ${\mathcal{Q}}$-distributor ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ to the ${\mathcal{Q}}$-functor $${\varphi}^:{{{\sf P}}Y}\to{{{\sf P}}X},\quad\psi\mapsto\psi\circ{\varphi};$$ hence, $$({\varphi}^(\psi))_x={\bigvee}_{y\in Y}\psi_y\circ{\varphi}(x,y)$$ for all $\psi\in{{{\sf P}}Y}$, $x\in X$. In particular, for a ${\mathcal{Q}}$-functor $f:X\to Y$ one has $$f^:=(f_{{\natural}})^:{{{\sf P}}Y}\to{{{\sf P}}X},\quad (f^(\psi))_x=\psi_{f(x)}.$$ Denoting by ${\widetilde{{\varphi}}}:Y\to{{{\sf P}}X}$ the transpose of ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ under the adjunction, determined by ${\widetilde{{\varphi}}}^{{\natural}}\circ({{\sf y}}_X)_{{\natural}}={\varphi}$, so that $({\widetilde{{\varphi}}}(y))_x={\varphi}(x,y)$ for all $x\in X$, $y\in Y$, we can now present ${\mathcal{Q}}$-Chu transforms, as follows: \[Chu\_transform\_presheaf\] A morphism $(f,g):{\varphi}\to\psi$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ (as in [(\[Chu\_transform\_def\_diagram\])]{}) may be equivalently presented as a commutative diagram $$\bfig \square/<-`<-`<-`<-/<700,500>[{{{\sf P}}X}`{{{\sf P}}Y}`W`Z;f^`{\widetilde{{\varphi}}}`{\widetilde{\psi}}`g] \efig$$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. Condition [(\[Chu\_transform\_def\])]{} then reads as $$({\widetilde{{\varphi}}}(g(z)))_x=({\widetilde{\psi}}(z))_{f(x)}$$ for all $x\in X$, $z\in Z$. For all $z\in Z$, $$f^({\widetilde{\psi}}(z))={\widetilde{\psi}}(z)\circ f_{{\natural}}={{\sf y}}_Z(z)\circ\psi\circ f_{{\natural}}={{\sf y}}_Z(z)\circ g^{{\natural}}\circ{\varphi}={{\sf y}}_Y(g(z))\circ{\varphi}={\widetilde{{\varphi}}}(g(z)).$$ \[dom\_initial\_lifting\] Let $D:{\mathcal{J}}\to{{\mathcal{Q}}\text{-}{{\bf Chu}}}$ be a diagram such that the colimit $W=\operatorname{colim}\operatorname{cod}D$ exists in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. Then any cone ${\gamma}:{\Delta}X\to\operatorname{dom}D$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ has a $\operatorname{dom}$-initial lifting ${\Gamma}:{\Delta}{\varphi}\to D$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ with ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}W$, $\operatorname{dom}{\Gamma}={\gamma}$. In particular, if ${\gamma}$ is a limit cone in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$, ${\Gamma}$ is a limit cone in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$. Considering the functors $${{\mathcal{Q}}\text{-}{{\bf Chu}}}\to^{\operatorname{dom}}{{\mathcal{Q}}\text{-}{{\bf Cat}}}\to^{(-)_{{\natural}}}{{\mathcal{Q}}\text{-}{{\bf Dis}}},\quad{{\mathcal{Q}}\text{-}{{\bf Chu}}}\to^{\operatorname{cod}}({{\mathcal{Q}}\text{-}{{\bf Cat}}})^{{{\rm op}}}\to^{(-)^{{\natural}}}{{\mathcal{Q}}\text{-}{{\bf Dis}}},$$ one has the natural transformation $${\kappa}:(\operatorname{dom}(-))_{{\natural}}{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}(\operatorname{cod}(-))^{{\natural}},\quad{\kappa}_{{\varphi}}:={\varphi}\ ({\varphi}\in\operatorname{ob}{{\mathcal{Q}}\text{-}{{\bf Chu}}}).$$ By the adjunction of Proposition \[cograph\_Kan\_adjunction\], ${\kappa}D:(\operatorname{dom}D)_{{\natural}}{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}(\operatorname{cod}D)^{{\natural}}$ corresponds to a natural transformation ${\widetilde{{\kappa}D}}:\operatorname{cod}D\to{{\sf P}}(\operatorname{dom}D)_{{\natural}}$, and the given cone ${\gamma}$ gives a cocone ${\gamma}^:{{\sf P}}(\operatorname{dom}D)_{{\natural}}\to{\Delta}{{{\sf P}}X}$. Forming the colimit cocone ${\delta}:\operatorname{cod}D\to{\Delta}W$ one now obtains a unique ${\mathcal{Q}}$-functor ${\widetilde{{\varphi}}}:W\to{{{\sf P}}X}$ making $$\bfig \square/<-`<--`<-`<-/<1000,500>[{\Delta}{{{\sf P}}X}`{{\sf P}}(\operatorname{dom}D)_{{\natural}}`{\Delta}W`\operatorname{cod}D;{\gamma}^`{\Delta}{\widetilde{{\varphi}}}`{\widetilde{{\kappa}D}}`{\delta}] \efig$$ commute in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ or, equivalently, making $$\bfig \square<1000,500>[{\Delta}X`(\operatorname{dom}D)_{{\natural}}`{\Delta}W`(\operatorname{cod}D)^{{\natural}};{\gamma}_{{\natural}}`{\Delta}{\varphi}`{\kappa}D`{\delta}^{{\natural}}] \place(500,0)[\circ] \place(500,500)[\circ] \place(0,250)[\circ] \place(1000,250)[\circ] \efig$$ commute in ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$, with ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}W$ corresponding to $\widetilde{{\varphi}}$. In other words, we have a cone ${\Gamma}:{\Delta}{\varphi}\to D$ with $\operatorname{dom}{\varphi}=X$, $\operatorname{dom}{\Gamma}={\gamma}$, namely ${\Gamma}=({\gamma},{\delta})$. Given a cone $\Theta:{\Delta}\psi\to D$ with $\psi:Y{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Z$ in ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$ and a ${\mathcal{Q}}$-functor $f:Y\to X$ with ${\gamma}\cdot{\Delta}f={\epsilon}:=\operatorname{dom}\Theta$, the cocone $\vartheta:=\operatorname{cod}\Theta:\operatorname{cod}D\to{\Delta}Z$ corresponds to a unique ${\mathcal{Q}}$-functor $g:W\to Z$ with ${\Delta}g\cdot{\delta}=\vartheta$ by the colimit property. As the diagram $$\bfig \square\|arra\|/<-`<-`<-`<-/<1200,600>[{\Delta}{{{\sf P}}X}`{{\sf P}}(\operatorname{dom}D)_{{\natural}}`{\Delta}W`\operatorname{cod}D;{\gamma}^``{\widetilde{{\kappa}D}}`{\delta}] \place(70,350)[\mbox{\scriptsize${\Delta}{\widetilde{{\varphi}}}$}] \morphism(0,600)<-500,-500>[{\Delta}{{{\sf P}}X}`{\Delta}{{{\sf P}}Y};] \place(-300,400)[\mbox{\scriptsize${\Delta}f^$}] \morphism(1200,600)\|b\|<-1700,-500>[{{\sf P}}(\operatorname{dom}D)_{{\natural}}`{\Delta}{{{\sf P}}Y};{\epsilon}^] \morphism<-500,-500>[{\Delta}W`{\Delta}Z;] \place(-300,-200)[\mbox{\scriptsize${\Delta}g$}] \morphism(1200,0)\|b\|<-1700,-500>[\operatorname{cod}D`{\Delta}Z;\vartheta] \morphism(-500,-500)\|l\|<0,600>[{\Delta}Z`{\Delta}{{{\sf P}}Y};{\Delta}{\widetilde{\psi}}] \efig$$ shows, the colimit property of $W$ also guarantees $f^{\widetilde{{\varphi}}}={\widetilde{\psi}}g$ (with ${\widetilde{\psi}}$ corresponding to $\psi$) which, by Corollary \[Chu\_transform\_presheaf\], means that $(f,g):\psi\to{\varphi}$ is the only morphism in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ with $\operatorname{dom}(f,g)=f$ and ${\Gamma}\cdot{\Delta}(f,g)=\Theta$. $\operatorname{dom}:{{\mathcal{Q}}\text{-}{{\bf Chu}}}\to{{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is small-topological; in particular, $\operatorname{dom}$ is a fibration with a fully faithful right adjoint which embeds ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ into ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ as a full reflective subcategory. $\operatorname{cod}:{{\mathcal{Q}}\text{-}{{\bf Chu}}}\to({{\mathcal{Q}}\text{-}{{\bf Cat}}})^{{{\rm op}}}$ has the dual properties. With the existence of small colimits guaranteed by Corollary \[QCat\_complete\], $\operatorname{dom}$-initial liftings to small $\operatorname{dom}$-structured cones exist by Theorem \[dom\_initial\_lifting\]. For the assertion on $\operatorname{cod}$, first observe that every ${\mathcal{Q}}$-category $X=(X,a)$ gives rise to the ${\mathcal{Q}}^{{{\rm op}}}$-category $X^{{{\rm op}}}=(X,a^{\circ})$, where $a^{\circ}(x,y)=a(y,x)$ $(x,y\in X)$. With the commutative diagram $$\bfig \square<1200,500>[({{\mathcal{Q}}\text{-}{{\bf Chu}}})^{{{\rm op}}}`{\mathcal{Q}}^{{{\rm op}}}\text{-}{{\bf Chu}}`{{\mathcal{Q}}\text{-}{{\bf Cat}}}`{\mathcal{Q}}^{{{\rm op}}}\text{-}{{\bf Cat}};(-)^{{{\rm op}}}`\operatorname{cod}^{{{\rm op}}}`\operatorname{dom}`(-)^{{{\rm op}}}] \efig$$ one sees that, up to functorial isomorphisms, $\operatorname{cod}^{{{\rm op}}}:({{\mathcal{Q}}\text{-}{{\bf Chu}}})^{{{\rm op}}}\to{{\mathcal{Q}}\text{-}{{\bf Cat}}}$ coincides with the small-topological functor $\operatorname{dom}:{\mathcal{Q}}^{{{\rm op}}}\text{-}{{\bf Chu}}\to{\mathcal{Q}}^{{{\rm op}}}\text{-}{{\bf Cat}}$. \[QChu\_complete\] ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ is complete and cocomplete, all small limits and colimits in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ are preserved by both $\operatorname{dom}$ and $\operatorname{cod}$. The $\operatorname{dom}$-initial lifting of a $\operatorname{dom}$-structured limit cone in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ is a limit cone in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$, which is trivially preserved. Having a right adjoint, $\operatorname{dom}$ also preserves all colimits. - Let us describe (small) products in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ explicitly: Given a family of ${\mathcal{Q}}$-distributors ${\varphi}_i:X_i{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y_i$ $(i\in I)$, one first forms the product $X$ of the ${\mathcal{Q}}$-categories $X_i=(X_i,a_i)$ as in Remark \[QCat\_limit\](1) with projections $p_i$ and the coproduct of the $Y_i=(Y_i,b_i)$ as in Remark \[QCat\_limit\](2) with injections $s_i$ $(i\in I)$. The transposes ${\widetilde{{\varphi}}}_i$ then determines a ${\mathcal{Q}}$-functor ${\widetilde{{\varphi}}}$ making the left square of $$\bfig \square/<-`<-`<-`<-/<700,500>[{{{\sf P}}X}`{{{\sf P}}X}_i`Y`Y_i;p_i^`{\widetilde{{\varphi}}}`{\widetilde{{\varphi}}}_i`s_i] \square(1500,0)<700,500>[X`X_i`Y`Y_i;(p_i)_{{\natural}}`{\varphi}`{\varphi}_i`(s_i)^{{\natural}}] \place(1500,250)[\circ] \place(2200,250)[\circ] \place(1850,0)[\circ] \place(1850,500)[\circ] \efig$$ commutative, while the right square exhibits ${\varphi}$ as a product of $({\varphi}_i)_{i\in I}$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ with projections $(p_i,s_i)$, $(i\in I)$; explicitly, $${\varphi}(x,y)=({\widetilde{{\varphi}}}(y))_x=({\widetilde{{\varphi}}}_i(y)\circ(p_i)_{{\natural}})_x=({\widetilde{{\varphi}}}_i(y))_{p_i(x)}={\varphi}_i(x_i,y)$$ for $x=((x_i)_{i\in I},q)$ in $X$ and $y=s_i(y)$ in $Y_i$, $i\in I$. - The coproduct of ${\varphi}_i:X_i{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y_i$ $(i\in I)$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ is formed like the product, except that the roles of domain and codomain need to be interchanged. Hence, one forms the coproduct $X$ of $(X_i)_{i\in I}$ and the product $Y$ of $(Y_i)_{i\in I}$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ and obtains the coproduct ${\varphi}:X{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}Y$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ as in $$\bfig \square<700,500>[X_i`X`Y_i`Y;(s_i)_{{\natural}}`{\varphi}_i`{\varphi}`({\varphi}_i)^{{\natural}}] \place(0,250)[\circ] \place(700,250)[\circ] \place(350,0)[\circ] \place(350,500)[\circ] \efig$$ so that ${\varphi}(x,y)={\varphi}_i(x,y_i)$ for $y=((y_i)_{i\in I},q)$ in $Y$ and $x=s_i(x)$ in $X_i$, $i\in I$. - The equalizer of $(f,g),({\overline{f}},{\overline{g}}):{\varphi}\to\psi$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ is obtained by forming the equalizer and coequalizer $$U\to^i X\two^f_{{\overline{f}}}Y\quad\text{and}\quad W\two^g_{{\overline{g}}}Z\to^p V$$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$, respectively. With ${\widetilde{\chi}}:V\to{{{\sf P}}U}$ obtained from the coequalizer property making $$\bfig \square/<-`<-`<-`<-/<700,500>[{{{\sf P}}U}`{{{\sf P}}X}`V`W;i^`{\widetilde{\chi}}`{\widetilde{{\varphi}}}`p] \efig$$ commutative, Theorem \[dom\_initial\_lifting\] guarantees that $$\chi\to^{(i,p)}{\varphi}\two^{(f,g)}_{({\overline{f}},{\overline{g}})}\psi$$ is an equalizer diagram in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$, where $$\chi(x,p(w))=({\widetilde{\chi}}(p(w))_x=(i^({\widetilde{{\varphi}}}(w)))_x=({\widetilde{{\varphi}}}(w)\circ i_{{\natural}})_x={\varphi}(i(x),w)$$ for all $x\in U$, $w\in W$. - Coequalizers in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ are formed like equalizers, except that the roles of domain and codomain need to be interchanged. We will now strengthen Corollary \[QChu\_complete\] and show total completeness and total cocompleteness of ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ with the help of Proposition \[total\_condition\]. To that end, let us observe that, since the limit and colimit preserving functors $\operatorname{dom}$ and $\operatorname{cod}$ must in particular preserve both monomorphisms and epimorphisms, a monomorphism $(f,g):{\varphi}\to\psi$ in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ must be given by a monomorphism $f$ and an epimorphism $g$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$, i.e., by an injective ${\mathcal{Q}}$-functor $f$ and a surjective ${\mathcal{Q}}$-functor $g$. Consequently, ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ is wellpowered, and so is its dual $({{\mathcal{Q}}\text{-}{{\bf Chu}}})^{{{\rm op}}}\cong{\mathcal{Q}}^{{{\rm op}}}\text{-}{{\bf Chu}}$. The main point is therefore for us to prove: \[QChu\_generator\] ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ contains a generating set of objects and, consequently, also a cogenerating set. With the notations explained below, we show that $$\{\eta_s:\varnothing{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}D_s\mid s\in\operatorname{ob}{\mathcal{Q}}\}\cup\{{\lambda}_t:\{t\}{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}{\widehat{C}}\mid t\in\operatorname{ob}{\mathcal{Q}}\}$$ is generating in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$. Here $D_s$ belongs to a generating set of ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ (see Remark \[QCat\_generator\]), and $$C=\coprod\limits_{t\in\operatorname{ob}{\mathcal{Q}}}{{\sf P}}\{t\}$$ is a coproduct in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ (see Remark \[QCat\_limit\](2)) of the presheaf ${\mathcal{Q}}$-categories of the singleton ${\mathcal{Q}}$-categories $\{t\}$ (see Proposition \[cograph\_Kan\_adjunction\]). From $C$ one obtains ${\widehat{C}}$ by adding an isomorphic copy of each object in $C$, which may be easily explained for a ${\mathcal{Q}}$-category $(X,a)$: simply provide the set ${\widehat{X}}:=X\times\{1,2\}$ with the structure $$\|(x,i)\|_{{\widehat{X}}}=\|x\|_{X}\quad\text{and}\quad{\widehat{a}}((x,i),(y,j))=a(x,y)$$ for all $x,y\in X$, $i,j\in\{1,2\}$. Noting that the objects of ${{\sf P}}\{t\}$ are simply ${\mathcal{Q}}$-arrows with domain $t$, we now define ${\lambda}_t:\{t\}{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}{\widehat{C}}$ by $${\lambda}_t(u,i)=\begin{cases} u & \text{if}\ \operatorname{dom}u=t,\\ \bot & \text{else} \end{cases}$$ for $i\in\{1,2\}$ and every object $t$ and arrow $u$ in ${\mathcal{Q}}$. For another element $(v,j)$ in ${\widehat{C}}$, if $\operatorname{dom}v=\operatorname{dom}u=t$ one then has $$[(u,i),(v,j)]\circ{\lambda}_t(u,i)=(v{\swarrow}u)\circ u\leq v={\lambda}_t(v,j),$$ and in other cases this inequality holds trivially. Hence, ${\lambda}_t$ is indeed a ${\mathcal{Q}}$-distributor. Let us now consider ${\mathcal{Q}}$-Chu transforms $(f,g)\neq({\overline{f}},{\overline{g}}):{\varphi}\to\psi$ as in $$\bfig \square\|alra\|/@{->}@<3pt>`->`->`@{->}@<3pt>/<800,500>[(X,a)`(Y,b)`(W,d)`(Z,c);f_{{\natural}}`{\varphi}`\psi`g^{{\natural}}] \morphism(0,500)\|b\|/@{->}@<-3pt>/<800,0>[(X,a)`(Y,b);{\overline{f}}_{{\natural}}] \morphism(0,0)\|b\|/@{->}@<-3pt>/<800,0>[(W,d)`(Z,c);{\overline{g}}^{{\natural}}] \place(0,250)[\circ] \place(800,250)[\circ] \place(400,-30)[\circ] \place(400,30)[\circ] \place(400,470)[\circ] \place(400,530)[\circ] \efig$$ [Case 1]{}: $X=\varnothing$ is the initial object of ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$ (and ${{\mathcal{Q}}\text{-}{{\bf Dis}}}$). Then $g\neq{\overline{g}}$, and we find $s\in\operatorname{ob}{\mathcal{Q}}$ and $h:W\to D_s$ with $hg\neq h{\overline{g}}$ in ${{\mathcal{Q}}\text{-}{{\bf Cat}}}$. Consequently, $(1_{\varnothing},h):\eta_s\to{\varphi}$ satisfies $(f,g)(1_{\varnothing},h)\neq({\overline{f}},{\overline{g}})(1_{\varnothing},h)$. [Case 2]{}: $f\neq{\overline{f}}$, so that $f(x_0)\neq{\overline{f}}(x_0)$ for some $x_0\in X$. Then, for $t:=\|x_0\|$, $e:\{t\}\to X$, $\|x_0\|\mapsto x_0$, is a ${\mathcal{Q}}$-functor with $fe\neq{\overline{f}}e$, and it suffices to show that $$h:W\to{\widehat{C}},\quad w\mapsto({\varphi}(x_0,w),1)$$ is a ${\mathcal{Q}}$-functor making $(e,h):{\lambda}_t\to{\varphi}$ a ${\mathcal{Q}}$-Chu transform. Indeed, $$\begin{aligned} &d(w,w')\leq{\varphi}(x_0,w'){\swarrow}{\varphi}(x_0,w)=[h(w),h(w')],\\ &{\lambda}_t(h(w))={\varphi}(x_0,w)={\varphi}(e(t),w)\end{aligned}$$ for all $w,w'\in W$. [Case 3]{}: $X\neq\varnothing$ and $g\neq{\overline{g}}$. Then $g(z_0)\neq{\overline{g}}(z_0)$ for some $z_0\in Z$, and with any fixed $x_0\in X$ we may alter the previous definition of $h:W\to{\widehat{C}}$ by $$h(w):=\begin{cases} ({\varphi}(x_0,w),2) & \text{if}\ w={\overline{g}}(z_0),\\ ({\varphi}(x_0,w),1) & \text{else}. \end{cases}$$ The verification for $h$ to be a ${\mathcal{Q}}$-functor and $(e,h):{\lambda}_t\to{\varphi}$ a ${\mathcal{Q}}$-Chu transform remain intact, and since $hg\neq h{\overline{g}}$, the proof is complete. A generating set in ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ may be alternatively given by $$\{{\lambda}_{\varnothing}:\varnothing{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}{\widehat{C}}\}\cup\{{\lambda}_t:\{t\}{{\to\hspace{-3.1ex}{\circ}\hspace{1.9ex}}}{\widehat{C}}\mid t\in\operatorname{ob}{\mathcal{Q}}\},$$ so that in Case 1 one may proceed exactly as in Case 3 only by replacing ${\varphi}(x_0,w)$ with $\top:q\to\|w\|$ for any fixed $q\in\operatorname{ob}{\mathcal{Q}}$. With Theorem \[QChu\_generator\] we obtain: ${{\mathcal{Q}}\text{-}{{\bf Chu}}}$ is totally complete and totally cocomplete and, in particular, hypercocomplete and hypercomplete. [10]{} J. Ad[á]{}mek. Colimits of algebras revisited. , 17:433–450, 1977. J. Ad[á]{}mek, H. Herrlich, and G. E. Strecker. . Wiley, New York, 1990. M. Barr. $$-[Autonomous]{} categories and linear logic. , 1:159–178, 1991. J. Barwise and J. Seligman. , volume 44 of [Cambridge Tracts in Theoretical Computer Science]{}. Cambridge University Press, Cambridge, 1997. J. B[é]{}nabou. Les distributeurs. Universit[é]{} Catholique de Louvain, Institute de Mat[é]{}matique Pure et Appliqu[é]{}e, Rapport no. 33, 1973. J. B[' e]{}nabou. Distributors at work. Lecture notes of a course given at TU Darmstadt, 2000. F. Borceux. , volume 50 of [Encyclopedia of Mathematics and its Applications]{}. Cambridge University Press, Cambridge, 1994. R. B[ö]{}rger and W. Tholen. Cantors [Diagonalprinzip]{} f[ü]{}r [Kategorien]{}. , 160(2):135–138, 1978. R. B[ö]{}rger and W. Tholen. Total categories and solid functors. , 42:213–229, 1990. B. Ganter. Relational [Galois]{} connections. In S. O. Kuznetsov and S. Schmidt, editors, [Formal Concept Analysis]{}, volume 4390 of [Lecture Notes in Computer Science]{}, pages 1–17. Springer, Berlin-Heidelberg, 2007. E. Giuli and W. Tholen. A topologist’s view of [Chu]{} spaces. , 15(5-6):573–598, 2007. M. Grandis. Weak subobjects and the epi-monic completion of a category. , 154(1-3):193–212, 2000. M. Grandis. On the monad of proper factorisation systems in categories. , 171(1):17–26, 2002. H. Heymans. . PhD thesis, Universiteit Antwerpen, Belgium, 2010. D. Hofmann, G. J. Seal, and W. Tholen, editors. , volume 153 of [Encyclopedia of Mathematics and its Applications]{}. Cambridge University Press, Cambridge, 2014. U. H[ö]{}hle and T. Kubiak. A non-commutative and non-idempotent theory of quantale sets. , 166:1–43, 2011. G. M. Kelly. , volume 64 of [ London Mathematical Society Lecture Note Series]{}. Cambridge University Press, Cambridge, 1982. F. W. Lawvere. Metric spaces, generalized logic and closed categories. , XLIII:135–166, 1973. S. G. Matthews. Partial metric topology. , 728(1):183–197, 1994. V. Pratt. Chu spaces and their interpretation as concurrent objects. In J. Leeuwen, editor, [Computer Science Today]{}, volume 1000 of [Lecture Notes in Computer Science]{}, pages 392–405. Springer, Berlin-Heidelberg, 1995. Q. Pu and D. Zhang. Preordered sets valued in a [GL]{}-monoid. , 187(1):1–32, 2012. K. I. Rosenthal. , volume 348 of [Pitman Research Notes in Mathematics Series]{}. Longman, Harlow, 1996. L. Shen and W. Tholen. Topological categories, quantaloids and [Isbell]{} adjunctions. , 2015. R. Street and R. F. C. Walters. Yoneda structures on 2-categories. , 50(2):350–379, 1978. I. Stubbe. Categorical structures enriched in a quantaloid: categories, distributors and functors. , 14(1):1–45, 2005. I. Stubbe. Categorical structures enriched in a quantaloid: tensored and cotensored categories. , 16(14):283–306, 2006. I. Stubbe. An introduction to quantaloid-enriched categories. , 256(0):95–116, 2014. Special Issue on Enriched Category Theory and Related Topics (Selected papers from the 33rd Linz Seminar on Fuzzy Set Theory, 2012). Y. Tao, H. Lai, and D. Zhang. Quantale-valued preorders: Globalization and cocompleteness. , 256(0):236–251, 2014. Special Issue on Enriched Category Theory and Related Topics (Selected papers from the 33rd Linz Seminar on Fuzzy Set Theory, 2012). W. Tholen. Semi-topological functors [I]{}. , 15(1):53–73, 1979. W. Tholen. Note on total categories. , 21:169–173, 1980. R. F. C. Walters. Sheaves and [Cauchy]{}-complete categories. , 22(3):283–286, 1981. [^1]: Partial financial assistance by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. [^2]: Here our terminology naturally extends Lawvere’s notion of metric and is synonymous with “generalized partial metric” as used by Pu-Zhang [@Pu2012] who dropped finiteness ($a(x,y)<\infty$), symmetry ($a(x,y)=a(y,x)$) and separation ($a(x,x)=a(x,y)=a(y,y)\iff x=y$) from the requirement for the notion of “partial metric” as originally introduced by Matthews [@Matthews1994].
--- abstract: \| We consider several models whose motivation arises from statistical mechanics. We begin by investigating some families of distributions of translation invariant subgraphs of some Cayley graphs, and in particular subgraphs of the square lattice. We then discuss some properties of the Spin-Glass model in that lattice. We continue in describing some properties of the Spin-Glass models in some other graphs. The last two parts of this work are devoted to the understanding of two dynamical processes on graphs. The first one is the well known zero-temperature Glauber dynamics on some families of graphs. The second dynamics, which we call the Loop Dynamics, is a natural generalization of the zero-temperature Glauber dynamics, which appears to have some interesting properties. We analyzed some of its properties for planar lattices, though the exact same techniques are applied for larger families of graphs as well. author: - 'Ran J. Tessler' bibliography: - 'bibli.bib' title: Geometry and Dynamics in Zero Temperature Statistical Mechanics Models --- To Yaki and Sara, my grandparents, and to Boris. Introduction ============ Background ---------- $~$\ This paper is divided into four main theoretical sections. The remaining parts of this paper are the introduction, the Hebrew abstract and most importantly the dedications. Each of the main theoretical sections contains subsections of background, notations and formulation of results, while the rest of each section is devoted to proving these results. The first section deals with properties of distributions of translation invariant families of subgraphs of amenable Cayley graphs. In particular we investigate some properties of such distributions in the square lattice. These results, in addition to some independent interest are used in proofs of some results from later sections. The Second part considers the Edwards-Anderson Ising spin glass model on the planar square lattice and other families of graphs. For the square lattice spin glass we obtain new results about the geometry of its ground states (to be defined below). We then continue in exploring the Edwards-Anderson Ising spin glass model on other graphs. We show that for wide families of graphs we have exactly one ground state, while for a large family of other graphs we have an infinite amount. The last two part parts are devoted to exploring two dynamical processes. We first consider zero-temperature Glauber dynamics on some families of graphs. Our main result in that section is that on regular trees with an even degree this dynamics does not freeze. This is a complement to a known result that for regular trees with an odd degree the dynamics does freeze. We then define and discuss a new dynamics $\emph{the Loop dynamics}$ which is a natural generalization of the zero-temperature Glauber dynamics. It is more difficult to show that this process is well defined, but it appears to be worth the effort, as weak limits of this dynamical process happen to have some interesting properties. The first two parts and the forth part are taken from a joint work with Noam Berger. The third is a joint work with Oren Louidor. Ergodic Theory Background and the Mass Transport Principle ---------------------------------------------------------- Let $(X,\mathfrak{B},\mu)$ be a finite measure space, and $T:X\rightarrow X$ a measurable map. 1. We say that $T$ is $\emph{measure~preserving}$ if for every $B\in\mathfrak{B}$ $$\mu(B)=\mu(T^{-1}(B))$$. 2. We say that a measure preserving map $T$ is $\emph{ergodic}$ if the only measurable sets $B\in\mathfrak{B}$ such that $$\mu(T^{-1}(B)\triangle B)=0$$ are either sets of measure zero or sets with complement $X\backslash B$ of measure zero. Let $(X,\mathfrak{B},\mu)$ be a finite measure space, and $G$ a discrete group acting on it, such that $\forall g \in G$ the action of $g$ on $X$ is a measurable map. 1. We say that $\mu$ is $\emph{invariant}~\emph{under}~\emph{the}~\emph{action}~\emph{of}~G$ if for every $B\in\mathfrak{B}$ and every $g \in G$ $$\mu(B)=\mu(g\cdot {B})$$. 2. We say that the action of $G$ is $\emph{ergodic}$ if the only measurable sets $B\in\mathfrak{B}$ such that $$\mu(g\cdot {B}\triangle B)=0$$ for every $g\in G$, are either sets of measure zero or sets with complement $X\backslash B$ of measure zero. $~$\ $\mathbf{The ~ Mass ~ Transport ~ Principle}$\ Let $G=(V,E)$ be a Cayley graph of some discrete group $\Gamma$ or more generally a transitive graph whose automorphism group $\Gamma$ is unimodular. Let $\mu$ be a $\Gamma$-invariant measure on $\{0,1\}^{E}$. A $\emph{mass}$ function $m=m(\omega;x,y)$ is a $\mu$-measurable function from $\{0,1\}^{E}\times{V}\times{V}$ to $\mathbb{R}$. We will also assume that it is invariant under the diagonal action of $\Gamma$, i.e. $m(\omega;x,y)=m(\gamma \omega;\gamma x,\gamma y)$, for any $\gamma\in\Gamma,\omega\in\{0,1\}^{E},x,y\in V$. A mass function should be thought as the mass transferred from $x$ to $y$ conditioned on the configuration $\omega$. We denote by $M(x,y)=\mathbb{E}_\mu m(\omega;x,y)$. The Mass-Transport principle ($\emph{MT}$) says that $$\label{eq:MT} \forall x\in V ~ \Sigma_{y\in V}M(x,y)=\Sigma_{y\in V}M(y,x)$$ In particular the left hand side is finite iff the right hand side is finite. A good reference for this principle can be found in [@lyonsperes] Translation Invariant Measures of Subgraphs in the planar lattice and other graphs {#sec:inv subgraphs} ================================================================================== $~$\ Background {#sec:Background-inv subgraphs} ---------- $~$\ This section explores some properties of translation invariant measures of subgraphs of some families of graphs. Our main interest is the planar lattice, where we investigate some families of translation invariant paths and forests and show both qualitative and quantitative results. These objects appear naturally in probability (e.g. [@zerner]), graph theory, mathematical physics and more. Some of these results will be used later on in the section on spin glasses. Preliminaries {#Prelim-inv subgraphs} ------------- This subsection is mainly devoted to definitions that of objects that we shall study throughout the section. \[def:direction\] Given a simple, two-sided-infinite path $P=(V(P),E(P))$, a $\emph{direction}~t:V(P)\rightarrow\mathbb{Z}$ is a bijective function from the path’s vertices to $\mathbb{Z}$, such that two vertices are neighbors in the path iff their images are consecutive numbers in $\mathbb{Z}$. Given such a direction, and a vertex $v\in V(P)$, we define the $\emph{past}$ of $v$ to be the set of vertices whose $t$-value is smaller than that of $v$, i.e. the set $\{u\in V(P)\| t(u)<t(v)\}$. Similarly one defines the $\emph{future}$ of a vertex. A $\emph{ray}$ is a half-infinite straight line segment which start at some point. We have the following trivial observation. \[obs:past is transitive\] Let $u,v,w$ be three vertices of a path with a given direction. Then if $v$ is in the past of $u$, and $w$ is in the past of $v$ then $w$ is in the past of $u$. Next we define the notions of a $\emph{cross}$ and a $\emph{snail}$, that use us for obtaining quantitative bounds on biinfinite simple lattice paths which are in the support of some translation invariant measure of paths. \[def:crosses\] Let $P$ be a bi-infinite path in the standard lattice graph $\mathbb{Z}^{2}$. Assume that $P$ intersects any translation, by integer shifts (in both directions),of the positive $x$-axis, negative $x$-axis, positive $y$-axis and negative $y$-axis infinitely many times. Let $p=(a,b)$ be a (lattice) vertex of $P$ and $n$ be a positive integer. We consider the intersection (lattice) points of the path with the half-line $\{(s,b)\| s > a\}$. We order these points by their distance from $p$ (or their $x$-coordinate). Denote by ${p_n}^{x_+}$ the $n^{th}$ closest (to $p$) point out of the above intersection points, and similarly define ${p_n}^{x_-}$, ${p_n}^{y_+}$, ${p_n}^{y_-}$. We define the $\emph{n}^{th}~\emph{cross}~\emph{of}~p$ as the union of two line segment, the one which connects ${p_n}^{x_+}$ and ${p_n}^{x_-}$, and the one which connects ${p_n}^{y_+}$ and ${p_n}^{y_-}$. We define the $\emph{n}^{th}~\emph{snail}~\emph{of}~p$, as the smallest (with respect to inclusions) connected subgraph of the path which contains $p$ and all the path points from the $n^{th}$-cross of $p$. The $\emph{length}$ of a snail is defined as the number of path edges included in it. We finish this subsection with some definitions concerning infinite trees and forests. Let $T$ be an infinite tree of bounded degree. We say that it is $\emph{single-infinite}$ if it doesn’t have two disjoint one-way infinite paths. We say it is $\emph{bi-infinite}$ if it has two disjoint one-way-infinite paths, but not three. Otherwise we say it is $\emph{multi-infinite}$. In a single-infinite tree, any vertex $v$ has a single one-way infinite path which starts at it. We call this path the $\emph{stem}$ of $v$, and denote it by $S(v)$. We denote by $R(v)$, and call it the $emph{roots}$ of $v$, the unique connected component of $T\backslash S(v)$ which contains $v$. Note that if $u$ is a vertex in $R(v)$ then $v$ is a vertex in $S(u)$. In a bi-infinite tree, there is a single two-way infinite path. We call the $\emph{path}$ of the tree $P(T)$, or simply $P$. In a multi-infinite tree, a vertex $v$ with the property that $T\backslash \{v\}$ has at least three infinite components is called an $\emph{encounter point}$. We denote by $\delta_T(x,y)$ the tree distance between the vertices $x,y$. We shall sometimes write only $\delta$, if it is clear to which tree we are referring. The graph distance function will be denoted by $d(\cdot,\cdot)$ \[rmk:rmk forests\] It follows from the famous König’s Lemma (e.g. [@konig]), that any infinite tree of bounded degree contains at least one one-way infinite path. If it has no two such disjoint paths, then unless the tree itself is a one-way infinite path (what is impossible in the translation invariant scenario), it has no unique or canonic one-way infinite path. On the other hand each vertex has a single such path starting from it. If the tree is bi-infinite is has exactly one bi-infinite path. A multi-infinite tree always has at least one encounter-point. Moreover, it is easy to see, following [@burton_keane], that the number of encounter points in some finite domain in the graph is never higher than the maximal number of disjoint one-way infinite paths which intersect the boundary of the domain (although we can decompose differently these paths, the maximal number of paths is finite and well defined). Main Results {#sec:Main Results-inv subgraphs} ------------ $~$\ In the subsection that handles translation invariant measure of bi-infinite paths in the planar lattice we prove two main results. The first one, which concerns intersection of paths from the support of the measure with rays is the following theorem: \[thm:past intersects a ray infinitely many times\] Let $\mu$ be a measure of infinite paths in the $\mathbb{Z}^{2}$ lattice. Assume that $\mu$ is ergodic with respect to the group of $\mathbb{Z}^{2}$-translations and invariant under the group of rotations of the plane around the origin by integer multiples of $\pi/2$. Let $v$ be a vertex of the path, then the past of $v$ intersects the ray which is parallel to the $y$-axis, starts at $v$ and whose direction is upwards (intersects the lower half lattice only finitely many times) infinitely many times. Similar propositions hold for the other rays and the future as well. The second theorem shows a quantitative result about the ’growth’ of paths in the support of a translation invariant measure. \[thm:lim inf snail is at least quadratic\] Let $\mu$ be a translation invariant measure of bi-infinite paths in the $\mathbb{Z}^{2}$-lattice. Let $P$ be a path in the support of the measure, and $p$ be a lattice point in the path. then there is a positive $c = c(p,P)$ such that for infinitely many values of $n\in \mathbb{N}$, the length of the $n^{th}$-snail of $p$ is at least $cn^{2}$. Next, we consider measures of forests in planar lattices, lattices in higher dimensions and in general amenable Cayley graphs. We show that the infinite connected components must be of some given shape (i.e. have no more than two infinite disjoint paths as subgraphs) and we analyze expected properties and sizes of subgraphs in these trees. The main result of that part is the theorem below: \[thm:single-infinite at least boundary sized or nlogn\] Let $\mu$ be an ergodic , translation-invariant and rotation-invariant (as described above) measure of single-infinite trees on the lattice $\mathbb{Z}^{2}$. Then there exist a positive constant $\rho$, such that for any vertex $v$, and an increasing sequence of boxes, $\{B_n\}_{n=1}^{\infty}$, centered at $v$, the following holds: Let $e_n$ be the number of edges of the tree which lay in the $n^{th}$ box and lay on a simple path in the tree which connects two boundary points of the box, then for all large enough $n$, $\mathbb{E}_\mu e_n > \rho nlog(n)$. Translation Invariant Measures of Infinite $\mathbb{Z}^{2}$ Paths {#sec:infinite paths-inv subgraphs} ----------------------------------------------------------------- $~$\ In this subsection we study translation-invariant measure supported on single two-sided infinite lattice paths. We consider the geometric properties of paths which are in the support of such measures. We prove some growth bounds on intersections of such paths with a fixed line, and we investigate intersections of such paths with other objects in the plain. We begin by proving Theorem \[thm:past intersects a ray infinitely many times\]. The proof is divided into several lemmas. Under the assumptions of the theorem, denote by $H^{+}$ the ray through $v$ in the direction of the upper half $y$-axis. Denote by $H^{-}$ the second ray which is parallel to the $y$-axis and starts at $v$, but is a translation of the lower $y$-axis. We now state and prove some lemmas, each assumes the conditions of the theorem. \[lem:one ray -> all rays\] Assume that the origin is in the path, and that its past intersects the positive $y$-axis only finitely many times. Then for any point $(a,b)$ in the path, its past intersects the ray $\{(a,t)\| t > b\}$ only finitely many times. Assume the opposite. It is clear that if a point $(a,b)$ in the path has the property that its past intersects the half ray above it which is parallel to the $y$-axis only finitely many times, then this is true for any point $(a,c)$ in the path. It is therefore follows that finite intersection is a property of lines which are parallel to the $y$-axis. Due to translation invariance, and to the assumptions, there must be infinitely many rays which are a parallel translation (in the $x$ direction) of the positive $y$-axis such that the past of a point in the path intersects infinitely many times, and infinitely many other rays which the past does not intersects infinitely many times. In addition it also follows that these types alternate infinitely many times. In particular, there must be five such rays $a_1,a_2,a_3, b_1,b_2$ which are translations of the positive $y$-axis such that the rays which are indexed $a_i$ intersect the past of any point only finitely many times, and those indexed by $b_j$ are intersected infinitely many times, and that if we order these rays according to the $x$-coordinate we have that $a_1$ is the leftmost, $b_1$ comes after it, then $a_2$,$b_2$, and $a_3$ (i.e. if ${a_i}^{x}$ is the $x$-coordinate of $a_i$, and similarly for $b_j$ then ${a_1}^{x}<{b_1}^{x}<{a_2}^{x}<{b_2}^{x}<{a_3}^{x}$). Let $v\in\mathbb{Z}^{d}$ be a point in the path. There is some $h\in\mathbb{Z}$ such that the past of $v$ does not intersect the $a_i$ rays at height (i.e. $y$-coordinate) higher than $h$, and we assume in addition that $h$ is higher than the $y$-coordinates of lowest point of each $a_i$. Define $T_j$ to be the times of intersections of the path with $b_j$ at points with $y$-coordinate bigger than $h$ (by ’time of intersection’ we mean precisely minus the path distance from the intersection point to $v$), $j\in\{1,2\}$. Since the past of $v$ intersects each $b_j$ infinitely many times, and in particular intersects in points with $y$-coordinate bigger than $h$, it follows that\ $\forall t_1 \in T_1~\exists\{t_i\}_{i\geq 2}\subseteq T_1,~\exists\{s_i\}_{i\geq 1}\subseteq T_2~t_1>s_1>t_2>s_2...$ Now, since between any two times $t_i$ and $s_i$ as above, the path must intersect the segment parallel to the $x$-axis, at height $h$, that is between $a_1$ and $a_2$, but there are only finitely many points on this segment, whence the past must intersect itself. A contradiction. ![Intersections of the past with rays - Lemma \[lem:one ray -> all rays\]](past_intersects_finitely){width="70.00000%"} \[lem:one ray -> not opposite ray\] If the past (of $v$) intersects $H^{+}$ only finitely many times, then it intersects $H^{-}$ infinitely many times. Assume this is not the case, then after going backwards in time there is a point $u\in P\bigcap\{H^{+}\bigcup H^{-}\}$ that all its past is either strictly to the left of $u$ or strictly to the right of $u$. The set of points that all their past is strictly to their left will be called $PL$, similarly we define the set $PR$ for the right, and for the future $FL$, $FR$. Let $A = FL\cup FR \cup PL \cup PR$ their union. The event that the origin belongs to $A$ does not depend on the notion of ’direction’, and hence is $\mu$-measurable (in Definition \[def:direction\], defining a direction required a choice, which may not be measurable). Let $\rho$ be the probability of that event. The ergodic theorem guarantees that for any large enough $N$, in the $N\times N$ square around the origin there are more than $\rho N^{2}/2$ points of $A$. On the other hand, the size of $PL$, and any of the other three sets is at most linear in $N$, since for any two points of $PL$ the $x$-coordinate must be different, as one must be strictly in the past of the other. Whence, $\rho$ must be $0$, and the lemma follows. \[lem:one ray past -> not future\] If the past (of $v$) intersects $H^{+}$ only finitely many times, then its future intersects it infinitely many times. Assume for contradiction this is not the case, then there is a point $u$ in the path, with the same $x$-coordinate as $v$, and with highest $y$-coordinate among all the points with that property. Let $\rho$ be the probability that the origin is the point in the path with the highest $y$ coordinate among those with $x$-coordinate $0$. An argument similar to the previous lemma shows that $\rho$ must be also $0$, but it can also be shown in a dozen different ways. We now continue to the proof of the theorem. Clearly all of the above lemmas can be proved for rays in the $x$-axis direction as well. In addition, due to the first lemma, if the past (future) intersects some ray which is parallel to the $y$-axis ($x$-axis) infinitely (only finitely) many times, it will intersect any other ray of the same direction infinitely (only finitely) many times. There are several ways to finish the proof from here. One way is the following. Let $\mathcal{S}$ be the set of line segments which connect $(0,0)$ to the eight points $(i,j)\neq (0,0)$ where $-1\leq i,j \leq 1$. We define a function from the paths in the support of our measure to subsets of $\mathcal{S}$, which is translation invariant yet in general not rotation invariant (in multiples of $\pi/2$). The function is defined as follows:\ Let $\mathcal{X} = \mathcal{X}_{x,+}\bigcup\mathcal{X}_{x,-}\bigcup\mathcal{X}_{y,+}\bigcup\mathcal{X}_{y,-}$, where\ $\mathcal{X}_{x,+} = \{(x,0)\in\mathbb{Z}^{2}\| x > 0\}$\ $\mathcal{X}_{x,-} = \{(x,0)\in\mathbb{Z}^{2}\| x < 0\}$\ and the sets $\mathcal{X}_{y,+}, \mathcal{X}_{y,-}$ are defined in a similar way. Given a path $P$, the past of any point in $P$ intersects at least two of the sets $\mathcal{X}_{x,\pm},\mathcal{X}_{y,\pm}$ infinitely many times, and in the case it intersects just two of them, then one of them is from $\mathcal{X}_{x,\pm}$ and the other is from $\mathcal{X}_{y,\pm}$. We now define the subset $S$ of $\mathcal{S}$, which is the image of $P$. In case the past of some point in $P$ intersects all the elements of $\mathcal{X}$ infinitely many times, it will not contribute an element of $\mathcal{S}$ to $S$. If it intersects three of $\mathcal{X}$’s elements infinitely many times, then it contribute to $S$ the segment which lays in the ray that it intersects only finitely many times (for example, if it were the negative $y$ axis, we would have put in $S$ the segment between the origin and $(0,-1)$). If it intersects two axis finitely many times many times we add the segment which lays on the (internal) bisector of these two rays (e.g. if the rays are the positive $x$ and $y$ axis, we add to $S$ the segment between the origin and $(1,1)$). We do the same procedure for the future. ![Possible diagrams for the set S (Up to symmetries) - Theorem \[thm:past intersects a ray infinitely many times\]](diagrams_s){width="70.00000%"} Note that the set $S$ does not depend on the direction of the path, i.e. on the choice of past and future. In addition, it is translation-invariant (due to the above arguments), but if we rotate the path in $\pi/2$ for example, we rotate (geometrically) the resulting set $S$ as well. If either the past (of some point) or the future does not intersect at least one ray with positive probability, then the set $S$ has at least one element, but at most two. Due to ergodicity, the set $S$ must be constant a.s., on the other hand, no such set, which contains one or two elements of $\mathcal{S}$ returns to itself under a rotation of $\pi/2$, in contradiction with the translation invariance. There is another proof to this claim, using a result of H. Kesten [@NOAM_KESTEN]. Similar techniques lead to the following generalization \[thm:past intersects a rational ray infinitely many times\] Let $\mu$ be a measure of infinite paths in the $\mathbb{Z}^{2}$ lattice. Assume that $\mu$ is ergodic with respect to the group of $\mathbb{Z}^{2}$-translations and invariant under the group of rotations of the plane around the origin by integer multiples of $\pi/2$. Let $v$ be a vertex of the path, then the past of $v$ intersects any ray with a rational slope which starts at $v$ infinitely many times (intersections need not to be at lattice points this time). Similar proposition hold for the future. Next we deal with measures of bi-infinite paths in the planar lattice, and try to achieve some growth bounds. The next lemma, geometric in its spirit, will be our main tool for proving quantitative bounds on the behavior of translation invariant lattice paths. \[lem:cross lemma\] Let $P$ be a path as in the above definition. Let $p,q$ be two points of $P$ and assume that the $n^{th}$-cross of $p$ and the $m^{th}$-cross of $q$ intersect (perpendicularly) in a point. Then their snails intersect , and in particular the path distance between $p$ and $q$ is not more than the sum of the lengths of the corresponding snails. Though it is easy to be convinced in the correctness of the Lemma, the proof is slightly tricky. We show that the snails intersect, and the rest of the claim follows immediately from this fact. Denote by $O$ the intersection point of crosses. Assume without loss of generality. that $O$ is on the segment between $p$ and ${p_n}^{x_+}$, and that $O$ is on the segment between $q$ and ${q_n}^{y_-}$. Denote by $p'$ the point ${p_n}^{x_+}$, and by $q'$ the point ${q_m}^{y_-}$. From now on (in this proof) when we talk about the $\emph{partial}~\emph{snail}$ of $p$ we mean the minimal connected subpath that connects the points $p, {p_1}^{x_+},...,{p_n}^{x_+}$, and similarly for $q$ (but in the direction of the lower $y$-axis). We show that even these partial snails intersect. If $O$ is on either partial snail, we are done, since in this case it is in the path, and it must be one of ${p_1}^{x_+},...,{p_n}^{x_+}$ and one of ${q_1}^{y_-},...,{q_m}^{y_-}$. In addition, if the partial snail of $p$ intersects the segment between $q$ and $q'$, we are done as well, as again this is a path point which belongs to the two snails. Thus, we may assume, towards contradiction, that the partial snail of $p$ does not intersect the segment between $q$ and $q'$ (and in particular not $O$), and the same assumption for the snail of $q$. Thus, if we consider the graphs of the union of the partial snail of $p$ with the segment between $p$ and $p'$, and the corresponding graph of $q$, their only intersection is $O$. ![Two intersecting snails - Lemma \[lem:cross lemma\]](intersection_snails){width="70.00000%"} We can deform the graph (and by ’graph’ we mean the picture of the graph, not a graph in the sense of graph theory yet) of $p$ slightly so that it remains a piecewise smooth graph (polygonal), such that the segment between $p$ and $p'$ remains in place, but the intersections in this graph are in a point, and there are not common segments to the deformed path and the $p$-$p'$ segment (e.g. if the path intersects the segment, lays on is for several edges and then cross to its other side - we leave one intersection point, and move slightly the rest of the curve. If after laying on the segment it does not cross, but returns to the previous side - we eliminate the intersection by moving that part of path slightly). We do it for the other graph as well, and in a way that no new intersections are created or destroyed (between the two graphs). Note that $p, p',q,q'$ remain in place, and no deformation occurs in a neighborhood of $O$. ![Deformation of the intersection of figure 3 - Lemma \[lem:cross lemma\]](deformation){width="70.00000%"} Now we apply a graph theoretic argument. Consider the deformed graph of $p$, define self intersection points and points of intersection between the segment and the rest of the graph as vertices. Curves between them will stand for edges. This is a planar graph (with possibly multiple edges between vertices). It is easy to verify that every degree in the graph is either $2$ or $4$, and in particular - even. Hence it has an Euler-cycle. It is easy to see that such a cycle, in a planar graph can be decomposed as union of disjoint planar loops, such that none of which intersects itself, and mutual intersections are only in vertices. A similar argument can be applied on the deformed graph of $q$. The total number of intersections of the two deformed graphs, is the sum over pairs of loops (one from each graph) of their intersections, as they have no common vertex. It is a well known fact that number of intersections of two transversal generic loops in the plain (e.g. polygonal closed paths, nowhere tangent to each other and whose intersection is made of a finite set of points, as in our case) is always an even number. Thus, their sum is even as well. But this is a contradiction to the fact that the only intersection point $O$ (and it is a generic intersection point). And the result follows. One may wonder about how the length of the path which connects $p$, a point on the path, and ${p_n}^{x_+}$ behaves. It is not difficult to verify that as a function of $n$, the expected (path) distance between an arbitrary vertex $p$ which is on the path and ${p_1}^{x_+}$, the closest intersection of the path with the positive $x$-ray from $p$ must be infinite, and hence the expected length of the path which connects $p$ and ${p_n}^{x_+}$ must be super linear. But can we say more? A natural guess would be that if we consider the length of the path which connects $p$ and ${p_n}^{x_+}$, it will always be at least $cn^2$, for some positive $c$, for large enough $n\in\mathbb{N}$. A construction of B.Weiss shows that this is not the case ([@Weiss]). Yet we prove that a weaker claim does hold, that infinitely many times, this inequality holds if we consider the $n^{th}$-snail. This is Theorem \[thm:lim inf snail is at least quadratic\] that we state again below. It is followed by several conclusions, which show, for example, that if we add some more restrictions on the paths we can show that the path-distance between $p$ and ${p_n}^{x_+}$ is at least $cn^2$ infinitely many times. \[thm:lim inf snail is at least quadratic\] Let $\mu$ be a translation invariant measure of bi-infinite paths in the $\mathbb{Z}^{2}$-lattice. Let $P$ be a path in the support of the measure, and $p$ be a lattice point in the path. then there is a positive $c = c(p,P)$ such that for infinitely many values of $n\in \mathbb{N}$, the length of the $n^{th}$-snail of $p$ is at least $cn^{2}$. Without loss of generality we assume that $\mu$ is ergodic, otherwise we apply ergodic decomposition. For any positive $\varepsilon$, we define the set $A_{\varepsilon}$ as the set of points in $P$ such that for only finitely many values of $n\in\mathbb{N}$, the length of the $n^{th}$-snail is at least $\varepsilon n^{2}$. We define $A_0$ as $\bigcap_{\varepsilon >0}A_\varepsilon$. The theorem would follow if we prove that $A_0$ is empty. Define $A_{\varepsilon , N}$ to be the set of points in $A_\varepsilon$ such that for any $n\geq N$, the length of their $n^{th}$-snail is not more than $2 \varepsilon n^{2}$. It is clear that $A_\varepsilon = \bigcup_{N\varepsilon\mathbb{N}}A_{\varepsilon,N} $, and that $A_{\varepsilon,N}$ is increasing in $N$. Hence, if we prove a uniform bound (in $N$) on the probability that the origin is in $A_{\varepsilon ,N}$, this bound will hold for $A_{\varepsilon}$ as well. If this bound tends to $0$ with $\varepsilon$, then the probability that the origin is in $A_0$ is $0$ as well. Thus, due to translation invariance, $A_0$ will be empty a.s. We now turn to prove the required bounds. \[lem:the density of A\_eps\] Under the above definitions and conditions, the probability that the origin is in $A_{\varepsilon , N}$ (given it is on the path) is bounded by $8 \varepsilon$. Denote by $\rho$ the probability that the origin is in $A_{\varepsilon , N}$. Due to the ergodic theorem, for large enough $M$ (that in particular we assume that is larger than $N$), in the $M\times M$ square around the origin there are at least $\rho M^{2}/2$ lattice points from $A_{\varepsilon , N}$. Since for any such two points, their $M$-crosses must intersect, it follows from Lemma \[lem:cross lemma\], that the path distance between these two points is no more than the sum of the lengths of their snails. As these two points are in $A_{\varepsilon , N}$, the sum of lengths of their snails is bounded by $4 \varepsilon M^{2}$. Thus, the diameter of the connected subgraph of the path that contains all the points of $A_{\varepsilon , N}$ which are in the $M \times M$ square is at most $4 \varepsilon M^{2}$. But any connected subgraph of a path is a path, and its diameter is its length. Thus, we have found a path of length at most $4 \varepsilon M^{2}$, which contains at least $\rho M^{2}/2$ vertices. Hence, $8 \varepsilon \geq \rho$ And thus, the theorem is proved. The above theorem and its way of proof lead to some conclusions. Under the above conditions, if we also assume that the measure is ergodic with respect to translation by every group element (not only ergodic with respect to the whole group’s action), then there is a positive $c = c(\mu)$ such that for $\mu$-almost every path $P$, any point $p$ on the path belong to $A_c$. Let $B_\varepsilon$ be the complement of $A_\varepsilon$. We define ${B_\varepsilon}^{x}$ as the set of points $p$ such that for infinitely many values of $n$ the minimal connected subpath that contains all the points $p,{p_1}^{x_+},..., {p_n}^{x_+},{p_1}^{x_-},...,{p_n}^{x_-}$ is of length at least $\varepsilon n^{2}$. Similarly one defines ${B_\varepsilon}^{y}$. We have the following simple properties:\ 1. If $p$ is in $B_\varepsilon$ then either $p$ is in ${B_{\varepsilon /2}}^{x}$ or in ${B_{\varepsilon /2}}^{y}$.\ 2. If $p$ is in ${B_\varepsilon}^{x}$ or ${B_\varepsilon}^{y}$ then $p$ is in ${B_\varepsilon}$.\ 3. If $p$ is in ${B_\varepsilon}^{x}$ then any other (path) point with the same $y$ coordinate is also in ${B_\varepsilon}^{x}$. If $p$ is in ${B_\varepsilon}^{y}$ then any other point with the same $x$ coordinate belongs to ${B_\varepsilon}^{y}$. Assume that the origin is in the path. Due to the above theorem, there is some $\varepsilon > 0$, for which the origin is in $B_\varepsilon$. Due to property 1, and without loss of generality the origin is in ${B_{\varepsilon /2}}^{x}$. Consider the event that the origin is in ${B_{\varepsilon /2}}^{x}$, it is invariant under translations of direction $x$. Thus, by ergodicity in each direction separately this is a $0-1$ event. But it has a positive probability (for some $\varepsilon$), and therefore it occurs a.s. Due to invariance to the other direction we see that a.s. any path point is in ${B_{\varepsilon /2}}^{x}$. And hence, due to property 2, a.s. any path point is in ${B_{\varepsilon /2}}$. Under the assumption of the previous conclusion, if in addition the distribution $\mu$ is invariant under rotations around the origin by angles which are integer multiples of $\pi /2$ then there exist a unique $\varepsilon = \varepsilon (\mu)$ such that for every point in the path, for any positive $\delta < \varepsilon$, for each $\alpha\in\{ x_+, x_-, y_+, y_- \}$ there are infinitely many values of $n$, such that the length of the minimal connected subpath which contains $p, {p_1}^{\alpha},...{p_n}^{\alpha}$ is at least $\delta n^{2}$, and such that no path point that property for any $\delta > \varepsilon$. The proof is similar to the above proof, and the rotation invariance guarantees that there is a uniform positive $\varepsilon$ to each of the four directions. Assume $\mu$ is rotation-invariant, as in the above conclusion, and ergodic with respect to the group’s action (and not necessarily to each direction separately), then almost surely, for any point $p$ in the path $P$ there exist positive ${\varepsilon}_x , {\varepsilon}_y$, which depend on the path and the point such that, in the notation of the first conclusion, $p$ belongs to ${B_{{\varepsilon}_x}}^{x} \bigcap {B_{{\varepsilon}_y}}^{y}$. $\mathbf{Sketch.}$ The event that there exist a point with no positive ${\varepsilon}_x$ for example, is translation invariant. Hence it must be a $0-1$ event. If it were an event of probability $1$, then the same would hold for points with no positive ${\varepsilon}_x$. As in the first conclusion, if $p$ has no positive $\varepsilon_x$ with the given property, then the same holds for any other point with the same $y$ coordinate as $p$. Thus, it is a property of lines in the lattice which are parallel to the $x$-axis. Thus, with probability $1$, there should be such a line. Similarly, there should be a line with a similar property which is parallel to the $x$ axis. Denote by $q$ their intersection. From property 1 in the first conclusion it follows that $q$ is not in any $B_\varepsilon$. But this contradicts Theorem \[thm:lim inf snail is at least quadratic\]. Translation Invariant Measures of Infinite Lattice Trees {#sec:infinite trees-inv subgraphs} -------------------------------------------------------- $~$\ This subsection investigates measures of trees and forests which are subgraph of a given graph, usually the graph $\mathbb{Z}^{d}$. Most of the results in this subsection are rather simple, but allow us to imagine the geometric picture trees which arise in translation invariant contexts. The last theorem of the subsection is more interesting and also slightly more sophisticated. We consider measures, invariant under the graph’s group of symmetries (or a subgroup of it), of infinite trees or forests whose connected components are infinite trees. Some of the results hold for a general unimodular transitive graph, some are proved for amenable Cayley graph and some hold for Euclidean lattices (and in particular the planar square lattice). Let $G$ be a connected Cayley graph of an amenable group. Let $\mu$ be a measure on forests which are unions of infinite trees in $G$. Assume that $\mu$ is invariant under the groups’s action. Then in almost every forest which is in the support of the measure, any tree component is either single-infinite or bi-infinite. By ergodic decomposition we may assume that $\mu$ is ergodic. Assume there are some multi infinite components, in that case there are encounter points as well. Let $F$ be any subgraph of the graph $G$. By the last part of Remark \[rmk:rmk forests\], $\mu$-a.s the number of encounter points inside $F$ cannot be larger than the number of boundary edges of $F$ (this is actually always true, not just $\mu-$a.s). Thus, the $\mu$-expectation of the number of encounter points inside $F$ is not more than the number of boundary edges. On the other hand, if we assume that the probability for having encounter points is positive, then there exists a positive number $\rho$ such that an arbitrary vertex $v$ is an encounter point with probability $\rho$. The linearity of expectation yields that the expected number of encounter points in $F$ is $\rho \|F\|$. Thus $\| \partial F\| \geq \rho \|F\|$, where $\partial F$ is the set of boundary edges of $F$. Let $\{ F_n \}$ be a F$\phi$lner sequence. The above argument shows that $\| \partial F_n\| / \|F_n\| \geq \rho$. But this contradict the definition of F$\phi$lner sequences, where the LHS of the inequality must tend to $0$. We turn examine the shape of single-infinite trees (or forests made of infinite trees), that are in the support of an invariant measure (under the group actions). Our assumption on the full graph are that it is a (unimodular) Cayley graph. Our main tool will be the Mass-Transport principle. \[clm:trees MT\] Let $G$ be a unimodular transitive graph (a Cayley graph of a discrete group). Let $\mu$ be a measure, invariant under the automorphism group of the graph (the group’s actions), of forests made of single-infinite trees. Let $f:\mathbb{N}\rightarrow \mathbb{R}$ be any nonnegative valued function. Then $$\forall x\in V~ \sum_{k\in\mathbb{N}}f(k) = \mathbb{E}_\mu[\sum_{y\in R(x)\backslash{x}}f(\delta(y,x))]$$ In particular, one side is infinite iff the second side is. It is a direct conclusion from Equation \[eq:MT\] if we define the mass function\ $m(\omega;x,y)=f(\delta_\omega(x,y))\times\mathbf{1}_{x\in {S(y)}}$\ where $\delta_\omega(\cdot,\cdot)$ is the tree distance between $x,y$ in configuration $\omega$, and it is defined to be $0$ if $x,y$ are not in the same tree (or not in a tree), and $\mathbf{1}_{x\in {S(y)}}$ is the characteristic function of the event that $x$ is in the stem of $y$ (in configuration $\omega$). We give some conclusions from the above claim. In all the conclusions we assume the assumptions of Claim \[clm:trees MT\]. \[con:infinite functions of root\] Let $f:\mathbb{N}\rightarrow \mathbb{R}$ be such that $\sum_{k\in\mathbb{N}}f(k)=\infty$, then\ $\mathbb{E}_\mu [\sum_{y\in R(x)\backslash{x}}f(\delta(y,x))]=\infty$ \[con:infinite expected root\] For an arbitrary vertex $v$ the $\mu$-expected size of $R(v)$ is infinite. This follows immediately from the above conclusion, just by taking the constant function $f\equiv 1$, but we give an equivalent proof, in order to improve the intuition for the scenario. If each vertex of the forest sends one unit of mass to each vertex in its stem, since any vertex in a tree sends an infinite amount, and any vertex has a positive probability to be in the tree, it follows that the expected total mass that each vertex in the forest sends is infinite. Thus, the MT principle tells us that any vertex receives on average an infinite amount of mass. But any vertex not in the forest receive nothing, and a vertex $v$ in the forest receives exactly the size of $R(v)\backslash{v}$. \[con:one expected root element of dist n\] Let $v$ be a vertex in the graph. The $\mu$-expected value of the number of vertices in $R(v)$ (conditioned on $v$ being in the forest), with tree-distance $n$, is $1$. This follows immediately from Claim \[clm:trees MT\]. If each vertex of the forest sends one unit of mass to the (unique) vertex in its stem that is in distance $n$ from it. Any vertex of the forest sends exactly one unit of mass. The MT principle tells us that a vertex, conditioned on being in the forest, receives an expected number of one unit of mass. This implies the claim. \[con:lower bounds for large root in amenable graphs\] Let $\alpha^{v}(n)$ be the number of vertices in a ball of radius $n$ in the graph $G$ around a given vertex $v$, i.e. the number of vertices in the graph, whose shortest path to $v$ is of length at most $n$. Let $\beta^{v}(n)$ be the size of the sphere of radius $n$ around $v$ in the graph, i.e. the set of vertices in the graph, whose shortest path to $v$ is of length exactly $n$. Then the $\mu$-probability that a given vertex $v$, conditioned on being in the forest, has at least one vertex in $R(v)$ of tree-distance $n$ is not less than $1/\alpha^{v}(n)$. The $\mu$-probability that a given vertex $v$, conditioned on being in the forest, has at least one vertex in $R(v)$ of graph-distance $n$ is not less than $1/\beta^{v}(n)$. Note that in $R(v)$ there is a vertex whose graph (tree) distance from $v$ is at least $n$, is exactly the probability there is a vertex in $R(v)$ whose graph (tree) distance from $v$ is exactly $n$, since in the path between $v$ and $u$ which is of length more than $n$, there is at least one vertex whose distance from $v$ is $n$. First note that since we always assume that $G$ is transitive $\alpha^{v}(n)$ does not depend on the vertex $v$, and hence we may denote it simply by $\alpha(n)$. Similarly to $\beta^{v}(n)$. Second observe that for any tree $T$ which is a subgraph of a graph $G$, the tree-distance of two tree vertices is at least the graph distance, i.e. $\delta_T(x,y)\geq d(x,y)$. Take a vertex $v$, and consider the ball of radius $n$ around it (in $G$). denote by $\alpha$ the number of the vertices in $B$. Any vertex in $R(v)$, which has a tree distance exactly $n$ from $v$, its graph distance from $v$ is at most $n$, and hence it belongs to $B$. Thus, the number of elements of $R(v)$ whose tree distance from $v$ is exactly $n$ is (nonnegative, and) bounded from above by $\alpha$ and by Conclusion \[con:one expected root element of dist n\] has expectation $1$. Now the Markov inequality guarantees that the probability there are vertices in $R(v)$ of tree-distance exactly $n$ is at least $1/\alpha$. We would like to achieve the corresponding result for graph-distances. For this first consider the following mass function: Each vertex in the tree sends one mass unit to each vertex in its stem whose graph distance from it is exactly $n$. It is clear that each vertex in the tree sends at least one mass unit, and hence, in average, according to Claim \[clm:trees MT\] receives at least $1$ unit of mass. This time we consider $\beta$, which is the number of vertices of the $n-$sphere centered in some given vertex $v$. Again $\beta$ bounds the number of vertices in $R(v)$ of graph distance $n$ from $v$, and again Markov’s inequality shows that the probability that there is at least one vertex in $R(v)$ of graph-distance $n$ from $v$ is at least $1/\beta$. \[con:lower bounds for large root in lattices\] If the graph $G$ is the $\mathbb{Z}^d$-lattice, then for a vertex $v$ in the forest the probability that there is some element of $R(v)$ with graph distance at least $n$ is $\Omega(n^{1-d})$. In addition, the probability that there is an element of $R(v)$ with tree-distance at least $n$ is $\Omega(n^{-d})$ The ball (box) of radius $n$ in the $\mathbb{Z}^d$-lattice has $\Theta(n^{d})$ vertices and its boundary has $\Theta(n^{d-1})$ vertices. Using Conclusion \[con:lower bounds for large root in amenable graphs\] we get the result immediately. Most of the above can be established in a similar way to a bi-infinite tree. In a bi-infinite tree $T$, there is a unique path between any vertex and $P(T)$. Thus, even though we do not have a canonic “forward” direction, as in the single-infinite case, we do have two “forward” directions, that may have a finite path in common. Indeed, we can move “forward” in the path which connects it to $P(T)$, after we reach $P(T)$ we have two legal directions towards infinity. Thus, one can construct mass functions as we did for this case as well, such that mass is again sent only forward. Our next results handle densities of bi-infinite and single-infinite trees which are in the support of some invariant measure. As before, due to ergodic decomposition we may assume the measure is ergodic. The main question we are interested in is the following - if we consider a large ball in the graph, and count edges which lay on simple paths in the tree which connect two of the ball’s boundary points, how many such edges are there? We show that in the double-infinite case the number is proportional to the volume of the ball. In the single infinite it is at least a fixed portion of the boundary (we consider only lattices for simplicity). The surprising result is that for the two dimensional lattice we show a superlinear bound. A generalization of this fact plays an important role in our study of spin systems. All these results can be easily generalized to forest of infinite trees as well. \[clm:bi-infinite have densities\] Let $\mu$ be an ergodic measure of bi-infinite trees on a connected Cayley graph of a group, invariant under the group actions. Then there exist a positive constant $\rho$, such that for any vertex $v$, and any finite subgraph $F$ the following holds: Let $S$ be the $\mu$-expected number of edges of the tree which belong to $F$ and lay on a simple path in the tree which connects two boundary points of $F$, and $E$ is the number of all edges of $F$, then $S \geq \rho E$. $\mathbf{Sketch.~}$The idea is that any edge in $P(T)$, the path of the tree, has the above property. And any edge has a positive probability to be in $P(T)$. Thus the expected number of edges that both of whose vertices are in $F$ and belong to $P(T)$ is a fixed, positive portion of the size of $E$. If we assume that the above Cayley graph is also amenable, an ergodic theorem due to Lindenstrauss ([@pointwise_erg]) guarantees that by replacing the single set $F$ with a F$\phi$lner sequence, we get that the above conclusion holds not only in expected value but also a.s. for some subsequence, i.e. there is an infinite subsequence of the given F$\phi$lner sequence such that for almost every set in the subsequence the number of edges which lay on simple paths that connect two boundary edges is at least a $\rho-$portion of all the edges in that set. Our last result is the main theorem of this subsection. A variant of this theorem will appear later as one of the key stones in the proof that in the ground-state of a planar Ising spin-glass system, there is no infinite component made of edges which are all unsatisfied. Note that we have stated a partial version of this theorem in Subsection \[sec:Main Results-inv subgraphs\], we now state and prove a slightly more general theorem. For simplicity the next result is formulated for the lattice $\mathbb{Z}^{d}$, and for measures which are also invariant under rotations (of multiples of $\pi/2$ in any integer axis). Yet, it can be be extended to larger families of graphs, and the rotations-invariance is, in fact, not necessary. \[thm:single-infinite at least boundary sized or nlogn\] Let $\mu$ be an ergodic , translation-invariant and rotation-invariant (as described above) measure of single-infinite trees on the lattice $\mathbb{Z}^{d}$. Then there exist a positive constant $\rho$, such that for any vertex $v$, and an increasing sequence of boxes, $\{B_n\}_{n=1}^{\infty}$, centered at $v$, the following holds: Let $e_n$ be the number of edges of the tree which lay in the $n^{th}$ box and lay on a simple path in the tree which connects two boundary points of the box, then for all large enough $n$, $\mathbb{E}_\mu e_n > \rho n$. Moreover, for $d=2$, we have $\mathbb{E}_\mu e_n > \rho nlog(n)$. We prove the result for the case $d=2$ and the case $d > 2$ together. Yet there is a simpler proof for the latter case (which is also less interesting). ![The intersection of an infinite lattice tree with a large square - Theorem \[thm:single-infinite at least boundary sized or nlogn\]](nlogn){width="70.00000%"} First we observe that similarly to Conclusion \[con:lower bounds for large root in lattices\] we can show that with probability of $\Omega(n^{1-d})$ for a vertex $v$ in the single-infinite tree there is a vertex $u\in R(v)$ with $\\| u-v \\|_\infty = n$, this is done in the standard way, using the mass function $m(u,v)$ which is always $0$ unless $v\in S(u)$ and $\\| u-v \\|_\infty = n$, and then it is $1$. Thus, there is a positive constant $c_1$, such that for a given vertex $v$ and $n\in\mathbb{N}$, the probability there is another vertex $u\in R(v)$ such that $\\| u-v \\|_\infty = n$ is at least $c_1 n^{1-d}$. for any $n$ with probability at least $c_1 n^{1-d}$ for a given vertex $v$ there is another vertex $u\in R(v)$ such that $\\| u-v \\|_\infty = n$. In particular, due to the rotation-invariance, if we denote by $c := c_1/2d$, then for any $n$ with probability at least $c_1 n^{1-d}$ for a given vertex $v$ there is another vertex $u\in R(v)$ such that $\\| u-v \\|_\infty = n$, and that $u^{1}-v^{1}=n$, where $x^{i}$ is the $i^{th}$ coordinate of $x$ (and of course the same probability bound holds for any $u^{i}-v^{i}=\pm n$). There is a constant $b>0$ such that for any $n$ the number of vertices in the boundary of a box of $l_\infty$ norm $n$ is at least $bn^{d-1}$. Every vertex $v$ from the tree, which lays in a given box, is a part of a simple path (contained in the tree) which connects two points on its boundary iff there is another vertex from $R(v)$ which is in that boundary. The reason is that there is always a “half-path” from $v$ to the boundary - $S(v)$ (the part from $v$ until first intersection with the boundary of the box). In addition, the number of tree edges which lay on simple paths which connect to boundary edges is clearly proportional to the number of vertices with the same property (as the degree in the graph is bounded by $2d$). Thus, it is enough to give a lower estimate for the expected number of vertices with that property. Denote by $p$ the probability a given vertex lays in $T$ (of course, $p$ does not depend on the vertex, it is an a.s. constant which depends only on $\mu$ and is the same for any point and any configuration in the support of $\mu$). Then the probability a given vertex lays in the tree, and in its roots set there is a vertex at $l_\infty$ distance $t$ from it, in a given direction is at least $pc n^{1-d}$. Consider a fixed vertex $v$, denote by $S_t$ the set of points with $l_\infty$ distance $t$ from $v$, $B_t$ be the box of vertices whose $l_\infty$ distance from $v$ is at most $t$. Combining all of the above gives that the expected number of tree vertices in a box, which lay on simple paths (simple paths which are contained in $T$) and connect two of $B_n$’s boundary points is at least $$pbc\sum_{1\leq k \leq n-1}(n-k)^{d-1}/{k^{d-1}} > (n/2)^{d-1}\sum_{1\leq k \leq \lfloor n/2 \rfloor}k^{1-d}$$ For $d=2$ this gives an $\Omega(nlog(n))$ bound. For higher $d$ is gives the expected linear bound. As required. Spin Glass Models {#sec:spinglasses} ================= $~$\ Background {#sec:Background-spinglass} ---------- $~$\ The understanding of spin glass models in large or infinite graphs has been the subject of many studies in physics, mathematics and neuroscience. Questions regarding the multiplicity of ground states in finite dimensional short-ranged systems, such as the Edwards-Anderson (EA) Ising spin glass, and in particular the 2D case were the subjects of several researches and simulations (e.g. [@EA], [@Newman_Stein_plain], [@Newman00natureof], [@Newman_Stein_halfplain],[@PhysRevLett.58.57],[@0022-3719-17-18-010],[@PhysRevLett.56.1601],[@PhysRevLett.83.5126],[@PhysRevB.46.973], [@HARTMANN]). Even less is known about the geometry of the ground states. Preliminaries {#sec:Prelim-spinglass} ------------- $~$\ This part of the thesis concerns the EA Ising spin glass with the Hamiltonian $$\mathcal{H}_\mathcal{J}(\sigma) = -\sum_{<x,y>}J_{xy}\sigma_x\sigma_y$$ Where $\mathcal{J}$ denotes a specific realization of the couplings $J_{xy}$ and $\sigma$ is a spin configuration, i.e. for any vertex $x$ in the graph, $\sigma_x = \pm 1$. The sum is taken over nearest neighbor pairs $<x,y>$ in a given graph G. We confine ourselves to the case where $J_{xy}$ are independently chosen from a mean zero Gaussian or any other symmetric, continuous distribution supported by the real line. The overall disorder measure is denoted by $\nu(\mathcal{J})$. A $\emph{Configuration}$ for some realization $\mathcal{J}$ is a choice of spin to each site. Although the Hamiltonian might not be defined, due to divergence, we can still compare the “Energy difference” (defined as the Hamiltonian’s difference) between two configurations which differ by a finite amount of spins. A $\emph{Ground}~\emph{State}$ is a configuration whose energy cannot be lowered by flipping any finite subset of spins. That is, all ground state spin configurations must satisfy the constraint $$-\sum_{<x,y>\in\mathcal{{\partial C}}}J_{xy}\sigma_x\sigma_y \leq 0$$ for any closed finite subset of vertices $\mathcal{C}$. One should note that if we restrict our attention to a planar graph, ${\partial C}$ is actually a union of loops in the dual lattice. One should note that if $\sigma$ is a ground state, so is $-\sigma$, the configuration which is the result of flipping all the spins in $\sigma$. Thus, it makes more sense to talk about $\emph{ground state pair}$ or $\emph{GSP}$. The amount $J_{xy}\sigma_x\sigma_y$ in some configuration is sometimes called the $\emph{value}$ of the edge $\{x,y\}$ (in this configuration). Let $\mu_\mathcal{J}$ be a general conditional (on $\mathcal{J}$) distribution which is translation-invariant and is supported on GSPs (or more precisely - the ground state representatives of them) for $\mathcal{J}$. Let $G$ be a graph, $J$ a realization of couplings. An edge $e= \{ x, y \} $ is said to be $\emph{fixed}$, if either $\|J_{xy}\|>\sum_{z\sim x}\|J_{xz}\|$ or $\|J_{xy}\|>\sum_{z\sim y}\|J_{yz}\|$. A subgraph $H$ is said to be $\emph{fixed}$ if there is a tree $T\subseteq G$ such that all the edges in $T$ are fixed and all the vertices of $H$ are vertices of $T$. Under some spin configuration the product of an edge’s interaction with its two vertices’ spins is called its $\emph{value}$. We say that a bond $e=\{x,y\}$ is $\emph{unsatisfied}$ (in some spin configuration $\sigma$) if $J_{xy}\sigma_x\sigma_y$ is negative. In this case we also say that the dual bond $e^{}$ is unsatisfied. In every GSP, the value of a fixed edge is positive and hence either in every GSP its vertices have the same spin, or in every GSP its vertices have opposite spins. In other wards, the products of its vertices’ spins is the same in any GSP. In a similar manner, all the edges of a fixed subgraph have the same value in each GSP and thus, the spins of $H$’s vertices equal each other. Let $G=(V,E)$ be a graph, $S\subseteq V$ a subset of vertices. We say that $S$ bounds a subgraph $H$ if $H$ is a connected component of $G\mid_{V\setminus {S}}$. Let $G=(V,E)$ be a graph, the $\emph{boundary}$ of a subgraph is the set of edges which connect it to its complement in $G$. We continue by defining several families of graphs that we explore. Let $H = (V_H,E_H)$ be a graph. We say that a graph $G=(V_G,E_G)$ is a $\emph{H-type Graph}$ if $V=\bigcup_{h\in V_H}{\{h\}}\times{V_h}$ and if there is an edge between $(h,v)$ and $(h',v')$ then either $h=h'$ or $h\sim_H h'$, where by $\sim_H$ we mean that $h,h'$ are neighbors in $H$. We denote by $G_h$ the subgraph of $G$ over the vertices with H-coordinate $h$. We call it the $\emph{the}~$ $h^{th}~$ $\emph{slice}$. For a vertex of the form $(h,v)$ we shall say that $h$ is its $\emph{level}$. The $\emph{width}$ of the $h^{th}$-slice is the size of $V_h$. The $\emph{width}$ of $G$ is defined as $sup_{h\in H} \|V_h\|$. A $\mathbb{Z}$-type graph is called a $\emph{deformed cylinder}$. An important special case is the product of graphs, defined as follows Let $H=(V,E),G=(U,F)$ be graphs. We define their $\emph{product}, G\times H$ as the graph whose vertex set is $V\times U$ and there is an edge between $(h,g), (h',g')$ iff exactly one of the following occurs - $h\sim_H h'$ and $g = g'$ or $g\sim_G g'$ and $h = h'$. $\mathbb{Z}\times G$ will be called a $\emph{G-cylinder}$ or simply a $\emph{cylinder}$. $K_n$ will denote the complete graph on $n$ vertices. $C_n$ will stand for the cycle of length $n$. Throughout this section, unless stated otherwise the coupling are picked from a product measure of some continuous symmetric distribution whose support is all the real line. In addition, we use notations from the previous section, when handling infinite trees or forests. Several writers analyzed the spin glass objects in several scenarios, but only few rigorous results were attained. Some of the most impressive results were obtained in [@Newman00natureof], [@Newman_Stein_halfplain]. It was shown that GSPs that belong to the support of some $\emph{metastate}$ in the plane (a metastate is a special type of distribution over GSPs), if there is more than one, must agree on all the bonds except for some bi-infinite path. They also showed that in the half plane, there is a unique metastate. Main Results {#sec:main res-spinglass} ------------ $~$\ Throughout most of this section we assume that the interactions are distributed according to some nontrivial product measure. Our first result, concerns the geometry of any translation invariant distribution of GSPs (not necessarily a metastate). We show that in any translation-invariant measure for GSPs, no GSP contains an infinite cluster of unsatisfied (dual) edges. This may be stated as \[thm:no infty unsat cluster\] In almost every GSP in the support of a translation-invariant measure of GSPs (and interactions) the dual unsatisfied edges do not percolate. Moreover, the collection of unsatisfied (dual) edges forms a forest of positive density. This is one of the first rigorous results about the GSPs of this model. We then consider the number of GSPs in an infinite graph (when the interactions are distributed according to a product measure of continuous distribution supported on the real line). We prove that cylinders and deformed cylinders (with some restriction) a.s. have a unique GSP. On the other hand, we show that regular trees (of degree at least $3$) have infinitely many GSPs, for almost every realization of interactions, and moreover a translation invariant measure of GSPs, supported on uncountably many configurations. Geometry of 2D GSPs {#sec:2D-spinglass} ------------------- $~$\ This section will be devoted to exploring the set of unsatisfied (dual) edges in GSPs in the 2D EA Ising spin glass model. More specifically, we restrict ourselves to GSPs which are in the support of some translation-invariant joint distribution of couplings and GSPs. We shall sometimes call such a distribution a translation-invariant $\emph{scheme}$ of GSPs. Throughout this section we consider $\mu_\mathcal{J}$, a translation-invariant scheme of GSPs (where the graph is the square lattice), and that $\mathcal{J}$ is the product measure of a symmetric distribution which is supported on all the real line. We begin with a couple of easy lemmas. \[lem:un-sat form a forest\] For any GSP in the support of $\mu_\mathcal{J}$, the set of dual edges whose primal edges are not satisfied form a forest. Indeed, if there was a closed cycle in this dual graph, then flipping all the spins in the finite plane region which is bounded by this cycle would reduce the energy. We denote by $\mathfrak{F} = (\mathfrak{V},\mathfrak{E})$ the forest made of the unsatisfied dual edges of some GSP in the support of the measure. \[lem:the forest has positive dens\] The edges (vertices) of $\mathfrak{F}$ have a positive density, i.e. the limit $\lim_{n\rightarrow\infty}\sharp(\mathfrak{E}\bigcap{E_n})/n^{2}$ exists and is greater than $0$, where $E_n$ is the set of edges of a $n\times n$ dual square which has a fixed center (e.g. $(0.5,0.5)$), and there is a similar expression for the vertices. Due to ergodic decomposition the existence of the limit is clear. Positiveness will follow if we can show that there are unsatisfied edges a.s. More generally, we show that a.s. there is a positive bound for the fraction of unsatisfied edges in any spin configuration, given the interactions. Consider a unit square in the lattice. With positive probability the product of the interactions of its edges is negative. But then under any choice of spins, the product of its edges’ values is negative, and hence at least one of these edges must be positive. We now reach to the main aim of this chapter, proving Theorem \[thm:no infty unsat cluster\]. This theorem is one of the main results of this work, and its proof is slightly more entangled then previous proofs. We formulate the theorem slightly different than in Subsection \[sec:main res-spinglass\], though the formulations are clearly equivalent. \[thm:no infty unsat cluster\] All the connected components of $\mathfrak{F}$ are finite a.s. From the results of Subsection \[sec:infinite trees-inv subgraphs\] it follows that any infinite connected component must be either a bi-infinite tree or a single-infinite tree. We begin by showing that no bi-infinite trees components exist in $\mathfrak{F}$. Without loss of generality. $\mu_\mathcal{J}$ is ergodic. For any bi-infinite component we can define, as in Subsection \[sec:infinite trees-inv subgraphs\] its $\emph{path}$, the single bi-infinite path which is contained in it. Let $\mathfrak{P}$ denote the union of these paths. It has a well defined density (in the sense of Lemma \[lem:the forest has positive dens\]), due to ergodicity. We would like to show this density is $0$. Assume the density equals some $\rho > 0$. Consider a large $N\times N$ square in the dual lattice. There are constants $A,B$ s.t. with probability $1$ we can choose $N$ large enough such that such that the sum of absolute values of the squares boundary interactions (of the primal edges) is not more that $AN$, the number of edges of $\mathfrak{P}$ which lay in the square is at least $(\rho/2)N^{2}$ and that the some of these edges interactions, in absolute value is at least $BN^{2}$. Indeed, standard analysis shows that for $N$ large enough, even if we consider the smallest (in absolute value) $(\rho/2)N^{2}$ interactions, their sum will be at least some fixed multiple of $N^{2}$. Moreover, we can choose $N$ to be so large that $AN-2BN^{2}<0$ But now note that these paths divide the square into disjoint regions. each edge of $\mathfrak{P}$ which is in the interior of the square appears in the boundaries of exactly two such regions. The rest of the regions’ boundaries are edges from the original square’s boundary, each appears exactly once. Thus, the sum of the values over the boundaries (with multiplicity) is at most $AN-2BN^{2}<0$ (as the edges of $\mathfrak{P}$ have all negative value). But this means that there exists at least one region that if we flip all its spins the Hamiltonian decreases. A contradiction. We now move to the more sophisticated part - showing that there are no single-infinite components. The crux is the following. We still can decompose the square into regions whose boundaries are either part of the square’s boundary or of the trees. But now we do not know whether the sum of boundary values is negative. We do know, due to Theorem \[thm:single-infinite at least boundary sized or nlogn\], that the number of edges from the trees that form these boundaries is superlinear (it was shown for a single tree - but the exact same consideration works for a forest). This would suffice if the absolute values of the interactions were bounded away from $0$. But in the continuous case we have to work harder, yet, the result will follow immediately from the next lemma. Let $\mathcal{J}$ be a product measure of a continuous distribution whose support is all the positive real line on the dual square lattice. Let $\tau_\mathcal{J}$ be a translation invariant scheme of forests of single-infinite trees (by scheme we mean, again, a joint distribution of couplings and trees). Then the expected sum of interactions of edges of the forest which lay on a simple path which starts and ends in the boundary of a $N \times N$ square around the origin is a super linear function of $N$. Knowing the lemma guarantees that we can divide the square into regions, bounded by the square’s boundary, and the trees’ edges, that at least one of the regions has a boundary whose sum of values is negative, and hence a flip reduces the Hamiltonian. Let $u,v$ be two vertices in the forest. Define the mass function $m_t(u,v)$ in the following manner. $m_t(u,v)=1$ if $v\in S(u)$, $\|v-u\|_\infty = t$ and the interaction of the single edge in the forest that connects $v$ to $S(v)$ is at least $\delta$, where $\delta$ is a constant to be chosen later on. Otherwise $m_t(u,v)=0$. Let $O$ be the “origin” of the dual graph, the point $(0.5,0.5)$. Denote by $E_t$ the value of $\mathbb{E}[\sum_{v\in\mathbb{Z}^{2}}m_t(O,v)\|O~is~in~the~forest]$, this value will not change if we replace $O$ by any other vertex $u$, of course. By the MT principle $E_t = \mathbb{E}[\sum_{u\in\mathbb{Z}^{2}}m_t(u,O)\|O~is~in~the~forest]$. Denote by $p_t$ the probability that for the origin $O$ (or any other vertex), which is conditioned to be in the forest, there is a vertex $u\in R(O)$, at $l_\infty$ distance $t$ from $O$ and that the edge which connects $O$ to $S(O)$ has an interaction at least $\delta$. Note that $$\label{eq:markov eq} p_t \geq E_t/4t$$ This is due to Markov’s inequality, as $\sum_{u\in\mathbb{Z}^{2}}m_t(u,O) \leq 4t$ ($4t$ is the total number of vertices at $l_\infty$-distance $t$ from $O$, similarly to Theorem \[thm:single-infinite at least boundary sized or nlogn\]). We now show that for a small enough $\delta$ there is a constant $C$ which satisfies the following condition $$\label{eq:perc ineq} \sum_{t\leq s\leq t+C\log{(t+1)}}E_s \geq 1$$ Indeed, choose some $\delta$ such that the edges with interaction smaller than $\delta$, if taken as open edges form a subcritical percolation. For a large enough $c$ the probability a given $m\times m$ square contains an open cluster of size greater than $c\log{(m)}$ decreases polynomially in $m$, and the degree of the polynomial in an increasing function of $c$ (which grows to $\infty$ as $c$ does), see [@grimmett]. Since the forest has a positive density, if we consider a given $m\times m$ square around a vertex, conditioned this vertex is in the forest, we get a similar bound. Take some vertex $u$, conditioned to be in the forest. since $S(u)$ is a single-infinite path, if we consider the intersection of $S(u)$ with the annulus $A(u,t,C) := \{v ~s.t.~ t\leq \|u-v\|_\infty \leq t+C\log{(t+1)}\}$, then there is some connected path in this intersection which connects the internal and external boundary. In particular it length is at least $C\log{(t+1)}$. Due to the above remarks, for large enough $C$, at least two of its edges will have an interaction of at least $\delta$, with very high probability. Thus, $C$ can be taken as such that Inequality \[eq:perc ineq\] is valid. Denote by $p_t ^{R}$ the probability that for the origin $O$ (or any other vertex), which is conditioned to be in the forest, there is a vertex $u\in R(O)$, at $l_\infty$ distance $t$ from $O$, and moreover $x(u)-x(0) = t$, where $x$ denotes the $x$-coordinate of the vertex (in words - the distance in achieved in the $x$-axis, from the $\mathbf{right}$), and that the edge which connects $O$ to $S(O)$ has an interaction at least $\delta$. Similarly we define $p_t ^{L}$, $p_t ^{U}$, $p_t ^{D}$ (L stands for left, U - up, D - down). It is trivial that $$\label{eq:trivial eq} p_t ^{R} + p_t ^{L} + p_t ^{U} + p_t ^{D} \geq p_t$$ We now finish the argument in a similar manner to that of Theorem \[thm:single-infinite at least boundary sized or nlogn\]. Consider a large $n\times n$ square around $O$, the expected sum of edges whose interaction is at least $\delta$ and that lay on a simple path that is contained in the forest and connects two boundary points of the square, which is a lower bound to the number we are chasing after, is at least $\Delta := \sum_{\gamma\in\{L,R,U,D\}}\sum_{t = 1}^{t=\lfloor(n-1)/2\rfloor} (n-2t)p_t^{\gamma}$. The reason is that $e={v,u}$ is an edge (with $u\in S(v)$) which lays on a simple path that connects two boundary edges, iff $R(v)$ intersects the boundary, as $S(v)$ always intersect it. Due to Equation \[eq:trivial eq\] $\Delta \geq \Delta_1 := \sum_{t = 1}^{t=\lfloor(n-1)/2\rfloor} (n-2t)p_t$. Due to Equation \[eq:markov eq\] we have $\Delta_1 \geq \Delta_2 := \sum_{t = 1}^{t=\lfloor(n-1)/2\rfloor} (n-2t)E_t/4t$. And due to Inequality \[eq:perc ineq\] we have $\Delta_2 \geq \sum_{m=1}^{N(n)} (n-2s_m)/4s_m$, where $s_m$ is the sequence that is defined by $s_1=1+C\log 2$, $s_k = s_{k-1} + C\log{(s_{k-1}+1)}$, and $N(n)$ is the last such $m$ with $s_m < n/3$. It is clear that $\Delta_2 \geq n\sum_{t=1}^{n/(C'\log(n))} (C''t\log(t+1))^{-1}$, and this is superlinear, as $\sum_{m=1}^{\infty}(m\log(m))^{-1})$ diverges. And the result follows. Thus, the forest is made only of finite components Families of Graphs with Unique GSPs {#sec:Graphs with Unique GSPs} ----------------------------------- $~$\ In this section we prove two main results. The first is that under a variety of conditions, cylinders and deformed cylinders have only one GSP. The second shows a sufficient condition for planar graphs to have only one GSP. In a future paper we generalize the planarity condition to wider families of graphs. Throughout the thesis we state the results and prove them assuming planarity of graphs, as it is easier both to formulate and to visualize. \[thm:one ground state in cylinders\] Let $G$ be a connected deformed cylinder that each of its slices is connected as well. Assume that there exist some $N\in\mathbb{N}$ such that there are both infinitely many positive levels and infinitely many negative levels whose slices have the following property: For each of these slices there are no more than $N$ edges touching it or contained in it. Then $G$ has exactly one GSP. Due to what was said above, we shall prove only the uniqueness. Assume, in order to reach a contradiction, that there are at least two GSPs. Remember that a GSP is actually a pair of configurations, which differ only by a global flip. Choose a representative to each GSP. Let these representatives be $\mathcal{C}_1,\mathcal{C}_2$. We can divide the vertices of $G$ into two sets - the set of vertices where $\mathcal{C}_1,\mathcal{C}_2$ agree, and the set where they do not agree. We can divide each of these two sets into connected clusters according to connectivity in the graph. Note that for any such a cluster from one set, all of its neighbors must belong to cluster from the other set, as if there where two different clusters from the same set, with an edge between them, since the division was according to connectivity in $G$, these two clusters must have been the same cluster. An immediate consequence is that an edge has a different value in $\mathcal{C}_1,\mathcal{C}_2$ if and only if it connects two different clusters. If there is only one cluster, we are done, and there is only one GSP. On the other hand, if there are more than one cluster, each cluster must be infinite. Indeed, if there were a finite cluster, the sum of the values of the edges which connect it to the other clusters (there are such edges, since the graph is connected) must have been non zero (since the coupling distribution is continuous). But then in one of the two GSPs it should have been negative (this amount in $\mathcal{C}_1$ is minus the amount in $\mathcal{C}_2$). Hence, in one of the two ground states we could have reduced the energy by flipping a finite set of spins. We should now notice that any vertex $v$ belongs to a finite connected subgraph of $G$ which is bounded by two slices, each of them has no more that $N$ edges, each edge with the property that one of its vertices belongs to the slice, where $N$ is the integer from the formulation of the theorem. Indeed, it follows from the assumptions of the theorem that if $v$ is of level $a$, there are integers $b,c$ with $b<a<c$ such that the slices in the levels $b,c$ have the above property. Note that any edge from this subset to its complement in the graph must have one vertex in one of these border slices, and one vertex “outside”. The main observation is that there exist a positive $\varepsilon=\varepsilon(N)$ such that any slice with the above property is fixed, with probability at least $\varepsilon$. Indeed, there are only finitely many isomorphism classes of graph to that slice. Consider such a class. Consider some specific spanning tree of it. Choose an order on its vertices, such that the vertex in the $i^{th}$ place, has all its neighbors (in the tree!) in places $j\leq i$, except at most one neighbor. Now, there is a positive probability such that for any $i$, if it has an edge in the tree, $e$, which is to a neighbor in higher place in the order, then the absolute value of the coupling in this edge is higher than the sum of all the edges which have at least one vertex in the slice, except for those in the spanning tree, together with the sum of absolute values of all couplings of the edges of $i$ in the tree (other than $e$). But in this case the tree is fixed. As there are only a finitely number of isomorphism classes of the slices’ graph (and we needed no information about the rest of $G$, except the bound for the number of edges which touch the slice), there exist a positive $\varepsilon=\varepsilon(N)$ such that any slice with the above property is fixed. It should be noted that the spanning tree and the order were crucial. If we had not have such an order, we could not have guaranteed a fixed spanning subgraph. If the tree were not spanning, it might have happened that only parts of the graph would have been fixed. Thus, standard arguments show that with probability $1$ (on the interactions) any vertex belongs to a finite subgraph which is bounded by two fixed slices (and the boundary edges of this subgraph are only those from these two subgraphs to the rest of the graph. A fixed edge has the same value in any GSP, and the same holds for a fixed slice. We can find an order preserving, injective and surjective map, $S$ from the set of fixed slices to $\mathbb{Z}$, where the order of the slices is induced from the order of their levels. We remember that any of the above clusters must be infinite. From the last remark it follows that any fixed slice must be contained fully in the interior of a cluster, and there is no cluster which is contained in a finite subgraph which is bounded from both sides by fixed slices (it must be infinite). In addition, due to the connectivity of clusters, the images under $S$ of the fixed slices which are contained in it must be in the form $\{z\in\mathbb{Z}\| a<z<b\}$ where $a,b\in\mathbb{Z}\bigcup\{-\infty,\infty\}$. From here it is immediate that there are at most two clusters. Indeed, if there were three, one of them must have contained only fixed slices whose image under $S$ is of the form $\{z\in\mathbb{Z}\| a_0<z<b_0\}$ where $a_0,b_0\in\mathbb{Z}$. But then this cluster is contained in the finite subgraph of $G$ which is bounded between the slices $S^{-1}(a_0),S^{-1}(b_0)$, and in particular be finite. ![Clusters of agreement$\backslash$ disagreement between two GSPs - Theorem \[thm:one ground state in cylinders\]](GSP_deformed_cyl){width="70.00000%"} We are left with case that there are two clusters. Assuming there were two, the sum of values of boundary edges between them (which is finite, as it belongs to a finite subgraph bounded between two fixed slices) must be positive in one GSP and negative in the other: denote by $h$ the value of that sum in one GSP, the corresponding sum in the other GSP will be $-h$. As the coupling distribution is continuous, $h\neq 0$ a.s. With probability $1$ there is a slice with at most $N$ edges which touches it or belongs to it, and has no edge in common with this boundary, and that each of its edges has an absolute value less then $\|h\|/(N+1)$, where $h$ is the sum of values of boundary edges in $\mathcal{C}_1$. Assume, without loss of generality. that $h<0$. Note that this boundary and the slice we have just describe are a boundary of some subgraph of $G$. But the sum of the values of this subgraph’s boundary edges must be at most $-\|h\|/(N+1)$, hence, flipping all of this subgraph’s spins reduces the Hamiltonian, in contradiction to the assumption that $\mathcal{C}_1$ is a ground state. Similar considerations lead to the next result (we formulate it for cylinders, for clarity, though it can be stated for deformed cylinders that satisfy conditions similar to those of the above theorem): \[thm:no infinite unsat cluster in cylinders\] Let $G$ be a connected cylinder. Then for almost every realization of couplings, in the GSP of $G$ there is no infinite connected cluster of unsatisfied edges. $\mathbf{Sketch}$. There is a positive probability that a pair of consecutive slices is fixed, and moreover, all the edges connecting these slices are fixed to be positive. Thus, any vertex belongs to a finite subgraph which is bounded by two such pairs of slices. But an infinite connected cluster which contains this vertex must contain at least on edge from connecting the two slices in one of these pairs. But this edge cannot be negative. The next theorem use a similar property to prove uniqueness of GSPs for many planar graphs. This result may be extended to much wider families of graphs. Let $N$ be some fixed integer and let $v$ be a vertex. A cycle (a simple closed path) $\mathcal{C}$ is said to $\emph{N-choke}$ (or simply $\emph{choke}$) $v$ if $\mathcal{C}$ surrounds $v$ and the number of edges which lay on it or touch a vertex in the it is smaller than $N$. A locally finite planar graph $G$ is said to be $\emph{choked}$ if for every vertex $v$ in the graph, there is a positive integer $N=N(v)$, and there is a sequence of cycles in the graph, $\{\mathcal{C}_n\}_{n\in\mathbb{N}}$, where each $\mathcal{C}_{n}$ ($N-$)chokes $v$ and such that for all $n\in\mathbb{N}$, $\mathcal{C}_{n+1}$ surrounds $\mathcal{C}_{n}$. In the above definition, we used the term ’locally finite graph’ for describing a planar graph such that in any compact subset of the plain there are at most finitely many of its vertices. By ’surrounded’ we meant that the surrounded object falls into the finite connected portion of the plane after the deletion of the surrounding object. \[clm:choked planar have fixed cycles\] In any choked graph $G$, with probability $1$, any vertex is surrounded by a sequence of infinitely many fixed cycles (by that we mean that the edges of the cycle are fixed) such that each cycle surrounds the cycles which come before in the sequence. Let $v$ be a vertex, $N=N(v)$ be as in the above definition, and $\mathcal{C}$ any cycle $N$-choking $v$. There is a positive probability, which is bounded from $0$ by a positive function of $N$ that $\mathcal{C}$ if fixed, this can be seen similarly to Theorem \[thm:one ground state in cylinders\]. Now standard considerations like Borel-Cantelly finish the argument as the event that $\mathcal{C}_n$ is fixed is independent of the event that $\mathcal{C}_m$ is fixed for far enough $n,m$. \[thm:one ground state in choked graphs\] Let $G$ be a choked graph then with probability $1$ it has only one GSP. Again, there is at least one, from compactness. Assuming there were at least two, consider as in Theorem \[thm:one ground state in cylinders\] one representative for each GSP and the division of the vertices to those which have the same spin under the two representatives and those which have opposite spins. Again we divide into connected clusters, and again each connected cluster must be infinite. Again no fixed edge can connect two different connected clusters. Due to planarity we may consider the dual edges to those which connect different clusters. These must divide the plane into several connected regions, each of which should contain an infinite number of dual faces - since it must contain an infinite number of vertices, and hence no subset of dual edges can form contain a closed cycle. Hence, any dual edge between two clusters lays on a bi-infinite simple path made of dual edges that are between two different clusters. We call such a path in the dual graph a $\emph{domain-wall}$. Note that two domain walls may intersect (the domain walls are not necessarily disjoint paths). Let $v$ be a vertex on an edge whose dual edge, $e^{}$ is in a domain wall (i.e. or the two representatives agree on $v$’s spin but disagree on the spin of one of its neighbors, or the opposite). This dual edge, $e^{}$, is surrounded by some fixed cycle, with probability $1$, by the previous lemma. Thus, any domain wall which contains this dual edge must either intersect the choked cycle or stay only in the finite part surrounded by it. It cannot intersect, since the intersection point would lay on a fixed edge, whose dual is in a domain wall, but this is impossible. The other option is impossible again, as this domain wall must be an infinite simple path, but there are only finitely many points in the region that is bounded by the fixed cycle. Regular Trees Have infinitely Many GSPs {#sec:inf GSPs in trees} --------------------------------------- $~$\ This section explores the nature of spin-glasses over trees. For convenience, we prove our results only for regular trees, although they can be trivially extended to much wider families of trees. We show that, under our usual model, regular trees of degree at least $3$ have uncountably many GSPs. This is surprising as it is ’clear’ there should be only one GSP, the one where every edge has a positive value (in trees, since there are no loops, this can always be done). The only obstruction is that GSP need not to be a global minimizer of energy, but only a local minimizer - it should be ’better’ then all the configurations which differ from it by finitely many spin choices. Our main technique is percolation theory for trees. \[thm:trees have infinitely many GSPs\] Under the model described in Subsection \[sec:Prelim-spinglass\], in the d-regular tree, for $d\geq 3$, for almost every coupling there are infinitely many GSPs. We begin with an ad-hoc definition. We say that an edge $f$ in an infinite graph is a $\emph{bridge}$, if after deleting it from the graph, but leaving its vertices in the graph, the graph is divided into two connected infinite subgraphs. Let $T$ be the regular tree, and let $\mathcal{C}$ be one of the two configurations which create the trivial GSP, the one where all the edges have positive value. We do the following: Take some light edge $e$ (the coupling has very small absolute value) with value $h$. Flip all the vertices in one connected cluster of $T\setminus \{e\}$. The only edge which changed its value was $e$. Because all the other edges have positive value, the new configuration will not be a ground state iff there are some edges $\{e_1,...,e_n\}$ whose sum of values is smaller than $h$, and such that if we delete only them from the tree, we have no finite connected cluster, but if we remove both them and $e$ then there is (exactly one) finite connected cluster. Flipping the finite cluster they bound, reduces the energy. Because all the other edges have positive values if the sum of the values of $e_i$ is smaller than h, each of them should be smaller than $h$ as well. Now, for $h$ small enough, percolation reasons show that there is a positive probability that the cluster which contains $e$ and only the edges of value at least $h$ is infinite and $e$ is a bridge in it. Indeed, if the degree of vertices in the tree is $d$, it is a common knowledge (e.g. [@grimmett]) that a percolation on the $d$-regular tree, with $p>1/(d-1)$, where $p$ is the probability we do not erase an edge from the graph, has infinitely many infinite connected clusters, and in particular, any vertex or edge has positive probability to be contained in one of them. Moreover, for any edge there is a positive probability to be a bridge in the cluster. For $h$ small enough, the probability an edge coupling has absolute value higher than $h$ is as high as we wish, and from the above it is clear there is a positive probability that in the graph obtained from deleting from the tree all the edges of interaction smaller than $h$ (except, maybe, the edge $e$), $e$ is a bridge in an infinite connected cluster. In this case, sets like $\{e_1,...,e_n\}$ with the property described above do not exist. Hence, if the described cluster happens to be infinite, $e$ happens to be a bridge in it, and we flip all the spins in one side of the edge $e$, we get a new ground state. There are infinitely many edges where the described event happens, and hence there are infinitely many GSPs (which are clearly different). With similar techniques one can even show the claim below. \[thm:trees have uncountably many different GSPs\] Under the model described in Subsection \[sec:Prelim-spinglass\], in the d-regular tree, for $d\geq 3$, for almost every coupling there are uncountably many GSPs. This is not surprising, as there is not much difference between Ising spin-glass on trees and ferromagnetic Ising on trees. And even in $\mathbb{Z}^{d}$-lattices, in the Ising ferromagnetic model, there are infinitely many GSPs. On the other hand, we show below an important difference between the ferromagnetic Ising on trees and on lattices: There is a translation-invariant measure, supported on uncountably infinitely many GSPs, for trees, while for lattices (and more generally for Cayley graphs of amenable groups) in any translation invariant measure in supported on one single GSP. The latter result will appear again in Section \[sec:dynamics-loops\] when we discuss the Loop dynamics. We shall use the following lemmas a couple of times in this work: \[lem:very biased perc cofinite\] Let $T$ be a $d$-regular tree, for some $d\geq 3$. Then there is some $1>p>0$ s.t. in the Bernoulli($p$) vertex (edge) percolation the complement in $T$ of the union of infinite clusters is made of finite clusters only. This is of course much stronger than saying that there are infinite clusters in the percolation. $\mathbf{Sketch.}$ We prove for site percolation, minor changes apply for bond percolation as well. We consider the percolation as coloring. A vertex is colored black with probability $p$ and otherwise white. The proof is based on the following simple facts that we do not prove:\ I. Given that a black vertex is in a finite cluster or that it is in a finite cluster and has (at least) one white neighbor, the expected size of this cluster is bounded by $1+\alpha (p)$, where $\alpha(p)\rightarrow 0$ as $p\rightarrow 1$ is a nonnegative, continuous function of $p$.\ II. Given that a vertex is white, the expected size of the white cluster containing it is (for $p$ close enough to $1$) $1+\beta (p)$, where again $\beta \rightarrow 0$ as $p \rightarrow 1$.\ III. Given that a black vertex $v$ has one white neighbor, the probability that the black component which contains it is finite is $\gamma (p)$, where $\gamma$ is a nonnegative function of $p$ which tends to $0$ as $p$ tends to $1$. Assume for example that a given vertex $v$ is black and lays in a black finite component. Then the size of this component is expected to be at most $1+\alpha$ (we do not write $p$ as it is fixed). All the neighboring clusters are white. There are at most (and actually, less than) $d ( 1+\alpha (p) )$ such clusters. The expected total sum of their sizes is no more than $d ( 1+\alpha (p) )(1 + \beta )$. And the last number is also a bound for the number of neighbors all these clusters have (together). All these neighbors are black, and hence, the expected number of finite clusters among these is no more than $d ( 1+\alpha (p) )(1 + \beta ) \gamma$. This number tends to $0$ as $p \rightarrow 1$, and hence, starting at some point, it is less then $1$. If we now continue the same analysis with any of the black neighbors which are parts of finite black components, we get a similar bound for the number of finite black components which are neighbors of the white components which are neighbors of this new black vertex (and we do not count clusters we have already reached), and we can continue like this ad infinitum. But what we have shown is that the complement of the infinite clusters is stochastically dominated by a sub-critical Galton-Watson process, and hence finite a.s. Only slight changes are needed if we start with a white vertex instead of black. \[lem: very biased - even majority cofinite\] Consider a Bernoulli $p$ site percolation on a $d$-regular tree $T$ as a coloring s.t. with probability $p$ a vertex is colored black and otherwise white (independently of the others). Color by red the vertices that the majority of their neighbors and them are colored black. Then if $p$ is close enough to $1$, the graph which is the result of removing the red vertices from $T$ (and edges which touch them) has only connected clusters of finite size. $\mathbf{Sketch.}$ In [@LiggettStaceySchonmann] it is shown that if we color vertices in two colors according to the product measure of Bernoulli(p) distributions and then recolor according to vertices which the majority of their neighbors and them is in some given color, then for any $1>q>0$, there is a $1>p>0$ s.t. the resulting recoloring, considered as site percolation, stochastically dominates the Bernoulli $q$ site percolation (Actually they show there something much more general then this majority process). Combining this with the above lemma finishes the argument. \[thm:translation invariant GSPs in trees\] Let $T$ be a $d$-regular tree for $d\geq 3$, and consider the model of Subsection \[sec:Prelim-spinglass\], where the distribution of the interactions is an i.i.d. continuous probability measure whose support is $\mathbb{R}_+$. Then there exist a translation-invariant measure $\mu$ supported on uncountably many GSPs. Of course, the positivity of the interactions is not important. $\mathbf{Sketch}$. First note that knowing which edge is satisfied is equivalent to knowing the GSP. There exists some small $\varepsilon > 0$ with the following property: Given an edge $e$ with interaction smaller than $\varepsilon$ in the tree, there is only a finite number of finite connected subgraph of the tree s.t. $e$ is in their boundary and less than half of the boundary edges have interaction smaller than $\varepsilon$. Indeed, there is an exponential number of connected subgraphs of the tree containing $e$ as a boundary edge having some given side. For small enough $\varepsilon$ the probability that one of them has more than half of its boundary edges having interaction smaller than $\varepsilon$, is exponentially small, with exponent base as small as we wish. In particular, all these probabilities are summable for small enough $\varepsilon$. A Borel-Cantelly argument shows that there are indeed only a finite number of finite connected subgraphs of the above type. Moreover, if $\varepsilon$ is so small such that the above sum (which is actually the expected number of “bad” finite connected subgraphs) is smaller than $1$, then with probability $1$ there are edges $e$ with no finite connected subgraphs of the above type. For any such edge flip a coin whether this edge is made satisfied or not. It is easily seen that we have created a translation invariant measure of GSPs. It is indeed a GSP as in every connected subgraph whose boundary contains an unsatisfied edge, most boundary edges have interaction at least $\varepsilon$. The remaining edges, even if all of them are flipped are dominated both in their number and in their sum (in absolute value). \[thm:single GSP in Z\^d if J=1\] Let $\mu$ be a translation invariant measure of ground states of the EA Ising spin glass system over $\mathbb{Z}^{d}$, with all the interactions being positive, then $\mu$ is supported on a single GSP, the one with all the spins the same. $\mathbf{Sketch}$. Assume there is another different GSP in the support of $\mu$, let $\sigma$ be one of the two representatives for it. In the dual complex $\mathbb{Z}^{d}$, construct the complex made of pieces of codimension $1$, which distinguish between areas of positive spins to those of negative spins in $\sigma$. By a dual piece of codimension $1$ we mean the hypercube of codimension $1$ and edges of length $1$, perpendicular to some edge of the graph $\mathbb{Z}^{d}$ and passes through the middle of that edge. We call those pieces which distinguish between vertices of different spin $\emph{domain walls}$. Consider a large box of side $N$ around the origin. Translation invariance implies that inside that box there are $\Theta(N^{d})$ pieces of the domain walls. Note that any such piece corresponds to an edge whose weight is negative (its two spins are opposite). The sum of all of them is expected to be $\Theta(N^{d})$ as well. The sum of all the pieces of the boundary of the box (not only those that may happen to be in domain walls) is $\Theta(N^{d-1})$. The domain walls divide the box into domains of same sign, such that any two neighboring (by a piece of the wall) domains have opposite spins. Any piece of the domain wall which is inside the box is hence counted exactly twice as boundary of a domain. But it follows that the sum of the weights of the boundaries of the domains (including the boundaries of the box, which are counted once) is negative. Thus, The Hamiltonian of one of these domains must be positive. Flipping all the spins in that domain reduces the Hamiltonian, in contradiction to $\sigma$ being a ground state. Actually one can achieve the above result for any amenable Cayley graph. Zero Temperature Glauber Dynamics in Statistical Mechanics Problems =================================================================== $~$\ Background {#sec:dynamics-background} ---------- $~$\ Dynamic evolution in statistical mechanics has become a subject of interest in the last decades. A description of the system at equilibrium and estimates on the time until such equilibrium is reached are of paramount importance here as it appears in many other scientific areas (e.g. biology, computer science, economics) where dynamical models are relevant as well. Common among such models in statistical mechanics, is the so-called $\emph{Glauber Dynamics}$, which can be used to describe the evolution in many spin-systems with a Boltzmann-Gibbs law at stationarity (such as Ising, Potts, etc.). This part of the work is devoted to obtaining new results about Glauber dynamics for the zero temperature Ising model on several infinite graphs. We introduce results and techniques which extend earlier works of other authors, e.g. [@NandaNewmanStein],[@GandolfiNewmanStein] [@CamiaSantisNewman]. Preliminaries {#sec:dynamics-prelim} ------------- $~$\ Let $G$ be a graph. $\mathcal{X}$ = $\{-1,1 \}^{G}$ will denote the space of all configurations (spin choices) over $G$. The Ising model on $G$ corresponds to a Gibbs distribution on configurations with Hamiltonian $$\label{eq:Hamiltonian} \mathcal{H}(\sigma) = -\sum_{x\thicksim y} \sigma_x\sigma_y$$ where summation is over all pairs of adjacent vertices. The $\emph{Glauber Dynamics}$, for the Ising model on $G$ in zero temperature is a Markov processes $\sigma_{t}$, on $\mathcal{X}$. Its evolution is governed by independent Poisson clocks with rate $1$ which are assigned to every vertex $x \in G$. When such a clock rings a spin flip is $\emph{considered}$. If the change in the energy $$\Delta\mathcal{H}_x(\sigma) = 2\sum_{y\thicksim x} \sigma_x\sigma_y$$ is negative (positive) a spin change in $x$ is done with probability 1 (0). In the case that the energy change is zero, a coin is tossed in order to decide whether to perform a flip or not. All coin toss outcomes are independent and independent of the initial configuration and the ring times. A flip which changes the energy is called an $\emph{energy-reducing flip}$. In other words, the configuration process $\{\sigma_t(v) ;\; t \geq 0, v \in V\}$ is almost surely completely determined, given the following: - $\sigma_0(v) ;\; v \in V$ - initial configuration - $\tau_k(v) ;\; k \geq 1, v \in V$ - rings times. - $\beta_k(v) ;\; k \geq 1, v \in V$ - coin toss outcomes. For convenience we will assume that our sample space $\Omega_V$ is the set of all possible assignments to these values. For instance, one can identify $\Omega_V$ with the space $$\big( \{-1,+1\} \times {{\mathbb{R}}}_+^{{\mathbb{N}}}\times \{-1,+1\}^{{\mathbb{N}}}\big)^V$$ and $\varpi = (\sigma_0(v), \tau_k(v), \beta_k(v) ; \; k \geq 1, v \in V) \in \Omega_V$. We denote by $\omega = (\tau_k(v), \beta_k(v) ; \; k \geq 1, v \in V)$. The standard Borel sigma algebra on $\Omega_V$ will be denoted ${{\mathcal{F}}}_V$. If $U \subset V$ we shall treat ${{\mathcal{F}}}_U$ as a subset of ${{\mathcal{F}}}_V$. A collection $\{\alpha(v) ; \; v \in V\}$ will often by viewed as a function $\alpha$ on $V$. If $U \subset V$, we shall write $\alpha\|_U$ for its restriction to $U$. If $\alpha_1, \alpha_2$ are functions on $U$ and $V \setminus U$ resp., we shall write $\alpha_1 \times \alpha_2$ for the function on $V$ which is given by their superposition. We shall employ this notation “point-wise” for time-evolving functions on $V$ (e.g. $\sigma_t$) as well. In the presence of boundary conditions on $U \subset V$, sites $v \in U$, have their values set to a prescribed value $$\sigma_t(v) \equiv \pm 1 \; \forall t \geq 0$$ and a prescription of these will be given as a configuration $\xi \in \{-1, +1\}^U$. Given the graph $G=(V,E)$ and possibly boundary conditions $\xi$ on $U \subset V$, the function which maps an $\varpi \in \Omega_V$ to the evolution it generates $(\sigma_t)_{t \geq 0}$ will be denoted by $\Sigma_G^\xi$. $$(\sigma_t)_{t \geq 0} = \Sigma_G^\xi(\varpi)$$ It will be denoted $\Sigma_G$ in the absence of boundary conditions. $\Sigma_G$ is in fact well defined only on a subset of $\Omega_V$, one which has measure one. This can be done using percolation techniques, and may be found in classical textbooks. We prove a much more general argument in Section \[sec:dynamics-loops\]. is standard (percolation arguments). It is more convenient to have the $k$-th coin toss outcome at vertex $v$, i.e. $\beta_k(v)$ be associated with the $k$-th ring at that vertex. Thus at the $k$-th ring, if a there’s a tie $\beta_k(v)$ is used and otherwise it is discarded. $\mathbf{P}_\omega$ denote the probability distribution on the realizations $\omega$ of the ring times and the coin tosses, and by $\mathbf{P}_{\sigma_{0}}$ the distribution of the initial configuration. Unless stated differently we assume that $\mathbf{P}_{\sigma_{0}}$ is just the product measure of symmetric Bernoully variables. The joint distribution is $\mathbf{P}_\varpi = \mathbf{P}_{\sigma_{0},\omega}=\mathbf{P}_{\sigma_{0}}\times\mathbf{P}_\omega$. We say that the dynamics over $G$ $\emph{freezes}$ if the limit $\sigma_{\infty}(\sigma_{0},\omega)$ = $\lim_{t\rightarrow\infty}\sigma_{t}(\sigma_{0},\omega)$ exists pointwise. In other words, each vertex flips finitely many times. If the above limit does not exist, but some vertices do freeze (flip at most finitely many times) and those which do not freeze do not form infinite connected clusters, we say that the dynamics $\emph{almost freezes}$. When some vertices freeze but those which do not freeze form infinite clusters we say that the dynamics $\emph{partially freezes}$. We say that a subset of vertices is $\emph{possibly self freezing (PSF)}$ if with probability $1$ no flip occurs in any of its vertices when they are initially all set to the same value (either all $+$ or all $-$). A cylinder is said to $\emph{freeze}$ $\emph{in}$ $\emph{slices}$ if it freezes and in the limit $\sigma_{\infty}$ all vertices in each slice have the same spin value. A vertex which stops flipping after some finite time is said to be, after that time $\emph{frozen}$. Main Results {#sec:dynamics-main res} ------------ $~$\ In [@NandaNewmanStein] it is shown that on many families of graphs the dynamics freezes a.s., when the initial configuration is i.i.d. symmetric $\pm 1$. In particular this includes all odd-degree regular trees. Our first result shows that this is not true for regular trees with even degree, i.e. with positive probability a site will flip forever. We also discuss the case of a biased i.i.d. initial configuration. Next we examine other graphs and show that under mild assumptions, for any bounded degree graph $G$ there exist some $N \in \mathbb{N}$ such that for any $n>N$, $G\times K_n$ freezes almost surely. We end this section with results and constructions which are related to cylinders. We show that any cylinder (with finite connected slices) almost freezes a.s. We also show that the $C_n$-cylinder and the $K_n$-cylinder freeze a.s. and moreover - freeze in slices a.s. We show, on the other hand that there are cylinders which do not freeze and some which freeze but not in slices (even when the slice is a transitive graph). Glauber Dynamics on Regular Trees with Even Degrees {#sec:dynamics-reg trees} --------------------------------------------------- $~$\ Here we investigate the zero-temperature Glauber dynamics on trees. It follows from [@NandaNewmanStein] that this dynamical process on a regular tree of odd degree freezes a.s. We show the complementary result that a.s. the dynamics does not freeze when the tree is regular of even degree (by that we mean that at least some vertex flips indefinitely), as long as our initial spin distribution is the product measure of symmetric $\pm 1$ distributions. We show that if the initial distribution is i.i.d. but non-symmetric, with probability $1$ - either the dynamics does not freeze, or it freezes to a configuration where all the spins are the same. In addition, we show that in the above scenario, if the initial distribution is biased enough, a.s. the dynamics freezes to the same-spin configuration. It is worth mentioning that a recent work of [@CAVITY] shows that if instead of Poisson clocks, we work in the simultaneous scenario (where all the clocks ring together at integer multiples of time units), then even if the bias in the initial spin distribution is small, the dynamics freezes to the same-spin configuration. This leads to the next conjecture (supported by simulations in [@CAVITY]). If the distribution of the initial spin configuration is the product distribution of biased Bernoulli variables then the zero-temperature Glauber dynamics on a regular tree of even degree freezes a.s. to a same spin configuration. The most inovative result of this subsection is the following. \[thm:even-regular trees do not freeze\] Consider the zero-temperature Glauber dynamics on a $d$-regular tree where $d$ is even. Assume that $\mathbf{P}_{\sigma_{0}}$ is the product measure of symmetric $\pm 1$ random variables. Then with $\mathbf{P}_{\sigma_{0},\omega}$-probability $1$ the dynamics does not freeze, i.e. a.s there exist a vertex which flips its spin indefinitely. Before proving this theorem we shall need several straight-forward propositions. We use the standard order for vectors and functions. \[prop:Monotonicity\] For any $G=(V,E)$, $U \subset V$ and $\xi \in \{-1, +1\}^U$. - $\Sigma_G^\xi$ is monotone increasing in $\sigma_0$, $(\beta_k)_{k \geq 1}$. - If $\xi_0 \geq \xi$ then $\Sigma_G^{\xi_0} \geq \Sigma_G^\xi$. and if $\xi \equiv 1$ on $U$ then $\Sigma_G^\xi\geq \Sigma_G$. If $G=(V,E)$ and $U \subset V$, then $G \setminus U$ is the graph obtained from $G$ by removing all vertices in $U$ and any edges the connect such vertices. For $v \in V$, we denote by $G(v)$ the connected component in $G$ which contains $v$. We shall write $G_U(v)$ as a short for $(G \setminus U)(v)$. If $G_0 = (V_0, E_0)$ is a sub-graph of $G$ then its closure $\overline{G_0}$ (with respect to $G$) is obtained from $G_0$ by adding all neighbors to vertices of $G_0$ which are not in $V_0$ and the edges that connect them. The following is immediate. \[prop:Independence\] Let $G=(V,E)$, $U \subset V$ and $\xi \in \{-1, +1\}^U$. Suppose that $G \setminus U$ breaks into $k$ connected components $G_1 = (V_1, E_1), \dots G_k = (V_k, E_k)$. Then $$\Sigma_G^\xi (\varpi) = \Sigma_{\overline{G_1}}^\xi (\varpi\|_{V_1}) \times \dots \times \Sigma_{\overline{G_k}}^\xi (\varpi\|_{V_k})$$ where ’$\times$’ has the obvious meaning. Let $x \in V$. If everything freezes with probability $1$, then with positive probability $\sigma_t(x) = 1$ for all $t \geq t_0$, where $t_0 > 0$ is a deterministic number. From Proposition \[prop:Monotonicity\], for all $\varpi$ such that this happens, also $\sigma^{y+}_t(x) = 1$ for all $t \geq t_0$, where $$(\sigma^{y+}_t)_{t \geq 0} = \Sigma_G^{\xi^{y+}} (\varpi),$$ $\xi^{y+}$ is the boundary condition $\xi^{y+}(y) = +1$ and $y$ is some neighbor of $x$. Now enumerate all the neighbors of $y$ as $x_1, \dots, x_{d-1}$ and $z$. Since we have just shown that events $\{\sigma^{y+}_t(x_i) = 1 ; \; \forall t \geq t_0\}$ happen with positive probability and since they are ${{\mathcal{F}}}_{G_y(x_i)}$-measurable by Proposition \[prop:Independence\], and therefore independent, the following has a positive probability as well $${{\mathcal{A}}}^+_z(y) = \left \{ \begin{array}{l} \sigma^{y+}_t(x_i) = 1; \; \forall t \geq t_0, i=1, \dots d-1 \\ \tau^1(y) > t_0 ,\, \sigma_0(y) = +1 \end{array} \right\}$$ But ${{\mathcal{A}}}^+_z(y) \subseteq \{\sigma_t(y) = +1 ;\; \forall t\geq 0\}$. Event ${{\mathcal{A}}}^-_z(y) \subseteq \{\sigma_t(y) = -1 ;\; \forall t\geq 0\}$ is defined similarly. Now suppose $z$ has neighbors $y_1, y_2, \dots y_d$. Since ${{\mathcal{A}}}^\pm_z(y_i)$ are ${{\mathcal{F}}}_{G_z(y_i)}$-measurable, they are independent and therefore we can have ${{\mathcal{A}}}^+_z(y_i)$ for all $i \leq d/2$ and ${{\mathcal{A}}}^-_z(y_i)$ for all $i > d/2$ occur at the same time with positive probability. But then $z$ must flip infinitely many times, as its neighbors are constantly in a tie. \[thm:tree-non symmetric case\] Consider the zero-temperature Glauber dynamics on a $d$-regular tree for even $d$. Assume that $\mathbf{P}_{\sigma_{0}}$ is the product measure of strictly-biased $\pm 1$ random variables. Then either the dynamics does not freeze with probability $1$ (a.s there exist a vertex which flips its spin indefinitely) or with probability $1$ it freezes to a same spin configuration. $\mathbf{Sketch.}$ The above proof shows, in fact, that if we assume a.s. freezing then either a.s. all the vertices freeze to $+1$ or a.s. all the vertices freeze to $-1$. Otherwise we could again find a vertex with half of its vertices frozen to $+1$ and the other half frozen to $-1$. Such a vertex would flip infinitely many times. If the bias is towards $+1$, then by monotonicity and standard coupling arguments, it is easy to see that the probability that all the vertices freeze to $+1$ is at least the probability they freeze to $-1$. Hence, in this case, either the configuration does not freeze a.s., or a.s. it freezes to the uniform configuration where all the spins are $+1$. \[thm:strongly biased freezes\] Consider the zero-temperature Glauber dynamics on a $d$-regular tree for even $d$. Then there exist some positive $p=p(d), 0<p<1$ such that if $\mathbf{P}_{\sigma_{0}}$ is the product measure of i.i.d. variables with probability $p$ for $+1$, then with $\mathbf{P}_{\sigma_{0},\omega}$-probability $1$ the dynamics freezes to the all $+1$ configuration. $\mathbf{Sketch.}$ There is a positive $p_1<1$ such that if $p > p_1$, then with $\mathbf{P}_{\sigma_{0}}$-probability $1$, any region of spins $-1$ is finite at time $t=0$. Moreover, there exist an even larger $p_2$, such that if $p > p_2$ then the complement of the set of all vertices whose spin is initially $+1$ and more than half of their neighbors are $+1$ has only finite connected components. This is due to Lemma \[lem: very biased - even majority cofinite\]. Clearly, any vertex in this $+1$ set will be $+1$ for all $t \geq 0$. On the other hand, any finite cluster in the complement is surrounded by by vertices whose spin is $+1$ for all $t \geq 0$ and therefore after finite time will have all its vertices $+1$ forever. Indeed, all the leaves of such components must turn permanently $+1$ after finite time and therefore, by induction, also the entire component. Glauber Dynamics on Product Graphs {#sec:dynamics-product} ---------------------------------- $~$\ We begin this section with some easy lemmas. \[lem:finite energy reducing\] Let $G$ be a finite graph, $\sigma_{0}$ any initial spins configuration, then $\mathbf{P}_\omega$-a.s. there is a time $T<\infty$, after which there are no more energy-reducing flips. Any flip can either reduce the total energy of the system $\mathcal{H}(\sigma_t)$ or leave it fixed. As $G$ is finite the total energy of the system is bounded from below and moreover any energy-reducing flip decreases the Hamiltonian by at least $2$. Therefore a.s. $\lim_{t \to \infty} \mathcal{H}(\sigma_t)$ exists and moreover is achieved after some finite time $T < \infty$. \[con:almost freezing finite energy reductions\] Under a dynamics which is almost-freezing with probability 1, any vertex has finitely many energy-reducing flips. Indeed, for any vertex $v$ which does not freeze there is a time $t=t_0$ such that the connected cluster of vertices which flip after this time and contains $v$ is finite. Clearly until this time $v$ had only finitely many flips. On the other hand, similar considerations to the ones which appeared in Lemma \[lem:finite energy reducing\] show that $v$ will have only finitely many energy-reducing flips from now on. The result follows. We shall now quote a simple lemma whose proof can be found in [@GandolfiNewmanStein]. Let $Z_t$ be a continuous-time time-homogeneous Markov process with state space $\mathcal{Z}$ and let ${Z_t}^{(\tau)}$ denote the time shifted process $Z_{t+\tau}$. If $A$ is a measurable subset of $\mathcal{Z}$, we say that $A$ $\emph{recurs}$ if $$\label{def:recur1} \{\tau > 0 \| Z_\tau \in A\} \qquad \text{is unbounded}.$$ If $B$ is an event, measurable with respect to the $\sigma$-field generated by $\{Z_t \| 0 \leq t \leq 1\}$, we say that $B$ recurs if $$\label{def:recur2} \{\tau > 0 \| Z^{(\tau)}\in B\} \qquad \text{is unbounded}.$$ Then, \[lem:from newman\] If $\inf_{z \in A} \mathbf{P}(B\|Z_0 = z) > 0$ then if $A$ recurs with positive probability, so does $B$. Given $\sigma$, a configuration on $G=(U,E)$ and $V \subseteq U$, we denote by $V^+(\sigma)$ the set of vertices whose spin is $+1$ and $V^-(\sigma)$ the set of vertices whose spin is $-1$. In the next definition $d$ is any positive constant. \[def:strongly freezing\] A family of finite graphs $\{G_n = (V_n, E_n)\}_n$ is $d$-$\emph{strongly freezing}$ if for any $0<p<1$ there exists $n_0$ such that for all $n>n_0$, if $\sigma_n$ is a configuration on $V_n$ chosen according to an i.i.d. symmetric $\pm 1$ product distribution, then with probability at least $p$ either all the vertices in $V_n$ have at least $d+1$ more neighbors in $V^+_n(\sigma_n)$ than in $V_n^-(\sigma_n)$ or all the vertices in $V_n$ have at least $d+1$ more neighbors in $V^-_n(\sigma_n)$ than in $V^+_n(\sigma_n)$. \[def:0-strongly freezing\] We say that a family of finite graphs $\{G_n\}_n$ is $\emph{strongly freezing}$ if it is $\emph{d-strongly freezing}$ for all $d\in\mathbb{N}$. Let $\{K_n\}$ be the sequence of complete graphs of $n$ vertices. $\{K_n\}$ is a d-strongly freezing family for any $d$. The following theorem shows that graph-products yield graphs which are more “regular” for the Glauber dynamics. For its proof, we use percolation techniques together with some of the above lemmas. \[thm:graph products freeze\] Let $\{G_n\}$ be a strongly freezing family of graphs. Then, for any $d\in\mathbb{N}$, there exist $N\in\mathbb{N}$ such that for any graph $G$ of bounded degree $d$ and countably many vertices, the zero-temperature Glauber dynamics over $G \times G_n$ almost freezes a.s. for all $n\geq N$. Moreover, at any vertex there are at most finitely many energy-reducing flips. This result may be generalized to graphs with a small number of vertices of higher degrees. It is well known (e.g. [@grimmett]) that for any graph $G$ of bounded degree $d$ and countably many vertices, the critical value for Bernoulli site percolation is at least $1 / (d-1)$. In particular, if the probability of “opening” a site is at most $1 / d$, then a.s. there are no infinite open cluster of sites. We pick $p=1 - 1 / d$, and find the $n_0$ as guaranteed from Definition \[def:strongly freezing\] for this $p$. Let $n>n_0$ be some integer. Let $U$ be the vertices of some slice of the graph. By definition, with probability $p$, either all the vertices in $U$ have at least $d+1$ more neighbors in $U^+(\sigma_{0})$ than in $U^-(\sigma_{0})$ or all the vertices in $U$ have at least $d+1$ more neighbors in $U^-(\sigma_{0})$ than in $U^+(\sigma_{0})$. Therefore these vertices will eventually freeze. Indeed, if, without loss of generality, any vertex in this slice has at least $d+1$ more neighbors in $U^+(\sigma_{0})$ than in $U^-(\sigma_{0})$, then clearly the vertices of $U^+(\sigma_{0})$ will never change their spin. Moreover, if we wait enough time, all the vertices of $U^-(\sigma_{0})$ will change their spin to $+1$, regardless of what are the spins of their other neighbors. We call a slice with the above property $\emph{good}$ and otherwise $\emph{bad}$. Due to the choice of $p$ bad slices form only finite clusters. Moreover, all the edges from these clusters to the rest of the graph are to good slices. Thus, for any cluster of bad slices, there is some time $T$ after which any edge from this cluster to its complement is to a frozen node. This implies. almost freezing a.s. The second part of the theorem follows from Conclusion \[con:almost freezing finite energy reductions\]. For any $d\in\mathbb{N}$, there exist a $N\in\mathbb{N}$ such that for any graph $G$ of bounded degree $d$ and countably many vertices, the zero-temperature Glauber dynamics over $G\times K_n$ almost freezes a.s. for all $n\geq N$ Moreover, at any vertex there are at most finitely many energy-reducing flips. Glauber Dynamics on Cylinders {#sec:dynamics-cylinder} ----------------------------- $~$\ In this subsection we treat cylinder-type graphs. \[thm:cylinders almost freeze\] Let $G$ be a finite graph with no isolated vertices (and at least two vertices). If $\mathbf{P}_{\sigma_{0}}$ is a product measure of i.i.d $\pm 1$ variables, then with $\mathbf{P}_{\sigma_{0},\omega}$-probability $1$ the $G$-cylinder almost freezes. Moreover, any vertex has only finitely many energy-reducing flips. Without loss of generality we may assume the graph is connected. With $\mathbf{P}_{\sigma_{0},\omega}$-probability 1 for any integer $l$ there are two other integers $i,j$ such that $i+1<l<j$ with the property that each pair of slices, in levels $i,i+1$, and $j,j+1$, are given the same spin. When $G$ is connected, any such pair of slices form a PSF. Indeed, each vertex in the pair has at least $2$ neighbors in the pair and $1$ neighbor not in the pair. Therefore, a.s. any vertex belongs to a connected subgraph whose boundary is frozen from time $t=0$ on. But this means that there are no infinite connected unfrozen subgraphs. The second part of the theorem follows from Lemma \[lem:finite energy reducing\]. A similar claim holds for bounded-width deformed cylinder with some condition on the edges between two successive slices. Yet, not every cylinder freezes, as the example below shows. \[ex:non freezing cylinder\] Let $G$ be the union of two copies of $K_n$ which share a single common vertex where $n\geq 4$. We now show that the $G$-cylinder does not freeze. Indeed, suppose that at time $t=0$ the vertices of the five successive slices at levels $-2,-1,0,1,2$ are assigned the following spins: - All the vertices of slices $-1,-2$ are assigned $-1$. - All the vertices of slices $+1,+2$ are assigned $-1$. - All vertices of one copy of $K_n$ in slice $0$ are given spin $-1$ and the rest are set to $+1$. This happens with positive probability. In this case all vertices except the vertex which is common to the two copies of $K_n$ in slice $0$, will not change their spin. This vertex, however, will change its spin infinitely many times, as it will always have an equal number of $+1$ and $-1$ neighbors. ![G-cylinder for Example \[ex:non freezing cylinder\]](ex4_10){width="70.00000%"} One the other hand, the dynamics on some cylinders do freeze. For any $n>1$ the dynamics over the $C_n$-cylinder and the $K_n$-cylinder freeze a.s. Moreover, it freezes in slices with probability $1$. As in Theorem \[thm:cylinders almost freeze\] each vertex lays between two all $+$ or all $-$ pairs of slices which are frozen from time $t=0$. Consider the subgraph between two such pairs. If it does not freeze in slices, either it does not freeze, or it freezes, but not in slices. In any case, as it is a finite graph, some spin configuration will recur in the sense of . If we show that from every such configuration, there is a positive probability to get to a configuration which is frozen in slices in one time unit, then by Lemma \[lem:from newman\] we will arrive to a contradiction. Indeed, this would imply that with probability one we will eventually get to a configuration which is frozen in slices, which is absorbing, by definition. We begin with the $C_n$ case. Without loss of generality we shall consider the sub-graph $C_n \times [-2,...,m+2]$ and assume that at some time $t$ the vertices in each of the two pairs of slices: $-1,-2$ and $m+1,m+2$ are either all $+$ or all $-$ (the spins can be different for each pair), while all other vertices have some prescribed spin. The “procedure” below produces a sequence of flips which, if occur in order and starting from the above configuration, will result in a configuration where all vertices in each slice have the same spin which is also the same as the spin of all vertices in the slice below or above it. Such configuration is frozen in slices, as desired. This sequence of flips will be feasible, i.e. it could be made to happen via a proper choice for clock-ring times and coin-tosses, which happens with positive probability. Suppose that the spins of the vertices in slice $-1$ are all $+1$. Consider the slice in level $0$: 1. If all its vertices are $+1$, do nothing and move to consider the slice at level $1$. 2. If they are not all $-1$, there is some vertex with a spin of $+1$. Then its neighbors in that slice have at least two (out of four) neighbors of the same spin and therefore they can be made $+1$. Iterate to set all vertices in this slice to $+1$. Move to consider the slice at level $1$. 3. If all the vertices have $-1$ spins, consider the slice at level $1$. If all its vertices are $-1$, do nothing and move to consider the slice at level $2$. 4. Otherwise, there is some vertex with $+1$ spin in the slice of level $1$. Its neighbor at level $0$ has again two out of four neighbors with spin $+1$ and can be made $+1$. Iterate until all the vertices in slice $0$ are $+1$. Then move to consider slice $1$. 5. Repeat these steps for the new considered slice. Stop when you reach level $m+1$. This shows the theorem for $C_n$. We now consider $K_n$. For $n=3$, $C_n = K_n$. Therefore we assume $n \geq 4$. Note that for such $K_n$-cylinder, a slice with all of its vertices having the same spin is frozen, since the majority of the neighbors of each vertex are in the slice itself. Therefore we can now consider a finite subgraph of the form $K_n \times [-1,...,m+1]$ and assume that at some time $t$ the configuration is such that all vertices in slices $-1$, $m+1$ have the same spin (maybe different for each slice) with some prescribed spins for the rest of the vertices. The “procedure” for obtaining a sequence of flips that yields a frozen-in-slices configuration is now: 1. Consider slice $0$. If more than half of its vertices have the same spin $s$, then any vertex with the opposite spin has at least the same number of neighbors with spins $s$ as the number of neighbors with the spin $1-s$. Indeed, if $n=2k$, then at least $k+1$ have spins $s$. Any vertex in the slice, with the opposite spin has at least $k+1$ neighbors with that spin and $2k-1+2 = 2k+1$ neighbors all-together. If $n=2k+1$, then there are at least $k+1$ vertices of spin $s$, and any vertex with the opposite spin has at least that numbers of neighbors with spin $s$ among $2k+1-1+2 = 2k+2$ neighbors all-together. Thus, for any vertex with a spin different from $s$, we flip to $s$. Then we move to consider the next slice. 2. If exactly half of the vertices in the slice have spin $s$, then in particular $n=2k$ for some positive integer $k$ and there are $k$ vertices with spin $+1$. Consider a vertex with spin $-1$, then it has at least $k+1$ neighbors with spin $+1$ (including slice $-1$), among $2k+1$ neighbors all-together. Thus, we can set all these vertices to $+1$ as well. We move to consider the next slice. 3. We stop when we reach slice $m+1$. This proves the theorem for $K_n$. It follows from [@NandaNewmanStein] that on any transitive unimodular graph with $\emph{odd}$ degrees, the dynamics freezes. A sufficient condition on the graph $G$ for the dynamics over $G$-cylinder not to freeze in slices is given in the proposition below. Let $G=(V,E)$ be a graph and suppose that there exists a partition of $V$ into $V_1, V_2$, such that every vertex in $V_1$ has more neighbors in $V_1$ than in $V_2$ and every vertex in $V_2$ has more neighbors in $V_2$ than in $V_1$. Then $\mathbf{P}_{\sigma_{0},\omega}$-a.s. the dynamics on the $G$-cylinder will not freeze in slices. We follow the construction in Example \[ex:non freezing cylinder\]. With positive probability slices $-2,-1$ can be initially given the spin $+1$ and the slices $1,2$ can be intially given spin $-1$. Then, still with positive probability, all vertices in slice $0$ with their second coordinate in $V_1$ are initialized with $+1$ and all vertices in slice $0$ with their second coordinate in $V_2$ are initialized with $-1$. This configuration is stable, i.e. there can be no flip which does not $\emph{increase}$ the energy. Let $G$ be $C_3 \times \mathbb{T}_n$, where $\mathbb{T}_n$ is the $n\times n$-torus or $\mathbb{Z}_n^{2}$, then the $G$-cylinder does not freeze in slices. Note that this is a (unimodular) transitive Cayley graph. Indeed, as in Example \[ex:non freezing cylinder\] with positive probability the vertices of slices $-2,-1$ can be initially given the spin $+1$ and the vertices of slices $1,2$ can be initially given the spin $-1$. Now the $0$-slice is composed of three tori. With positive probability two of them are set to $+1$ and one is set to $-1$. It is easy to verify that this is a stable state, i.e. no vertex will flip its spin. Thus, with probability $1$ even if the cylinder freezes, it will not freeze in slices. The Loop dynamics and its Applications {#sec:dynamics-loops} ====================================== $~$\ Background and Preliminaries {#sec:loops-background} ---------------------------- $~$\ In this section we define and explore a different evolution model, which is related to the GSPs of planar spin system. Much like the Glauber dynamics, transitions are local, i.e. allowed only between two configurations which differ at a finite number of vertices. While the Glauber dynamics for Ising yields the Ising Gibbs distribution in equilibrium, under the process which we define here, the weak limits of the distribution of the spins at time $t$ as $t \to \infty$, are all supported on the set of ground states configurations. All the results here are stated for planar lattices, but can be readily extended to all amenable Cayley graphs. Let $G$ be a planar graph embedded in the plane is such a way that any compact subset of the plane contains only finitely many vertices. Let $\Gamma$ be the set of all simple loops in the dual graph $G^{\ast}$. A set of positive numbers indexed by $\gamma \in \Gamma$, $\{f_\gamma\}_{\gamma\in\Gamma}$, will be called a set of $\emph{loop frequencies}$. An assignment $J_{xy}$ to each edge $x \thicksim y$ will be called a $\emph{coupling}$ or $\emph{interaction}$ as before. As before, $\mathcal{X}$ = $\{-1,1 \}^{G}$ will denote the space of all configurations over $G$, the Hamiltonian of the system is as before $$\mathcal{H}(\sigma) = -\sum_{x\thicksim y} J_{xy} \sigma_x\sigma_y.$$ The $\emph{loop-dynamics}$ with frequencies $\mathbf{f}=\{f_\gamma\}_{\gamma\in\Gamma}$ is a Markov process over the $\mathcal{X}$. With each loop $\gamma$, we associate an independent Poisson clock with rate $f_\gamma$. Given all couplings $J_{xy}$ and an initial configuration, at every clock ring, say of $\gamma$, the next configuration depends on the total energy contribution from edges whose dual is in $\gamma$, i.e. on $$\label{eq:Hamiltonian-loop} \mathcal{H}^\gamma_\mathcal{J}(\sigma) = -\sum_{<x,y>\in\gamma^{*}}J_{xy}\sigma_x\sigma_y$$ If it is positive, the spins of all vertices within (the finite subset of the plane bounded by) $\gamma$ are flipped. If it is zero, the same spins are flipped with probability $1/2$ independent of everything else. If it is negative, we do nothing. Clearly, in the first case, the total energy of the system is lowered by $2\mathcal{H}^\gamma_\mathcal{J}(\sigma)$ and we call such a step an $\emph{energy-reducing step}$. Note that it is not clear, yet, if this dynamical process is well defined. This will be the first main result of this chapter. As usual, we denote by $\mathbf{P}_\omega$ the underlying probability measure for all Poisson clocks and coins variables, $\mathbf{P}_{\sigma_{0}}$ for the initial configuration $\sigma_{0}$ and $\mathbf{P}_{\mathcal{J}}$ for the coupling variables $J_{xy}$. We assume as usual that $\mathbf{P}_{\sigma_{0}}$ is the product measure of some distribution on $\{-1, +1\}$ and in particular, unless stated differently, that this is the symmetric $\pm 1$ distribution. Similarly under $\mathbf{P}_{\mathcal{J}}$ the couplings $J_{xy}$ are i.i.d. and we shall further suppose that their distribution has a first moment. The joint distribution is $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega}=\mathbf{P}_\mathcal{J}\times\mathbf{P}_{\sigma_{0}}\times\mathbf{P}_\omega$. For what follows we set $G = \mathbb{Z}^{2}$. We shall say that $v$ is inside $\gamma$ or that $\gamma$ contains $v$ if $v$ is inside the finite subset of the plane which is bounded by $\gamma$. We denote by $V_\gamma$ the set of all vertices which are either inside $\gamma$ or on an edge whose dual is in $\gamma$. Let $l_\gamma$ be the number of dual edges on $\gamma$ and set $S_\gamma = \|V_\gamma\|$. Two loops $\gamma_1$ and $\gamma_2$ are congruent $\gamma_1 \sim \gamma_2$ if they can be obtained from one another by a composition of translations, rotation by 90 degrees and reflections. The $\emph{type}$ of a loop $[\gamma]$ is the equivalence class to which it belongs, under the above congruency relation. $[\Gamma]$ will stand for the collection of all loop types. A set of loop frequencies $f$ is called $\emph{proper}$ if it constant on each $[\gamma]$. In this case, we shall also treat $f$ as a function of $[\Gamma]$. In the same vein, we define $[\Gamma]$-versions of $l$ and $S$ as $l_{[\gamma]} = l_\gamma$ and $S_{[\gamma]} = S_\gamma$. Finally, $n_{[\gamma]}$ is the number of loops of type $[\gamma]$ which contain the origin. Main Results {#sec:loops-main res} ------------ The first thing we show is that the loop-dynamics is well defined, is the rates of the clocks satisfy some inequality, this is Theorem \[thm:loop dynamics well defined\]. We then prove, in Theorem \[thm:weak limits in loop dynamics\] that any subsequential limit of the dynamical process is supported only on GSPs. In case of Ferromagnetic system we show that the dynamics has a weak limit, which is supported on the GSP with all the spins being the same. Properties of the Loop Dynamics {#sec:loops-proofs} ------------------------------- The following theorem gives conditions on $f$ such that the loop-dynamics process is well defined \[thm:loop dynamics well defined\] If $$\label{eqn:WellDefinedness} \sum_{[\gamma]\in \Gamma}n_{[\gamma]} f_{[\gamma]} S_{[\gamma]} < \infty$$ then the loop dynamics is $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega}$-a.s well defined. To show that the process is well defined it suffices to find $\tau > 0$ such that the spin configuration can be computed up to time $\tau$ given the initial configuration and the ring-times of all clocks and coins. This is clearly the case for a finite graph, but for infinite graph, it is a priori possible to have an infinite sequence of vertices, with the spin of each vertex (up to time $\tau$) depending (due to a possible spin-flip) on the spin of the next vertex in the sequence. The condition in the theorem will ensure that with probability $1$, this dependency sequence is finite for all vertices. We may then compute the configuration at each time $t$, by progressing in units of $\tau$ time. Let $D$ be the smallest set of vertices, that contains the origin and satisfies the following property: If $v\in D$ and $\gamma$ is a loop for which $v\in V_\gamma$ and whose clock rang no later than time $\tau$, then $V_\gamma$ is also contained in $D$. Since equation guarantees that for any vertex $v$ and time $t$, the total number of rings up to time $t$ at the clocks of all loops containing $v$ is finite, it is enough to show that $D$ above is almost surely finite. Indeed, it is not difficult to see that the number of vertices in $D$ is stochastically dominated by the total number of nodes in a Galton-Watson tree, where the distribution of the number of decedents is Poisson with rate $\tau \sum_{[\gamma]\in \Gamma}n_{[\gamma]} f_{[\gamma]} S_{[\gamma]}$. By choosing $\tau$ small enough, we can make the latter smaller than $1$, in which case the size of the tree is finite almost surely as desired. The loop dynamics, as defined, may be considered as an extension of the zero-temperature Glauber dynamics. In the same manner, one may consider the analogous extension of the regular Glauber dynamics (with positive temperature). In that case when the clock of some loop rings, the probability of a flip depends on the relevant Boltzmann-Gibbs law for that temperature. The exact same analysis can be applied to show that this dynamical process is also well-defined. Next, we investigate the possible weak limits of $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega} (\sigma_t \in \cdot)$ as $t \to \infty$. We shall show that any subsequential limit must be supported on ground state configurations. However, first we need the following. \[lem:finite loop flips\] Under the above assumptions, if in addition the rates satisfy the inequality (for every edge $e$ and $l\in\mathbb{N}$) $$\sum_{[\gamma]\in \Gamma_{e}(l)} f_{[\gamma]} < e^{-10 l}$$ where $\Gamma_{e}(l)$ is the collection of loops of length at least $l$ which $e$ lays on, then for any loop $\gamma$ there are only finitely many energy reducing steps. In particular, if the couplings have an absolutely-continuous distribution (w.r.t. Lebesgue measure), then every loop flips only finitely many times. Denote by $E_\gamma (t)$ the expectation w.r.t. $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega}$ of the accumulated energy change in $\mathcal{H}^\gamma_\mathcal{J}(\sigma)$ up to time $t$. Note that this is always non-positive, almost surely finite (due to and the first moment assumption) and the same for all loops of the same type. Fix $N > 0$. For any type $[\gamma]$ there are $cN^2+O(N)$ loops of that type, which are contained in an $N \times N$ square around the origin. Therefore the expected accumulated energy change (in the $N \times N$ square) due to such loops is bounded above by $(1-\varepsilon)cN^2E_\gamma (t)$ where $\varepsilon = \varepsilon(\gamma, N) > 0$, is a constant that depends on $N$ and on the structure of the loop $\gamma$, and which tends to $0$ as $N\rightarrow \infty$. For such a square, the total energy change up to time $t$ (by which we mean the difference in the Hamiltonians restricted only to the square and its boundary between time $0$ and time $t$) is due to two sources: A negative contribution from spin-flips inside loops which are strictly contained in the square and (negative or positive) contribution from spin-flips inside loops which intersect the boundary of the square - both up to time $t$. We claim that there is a constant $L$ such the latter is bounded from above, in expectation, by $tLN\log (N)$, for large enough $N$. Indeed, it is enough to bound the expected total number triplets $(e,\gamma, t_i)$ such that $\gamma$ is a loop which intersects the boundary of the square, $e$ is an edge in the square which lays on $\gamma$ and the clock of $\gamma$ rang at time $t_i \leq t$. This number is clearly bounded from above by a constant multiple of $t \sum_{0\leq l \leq \frac{N}{2}}(N-l)\sum_{[\gamma]\in \Gamma_{e}(l)} f_{[\gamma]}$, since if a loop contains an edge of distance $k$ to the boundary, and also touches the boundary - its length must be at least $k$ (actually at least $2 k$). The last sum is bounded from above by $t (N\log (N) + O(N^{-9}))$, as can be seen if we divide the sum into summation over $l < \log (N)$, and summation over $\log (N) \leq l$. Hence the total contribution of the boundary is no more than $tLN\log (N)$. On the other hand, the expected total energy change in the box is bounded by the difference between the minimal and maximal total energy inside the square, which is at least $-MN^2$ where $M$ is some positive constant. Putting the above observations together gives $$(1-\varepsilon)cN^2 E_\gamma (t) + tN\log (N)L \geq -MN^2$$ Contributions from other internal loops are ignored as they can only sharpen the inequality (they are a non-positive amount which is added to the left hand side). Dividing by $N^2$ and letting $N\rightarrow\infty$, and then $\varepsilon\rightarrow 0$, keeping $t$ fixed, we obtain $E_\gamma (t) \geq -M$. Then, letting $t\rightarrow\infty$ and using the monotonicity of $E_\gamma (t)$, we get that the total energy change which is made by any loop $\gamma$ is finite almost surely. Finally, given a specific loop $\gamma$, if we assume that the couplings have an absolutely-continuous distribution w.r.t. Lebesgue measure, then there is no spin choice $\sigma$ for which $\mathcal{H}^\gamma_\mathcal{J}(\sigma) = 0$ (no zero-energy-changes can occur). Hence, each energy reducing spin-flip decreases the energy by at least some positive constant. Thus we obtain that the number of spin-flips of $\gamma$ (spin flips inside $\gamma$ which are caused by clock rings of $\gamma$) is finite almost surely. Actually the additional inequality for the rates was added only for simplicity, and is unnecessary, as the contribution of the boundary is also expected to be nonpositive. \[lem:no loop is negative afterwhile\] Under the above assumptions, if in addition the rate of every loop is positive, then for any loop there is some time $t$ such that after that time the value of the Hamiltonian over the edges dual to that loop’s edges is non-positive. $\mathbf{Sketch.}$ Consider a fixed loop $\gamma$, and take a large enough box around it. After some time no loop which is contained in that box makes an energy-reducing step. But it still may happen that that spin-flips inside larger loops which are not contained in the box will change the Hamiltonian of $\gamma$. We would like to show that after some time those changes will not make that Hamiltonian positive. The idea is that the total possible flip rate of long enough loops (such as loops that intersect the large box, yet not contained in it), must be smaller than that of $\gamma$ as large loops have extremely slow rates, and large enough loops with a common edge have small total flip rate, due to convergence. We would like to show that if there were infinitely many such changes, $\gamma$, whose clock has much more often rings, must make an energy-reducing step again. This follows immediately from Lemma \[lem:from newman\]. Applying the lemma for $A$ as being all the configurations that when reducing them to the large box around $\gamma$ leaves the Hamiltonian of $\gamma$ positive, and $B$ be the event that $\gamma$ flips (given any initial setting and in at most one time unit), gives us the result. We can now state: \[thm:weak limits in loop dynamics\] Any subsequential weak limit of $\sigma_{t}$ (for a sequence of times which tend to infinity) under $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega}$ must be a translation invariant measure, supported only on ground state configurations. Translation invariance is clear. By the above lemma, for any loop $\gamma$ we have $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega} (\mathcal{H}^\gamma_\mathcal{J}(\sigma_t) > 0) \to 0$ as $t \to \infty$. Therefore in any subsequential limit $\sigma_{\infty}$ we have $\mathcal{H}^\gamma_\mathcal{J}(\sigma_{\infty}) \leq 0$ for all $\gamma$ a.s.. Therefore $\sigma_{\infty}$ is a ground state configuration. Next we show that if $J_{xy} = 1$ for all $x \thicksim y$ with probability $1$, then under the loop dynamics there is a weak limit to $\sigma_t$ as $t\rightarrow\infty$ and it is supported on two configurations, those with all spins $+1$ or all spins $-1$. Yet, vertices flip their spin infinitely many times, i.e. there is no strong pointwise limit for $\sigma_t$. The reason is that having a strong limit is a translation-invariant event, which may be approximated by knowing the initial spins, in some neighborhood of the origin and by knowing the rings times and coins for loops in that neighborhood, up to some time. Thus this is a $0-1$ event. In the same manner, freezing to a configuration where all the spins are $+1$, is also a $0-1$ event. But now symmetry (with the other spin) shows it must be a $0$ event. It should be noted that such a result is still open for zero-temperature Glauber dynamics in the square lattice and an understanding of the support of the corresponding subsequential weak limit is lacking. See ([@CamiaSantisNewman]). \[clm:weak limit in loops +1\] If $\mathcal{J} \equiv 1$, then $\sigma_t \Longrightarrow \frac{1}{2}\delta_+ + \frac{1}{2}\delta_-$ where $\delta_+$ ($\delta_-$) is the Dirac mass on the all $+$ ($-$) configuration. Hence, for almost every $\sigma_0$, modulo the global flip, we have convergence in probability to the dirac mass on the GSP of a single spin (all $+$ or all $-$). It follows from Theorem \[thm:weak limits in loop dynamics\] that subsequential weak limits for $\sigma_{t}$ are translation invariant measures supported on ground states. Theorem \[thm:single GSP in Z\^d if J=1\] shows that the support of these measures contains only monochromatic configurations, i.e. any subsequential weak limit must be of form $p\delta_+ + (1-p)\delta_-$ for some $p \in [0,1]$ which depends on the given sequence. Invariance under $\mathbf{P}_{\mathcal{J},\sigma_{0},\omega}$ of the distribution of $\sigma_t$ w.r.t. to global spin-flip implies that $p$ must be $1/2$. $ $\ 1. The above analysis carries through for any product measure for the couplings, as long as their distribution is strictly positive. These models are called $\emph{Ferromagnetic}$. 2. The loop dynamics can be generalized to lattices of higher dimensions, where instead of loops we consider closed surfaces of co-dimension $1$. Similar bounds for the rates are needed to make the process well defined and all statements carry through with minor modifications (in particular the case $J=+1$). In fact, everything can be generalized to amenable Cayley graphs. We end this section with a result concerning the “stability” of a sequential limit under small perturbations of the couplings. Roughly speaking we show that there exist couplings, such that changing their value a small enough amount, without changing the order of clocks rings does not change evolution of spin configurations. We assume henceforth that the couplings are chosen from a product measure with marginals which are supported on the entire real line and with finite first moment. \[prop:walls between weak limits in dynamics\] Under the above assumptions, let $\sigma_1,\sigma_2$ be two spin configurations which are in the support of some subsequential weak limit. Consider the subgraph consisting of all edges which are dual to edges $(u,v)$ where $\sigma_1(u) = \sigma_2(u)$ but $\sigma_1(v) = -\sigma_2(v)$, then this subgraph is a collection of two-sided infinite dual paths. As the subsequential limits are translation invariant measures, this proposition is a direct result of [@Newman00natureof]. These dual paths are called $\emph{domain walls}$. Note that dual edges whose primal edges are $\emph{fixed}$ (according to the terminology of Section \[sec:spinglasses\]) will be in no domain wall, and hence will flip only finitely many times. \[clm:flexibility out of walls\] Given the couplings, the initial spin configuration, and the ring times, almost surely there exists an edge $e$ and a positive number $\varepsilon$ such that changing the coupling of $e$ by any number whose absolute value is smaller than $\varepsilon$ does not change $\{\sigma_t \}_{t\geq 0}$. $\mathbf{Sketch.}$ Consider any fixed edge $e$. In any weak limit it must be satisfied. Thus, there must be a time after which both of its spins flip together (or maybe do not flip at all). Up to that time, $t_0$, for only finitely many dual loops which intersect it (and hence - contain its dual edge) - their Poisson clock had ringed (we count multiplicity). Due to continuity, there exist some positive $\delta$ such that changing the value of this edge’s interaction by not more than $\delta$ would not change the sign of any of the dual loops whose clock rang (at the corresponding times). Note that if we increase the absolute value of the interaction by that $\delta$, we do not change the process up to time $t_0$. After time $t_0$, as our edge is always satisfied, we increased the value of any dual loop which intersects it, and hence we clearly did not cause any new flip. Thus, in this case the process does not change.\ What about lowering the absolute value of the interaction? Assume that we fix all the interactions other then that of $e$, and all the clock times (and coins, if needed), and we condition on $e$ being a fixed edge. From the above it follows that a.s. there are at most countably many distinct options for the series of loop flips. Indeed, for any value of the interaction of $e$ (which is conditioned to be a fixed edge), there is some interval with positive length containing this value (maybe as boundary), s.t. changing the value of $e$ within that interval does not change the evolution of flips. If $x,y$ are two possible interaction values for $e$, for which it is a fixed edge, either the interiors of their corresponding intervals are distinct or they coincide. Moreover the flip series depends on the interval and not on the interaction itself. As every interval contains a rational number - there are at most countably many such intervals. Thus, to any sequence of ring times, and interactions of all edges except $e$ there corresponds a unique a.s. countable set of interval boundaries. For any value of interaction other then these boundaries (which is in the half line of values where $e$ is fixed) there is some $\varepsilon > 0$ such that changing the value of the interaction in no more than $\varepsilon$ (to either side) leaves the flip sequence unchanged. As the set of boundaries is of measure $0$ we are done.
--- abstract: \| The Neumann initial-boundary problem for the chemotaxis system $$\begin{aligned} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa(\|x\|) u - \mu(\|x\|) u^p, \\ 0 = \Delta v - \frac{m(t)}{\|\Omega\|} + u, \quad m(t) {\coloneqq}{\int_\Omega}u(\cdot, t) \end{cases} \end{aligned}$$ is studied in a ball $\Omega = B_R(0) \subset {\mathbb{R}}^2$, $R {>}0$ for $p \ge 1$ and sufficiently smooth functions $\kappa, \mu: [0, R] {\rightarrow}[0, \infty)$.\ We prove that whenever $\mu', -\kappa' \ge 0$ as well as $\mu(s) \le \mu_1 s^{2p-2}$ for all $s \in [0, R]$ and some $\mu_1 {>}0$ then for all $m_0 {>}8 \pi$ there exists $u_0 \in {{C^{0}({{\overline}\Omega})}}$ with ${\int_\Omega}u_0 = m_0$ and a solution $(u, v)$ to with initial datum $u_0$ blowing up in finite time. If in addition $\kappa \equiv 0$ then all solutions with initial mass smaller than $8 \pi$ are global in time, displaying a certain critical mass phenomenon.\ On the other hand, if $p {>}2$, we show that for all $\mu$ satisfying $\mu(s) \ge \mu_1 s^{p-2-{\varepsilon}}$ for all $s \in [0, R]$ and some $\mu_1, {\varepsilon}{>}0$ the system admits a global classical solution for each initial datum $0 \le u_0 \in {{C^{0}({{\overline}\Omega})}}$.\ Key words: [chemotaxis, critical mass, finite-time blow-up, logistic source]{}\ AMS Classification (2010): [35B44 (primary); 35B33, 35K65, 92C17 (secondary)]{} author: - \| Mario Fuest[^1]\ [Institut für Mathematik, Universität Paderborn,]{}\ [33098 Paderborn, Germany]{} title: 'Finite-time blow-up in a two-dimensional Keller–Segel system with an environmental dependent logistic source' --- Introduction ============ We live in a heterogeneous environment and the fact that for instance growth or death rates may depend on spatial features has been incorporated into several models describing population dynamics. Among the more famous examples is the system $$\begin{aligned} \label{prob:competition} \begin{cases} u_t = d_1 \Delta u + u [{\kappa}(x) - u - v], \\ v_t = d_2 \Delta v + v [{\kappa}(x) - u - v] \end{cases}\end{aligned}$$ with $d_1, d_2 {>}0$ and ${\kappa}: \Omega {\rightarrow}[0, \infty)$, $\Omega \subset {\mathbb{R}}$, $n \in {\mathbb{N}}$, being a smooth, bounded domain, modelling two species $u$ and $v$ competing for a common resource, where ${\kappa}$ represents a reproduction rate influenced by the environment. It has the remarkable property that whenever $d_1 {<}d_2$, then there exists $u_\infty({\kappa}) {>}0$ such that for any initial data $u_0, v_0 \in {{C^{0}({{\overline}\Omega})}}$ with $u_0, v_0 \ge 0$ and $v_0 \not\equiv 0$ the corresponding solution $(u, v)$ converges to $(u_\infty({\kappa}), 0)$ – provided ${\kappa}$ is not constant, which reflects spatial heterogeneity ([@DockeryEtAlEvolutionSlowDispersal1998]). If, however, ${\kappa}$ is constant then $(\lambda {\kappa}, (1-\lambda) {\kappa}))$ is a steady state of for all $\lambda \in [0, 1]$ implying that species with different diffusion rates may coexist in homogeneous environments. Furthermore, there is considerable activtiy in the analysis of systems similar to ; for instance, convections terms have been added to these equations ([@LouEtAlGlobalDynamicsLotka2018]) and the case of weak competition ([@HeNiGlobalDynamicsLotka2016; @LouEffectsMigrationSpatial2006]) has been studied in great detail as well. These results (among others) may arouse interest to consider environmental depending functions in other models as well: The system $$\begin{aligned} \label{prob:log_pp} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa u - \mu u^p, \\ v_t = \Delta v - v + u, \end{cases}\end{aligned}$$ in $\Omega \times (0, T)$, where $\Omega \subset {\mathbb{R}}^n$, $n \in {\mathbb{N}}$, is a smooth, bounded domain, $T \in (0, \infty]$ and $\kappa, \mu {>}0$ and $p \ge 1$ are given parameters, is relevant in the modeling of, for instance, micro- and macroscopic population dynamics ([@HillenPainterUserGuidePDE2009], [@ShigesadaEtAlSpatialSegregationInteracting1979]) or tumor invasion processes ([@ChaplainLolasMathematicalModellingCancer2005]). For these so-called chemotaxis systems, at first introduced by Keller and Segel ([@KellerSegelTravelingBandsChemotactic1971]) even questions of global existence and boundedness are of great interest. After all, if one chooses $\kappa = \mu \equiv 0$ in in space-dimensions two ([@HorstmannWangBlowupChemotaxisModel2001; @SenbaSuzukiParabolicSystemChemotaxis2001]) and higher ([@WinklerFinitetimeBlowupHigherdimensional2013]) there are initial data leading to blow-up. For a more broad introduction to Keller–Segel models, which have been intensively studied in the past decades, we refer to the survey [@BellomoEtAlMathematicalTheoryKeller2015]. Intuitively, the superlinear degrading term $\mu u^p$ (with $\mu {>}0$ and $p {>}1$) in ) should somewhat decrease the possibility of (finite-time) blow-up. However, exactly how large $\mu$ and $p$ need to be in order to guarantee global existence seems to be an open question, even for constant $\kappa, \mu \ge 0$. If $n = 2$ and $\mu {>}0$ all classical solutions to exist globally in time ([@OsakiEtAlExponentialAttractorChemotaxisgrowth2002]). One may even replace $u^2$ by a function growing slightly slower than $s \mapsto s^2$ ([@XiangSublogisticSourceCan2018]). The same holds true in higher dimensions, provided $p {>}2$ or $p = 2$ and $\mu {>}\frac{n}{4}$ ([@WinklerBoundednessHigherDimensionalParabolicParabolic2010]), while for $p = 2$ and any $\mu {>}0$ at least global weak solutions have been constructed, which become smooth after finite time provided $\kappa$ is small enough ([@LankeitEventualSmoothnessAsymptotics2015]). As chemicals can be assumed to diffuse much faster than cells a typical simplification of is the parabolic-elliptic system $$\begin{aligned} \label{prob:log_pe} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa u - \mu u^p, \\ 0 = \Delta v - v + u. \end{cases}\end{aligned}$$ For $n = 2$ the conditions $p \ge 2$ and $\mu {>}0$ suffice to ensure global existence while for $n \ge 3$, $p = 2$ and $\mu \ge \frac{n-2}{n}$ or $n \ge 3$, $p {>}2$ and arbitrary $\mu {>}0$ the same can be achieved ([@KangStevensBlowupGlobalSolutions2016; @TelloWinklerChemotaxisSystemLogistic2007]). On the other hand, any thresholds may be surpassed, if $p = 2$, $\mu \in (0, 1)$ and the diffusion is sufficiently weak, that is, $\Delta u$ in the first equation in is replaced by ${\varepsilon}\Delta u$ for suitable ${\varepsilon}{>}0$ ([@WinklerHowFarCan2014; @LankeitChemotaxisCanPrevent2015]). This stays in contrast to the case without cross-diffusion as then ${\overline}u {\coloneqq}\max\{\\|u_0\\|_{{{L^{\infty}(\Omega)}}}, \frac{\kappa}{\mu}\}$ always forms a supersolution and thus indicates that in chemotaxis systems with logistic source nontrivial structures may emerge at least on intermediate time scales. Even more drastic formations are known to form if $p$ is chosen close to (but sill larger than) $1$. After initial data causing finite-time blow-up have been constructed in dimensions five and higher for certain $p {>}\frac32$ in a system closely related to in [@WinklerBlowupHigherdimensionalChemotaxis2011], in [@WinklerFinitetimeBlowupLowdimensional2018] finite-time blow-up has also been shown to occur in for any $n \ge 3$ and $$\begin{aligned} \begin{cases} p {<}\frac76, & n \in \{3, 4\}, \\ p {<}1 + \frac{1}{2(n-1)}, & n \ge 5. \end{cases}\end{aligned}$$ Hence, at least in space-dimensions three and higher even superlinear degegration terms do not always ensure global existence. The case of $\mu$ and $\kappa$ depending on space (and time) has also been studied. In their three-paper series [@SalakoShenParabolicellipticChemotaxisModel2018; @SalakoShenParabolicellipticChemotaxisModel2018a; @SalakoShenParabolicellipticChemotaxisModel2018b] Salako and Shen showed inter alia global existence of solutions to with $\Omega = {\mathbb{R}}$ provided $\inf_{x \in \Omega} \mu(x) {>}1$. #### Main results Apparently, rigorously proving blow-up in Keller–Segel systems is a difficult problem. Known proofs for parabolic-parabolic chemotaxis systems strongly rely on certain energy structures ([@CieslakLaurencotFiniteTimeBlowup2010; @HorstmannWangBlowupChemotaxisModel2001; @WinklerAggregationVsGlobal2010]) while in the parabolic-elliptic setting additional approaches are moment-type arguments ([@BilerLocalGlobalSolvability1998; @NagaiBlowupNonradialSolutions2001]) However, all these methods appear inadequate for chemotaxis systems with logistic source. In this paper we further simplify and consider $$\begin{aligned} \label{prob:p} \tag{P} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa(\|x\|) u - \mu(\|x\|) u^p, & \text{in $\Omega \times (0, T)$}, \\ 0 = \Delta v - \frac{m(t)}{\|\Omega\|} + u, \quad m(t) {\coloneqq}{\int_\Omega}u(\cdot, t), & \text{in $\Omega \times (0, T)$}, \\ \partial_\nu u = \partial_\nu v = 0, & \text{on $\partial \Omega \times (0, T)$}, \\ u(\cdot, 0) = u_0, & \text{in $\Omega$} \end{cases}\end{aligned}$$ for given functions $\kappa, \mu, u_0: \Omega {\rightarrow}{\mathbb{R}}$ and $T \in (0, \infty]$ where we henceforth fix $R {>}0$ and $\Omega {\coloneqq}B_R(0) \subset {\mathbb{R}}^2$. Our main results are the following. \[th:blow\_up\] Let $p \ge 1$, $\alpha \ge 2(p - 1)$, $\mu_1 {>}0$ and suppose that $\kappa, \mu \in C^0([0, R]) \cap C^1((0, R))$ satisfy $$\begin{aligned} \label{eq:blow_up:cond_kappa_mu} \kappa, -\kappa', \mu, \mu' \ge 0 \quad \text{in $(0, R)$} \end{aligned}$$ as well as $$\begin{aligned} \label{eq:blow_up:cond_mu} \mu(s) \le \mu_1 s^\alpha \quad \text{for all $s \in [0, R]$}. \end{aligned}$$ For any $m_0 {>}8\pi$ there exist $r_1 \in (0, R)$ and $\tilde m \in (0, m_0)$ such that if $$\begin{aligned} \label{eq:blow_up:cond_u0} 0 \le u_0 \in C^0({{\overline}\Omega}) \quad \text{is radially symmetric and radially decreasing} \end{aligned}$$ with $$\begin{aligned} \label{eq:blow_up:mass_concentration} {\int_\Omega}u_0 = m_0 \quad \text{and} \quad \int_{B_{r_1}(0)} u_0 \ge \tilde m, \end{aligned}$$ then there exists a classical solution $(u, v)$ to with initial datum $u_0$ blowing up in finite time; that is, there exists ${T_{\max}}\in (0, \infty)$ such that $$\begin{aligned} \label{eq:blow_up:limsup_u} \limsup_{t {\nearrow}{T_{\max}}} \\|u(\cdot, t)\\|_{L^\infty(\Omega)} = \infty. \end{aligned}$$ To give a more concrete example, the conditions and are for instance fulfilled if $p = 2$, $\kappa \ge 0$ is a constant and $\mu(r) = r^2, r \in [0, R]$. This result will be complemented by two statements on global solvability. Firstly, we show at least in the case $\kappa \equiv 0$ the value $8\pi$ – which does not, as one could have expected, depend on $\alpha$ or $p$ – is essentially optimal. \[prop:critical\_mass\] Let $\kappa \equiv 0$, $0 \le \mu \in C^0([0, R]) \cap C^1((0, R))$ and $p \ge 1$. For any nonnegative radially symmetric $u_0 \in C^0({{\overline}\Omega})$ with ${\int_\Omega}u_0 {<}8\pi$ there exists a global classical solution $(u, v)$ to with initial datum $u_0$. Secondly, if $p {>}2$, we prove that for arbitrary initial data global classical solutions exist provided $\mu$ does not grow too fast. \[prop:global\_ex\] Let $p {>}2$, $\alpha {<}p - 2$, $\mu_1 {>}0$ and $\kappa, \mu \in C^0([0, R]) \cap C^1((0, R))$. If $$\begin{aligned} \label{eq:global_ex:cond_mu} \mu(s) \ge \mu_1 s^\alpha \quad \text{for all $s \in [0, R]$} \end{aligned}$$ then admits a global classical solution for any nonnegative initial datum $u_0 \in {{C^{0}({{\overline}\Omega})}}$. #### Plan of the paper For the proof of Theorem \[th:blow\_up\] we will rely on a transformation introduced by Jäger and Luckhaus in [@JagerLuckhausExplosionsSolutionsSystem1992]. As will be seen in Lemma \[lm:pde\_w\] below the function $w: [0, R]^2 \times [0, {T_{\max}}) {\rightarrow}{\mathbb{R}}$ defined by $$\begin{aligned} w(s, t) {\coloneqq}\int_0^{\sqrt s} \rho u(\rho, t) {\,\mathrm{d}}\rho, \quad s \in [0, R^2], t \in [0, {T_{\max}}),\end{aligned}$$ solves the scalar PDI $$\begin{aligned} \label{eq:intro:p_pdi} w_t &\ge 4s w_{ss} + 2 w w_s - \frac{m(t)}{\|\Omega\|} s w_s - 2^{p-1} \int_0^s \mu(\sqrt \sigma) w_s^p(\sigma, \cdot) {\,\mathrm{d}\sigma}\quad \text{in $(0, R^2) \times (0, {T_{\max}})$}.\end{aligned}$$ In similar – but higher dimensional – settings for certain $s_0, \gamma {>}0$ the function $$\begin{aligned} \phi: [0, {T_{\max}}) {\rightarrow}{\mathbb{R}}; \quad t \mapsto \int_0^{s_0} s^{-\gamma} (s-s_0) w(s, t) {\,\mathrm{d}s},\end{aligned}$$ where $w$ denotes a similar transformed quantity, has been shown to solve a certain ODI implying finite-time blow-up ([@WinklerCriticalBlowupExponent2018], [@WinklerFinitetimeBlowupLowdimensional2018]). However, these techniques seem to be insufficient to provide any insights in the two dimensional setting, as the term stemming from the diffusion can apparently not be dealt with anymore. Therefore, we follow a different approach. In order to show finite-time blow-up for with $\kappa = \mu \equiv 0$ in the planar setting Winkler ([@WinklerHowUnstableSpatial2018]) has recently utilized the function $$\begin{aligned} \phi: [0, {T_{\max}}) {\rightarrow}{\mathbb{R}}; \quad t \mapsto \int_0^{s_0} (s-s_0)^\beta w(s, t) {\,\mathrm{d}s}\end{aligned}$$ for certain $s_0, \beta {>}0$ instead. Most terms in can be dealt similarly as in [@WinklerHowUnstableSpatial2018] – except for the nonlocal term $\int_0^s \mu(\sqrt \sigma) w_s^p(\sigma, \cdot) {\,\mathrm{d}\sigma}$ which is, of course, not present if $\mu \equiv 0$. The main idea for dealing with this integral is to derive a pointwise bound for $w_s$ (Lemma \[lm:ws\_bdd\]) and then integrate by parts, where the condition $\alpha \ge 2(p-1)$ is apparently needed in order to able to handle the remaining terms (Lemma \[lm:i4\]). Finally, we will then see by an ODI comparison argument that for suitably chosen initial data $\phi$ (and hence $u$) cannot exist globally in time. Preliminaries ============= The following statement on local existence, in its essence based on a fixed point argument, is standard. Hence we may omit a proof here and just refer to, for instance, [@CieslakWinklerFinitetimeBlowupQuasilinear2008] or [@TelloWinklerChemotaxisSystemLogistic2007] for more detailed arguments in similar frameworks. \[lm:local\_ex\] Let $0 \le u_0 \in C^0({{\overline}\Omega})$ and $\kappa, \mu \in C^0([0, R]) \cap C^1((0, R))$. Then there exist ${T_{\max}}\in (0, \infty]$ and a classical solution $(u, v)$ to uniquely determined by $$\begin{aligned} u &\in C^0({{\overline}\Omega}\times [0, {T_{\max}})) \cap C^{2, 1}({{\overline}\Omega}\times (0, {T_{\max}})), \\ v &\in \bigcap_{q {>}2} C^0([0, {T_{\max}}); W^{1, q}(\Omega)) \cap C^{2, 0}({{\overline}\Omega}\times (0, {T_{\max}})) \end{aligned}$$ and $$\begin{aligned} {\int_\Omega}v(\cdot, t) = 0 \quad \text{for all $t \in (0, {T_{\max}})$}. \end{aligned}$$ Moreover, this solution is nonnegative in the first component, radially symmetric if $u_0$ is radially symmetric and such that if ${T_{\max}}{<}\infty$ then $$\begin{aligned} \limsup_{t {\nearrow}{T_{\max}}} \\|u(\cdot, t)\\|_{L^\infty(\Omega)} = \infty. \end{aligned}$$ Unless otherwise stated we henceforth fix $u_0 \in {{C^{0}({{\overline}\Omega})}}$ satisfying as well as $\kappa, \mu \in C^0([0, R]) \cap C^1((0, R))$ fulfilling and denote the corresponding solution provided by Lemma \[lm:local\_ex\] by $(u, v)$ as well as the maximal existence time by ${T_{\max}}$. Finally, we set $m_0 {\coloneqq}m(0)$ and $\kappa_1 {\coloneqq}\\|\kappa\\|_{L^\infty((0, R))}$. \[lm:mass\] For all $t \in (0, {T_{\max}})$ the inequalities $$\begin{aligned} 0 \le m(t) \le m_0 {{\mathrm{e}}}^{\kappa_1 t} \end{aligned}$$ hold. Nonnegativity of $u$ implies $m \ge 0$ while an ODI comparison argument yields $m(t) \le m_0 {{\mathrm{e}}}^{\kappa_1 t}$ for $t {>}0$ due to $m' \le \kappa_1 m$ in $(0, {T_{\max}})$. As mentioned in the introduction the proof of Theorem \[th:blow\_up\] will rely on transforming into a scalar equation. \[lm:pde\_w\] Define $$\begin{aligned} w(s, t) {\coloneqq}\int_0^{\sqrt{s}} \rho u(\rho, t) {\,\mathrm{d}\rho}, \quad s \in [0, R^2], t \in [0, {T_{\max}}). \end{aligned}$$ Then $$\begin{aligned} \label{eq:pde_w:w_s_eq_u} w_s(s, t) = \tfrac12 u(\sqrt s, t) \end{aligned}$$ and $$\begin{aligned} \label{eq:pde_w:pde} w_t(s, t) &= 4s w_{ss}(s, t) + 2 w(s, t) w_s(s, t) - \frac{m(t)}{\|\Omega\|} s w_s(s, t) \notag \\ &{\mathrel{{\hphantom}{=}}}+ \int_0^s \left(\kappa(\sqrt \sigma) w_s(\sigma, t) - 2^{p-1} \mu(\sqrt \sigma) w_s^p(\sigma, t) \right) {\,\mathrm{d}\sigma}\end{aligned}$$ for $s \in (0, R^2)$ and $t \in (0, {T_{\max}})$. The first two equations in read in radial form $$\begin{aligned} u_t &= \frac1r (r u_r - r u v_r)_r + \kappa(r) u - \mu(r) u^p \quad \text{and} \\ 0 &= \frac1r (r v_r)_r -\frac{m(t)}{\|\Omega\|} + u, \end{aligned}$$ that is $$\begin{aligned} r v_r(r, \cdot) = \int_0^r \left( \frac{m(t)}{\|\Omega\|} \rho - \rho u(\rho, \cdot) \right) {\,\mathrm{d}\rho}= \frac{m(t)}{2\|\Omega\|} r^2 - w(r^2, \cdot). \end{aligned}$$ Thus, a direct calculation yields $$\begin{aligned} w_s(s, t) &= \frac{1}{2\sqrt s} \cdot \sqrt s u(\sqrt s, t) = \frac12 u(\sqrt s, t), \\ w_{ss}(s, t) &= \frac12 u_r(\sqrt s, t) \cdot \frac1{2 \sqrt s} = \frac1{4 \sqrt s} u_r(\sqrt s, t) \quad \text{and} \\ w_t(s, t) &= \int_0^{\sqrt s} \frac{\rho}{\rho} [\rho u_r(\rho, t) - \rho u(\rho, t) v_r(\rho, t)]_r {\,\mathrm{d}\rho}+ \int_0^{\sqrt s} \rho [\kappa(\rho) u(\rho, t) - \mu(\rho) u^p(\rho, t) ] {\,\mathrm{d}\rho}\\ &= \sqrt s u_r(\sqrt s, t) - u(\sqrt s, t) \left[ \frac{m(t)}{2\|\Omega\|} s - w(s, t) \right] - \frac12 \int_0^s \left( \kappa(\sqrt \sigma) u + \mu(\sqrt \sigma) u^p(\sqrt \sigma, t) \right) {\,\mathrm{d}\sigma}\\ &= 4 s w_{ss}(s, t) + 2 w(s, t) w_s(s, t) - \frac{m(t)}{\|\Omega\|} s w_s(s, t) - \int_0^s \left(\kappa(\sqrt \sigma) w_s + 2^{p-1} \mu(\sqrt \sigma) w_s^p(\sigma, t) \right) {\,\mathrm{d}\sigma}\end{aligned}$$ for $s \in (0, R^2)$ and $t \in (0, {T_{\max}})$. Supercritical mass allows for blow-up ===================================== Crucially relying on transforming into the scalar equation we will prove Theorem \[th:blow\_up\] at the end of this section. The function $\phi$ ------------------- \[lm:phi\_first\_ode\] Let $\beta {>}-1$ and $s_0 \in (0, R^2)$. The function $$\begin{aligned} \phi: [0, {T_{\max}}) {\rightarrow}{\mathbb{R}}, \quad t \mapsto {\int_0^{s_0}}(s_0-s)^\beta w(s, t) {\,\mathrm{d}s}\end{aligned}$$ belongs to $C^0([0, {T_{\max}})) \cap C^1((0, {T_{\max}}))$ and satisfies $$\begin{aligned} \label{eq:phi_first_ode:ode} \phi'(t) &\ge 4 {\int_0^{s_0}}(s_0-s)^\beta s w_{ss}(s, t) {\,\mathrm{d}s}\notag \\ &{\mathrel{{\hphantom}{=}}}+ 2 {\int_0^{s_0}}(s_0-s)^\beta s w(s, t) w_s(s, t) {\,\mathrm{d}s}\notag \\ &{\mathrel{{\hphantom}{=}}}- \frac{m(t)}{\|\Omega\|} {\int_0^{s_0}}(s_0-s)^\beta s w_s(s, t) {\,\mathrm{d}s}\notag \\ &{\mathrel{{\hphantom}{=}}}- 2^{p-1} {\int_0^{s_0}}\int_0^s (s_0-s)^\beta \mu(\sqrt{\sigma}) w_s^p(\sigma, t) {\,\mathrm{d}\sigma}{\,\mathrm{d}s}\notag \\ &{\eqqcolon}I_1(t) + I_2(t) + I_3(t) + I_4(t) \end{aligned}$$ for all $t \in (0, {T_{\max}})$. As $w \in C^0({{\overline}\Omega}\times [0, {T_{\max}})) \cap C^1({{\overline}\Omega}\times (0, {T_{\max}}))$ by Lemma \[lm:pde\_w\], the asserted regularity of $\phi$ follows from standard Lebesgue integration theory, while is then a direct consequence of Lemma \[lm:pde\_w\] and nonnegativity of $u$ and $\kappa$. Our goal is to show that after an appropriate choice of parameters $\phi$ satisfies a certain ODI, which then implies finiteness of ${T_{\max}}$. \[lm:ode\_blow\_up\] Let $T, \tilde T, c_1, c_2, c_3 {>}0$. If $y \in C^0([0, T)) \cap C^1((0, T))$ satisfies $$\begin{aligned} \begin{cases} y' \ge c_1 y^2 - c_2 y - c_3, \\ y(0) \ge y_0 \end{cases} \end{aligned}$$ in $(0, T)$ with $$\begin{aligned} y_0 \ge \frac{c_2 + \sqrt{c_1 c_3}}{c_1} + \frac1{c_1 \tilde T}, \end{aligned}$$ then necessarily $T \le \tilde T$. As $$\begin{aligned} c_1 s^2 - c_2 s - c_3 = 0 \quad \text{if and only if} \quad s = \frac{c_2 \pm \sqrt{c_2^2 + 4 c_1 c_3}}{2 c_1} {\eqqcolon}\lambda_\pm \end{aligned}$$ the ODI implies that $y$ is increasing if and only if $y \le \lambda_-$ or $y \ge \lambda_+$. Since $$\begin{aligned} \lambda_+ \le \frac{c_2 + \sqrt{c_1 c_3}}{c_1} {<}y_0 \end{aligned}$$ and $$\begin{aligned} (s-\lambda_-)(s-\lambda_+) \ge (s-\lambda_+)^2 \quad \text{for all $s \ge \lambda_+$} \end{aligned}$$ we conclude that $y$ is indeed increasing in $(0, T)$ and satisfies $$\begin{aligned} y' \ge c_1 (y - \lambda_+)^2 \end{aligned}$$ in $(0, T)$. Hence by integrating we obtain $$\begin{aligned} t = \int_0^t 1 {\,\mathrm{d}s}\le \int_{y(0)}^{y(t)} \frac1{c_1 (y - \lambda_+)^2} \le \frac1{c_1 (y_0 - \lambda_+)} - \frac1{c_1 (y(t) - \lambda_+)} {<}\tilde T - 0 = \tilde T \quad \text{for all $t \in (0, T)$}, \end{aligned}$$ which is absurd for $T {>}\tilde T$. Apart from the nonlocal term in all integrals therein as well as $\phi(0)$ can be estimated as in [@WinklerHowUnstableSpatial2018 Lemma 3.2]. For sake of completeness we nonetheless give short proofs for the following lemmata. \[lm:phi\_0\] Let $\beta {>}-1$ and $s_0 \in (0, R^2)$ as well as $\tilde m \in (0, m)$ and $\lambda \in (0, 1)$. If $$\begin{aligned} \int_{B_{r_1}(0)} u_0 \ge \tilde m \end{aligned}$$ with $r_1 {\coloneqq}(\lambda s_0)^2$, then $$\begin{aligned} \phi(0) \ge \frac{\tilde m}{2 \pi (\beta+1)} ((1-\lambda) s_0)^{\beta+1}. \end{aligned}$$ Set $s_1 {\coloneqq}\lambda s_0$. As $w_0$ is increasing (due to $u_0 \ge 0$) we have $$\begin{aligned} \phi(0) &= {\int_0^{s_0}}(s_0-s)^\beta w_0(s) {\,\mathrm{d}s}\\ &\ge \int_{s_1}^{s_0} (s_0-s)^\beta w_0(s_1) {\,\mathrm{d}s}\\ &= \int_0^{\sqrt{s_1}} \rho u_0(\rho) {\,\mathrm{d}\rho}\int_{s_1}^{s_0} (s_0-s)^\beta {\,\mathrm{d}s}\\ &\ge \frac{\tilde m}{2 \pi} \cdot \frac{((1-\lambda) s_0)^{\beta+1}}{\beta+1}. \qedhere \end{aligned}$$ \[lm:i1\] Let $\beta {>}1$ and $s_0 \in (0, R^2)$. Then for all $t \in (0, {T_{\max}})$ $$\begin{aligned} \label{eq:i1:statement} I_1(t) \ge -\frac{2}{\pi} s_0^\beta m_0 {{\mathrm{e}}}^{\kappa_1 t} \end{aligned}$$ holds, where $I_1$ is defined in . By integrating by parts twice we obtain for $t \in (0, {T_{\max}})$ $$\begin{aligned} I_1(t) &= 4 {\int_0^{s_0}}(s_0-s)^\beta s w_{ss}(s, t) {\,\mathrm{d}s}\\ &= 4 {\int_0^{s_0}}\left( \beta (s_0-s)^{\beta-1} s - (s_0-s)^\beta \right) w_s(s, t) {\,\mathrm{d}s}+ 0 \\ &= 4 {\int_0^{s_0}}(s_0-s)^{\beta-1} \left( (\beta+1) s - s_0 \right) w_s(s, t) {\,\mathrm{d}s}\\ &= 4 {\int_0^{s_0}}(s_0-s)^{\beta-2} \left[ (\beta-1) \left( (\beta+1) s - s_0 \right) - (\beta+1) (s_0-s) \right] w(s, t) {\,\mathrm{d}s}\\ &= - 8 \beta {\int_0^{s_0}}(s_0-s)^{\beta-2} \left(s_0 - \frac{\beta+1}{2} s\right) w(s, t) {\,\mathrm{d}s}. \intertext{As $w(\cdot, t)$ is nonnegative and increasing by \eqref{eq:pde_w:w_s_eq_u} and Lemma~\ref{lm:mass}, noting that $s_0 - \frac{\beta+1}{2} s \le 0$ if and only if $s \ge s_1 {\coloneqq}\frac{2s_0}{\beta+1} \in (0, s_0)$, we conclude } I_1(t) &\ge - 8 \beta {\int_0^{s_0}}(s_0-s)^{\beta-2} \left(s_0 - \frac{\beta+1}{2} s\right) w(s_1, t) {\,\mathrm{d}s}\\ &= - 4 s_0^\beta w(s_1, t). \end{aligned}$$ Because the definition of $w$ and Lemma warrant that $$\begin{aligned} w(s_1, t) \le w(R^2, t) = \frac{m(t)}{2 \pi} \le \frac{m_0 {{\mathrm{e}}}^{\kappa_1 t}}{2 \pi} \quad \text{for $t \in (0, {T_{\max}})$} \end{aligned}$$ a consequence thereof is . \[lm:i2\_i3\] Let $\beta {>}0$, $s_0 \in (0, R^2)$ and $\eta \in (0, 1)$. With $I_2$ and $I_3$ as in $$\begin{aligned} \label{eq:i2_i3:statement} I_2(t) + I_3(t) \ge (1-\eta) \frac{\beta(\beta+2)}{s_0^{\beta+2}} \phi^2(t) - \frac{m_0^2 {{\mathrm{e}}}^{2\kappa_1 t}}{2 \eta (\beta+1)(\beta+2) \|\Omega\|^2 } s_0^{\beta+2} \end{aligned}$$ holds then for all $t \in (0, {T_{\max}})$. Let $t \in (0, {T_{\max}})$. An integration by parts yields $$\begin{aligned} I_2(t) &= 2{\int_0^{s_0}}(s_0-s)^\beta w(s, t) w_s(s, t) {\,\mathrm{d}s}\\ &= {\int_0^{s_0}}(s_0-s)^\beta (w^2)_s(s, t) {\,\mathrm{d}s}\\ &= \beta {\int_0^{s_0}}(s_0-s)^{\beta-1} w^2(s, t) {\,\mathrm{d}s}+ \left[ (s_0-s)^\beta w^2(s, t) \right]_0^{s_0} \\ &= \beta {\int_0^{s_0}}(s_0-s)^{\beta-1} w^2(s, t) {\,\mathrm{d}s}\end{aligned}$$ while by another integration by parts and Young’s inequality we have $$\begin{aligned} I_3(t) &= - \frac{m(t)}{\|\Omega\|} {\int_0^{s_0}}(s_0-s)^\beta s w_s(s, t) {\,\mathrm{d}s}\\ &= \frac{m(t)}{\|\Omega\|} {\int_0^{s_0}}(s_0-s)^\beta w(s, t) {\,\mathrm{d}s}- \frac{\beta m(t)}{\|\Omega\|} {\int_0^{s_0}}(s_0-s)^{\beta-1} s w(s, t) {\,\mathrm{d}s}+ 0 \\ &\ge 0 - \eta \beta {\int_0^{s_0}}(s_0-s)^{\beta-1} w^2(s, t) {\,\mathrm{d}s}- \frac{\beta m^2(t)}{4\eta \|\Omega\|^2} {\int_0^{s_0}}(s_0-s)^{\beta-1} s^2 {\,\mathrm{d}s}\\ &\ge - \eta \beta {\int_0^{s_0}}(s_0-s)^{\beta-1} w^2(s, t) {\,\mathrm{d}s}- \frac{m_0^2 {{\mathrm{e}}}^{2\kappa_1 t}}{2 \eta (\beta+1)(\beta+2) \|\Omega\|^2 } s_0^{\beta+2}. \end{aligned}$$ As also by Hölder’s inequality $$\begin{aligned} \phi(t) &= {\int_0^{s_0}}(s_0-s)^\beta w(s, t) {\,\mathrm{d}s}\le \left({\int_0^{s_0}}(s_0-s)^{\beta+1} {\,\mathrm{d}s}\right)^\frac12 \left({\int_0^{s_0}}(s_0-s)^{\beta-1} w^2(s, t) {\,\mathrm{d}s}\right)^\frac12, \end{aligned}$$ that is, $$\begin{aligned} \phi^2(t) &\le \frac{s_0^{\beta+2}}{\beta+2} {\int_0^{s_0}}(s_0-s)^\beta w^2(s, t) {\,\mathrm{d}s}, \end{aligned}$$ we conclude . The fourth integral ------------------- In order to be able to advantageously integrate by parts in the nonlocal term in we first derive a pointwise bound for $w_s$, which in turn is prepared by the following two lemmata. \[lm:v\_rr\_le\_u\] In $(0, R) \times (0, {T_{\max}})$ the inequality $-v_{rr} \le u$ holds. As $u \ge 0$ we have by the second equation in $$\begin{aligned} (r v_r(r, t))_r \le r \frac{m(t)}{\|\Omega\|} \quad \text{for $(r, t) \in (0, R) \times (0, {T_{\max}})$}, \end{aligned}$$ hence upon integrating $$v_r(r, t) \le \frac{r}{2} \frac{m(t)}{\|\Omega\|} \quad \text{for $(r, t) \in (0, R) \times (0, {T_{\max}})$}.$$ Again by the second equation in we have $v_{rr} = \frac{m(t)}{\|\Omega\|} - u - \frac1r v_r$ such that a direct consequence thereof is $v_{rr} \ge -u$. \[lm:ur\_le\_0\] Throughout $(0, R) \times (0, {T_{\max}})$ we have $u_r \le 0$. Without loss of generality we may assume that $u_0 \in C^2({{\overline}\Omega})$ with $\partial_\nu u_0 = 0$ on $\partial \Omega$, as for less regular initial data the statement follows by an approximation procedure as in [@WinklerCriticalBlowupExponent2018 Lemma 2.2]. Since additionally $\sup_{(x, t) \in [0, R] \times [0, T]} \|\nabla v(x, t)\| {<}\infty$ by elliptic regularity theory (cf. [@FriedmanPartialDifferentialEquations1976 Theorem 19.1]) for all $T \in (0, {T_{\max}})$ we may invoke [@LiebermanHolderContinuityGradient1987 Theorem 1.1] to obtain $$\begin{aligned} u \in C^{1, 0}({\overline}\Omega \times [0, {T_{\max}})) \cap C^{3, 1}({\overline}\Omega \times (0, {T_{\max}})). \end{aligned}$$ Hence, fixing $T \in (0, {T_{\max}})$ and letting $Q_T {\coloneqq}[0, R] \times [0, T]$, the function $z {\coloneqq}u_r\|_{Q_T}$ belongs to $C^{0}(Q_T)$ as well as to $C^{2, 1}([0, R] \times (0, T))$ and satisfies, due to $u_t = u_{rr} + \frac1r u_r - u_r v_r - u \left(\frac{m(t)}{\|\Omega\|} - u\right) + \kappa(r) u - \mu(r) u^p$ in $Q_T$, $$\begin{aligned} z_t = z_{rr} + a(r, t) z_r + b(r, t) z + c(r, t) \quad \text{in $Q_T$}, \end{aligned}$$ wherein $$\begin{aligned} a(r, t) &{\coloneqq}\frac1r - v_r(r, t), \\ b(r, t) &{\coloneqq}-\frac1{r^2} - v_{rr}(r, t) - \frac{m(t)}{\|\Omega\|} + 2u(r, t) + \kappa(r) - p\mu(r) u^{p-1} \quad \text{and} \\ c(r, t) &{\coloneqq}\kappa'(r) u - \mu'(r) u^p(r, t) \end{aligned}$$ for $(r, t) \in Q_T$. As $\kappa' \le 0$ and $\mu' \ge 0$ by , $u_r(0, \cdot) = 0$ due to radial symmetry, $u_r(R, \cdot) \le 0$ since $u {>}0$ in $(0, R)$ and $u_{0r} \le 0$ because of we have $$\begin{aligned} \begin{cases} z_t \le z_{rr} + a(r, t) z_r + b(r, t) z & \text{in $(0, R) \times (0, T)$}, \\ z \le 0, & \text{on $\{0, R\} \times (0, T)$}, \\ z(\cdot, 0) \le 0, & \text{in $(0, R)$}. \end{cases} \end{aligned}$$ Lemma \[lm:v\_rr\_le\_u\] warrants that $-v_{rr} \le u$ in $Q_T$, hence $\sup_{(r, t) \in Q_T} b(x, t) \le 3u(r, t) + \kappa(r) {<}\infty$, such that the comparison principle [@QuittnerSoupletSuperlinearParabolicProblems2007 Proposition 52.4] becomes applicable and yields $z \le 0$. The statement follows then upon taking $T {\nearrow}{T_{\max}}$. \[lm:ws\_bdd\] We have $$\begin{aligned} w_s(s, t) \le \frac{m_0 {{\mathrm{e}}}^{\kappa_1 t}}{2 \pi s} \end{aligned}$$ for all $s \in (0, R^2), t \in (0, {T_{\max}})$. Let $r \in (0, R)$ and $t \in (0, {T_{\max}})$. On the one hand we have by Lemma \[lm:ur\_le\_0\] $$\begin{aligned} {\int_\Omega}u(\cdot, t) = 2\pi \int_0^R \rho u(\rho, t) {\,\mathrm{d}\rho}\ge 2\pi \int_0^r \rho u(r, t) {\,\mathrm{d}\rho}= \pi r^2 u(r, t) \end{aligned}$$ and one the other hand by Lemma \[lm:mass\] $$\begin{aligned} {\int_\Omega}u(\cdot, t) \le m_0 {{\mathrm{e}}}^{\kappa_1 t} \end{aligned}$$ such that $$\begin{aligned} u(r, t) \le \frac{m_0 {{\mathrm{e}}}^{\kappa_1 t}}{\pi r^2}. \end{aligned}$$ The statement follows due to $w_s(s, t) = \frac12 u(s^\frac12, t)$ for $s \in (0, R^2)$ and $t \in (0, {T_{\max}})$. The exponent $-1$ in Lemma \[lm:ws\_bdd\] is essentially optimal. Indeed, if we were able to show $w_s(s, t) \le f(t) s^{-q}$ for some $f \in C^0([0, \infty))$ and $q {<}1$ and all $(s, t) \in (0, R^2) \times (0, {T_{\max}})$, then also $u(r, t) \le 2 f(t) r^{-2q}$ for $r \in (0, R)$ and $t \in (0, {T_{\max}})$. However, this would yield $\sup_{t \in (0, T)} \\|u(\cdot, t)\\|_{{{L^{\lambda}(\Omega)}}} {<}\infty$ for some $\lambda {>}1$ and all finite $T \in (0, {T_{\max}}]$, which in turn would rapidly imply ${T_{\max}}= \infty$, confer the proof of Proposition \[prop:global\_ex\] below. With these preparations at hand we are finally able to deal with the fourth integral on the right-hand side of . \[lm:i4\] Let $\beta {>}-1$, $s_0 \in (0, \min\{1, R^2\})$ and suppose that $\mu$ satisfies for some $\mu_1 {>}0$ and $\alpha \ge 2(p-1)$. Then $$\begin{aligned} 2^{p-1} {\int_0^{s_0}}\int_0^s (s_0-s)^\beta \mu(\sqrt{\sigma}) w_s^p(\sigma, t) {\,\mathrm{d}\sigma}{\,\mathrm{d}s}\le C \phi(t) \end{aligned}$$ for all $t \in (0, {\hat T_{\max}})$, where $C {\coloneqq}\left(\frac{m_0 {{\mathrm{e}}}^\kappa}{\pi}\right)^{p-1} \mu_1$ and ${\hat T_{\max}}{\coloneqq}\min\{1, {T_{\max}}\}$. Let $\alpha ' := \frac{\alpha}{2} - (p-1)$. Due to we see that $\alpha' \ge 0$, such that an application of Lemma \[lm:ws\_bdd\] and an integration by parts yield $$\begin{aligned} &{\mathrel{{\hphantom}{=}}}2^{p-1} {\int_0^{s_0}}(s_0-s)^\beta \int_0^s \mu(\sqrt{\sigma}) w_s^p(\sigma, t) {\,\mathrm{d}\sigma}{\,\mathrm{d}s}\\ &\le \left(\frac{2m_0 {{\mathrm{e}}}^\kappa}{2\pi}\right)^{p-1} \mu_1 {\int_0^{s_0}}(s_0-s)^\beta \int_0^s \sigma^{\alpha'} w_s(\sigma, t) {\,\mathrm{d}\sigma}{\,\mathrm{d}s}\\ &= C {\int_0^{s_0}}(s_0-s)^\beta \left[ - \alpha' \int_0^s \sigma^{\alpha'-1} w(\sigma, t) {\,\mathrm{d}\sigma}+ s^{\alpha'} w(s, t) \right] {\,\mathrm{d}s}\\ &\le C {\int_0^{s_0}}(s_0-s)^\beta w(s, t) = C \phi(t) \end{aligned}$$ for $t \in (0, {\hat T_{\max}})$. Conclusion. Proof of Theorem \[th:blow\_up\] -------------------------------------------- As it turns out, for any initial mass $m_0 {>}8\pi$ we are able to find a suitable initial datum $u_0$ with ${\int_\Omega}u_0 = m_0$ as well as sufficiently small $s_0$ and sufficiently large $\beta$ such that a combination of the estimates above makes Lemma \[lm:ode\_blow\_up\] applicable – implying that $\phi$ and hence $u$ must blow up in finite time. Let $m_0 {>}8\pi$ and $\mu_1 {>}0$. The function $$f: (0, m_0] \times [0, {T_{\max}}) \times (1, \infty) \times [0, 1) \times [0, 1) {\rightarrow}{\mathbb{R}}$$ defined by $$(\tilde m, \tilde T, \beta, \lambda, \eta) \mapsto (1-\eta) \beta (\beta+2) \cdot \frac{\tilde m^2}{4\pi^2(\beta+1)^2} (1-\lambda)^{2\beta + 2} \cdot \frac{\pi}{2 m_0 {{\mathrm{e}}}^{\kappa \tilde T}}$$ is continuous and satisfies $$\begin{aligned} \lim_{\beta {\nearrow}\infty} f(m_0, 0, \beta, 0, 0) = \frac{m_0}{8\pi}. \end{aligned}$$ Thus, due to our assumption that $m_0 {>}8\pi$ we may first choose $\beta \in (1, \infty)$ and then $\tilde m \in (0, m)$, $\tilde T \in (0, \min\{1, {T_{\max}}\})$, $\lambda \in (0, 1)$ and $\eta \in (0, 1)$ as well as ${\varepsilon}\in (0, 1)$ such that $$\label{eq:blow_up:f_ge_1} (1-{\varepsilon})^2 f(\tilde m, \tilde T, \beta, \lambda, \eta) \ge 1.$$ For $s_0 {>}0$ let $$\begin{aligned} c_1(s_0) &{\coloneqq}(1-\eta) \frac{\beta(\beta+2)}{s_0^{\beta+2}}, \\ c_2(s_0) &{\coloneqq}\left(m_0 {{\mathrm{e}}}^\kappa \pi^{-1}\right)^{p-1} \mu_1, \\ c_{3,1}(s_0) &{\coloneqq}\frac{m_0^2 {{\mathrm{e}}}^{2\kappa_1 t}}{2 \eta (\beta+1)(\beta+2) \|\Omega\|^2 } s_0^{\beta+2} \quad \text{and} \\ \phi_0(s_0) &{\coloneqq}\frac{\tilde m}{2 \pi (\beta+1)} ((1-\lambda) s_0)^{\beta+1}, \end{aligned}$$ then there exist $d_1, d_2, d_3 {>}0$ such that $$\begin{aligned} \frac{c_2(s_0)}{c_1(s_0) \phi_0(s_0)} = d_1 s_0, \quad \frac{c_{3,1}(s_0)}{c_1(s_0) \phi_0^2(s_0)} = d_2 s_0^2 \quad \text{and} \quad \frac{1}{c_1(s_0) \phi_0(s_0)} = d_3 s_0 \end{aligned}$$ for all $s_0 {>}0$. Hence we may choose $s_0 \in (0, \min\{1, R^2\})$ small enough such that $$\begin{aligned} \frac{{\varepsilon}}{3} \phi_0(s_0) \ge \frac{c_2(s_0)}{c_1(s_0)}, \quad \left( \frac{{\varepsilon}}{3} \phi_0(s_0) \right)^2 \ge \frac{c_{3, 2}(s_0)}{c_1(s_0)} \quad \text{and} \quad \frac{{\varepsilon}}{3} \phi_0(s_0) \ge \frac{2}{c_1(s_0) \tilde T}. \end{aligned}$$ Set also $$\begin{aligned} c_{3,2}(s_0) &{\coloneqq}\frac{2 s_0^\beta m_0 {{\mathrm{e}}}^{\kappa \tilde T}}{\pi}, \end{aligned}$$ then $$\begin{aligned} \left( (1-{\varepsilon}) \phi_0(s_0) \right)^2 \ge \frac{c_{3, 2}(s_0)}{c_1(s_0)} \end{aligned}$$ by . Suppose now that $\kappa, \mu \in C^1([0, R])$ comply with and and that $u_0$ satisfies and with $r_1 {\coloneqq}(\lambda s_0)^2$, but that the corresponding solution $(u, v)$ given by Lemma \[lm:local\_ex\] is global in time. Due to the lemmata above the function $\phi$ defined in Lemma \[lm:phi\_first\_ode\] would then fulfill $$\begin{aligned} \begin{cases} \phi'(t) \ge c_1 \phi^2(t) - c_2 \phi(t) - c_{3,1} - c_{3, 2}, \quad t \in (0, \tilde T), \\ \phi(0) \ge \phi_0, \end{cases} \end{aligned}$$ where we abbreviated $c_i {\coloneqq}c_i(s_0)$ and $\phi_0 {\coloneqq}\phi_0(s_0)$. However, as $$\begin{aligned} \phi_0 &= \frac{{\varepsilon}}{3} \phi_0 + \frac{{\varepsilon}}{3} \phi_0 + (1-{\varepsilon}) \phi_0 + \frac{{\varepsilon}}{3} \phi_0 \\ &\ge \frac{c_2}{c_1} + \sqrt{\frac{c_{3,1}}{c_1}} + \sqrt{\frac{c_{3,2}}{c_1}} + \frac{2}{c_1 \tilde T} \\ &\ge \frac{c_2 + \sqrt{c_1 (c_{3,1} + c_{3,2})}}{c_1} + \frac{2}{c_1 \tilde T}, \end{aligned}$$ where we have again set $\phi_0 {\coloneqq}\phi_0(s_0)$, Lemma \[lm:ode\_blow\_up\] would imply $\tilde T \le \frac12 \tilde T$, hence our assumption that ${T_{\max}}= \infty$ must be false. Finally, is a direct consequence of Lemma \[lm:local\_ex\]. Notes on global solvability =========================== Finally, we include short proofs for Proposition \[prop:critical\_mass\] and Proposition \[prop:global\_ex\]. This proof is based on a comparison principle for the scalar equation . A similar idea (with a similar supersolution) has been employed in [@TaoWinklerCriticalMassInfinitetime2017 Lemma 5.2]. Let $u_0 \in {{C^{0}({{\overline}\Omega})}}$ be nonnegative and radially symmetric with $m_0 {\coloneqq}{\int_\Omega}u_0 {<}8 \pi$ as well as $(u, v)$ and $w$ be as constructed in Lemma \[lm:local\_ex\] and defined in Lemma \[lm:pde\_w\], respectively. Then we may choose $a \in (\frac{m_0}{2\pi}, 4)$ and as $w(\cdot, 0) \le \frac{m_0}{2\pi}$ in $(0, R^2)$ and $$\begin{aligned} \lim_{b {\searrow}0} \sup_{s \in (0, R^2)} \left\| \frac{a s}{b + s} - a \right\| = 0 \end{aligned}$$ we may also choose $b {>}0$ such that $$\begin{aligned} {\overline}w: [0, R^2] \times [0, \infty) {\rightarrow}{\mathbb{R}}, \quad (s, t) \mapsto \frac{a s}{b + s} \end{aligned}$$ fulfills ${\overline}w(\cdot, 0) \ge w(\cdot, 0)$ in $(0, R^2)$. Furthermore, by a direct computation $$\begin{aligned} {\overline}w_s(s, t) = \frac{ab}{(b + s)^2} \quad \text{and} \quad {\overline}w_{ss}(s, t) = -\frac{2ab}{(b + s)^3} \end{aligned}$$ for all $(s, t) \in (0, R^2) \times (0, \infty)$, hence $$\begin{aligned} &{\mathrel{{\hphantom}{=}}}{\overline}w_t(s, t) - 4s {\overline}w_{ss}(s, t) - 2 {\overline}w(s, t) {\overline}w_s(s, t) + \frac{m(t)}{\|\Omega\|} s {\overline}w_s(s, t) + 2^{p-1} \int_0^s 2^{p-1} \mu(\sqrt \sigma) {\overline}w_s^p(\sigma, t) {\,\mathrm{d}\sigma}\\ &\ge \frac{8 a b s}{(b + s)^3} - \frac{2 a^2 b s}{(b + s)^3} \ge 0 \end{aligned}$$ for all $(s, t) \in (0, R^2) \times (0, \infty)$ because of $a \le 4$. Therefore, ${\overline}w$ is a supersolution of , fulfills ${\overline}w(0, \cdot) \ge 0$ and ${\overline}w(R^2, \cdot) \ge \frac{m_0}{2\pi}$ as well as ${\overline}w(\cdot, 0) \ge w(\cdot, 0)$ such that the comparison principle warrants ${\overline}w \ge w$ in $(0, R^2) \times (0, {T_{\max}})$. As $w(0, t) = {\overline}w(0, t) = 0$ for all $t \in (0, {T_{\max}})$ this implies $$\begin{aligned} \limsup_{t {\nearrow}{T_{\max}}} w_s(0, t) = \limsup_{t {\nearrow}{T_{\max}}} \lim_{h {\searrow}0} \frac{w(h, t) - w(0, t)}{h} \le \limsup_{t {\nearrow}{T_{\max}}} \lim_{h {\searrow}0} \frac{{\overline}w(h, t) - {\overline}w(0, t)}{h} {<}\infty. \end{aligned}$$ Due to non-degeneracy of outside of the origin and boundedness of $w$ parabolic regularity ensures $\limsup_{t {\nearrow}{T_{\max}}} \\|w_s(\cdot, t)\\|_{L^\infty((0, R^2))} {<}\infty$ implying ${T_{\max}}= \infty$ by Lemma \[lm:local\_ex\]. Let $p {>}2$, $\alpha {<}p - 2$, $\mu_1 {>}0$, $\kappa, \mu \in C^1([0, R])$ be such that holds, $0 \le u_0 \in {{C^{0}({{\overline}\Omega})}}$ and denote the corresponding solution given by Lemma \[lm:local\_ex\] by $(u, v)$. By our assumption on $\alpha$ there exists $q \in (1, \min\{\frac{2p - 4 - \alpha}{\alpha}, 2\})$. Testing the first equation with $u^{q-1}$ gives $$\begin{aligned} \frac1q {\frac{\mathrm{d}}{\mathrm{d}t}}{\int_\Omega}u^q = - \frac{4(q-1)}{q^2} {\int_\Omega}\|\nabla u^\frac{q}{2}\|^2 + \frac{q-1}{q} {\int_\Omega}\nabla u^q \cdot \nabla v + {\int_\Omega}\kappa u^q - {\int_\Omega}\mu u^{p+q-1} \end{aligned}$$ in $(0, {T_{\max}})$, wherein $$\begin{aligned} {\int_\Omega}\nabla u^q \cdot \nabla v &= {\int_\Omega}u^{q+1} - {\int_\Omega}u^q \frac{m(t)}{\|\Omega\|} \le \frac{q}{\mu_1 (q-1)} {\int_\Omega}\|x\|^\alpha u^{p+q-1} + c_1 \int_0^R r^{1 - \alpha \frac{q+1}{p-2}} {\,\mathrm{d}r}\end{aligned}$$ in $(0, {T_{\max}})$ for some $c_1 {>}0$ by Young’s inequality (with exponents $\frac{p+q-1}{q+1}$ and $\frac{p+q-1}{p-2}$). By the definition of $q$ we have $1 - \alpha \frac{q+1}{p-2} {>}-1$, hence the function $y: [0, {T_{\max}}) {\rightarrow}{\mathbb{R}}, t \mapsto \frac1q {\int_\Omega}u^q$ satisfies $y' \le c_2$ in $(0, {T_{\max}})$ for some $c_2 {>}0$. Assuming for the sake of contradiction ${T_{\max}}{<}\infty$, this implies $\sup_{t \in (0, {T_{\max}})} {\int_\Omega}u^q(\cdot, t) {<}\infty$, hence by elliptic regularity theory (cf. [@FriedmanPartialDifferentialEquations1976 Theorem 19.1]) $\sup_{t \in (0, {T_{\max}})} \\|v(\cdot, t)\\|_{{{W^{2, q}(\Omega)}}}$ is finite as well. Therefore, the Sobolev embedding theorem warrants finiteness of $\sup_{t \in (0, {T_{\max}})} {\int_\Omega}\|\nabla v(\cdot, t)\|^\frac{2q}{2-q}$. Finally, as $\frac{2q}{2-q} {>}2$, a semi-group argument as in [@FuestBoundednessEnforcedMildly2019 Lemma 4.1] shows boundedness of $\{u(\cdot, t): t \in (0, {T_{\max}})\}$ in ${{L^{\infty}(\Omega)}}$ – contradicting Lemma \[lm:local\_ex\]. Acknowledgments {#acknowledgments .unnumbered} =============== The author is partially supported by the German Academic Scholarship Foundation and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140. [10]{} N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler. Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues. , 25(09):1663–1763, 2015. [](http://dx.doi.org/10.1142/S021820251550044X). P. Biler. Local and global solvability of some systems modelling chemotaxis. , 8, 1998. M. A. J. Chaplain and G. Lolas. Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. , 15(11):1685–1734, 2005. [](http://dx.doi.org/10.1142/S0218202505000947). T. Cieślak and P. Lauren[ç]{}ot. Finite time blow-up for a one-dimensional quasilinear parabolicparabolic chemotaxis system. , 27(1):437–446, 2010. [](http://dx.doi.org/10.1016/j.anihpc.2009.11.016). T. Cieślak and M. Winkler. Finite-time blow-up in a quasilinear system of chemotaxis. , 21(5):1057–1076, 2008. [](http://dx.doi.org/10.1088/0951-7715/21/5/009). J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski. The evolution of slow dispersal rates: a reaction diffusion model. , 37(1):61–83, 1998. [](http://dx.doi.org/10.1007/s002850050120). A. Friedman. . , Huntington, N.Y, 1976. M. Fuest. Boundedness enforced by mildly saturated conversion in a chemotaxis-MayNowak model for virus infection. , 472(2):1729–1740, 2019. [](http://dx.doi.org/10.1016/j.jmaa.2018.12.020). X. He and W.-M. Ni. Global dynamics of the LotkaVolterra competition-diffusion system: diffusion and spatial heterogeneity I. , 69(5):981–1014, 2016. [](http://dx.doi.org/10.1002/cpa.21596). T. Hillen and K. J. Painter. A user’s guide to PDE models for chemotaxis. , 58(1-2):183–217, 2009. [](http://dx.doi.org/10.1007/s00285-008-0201-3). D. Horstmann and G. Wang. Blow-up in a chemotaxis model without symmetry assumptions. , 12(02), 2001. [](http://dx.doi.org/10.1017/S0956792501004363). W. Jäger and S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis. , 329(2):819 – 824, 1992. [](http://dx.doi.org/10.2307/2153966). K. Kang and A. Stevens. Blowup and global solutions in a chemotaxisgrowth system. , 135:57–72, 2016. [](http://dx.doi.org/10.1016/j.na.2016.01.017). E. F. Keller and L. A. Segel. Traveling bands of chemotactic bacteria: A theoretical analysis. , 30(2):235–248, 1971. [](http://dx.doi.org/10.1016/0022-5193(71)90051-8). J. Lankeit. Chemotaxis can prevent thresholds on population density. , 20(5):1499–1527, 2015. [](http://dx.doi.org/10.3934/dcdsb.2015.20.1499). J. Lankeit. Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. , 258(4):1158–1191, 2015. [](http://dx.doi.org/10.1016/j.jde.2014.10.016). G. M. Lieberman. Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions. , 148(1):77–99, 1987. [](http://dx.doi.org/10.1007/BF01774284). Y. Lou. On the effects of migration and spatial heterogeneity on single and multiple species. , 223(2):400–426, 2006. [](http://dx.doi.org/10.1016/j.jde.2005.05.010). Y. Lou, X.-Q. Zhao, and P. Zhou. Global dynamics of a LotkaVolterra competitiondiffusionadvection system in heterogeneous environments. , 2018. [](http://dx.doi.org/10.1016/j.matpur.2018.06.010). T. Nagai. Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. , 6(1):37–55, 2001. [](http://dx.doi.org/10.1155/S1025583401000042). K. Osaki, T. Tsujikawa, A. Yagi, and M. Mimura. Exponential attractor for a chemotaxis-growth system of equations. , 51(1):119–144, 2002. [](http://dx.doi.org/10.1016/S0362-546X(01)00815-X). P. Quittner and P. Souplet. . Birkhäuser Advanced Texts / Basler Lehrbücher. [Birkhäuser Basel]{}, Basel, 2007. [](http://dx.doi.org/10.1007/978-3-7643-8442-5). R. B. Salako and W. Shen. Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading. , 28(11):2237–2273, 2018. [](http://dx.doi.org/10.1142/S0218202518400146). R. B. Salako and W. Shen. Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. II. Existence, uniqueness, and stability of strictly positive entire solutions. , 464(1):883–910, 2018. [](http://dx.doi.org/10.1016/j.jmaa.2018.04.034). R. B. Salako and W. Shen. Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. III. Transition fronts. preprint, https://arxiv.org/abs/1811.01525, 2018. T. Senba and T. Suzuki. Parabolic system of chemotaxis: Blowup in a finite and the infinite time. , 8(2):349–368, 2001. [](http://dx.doi.org/10.4310/MAA.2001.v8.n2.a9). N. Shigesada, K. Kawasaki, and E. Teramoto. Spatial segregation of interacting species. , 79(1):83–99, 1979. [](http://dx.doi.org/10.1016/0022-5193(79)90258-3). Y. Tao and M. Winkler. Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production. , 19(12):3641–3678, 2017. [](http://dx.doi.org/10.4171/JEMS/749). J. I. Tello and M. Winkler. A chemotaxis system with logistic source. , 32(6):849–877, 2007. [](http://dx.doi.org/10.1080/03605300701319003). M. Winkler. Aggregation vs. global diffusive behavior in the higher-dimensional KellerSegel model. , 248(12):2889–2905, 2010. [](http://dx.doi.org/10.1016/j.jde.2010.02.008). M. Winkler. Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source. , 35(8):1516–1537, 2010. [](http://dx.doi.org/10.1080/03605300903473426). M. Winkler. Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. , 384(2):261–272, 2011. [](http://dx.doi.org/10.1016/j.jmaa.2011.05.057). M. Winkler. Finite-time blow-up in the higher-dimensional parabolicparabolic KellerSegel system. , 100(5):748–767, 2013. [](http://dx.doi.org/10.1016/j.matpur.2013.01.020). M. Winkler. How far can chemotactic cross-diffusion enforce exceeding carrying capacities? , 24(5):809–855, 2014. [](http://dx.doi.org/10.1007/s00332-014-9205-x). M. Winkler. A critical blow-up exponent in a chemotaxis system with nonlinear signal production. , 31(5):2031–2056, 2018. [](http://dx.doi.org/10.1088/1361-6544/aaaa0e). M. Winkler. Finite-time blow-up in low-dimensional KellerSegel systems with logistic-type superlinear degradation. , 69(2), 2018. [](http://dx.doi.org/10.1007/s00033-018-0935-8). M. Winkler. How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases. , 2018. [](http://dx.doi.org/10.1007/s00208-018-1722-8). T. Xiang. Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system. , 59(8):081502, 2018. [](http://dx.doi.org/10.1063/1.5018861). [^1]: fuestm@math.uni-paderborn.de
--- abstract: 'The neutron and proton drip lines represent the limits of the nuclear landscape. While the proton drip line is measured experimentally up to rather high $Z$-values, the location of the neutron drip line for absolute majority of elements is based on theoretical predictions which involve extreme extrapolations. The first ever systematic investigation of the location of the proton and neutron drip lines in the covariant density functional theory has been performed by employing a set of the state-of-the-art parametrizations. Calculated theoretical uncertainties in the position of two-neutron drip line are compared with those obtained in non-relativistic DFT calculations. Shell effects drastically affect the shape of two-neutron drip line. In particular, model uncertainties in the definition of two-neutron drip line at $Z\sim 54, N=126$ and $Z\sim 82, N=184$ are very small due to the impact of spherical shell closures at $N=126$ and 184.' address: - 'Department of Physics and Astronomy, Mississippi State University, MS 39762' - 'Fakultät für Physik, Technische Universität München, D-85748 Garching, Germany' author: - 'A. V. Afanasjev' - 'S. E. Abgemava' - 'D. Ray' - 'P. Ring' title: 'Nuclear landscape in covariant density functional theory.' --- Proton and neutron drip lines ,covariant density functional theory ,two-particle separation energies At present, the nuclear masses of approximately 3000 out of roughly 7000 nuclei expected between nuclear drip lines are known [@AME2012]. Nuclear existence ends at the drip lines. While the proton drip line has been delineated in experiment up to protactinium ($Z=91$), the position of the neutron drip line beyond $Z=8$ is determined only in model calculations. Different models and different parameterizations show rather large variations in predictions of the neutron drip line. Moreover, because of experimental limitations even in foreseeable future it will be possible to define the location of neutron-drip line for the majority of elements only in model calculations. In such a situation it is important to estimate the errors in the location of the predicted neutron drip line introduced by the use of the various calculations. In this context we have to distinguish the results and related theoretical uncertainties obtained within the same model, but with different parameterizations and the results and uncertainties obtained with different models. Theoretical uncertainties(errors) in the prediction of physical observables have several sources of origin. Within one class of models they are the consequences of specific assumptions and the optimization protocols. The differences in the basic assumptions of different model classes is another source. They lead to theoretical uncertainties which can be revealed only by a systematic comparison of a variety of models. The first attempt to estimate theoretical uncertainties in the definition of two-neutron drip line within one class of models has been performed within the Skyrme density functional theory (SDFT) in Ref. [@Eet.12] employing the set of six parametrizations. These results were compared with those obtained in other classes of non-relativistic models such as the microscopic-macroscopic finite range droplet model (FRDM) [@MNMS.95] and the Skyrme Hartree-Fock-Bogoliubov (HFB) calculations of Ref. [@GCP.10] with the HFB-21 parametrization. It turns out that the two-neutron drip lines of the FRDM and Skyrme-HFB calculations are located either within the SDFT error band or very close to it. Similar calculations exist also for non-relativistic DFT models based on the finite range Gogny forces D1S [@DGLGHPPB.10] and D1M [@GHGP.09]. The question of theoretical errors in the definition of the neutron drip line is still not resolved since the important class of nuclear structure models known under name covariant density functional theory (CDFT) [@Serot1986_ANP16-1; @Reinhard1989_RPP52-439; @Ring1996_PPNP37-193; @VALR.05; @Meng2006_PPNP57-470] has not been applied so far in a reliable way to the study of this quantity. Typically, non-relativistic and relativistic DFT differ significantly in the prediction of separation energies close to the drip lines and, in general, of isovector properties far from stability [@V.05]. This may lead to neutron drip lines which differ substantially from non-relativistic models. The goals of the present manuscript are (i) the systematic study of two-proton- and two-neutron-drip lines within the relativistic Hartree-Bogoliubov (RHB) framework [@Kucharek1991_ZPA339-23; @GonzalesLlarena1996_PLB379-13] using several state-of-the-art CDFT parametrizations, (ii) the estimate of theoretical errors in the location of the drip lines within CDFT framework, and (iii) the comparison of the drip lines obtained in relativistic and non-relativistic DFT and thus the estimate of global theoretical errors. To our knowledge, there were only two previous attempts to study the neutron-drip line in the CDFT frawework [@HSet.97; @GTM.05]. However, both of them employ quite crude approximations to the physics of drip line nuclei with a rather limited validity. For example, the pairing correlations have been completely ignored in the studies of Ref. [@HSet.97] and the treatment of pairing via BCS approximation in Ref. [@GTM.05] is questionable in the region of drip line since this approximation does not take into account the continuum properly and leads to the formation of a neutron gas [@DFT.84] in nuclei near neutron-drip line. In addition, these calculations use at most 14 fermionic shells for the harmonic oscillator basis, which according to our study and the one of Ref. [@RA.11] is not sufficient for a correct description of binding energies of actinides and superheavy nuclei and the nuclei in the vicinity of neutron-drip line. The RHB framework with a finite range pairing force is a proper tool for that purpose. It has been applied very successfully with the parameter set NL3 [@LVR.01; @LVR.04] and the parameter set DD-PC1 [@Ferreira2011_PLB701-508] at the proton drip line and it has the proper coupling to the continuum at the neutron drip line. In the present manuscript, the RHB framework is used for a systematic studies of ground state properties of all even-even nuclei from the proton- to neutron drip line. The separable version [@TMR.09; @Tian2009_PRC80-024313] of the finite range Brink-Booker part of the Gogny D1S force is used in the particle-particle channel; its strength variation across the nuclear chart is defined by means of the fit of rotational moments of inertia calculated in the cranked RHB framework to experimental data via the procedure of Ref. [@AA.13]. The need for such $A$-dependent variation of the strength of the Brink-Booker part of the Gogny D1S force in the CDFT application has recently been discussed in Refs. [@AA.13; @WSDL.13]. As the absolute majority of nuclei are known to be axially and reflection symmetric in their ground states, we consider only axial and parity-conserving intrinsic states and solve the RHB-equations in an axially deformed oscillator basis [@Gambhir1990_APNY198-132; @Ring1997_CPC105-77]. The truncation of the basis is performed in such a way that all states belonging to the shells up to $N_F = 20$ fermionic shells and $N_B = 20$ bosonic shells are taken into account. This provides sufficient numerical accuracy. As the absolute majority of nuclei are known to be axially and reflection symmetric in their ground states, we consider only axial and parity-conserving intrinsic states. For each nucleus the potential energy curve in large deformation range from $\beta_2=-0.4$ up to $\beta_2=1.0$ is obtained by means of constraint on the quadrupole moment $Q_{20}$. Then, the correct ground state configuration and its energy are defined; this procedure is especially important for the cases of shape coexistence. In axial reflection-symmetric calculations for superheavy nuclei with $Z\geq 100$, the superdeformed minimum is frequently lower in energy than the normal deformed one [@AAR.12]. As long as triaxial and octupole deformations are not included, this minimum is stabilized by the presence of an outer fission barrier. Including such deformations, however, it often turns out that this minimum either disappears or becomes a saddle point, unstable against fission [@AAR.12]. Since these deformations are not included in the present calculations, we restrict our consideration to spherical or normal-deformed ground states in the $Z\geq 100$ nuclei. This also facilitates the comparison with non-relativistic results which favor such ground states for these nuclei. Three existing classes of covariant density functional models are used throughout this paper: the nonlinear meson-nucleon coupling model (NL), the density-dependent meson-exchange model (DD-ME), and a density-dependent point coupling model (DD-PC); see their comparison in Ref. [@AAR.12]. The main differences among them lay in the treatment of the range of the interaction, the mesons, and the density dependence. The interaction in the first two classes has a finite range, while the third class uses a zero-range interaction with one additional gradient term in the scalar-isoscalar channel. The mesons are absent in the density-dependent point coupling model. The density dependence is explicit in the last two models, while it shows up via the nonlinear meson-couplings in the first case. Each of these model classes is represented here by the energy density functional (EDF) that is considered to be the state-of-the-art. The NL model is represented here by the NL3\* [@NL3] EDF which has the smallest number of parameters amongst considered EDF fitted to data. The DD-ME model is represented by the DD-ME2 [@DD-ME2] and the DD-ME$\delta$ [@DD-MEdelta] EDFs. The DD-ME$\delta$ EDF differs from others by the inclusion of the $\delta$-meson, which leads to different proton and neutron effective masses. In addition, the parameters of the DD-ME$\delta$ EDF are largely based on microscopic [ab initio]{} calculations in nuclear matter; only four of its parameters are fitted to finite nuclei. On the contrary, all parameters of other EDF were adjusted to experimental data based on the properties of finite nuclei. The DD-PC model is represented by the DD-PC1 [@DD-PC1] EDF. In contrast to the other functionals, which are fitted to spherical nuclei, this EDF is fitted to a large set of deformed nuclei. ![image](prot-drip-all.eps){width="18cm"} Fig. \[chart\] shows the nuclear landscape as obtained with these CDFT parametrizations. The particle stability (and, as a consequence, a drip line) of a nuclide is specified by its separation energy, namely, the amount of energy needed to remove particle(s). Since our investigation is restricted to even-even nuclei, we consider two-neutron $S_{2n}=B(Z,N-2)-B(Z,N)$ and two-proton $S_{2p}=B(Z-2,N)-B(Z,N)$ separation energies. Here $B(Z,N)$ stands for the binding energy of a nucleus with $Z$ protons and $N$ neutrons. If the separation energy is positive, the nucleus is stable against two-nucleon emission; conversely, if the separation energy is negative, the nucleus is unstable. Thus, two-neutron and two-proton drip lines are reached when $S_{2n}\leq 0$ and $S_{2p}\leq 0$, respectively. EDF measured --------------- ------------------ ------------------ ------------------------- ------------------------- $\Delta E_{rms}$ $\Delta E_{rms}$ $\Delta (S_{2n})_{rms}$ $\Delta (S_{2p})_{rms}$ NL3\ 2.97 3.01 1.21 1.28 DD-ME2 2.42 2.48 1.09 0.99 DD-ME$\delta$ 2.31 2.42 1.11 1.11 DD-PC1 2.02 2.17 1.25 1.13 : The rms-deviations $\Delta E_{rms}$, $\Delta (S_{2n})_{rms}$ ($\Delta (S_{2p})_{rms}$) between calculated and experimental binding energies $E$ and two-neutron(-proton) separation energies $S_{2n}$ ($S_{2p}$), respectively. They are given in MeV for indicated CDFT parametrizations with respect of “measured” and “measured+estimated” sets of experimental masses. \[deviat\] The accuracy of the description of separation energies depend on the accuracy of the description of mass differences. The global RHB calculations of masses with employed parametrizations lead to the rms-deviations $\Delta E_{rms}$ between calculated and experimental binding energies which are listed in Table \[deviat\]. The detailed results of these calculations will be presented in a forthcoming manuscript [@AARR.13]. The masses given in the AME2012 mass evaluation [@AME2012] can be separated into two groups; one represents nuclei with masses defined only from experimental data, the other contains nuclei with masses depending in addition on either interpolation or extrapolation procedures. For simplicity, we call the masses of the nuclei in the first and second groups as measured and estimated. There are 640 measured and 195 estimated masses of even-even nuclei in the AME2012 mass evaluation. One can see in Table \[deviat\] that the addition of estimated masses leads only to a slight decrease of the accuracy of the description of experimental data. Two-neutron $S_{2n}$ and two-proton $S_{2p}$ separation energies are described with typical accuracy of 1 MeV (Table \[deviat\]). One can see that not always the parametrization which provides the best description of masses gives the best description of two-particle separation energies. This is because the separation energies are related to the derivatives of binding energies with respect of particle number. Fig. \[chart-odd\] shows that theoretical uncertainties (i. e. the spread of the predictions due to different EDF) are rather small for two-proton drip line. In addition, the results of the calculations are very close to experimental data. This is because the proton-drip line lies close to the valley of stability, so that extrapolation errors towards it are small. Another reason is the fact the Coulomb barrier provides a rather steep potential reducing considerably the coupling to the proton continuum. This leads to a relatively low density of the single-particle states in the vicinity of the Fermi level. The situation is different for the two-neutron drip line. In the majority of the cases, the theoretical uncertainties in the location of this line are much larger than for the two-proton drip one and they are generally increasing with the increase of mass number. This is commonly attributed to poorly known isovector properties of EDF [@Eet.12]. Although this factor contributes, such an explanation is somewhat oversimplified from our point of view. That is because for some combinations of $Z$ and $N$ there is basically no (or very little) dependence of the predictions for the location of the two-neutron drip line on the CDFT parametrization. Such a weak (or vanishing) dependence is especially pronounced at spherical neutron shell closures with $N=126,184$ and 258 around proton numbers $Z=54,80$ and 110. It is interesting that the impact of shell structure at these particle numbers on the shape of the two-neutron drip line is more pronounced than that for the two-proton drip line due to $Z=50$ and 82 proton shell gaps. However, moving away from these spherical shell closures the spread of theoretical predictions for the two-neutron drip line increases. This move also induces the deformation in the nuclei. Thus, there is a close correlation between the nuclear deformation at the neutron-drip line and the uncertainties in the prediction of neutron-drip line; the regions of large uncertainties corresponds to transitional and deformed nuclei. This is caused by the underlying densities of the single-particle states. The spherical nuclei under discussion are characterized by large shell gaps and a clustering of highly degenerate single-particle states around them. Deformation removes this high degeneracy of single-particle states and leads to a more equal distribution of the single-particle states with energy. Moreover, the density of bound neutron single-particle states close to the neutron continuum is substantially larger than that on the proton-drip line. As a consequence, inevitable inaccuracies in the DFT description of the deformed single-particle state energies which are present even in the valley of beta-stability [@AS.11] will lead to larger uncertainties in the predictions of the neutron-drip line. ![Two-neutron separation energies $S_{2n}$, neutron chemical potentials $\lambda_n$, and quadrupole deformations $\beta_2$ of the Th$(Z=90)$ isotopes obtained in the RHB(DD-ME2) calculations.[]{data-label="reemer"}](reemergence.eps){width="8cm"} For some isotope chains, there are regions of two-neutron stability (not shown in Fig. \[chart\]) at neutron numbers beyond the primary two-neutron drip line. The physical mechanism behind the appearance of these regions is illustrated in Fig. \[reemer\] on the example of the Th isotope chain. Two-neutron separation energies $S_{2n}$ and the neutron chemical potential $\lambda_{2n}$ are positive and negative in two-neutron bound nuclei ($N\leq 184$), respectively. The $S_{2n}$ and $\lambda_{2n}$ values become negative and positive for two-neutron unbound nuclei ($186\leq N\leq 192$), respectively. A further increase of the neutron number triggers an increase of quadrupole deformation $\beta_2$ leading to a lowering of the neutron chemical potential $\lambda_n$ which again becomes negative. As a consequence, two-neutron binding reappears ($S_{2n}>0$) at $N=194-206$. Further increase of $N$ beyond 206 leads to two-neutron unbound nuclei. The appearance of these regions, however, strongly depends on the CDFT parametrization. For example, such regions exist at $(Z=62,N=132-146)$, $(Z=88,N=194-206)$ for DD-PC1, at $(Z=74,N=176-184)$, $(Z=90,N=194-206)$ for DD-ME2 and at $(Z=62,N=132-142)$, $(Z=74,N=178-184)$ and $(Z=90,N=204-206)$ for DD-ME$\delta$. However, the regions of stability beyond the primary drip line are absent in the RHB(NL3\) calculations. A similar reappearance of two-neutron binding with increasing neutron number beyond primary two-neutron drip line exists also in many SDFT parametrizations [@Eet.12]. Both in CDFT and SDFT, the regions of two-neutron binding reappearance represent the peninsulas emerging from the nuclear mainland. Ref. [@Eet.12] suggested that such behavior is due to the presence of shell effects at neutron closures that tend to lower binding energy along the localized bands of stability. This is certainly true in some cases. However, our analysis presented above suggests that local changes of the shell structure induced by deformation changes play also an important role. Similar to the CDFT(NL3\) results, there are also some Skyrme EDF which do not show the reappearance of two-neutron binding [@Eet.13]. It is interesting to compare theoretical CDFT uncertainties in the definition of the two-proton and two-neutron drip lines with the ones obtained in non-relativistic calculations. Fig. \[chart-shade\] presents such a comparison. We use so-called ’2012 Benchmark uncertainties” [@Eet.13] obtained in Ref. [@Eet.12] for Skyrme DFT employing six parametrizations; these uncertainties are shown by the combination of yellow and blue shaded areas in Fig. \[chart-shade\]. The CDFT uncertainties are represented by the combination of the plum and blue shaded areas. One can see that the CDFT and SDFT uncertainties in the definition of two-proton drip line are small; they tightly overlap at $Z\leq 70$ while for higher $Z$ the CDFT uncertainties are shifted slightly towards neutron deficient nuclei as compared with the SDFT ones. The uncertainties for two-neutron drip line are larger but still they are similar in two models in many regions. In particular, the two-neutron drip line at $Z\sim 54, N=126$ and $Z\sim 82, N=184$ is well defined not only in the CDFT and SDFT calculations, but also in the mic+mac (FRDLM) and Gogny D1S calculations. This uniqueness is due to corresponding well pronounced spherical shell closures in the model calculations. The predictions of the DD-ME2, DD-ME$\delta$ and DD-PC1 parametrizations are close to each other (Fig. \[chart\]) and are within the ’2012 Benchmark uncertainties’. The NL3\* parametrization typically shifts the two-neutron drip line to a higher $N$-value exceeding in some regions ’2012 Benchmark uncertainties’. However, the same is true for recently fitted Skyrme TOV-min parametrization [@Eet.13], the two-neutron drip line of which is very similar to the one obtained in the RHB(NL3\) calculations. The biggest difference between CDFT and Skyrme DFT calculations appears at $N=258,Z\sim 110$ (see Fig. \[chart-shade\]) where the two-neutron drip line is uniquely defined in the CDFT calculations due to large spherical gap at $N=258$. This gap is also present in many Skyrme EDF but it does not prevent a significant spread of Skyrme DFT predictions for the two-neutron drip line in this region. This again underlines the importance of shell structure in the predictions of the details of the two-neutron drip line. A similar difference between CDFT and SDFT exists also in superheavy nuclei with $Z\approx 120-126, N\approx 172-184$ where different centers of islands of stability are predicted by these models [@BRRMG.98; @AKF.03]. These results are contrary to the fact that both models generally agree for lighter $Z\leq 100$ nuclei. The DD-\ CEDF predict two-neutron drip line at lower $N$ as compared with the NL3\* one (see Fig. \[chart\]). It is tempting to associate this feature with different symmetry energies $J$ ($J\sim 32$ MeV for DD\* and $J\sim 39$ MeV for NL3\). However, a detailed analysis of 14 two-neutron drip lines obtained in relativistic and non-relativistic calculations does not reveal clear correlations between the location of two-neutron drip line and the nuclear matter properties of the employed force. In conclusion, a detailed analysis of two-neutron drip lines in covariant and non-relativistic DFT has been performed. These results clearly indicate that the shell structure is not washed near or at two-neutron drip line. In particular, model uncertainties in the definition of two-neutron drip line at $Z\sim 54, N=126$ and $Z\sim 82, N=184$ are very small due to the impact of spherical shell closures at $N=126$ and 184. The largest difference between covariant and Skyrme DFT exist in superheavy nuclei, where the first model (contrary to second) predicts significant impact of the $N=258$ spherical shell closure. The spread of theoretical predictions grows up on moving away from these spherical closures. The development of deformation causes it. Both poorly known isovector properties of the forces and inevitable inaccurcies in the description of deformed single-particle states in the DFT framework contribute to that. The number of particle-bound even-even $Z\leq 120$ nuclei is 2040, 2050, 2057 and 2216 in the DD-PC1, DD-ME2, DD-ME$\delta$ and NL3\ parametrizations, respectively. This is close to the numbers obtained in SDFT. Thus, our calculations support the estimate of Ref. [@Eet.12] that around 7000 different (including odd- and odd-odd ones) nuclides have to exist. The authors would like to thank J. Erler for valuable discussions. This work has been supported by the U.S. Department of Energy under the grant DE-FG02-07ER41459 and by the DFG cluster of excellence Origin and Structure of the Universe (www.universe-cluster.de). This research was also supported by an allocation of advanced computing resources provided by the National Science Foundation. The computations were partially performed on Kraken at the National Institute for Computational Sciences (http://www.nics.tennessee.edu/). [99]{} M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCormick, X. Xu and B. Pfeiffer, Chinese Physics C [36]{}, 1603 (2012). J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac, M. Stoitsov, Nature [486]{}, 509 (2012). P. M[ö]{}ller, J. R. Nix, W. D. Myers, W. J. Swiatecki, At. Data Nucl. Data Tabl. [59]{}, 185 (1995). S. Goriely, N. Chamel and J. M. Pearson, Phys. Rev. C [82]{}, 035804 (2010). J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Peru, N. Pillet, and G. F. Bertsch, Phys. Rev. C [81]{}, 014303 (2010). S. Goriely, S. Hilaire, M. Girod, and S. P[é]{}ru, Phys. Rev. Lett. [102]{}, 242501 (2009). B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. [16]{}, 1 (1986). P.-G. Reinhard, Rep. Prog. Phys. [52]{}, 439 (1989). P. Ring, Prog. Part. Nucl. Phys. [37]{}, 193 (1996). D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Phys. Rep. [409]{}, 101 (2005). J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H. Long, and L. S. Geng, Prog. Part. Nucl. Phys. [57]{}, 470 (2006). D. Vretenar, Nucl. Phys. [A 751]{}, 264c (2005). H. Kucharek and P. Ring, Z. Phys. A [339]{}, 23 (1991). T. Gonzalez-Llarena, J. L. Egido, G. A. Lalazissis, and P. Ring, Phys. Lett. B [339]{}, 23 (1991). D. Hirata, K. Sumiyoshi, I. Tanihata, Y. Sugahara, T. Tachibana and H. Toki, Nucl. Phys. [A616]{}, 438c (1997). L. Geng, H. Toki, and J. Meng, Prog. Theor. Phys., [113]{}, 785 (2005). J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys. [A422]{}, 103 (1984). P.-G. Reinhard, B. K. Agrawal, Int. Jour. Mod. Phys. [E20]{}, 1379 (2011). G. A. Lalazissis, D. Vretenar, P. Ring, Nucl. Phys. [A679]{}, 481 (2001). G. A. Lalazissis, D. Vretenar, P. Ring, Phys. Rev. [C69]{}, 173011 (2004). L. S. Ferreira and E. Maglione and P. Ring, Phys. Lett. B [701]{}, 508 (2011). Y. Tian, Z. Y. Ma, and P. Ring, Phys. Lett. [B676]{}, 44 (2009). Y. Tian, Z. Y. Ma, and P. Ring, Phys. Rev. C [80]{}, 024313 (2009). A. V. Afanasjev and O. Abdurazakov, Phys. Rev. C [88]{}, 014320 (2013). L. J. Wang, B. Y. Sun, J. M. Dong, and W. H. Long, Phys. Rev. C [87]{}, 054331 (2013). Y. K. Gambhir, P. Ring, and A. Thimet, Ann. Phys. (N. Y.) [198]{}, 132 (1990). P. Ring, Y. K. Gambhir, and G. A. Lalazissis, Comp. Phys. Comm. [105]{}, 77 (1997). H. Abusara, A. V. Afanasjev, and P. Ring, Phys. Rev. [C85]{}, 024314 (2012). G. A. Lalazissis, S. Karatzikos, R. Fossion, D. Pe[ñ]{}a Arteaga, A. V. Afanasjev, and P. Ring, Phys. Lett. [B671]{}, 36 (2009). G. A. Lalazissis, T. Nik[š]{}i[ć]{}, D. Vretenar, and P. Ring, Phys. Rev. [C71]{}, 024312 (2005). X. Roca-Maza, X. Viñas, M. Centelles, P. Ring, and P. Schuck, Phys. Rev. C 84, 054309 (2011). T. Nikšić, D. Vretenar, and P. Ring, Phys. Rev. [C78]{}, 034318 (2008). S. E. Abgemava, A. V. Afanasjev, D. Ray, and P. Ring, in preparation. A. V. Afanasjev and S. Shawaqfeh, Phys. Lett. [B706]{}, 177 (2011). J. Erler, C. J. Horowitz, W. Nazarewicz, M. Rafalski, P.-G. Reinhard, Phys. Rev. c [87]{}, 044320 (2013). M. Bender, K. Rutz, P.-G. Reinhard, J. A. Maruhn, and W. Greiner, Phys. Rev. [C58]{}, 2126 (1998). A. V. Afanasjev, T. L. Khoo, S. Frauendorf, G. A. Lalazissis, and I. Ahmad, Phys. Rev. [C67]{}, 024309 (2003).
--- author: - 'A. Hanslmeier , R. Brajša, J. Čalogović, B. Vršnak' - 'D. Ruždjak' - 'F. Steinhilber' - 'C.L. MacLeod' - 'Ž. Ivezić$^{\rm ,}$' - 'I. Skokić' date: 'Received May 31, 2010; accepted ......' subtitle: 'II. Analysis of cosmogenic $^{10}\rm Be$ data' title: The chaotic solar cycle --- [The variations of solar activity over long time intervals using a solar activity reconstruction based on the cosmogenic radionuclide $^{10}$Be measured in polar ice cores are studied.]{} [The periodicity of the solar activity cycle is studied. The solar activity cycle is governed by a complex dynamo mechanism. Methods of nonlinear dynamics enable us to learn more about the regular and chaotic behavior of solar activity. In this work we compare our earlier findings based on $^{14}$C data with the results obtained using $^{10}$Be data.]{} [By applying methods of nonlinear dynamics, the solar activity cycle is studied using solar activity proxies that have been reaching into the past for over 9300 years. The complexity of the system is expressed by several parameters of nonlinear dynamics, such as embedding dimension or false nearest neighbors, and the method of delay coordinates is applied to the time series. We also fit a damped random walk model, which accurately describes the variability of quasars, to the solar $^{10}$Be data and investigate the corresponding power spectral distribution. The periods in the data series were searched by the Fourier and wavelet analyses.]{} [The solar activity on the long-term scale is found to be on the edge of chaotic behavior. This can explain the observed intermittent period of longer lasting solar activity minima. Filtering the data by eliminating variations below a certain period (the periods of 380 yr and 57 yr were used) yields a far more regular behavior of solar activity. A comparison between the results for the $^{10}$Be data with the $^{14}$C data shows many similarities. Both cosmogenic isotopes are strongly correlated mutually and with solar activity. Finally, we find that a series of damped random walk models provides a good fit to the $^{10}$Be data with a fixed characteristic time scale of 1000 years, which is roughly consistent with the quasi-periods found by the Fourier and wavelet analyses.]{} [The time series of solar activity proxies used here clearly shows that solar activity behaves differently from random data. The unfiltered data exhibit a complex dynamics that becomes more regular when filtering the data. The results indicate that solar activity proxies are also influenced by other than solar variations and reflect solar activity only on longer time scales.]{} Introduction ============ The study of the periodic, chaotic, or stochastic nature of solar activity requires the analysis of long-lasting time series of solar activity indices. Since direct solar activity observations have been available only since the beginning of telescopic observations and cover a time span of several hundred years, proxies of solar activity have to be used. Such proxies could be observations of aurorae, cosmogenic isotopes, growth of certain plants like corals, etc. They have to be used to obtain a longer time series of the solar activity. An overview of these proxies can be found, e.g., in Hanslmeier (2007). The theory of deterministic chaos has been successfully applied to many areas of physics, including geophysics, astrophysics, and meteorology (Peitgen et al. 1994, 2004; Cvitanović et al. 2009). Chaotic phenomena in astrophysics and cosmology, mainly for dynamics in the solar system and galactic dynamics as well as for such applications to cosmology as properties of cosmic microwave background radiation, were reviewed by Gurzadyan (2002), while the phenomena showing evidence of nonlinear dynamics in solar physics are summarized by Wilson (1994) and by Hanslmeier (1997). The main motivation for this work is to address the question whether solar behavior is chaotic or random. The answer to this question is important for at least two reasons. First, it has implications for the dynamo models, which according to present ideas should describe the physical processes that govern the observed manifestations of solar magnetic activity. Second, this distinction is important for procedures of predicting and reconstructing solar activity on different temporal scales. Chaos is a special kind of complex behavior of dynamic systems described by nonlinear equations. The dynamic variables that describe the properties of the system and its time evolution are in the nonlinear form, i.e., they have higher order terms. Indications of chaotic behavior in solar dynamo models are present in a number of cases. The solar activity cycle is modulated by several quasi-periodic cycles showing period-doubling characteristics and aperiodic grand minima with a characteristic time scale exceeding several tens of cycle periods (Ruzmaikin et al. 1992; Rüdiger & Hollerbach 2004). The impossibility of representing long-term sunspot cycle as a periodic process, period-doubling oscillations, and positive Kolmogorov entropy found in $^{14}$C data (Gizzatullina et al. 1990) point to deterministic chaos (Ruzmaikin et al. 1992). Possible mechanisms include nonlinear back-reaction of the magnetic field visible in a modulation of the differential rotation; stochastic fluctuation of the $\alpha$-effect; variation in the meridional circulation; on-off intermittency due to a threshold field strength for dynamo action. Nonlinearity is present in equations of motion, where the Lorenz force is quadratic in a magnetic field, and possibly in dynamo equations, where $\alpha$ may also be quadratic in a magnetic field (Hoyng 1992; Rosner & Weiss 1992; Stix 2002; Mestel 2003; Ossendrijver 2003; Schüssler & Schmitt 2004). This paper is a continuation of our study about the chaotic solar cycle using $^{14}$C measurements (Hanslmeier & Brajša 2010), which is referred to as Paper I in this work. In Paper I, references to studies of nonlinear effects in the solar activity in theoretical models are given. In addition, fluctuations in solar dynamo parameters were considered to explain variability of the solar cycle (Hoyng 1993; Ossendrijver & Hoyng 1996; Moss et al. 2008). Chaos and intermittency in the solar cycle were reviewed by Spiegel (2009). Nonlinear dynamics methods have also been applied to predicting solar activity cycles (see, e.g., Sello 2001; Aguirre et al. 2008; Kitiashvili & Kosovichev 2008), and one has to distinguish between stochastic and chaotic behavior. The reader is again referred to the literature cited in Paper I. Steinhilber et al. (2008) used the results of Vonmoos et al. (2006) and reconstructed solar activity for about the past 9300 years using the cosmogenic radionuclide $^{10}$Be measured in polar ice cores. In that paper, solar activity is expressed as solar modulation potential $\Phi$. This $\Phi$ record has been used recently to obtain other records of solar activity, such as interplanetary magnetic field (Steinhilber et al. 2010) and total solar irradiance (Steinhilber et al. 2009). The abundance depends on the intensity of the cosmic ray flux, which can enter the Earth’s atmosphere. The intensity of cosmic rays is lower when the Sun is very active and vice versa. Thus, there is an anticorrelation between the production of cosmogenic isotopes and solar activity. Besides solar activity, the $^{10}\rm Be$ signal is also influenced by geomagnetic field intensity variations and system (climate) effects. However, when the data is averaged over 22 or more years, the system effect component contributes less than 10% to the total signal (McCracken 2004). We note that in Vonmoos et al. (2006) and in Steinhilber et al. (2008) the effect by geomagnetic field intensity variations has already been considered. However, there is large uncertainty in the reconstructions of geomagnetic field intensity. In addition, because the geomagnetic field intensity has to be considered when using cosmogenic radionuclides (e.g., $^{10}\rm Be$, $^{14}\rm C$), there is also some uncertainty in the $\Phi$ reconstruction. To estimate geomagnetic influence, we follow the same approach as in Paper I and add random noise with different amplitudes to the $\Phi$ record. While the usual, well-known random walk model has been widely used in solar physics (e.g., Leighton 1964; Sheeley et al. 1987; Wang & Sheeley 1994; Hagenaar et al. 1999; Hathaway 2005; Brajša et al. 2008), this is not the case with the damped random walk (DRW) model. In the present work, we use a DRW model to analyze the time series of reconstructed solar activity. In a non-solar context, Kelly et al. (2009) introduced a model where the optical variability of a given quasar is described by a DRW. The difference with respect to the well-known random walk is that an additional self-correcting term pushes any deviations back towards the mean flux on a time scale $\tau$. It has been established by Kelly et al. (2009), Koz[ł]{}owski et al. (2010), and MacLeod et al. (2010) that a DRW can statistically explain the observed light curves of quasars at an impressive fidelity level (0.01-0.02 mag). The model has only three free parameters: the mean value of the light curve ($\mu$), the driving amplitude of the stochastic process, and the damping (or characteristic) time scale $\tau$. The predictions are only statistical, and the random nature reflects our uncertainty about the details of the physical processes. The paper is structured as follows. In Section 2, the data and the applied method to analyze the data are described. Section 3 gives the results, and in Section 4, we summarize our main results and draw the conclusions. Data and data analysis ====================== Data ---- The solar modulation function $\Phi$ during most of the Holocene (last 9300 years) was reconstructed based on the cosmogenic $^{10}\rm Be$ data, as described by Vonmoos et al. (2006) and by Steinhilber et al. (2008). The data used, which were analyzed by different methods of nonlinear dynamics, are shown in Figs. \[c14\_be\] and \[15snnr1\]. The $^{10}\rm Be$ data are limited to the last 9300 years due to some missing measurements corresponding to the time prior to that period. Also, a climatic change occurred at the end of the last glacial period about 11700 years ago. Such climatic changes could be connected to the change in the precipitation rate and thus to the $^{10}\rm Be$ concentration in ice (system effects). However, during the Holocene period, there are no indications for strong changes in the precipitation source for central Greenland (Johnsen et al. 1989; Mayewski et al. 1997). As stated in the Introduction, the variations seen in the data should be due to changes in solar activity. However, even though the data are filtered, there could be still some influence/error due to system effects (climate) and uncertainty in the reconstructions of geomagnetic field intensity. To test the robustness of our results, we followed our approach in Paper I, i.e., stimulating the field by adding random data of different amplitudes to the original values and then studying how the results changed. Methods of nonlinear dynamics and time series analysis ------------------------------------------------------ We present the calculation of the same parameters that were studied in Paper I: - mutual information - embedding dimension - false nearest neighbors For a more detailed description of these parameters, see Paper I or the papers by Takens (1980), Sauer et al. (1991), Kennel et al. (1992), Kantz & Schreiber (1997), Rhodes & Morari (1997), and Schreiber (1999). The method of delayed coordinates (also described in Paper I) was used for the $^{10}\rm Be$ data, and a comparison with the $^{14}\rm C$ data is given. Damped random walk ------------------ The time variability is modeled as a stochastic process described by the exponential covariance matrix $$S_{ij} = \sigma^2 \exp(-\|t_i-t_j\|/\tau) \label{eqn:cfunc}$$ between times $t_i$ and $t_j$. As explained by Kelly et al. (2009) and Koz[ł]{}owski et al. (2010), this corresponds to a DRW with a damping, or characteristic time scale $\tau$ and a long-term standard deviation of variability $\sigma$. The DRW model used here is more specifically an Ornstein-Uhlenbeck process. Following Koz[ł]{}owski et al. (2010), we model the time series of solar activity and estimate the parameters and their uncertainties using the method of Press et al. (1992), its generalization in Rybicki & Press (1992), and the fast computational implementation described in Rybicki & Press (1995). As in MacLeod et al. (2010), we express the long-term variability in terms of the structure function (SF), where the SF is the root mean square (rms) magnitude difference as a function of the time lag ($\Delta t$) between measurements. The characteristic time scale for the SF to reach an asymptotic value $\rm{SF}_{\infty}$ is the damping time scale, $\tau$. The SF for a DRW is $$SF(\Delta t) = \rm{SF}_{\infty}(1-e^{-\|\Delta t\|/\tau})^{1/2}, \label{eq:sfdt}$$ and the asymptotic value at large $\Delta t$ is $$\begin{aligned} SF(\Delta t >> \tau) \equiv \rm{SF}_{\infty} = \sqrt{2} \sigma. \label{eq:sfinf}\end{aligned}$$ In addition to the $\chi^2$ per degree of freedom ($\chi^2/N_{dof}$), information on the goodness of fit is provided by the parameters $\Delta L_{\rm{noise}}$ and $\Delta L_{\infty}$. We define $\Delta L_{\rm{noise}} \equiv \ln{(L_{\rm{best}}/L_{\rm{noise}})}$, which was used in Koz[ł]{}owski et al. (2010) and MacLeod et al. (2010) to select quasar light curves that are better described by a DRW than by pure white noise. Here, $L_{\rm{best}}$ is the likelihood of the best-fit stochastic model and $L_{\rm{noise}}$ is that for a white noise solution ($\tau \equiv 0$). We define $\Delta L_{\infty} \equiv \ln{(L_{\rm{best}}/L_{\infty})}$, where $L_{\infty}$ is the likelihood that $\tau \rightarrow \infty$, indicating that the length of the time series under consideration is too short to accurately measure $\tau$. Period analysis --------------- To find out if there was some periodicity in the analyzed time series, we applied the Fourier analysis according to the method of Deeming (1975) and its’ later upgraded version of Lenz & Breger (2005). Furthermore, we used the wavelet analysis of Torrence & Compo (1998). This method made it possible to identify especially pronounced quasi-periods (see, e.g., Temmer et al. 2004). Results ======= First we give a comparison between the $^{14}\rm C $ and $^{10}\rm Be$ measurements (Fig. \[c14\_be\]). The carbon measurements give the reconstructed sunspot index; these values were scaled to the $\Phi$ measurements from the $^{10}\rm Be$ data as earlier described. It is seen that the general trend of the data is quite similar, in agreement with Beer et al. (2007) and Usoskin et al. (2009). The $^{14}\rm C$ data are sampled every ten years, the $^{10}\rm Be$ data every 25 years. Nonlinear dynamics ------------------ In the upper panel of Fig. \[15snnr1\], the reconstructed $\Phi$ record is shown. To suppress noise, a Fourier filter of 380 years was applied to eliminate the shorter time scale fluctuations seen in the data shown in the lower panel in Fig. \[15snnr1\]. ![image](c14f_del.ps){width="12cm"} ![image](be10f_del.ps){width="12cm"} We simulated a possible variation in the geomagnetic field to demonstrate the robustness of the results. In Fig. \[15snnr1\] results are also shown for a simulation of random geomagnetic variations of a period twice the filtering period of 380 years (dotted line). The amplitudes were chosen to simulate a worst case, which means they are of comparable amplitude to the original data. These assumptions correspond to statements by Solanki et al. (2004, Fig. 3c), who reported a magnetic dipole variation of a factor of two at the maximum and of a period that is at least longer than the value used for our filtered data. Next, the delay method was applied. This method is also based on the theorem of Takens (1980) and Sauer et al. (1991). More details on this method can be found in Paper I. In Figs. \[c14f\] and \[Be10f\], a study of delayed data for the $^{14}\rm C$ and the $^{10}\rm Be$ is given. The delay values were taken as $\Delta t=1, 5, 10, 15, 25$, which means the data values $x_i$ are plotted versus $x_{i\Delta t}$. We note that the value of $\Delta t=1$ means a time gap of 10 yr in the case of the $^{14}\rm C$ data and 25 yr in the case of the $^{10}\rm Be$ data. Since we only want to demonstrate the chaotic or nonchaotic behavior for the given data sets, this difference does not have any significance here. The results shown are for the filtered data, which distinctly show some regular behavior. The results given in Figs. \[c14f\] and \[Be10f\] clearly demonstrated that the dynamics becomes more complex when going to variations on a shorter timescale. The behavior of the attractor given by the delay coordinates does not show a significant difference with respect to the original filtered data. By such a filter, variations below a time scale of 57 years were eliminated (not shown here). In Fig. \[mut\] the mutual information is shown as a function of delay (upper panel) and the false nearest neighbors (lower panel) for the the different datasets: unfiltered data (full line), filtered data (dotted line), and random data (dashed line). For the filtered data, the mutual information curve decreases less steeply than for the original data. For the random data, the graph immediately declines to zero. To investigate what effect the length of the time series could have on the results, we split the $^{10}\rm Be$ data into two halves and compare the results for each dataset (Figs. \[plot\_mut\_fnn\] and \[plot\_del5\]). The results concerning false nearest neighbor and mutual information were found to be quite similar. Therefore, our results are not influenced by the limits of the available time series. Damped random walk ------------------ Table \[tab:DRWresults\] lists the best-fit DRW parameters for the solar $^{10}\rm Be$ data, assuming an uncertainty of 80 megavolts (MV) for all data points. A relative likelihood of $\Delta L_{noise} = 29$ indicates that the best-fit DRW model provides a better fit to the $^{10}$Be data than pure white noise. However, a single DRW model cannot accurately reproduce the data and yields a reduced $\chi^2$ of 1.9. A series of DRW models with $\tau$ fixed to 1000 years and varying SF$_{\infty}$ produces an average model that is in perfect agreement with the data to within the adopted errors, whereas a series of DRW models with varying $\tau$ and fixed SF$_{\infty} = 165$ MV cannot reproduce the data. Therefore, the best-fit characteristic time scale of 1000 years seems to be well constrained. However, we note that the time scale of 1000 years has a large uncertainty when measured using a DRW analysis due to the limited light curve length for the solar data. The best-fit DRW model gives $1\sigma$ Bayesian upper and lower limits on $\tau$ of $10^{4.4}$ and $10^{2.9}$ years, respectively. The data and best-fit DRW models are shown in Fig. \[fig:DRW\]. ------------------------- -------- $\mu$ (MV) 390.68 $\tau$ (years) 1000 $\rm{SF}_{\infty}$ (MV) 165.52 $\chi^2/N_{dof}$ 1.9 $\Delta L_{\rm{noise}}$ 29 $\Delta L_{\infty}$ 192 ------------------------- -------- : Best-fit DRW parameters for solar $^{10}\rm Be$ data. \[tab:DRWresults\] The inability of a single DRW process to describe the solar $^{10}$Be data is due to an inaccurate power spectral distribution (PSD) description on short time scales. In Fig. \[psd\_drw\], the PSD for the solar $^{10}$Be data is compared to the PSD for a DRW model. The $x$-axis is log frequency in \[1/years\]. The $y$ axis has arbitrary units. The smaller dots show the PSD for a DRW process with a characteristic time scale arbitrarily set to $\tau = 10^{2.7}$ years (indicated by the vertical dotted line), and the larger dots show the PSD for the solar $^{10}\rm Be$ data. The dashed line shows a spectral index of -2, which corresponds to a DRW process on time scales shorter than $\tau$. However, the solar data show a steeper PSD slope than a DRW at frequencies larger than about $10^{-0.1}$ yr$^{-1}$. The dotted-dashed line shows the average PSD for many DRW models with $\tau=1000$ years, which is still too shallow to accurately describe the solar data. This discrepancy between the data and DRW models can be seen in Fig. \[fig:DRW\], where some of the observed points fall outside of the $1\sigma$ range of model light curves (shown as dotted lines), but the data are still consistent with the models, given their uncertainty of 80 MV. In other words, the data generally show a larger scatter on short time scales with respect to the DRW model. Period analysis --------------- The solar $^{10}\rm Be$ time series was also analyzed by Period04 software (Lenz & Breger 2005). Table \[tab:Be04periods\] lists several strongest quasi-periods as seen in Fourier spectrum (Fig. \[Be04power\]) with a dominant approximatively 1000-year cycle. The wavelet analysis results obtained are presented in Fig \[wavelet\_10Be3\]. We see that by applying the wavelet analysis a quasi-period of about 1000 yr can also be identified, but the significance is not very high and varies during the analyzed time interval. No. Period (years) ----- ---------------- 1. 10970 2. 976 3. 2169 4. 207 5. 715 6. 351 7. 790 8. 500 9. 365 10. 3454 : Strongest periods from Fig. \[Be04power\] for solar $^{10}\rm Be$ data as calculated by Period04 software. \[tab:Be04periods\] ![image](wavelet_10Be3.eps){width="17cm"} Discussion and conclusions ========================== We applied standard methods of nonlinear dynamics (the time series analysis including calculation of the embedding dimension and delay, the false nearest neighbor estimation, and the investigation of the mutual information; for details see Paper I) to study the behavior of solar activity. It must be stressed that the data consist of averaged values, where the averaging was done over a 25-year interval. Therefore, the results are only indicative of longer term variations and do not include and represent the 11-year solar activity cycle because of the short-term noise and limited sampling rate in the $^{10}\rm Be$ data. Hence, the conclusions are valid for longer time scale modulations of solar activity. To make a further distinction between shorter time scale and longer time scale variations, we applied a Fourier low-pass filtering. The mutual information and false nearest neighbor methods applied give insight into the complexity of the underlying system. From this information, we estimated the embedding dimension $m$. In the case of the unfiltered data, we obtained the value $m=15$. At this value, the number of false nearest neighbors becomes very small (Fig. \[mut\]). In the case of filtered data, this value is lower, while for the elimination of fluctuations with $t\le 380 \,\rm yr$, the value is about five. Both values are similar to the values obtained in Paper I. The complexity of the system therefore strongly increases on shorter time scales. Also, we repeated the method of delay coordinates for different delay values. The results for the $^{10}$Be data are similar to the ones obtained with the $^{14}$C data (see Paper I). The topological structure is quite complex when considering the unfiltered original data. The structure, however, becomes fairly simple when considering the filtered data, eliminating $t\le 380 \,\rm yr$ fluctuations.This can be interpreted that solar activity proxies seem to exhibit a more regular and predictable behavior when variations at larger time scales are considered. These larger time scales are well above 100 years. Concerning the methods of nonlinear dynamics, we have analyzed the time series, estimated the false nearest neighbors, and calculated the mutual information. The mutual information is a generalization of the autocorrelation function. Also, the method of delays and the calculation of the embedding dimension $m$ were applied. The embedding dimension can help to answer the question whether the topological structure of a system in a phase space is preserved by transformation (mapping). The topological structure is preserved if neighbors are mapped into neighbors, and it is not preserved if this is not the case. Moreover, it is well known that the Lyapunov exponents determine the evolution of a dynamical system. The topological structure of a system is preserved if $m > m_0$, and it is not preserved if $m < m_0$, where $m_0$ is the minimal embedding dimension for a time series. In that last case, the neighbors are mapped into false neighbors. For the data used in the present analysis, we can estimate the value of $m_0$ from the upper panel of Fig. \[mut\] (mutual information as a function of delay). All curves become fairly constant at the y-value of about one or less. As a result, $m_0 = 1$. Now from the lower panel of Fig. \[mut\] (fraction of false nearest neighbors as a function of $m$), we can estimate $m$ for the unfiltered and filtered data. For the unfiltered data, $m = 15$, and for filtered data, $m$ has a smaller value of 7. So we can conclude that the criterion $m > m_0$ holds for both the filtered and the unfiltered data, implying that the topological structure is preserved, i.e., that the neighbors are fairly well mapped into neighbors after the transformation. The application of false nearest neighbors and embedding dimension in the astrophysical (and especially in the solar context) are discussed in detail by Regev (2006) and by Gilmore and Letellier (2007). In order to test our analysis against the dependence on the number of data points used, we split the $^{10}\rm Be$ data into two halves and compared the results for each subset. As the results are quite similar, we conclude that they are not influenced by those limits. Finally, we fit a DRW model, which accurately describes the variability of quasars, to the solar $^{10}$Be data. In this way, we find that a series of DRW models provides a good fit to the $^{10}$Be data with a characteristic time scale of 1000 years. The best-fit $\tau$ from the DRW analysis (1000 years) seems to be real, although the power spectrum of the solar $^{10}$Be data has a different shape than that for a DRW process (the latter has a shallower power-law slope). Although the DRW process is linear and stochastic, this does not necessarily mean that the solar $^{10}$Be data are not consistent with a chaotic, nonlinear, and deterministic process, even if the data were perfectly described by the DRW model. Stochastic processes are commonly used to approximate an astrophysical time series that is truly chaotic in nature, and our analysis in Section 2.3. may indeed represent a similar approximation. In addition, we note that Ma (2007) found a 1000-year cycle in solar activity by investigating long-term fluctuations of reconstructed sunspot number series. It is interesting that a quasi-period of about 1000 years was also identified in the present work using Fourier and wavelet analyses, though there were equally strong quasi-periods at 207 and 2169 years. Concerning the influence of geomagnetic field variations, one can estimate that variations on a time scale of the order of 1 ky are of solar origin, but longer periodicities $>3\,\rm ky$ may be caused by magnetic modulation. Our results on the stochastic and chaotic properties of solar activity based on $^{10}$Be data are quite similar to the results obtained with the $^{14}$C time series discussed in Paper I, confirming the findings of Beer et al. (2007) and of Usoskin et al. (2009). This demonstrates the robustness of our method and indicates that such an analysis tool is suited to study the complex time behavior of such systems. Both cosmogenic isotopes are strongly correlated mutually and with solar activity. The cosmogenic radionuclides $^{14}$C and $^{10}$Be as long-term indices for solar activity were studied by Beer (2000a, 2000b) and Steinhilber et al. (2008). They stress that these data are produced in a similar way, but that their geochemical behavior is different. The $^{10}$Be production rate (0.018 cm$^{-2}{\rm s}^{-1}$) is more than 100 times lower than that of $^{14}$C because it is only produced by high-energy spallation processes, where $^{14}$C is produced by thermal neutrons interacting with nitrogen. After production, $^{10}$Be gets attached to aerosols and is immediately removed from the atmosphere after about one year. It is then transported to the ground, where it is archived, e.g., in polar ice. In contrast, $^{14}$C after production forms $^{14}$CO$_2$ and is included in the global carbon cycle, i.e., it is exchanged between the reservoirs of CO$_2$, such as atmosphere, biosphere, and the oceans. Consequently, it has much longer residence times, depending on the reservoir (atmosphere: ten years, biosphere: 60 years, ocean: 1000 years). Thus, the $^{14}$C signal (called $\Delta$ $^{14}$C) is not the $^{14}$C production rate at one timepoint, but reflects the $^{14}$C production integrated over several millennia. In addition, one has to consider that the halflife of $^{14}$C is about 5730 years, which is of the same order as the exchange processes between atmosphere, biosphere, and the ocean. As a result, the carbon cycle and the radioactive decay of $^{14}$C have to be considered in order to get the $^{14}$C production rate out from the $\Delta$ $^{14}$C. Both have been considered in Solanki et al. (2004), whose record was the basis for Paper I. The carbon cycle itself has no influence on the transport of $^{10}$Be to the polar ice, and thus we can conclude here that the similarities we found between these two radionuclides indicate that the dominant signal in the data is the production signal due to solar activity and geomagnetic field intensity variations rather than a climate signal. Our findings agree with the study by Beer et al. (2007), who applied principal component analysis to both radionuclides. They found that most of the variation is described by the first principal component, which reflects production. In a continuation of this work, we plan to perform the Hurst analysis of the $^{14}$C and $^{10}$Be data. The research leading to the results presented in this paper received partial funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreements nos. 218816 (SOTERIA) and 263252 (COMESEP) and from the Alexander von Humboldt Foundation. F. Steinhilber acknowledges financial support by NCCR Climate - Swiss climate research. Ž. Ivezić acknowledges support by the Croatian National Science Foundation grant O-1548-2009. The authors also acknowledge the support from the Austrian-Croatian Bilateral Scientific Project (2010/11) for financing the exchange of scientists and would like to thank B. Kelly for helpful insight regarding the DRW process as well as J. Beer and the anonymous referee for helpful comments and suggestions. Aguirre, L.A., Letellier, C., & Maquet, J., 2008, Sol. Phys., 249, 103 Beer, J., 2000a, SSRV, 93, 107 Beer, J., 2000b, SSRV, 94, 53 Beer, J., McCracken, K. G., Abreu, J., Heikkilä, U., & Steinhilber, F., 2007, Proc. 30th International Cosmic Ray Conference, Merida, Mexico Brajša, R. et al., 2008, Cent. Eur. Astrophys. Bull., 32, 165 Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., & Vattay, G., 2009, Chaos: Classical and Quantum, ChaosBook.org Deeming, T. J., 1975, Ap&SS, 36, 137 Gilmore, R., Letellier, C., 2007, The Symmetry of Chaos, Oxford University Press, New York Gizzatullina, S. M., Rukavishnikov, V. D., Ruzmaikin, A. A., Tavastsherna, K. S., 1990, Sol. Phys. 127, 281 Gurzadyan, V. G., 2002, in The Ninth Marcel Grossmann Meeting, eds. V. G. Gurzadyan, R. T. Jantzen, & R. Ruffini, 182 Hagenaar, H. J., Schrijver, C. J., Title, A. M., Shine, R. A., 1999, ApJ, 511, 932 Hanslmeier, A., 1997, Hvar Obs. Bull., 21, 77 Hanslmeier, A., 2007, The Sun and Space Weather, 2nd Ed. (Dordrecht: Springer-Verlag) Hanslmeier, A., & Brajša, R., 2010, A&A, 509, A5 (Paper I) Hathaway, D. H., 2005, ASP Conf. Ser., 346, 19 Hoyng, P., 1992, in J. T. Schmelz and J. C. Brown (eds.) The Sun, A Laboratory for Astrophysics, NATO ASI Ser. 373, 99 Hoyng, P., 1993, A&A, 272, 321 Johnsen, S. J., Dansgaard, W., and White, J. W. C., 1989, Tellus, Ser. B, 41, 452 Kantz, H., & Schreiber, T., 1997, Nonlinear Time Series Analysis (Reading, Massachusetts: Cambridge University Press) Kelly, B. C., Bechtold, J., & Siemiginowska, A. 2009, , 698, 895 Kennel, M. B., Brown, R., & Abarbanel, H. D. I., 1992, Phys. Rev. A, 45, 3403 Kitiashvili, I. & Kosovichev, A. G., 2008, ApJ, 688, L49 Koz[ł]{}owski, S., et al. 2010, , 708, 927 Leighton, R. B., 1964, ApJ, 140, 1547 Lenz P., Breger M. 2005, CoAst, 146, 53 Ma, L. H., 2007, Sol. Phys., 245, 411 MacLeod, C. L., et al. 2010, , 721, 1014 Mayewski, P. A., Meeker, L. D., Twickler, M. S., Whitlow, S., Yang, Q., Lyons, W. B., and Prentice, M., 1997, J. Geophys. Res., 102, 26345 McCracken, K. G., 2004, J. Geophys. Res., 109, A04101 Mestel, L., 2003, Stellar Magnetism, Clarendon Press, Oxford, 195 Moss, D., Sokoloff, D., Usoskin, I., & Tutubalin, V., 2008, Sol. Phys., 250, 221 Ossendrijver, A.J.H. & Hoyng, P., 1996, A&A, 313, 959 Ossendrijver, M., 2003, A&A Rev, 11, 287 Peitgen, H.-O., Jürgens, H., & Saupe, D., 1994, CHAOS, Bausteine der Ordnung, Springer-Verlag, Klett-Cotta, Berlin, Stuttgart Peitgen, H.-O., Jürgens, H., & Saupe, D., 2004, Chaos and Fractals, Springer-Verlag, Berlin, 2nd ed. Press, W. H., Rybicki, G. B., & Hewitt, J. N. 1992, , 385, 404 Regev, O., 2006, Chaos and Complexity in Astrophysics, Cambridge University Press, New York Rhodes, C. & Morari, M., 1997, [PhysRevE]{}, 55, 6162 Rosner, R., Weiss, N., 1992, in K. L. Harvey (ed.), The Solar Cycle, ASP Conf. Ser., 27, 511 Ruzmaikin, A., Feynman, J., Kosacheva, V., 1992, in K. L. Harvey (ed.), The Solar Cycle, ASP Conf. Ser., 27, 547 Rüdiger, G., Hollerbach, R., 2004, The Magnetic Universe, Wiley-VCH, Weinheim, 95 Rybicki, G. B., & Press, W. H., 1992, , 398, 169 Rybicki, G. B., & Press, W. H., 1995, PhysRevLett, 74, 1060 Sauer, T., Yorke, J.A. & Casdagli, M., 1991, J. Stat. Phys., 65, 579 Schreiber, Th., 1999, Physics Reports 308, 1 Sello, S., 2001, A&A, 377, 312 Sheeley, N. R., Jr., Nash, A. G., Wang, Y.-M., 1987, ApJ, 319, 481 Solanki, S. K., Usoskin, I. G., Kromer, B., Sch[ü]{}ssler, M., & Beer, J., 2004, , 431, 1084 Spiegel, E. A., 2009, SSRV, 144, 25 Steinhilber, F., Abreu, J.A., & Beer, J., 2008, Astrophys. and Space Science Transactions, 4, 1 Steinhilber, F., Beer, J., & Fröhlich, C., 2009, Geophys. Res. Lett., 36, L19704 Steinhilber, F., Abreu, J.A., Beer, J., & McCracken, K.G., 2010, J. Geophys. Res., 115, A01104 Schüssler, M., Schmitt, D., 2004, in Geophysical Monograph Ser., 141, J. M. Pap & P. Fox (eds.), AGU, p.33 Stix, M., 2002, The Sun, Springer-Verlag, Berlin, 2nd ed., 305 Takens, F., 1980, Detecting Strange Attractors in Turbulence, in Dynamical Systems and Turbulence, Warwick, ed. D.A. Rand and L.-S. Young, LNM898, Springer Temmer, M., Veronig, A., Rybák, J., Brajša, R., Hanslmeier, A., Sol. Phys., 221, 325 Torrence, C., Compo, G. P., 1998, Bull. Amer. Meteor. Soc, 79, 61 Usoskin, I.G., Horiuchi, K., Solanki, S., Kovaltsov, G.A., & Bard, E., 2009, J. Geophys. Res., 114, A03112 Vonmoos M., Beer J., Muscheler R., 2006, J. Geophys. Res, 111, A10105 Wang, Y.-M., Sheeley, N. R., Jr., 1994, ApJ, 430, 399 Wilson, P. R., 1994, Solar and Stellar Activity Cycles, Cambridge University Press, Cambridge, Ch. 13 (researched and drafted by H. B. Snodgrass), p. 228
--- abstract: 'In this work, we examine effects of large permanent charges on ionic flow through ion channels based on a quasi-one dimensional Poisson-Nernst-Planck model. It turns out large [positive]{} permanent charges inhibit the flux of cation as expected, but strikingly, as the transmembrane electrochemical potential for anion increases in a particular way, the flux of anion decreases. The latter phenomenon was observed experimentally but the cause seemed to be unclear. The mechanisms for these phenomena are examined with the help of the profiles of the ionic concentrations, electric fields and electrochemical potentials. The underlying reasons for the near zero flux of cation and for the decreasing flux of anion are shown to be different over different regions of the permanent charge. Our model is oversimplified. More structural detail and more correlations between ions can and should be included. But the basic finding seems striking and important and deserving of further investigation.' author: - 'Liwei Zhang[^1], Bob Eisenberg[^2], and Weishi Liu[^3]' title: 'An effect of large permanent charge: Decreasing flux to zero with increasing transmembrane potential to infinity' --- Introduction ============ Membranes define biological cells. They provide the barrier that separates, and defines the inside of a cell from the rest of the world. Membranes are much more than just a barrier. They provide pathways for selected molecules to enter and leave cells. The barriers must not be perfect or cells would soon die from lack of energy or drown in their waste. Biological cells need energy to survive and that is provided (in almost all cases) by substances that must cross the membrane. Substances cross membranes through proteins specialized for the task. For a very long time ([@Hod51; @Uss49]) these proteins have been separated into two classes, channels and transporters ([@Tos89]), and studied in two traditions, one of electrophysiology ([@Hille01; @ZT15]), the other of enzymology ([@Hille89; @SL14]), although the distinction between the two approaches was less than clearcut ([@Eis90]). Channels have been viewed fundamentally as charged ‘holes in the wall’ (created by the membrane) that could open and close ([@SN95]) but, once open, the channel followed simple laws of electrodiffusion ([@Eis11]). Transporters were viewed as more biological devices, involving conformation changes, coupling to energy sources (either ATP hydrolysis or the movement of other ions), with quantitative description much more difficult, particularly if the description was to be transferrable with parameters that were independent of conditions. The enormous literature can be sampled in [@ASAG98; @BBHS69; @BL99; @SL14; @Tos89]. Structural biology has shown that transporters and channels have very similar structures ([@CMWSFRCBZ; @LSRKGNK; @SKSKKMN; @WM17]). Biophysics has shown that the processes that open and close channels (‘activation’ and ‘inactivation’) can be coupled to give properties rather like transporters. Physics has shown that the electric fields assumed to be constant in classical electrophysiology must depend on the distribution of charge in the channel and surrounding solutions ([@Eis96]) and so must change with experimental conditions and with location. The detailed properties of open channels have not been compared with transporters in the modern literature, as far as we know, certainly not using models that satisfy the physical requirement that potential profiles (i.e., electric fields) be computed from (and thus be consistent with) all the charges in the system. Here we consider a simple model of a permanently open ion channel. (We leave gating for later consideration.) Most biologists imagine that if the driving force for electrodiffusion is increased–that is to say, if the gradient of electrochemical potential across the channel is increased in magnitude–the flux through the channel should increase. We show here that is not always the case. Consider a channel with large permanent charge and the flux of ions with the opposite sign as the permanent charge (called counter ions in the ion exchange literature ([@Hel62]) or majority charge carriers in the semiconductor literature ([@Pie96; @Sze81; @VGK10]). The flux of ions with the opposite sign as the permanent charge in a channel can decrease dramatically as the driving force increases – we term this phenomenon as [the declining phenomenon]{}. More precisely, if the concentration of the ion is held fixed on one side of the channel, and the concentration decreased on the other (‘trans’) side of the channel, the flux of the counter ions can decrease if the permanent charge density is large, as we show here. A depletion zone can form that prevents flow even though the driving force increases to large values. It is worthwhile to emphasize that if one increases the transmembrane electrochemical potential in a different manner, such as, by increasing the transmembrane electric potential or the concentration of the ion at one side of the channel, then one does not have the declining phenomenon (see Remark \[largemu\]). The decline of flux with [trans]{} concentration has been considered a particular, even defining proerties of transporters, involving conformation changes of state and other properties of proteins less well defined (physically) than electrodiffusion ([@KS59; @Uss47]). Declining flux has been called exchange diffusion (in the early transport literature) and self-exchange more recently and is an example of obligatory exchange. Obligatory exchange is a wide spread, nearly universal property of the nearly eight hundred transporters known twenty years ago ([@ASAG98; @GS97]) with many more known today ([@TK15]). Obligatory exchange is ascribed widely to changes in the structure of transport proteins, to conformation changes in the spatial distribution of the mass of the protein ([@FG86]). Obligatory exchange is often thought to be a special property of transporters not found in channels. The structure of many transporters is now known thanks to the remarkable advances of cryo-electron microscopy, recognized in the 2018 Nobel Prize. A transporter (of one amino acid sequence and thus of a perfectly homogeneous molecular type) exists in different states. Each state is said to have a different conformation meaning, in physical language, that the spatial distribution of mass is different in different states, and the distributions of the different states form disjoint sets, with no overlap. The movement of ions is not directly controlled or driven by the conformation of mass, however. Rather the distribution of mass produces a distribution of steric repulsion forces, and a spatial distribution of electrical forces (because the mass is associated with charge, mostly permanent charge of acid and base groups of the protein, but also significant polarization charge, as well). It is the conformation of these forces that determines the movement of ions. The spatial distribution of forces contributes to the potential of mean force reported in simulations of molecular dynamics. This paper shows that channels with one spatial distribution of mass can have properties of self-exchange (for majority charge carrier counter ions) if the density of permanent charge is large. The spatial distribution of electrical forces can change and create a depletion zone that controls ion movement, while the spatial distribution of the mass of the protein does not change. The conformation governing current flow is the conformation of the electric field – the depletion zone – more than the conformation of mass, in the model considered here. The current flow of counter ions is much greater than the current flow of co-ions because there are many more counter ions than co-ions near the permanent charge. Transporters almost always allow much less current flow than channels. It should be emphasized that the depletion zone considered here arises from the self-consistent solution of the Poisson-Nernst-Planck (PNP) equations of a specific model (large permanent charge, counter ion transport) and that parameters are not adjusted in any way to create or modify the phenomena. This is in stark contrast to calculations of chemical kinetic models that involve many adjustable parameters, without clear physical meaning, and equations that do not conserve current ([@Eis14]). Depletion zones play crucial roles in the behavior of nonlinear semiconductor devices although there they usually arise at locations in PN diodes where permanent charge (called doping in that literature) changes sign. Depletion zones of the type studied here are likely to occur in semiconductors but have received little attention because they have less dramatic effects than those in diodes ([@Pie96; @Sze81; @VGK10]) that follow drift diffusion and PNP equations rather like those of open ionic channels ([@Eis12; @Eis96]). The possible role of depletion zones in channel function has been the source of speculation and experimental verification ([@Eis96-1; @MVWMRE]). It is striking that depletion zones can change the conformation of the electric field and mimic the obligatory exchange traditionally thought to occur only in transporters. Depletion zones can create plastic electric fields whose change in shape dominate the flux through a channel of fixed structure. Our model is of course oversimplified as are any models, or even simulations in apparent atomic detail, of condensed phases. More structural detail and more correlations between ions can and should be included. But the basic finding that large permanent charge can produce depletion zones and those regions can produce a decline of counter ion flux when driving forces increase seems striking and important and deserving of further investigation. The rest of the paper is organized as follows. In Section \[setup\], we describe the quasi-one-dimensional Poisson-Nernst-Planck type model and its dimensionless form for ionic flow. Assumptions are specified with two key assumptions: a dimensionless parameter $\varepsilon$ defined in (\[rescale\]) is small and a dimensionless parameter $Q_0>0$ defined in (\[Q\]) from the permanent charge is large. In Section \[formulas\], approximation formulas (in small $\varepsilon$ and $\nu=1/{Q_0}$) for fluxes are provided, which have a number of non-trivial consequences. One of the apparent non-trivial consequences is that the leading order term $J_{10}$ of cation flux is zero, independent of the values of transmembrane electrochemical potential of the cation. The mechanism of this distinguished effect of large (positive) permanent charge is examined in details in Section \[Dyn4J10\] with the help of the internal dynamics (profiles of the electric field, cation concentrations and electrochemical potential). It turns out the mechanism for $J_{10}=0$ is different over different regions of permanent charge. In Section \[Dyn4declin\], a rather counter-intuitive [declining phenomenon]{} – increasing of anion transmembrane electrochemical potential leads to decreasing of anion flux – is shown to be consistent with our analytical result. Thus, for the first time (to the best of our knowledge), a mechanism of [large]{} permanent charge for such a phenomenon is revealed. The mechanism is then examined in details again with the help of the internal dynamics. We conclude this paper with a general remark in Section \[conclude\]. Classical PNP with large (positive) permanent charge {#setup} ==================================================== A quasi-one-dimensional PNP model --------------------------------- Our study is based on a quasi-one-dimensional PNP model for open channels, first proposed in [@NE] and, for a special case, rigorously justified in [@LW]. For a mixture of $n$ ion species, a quasi-one-dimensional PNP model is $$\begin{aligned} \begin{split}\label{ssPNP} &\frac{1}{A(X)}\frac{d}{dX}\Big(\varepsilon_r(X)\varepsilon_0A(X)\frac{d\Phi}{dX}\Big)=-e_0\Big(\sum_{s=1}^nz_sC_s+{\cal Q}(X)\Big),\\ & \frac{d{\cal J}_k}{dX}=0, \quad -{\cal J}_k=\frac{1}{k_BT}{\cal D}_k(X)A(X)C_k\frac{d\mu_k}{d X}, \quad k=1,2,\cdots, n \end{split}\end{aligned}$$ where $X\in [a_0,b_0]$ is the coordinate along the axis of the channel and baths, $A(X)$ is the cross-sectional area of the channel at the location $X$, $e_0$ is the elementary charge, $\varepsilon_0$ is the vacuum permittivity, $\varepsilon_r(X)$ is the relative dielectric coefficient, ${\cal Q}(X)$ is the permanent charge density, $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, $\Phi$ is the electrical potential, and, for the $k$th ion species, $C_k$ is the concentration, $z_k$ is the valence, ${\cal D}_k(X)$ is the diffusion coefficient, $\mu_k$ is the electrochemical potential, and ${\cal J}_k$ is the flux density. Equipped with system (\[ssPNP\]), a meaningful boundary condition for ionic flow through ion channels (see, [@EL07] for a reasoning) is, for $k=1,2,\cdots, n$, $$\Phi(a_0)={\cal V}, \ \ C_k(a_0)={\cal L}_k>0; \quad \Phi(b_0)=0, \ \ C_k(b_0)={\cal R}_k>0. \label{ssBV}$$ Mathematically, we will be interested in solutions of the boundary value problem (BVP) (\[ssPNP\]) and (\[ssBV\]). An important measurement for properties of ion channels is the [I-V (current-voltage) relation]{} where the current ${\cal I}$ depends on the transmembrane potential (voltage) ${\cal V}$ and is given by $${\cal I}=\sum_{s=1}^nz_s{\cal J}_s({\cal V})$$ where ${\cal J}_k({\cal V})$’s are determined by the BVP (\[ssPNP\]) and (\[ssBV\]) for fixed ${\cal L}_k$’s and ${\cal R}_k$’s. Of course, the relations of individual fluxes ${\cal J}_k$’s to ${\cal V}$ contain more information ([@JEL17]) but it is much harder to experimentally measure the individual fluxes ${\cal J}_k$’s. ### Electroneutrality boundary conditions In relation to typical experimental designs, the positions $X=a_0$ and $X=b_0$ are located in the baths separated by the channel. They are the locations of two electrodes that are applied to control or drive the ionic flow through the ion channel. Ideally, the experimental designs should not affect the intrinsic ionic flow properties so one would like to design the boundary conditions to meet the so-called electroneutrality $$\begin{aligned} \label{electroneutral} \sum_{s=1}^nz_s{\cal L}_s=0=\sum_{s=1}^nz_s{\cal R}_s.\end{aligned}$$ The reason for this is that, otherwise, there will be sharp boundary layers which cause significant changes (large gradients) of the electric potential and concentrations near the boundaries so that a measurement of these values has non-trivial uncertainties. One clever design to remedy this potential problem is the “four-electrode-design": two “outer-electrodes” in the baths far away from the ends of the ion channel to provide the driving force and two “inner-electrodes” in the baths near the ends of the ion channel to measure the electrical potential and the concentrations as the “real” boundary conditions for the ionic flow. At the inner electrodes locations, the electroneutrality conditions are reasonably satisfied, and hence, the electric potential and concentrations vary slowly and a measurement of these values would be robust. We point out that the geometric singular perturbation framework for PNP type models developed in [@EL07; @ELX15; @JL12; @LLYZ13; @Liu05; @Liu09] can treat the case without electroneutrality assumption; in fact, the boundary layers can be determined by the boundary conditions directly. ### Electrochemical potentials The electrochemical potentials $\mu_k(X)$ consists of the ideal component $\mu_k^{id}$ and the excess component $\mu_k^{ex}$ where the ideal component $$\begin{aligned} \label{idealComp} \mu_k^{id}(X)=z_ke_0\Phi(X)+k_BT\ln \frac{C_k(X)}{C_0}\end{aligned}$$ is the point-charge contribution where $C_0$ is a characteristic concentration, and the excess component $\mu_k^{ex}(x)$ accounts for ion size effects. As explained above, although not totally physical for ion channel problems in general, we will consider only the ideal component in this work that can act as guidance for further studies of more accurate models with excess component. ### Permanent charges and channel geometry The permanent charge ${\cal Q}(X)$ is a simplified mathematical model for ion channel (protein) structure. It is determined by the spatial distribution of amino acids in the channel wall, the acid (negative) and base (positive) side chains, more than anything else ([@Eis96]). We will assume ${\cal Q}(X)$ is known and take an oversimplified description to capture some essence of its effects. For this paper, we take it to be as in ([@EL07]), for some $a_0<A<B<b_0$, $$\begin{aligned} \label{calQ} {\cal Q}(X)=\left\{\begin{array}{cc} 0, & X\in [a_0,A)\cup (B,b_0]\\ 2{\cal Q}_0, &X\in (A,B). \end{array}\right.\end{aligned}$$ We will be interested in the case where $\|{\cal Q}_0\|$ is large relative to ${\cal L}_k$’s and ${\cal R}_k$’s. The cross-sectional area $A(X)$ typically has the property that $A(X)$ is much smaller for $X\in (A,B)$ (the neck region) than that for $X\not\in (A,B)$. It is interesting to note that, in [@JLZ15], the authors showed that the neck of the channel should be “narrow" (small $A(X)$ for $X\in (A,B)$) and “short" (small $B-A$) to optimize an effect of permanent charge. ### Dielectric coefficient and diffusion coefficients We assume that $$\begin{aligned} \label{diffusionCoeff} \varepsilon(X)=\varepsilon_r\,\mbox{ is a constant, and }\; {\cal D}_k(X)= {\cal D}(X){\cal D}_k \end{aligned}$$ for some dimensionless function ${\cal D}(X)$ (same for all $k$) and dimensional constant ${\cal D}_k$. Note that the assumption ${\cal D}_k(X)= {\cal D}(X){\cal D}_k$ is equivalent to the statement that ${\cal D}_k(X)/{{\cal D}_j(X)}$ is a constant for $k\neq j$. Roughly speaking, the assumption says that, as the environment varies from location to location, its influences on the two diffusion coefficients ${\cal D}_k(X)$ and ${\cal D}_j(X)$ at the same location $x$ are the same; that is, the two diffusion coefficients vary from one common environment to another common environment in a way so that their ratio is independent of locations. This is not a justification of this assumption but only an explanation of what it reflects. ### Main assumptions In the sequel, we assume [the boundary electroneutrality condition in (\[electroneutral\]), the form of the permanent charge in (\[calQ\]), constant dielectric coefficient and diffusion coefficient property in (\[diffusionCoeff\]), and the electrochemical potential is ideal $\mu_k=\mu_k^{id}$ in (\[idealComp\]).]{} There are two key assumptions for this work to be discussed in terms of dimensionless variables below. Dimensionless form of the quasi-one-dimensional PNP model --------------------------------------------------------- The following rescaling (see [@Gil99]) or its variations have been widely used for convenience of mathematical analysis. Let $C_0$ be a characteristic concentration of the ion solution. We now make a dimensionless re-scaling of the variables in system (\[ssPNP\]) as follows. $$\begin{aligned} \label{rescale}\begin{split} &\varepsilon^2=\frac{\varepsilon_r\varepsilon_0k_BT}{e_0^2(b_0-a_0)^2C_0},\; x=\frac{X-a_0}{b_0-a_0},\; h(x)=\frac{A(X)}{(b_0-a_0)^2}, \; Q(x)=\frac{{\cal Q}(X)}{C_0}, \\ &D(x)={\cal D}(X),\; \phi(x)=\frac{e_0}{k_BT}\Phi(X), \; c_k(x)=\frac{C_k(X)}{C_0}, \; J_k=\frac{{\cal J}_k}{(b_0-a_0)C_0 {\cal D}_k}. \end{split}\end{aligned}$$ The dimensionless quantity $Q(x)$ from the permanent charge ${\cal Q}$ in (\[calQ\]) becomes $$\begin{aligned} \label{Q} Q(x)=\left\{\begin{array}{cc} 0, & x\in [0,a)\cup (b,1]\\ 2Q_0, &x\in (a,b), \end{array}\right.\end{aligned}$$ where $$0<a=\frac{A-a_0}{b_0-a_0}<b=\frac{B-a_0}{b_0-a_0}<1,$$ and, in terms of the dimensionless quantities, the subinterval $(a,b)\subset (0,1)$ corresponds to the neck region $[A,B]$. [Key assumptions.]{} [The parameter $\varepsilon$ is small and the parameter $Q_0>0$ is large.]{} The case where $Q_0<0$ with $\|Q_0\|$ large can be treated in the same way. The assumption of smallness of the dimensionless parameter $\varepsilon$ is reasonable particularly because we are interested in the limit of large $\|Q_0\|$. Thus, we can take large but fixed characteristic concentration $C_0$ in the definition of $\varepsilon$ (\[rescale\]). For example, if $b_0-a_0=25nm$ and $C_0=10 M$, then the dimensionless parameter $\varepsilon\approx 10^{-3}$ ([@EL17]). The mathematical consequence of smallness of $\varepsilon$ is that the BVP (\[PNP\]) and (\[BVO\]) can be treated as a [singularly perturbed problem]{}. A general geometric framework for analyzing the singularly perturbed BVP of PNP type systems has been developed in [@EL07; @Liu05; @Liu09; @LX15] for classical PNP systems and in [@JL12; @LLYZ13; @LTZ12] for PNP systems with finite ion sizes. In terms of the new variables in (\[rescale\]), the BVP (\[ssPNP\]) and (\[ssBV\]) becomes $$\begin{aligned} \label{PNP}\begin{split} &\frac{\varepsilon^2}{h(x)}\frac{d}{dx}\left(h(x)\frac{d\phi}{dx}\right)=-\sum_{s=1}^nz_s c_s -Q(x),\\ &\frac{d J_k}{dx}=0, \quad - J_k=\frac{1}{k_BT}D(x)h(x)c_k\frac{d \mu_k}{d x}, \end{split}\end{aligned}$$ with boundary conditions at $x=0$ and $x=1$ $$\begin{aligned} \label{BVO}\begin{split} \phi(0)=&V,\; c_k(0)=L_k; \; \phi(1)=0,\; c_k(1)=R_k, \end{split}\end{aligned}$$ where $$V:=\frac{e_0}{k_BT}{\cal V},\quad L_k:=\frac{{\cal L}_k}{C_0},\quad R_k:=\frac{{\cal R}_k}{C_0}.$$ In this work, we will consider the BVP (\[PNP\]) and (\[BVO\]) for ionic mixtures with one cation of valence $z_1=1$ and an anion of valence $z_{2}=-1$. We will be interested in properties of ionic flow for large $\|Q_0\|$. It turns out $$\alpha=\frac{H(a)}{H(1)}\;\mbox{ and }\; \beta=\frac{H(b)}{H(1)}\;\; \mbox{ where }\; H(x)=\int_0^x\frac{1}{D(s)h(s)}ds$$ are the key parameters that characterize the effect of channel geometry and location of permanent charge. The physical meanings of these parameters could be seen from the special case when $h(x)=h_0$ and $D(x)=D_0$ are constants. In this case, $$H(x)=\frac{x}{D_0h_0},$$ which is [proportional]{} to the (scaled) length $x$ of the region over $[0,x]$ of the channel (conductive material), [inversely proportional]{} to the (scaled) cross-sectional area $h_0$ of the channel and to the (scaled) diffusion coefficient or electric conductivity $D_0$; that is, $H(x)$ is the [resistance]{} of the portion of the channel over $[0,x]$. Approximations of fluxes [@ZL17]. {#formulas} ================================= We now recall the results on approximations of $(\phi, c_1,c_2,J_1,J_2)$ and $\mu_k$’s from [@ZL17] for the case where $z_1=1$ and $z_2=-1$ with $L_1=L_2=L$ and $R_1=R_2=R$. Based on the assumptions that $\varepsilon>0$ is SMALL and $Q_0>0$ is LARGE, one has expansions of fluxes in $\varepsilon$ and in $\nu=1/{Q_0}$ near $\varepsilon=0$ and $\nu=0$. First, one expands the fluxes in $\varepsilon$ as $$J_1(\varepsilon;\nu)=J_1(\nu)+O(\varepsilon)\;\mbox{ and }\; J_2(\varepsilon)=J_2(\nu)+O(\varepsilon),$$ where $J_k(\nu)$, depending also on the parameters $(V, L, R, H(1), \alpha,\beta)$, are the zeroth order in $\varepsilon$ terms of the fluxes. Then, one expands $J_k(\nu)$ about $\nu$ as $$\begin{aligned} J_1(\nu)=J_{10}+J_{11}\nu+O(\nu^2)\;\mbox{ and }\; J_2(\nu)=J_{20}+J_{21}\nu+O(\nu^2).\end{aligned}$$ Thus, $$\begin{aligned} \label{expinnu} J_1(\varepsilon;\nu)=J_{10}+J_{11}\nu+O(\nu^2,\varepsilon)\;\mbox{ and }\; J_2(\varepsilon;\nu)=J_{20}+J_{21}\nu+O(\nu^2,\varepsilon).\end{aligned}$$ Note that $J_{10}$ and $J_{20}$, depending on $(V, L, R, H(1),\alpha,\beta)$, contain the leading order effect of the [small]{} $\nu$ (or equivalently, [large]{} $Q_0$). The following result is established in [@ZL17]. \[expJs\] One has $$\begin{aligned} \label{ejJ}\begin{split} J_{10}&=0,\\ J_{11}&=\frac{1}{2H(1)(\beta-\alpha)}\left(\frac{(1-\beta)L+\alpha R}{(1-\beta)\sqrt{e^{V}L}+\alpha\sqrt{R}}\right)^2(e^{V}L-R); \\ J_{20}&=\frac{2\sqrt{LR}}{H(1)}\frac{1}{(1-\beta)\sqrt{L}+\alpha\sqrt{e^{-V}R}}(\sqrt{e^{-V}L}-\sqrt{R}),\\ J_{21}&=\frac{(1-\beta)L+\alpha R}{2H(1)(\beta-\alpha)(\sqrt{e^{-V}L}-\sqrt{R})((1-\beta)\sqrt{L}+\alpha\sqrt{e^{-V}R})^3} \\ &\quad\left\{-2(\beta-\alpha)^2(\sqrt{e^{-V}L}-\sqrt{R})^2\sqrt{e^{-V}}LR\right.\\ &\quad\left.+((1-\beta)L+\alpha R)(L-Re^{-V})\left[\frac{(1-\beta)L+\alpha R}{2}\sqrt{e^{-V}}\ln{\frac{L}{e^VR}}\right.\right.\\ &\qquad\qquad\left.\left.-((1-\beta)\sqrt{L}+\alpha\sqrt{e^{-V}R})(\sqrt{e^{-V}L}-\sqrt{R})\right]\right\}. \end{split}\end{aligned}$$ A distinct implication of $J_{10}=0$ in Proposition \[expJs\] is that large (positive) permanent charge $Q_0$ (or small $\nu=1/{Q_0}$) inhibits the cation flux. We will provide a detailed discussion in Section \[Dyn4J10\] on what happens to the internal dynamics that is consistent with this conclusion. To this end, we recall another immediate consequence of (\[ejJ\]) (see [@ZL17] for more). \[Saturate\] \[Current Saturation\] For large permanent charge $Q_0$ (small ${\nu}=1/{Q_0}$) and to the leading order terms in $\nu$, each individual fluxes $J_k$’s, and hence, the total current $I$ saturate as $\|V\|\to \infty$; more precisely, one has $$\begin{aligned} \label{saturation}\begin{split} \lim_{V\to +\infty} J_{20}=&-\frac{1}{1-\beta}\frac{ 2R}{H(1)}, \quad \lim_{V\to -\infty} J_{20}=\frac{1}{\alpha}\frac{2L}{H(1)},\\ \lim_{V\to +\infty} J_{11}=&-\lim_{V\to +\infty}J_{21}=\frac{1}{(1-\beta)^2}\frac{((1-\beta)L+\alpha R)^2}{2H(1)(\beta-\alpha) },\\ \lim_{V\to -\infty} J_{11}=&- \lim_{V\to -\infty}J_{21}=-\frac{1}{\alpha^2}\frac{((1-\beta)L+\alpha R)^2}{2H(1)(\beta-\alpha)}. \end{split}\end{aligned}$$ All the above (finite) limits can be derived from (\[ejJ\]) easily. Note this is NOT the case if the permanent charge is small ([@JLZ15]). Internal dynamics for $J_{10}=0$. {#Dyn4J10} ================================= It follows from the Nernst-Planck equation in (\[PNP\]) that $$J_1\int_0^1\frac{k_BT}{ D_1(x)h(x) c_1(x)}dx=\mu_1(0)-\mu_1(1).$$ Thus, $J_1$ has the same sign as that of $\mu_1(0)-\mu_1(1)$; in particular, if $\mu_1(0)-\mu_1(1)\neq 0$, then $J_1\neq 0$. For the setup of this paper, there are three regions of permanent charge $Q(x)$: $Q(x)=0$ for $x\in [0,a)$ and $x\in (b,1]$ and $Q(x)=2Q_0$ for $x\in [a,b]$ with [large]{} $Q_0$ or [small]{} $\nu$ with ${\nu}=1/{Q_0}$. A major consequence of large $Q_0$ is, from Proposition \[expJs\], up to the leading order in $\nu=1/{Q_0}$ and in $\varepsilon$, the flux of cation is $J_{10}=0$, [even if the transmembrane potential $\mu_1(0)-\mu_1(1)\neq 0$]{}. We will reveal the internal dynamics that lead to this conclusion. To do so, we will discuss what happens over each subinterval based on the approximated (of zeroth order in $\varepsilon$) functions of profiles. Let $$\begin{aligned} \label{leadv} (\phi(x;\varepsilon,\nu),c_k(x;\varepsilon,\nu), J_k(\varepsilon,\nu))=(\phi(x;\nu),c_k(x;\nu) , J_k(\nu))+O(\varepsilon)\end{aligned}$$ be the solution of the BVP (\[PNP\]) and (\[BVO\]). For $\nu>0$ small, one has the following expansions $$\begin{aligned} J_1(\nu)=&J_{10}+J_{11}\nu+O(\nu^2), \end{aligned}$$ where $J_{10}=0$ and $J_{11}$ are given in (\[ejJ\]). We will also provide figures for the profiles of rescaled electrical potential $\phi(x;\nu)$, cation concentration $c_1(x;\nu)$ and electrochemical potential $\mu_1(x;\nu)$ of cation, respectively. The parameter values used are $$\begin{aligned} \label{parameters}\begin{split} e_0=&1.6022 \times 10^{-19}\ (C),\ k_B=1.3806\times 10^{-23}\ (JK^{-1}),\ T=273.16\ (K),\\ \mathcal{V}=& 0.01\ (V), \ {\cal L}=0.5\ (M), \;{\cal R}=10^{-5}\ (M), \; {\cal Q}_0=10^8\ (M),\; C_0=1\ (M),\\ a_0=&0, \; b_0=10\ (nm),\; A=1/3\ (nm), \; B=1/2\ (nm). \end{split} \end{aligned}$$ In terms of the dimensionless parameters, $$V \approx 0.425,\; L=0.5,\; R=10^{-5}\approx 0,\; \;a=\frac{1}{3},\;b=\frac{1}{2},\; \ \nu=10^{-8}\approx 0,$$ The scaled cross-sectional area $h(x)$ is taken to be $$\begin{aligned} h(x)=\left\{\begin{array}{ccc} \pi(-x+r_0+a)^2, & x\in(0,a)\\ \pi r_0^2, & x\in(a,b)\\ \pi(x+r_0-b)^2, & x\in(b,1). \end{array}\right.\end{aligned}$$ The same setup will be used for figures in Section \[Dyn4declin\]. Notice that the whole interval is $(0,1)$, and it is divided into three subintervals $(0,a), (a,b)$ and $(b,1)$. Permanent charge is located over $(a,b)$. The radius of the neck of ion channel is $c$. In the figures, we let $r_0=0.5$. We make plots of each function in interval $(0,1)$, also in every subintervals. Internal dynamics over the interval $(0,a)$ ------------------------------------------- The leading order terms of $(\phi, c_1)$ in (\[leadv\]) are derived in [@ZL17]. One has \[expProfile11\] For $x\in (0,a)$, $$\begin{aligned} \phi(x;\nu)=&\phi_0(x)+\phi_1(x)\nu+O(\nu^2) \mbox{ with }\\ \phi_0(x)=& V- \ln\Big(1- \frac{J_{20}}{2L}H(x)\Big),\quad \phi_1(x)=-\frac{c_{11}(x)}{c_{10}(x)}+\frac{2J_{11}}{J_{20}}\ln{\frac{c_{10}(x)}{L}};\\ c_{10}(x)=& L- \frac{J_{20}}{2}H(x),\quad c_{11}(x)= -\frac{J_{11}+J_{21}}{2}H(x);\\ \mu_{10}(x)=&\mu_{10}(0)=V+\ln{L}, \quad \mu_{11}(x)=2\frac{J_{11}}{J_{20}}\ln{\frac{c_{10}(x)}{L}}. \end{aligned}$$ Figure 1 shows the profiles of $c_1(x;\nu)$, $\phi(x;\nu)$ and $\mu_1(x;\nu)$ over the interval $(0,a)$. \[c1one\] ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ over interval $[0,a]$. Note that $\mu_{10}'(x)=0$ but $\mu_1'(x;\nu)= \mu_{11}'(x)\nu+O(\nu^2)\neq 0$ as shown in the figure.](c1_0a.eps){width="2.5in"} ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ over interval $[0,a]$. Note that $\mu_{10}'(x)=0$ but $\mu_1'(x;\nu)= \mu_{11}'(x)\nu+O(\nu^2)\neq 0$ as shown in the figure.](phi_0a.eps){width="2.5in"} ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ over interval $[0,a]$. Note that $\mu_{10}'(x)=0$ but $\mu_1'(x;\nu)= \mu_{11}'(x)\nu+O(\nu^2)\neq 0$ as shown in the figure.](mu1_0a.eps){width="2.5in"} The following is then a direct consequence. \[claim11\] Over the interval $(0,a)$, $c_{10}(x)=O(1)$ and $\mu_{10}'(x)=0$, and hence, one has $J_{10}=0$. Internal dynamics over the interval $(a,b)$ ------------------------------------------- It follows again from [@ZL17] that one has the following approximations. \[expProfile21\] For $x\in (a,b)$, $$\begin{aligned} \phi(x;\nu)=&-\ln \nu+\phi_0(x)+\phi_1(x)\nu +O(\nu^2)\;\mbox{ with }\\ \phi_0(x)=&\ln{R}-2\ln{B_0}+\ln{2},\\ \phi_1(x)=&\phi_1^a-A_0+\frac{J_{20}}{2}(H(x)-H(a));\\ c_{10}(x)=&0, \quad c_{11}(x)=\frac{1}{2}A_0^2-J_{11}(H(x)-H(a)),\\ \mu_{10}(x)=&\ln{R}-2\ln{B_0}+\ln{2}+\ln{\left(\frac{1}{2}A_0^2-J_{11}(H(x)-H(a))\right)},\\ \mu_{11}(x)=&\phi_1^a-A_0+\frac{J_{20}}{2}(H(x)-H(a)), \end{aligned}$$ where $$\begin{aligned} A_0=&\frac{\sqrt{e^{V}L}}{(1-\beta)\sqrt{e^{V}L}+\alpha\sqrt{R}}((1-\beta)L+\alpha R),\\ B_0=&\frac{\sqrt{R}}{(1-\beta)\sqrt{e^{V}L}+\alpha\sqrt{R}}((1-\beta)L+\alpha R),\\ A_1=&\frac{2(\beta-\alpha)^2(L-A_0)^2-\alpha^2(A_0^2-B_0^2)\ln{\frac{B_0L}{A_0R}}}{4(\beta-\alpha)((1-\beta)L+\alpha R)(L-A_0)}A_0B_0,\\ B_1=&-\frac{(1-\beta)\left(2(\beta-\alpha)^2(L-A_0)^2-\alpha^2(A_0^2-B_0^2)\ln{\frac{B_0L}{A_0R}}\right)}{4\alpha(\beta-\alpha)((1-\beta)L+\alpha R)(L-A_0)}A_0B_0,\\ \phi_1^a=&\frac{\ln{\frac{B_0}{R}}}{\ln{\frac{B_0L}{A_0R}}}\left(2\Big(\frac{B_1}{B_0}-\frac{A_1}{A_0}\Big)+\frac{\beta-\alpha}{\alpha}(L-A_0)\right)-2\frac{B_1}{B_0}-\frac{\beta}{\alpha}(L-A_0)+L. \end{aligned}$$ Figure 2 includes the profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ over interval $[a,b]$. One has the following immediate consequence. \[claim21\] Over the interval $(a,b)$, $c_{10}(x)=0$, and hence, $J_{10}=0$. Furthermore, $\mu_{10}(a)=V+\ln L$ and $\mu_{10}(b)=\ln R$. As $R\to 0$, $\mu_{10}(a)-\mu_{10}(b)\to \infty$. It follows from Proposition \[expProfile21\] directly. \[c2one\] ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, $\mu_1(x;\nu)$ over interval $[a,b]$. Note that $c_{10}(x)=0$ but $c_1(x;\nu)=c_{10}(x)+c_{11}(x)\nu+O(\nu^2)=c_{11}(x)\nu+O(\nu^2)\neq 0$ as shown in the figure. Also note the large drop of $\mu_1(x)$ over this interval due to that $R$ is taken to be small.](c1_ab.eps){width="2.5in"} ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, $\mu_1(x;\nu)$ over interval $[a,b]$. Note that $c_{10}(x)=0$ but $c_1(x;\nu)=c_{10}(x)+c_{11}(x)\nu+O(\nu^2)=c_{11}(x)\nu+O(\nu^2)\neq 0$ as shown in the figure. Also note the large drop of $\mu_1(x)$ over this interval due to that $R$ is taken to be small.](phi_ab.eps){width="2.5in"} ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, $\mu_1(x;\nu)$ over interval $[a,b]$. Note that $c_{10}(x)=0$ but $c_1(x;\nu)=c_{10}(x)+c_{11}(x)\nu+O(\nu^2)=c_{11}(x)\nu+O(\nu^2)\neq 0$ as shown in the figure. Also note the large drop of $\mu_1(x)$ over this interval due to that $R$ is taken to be small.](mu1_ab.eps){width="2.5in"} Internal dynamics over the interval $(b,1)$ ------------------------------------------- \[expProfile31\] For $x\in (b,1)$, $$\begin{aligned} \phi(x;\nu)=& \phi_0(x)+ \phi_1(x)\nu+O(\nu^2)\;\mbox{ with }\\ \phi_0(x)=&\ln R-\ln c_{10}(x),\\ \phi_1(x)=&\phi_1^b-\frac{1}{2}\left(B_0^2-\frac{2B_1-B_0^2}{B_0}\right)-\frac{B_0c_{11}(x)-B_1c_{10}(x)}{B_0c_{10}(x)}+\frac{2J_{11}}{J_{20}}\ln{\frac{c_{10}(x)}{B_0}};\\ c_{10}(x)=& B_0-\frac{J_{20}}{2}(H(x)-H(b)),\quad c_{11}(x)= B_1-\frac{J_{11}+J_{21}}{2}(H(x)-H(b));\\ \mu_{10}(x)=&\mu_{10}(1)=\ln{R}, \quad \mu_{11}(x)=\phi_1(x)+\frac{c_{11}(x)}{c_{10}(x)}, \end{aligned}$$ where $$\begin{aligned} \phi_1^{b}=&\frac{\ln{\frac{B_0}{R}}}{\ln{\frac{B_0L}{A_0R}}}\left(2\Big(\frac{B_1}{B_0}-\frac{A_1}{A_0}\Big)+\frac{\beta-\alpha}{\alpha}(L-A_0)\right)-2\frac{B_1}{B_0}+B_0. \end{aligned}$$ Figure 3 shows the profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ over interval $[b,1]$. \[c1three\] ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ on interval $[b,1]$. Note that $\mu_{10}'(x)=0$ but $\mu_1'(x;\nu)=\mu_{10}(x)+ \mu_{11}'(x)\nu+O(\nu^2)= \mu_{11}'(x)\nu+O(\nu^2)\neq 0$.](c1_b1.eps){width="2.5in"} ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ on interval $[b,1]$. Note that $\mu_{10}'(x)=0$ but $\mu_1'(x;\nu)=\mu_{10}(x)+ \mu_{11}'(x)\nu+O(\nu^2)= \mu_{11}'(x)\nu+O(\nu^2)\neq 0$.](phi_b1.eps){width="2.5in"} ![Profiles of $c_1(x;\nu)$, $\phi(x;\nu)$, and $\mu_1(x;\nu)$ on interval $[b,1]$. Note that $\mu_{10}'(x)=0$ but $\mu_1'(x;\nu)=\mu_{10}(x)+ \mu_{11}'(x)\nu+O(\nu^2)= \mu_{11}'(x)\nu+O(\nu^2)\neq 0$.](mu1_b1.eps){width="2.5in"} \[claim31\] Over the interval $(b,1)$, noting $B_0=O(\sqrt{R})$, $c_{10}(x)=O(1)$ and $\mu_1'(x)=0$, and hence, $J_{10}=0$. Summary of mechanism for $J_{10}=0$. ------------------------------------ From the above discussion, we conclude that the mechanisms for $J_{10}=0$ are different over each subintervals. More precisely, one has that, - over the first interval $(0,a)$, $J_{10}=0$ is the result of [constant]{} electrochemical potential $\mu_{10}(x)=\mu_{10}(0)=\mu_{10}(a)$ (so that $\mu_{10}'(x)=0$) while $c_{10}(x)\neq 0$; - over the last interval $(b,1)$, similar to that over the first interval $(0,a)$, $\mu_{10}(x)=\mu_{10}(1)$ so that $\mu_{10}'(x)=0$ while $c_{10}(x)\neq 0$, and, for $R$ small, $c_{10}(x)=O(\sqrt{R})$ over this interval; - over the middle interval $(a,b)$ where permanent charge is not zero and large, the electrochemical potential $\mu_{10}(x)$ is not a constant so that $\mu_{10}'(x)\neq 0$, however, $c_{10}(x)=0$, in particular, the drop of $\mu_{10}(x)$ over this interval equals the transmembrane electrochemical potentials, that is, $\mu_{10}(a)-\mu_{10}(b)=\mu_1(0)-\mu_1(1)$. Here we provide the profiles of concentration (Fig. 4), electrical potential (Fig. 5) and electrochemical potential (Fig. 6) over the whole interval $[0,1]$. The figures of $c_1(x)$ and $\phi(x)$ over interval $(0,x_0)$ are not continue, because we make the plots of system with $\varepsilon=0$. For the limiting system at $x=a$ and $x=b$, there are two fast layers, $c_1(x)$ and $\phi(x)$ changes very fast, but $\mu_1(x)$ keeps the same value in fast layers. Recall that $\mu_1^\delta=\mu_1(0)-\mu_1(1)=k_B T(V+\ln{L}-\ln{R})$, if $R=10^{-5}$, $\mu_1^\delta \sim 2.5902\times 10^{-23}$. \[c1\] ![Profile of $c_1(x;\nu)$ over $[0,1]$. Note that $c_{10}(x)=0$ for $x\in (a,b)$ and, for $x\in (b,1)$, $c_{10}(x)=O(\sqrt{R})$ is not zero but small for $R=10^{-5}$ chosen for the numerics.](c1_01.eps){width="3in"} \[phi\] ![Profile of $\phi(x;\nu)$ over $[0,1]$ ](phi_01.eps){width="3in"} \[mu1\] ![Profile of $\mu_1(x;\nu)$ over $[0,1]$. Note the large drop of $\mu_1$ over the interval $(a,b)=(1/3,1/2)$ and the nearly constant values over the other two subintervals.](mu1_01.eps){width="3in"} Declining phenomenon and internal dynamics {#Dyn4declin} ========================================== In this section, we will show that [large permanent charge is a mechanism for the declining phenomenon]{} described in the introduction. Recall that, by [the declining phenomenon]{}, we mean the following observed experimentally. [For fixed $V$ and $L_1=L_2=L$, as $R_1=R_2=R$ decreases to zero, the flux of counterion ($J_2$ in the setting since $Q_0>0$) decreases monotonically to zero.]{} This phenomenon is rather counterintuitive since the transmembrane electrochemical potential for the counter-ion $$\mu_2(0)-\mu_2(1)=z_2V+\ln L_2-\ln R_2\to +\infty\;\mbox{ as }\; R_2=R \to 0.$$ Experimental phenomena are consistent with our analysis ------------------------------------------------------- The result in (\[ejJ\]) actually justifies the observation, up to the leading order $J_{20}$ in $\nu$ for $\nu$ near $0$ (or for $Q_0\to +\infty$); that is, \[declining\] The leading order term $J_{20}$ of the flux is monotone and concave downward as a function of $R$ and, as $R\to 0$, $J_{20}\to 0$. Indeed, from the expression of $J_{20}$ in (\[ejJ\]) and treating $J_{20}$ as a function of $w$ where $R=w^2$ for convenience, one has $$J_{20}(w)=\frac{\sqrt{L}}{2H(1)}\frac{\sqrt{e^{-V}L}w-w^2}{(1-\beta)\sqrt{L}+\alpha\sqrt{e^{-V}}w}.$$ It is clear that $J_{20}(w)\to 0^+$ as $w\to 0^+$. Note that the derivative of $J_{20}$ in $w$ is $$J_{20}'(w)=\frac{1}{2H(1)}\frac{(1-\beta)L\sqrt{e^{-V}}-2(1-\beta)\sqrt{L}w-\alpha \sqrt{e^{-V}} w^2}{[(1-\beta)\sqrt{L}+\alpha\sqrt{e^{-V}}w]^2}.$$ Thus, from the expression of the numerator, if $w$ is smaller than some $w_0$, then $J_{20}'(w)>0$, and hence, as $w\to 0^+$ (or equivalently, $R_2=R\to 0$) monotonically over the interval $[0,w_0]$, $J_{20}(w)\to 0$ monotonically. It is not hard to show that for $R_2=R\ge 0$ but smaller than some positive value, the graph of $J_{20}$ as a function of $R$ is concave downward. In Figure 7, the horizontal axis is for $R$ and the vertical for $J_{20}$. We fix $V=0.425$ and $L=0.5$ as in (\[parameters\]), and vary $R\in (0,10^{-3}]$. The monotonicity and concave downward features of the graph are apparent. ![Declining curve: $J_{20}$ vs $R$ for $R\in (0, 10^{-3}]$ with $L=0.5$ and $V=0.425$.](j20_R.eps){width="3in"} \[largemu\] We comment that, for large permanent charge, the declining curve phenomenon occurs when the transmembrane electrochemical potential $\mu_2(0)-\mu_2(1)$ is increasing to infinity in a particular way; that is, as $R\to 0$. If one increases the transmembrane electrochemical potential $\mu_2(0)-\mu_2(1)$ in a different manner, for example, as $\|V\|\to \infty$ or as $L\to \infty$, Corollary \[Saturate\] shows that the declining curve phenomenon does not happen. It is also important to note that, when the next order term $J_{21}\nu$ is considered, then, as $R_2=R\to 0$, $J_{21}\to \infty$. But, if $R\to 0$ and $\nu\ln R\to 0$, then $J_{21}\nu\to0$. Thus, only when $\nu$ is very small ($Q_0$ is very large), is the term $J_{21}\nu$ not significant, and hence, the term $J_{20}$ dominates the described behavior. Mechanism of declining phenomena from the profiles -------------------------------------------------- Recall, from the Nernst-Planck equation in (\[PNP\]) that $$-J_2=\frac{1}{k_BT}D_2(x)h(x) c_2(x)\frac{d}{dx}\mu_2(x).$$ Since $D_2(x)$ and $h(x)$ are fixed, we will treat them as of order $O(1)$ quantities so that they do not contribute much to the near zero flux scenario as $R\to 0$. Thus, as far as the near zero flux mechanism is concerned, one has $$\begin{aligned} \label{approNP} -J_2\approx c_2(x)\frac{d}{dx}\mu_2(x).\end{aligned}$$ One sees that the gradient of the electrochemical potential $\frac{d}{dx}\mu_2(x)$ is the main driving force for the flux $J_2$. Intuitively, large drop of (or transmembrane) electrochemical potential $\mu_2(0)-\mu_2(1)$ of $\mu_2$ produces large flux $J_2$. In this sense, the declining curve phenomenon is rather counterintuitive. A careful look at (\[approNP\]) reveals that there is only one possibility for the declining curve phenomenon; that is, whenever $\frac{d}{dx}\mu_2(x)$ is large, $c_2(x)$ has to be much smaller in order to produce a small flux $\|J_2\|$. We will apply the analytical results of the internal dynamics from ([@ZL17]) to show that this is indeed the case. For our setup, there are three regions of permanent charge $Q(x)$: $Q(x)=0$ for $x\in [0,a)$ and $x\in (b,1]$ and $Q(x)=2Q_0$ for $x\in [a,b]$ with [large]{} $Q_0$ or [small]{} $\nu$ with ${\nu}=1/{Q_0}$. For fixed $V$ and $L$, $\mu_2(0)-\mu_2(1)=z_2V+\ln L-\ln R\approx -\ln R\gg 1$ for small $R$. We need to understand \(i) HOW the electrochemical potential $\mu_2$ drops an order $O(-\ln R)\gg 1$ over the interval $x\in [0,1]$; \(ii) HOW $J_2$ can be small, for small $\nu$ and $R$, at every $x\in [0.1]$; \(iii) Most importantly, HOW the above two things, with the constraint (\[approNP\]), can happen simultaneously. There are two small parameters $\nu$ and $R$ in our considerations. The relative sizes of these two parameters is relevant for the result. We will assume $\nu\ln R\ll 1$. We now discuss what happens over each subinterval based on the approximated (of zeroth order in $\varepsilon$) functions of profiles. To do so, let $$(\phi(x;\varepsilon,\nu),c_k(x;\varepsilon,\nu), J_k(\varepsilon,\nu))=(\phi(x;\nu),c_k(x;\nu) , J_k(\nu))+O(\varepsilon)$$ be the solution of the boundary value problem. For $\nu>0$ small, one has the following expansions $$\begin{aligned} J_2(\nu)=J_{20}+J_{21}\nu+O(\nu^2), \end{aligned}$$ where $J_{20}$ and $J_{21}$ are given in (\[ejJ\]). The expansions in $\nu$ for $\phi(x;\nu)$, $c_1(x;\nu)$, and $c_2(x;\nu)$ are not [regular]{} and are qualitatively different over the subintervals $(0,a)$, $(a,b)$ and $(b,1)$. They will be given explicitly in each subsection below for us to understand what happens over each subinterval. ### Internal dynamics over the interval $(0,a)$ The leading order terms of $(\phi,c_2)$ are derived in [@ZL17]. One has \[expProfile12\] For $x\in (0,a)$, $$\begin{aligned} \phi(x;\nu)=&\phi_0(x)+\phi_1(x)\nu+O(\nu^2) \mbox{ with }\\ \phi_0(x)=& V- \ln\Big(1- \frac{J_{20}}{2L}H(x)\Big),\quad \phi_1(x)=-\frac{c_{21}(x)}{c_{20}(x)}+\frac{2J_{11}}{J_{20}}\ln{\frac{c_{20}(x)}{L}};\\ c_{20}(x)=& L- \frac{J_{20}}{2}H(x),\quad c_{21}(x)= -\frac{J_{11}+J_{21}}{2}H(x);\\ \mu_{20}(x)=&-V+2\ln{c_{20}(x)}-\ln{L},\quad \mu_{21}(x)=2\left(\frac{c_{21}(x)}{c_{20}(x)}-\frac{J_{11}}{J_{20}}\ln{\frac{c_{20}(x)}{L}}\right). \end{aligned}$$ Figure 8 shows profiles of $c_2(x;\nu)$, $\phi(x;\nu)$ and $\mu_2(x;\nu)$ over the interval $(0,a)$. \[claim12\] Over the interval $(0,a)$, $c_2(x;\nu)=O(1)$ BUT $$\mu_2'(x;\nu)=-\frac{J_{20}}{h(x)c_{20}(x)}+O(\nu\ln R)=O(\sqrt{R},\nu\ln R).$$ Therefore, from (\[approNP\]), $$J_2\approx c_2(x) \frac{d}{dx}\mu_2(x;\nu)=O(1)O(\sqrt{R}, \nu\ln R)=O(\sqrt{R},\nu\ln R)$$ and $\mu_2(x)$ drops an order of $O(\sqrt{R},\nu\ln R)$ over the interval $(0,a)$. In particular, for zeroth order in $\nu$, $J_{2}\approx J_{20}=O(\sqrt{R})$ and $\mu_2'(x)=O(\sqrt{R})$ over the interval $(0,a)$. \[c2one\] ![Profiles of $c_k(x;\nu)$, $\phi(x;\nu)$, and $\mu_k(x;\nu)$ over interval $[0,a]$.](c2_0a.eps){width="2.5in"} ![Profiles of $c_k(x;\nu)$, $\phi(x;\nu)$, and $\mu_k(x;\nu)$ over interval $[0,a]$.](phi_0a.eps){width="2.5in"} ![Profiles of $c_k(x;\nu)$, $\phi(x;\nu)$, and $\mu_k(x;\nu)$ over interval $[0,a]$.](mu2_0a.eps){width="2.5in"} ### Internal dynamics over the interval $(a,b)$ It follows from [@ZL17] that \[expProfile22\] For $x\in (a,b)$, $$\begin{aligned} \phi(x;\nu)=&-\ln \nu+\phi_0(x)+\phi_1(x)\nu +O(\nu^2)\;\mbox{ with }\\ \phi_0(x)=&\ln{R}-2\ln{B_0}+\ln{2},\\ \phi_1(x)=&\phi_1^a-A_0+\frac{J_{20}}{2}(H(x)-H(a));\\ c_{2}(x;\nu)=&\frac{1}{\nu}+\left(\frac{1}{2}A_0^2-J_{11}(H(x)-H(a))\right)\nu+O(\nu^2);\\ \mu_{20}(x)=&-\ln{R}+2\ln{B_0}-\ln{2},\\ \mu_{21}(x)=&-\phi_1^a+A_0-\frac{J_{20}}{2}(H(x)-H(a)),\\ \mu_{22}(x)=&\frac{1}{2}A_0^2-J_{11}(H(x)-H(a)); \end{aligned}$$ where $$\begin{aligned} A_0=&\frac{\sqrt{e^{V}L}((1-\beta)L+\alpha R)}{(1-\beta)\sqrt{e^{V}L}+\alpha\sqrt{R}},\quad B_0=\frac{\sqrt{R}((1-\beta)L+\alpha R)}{(1-\beta)\sqrt{e^{V}L}+\alpha\sqrt{R}},\\ A_1=&\frac{2(\beta-\alpha)^2(L-A_0)^2-\alpha^2(A_0^2-B_0^2)\ln{\frac{B_0L}{A_0R}}}{4(\beta-\alpha)((1-\beta)L+\alpha R)(L-A_0)}A_0B_0,\\ B_1=&-\frac{(1-\beta)\left(2(\beta-\alpha)^2(L-A_0)^2-\alpha^2(A_0^2-B_0^2)\ln{\frac{B_0L}{A_0R}}\right)}{4\alpha(\beta-\alpha)((1-\beta)L+\alpha R)(L-A_0)}A_0B_0,\\ \phi_1^a=&\frac{\ln{\frac{B_0}{R}}}{\ln{\frac{B_0L}{A_0R}}}\left(2\Big(\frac{B_1}{B_0}-\frac{A_1}{A_0}\Big)+\frac{\beta-\alpha}{\alpha}(L-A_0)\right)-2\frac{B_1}{B_0}-\frac{\beta}{\alpha}(L-A_0)+L. \end{aligned}$$ The profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ are shown in Figure 9. ![Profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over interval $[a,b]$](c2_ab.eps){width="2.5in"} ![Profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over interval $[a,b]$](phi_ab.eps){width="2.5in"} ![Profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over interval $[a,b]$](mu2_ab.eps){width="2.5in"} Note that, over this interval, $c_2(x;\nu)$ is [singular]{} in $\nu$. To understand what happens to the internal dynamics for anions over the interval $(a,b)$, we need to resolve this singularity by considering at least $O(\nu)$-terms. \[claim22\] Over the interval $(a,b)$, $c_2(x;\nu)=O(1/ \nu)\gg 1$ and $\; \mu_{2}(x;\nu)=O(\ln R)\gg 1$ BUT $$\mu_2'(x;\nu)\approx \mu_{21}'(x)\nu=-\frac{J_{20}}{2h(x)}\nu =O(\nu\sqrt{R}).$$ Therefore, from (\[approNP\]), $$\begin{aligned} J_2\approx O(1/\nu)O(\nu\sqrt{R})=O(\sqrt{R})\;\mbox{ and }\; \mu_2(x;\nu) \;\mbox{ only drops }\; O(\nu\sqrt{R})\; \mbox{ over }\; (a,b). \end{aligned}$$ \[lastdrop\] Note that $\mu_2(x)$ drops much less over the interval $(a,b)$ than its drop over the interval $(0,a)$. But both drops are small and contribute nearly nothing to the total drop $\mu_2(0)-\mu_2(1)=O(-\ln R)\gg 1$. The only way to realize the large total drop $\mu_2(0)-\mu_2(1)$ is that $\mu_2(x)$ drops A LOT over the subinterval $(b,1)$. Indeed, this is the case as claimed below. ### Internal dynamics over the interval $(b,1)$ \[expProfile32\] For $x\in (b,1)$, $$\begin{aligned} \phi(x;\nu)=& \phi_0(x)+ \phi_1(x)\nu+O(\nu^2)\;\mbox{ with }\\ \phi_0(x)=&\ln R-\ln c_{20}(x),\\ \phi_1(x)=&\phi_1^b-\frac{1}{2}\left(B_0^2-\frac{2B_1-B_0^2}{B_0}\right)-\frac{B_0c_{21}(x)-B_1c_{20}(x)}{B_0c_{20}(x)}+\frac{2J_{11}}{J_{20}}\ln{\frac{c_{20}(x)}{B_0}};\\ c_{20}(x)=&B_0-\frac{J_{20}}{2}(H(x)-H(b)),\quad c_{21}(x)=B_1-\frac{J_{11}+J_{21}}{2}(H(x)-H(b));\\ \mu_{20}(x)=&2\ln{c_{20}(x)}-\ln R,\quad \mu_{21}(x)=\frac{c_{21}(x)}{c_{20}(x)}-\phi_1(x); \end{aligned}$$ where $$\begin{aligned} \phi_1^{b}=&\frac{\ln{\frac{B_0}{R}}}{\ln{\frac{B_0L}{A_0R}}}\left(2\Big(\frac{B_1}{B_0}-\frac{A_1}{A_0}\Big)+\frac{\beta-\alpha}{\alpha}(L-A_0)\right)-2\frac{B_1}{B_0}+B_0. \end{aligned}$$ The profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over $(b,1)$ are shown in Figure 10. \[c2three\] ![Profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over interval $[b,1]$](c2_b1.eps){width="2.5in"} ![Profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over interval $[b,1]$](phi_b1.eps){width="2.5in"} ![Profiles of $c_2(x;\nu)$, $\phi(x;\nu)$, and $\mu_2(x;\nu)$ over interval $[b,1]$](mu2_b1.eps){width="2.5in"} \[claim32\] Over the interval $(b,1)$, noting $B_0=O(\sqrt{R})$, $c_{20}(x)$ changes from $c_{20}(b)=B_0=O(\sqrt{R})$ to $c_{20}(1)=R$ monotonically, and $\mu_2(x)$ changes from $\mu_2(b)=O(1)$ to $\mu_2(1)=\ln R$. Therefore, for $x\in (b,1)$, from (\[approNP\]), $$\begin{aligned} J_2\approx O (\sqrt{R}) \;\mbox{ and }\; \mu_2(x) \;\mbox{ drops }\; O(-\ln R)\; \mbox{ over }\; (b,1). \end{aligned}$$ \[difflast\] Note that, over this interval, the order of $\mu'_2(x)$ varies in $x$ from $\mu'_2(b)=O(1)$ to $\mu'_2(1)=O(1/{\sqrt{R}})$ but overall drops is $O(-\ln R)$. This is different from what happened over the intervals $(0,a)$ and $(a,b)$. Summary of mechanism for declining phenomenon. ---------------------------------------------- In summary, with the technical assumption that $\nu\ln R\le \sqrt{R}$, we have, as $R\to 0$, $J_2=O(\sqrt{R})$ over the whole interval $(0,1)$ but with completely DIFFERENT scenarios over different subintervals $(0,a)$, $(a,b)$ and $(b, 1)$. More precisely, - over the subinterval $(0,a)$, one has $c_2(x)=O(1)$ but $\mu_2'(x)=O(\sqrt{R})\ll 1$ so that $J_2=O(\sqrt{R})$; ([Note that the drop of $\mu_2$ over the interval $(0,a)$ is of order $O(\sqrt{R})$, which has nearly no contribution to the drop of $\mu_2$ over the whole interval $(0,1)$.]{}) - over the subinterval $(a,b)$, $c_2(x)=O(1/{\nu})$ but $\mu_2'(x)=O(\nu\sqrt{R})\ll 1$ so that $J_2=O(\sqrt{R})$; ([Note that the drop of $\mu_2$ over the interval $(a,b)$ is of order $O(\nu\sqrt{R})$, which is even smaller than that over the subinterval $(0,a)$ and, of course, has nearly no contribution to the drop of $\mu_2$ over the whole interval $(0,1)$.]{}) - over the subinterval $(b,1)$, different from what happened over each of the previous two subintervals, the orders of $c_2(x)$ and $\mu_2'(x)$ are NOT uniform in $x\in (b,1)$ but the drop of $\mu_2(x)$ of $O(\ln R)$ is fully realized over this subinterval (see Remark \[difflast\]). Here we provide the profiles of concentration (Fig. 11), electric field (Fig. 12) and electrochemical potential (Fig. 13) of the anion over the whole interval $[0,1]$. The figures of $c_k(x)$ and $\phi(x)$ over interval $(0,x_0)$ are not continue, because we make the plots of system with $\varepsilon=0$. For the limiting system at $x=a$ and $x=b$, there are two fast layers, $c_k(x)$ and $\phi(x)$ changes very fast, but $\mu_k(x)$ keeps the same value in fast layers. Recall that $\mu_2^\delta=\mu_2(0)-\mu_2(1)=k_B T(-V+\ln{L}-\ln{R})$, if $R=10^{-5}$, then $\mu_2^\delta \sim 2.4780\times 10^{-31}$, that’s why in the last figure, $ \mu_2(x)$ can not reach the infinity. \[c2\] ![Profile $c_2(x;\nu)$ over $[0,1]$](c2_01.eps){width="3in"} \[phi\] ![Profile of $\phi(x;\nu)$ over $[0,1]$ ](phi_01.eps){width="3in"} \[mu2\] ![Profile of $\mu_2(x;\nu)$ over $[0,1]$](mu2_01.eps){width="3in"} Concluding remarks {#conclude} ================== In this work, we examine effects of large permanent charges on ionic flow through ion channels based on a quasi-one dimensional Poisson-Nernst-Planck model. We show that one of the defining properties of transporters, obligatory exchange, can arise in an open channel with just one structure. When the permanent charge is large, the current carried by counter ions, majority charge carriers with the opposite sign as the permanent charge, can decline, even to zero, as the driving force (the gradient of electrochemical potential) increases. We also show that large permanent charges essentially inhibit the flux of co-ions, regardless of the magnitude of transmembrane electrochemical potential. [Acknowledgements.]{} It is a pleasure to thank Mordy Blaustein, Don Hilgemann and Ernie Wright for help with the literature of transporters and Chris Miller for help with the original formulation of the declining phenomenon in Section \[Dyn4declin\]. LZ thanks the University of Kansas for its hospitality during her visit from Oct. 2016-Oct. 2017 when this research is conducted. LZ is partially supported by NNSF of China grants no. 11431008 and no. 11771282, and the Joint Ph.D. Training Program sponsored by the Chinese Scholarship Council. [999]{} J. N. Abelson, M. I. Simon, S. V. Ambudkar, and M. M. Gottesman, [ABC Transporters: Biochemical, Cellular, and Molecular Aspects.]{} Academic Press, 1998. P. F. Baker, M. P. Blaustein, A. L. Hodgkin, and R. A. Steinhardt, The influence of calcium on sodium efflux in squid axons. [J. Physiol.]{} [200]{} (1969), 431-458. M. P. Blaustein and W. J. Lederer, Sodium/Calcium Exchange: Its Physiological Implications. [Physiol. Rev.]{} [79]{} (1999), 763-854. P. Chlanda, E. Mekhedov, H. Waters, C. L. Schwartz, E. R. Fischer, R. J. Ryham, F. S. Cohen, P. S. Blank, and J. Zimmerberg, The hemifusion structure induced by influenza virus haemagglutinin is determined by physical properties of the target membranes. [Nat. Microbiol.]{} [1]{} (2016), 16050. B. Eisenberg and W. Liu, Relative dielectric constants and selectivity ratios in open ionic channels. [Mol. Based Math. Biol.]{} [5]{} (2017), 125-137. B. Eisenberg, Shouldn’t we make biochemistry an exact science? [ASBMB Today]{} [13]{} (2014), 36-38. B. Eisenberg, Ions in Fluctuating Channels: Transistors Alive. [Fluctuation and Noise Letters]{} [11]{} (2012), 1240001. B. Eisenberg, Crowded Charges in Ion Channels. [Advances in Chemical Physics]{} [148]{} (ed. Stuart A. Rice and Aaron R. Dinner), 77-223. John Wiley & Sons, Inc., 2012. R. S. Eisenberg, Computing the field in proteins and channels. [J. Memb. Biol.]{} [150]{} (1996), 1-25. R. S. Eisenberg. Atomic Biology, Electrostatics and Ionic Channels. [In New Developments and Theoretical Studies of Proteins (ed. R. Elber)]{}, 269-357. World Scientific: Philadelphia, 1996. R. S. Eisenberg. Channels as enzymes: Oxymoron and Taytology. [J. Memb. Biol.]{} [115]{} (1990), 1-12. B. Eisenberg and W. Liu, Poisson-Nernst-Planck systems for ion channels with permanent charges. [SIAM J. Math. Anal.]{} [38]{} (2007), 1932-1966. B. Eisenberg, W. Liu, and H. Xu, Reversal permanent charge and reversal potential: case studies via classical Poisson-Nernst-Planck models. [Nonlinearity]{} [28]{} (2015), 103-128. O. Frohlich and R. B. Gunn, Erythrocyte anion transport: the kinetics of a single-site obligatory exchange system. [Biochem. Biophys. Acta]{} [864]{} (1986), 169-194. D. Gillespie, A singular perturbation analysis of the Poisson-Nernst-Planck system: Applications to Ionic Channels. [Ph.D. Dissertation]{}, Rush University at Chicago, 1999. J. Griffiths and C. Sansom, [The Transporter Facts Book.]{} Academic Press, 1997. F. Helfferich, [Ion Exchange]{} (1995 Reprint). McGraw Hill reprinted by Dover, 1962. B. Hille, [Ion Channels of Excitable Membranes.]{} (3rd ed.) Sinauer Associates Inc., 2001. B. Hille, Transport Across Cell Membranes: Carrier Mechanisms, Ch. 2 in [Textbook of Physiology]{} [Vol. 1]{} (eds H.D. Patton et al.), 24-47. Saunders, 1989. A. L. Hodgkin, The ionic basis of electrical activity in nerve and muscle. [Biol. Rev.]{} [26]{} (1951), 339-409. S. Ji, B. Eisenberg, and W. Liu, Flux Ratios and Channel Structures. [J. Dyn. Diff. Equat.]{} (2017). https://doi.org/10.1007/s10884-017-9607-1. S. Ji and W. Liu, Poisson-Nernst-Planck systems for ion flow with Density Functional Theory for hard-sphere potential: I-V relations and critical potentials. Part I: Analysis. [J. Dyn. Diff. Equat.]{} [24]{} (2012), 955-983. S. Ji, W. Liu, and M. Zhang, Effects of (small) permanent charge and channel geometry on ionic flows via classical Poisson-Nernst-Planck models, [SIAM J. Appl. Math.]{} [75]{} (2015), 114-135. R. D. Keynes and R. C. Swan, The effect of external sodium concentration on the sodium fluxes in frog skeletal muscle. [J. Physiol.]{} [147]{} (1959), 591-625. G. Lin, W. Liu, Y. Yi, and M. Zhang, Poisson-Nernst-Planck systems for ion flow with a local hard-sphere potential for ion size effects. [SIAM J. Appl. Dyn. Syst.]{} [12]{} (2013), 1613-1648. W. Liu. Geometric singular perturbation approach to steady-state Poisson-Nernst-Planck systems. [SIAM J. Appl. Math.]{} [65]{} (2005), 754-766. W. Liu, One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species. [J. Differential Equations]{} [246]{} (2009), 428-451. W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels. [J. Dyn. Diff. Equat.]{} [22]{} (2010), 413-437. W. Liu, X. Tu, and M. Zhang, Poisson-Nernst-Planck systems for ion flow with Density Functional Theory for hard-sphere potential: I-V relations and critical potentials. Part II: Numerics. [J. Dyn. Diff. Equat.]{} [24]{} (2012), 985-1004. W. Liu and H. Xu, A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow. [J. Differential Equations]{} [258]{} (2015), 1192-1228. M. Lu, J. Symersky, M. Radchenko, A. Koide, Y. Guo, R. Nie, and S. Koide, Structures of a Na$^+$-coupled, substrate-bound MATE multidrug transporter. [Proceedings of the National Academy of Sciences]{} [110]{} (2013), 2099-2104. H. Miedema, M. Vrouenraets, J. Wierenga, W. Meijberg, G. Robillard, and B. Eisenberg, A Biological Porin Engineered into a Molecular, Nanofluidic Diode. [Nano Lett.]{} [7]{} (2007), 2886-2891. W. Nonner and R. S. Eisenberg, Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type Calcium channels. [Biophys. J.]{} [75]{} (1998), 1287-1305. R. F. Pierret, [Semiconductor Device Fundamentals.]{} Addison Wesley, 1996. B. Sakmann and E. Neher, [Single Channel Recording.]{} (2nd ed.), Plenum, 1995. W. D. Stein and T. Litman, [Channels, carriers, and pumps: an introduction to membrane transport.]{} Elsevier, 2014. R. B. Stockbridge, L. Kolmakova-Partensky, T. Shane, A. Koide, A. Koide, C. Miller, and S. Newstead, Crystal structures of a double-barrelled fluoride ion channel. [Nature]{} [525]{} (2015), 548-551. S. M. Sze, [Physics of Semiconductor Devices.]{} John Wiley & Sons, 1981. F. L. Theodoulou and I. D. Kerr, ABC transporter research: going strong 40 years on. [Biochemical Society Transactions]{} [43]{} (2015), 1033-1040. D. Tosteson, [Membrane Transport: People and Ideas.]{} American Physiological Society, 1989. H. H. Ussing, The distinction by means of tracers between active transport and diffusion. [Acta physiologica Scandinavica]{} [19]{} (1949), 43-56. H. H. Ussing, Interpretation of the exchange of radio-sodium in isolated muscle. [Nature]{} [160]{} (1947), 262-263. D. Vasileska, S. M. Goodnick, and G. Klimeck, [Computational Electronics: Semiclassical and Quantum Device Modeling and Simulation.]{} [Vol. 764]{}, CRC Press, 2010. W. Wang and R. MacKinnon, Cryo-EM Structure of the Open Human Ether-a-go-go-Related K+ Channel hERG. [Cell]{} [169]{} (2017), 422-430 e410. L. Zhang and W. Liu, Poisson-Nernst-Planck systems for ion channels with large permanent charges. [Preprint.]{} J. Zheng and M. C. Trudeau, [Handbook of ion channels.]{} CRC Press, 2015. [^1]: School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai 200240, P. R. China ([zhangliwei01@sjtu.edu.cn]{}). [^2]: Department of Molecular Biophysics and Physiology, Rush Medical Center, 1759 Harrison St., Chicago, Illinois 60612 ([beisenbe@rush.edu]{}). [^3]: Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd., Lawrence, Kansas 66045 ([wsliu@ku.edu]{}).
--- abstract: 'Plasmon resonance, with strong coupling of light to electrons at a metal-dielectric interface, allows light confinement and control at subwavelength scale. It’s fundamentally limited by the inherent mobility of the electrons, leading to the corresponding non-locality of the electromagnetic response.[@Ciraci2012; @Mortensen2015] We report that this non-locality also results in the formation of a hyperbolic layer near the metal-dielectric interface, with a strong anisotropy of its electromagnetic response. While the resulting “hyperbolic blockade" leads to the suppression of the conventional plasmon resonance, the hyperbolic layer also supports an entirely new class of surface waves, that offer longer propagation distance and stronger field confinement, simultaneously. Furthermore, these “hyper-plasmons" are not limited to the proximity of the plasmon resonance, which dramatically extends the operational bandwidth of plasmonic devices.' author: - 'Evgenii E. Narimanov' title: 'Hyper-Plasmonics: hyperbolic modes of a metal-dielectric interface.' --- With the ultimate goal of controlling light on a subwavelength scale, the field of nanophotonics generally relies on two main ideas – the plasmon resonance and the use of hyperbolic media. In the former approach, the subwavelength confinement of the electromagnetic field is achieved via the resonant coupling to free charge carriers in a conducting medium,[@Maier-book] while in the latter its the result of the extreme anisotropy of the material response that qualitatively changes the nature of the propagating fields.[@ENprbrc] These are generally considered as fundamentally distinct concepts, with their inherent advantages and drawbacks: e.g. plasmonic systems that rely on the properties of a single metal-dielectric interface are generally simpler to fabricate, but are generally limited to the proximity of the corresponding resonance frequency,[@Maier-book] while the approach based on hyperbolic media offers a broad bandwidth at the expense of highly nontrivial fabrication when the required anisotropy is due to the nanostructuring of the material.[@hyperbolic-3D] However, it is now well understood that the fundamental limits on the light confinement in both cases are defined by the inherent non-locality of the electromagnetic response in the constituent materials, due to e.g. the mobility of the free carriers in conducting materials,[@Ciraci2012; @Mortensen2015] In this work, we demonstrate that electromagnetic non-locality leads to an even deeper connection between these two seemingly different concepts of plasmon resonance and hyperbolic media: the inherent mobility of the free charge carriers in a plasmonic material leads to a strong dielectric anisotropy near the metal-dielectric interface, where the corresponding electromagnetic response becomes effectively hyperbolic. The resulting hyperbolic layer near the metal-dielectric interface supports a new type of surface waves that, compared to the conventional surface plasmons, offer both longer propagation distance and stronger field localization, [as the same time]{}. This behavior is not limited to the proximity of the plasmon resonance, but – in agreement with the generally broad bandwidth response in hyperbolic media [@ENprbrc] – persists well above the corresponding resonance frequency. Not only does this leads to a dramatic change in the resulting photonic density of states (by several order of magnetite) and consequently in all associated phenomena – from quantum electrodynamics to nonlinear optics to near-field thermal transport, but – by virtue of freeing plasmonics from the proximity of the corresponding plasmon resonance frequency – opens the field to a large class of materials that were never before considered in the context of plasmonics. ![The classical trajectory of an electron in a thin film, with the oscillatory external field parallel (a) and perpendicular (b) to the surface. Panel (c) shows the electric field (orange) and the resulting electron dipole moment, in the tangential (blue) and normal (red) directions. Note the opposite sign (relative phase $\pi$) of the electronic polarization in tangential direction and $\pi/2$ phase delay when the field is normal to the surface. The vector ${\bf v}_0$ in panels (a) and (b) represents the initial velocity of the electron. Note that higher harmonics visible in the red curve in (c), are removed by averaging over the actual distribution of the directions of the initial velocity.[]{data-label="fig:1"}](pic000){width="3.5in"} In the local approximation, the electromagnetic response of free carriers to a time-dependent electric field depends on the corresponding frequency and the carrier scattering time, and can be defined in terms of the momentum transfer between the field and the free carriers. However, in a close proximity to a high quality metal-dielectric interface that can be considered locally flat, the electron surface reflection will reverse normal to the surface component of the momentum, while leaving its tangential projection intact. As a result, while the specular reflection at the interface will not strongly affect the electromagnetic response in the tangential direction, its component that is normal to the metal surface, will be substantially altered – leading to a strong anisotropy in this interfacial layer. ![Gaussian beam incident on a metal-dielectric interface. Panel (a) shows the magnitude of the tangential component of the electric field. Panels (b)-(d) show the corresponding time-averaged energy density of the tangential electric field $w_x$ (panel (b)), the energy density of the normal to the interface electric field $w_z$ (panel (c)) and the product $w_x w_z$ (panel (d)). The vertical white line indicates the interface $z=0$. Note clearly visible dielectric region $z < 0$ ($w_x > 0$, $w_z > 0$), metallic region $z \gtrsim 0.01 c/\omega_p$, and the hyperbolic layer $0 < z \lesssim 0.01 c/\omega_p$. The frequency of the incident beam $\omega = 0.5 \omega_p$, the electron scattering time $\tau = 18.84 / \omega_p$, the crystal lattice permittivity of the conductor $\epsilon_\infty = 12.15$, the permittivity of the dielectric $\epsilon_d = 10.23$, and the Fermi velocity $v_F = 0.00935$; for the plasma wavelength $\lambda_p \equiv 2 \pi c/\omega_p = 10 \ \mu{\rm m}$ these parameters correspond to the ${\rm AlInAs}/{\rm InGaAs} $ material system of Ref. [@ref:nmat]. Note that in this case the electron de Broglie wavelength $\lambdabar \simeq 1 \ {\rm nm}$, well below the thickness of the hyperbolic layer $(\sim 20 \ {\rm nm}$).[]{data-label="fig:2"}](pic01.pdf){height="6.55in"} In the presence of surface roughness the free carrier reflection is no longer specular,[@Ziman] however the effect of the surface scattering on the momentum transfer from the free carriers to the interface (and thus the entire sample as a whole) is still very different in the normal and tangential directions. As a result, the free carrier electromagnetic response near the conductor - dielectric interface retains its strong anisotropy. This behavior is illustrated in Fig. \[fig:1\], where we consider the example of an electron that was originally moving parallel to the interface, under the parallel to the surface and perpendicular to the surface electric fields. For the field parallel to the surface, the response is similar to that in the bulk medium, and the resulting contribution to the effective dipole moment ${\bf p}\left( t\right)$ and the corresponding polarization of the medium, is opposite to the field - see Fig. \[fig:1\](a,c), just as in the bulk of the material. However, when the field is driving the electron towards the surface (see Fig. \[fig:1\](b)), the resulting reflection from the interface reverses the sign of the normal to the interface component of its velocity – and the momentum initially given to the electron by the field, at the reflection is transferred to the crystal as the whole. As a result, compared to the bulk of the material, the electron response in the normal-to-the-interface direction is strongly suppressed. As seen in Fig. \[fig:1\](c), the phase difference between the resulting dipole moment and the field, is now $\pi/2$, instead of the original value of $\pi$. The corresponding contribution to the permittivity is no longer negative, but imaginary, which represents the effective loss that accounts for the transfer of the momentum from the electron to the crystal as the whole at each reflection. Without the negative contribution of the free electrons, the real part of the permittivity in the normal to the interface direction is now effectively positive – and the thin layer near the surface behaves as if it had negative permittivity parallel to the interface and positive permittivity normal to the interface. A high-quality metal-dielectric surface therefore supports a [hyperbolic layer]{}. ![The dispersion of the surface waves at the metal - dielectric interface, with the in-plane momentum $k_\tau$ in units of $\omega_p / c$ and the frequency $\omega$ in units of $\omega_p$. Panel (a) corresponds to the standard result for the Drude metal, with the permittivity $\epsilon_m = \epsilon_\infty \left(1 - \omega_p^2 /\left( \omega \left(\omega + i/\tau\right) \right) \right)$, in logarithmic scale (main panel) and linear coordinates (the inset). Panels (b) - (c) show the results for the exact solution, with the ratio of the Fermi velocity to the speed of light in vacuum, $v_F/c = 0.05$ (b), $0.0063$ (c), and $0.00935$ (d). The material parameters ($\epsilon_\infty$, $\tau$, $\epsilon_d$) are the same as in Fig. \[fig:2\]. The red line corresponds to the conventional plasmon, blue – to the hyperbolic mode, green – to the hybrid hyper-plasmon, and magenta curve – to the suppressed resonant plasmon. With the plasma wavelength $\lambda_p = 10 \ \mu{\rm m}$, the doped semiconductor system ${\rm AlInAs}/{\rm InGaAs}$ corresponds to the panel (d). Note that in all cases, the wavenumber $k_\tau$ is below the Landau damping limit $\omega / v_F$.[@LL:physical_kinetics][]{data-label="fig:3"}](pic04m3.pdf){height="6.7in"} The formation of the hyperbolic layer relies on high quality of the interface that supports it. While the hyperbolic layer will adiabatically follow a smooth variation of the surface geometry, short-range surface roughness amplitude $h$ that exceeds the characteristic scale of $v_F/\omega$, where $v_F$ is the Fermi velocity of the electrons in the metal, will suppress it. At optical frequencies, this length scale can be on the order of a few nanometers or below, and the formation of the hyperbolic layers that’s predicted in the present work, is only expected in high-quality samples with sub-nanometer surface roughness. At lower frequencies however this surface quality requirement is proportionally relaxed – e.g. for mid-IR “designer metals” [@ref:nmat; @Wasserman2015] one needs $h \lesssim 10 \ {\rm nm}$. Note that the conventional hydrodynamical models recently used to account for the free carrier non-locality, generally rely on the material parameters (such as e.g. the phenomenological parameter $\beta$ in Refs. [@Ciraci2012; @Mortensen2015; @BoardmanBook; @Eguiluz1976; @PendryHydro] ) that are taken from the [bulk]{} electromagnetic response of the conduction electrons. This approximation does not allow to describe the inherent anisotropy of the electromagnetic response in the hyperbolic layer near the metal-dielectric interface. Since the electromagnetic response of free charge is essentially nonlocal, the definition of hyperbolic vs. dielectric vs. metallic response does not involve local tensor of the dielectric permittivity. Instead, we rely on the general expression of the electromagnetic energy density[@footnote0; @LLcm] in terms of the magnetic field ${\bf B}$, electric field ${\bf E}$ and the electric displacement vector ${\bf D}$: $$\begin{aligned} w & = & \frac{B^2 + {\bf E}\cdot{\bf D}}{8 \pi} \equiv w^B + w_{x,y}^E + w_z^E,\end{aligned}$$ where $w^B$ is the magnetic field energy density, $$\begin{aligned} w^B & = & \frac{B^2}{8 \pi},\end{aligned}$$ and $$\begin{aligned} w_{x,y}^E = \frac{E_x D_x + E_y D_y}{8 \pi}, \ \ \ w_z^E = \frac{E_z D_z}{8 \pi}\end{aligned}$$ correspond to the energy densities of the electric field components that are parallel and normal to the surface, respectively. Therefore, by definition, in a dielectric $w_{x,y}^E > 0$ and $w_{z}^E>0$, in a metal $w_{x,y}^E < 0$ and $w_{z}^E<0$, while in a hyperbolic medium $w_{x,y}^E$ and $w_{z}^E$ have opposite signs. For a local medium where ${\bf D} = \epsilon {\bf E}$, this reduces to the conventional definition of the dielectric, metallic and hyperbolic median in terms of $\epsilon_{x,y}$ and $\epsilon_z$. A direct calculation or a measurement of the local electromagnetic energy density will therefore immediately uncover the specific type of the response. ![image](pic045.pdf){width="6.5in"} In Fig. \[fig:2\] we consider a gaussian electromagnetic beam incident onto a half-infinite metal with an atomically flat boundary at $z=0$, and calculate the actual distribution of the electromagnetic energy density that takes full account of the non-locality of the electron response in the metal. Here, the numerical values for the plasma frequency, electron scattering time etc. correspond to the high-quality interface of doped semiconductor GaInAs with the dielectric AlInAs, the material platform which over the last decade became the system of choose for plasmonic systems in mid-IR range. [@ref:nmat; @Wasserman2015] While the magnitude of the electric field (see Fig. \[fig:2\](a)) displays the conventional intensity pattern of the reflected wave, the plots of the local energy density (Fig. \[fig:2\](b)-(d)) clearly show the presence of the hyperbolic layer at $0 < z \lesssim 0.01 c/\omega_p$, where $\omega_p$ is the plasma frequency, determined from the metal’s bulk response. Note that, the thickness of the hyperbolic layer in this example exceeds the electron de Broglie wavelength by more than an order of magnitude – so that the formation of the hyperbolic layer can be treated within the semiclassical framework. The actual response to the time-dependent electromagnetic field is defined by the electronic density matrix $\rho_{{\bf p} {\bf p}'}$, governed by the Liouville - von Neumann equation [@ref:density-matrix] that in the linear response regime reduces to [@UstinovOkulov1975; @UstinovOkulov1979] $$\begin{aligned} \frac{\varepsilon_{\bf p} - \varepsilon_{{\bf p}'} + \hbar \omega}{i\hbar} \ \rho_{{\bf p} {\bf p}'} + \frac{f^{(0)}_{\bf p} -f^{(0)}_{{\bf p}'} }{\varepsilon_{\bf p} - \varepsilon_{{\bf p}'} } V^E_{{\bf p} {\bf p}'} & = & I_{{\bf p} {\bf p}'}\left[ \rho \right], \label{eq:density_matrix}\end{aligned}$$ where $ I_{{\bf p} {\bf p}'}\left\{ \rho \right\}$ is the collision integral that includes the contributions from both the bulk and the surface scattering of the free carriers, $f^{(0)}_{\bf p} \equiv f_0(\varepsilon_{\bf p})$ is the equilibrium (Fermi-Dirac) distribution function, and $V_{{\bf p} {\bf p}'}$ is the matrix element of the spatially dependent amplitude of the electric field ${\bf E}\left({\bf r}, t\right) = {\bf E}\left({\bf r}\right) \exp\left(- i \omega t\right)$ that is given by $$\begin{aligned} V^E_{{\bf p} {\bf p}'} & = & \int d{\bf r} \ \ {\bf j}_{{\bf p} {\bf p}'}\cdot{\bf E}\left({\bf r}\right),\end{aligned}$$ where ${\bf j}_{{\bf p} {\bf p}'}$ is the matrix element of the charge carrier current density.[@ref:LL] When the relevant “classical” parameters such as the mean-free path $\ell \equiv v_F \tau$ and $v_F / \omega$ are well above the free carrier de Broglie wavelength $\lambdabar$, the Wigner transformation [@Wigner1932; @ref:LL] of the density matrix reduces [@KohnLuttinger; @UstinovOkulov1975; @UstinovOkulov1979] Eqn. (\[eq:density\_matrix\]) to the Boltzmann equation for the charge carrier distribution function $f_{\bf p}\left({\bf r}\right)$ $$\begin{aligned} -i\omega f_{\bf p}\left({\bf r}\right) + {\bf v}_{\bf p} \cdot \nabla f_{\bf p}\left({\bf r}\right) + e {\bf E}\cdot \frac{\partial f^{(0)}_{\bf p}}{\partial {\bf p}} & = & \hat{I}\left[ f_{\bf p} \right], \label{eq:Boltzmann-Wigner}\end{aligned}$$ where ${\bf v}_{\bf p} \equiv \partial\varepsilon_{\bf p}/\partial{\bf p}$ is the charge carrier group velocity for the Bloch momentum ${\bf p}$, the collision integral $\hat{I}\left[ f_{\bf p}\right]$ includes both the bulk and the surface scattering contributions, and has a highly nontrivial form. However, if the surface roughness $h$ is substantially smaller than $v_F/\omega$ and the electron mean free path $\ell = v_F \tau$,[@footnote1] the kinetic equation (\[eq:Boltzmann-Wigner\]) can be expressed in the conventional form[@KohnLuttinger; @UstinovOkulov1975; @UstinovOkulov1979] $$\begin{aligned} -i\omega f_{\bf p} + {\bf v}_{\bf p} \cdot \nabla f_{\bf p} + e {\bf E}\cdot {\bf v}_{\bf p} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} & = & - \frac{f_{\bf p} - f_0}{\tau} \label{eq:Boltzmann}\end{aligned}$$ where the effective relaxation time $\tau$ is defined by the bulk scattering, while the effect of the surface is described by the the boundary condition on the distribution function at the interface [@UstinovOkulov1975; @UstinovOkulov1979] – see [ Appendix A]{}. For a high-quality interface along one of the symmetry planes of the crystal, the latter reduces to the specular reflection boundary condition at the surface[@Ziman; @Andreev1971; @Soffer1967; @Fuchs1938; @Sondheimer1950; @Kaner1980; @ReuterSondheimer1948] $$\begin{aligned} f_{{\bf p}^-}\left({\bf r}_s\right) & = &f_{{\bf p}^+}\left({\bf r}_s\right), \label{eq:ss_specular}\end{aligned}$$ where ${\bf p}^+$ and ${\bf p}^-$ are connected by the specular reflection condition, with equal tangential to the surface components $p_\tau^+ = p_\tau^-$, and positive and negative group velocity components in the normal to the interface direction: $\left({\bf v}_{\bf p^+}\right)_{n_s} > 0$, $\left({\bf v}_{\bf p^-}\right)_{n_s} < 0$, respectively.[@footnoteReflection] Note that while the standard derivation [@BoardmanBook; @Eguiluz1976] of the hydrodynamic models [@BoardmanBook; @PendryHydro] for the electromagnetic response of free charge carriers usually follows the application of the Hamilton’s principle to the Hohenberg-Kohn ground state Hamiltonian,[@HohenbergKohn] the hydrodynamic model can also be derived as an approximation for the solution of the kinetic equation (\[eq:Boltzmann-Wigner\]) based on the method of moments.[@ChristensenThesisBook] Such an approximation however neglects the essential anisotropy of the free carrier surface scattering, and the resulting hydrodynamic approach is therefore unable to capture the formation of the hyperbolic surface layer, as well as its implications. ![The “figure-of-merit” ${\rm Re } k_\tau / {{\rm Im} k_\tau}$ of the surface wave vs. the compression factor $k_\tau / (\omega/c)$, at the ${\rm AlInAs}/{\rm InGaAs} $ interface. Red line corresponds to the conventional plasmon and is calculated using the Drude theory (see also Fig. \[fig:3\](a)), while the green curve corresponds to the exact solution for the hyper-plasmon (see Fig. \[fig:3\](e)). []{data-label="fig:5"}](pic06m.pdf){width="3.in"} The electromagnetic field at the interface of a dielectric with a conducting medium is defined by the self-consistent solution of the kinetic equation and the surface scattering boundary condition together with the Maxwell equations, where the electron charge and current densities are given by $$\begin{aligned} \rho\left({\bf r}\right) & = & 2 \int \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \cdot \left(f_{\bf p}\left({\bf r}\right) -f_0\left(\varepsilon_{\bf p} \right) \right) , \label{eqn:rho} \\ {\bf j}\left({\bf r}\right) & = & 2 \int \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \cdot e \ {\bf v}_{\bf p} f_{\bf p}\left({\bf r}\right), \label{eqn:current}\end{aligned}$$ For a high-quality planar surface, [@footnote2] the corresponding mathematical problem can be reduced to the system of two coupled linear integro-differential equations (see [Appendix B]{}) that allows an exact analytical solution. For the the electric field in the conducting medium we obtain $$\begin{aligned} {\bf E}_k\left(z > 0\right) & = & \int_{- \infty}^{\infty} \frac{dq}{2\pi} \ {\bf e}\left(k,q\right) \ \exp\left(i k x - i q z\right), \ \label{eq:e_Fourier}\end{aligned}$$ where $$\begin{aligned} {\bf e}\left(k,q\right) & = & \frac{2}{D\left(k,q\right)} \left( \left. \frac{\partial E_x}{\partial z}\right\|_{z = +0} - i k \left. E_z\right\|_{z = +0} \right) \nonumber \\ & \times & \left(\epsilon_{zz}\left(k,q\right) \frac{\omega^2}{c^2} - k^2, 0,\nu_{xz}\left(k,q\right)\right), \end{aligned}$$ and $$\begin{aligned} D\left(k,q\right) & = & \left(\epsilon_{xx}\left(q\right) \frac{\omega^2}{c^2} - q^2\right)\nonumber \\ & \times & \left(\epsilon_{zz}\left(q\right) \frac{\omega^2}{c^2} - k^2 \right) - \nu^2_{xz}\left(k,q\right), \end{aligned}$$ with $$\begin{aligned} \epsilon_{xx}\left(k,q\right) & = &\epsilon_\infty - \frac{16 \pi i e^2 \tau}{\omega} \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & v_x^2 \ \frac{ 1 - i \omega \tau + i k v_x \tau }{ \left( 1 - i \omega \tau + i k v_x \right)^2 +q^2 v_z^2 \tau^2 },\end{aligned}$$ and $$\begin{aligned} \epsilon_{zz}\left(k,q\right) & = &\epsilon_\infty - \frac{16 \pi i e^2 \tau}{\omega} \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & v_z^2 \ \frac{ 1 - i \omega \tau + i k v_x \tau}{ \left( 1 - i \omega \tau + i k v_x \tau \right)^2 +q^2 v_z^2 \tau^2 },\end{aligned}$$ and $$\begin{aligned} \nu_{xz}\left(k,q\right) & = &k q - \frac{16 \pi e^2 \tau^2 \omega q}{c^2} \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & v_x v_z^2 \ \frac{ 1 - i \omega \tau + i k v_x \tau}{ \left( 1 - i \omega \tau + i k v_x \tau \right)^2 +q^2 v_z^2 \tau^2 }.\end{aligned}$$ ![image](pic07m.pdf){width="6.5in"} For a degenerate electron gas [@footnote3] we reduce these expressions to $$\begin{aligned} \epsilon_{xx}\left(k,q\right) & = & \epsilon_\infty - \frac{3 \epsilon_\infty}{2} \frac{\omega_p^2}{\omega \left(\omega + i/\tau\right)} \left\{ \frac{q^2 - 2 k^2}{\left(k^2 + q^2\right)^2} \frac{\left(\omega + i/\tau\right)^2}{v_F^2} \right. \nonumber \\ & +& \left(\frac{q^2}{k^2 + q^2} + \frac{2 k^2 - q^2}{\left(k^2 + q^2\right)^2} \cdot \frac{\left(\omega + i/\tau\right)^2}{v_F^2}\right) \nonumber \\& \times & \left. {\cal F}_0\left(\frac{v_F \sqrt{k^2 + q^2}}{\omega + i/\tau}\right) \right\}, \nonumber \\ \epsilon_{zz}\left(k,q\right) & = & \epsilon_\infty - \frac{3 \epsilon_\infty}{2} \frac{\omega_p^2}{\omega \left(\omega + i/\tau\right)} \left\{ \frac{k^2 - 2 q^2}{\left(k^2 + q^2\right)^2} \frac{\left(\omega + i/\tau\right)^2}{v_F^2} \right. \nonumber \\ & +& \left(\frac{k^2}{k^2 + q^2} + \frac{2 q^2 - k^2}{\left(k^2 + q^2\right)^2} \cdot \frac{\left(\omega + i/\tau\right)^2}{v_F^2}\right) \nonumber \\& \times & \left. {\cal F}_0\left(\frac{v_F \sqrt{k^2 + q^2}}{\omega + i/\tau}\right) \right\}, \\ \nu_{xz}\left(k,q\right) & = & k q \left\{ 1 + \frac{9 \epsilon_\infty}{2} \frac{ \omega}{\omega + i/\tau} \frac{\omega_p^2}{\left(k^2 + q^2\right) c^2 } \right. \nonumber \\ & \times & \left. {\cal F}_{1}\left(\frac{v_F \sqrt{k^2 + q^2}}{\omega + i/\tau}\right) \right\},\end{aligned}$$ where $$\begin{aligned} {\cal F}_{0}\left(x\right) & = & \frac{1}{2 x} \log\frac{1+x}{1-x},\end{aligned}$$ and $$\begin{aligned} {\cal F}_{1}\left(x\right) & = & \frac{1}{x} \left\{ \frac{1}{x} + \frac{1}{2} \left( \frac{1}{3} - \frac{1}{x^2}\right) \log\frac{1+x}{1-x} \right\}.\end{aligned}$$ The results presented in Fig. \[fig:2\], were obtained using this solution (see Appendix C). For a surface state at the metal-dielectric interface, matching the tangential electric field and the normal component of the electric displacement at the interface yields (see [Appendix D]{}) $$\begin{aligned} \frac{1}{\pi} \int_0^\infty dq \ \frac{\epsilon_{zz}\left(k,q\right) \omega^2 / c^2 - k^2}{D\left(k,q\right)} & = & - \frac{c^2}{\omega^2} \frac{\kappa_d}{\epsilon_d}, \label{eq:sw1}\end{aligned}$$ where $\epsilon_d$ is the permittivity of the dielectric medium, and $$\begin{aligned} \kappa_d & = & \sqrt{k^2 - \epsilon_d \omega^2 / c^2}\end{aligned}$$ is the corresponding field decay rate. Eqn. (\[eq:sw1\]) generally has two distinct solutions. For a sufficiently small value of the ratio of the Fermi velocity to the speed of light in vacuum, these correspond to (i) the conventional surface plasmon, and (ii) the hyperbolic wave that is primarily supported by the hyperbolic layer [@footnoteMultiHyper] – see Fig. \[fig:3\](b). Note that the hyperbolic surface wave is only present above the cut-off frequency that is close to that of the standard surface plasmon resonance at the plant interface $\omega_{sp}$, when the bulk (Drude) metal permittivity $$\begin{aligned} \epsilon_m\left(\omega\right) & = & \epsilon_\infty \left( 1 - \frac{\omega_p^2}{\omega \left(\omega + i/\tau\right)}\right) \label{eq:epsilon_Drude}\end{aligned}$$ satisfies the resonance condition[@Maier-book] $$\begin{aligned} \epsilon_m\left(\omega_{sp} \right) = - \epsilon_d. \label{eq:sp} \end{aligned}$$ With the increase of the ratio $v_F/c$ (by e.g. increasing the doping density in a semiconductor) beyond its critical value $(v_F/c)_$, these two branches of the dispersion diagram undergo an avoided crossing (see [Appendix E]{}), so that the “conventional” surface plasmon continuously evolves into the hyperbolic mode (green curve in Fig. \[fig:4\](c),(d)), while the plasmon resonance, with its peak in the frequency dependence of the in-plane wavenumber (and the corresponding photonic density of states), is strongly suppressed (magenta curve in Fig. \[fig:3\](c),(d)). The physical origin of this suppression originates from the fact that plasmonic resonance relies on the resonant coupling between the electromagnetic field to the free charges in the immediate vicinity of the interface, The formation of the hyperbolic layer with strongly anisotropic electromagnetic response, no longer allows the resonance condition near the interface, and the conventional plasmon resonance is rapidly suppressed. One of the main challenges in nanoplasmonics is the inherent trade-off between the contradictory requirements of the surface plasmon propagation and field confinement.[@Maier-book] In a conventional surface plasmon, an improvement of the “compression factor” [@Marin] $k_\tau / k_0$ (that defines the field confinement) can be generally achieved only at the expense of the smaller propagation distance. This is illustrated by the red curve in Fig. \[fig:5\], which plots the “figure of merit" ${\rm Re } \left[k_\tau\right] / {\rm Im}\left[ k_\tau\right]$ that represents the propagation distance in units of the plasmon’s own wavelength, vs. the compression factor. However, the new “hyper-plasmon” surface wave that is supported by the hyperbolic layer (green curve in Fig. \[fig:5\]) greatly exceeds the these values, for both the propagation distance and the compression factor, [simultaneously]{}. Due to the inherent singularity in the density of states of a hyperbolic medium,[@EN_prl] the formation of the hyperbolic layer dramatically changes the photonic density of states near a high-quality metal-dielectric interface, with the resulting effect on all related phenomena – from radiative heat transfer to quantum-electrodynamic effects to Förster energy transfer to nonlinear optics. As an example of this behavior, in Fig. \[fig:5\] we plot the spontaneous emission rate near the metal-dielectric interface, as a function of frequency (Fig. \[fig:5\](a)) and the distance to the interface (Fig. \[fig:5\](b)). Note the dramatic suppression of the conventional plasmon resonance, and the enhancement of the emission rate above the plasmon resonance frequency. While the Drude theory predicts positive permittivity tensor above the plasma frequency, the inherent non-locality of the electronic response near the metal-dielectric interface dramatically modifies this simple picture. Above the plasma frequency the hyper-plasmon surface wave propagates with the in-plane wavenumber $k_\tau \gg \omega/c$, corresponding to the phase velocity $v_{\rm ph} \ll c$. For the electrons in the metal, the characteristic velocity $v \sim v_F$ can therefore be on the order of $v_{\rm ph}$, which results in the Doppler phase shift that is comparable to the actual frequency $\omega$. As a result, even with $\omega > \omega_p$, for an electron that is propagating in the direction close to that of the surface wave, the resulting Doppler-shifted frequency $$\begin{aligned} \omega' & = & \omega - {\bf k} \cdot {\bf v}\end{aligned}$$ can be well below $\omega_p$, thus increasing its negative contribution to the total permittivity $$\begin{aligned} \epsilon \simeq \epsilon_\infty \left(1 - \frac{\omega_p^2}{\left(\omega'\right)^2}\right).\end{aligned}$$ Therefore, even when in the stationary frame of reference the frequency $\omega$ is well above $\omega_p$, the apparent dielectric permittivity parallel to the surface that corresponds to electromagnetic waves with large wavenumbers, is still negative. As a result, for $k \gg \omega/c$ the hyperbolic layer is still present above $\omega_p$ – which explains the continued existence of the hyper-plasmon surface wave at higher frequencies. The [hyperbolic blockade]{} – the suppression of the plasmon resonance due to the formation of the hyperbolic layer at the metal surface, caused by the inherent non-locality of the free electron electromagnetic response – is the general feature of a high-quality metal-dielectric interface. However, a finite surface roughness leads to an effective averaging of the polarization anisotropy in the hyperbolic layer, and reduces the effect of the hyperbolic blockade. Quantitatively, this corresponds to the [short-range]{} roughness[@roughness] amplitude $h$ that exceeds the thickness of the hyperbolic layer, $\sim v_F/\omega$. Near the plasma frequency, the hyperbolic layer thickness $v_F/\omega_p$ is within a single order of magnitude from the Thomas-Fermi screening length, $v_F/\omega_p = \left( \sqrt{3 \epsilon_\infty}/\sqrt[3]{\pi} \right) \ R_{\rm TF}$. In good plasmonic metals such as silver or gold, the hyperbolic layer thickness can therefore be on the order of a fraction of a nanometer, and the effect of the hyperbolic blockade in all but the highest-quality samples will be negligible. The situation however is dramatically different in other conductors, such as e.g. transparent conducting oxides[@TCO] or doped semiconductors.[@ref:nmat; @Wasserman2015] E.g. in the latter, the thickness of the hyperbolic layer is in the range between $10 \ {\rm nm}$ and $100 \ {\rm nm}$, and exceeds both the typical roughness in high-quality MBE- or MOCVD-grown samples (generally on the order of a fraction of a nanometer) and the corresponding electron de Broglie wavelength $\lambdabar$ by almost two orders of magnitude – see the caption of Fig. \[fig:2\]. Experiments on doped semiconductor materials should therefore show clear manifestations of the hyperbolic blockade, predicted in the present work. The formation of the hyperbolic layers near the metal-dielectric interface both below [and]{} above the plasma frequency, also offers an entirely new approach for the search of new plasmonic materials. With the requirement for the operation in the proximity to the surface plasmon resonance frequency, the material options for nanoplasmonics remain fairly limited.[@TCO; @Sasha_review] Although plasmonic bandwidth can be improved by using the metamaterial approach,[@meta-book] where one can design and fabricate a metal-dielectric composite that extends the plasmonic behavior to a broader frequency range in a variety of form-factors, from planar metamaterials[@metamaterial-plasmons] to core-shell plasmonic particles,[@core-shells] this comes at the cost of an increased fabrication complexity,[@meta-book] with resulting “hit” in performance due to inevitable disorder at each interface.[@metal-losses] In contrast to this behavior, the hybrid “hyper-plasmons” introduced in the present work, offer high field compression factors that are not limited to the proximity to the resonance frequency $\omega_{\rm sp}$ – and exist well above its value (see Fig. \[fig:3\]). To put it in the context of an actual material platform, the high-quality doped semiconductors originally introduced as plasmonic materials for mid- and far-infrared frequencies,[@ref:nmat] support hyper-plasmons well into the near-IR range. In conclusion, we introduced the concept of the hyper-plasmonic surface wave, supported by hyperbolic layers near any high-quality metal-dielectric interface. We presented the theory of this effect based on first-principles approach that takes full account of the mobility of free charge carriers in plasmonic materials and the corresponding non-locality of the electromagnetic response. For a high-quality planar interface, we obtained the exact solution of the resulting system of coupled integro-differential equations. We demonstrated that hyper-plasmonic surface waves with simultaneously high compression factors and long propagation distance can be supported by an interface of a dielectric with conducting material, well above the corresponding plasma frequency – thus opening the field of plasmonics to many new materials, or extending the applications of existing materials in nanophotonics to shorter wavelength. Acknowledgments {#acknowledgments .unnumbered} =============== This work was partially supported by the National Science Foundation (grant 1629276-DMR), Army Research Office (grant W911NF-14-1-0639) and Gordon and Betty Moore Foundation. Boundary condition for the charge carriers distribution function. ================================================================= The effect of the surface can be describes by the the boundary condition on the distribution function at the interface, [@UstinovOkulov1975; @UstinovOkulov1979] which in the general case can be expressed as $$\begin{aligned} f_{{\bf p}^-}\left({\bf r}_s\right) & = & \int d{\bf p}^+ \ W\left({\bf p}^-,{\bf p}^+\right) \ f_{{\bf p}^+}\left({\bf r}_s\right), \label{eq:ss_indicatrix}\end{aligned}$$ where the coordinate ${\bf r}_s$ corresponds to the surface, the momenta ${\bf p}^+$ and ${\bf p}^-$ correspond to the electron momenta with respectively positive and negative group velocity components in the normal to the interface direction: $\left({\bf v}_{\bf p^+}\right)_{n_s} > 0$, $\left({\bf v}_{\bf p^-}\right)_{n_s} < 0$, and the surface scattering indicatrix $W\left({\bf p}^-,{\bf p}^+\right) $ can be calculated from first principles.[@UstinovOkulov1975; @UstinovOkulov1979] When the characteristic surface roughness is smaller then both the mean free path $\ell \equiv v_F \tau$ and $v_F/\omega$, Eqn. (\[eq:ss\_indicatrix\]) can be represented in terms of the specular reflection probability[@Fuchs1938; @Sondheimer1950] ${\cal P}$ as $$\begin{aligned} f_{{\bf p}^-}\left({\bf r}_s\right) & = &{\cal P} \ f_{{\bf p}^+}\left({\bf r}_s\right) + \left(1 + {\cal P} \right) \Phi_\varepsilon\left(\varepsilon_{{\bf p}^-} \right), \label{eq:ss_diffuse}\end{aligned}$$ with ${\cal P} = 1$ corresponding to the ideal interface (\[eq:ss\_specular\]) with specular reflection and ${\cal P} = 0$ for the opposite limit of diffuse (Lambertian) scattering of the free charge carriers. Here, the function $\Phi_\varepsilon$ is obtained from the conservation of the electron flux to and from the boundary. The specular reflection probability ${\cal P}$ may be treated as a phenomenological parameter, or alternatively calculated quantum-mechanically from the statistical properties of the surface roughness, [@Ziman; @UstinovOkulov1975; @UstinovOkulov1979] e.g. when the surface roughness correlation length is smaller than electron be Broglie wavelength $\lambdabar$ we find[@Ziman] $$\begin{aligned} {\cal P} & = & \exp\left( - \frac{16 \pi^2 h^2}{\lambdabar^2}\right). \label{eq:phL}\end{aligned}$$ For a high-quality interface along one of the symmetry planes of the crystal, Eqn. (\[eq:ss\_diffuse\]) reduces to the specular reflection boundary condition at the surface [@Ziman; @UstinovOkulov1975; @UstinovOkulov1979] $$\begin{aligned} f_{{\bf p}^-}\left({\bf r}_s\right) & = &f_{{\bf p}^+}\left({\bf r}_s\right), \label{eq:ss_specular_SM}\end{aligned}$$ where ${\bf p}^+$ and ${\bf p}^-$ are now connected by the specular reflection condition, with equal tangential to the surface components $p_\tau^+ = p_\tau^-$. Electromagnetic field and charge carrier distribution at the metal-dielectric interface. ======================================================================================== The electromagnetic field, and charge and carrier densities near the metal dielectric interface are defined by the self-consistent solution of the system of Maxwell’s equations, $$\begin{aligned} {\rm div}\ {\bf D} & = & 4 \pi \rho\left({\bf r}, t\right) \label{eq:divD} \\ {\rm div} \ {\bf B} & = & 0 \label{eq:divB} \\ {\rm curl} \ {\bf E} & = & - \frac{1}{c} \frac{\partial {\bf E}}{\partial t}\label{eq:curlE} \\ {\rm curl} \ {\bf B} & = & - \frac{4 \pi }{c} {\bf j}\left({\bf r}, t\right) + \frac{1}{c} \frac{\partial {\bf D}}{\partial t}, \label{eq:curlB} \end{aligned}$$ where the displacement field $$\begin{aligned} {\bf D} & = & \epsilon {\bf E} = \left\{ \begin{array}{lc} \epsilon_d \ {\bf E}, & z < 0 \\ \epsilon_\infty \ {\bf E}, & z > 0 \end{array} \right. ,\end{aligned}$$ $\epsilon_d$ is permittivity of the dielectric and $\epsilon_\infty$ is the “background” permittivity of the crystal lattice in the conductor, while the free charge density $ \rho\left({\bf r}, t\right)$ and the free current density $ {\bf j}\left({\bf r}, t\right)$ are defined by the charge carrier distribution function $f_{\bf p}\left({\bf r}, t\right)$ via $$\begin{aligned} \rho\left({\bf r},t\right) & = & 2 \int \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \cdot \left(f_{\bf p}\left({\bf r}\right) -f_0\left(\varepsilon_{\bf p} \right) \right) , \label{eq:rhoSM} \\ {\bf j}\left({\bf r}, t\right) & = & 2 \int \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \cdot e {\bf v}_{\bf p} f_{\bf p}\left({\bf r}, t\right). \label{eq:currentSM}\end{aligned}$$ In the liner response regime, the charge carrier distribution function $f_{\bf p}\left({\bf r}, t\right)$ satisfies the Boltzmann kinetic equation $$\begin{aligned} \frac{\partial f_{\bf p}}{\partial t} + {\bf v}_{\bf p} \cdot \nabla f_{\bf p} + e {\bf E}\cdot {\bf v}_{\bf p} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} & = & - \frac{f_{\bf p} - f_0}{\tau}, \label{eq:Boltzmann_SM}\end{aligned}$$ with the boundary condition at the metal-dielectric interface (see also Eqn. (\[eq:ss\_indicatrix\]) $$\begin{aligned} \left. f_{{\bf p}} \right\|_{z = 0, v_z < 0} & = & \int_{v'_z > 0} d{\bf p}' \ W\left({\bf p}, {\bf p}'\right) \ \left. f_{{\bf p}'} \right\|_{z = 0}. \label{eq:general_bc_SM}\end{aligned}$$ When the surface roughness is much smaller than the charge carrier de Broglie wavelength, $h \ll \lambdabar$, or if $h \simeq \lambdabar$ and surface roughness correlation length $L \gg \lambdabar$, Eqn. (\[eq:general\_bc\_SM\]) reduces to the specular reflection boundary condition (see also Eqns. (\[eq:ss\_specular\]) and (\[eq:ss\_specular\_SM\])) $$\begin{aligned} \left. f\left(v_x, v_y, v_z\right)\right\|_{ z = 0} & = &\left. f\left(v_x, v_y, - v_z\right)\right\|_{ z = 0} . \label{eq:specular_bc_SM}\end{aligned}$$ For a harmonic wave with the in-plane momentum $k$ in the $x$-direction, $$\begin{aligned} {\bf E}\left({\bf r}, t\right) & = & \left( E_x\left(z\right), 0, E_z\left(z\right)\right) \ \exp\left(i k x - i \omega t\right), \label{eq:E_SM} \\ {\bf B}\left({\bf r}, t\right) & = & \left( 0, B\left(z \right),0 \right) \ \exp\left(i k x - i \omega t\right), \label{eq:B_SM} \\ {f_{\bf p}}\left( {\bf r}, t\right) & = &f_0\left(\varepsilon\right) + {f}\left({\bf v}, z \right) \ \exp\left(i k x - i \omega t\right),\label{eq:f_SM}\end{aligned}$$ Note that in the harmonic representation (\[eq:E\_SM\]),(\[eq:B\_SM\]),(\[eq:f\_SM\]), Eqns. (\[eq:divD\]),(\[eq:divB\]) directly follow from (\[eq:curlE\]),(\[eq:curlB\]), and therefore do not represent independent constrains onto the electromagnetic field and the charge carrier distribution function.[@RamoBook] Applying ${\rm curl}$ to (\[eq:curlE\]), and using (\[eq:curlB\]), (\[eq:currentSM\]), (\[eq:E\_SM\]), (\[eq:f\_SM\]), for $z > 0$ we obtain $$\begin{aligned} - \frac{\partial^2 E_x}{\partial z^2} + i k \frac{\partial E_z}{\partial z} & = & \frac{4 \pi i \omega}{c^2} j_x + \epsilon_\infty \left( \frac{\omega}{c}\right)^2 E_x, \label{eq:E1_SM} \\ i k \frac{\partial E_x}{\partial z} + k^2 E_z & = & \frac{4 \pi i \omega}{c^2} j_z + \epsilon_\infty \left( \frac{\omega}{c}\right)^2 E_z, \label{eq:E2_SM}\end{aligned}$$ where $$\begin{aligned} j_{x,z} & = & 2 e \int \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3}\ v_{x,z} \ f\left({\bf v}, z\right). \label{eq:jxz_SM}\end{aligned}$$ Substituting (\[eq:f\_SM\]) into the kinetic equation (\[eq:Boltzmann\_SM\]) and the boundary condition (\[eq:specular\_bc\_SM\]), we obtain $$\begin{aligned} f\left({\bf v},z\right) & = & - e\ \frac{ \theta\left(v_z\right)}{v_z} \frac{\partial f_0}{\partial\varepsilon} \int_0^\infty d\zeta \ \left( v_x E_x\left(\zeta\right) + v_z E_x\left(\zeta\right) \right) \nonumber \\ & \times & \exp\left( - \frac{\zeta + z}{v_z}\left(\frac{1}{\tau} - i\omega + i k v_z\right) \right) \nonumber \\ & - & e\ \frac{ \theta\left(v_z\right)}{v_z} \frac{\partial f_0}{\partial\varepsilon} \int_0^z d\zeta \ \left( v_x E_x\left(\zeta\right) + v_z E_x\left(\zeta\right) \right) \nonumber \\ & \times & \exp\left[ \frac{\zeta - z}{v_z}\left(\frac{1}{\tau} - i\omega + i k v_z\right) \right] \nonumber \\ & + & e \ \frac{\theta\left(- v_z\right)}{v_z} \frac{\partial f_0}{\partial\varepsilon} \int_z^\infty d\zeta \ \left( v_x E_x\left(\zeta\right) + v_z E_x\left(\zeta\right) \right) \nonumber \\ & \times & \exp\left[ \frac{\zeta - z}{v_z}\left(\frac{1}{\tau} - i\omega + i k v_z\right) \right] .\label{eq:Boltzmann_solution_SM}\end{aligned}$$ Following the approach of Ref. [@ReuterSondheimer1948], originally developed in the context of the calculation of surface impedance of metals at microwave frequencies, we introduce the auxiliary fields $$\begin{aligned} {\cal E}_x\left(z\right) & = & E_x\left(\left\|z\right\|\right), \label{eq:newEx} \end{aligned}$$ and $$\begin{aligned} {\cal E}_z\left(z\right) & = & E_z\left(\left\|z\right\|\right) {\rm sign} \left(z\right), \label{eq:newEz} \end{aligned}$$ that represent respectively even- and odd “extension” of the electric field in the conductor ($z>0$) to the entire range $-\infty < z < \infty$. Substituting (\[eq:newEx\]) and (\[eq:newEz\]) together with (\[eq:jxz\_SM\]) and (\[eq:Boltzmann\_solution\_SM\]) into (\[eq:E1\_SM\]) and (\[eq:E2\_SM\]), we obtain $$\begin{aligned} \frac{\partial^2 {\cal E}_x}{\partial z^2} & + & \epsilon_\infty \left(\frac{\omega}{c}\right)^2 {\cal E}_x - i k \frac{\partial {\cal E}_z}{\partial z} \nonumber \\ = & - & \frac{4 \pi i \omega}{c^2} \int_{-\infty}^\infty d\zeta \ K_{xx}\left(z - \zeta\right) \ {\cal E}_x\left(\zeta\right) \nonumber \\ & - & \frac{4 \pi i \omega}{c^2} \int_{-\infty}^\infty d\zeta \ K_{xz}\left(z - \zeta\right) \ {\cal E}_z\left(\zeta\right), \label{eq:eqn1_SM} \end{aligned}$$ and $$\begin{aligned} - i k \frac{\partial {\cal E}_x}{\partial z} & + & \left(\epsilon_\infty \left(\frac{\omega}{c}\right)^2 - k^2 \right) \ {\cal E}_z \nonumber \\ = & - & \frac{4 \pi i \omega}{c^2} \int_{-\infty}^\infty d\zeta \ K_{zx}\left(z - \zeta\right) \ {\cal E}_x\left(\zeta\right) \nonumber \\ & - & \frac{4 \pi i \omega}{c^2} \int_{-\infty}^\infty d\zeta \ K_{zz}\left(z - \zeta\right) \ {\cal E}_z\left(\zeta\right), \label{eq:eqn2_SM}\end{aligned}$$ where $$\begin{aligned} K_{xx}\left( u \right) & = & 2 \int_{v_z > 0} \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \left(- \frac{\partial f_0}{\partial \varepsilon}\right) \frac{v_x^2}{v_z} \nonumber \\ & \times & \exp\left( - \left( 1 - i \omega \tau + i k v_x \tau \right) \frac{ \left\| u \right\| }{v_z \tau} \right), \label{eq:kxx} \\ K_{xz}\left( u \right) & = & K_{zx}\left( u \right) = 2 \ {\rm sign}\left(u\right) \int_{v_z > 0} \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \left(- \frac{\partial f_0}{\partial \varepsilon}\right) v_x \nonumber \\ & \times & \exp\left( - \left( 1 - i \omega \tau + i k v_x \tau \right) \frac{ \left\| u \right\| }{v_z \tau} \right), \label{eq:kxz} \\ K_{zz}\left( u \right) & = & 2 \int_{v_z > 0} \frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \left(- \frac{\partial f_0}{\partial \varepsilon}\right) {v_z} \nonumber \\ & \times & \exp\left( - \left( 1 - i \omega \tau + i k v_x \tau \right) \frac{ \left\| u \right\| }{v_z \tau} \right). \label{eq:kzz} \end{aligned}$$ Despite its relative complexity, the system of coupled linear integro-differential equations (\[eq:eqn1\_SM\]),(\[eq:eqn2\_SM\]) only has difference kernels, and by means of the Fourier transform $$\begin{aligned} e_x\left(k, q\right) & = & \int_{-\infty}^\infty dz \ {\cal E}_x \exp\left( i q z \right) \label{eq:ex_SM}, \\ e_z\left(k, q\right) & = & \int_{-\infty}^\infty dz \ {\cal E}_z \exp\left( i q z \right) \label{eq:ez_SM},\end{aligned}$$ can be reduced to a system of linear algebraic equations.[@Morse] We therefore obtain $$\begin{aligned} { e_x}\left(k,q\right) & = & \frac{2 \ A\left( k \right) }{D\left(k,q\right)} \left(\epsilon_{zz}\left(k,q\right) \frac{\omega^2}{c^2} - k^2\right), \label{eq:ex_kq_SM} \\ { e_z}\left(k,q\right) & = & \frac{2 \ A\left( k \right) }{D\left(k,q\right)} \ \nu_{xz}\left(k,q\right), \label{eq:ez_kq_SM} \end{aligned}$$ where $$\begin{aligned} A\left( k \right) & = & \left. \frac{\partial E_x}{\partial z}\right\|_{z = +0} - i k \left. E_z\right\|_{z = +0}, \label{eq:A_SM} \\ D\left(k,q\right) & = & \left(\epsilon_{xx}\left(q\right) \frac{\omega^2}{c^2} - q^2\right)\nonumber \\ & \times & \left(\epsilon_{zz}\left(q\right) \frac{\omega^2}{c^2} - k^2 \right) - \nu^2_{xz}\left(k,q\right), \label{Eq:D_SM}\end{aligned}$$ and $$\begin{aligned} \epsilon_{xx}\left(k, q\right) & = & \epsilon_\infty - \frac{16 \pi i e^2 }{\omega} \int_{0}^\infty du \cos\left(qu\right) \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}}\nonumber \\ & \times & \frac{v_x^2}{v_z} \ \exp\left( - \left( 1 - i \omega \tau + i k v_x \tau \right) \frac{ u}{v_z \tau} \right), \label{eq:eps_xx_SM} \\ \epsilon_{zz}\left(k, q\right) & = & \epsilon_\infty - \frac{16 \pi i e^2}{\omega} \int_{0}^\infty du \cos\left(qu\right) \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}}\nonumber \\ & \times & {v_z} \ \exp\left( - \left( 1 - i \omega \tau + i k v_x \tau \right) \frac{ u}{v_z \tau} \right), \label{eq:eps_zz_SM} \\ \nu_{xz}\left(k,q\right) & = &k q - \frac{16 \pi e^2}{\omega} \int_{0}^\infty du \ \sin\left(qu\right) \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & {v_x} \ \exp\left( - \left( 1 - i \omega \tau + i k v_x \tau \right) \frac{ u}{v_z \tau} \right). \label{eq:nu_xz_SM}\end{aligned}$$ For $z>0$, the auxiliary field ${\cal \bf E}$ is identical with the the actual electric field ${\bf E}$, and Eqns. (\[eq:ex\_SM\]) - (\[eq:nu\_xz\_SM\]) therefore offer the exact analytical solution for the electric field in the metal: $$\begin{aligned} {\bf E}\left(z>0\right) & = & \int_{-\infty}^\infty \frac{dq}{2 \pi} \ {\bf e}\left(k,q\right) \ \exp\left( - i q z\right). \label{eq:e_Fourier_SM}\end{aligned}$$ The amplitude $A(k)$ in Eqn. (\[eq:A\_SM\]) is defined by the values of the normal component of the electrical field $\left. E_z\right\|_{z = +0}$ and the normal derivative of the tangential electric field $ \left. {\partial E_x}/{\partial z}\right\|_{z = +0}$ at the boundary. These magnitudes depend of the electric field in the dielectric ($z < 0$), and are obtained from the continuity of the tangential components of the electrical field and the normal components of the displacement vector $$\begin{aligned} \left. E_x\right\|_{z = -0} & = & \left. E_x\right\|_{z= + 0}, \label{eq:E_bc_SM} \\ \epsilon_d \left. E_z\right\|_{z = -0} & = & \epsilon_\infty \left. E_z\right\|_{z= + 0}, \label{eq:D_bc_SM} \end{aligned}$$ where $\epsilon_d$ is the permittivity of the dielectric. Finally, the $u$-integration in Eqns. (\[eq:eps\_xx\_SM\]),(\[eq:eps\_zz\_SM\]),(\[eq:nu\_xz\_SM\]) can be performed analytically, which yields $$\begin{aligned} \epsilon_{xx}\left(k,q\right) & = &\epsilon_\infty - \frac{16 \pi i e^2 \tau}{\omega} \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & v_x^2 \ \frac{ 1 - i \omega \tau + i k v_x \tau }{ \left( 1 - i \omega \tau + i k v_x \right)^2 +q^2 v_z^2 \tau^2 }, \\ \epsilon_{zz}\left(k,q\right) & = &\epsilon_\infty - \frac{16 \pi i e^2 \tau}{\omega} \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & v_z^2 \ \frac{ 1 - i \omega \tau + i k v_x \tau}{ \left( 1 - i \omega \tau + i k v_x \tau \right)^2 +q^2 v_z^2 \tau^2 }, \\ \nu_{xz}\left(k,q\right) & = &k q - \frac{16 \pi e^2 \tau^2 \omega q}{c^2} \int_{v_z > 0}\frac{d{\bf p}}{\left(2 \pi \hbar\right)^3} \frac{\partial f_0}{\partial \varepsilon_{\bf p}} \nonumber \\ & \times & v_x v_z^2 \ \frac{ 1 - i \omega \tau + i k v_x \tau}{ \left( 1 - i \omega \tau + i k v_x \tau \right)^2 +q^2 v_z^2 \tau^2 }. \ \ \ \end{aligned}$$ The reflection amplitude at the planar metal-dielectric boundary. ================================================================= For a given in-plane momentum $k$, the electric electric field in the dielectric ($z<0$) with the permittivity $\epsilon_d$ can be expressed as $$\begin{aligned} {\bf E}\left({\bf r}, t\right) & = & E_+ \left(1, 0, - \frac{k }{\sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2}} \right) \nonumber \\ & \times & \exp\left(i k x + i \sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2} \ z - i \omega t\right) \nonumber \\ & + & E_- \left(1, 0, \frac{k }{\sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2}} \right) \nonumber \\ & \times & \exp\left(i k x - i \sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2} \ z - i \omega t\right), \ \ \ \ \ \ \ \label{eq:e_field_SM}\end{aligned}$$ leading to the corresponding magnetic field $$\begin{aligned} {\bf B}\left({\bf r}, t\right) & = & \frac{c}{i \omega} \ {\rm curl} \ {\bf E} \nonumber \\ & = & \hat{\bf y} \left[E_+ \exp\left(i \sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2} z\right) \right. \nonumber \\ & &\ \ \ \ - \left. E_+ \exp\left(i \sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2} z\right) \right] \nonumber \\ & \times & \frac{\epsilon_d\ \omega/c}{\sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2}} \exp\left( i k x - i \omega t \right).\end{aligned}$$ Therefore the electromagnetic wave impedance[@RamoBook; @Schelkunoff1938] in the $z = -0$ plane $$\begin{aligned} \left. Z \right\|_{z = - 0} & \equiv & \left. \frac{E_x}{B_y}\right\|_{z = +0} = \frac{ r+1}{r-1} \ \frac{\sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2}}{\epsilon_d\ \omega/c}, \ \ \ \label{eq:Z_minus_SM}\end{aligned}$$ where the reflection coefficient $$\begin{aligned} r \equiv \frac{E_+}{E_-}.\end{aligned}$$ On the other hand, from Eqns. (\[eq:ex\_kq\_SM\])-(\[eq:e\_Fourier\_SM\]) the tangential electric field at the metal side of the interface $$\begin{aligned} \left. E_x\right\|_{z = +0} & = & \left( \left. \frac{\partial E_x}{\partial z}\right\|_{z = +0} - i k \left. E_z\right\|_{z = +0} \right) \nonumber \\ & \times & \frac{1}{\pi} \int_{-\infty}^\infty dq \ \frac{\epsilon_{zz}\left(k, q\right) \left(\omega/c\right)^2 - k^2}{D\left(k, q\right)},\end{aligned}$$ while the magnetic field $$\begin{aligned} \left. {\bf B}\right\|_{z = +0} & = & \left. \frac{c}{i \omega} \ {\rm curl} \ {\bf E} \ \right\|_{z = +0} \nonumber \\ & = & \frac{c}{i \omega} \ \hat{\bf y} \left( \left. \frac{\partial E_x}{\partial z}\right\|_{z = +0} - i k \left. E_z\right\|_{z = +0} \right), \label{eq:B_plus_SM}\end{aligned}$$ so that the corresponding wave impedance in the $z = +0$ plane $$\begin{aligned} \left. Z \right\|_{z = + 0} & \equiv & \left. \frac{E_x}{B_y}\right\|_{z = +0} \nonumber \\ & = & \frac{ i \omega}{\pi c}\int_{-\infty}^\infty dq \ \frac{\epsilon_{zz}\left(k, q\right) \left(\omega/c\right)^2 - k^2}{D\left(k, q\right)}. \label{eq:Z_plus_SM}\end{aligned}$$ From Eqns. (\[eq:Z\_minus\_SM\]) and (\[eq:Z\_plus\_SM\]) for the reflection coefficient $r$ we therefore obtain $$\begin{aligned} r & = & - 1 + 2 \left\{ 1 + i\ \frac{\epsilon_d \left(\omega/c\right)^2 }{\sqrt{\epsilon_d \left(\omega/c\right)^2 - k^2}} \right. \nonumber \\ & \times & \left. \frac{1}{\pi} \int_{-\infty}^\infty dq \ \frac{\epsilon_{zz}\left(k, q\right) \left(\omega/c\right)^2 - k^2}{D\left(k, q\right)} \right\}^{-1}.\end{aligned}$$ ![image](pic05m2.pdf){width="6.5in"} Surface waves at the metal-dielectric interface. ================================================ For a surface wave at the metal-dielectric interface with the in-plane momentum $k > \sqrt{\epsilon_d} \omega/c$, the electric field in the dielectric half-space $z<0$ is given by $$\begin{aligned} {\bf E}\left({\bf r}, t\right) & = & E_0 \left(1, 0, - \frac{i k }{\sqrt{k^2 - \epsilon_d \left(\omega/c\right)^2}} \right) \nonumber \\ & \times & \exp\left(i k x + \sqrt{k^2 - \epsilon_d \left(\omega/c\right)^2 } \ z - i \omega t\right), \ \ \ \ \ \ \ \label{eq:e_field_sw_SM}\end{aligned}$$ while the corresponding magnetic field $$\begin{aligned} {\bf B}\left({\bf r}, t\right) & = &\hat{\bf y} \ E_0 \ \frac{i \epsilon_d\ \omega/c}{\sqrt{k^2 - \epsilon_d \left(\omega/c\right)^2 }} \nonumber \\ & \times & \exp\left(i k x + \sqrt{k^2 - \epsilon_d \left(\omega/c\right)^2 } \ z - i \omega t\right). \ \ \ \ \ \ \ \end{aligned}$$ The wave impedance at $z = -0$ is therefore given by $$\begin{aligned} \left. Z \right\|_{ z = - 0 } & \equiv & \left. \frac{E_x}{B_y}\right\|_{z = -0} = \frac{\sqrt{k^2 - \epsilon_d \left(\omega/c\right)^2 }}{i \ \epsilon_d\ \omega/c}. \label{eq:Z_sw_SM}\end{aligned}$$ From Eqns. (\[eq:Z\_plus\_SM\]) and (\[eq:Z\_sw\_SM\]) $$\begin{aligned} \frac{1}{\pi }\int_{-\infty}^\infty dq \ \frac{\epsilon_{zz}\left(k, q\right) \left(\omega/c\right)^2 - k^2}{D\left(k, q\right)} \nonumber \\ = - \frac{\epsilon_d \left( \omega/c \right)^2}{\sqrt{k^2 - \epsilon_d \left(\omega/c\right)^2 }},\end{aligned}$$ which defines the dispersion law of the surface wave $\omega\left(k\right)$. “Crossing” to “Avoided Crossing” crossover ========================================== The dispersion equation for the surface modes at the conductor-dielectric interface, Eqn. (\[eq:sw1\]) generally has two distinct solutions. For a sufficiently small value of the ratio of the Fermi velocity to the speed of light in vacuum, these correspond to the conventional surface plasmon (red curve in Fig. \[fig:S\] (a),(b) and (e),(f)), and the hyperbolic wave that is primarily supported by the hyperbolic layer (blue curve in see Fig. \[fig:S\] (a),(b) and (e),(f)). In this regime, there is a large difference in the lifetimes of the “plasmonic" and the “hyperbolic" surface waves, so the seemingly un-avoided crossing in the plot of the real parts of the wavenumber and the frequency in Fig. \[fig:S\] (a),(b) is a direct consequence of this behavior – in the full phase space (see Fig. \[fig:S\] (e),(f)) these two modes actually stay far apart from each other. With the increase of the ratio $v_F/c$ (by e.g. increasing the doping density in a semiconductor) however, the corresponding lifetimes approach each other, and at the critical value of $v_F/c$ the “plasmonic” and the “hyperbolic” modes finally approach degeneracy and undergo an avoided crossing. Fig. \[fig:S\] (c),(g) corresponds to the value of $v_F/c$ just above this critical point. From now on, with an increase of the frequency, the “conventional” surface plasmon continuously evolves into the hyperbolic mode – see the evolution of the magenta curve in Fig. \[fig:S\] (c),(d) and (g),(h). At the same time, the standard plasmonic resonance, which generally manifests itself by the peak in the frequency dependence of the in-plane wavenumber (and the corresponding photonic density of states), is strongly suppressed – see the behavior of the magenta curve in Fig. \[fig:S\](c),(d) and note the use of the logarithmic scale. [10]{} C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science [337]{} (6098), 1072 - 1074 (2012). G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature communications [6]{}, 7132 (2015). S. A. Maier, “Plasmonics: Fundamentals and Applications,” (Springer; 1st edition, 2007). V. A. Podolskuiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B [71]{}, 201101(R) (2005). S. S. Kruk, Z. J. Wong, E. Pshenay-Severin, K. O’Brien, D. N. Neshev, Yu. S. Kivshar and X. Zhang, “Magnetic hyperbolic optical metamaterials,” Nature Communications [7]{}, 11329 (2016). J. Ziman, “Electrons and Phonons: The Theory of Transport Phenomena in Solids ,” (Oxford University Press; reprint edition, 2001) A. J. Hoffman, L. V. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco and C. Gmachl, “Negative refraction in semiconductor metamaterials," Nature Materials [6]{}, 948 (2007). Y. Zhong, S. Malagari, T. Hamilton, and D. Wasserman, “Mid-Infrared Plasmonic Materials,” J. Nanophotonics [9]{}(1), 093791 (2015). A. D. Boardman, “Hydrodynamic theory of plasmon-polaritons on plane surfaces,” in [Electromagnetic Surface Modes]{}, (Wiley, 1982). A. Eguiluz, J. J. Quinn, “Hydrodynamics model for surface plasmons in metals and degenerate semiconductors,” Phys. Rev. B [14]{}, 1347 (1976). Y. Luo, A. I. Fernandez-Dominguez, A. Wiener, S. A. Maier, and J. B. Pendry, “Surface Plasmons and Nonlocality: A Simple Model,” Phys. Rev. Lett. [111]{}, 093901 (2013). As we’re primarily interested in optical frequencies, the metal itself can be considered as a non-magnetic medium.[@LLcm] However, the extension of our approach to the case of magnetic materials, with the energy density written in terms of the scalar product ${\bf B}\cdot {\bf H}$,[@LLcm] is straightforward. L. D. Landau, L. P. Pitaevskii, and E.M. Lifshitz, “Electrodynamics of Continuous Media,” (Butterworth-Heinemann; 2nd edition, 1984). H. Breuer, F. Petruccione, “The theory of open quantum systems,” (Oxford University Press; 1st edition, 2002). V.I. Okulov, V. V. Ustinov, “Boundary condition for the distribution function of conduction electrons scattered by a metal surface,” Sov. Phys. JETP [40]{} (3), 584 - 590 (1975). V. I. Okulov, and V. V. Ustinov, “Surface Scattering of Conductivity Electrons and Kinetic Phenomena in Metals,” Fizika Nizkikh Temperatur [5]{} (3), 213 - 251 (1979). E. Wigner, “On the Quantum Correction for Thermodynamic Equilibrium," Physical Review [40]{} (5), 749 (1932). L. D. Landau and L. M. Lifshitz, “Quantum Mechanics,” (Butterworth-Heinemann; 3rd edition, 1981). W. Kohn and J. M. Luttinger, “Quantum Theory of Electrical Transport Phenomena,” Phys. Rev. [108]{}, 590 - 611 (1957). Here there is no restriction on the relative value of the surface roughness $h$ as compared to the de Broglie wavelength $\lambdabar$. A. F. Andreev, “Interaction of Conducting Electrons with the Metal Surface,” Uspekhi Fiz. Nauk [105]{} (1), 113 - 124 (1971). S.B. Soffer, “Statistical Model for the Size Effect in Electrical Conduction,” J. Appl. Phys. [38]{} (4), 1710 - 1715, (1967). K. Fuchs, “The Conductivity of Thin Metallic Films According To the Electron Theory of Metals,” Proc Cambridge Phil. Soc. [34]{} (1), 100 - 108 (1938). E.H. Sondheimer, “The Mean Free Path of Electrons in Metals,” Advances in Physics [bf 50]{} (6), 499 - 537 (2001). E. A. Kaner, A. A. Krokhin, N. M. Makarov, and V. A. Yampolskii, “Surface absorption of electromagnetic waves in metals by random boundary inhomogeneities,” Sov. Phys. JETP [52]{} (5), 938 - 944 (1980). G. E. H. Reuter and E. H. Sondheimer, “The Theory of the Anomalous Skin Effect in Metals,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences [195]{} (1042) , 336 - 364 (1948). Note that for an anisotropic Fermi surface, ${\bf p}^-_n$ is not necessarily equal to $ - {\bf p}_n^+$. [@Ziman] P. Hohenberg, and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev. [136]{}, 864 (1964). T. Christensen, “From Classical to Quantum Plasmonics in Three and Two Dimensions,” (Springer, 2017). An exact solution can be also obtained for the opposite case of diffuse scattering. While qualitatively similar, the resulting expressions show a higher degree of algebraic complexity, and will be given elsewhere. The inequality $\varepsilon_F \gg k_B T$ is generally well satisfied for plasmonic materials even at room temperature: while $k_B T \simeq 20 \ {\rm meV}$, the Fermi energy is usually in the excess of an electron-volt. Note that, while a conventional hyperbolic waveguide supports multiple guided modes, [@ENprbrc] these correspond to progressively higher wavenumbers $k_\tau$. In a conducting medium however, these higher-order modes with $k \gg v_F / \omega$ are strongly suppressed due to Landau damping. [@LL:physical_kinetics] L. P. Pitaevskii, and E.M. Lifshitz, “Physical Kinetics,” (Butterworth-Heinemann; 1st edition, 1981). N. Rivera, I. Kaminer, B. Zhen, J. D. Joannopoulos, M. Soljacić, “Shrinking light to allow forbidden transitions on the atomic scale,” Science [353]{} (6296), 263 - 269 (2016). I. I. Smolyaninov, and E. E. Narimanov, “Metric Signature Transitions in Optical Metamaterials,” Phys. Rev. Lett. 105, 067402 (2010). A slowly-varying (on the scales of the electron de Broglie wavelength and the Thomas-Fermi screening length $R_{\rm TF}$) surface roughness will simply lead to the hyperbolic layer adiabatically following its landscape, and will not suppress the hyperbolic blockade. J. Kim, G. V. Naik, N. K. Emani, U. Guler, A. Boltasseva, “Plasmonic Resonances in Nanostructured Transparent Conducting Oxide Films,” IEEE Journal of Selected Topics in Quantum Electronics [19]{}, 4601907, (2013). G. V. Naik, V. M. Shalaev, A. Boltasseva, “Alternative Plasmonic Materials: Beyond Gold and Silver,” Advanced Materials [25]{} (24), 3264Ð3294 (2013). W. Cai and V. Shalaev, “Optical Metamaterials: Fundamentals and Applications,” (Springer, 1st edition, 2009). Z. Liu, M. D. Thoreson, A. V. Kildishev, and V. M. Shalaev, “Translation of nanoantenna hot spots by a metal-dielectric composite superlens,” Appl. Phys. Lett. [95]{}, 033114 (2009). S. J. Oldenburg, R.D. Averitt, S.L. Westcott and N.J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. [288]{}, 243 - 247 (1998). H. Reddy, U. Guler, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Temperature-dependent optical properties of gold thin films,” Optical Materials Express [6]{} (9), 2776 - 2802 (2016). S. Ramo, J. R. Whinnery, T. Van Duzer, “Fields and Waves in Communication Electronics,” (John Wiley & Sons.; 3rd edition, 1994). S. A. Schelkunoff, “The Impedance Concept and Its Application to Problems of Reflection, Refraction, Shielding and Power Absorption,” Bell Syst. Tech. J. [17*]{} (1), 17 - 48 (1938). P. M. Morse and H. Feshbach, “Methods of Theoretical Physics,” part II (Feshbach Publishing, 2004).
--- abstract: \| A [Reeb space]{} is defined as the space of all the connected components of inverse images of a smooth map and it is a fundamental tool in the studies of differentiable manifolds using generic smooth maps whose codimension is not positive such as Morse functions, their higher dimensional versions including [fold maps]{} and general [stable]{} maps. A [special generic]{} map is a fold map and regarded as a generalization of Morse functions with just $2$ singular points on homotopy spheres and the Reeb space is a compact manifold whose dimension is equal to that of the target manifold and which can be immersed into the target manifold. In this paper, we generalize a quotient map onto a Reeb space of a special generic map. We define a map onto a polyhedron locally a quotient map induced from a special generic map. In fact, there have been such generalizations. For example, in 1996, Kobayashi and Saeki generalized stable maps into the plane of closed manifolds of dimensionl larger than $3$ to study geometry of such maps systematically and the author recently defined such tools. We take advantage of the generalized maps to construct [lifts]{} of Morse functions of a certain class; the composition of the lift and the canonical projection is the original function. It is an answer of an explicit problem in the studies of lifts or desingularizations of smooth maps, or maps such that the compositions of the found maps and the canonical projections are original maps, which are fundamental and important in the studies of smooth maps and applications to algebraic and differential topology of manifolds. address: '19-9-606 Takayama, Tsuzuki-ku, Yokohama, Kanagawa 224-0065, JAPAN' author: - Naoki Kitazawa title: Generalizations of Reeb spaces of special generic maps and applications to a problem on lifts of Morse functions --- Introduction. {#sec:1} ============= Fundamental tools. ------------------ A [Reeb space]{} is defined as the quotient space of all the connected components of inverse images of a smooth map whose codimension is minus. It is a fundamental important tool in the studies of Morse functions, [fold maps]{}, which are higher dimensional versions of Morse functions and general generic maps including [stable]{} maps and applications to geometry of differentaible manifolds. For Reeb spaces, see [@reeb] for example. As a fundamental fact, the Reeb space of a Morse functions is a graph and generally, for maps mentioned here, the Reeb spaces are polyhedra respecting the canonical triangulations of manifolds and the maps. For such facts, see [@shiota] and see also [@hiratukasaeki] and [@hiratukasaeki2] for example. In addition, for Morse functions, fold maps and stable maps etc., see [@golubitskyguillemin] for example. We review [fold maps]{} and [special generic maps]{}. A [fold map]{} is a smooth map such that each singular point $p$ is of the form $$(x_1,\cdots,x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m-i(p)}{x_k}^2-\sum_{k=m-i(p)+1}^{m}{x_k}^2)$$ for some integers $m,n,i(p)$ satisfying $m \geq n \geq 1$ and $0 \leq i(p) \leq \frac{m-n+1}{2}$. Note that a Morse function is a fold map. The integer $i(p)$ is taken as a non-negative integer not larger than $\frac{m-n+1}{2}$ uniquely and we call $i(p)$ the [index]{} of $p$. The set of all the singular points of an index is a smooth submanifold of dimension $n-1$ and the restriction of the fold map to the singular set is an immersion and it is transversal if the fold map is [stable]{} (stable Morse functions exist densely on smooth closed manifolds and fold maps are generically stable). A [special generic map]{} is a fold map such that for all the singular points, the indices are $0$. Morse functions with just $2$ singular points on homotopy spheres and the canonical projections of unit spheres are simplest examples of special generic maps. We can know fundamental properties and important works on special generic maps in [@saeki], [@saekisakuma] and [@wrazidlo] for example. Contents of this paper. ----------------------- The Reeb space of a special generic map is regarded as an immersed compact manifold with a non-empty boundary whose dimension is same as that of the target manifold. In this paper, we generalize the quotient maps onto the Reeb spaces induced from special generic maps into Euclidean spaces as smooth maps onto compact smooth manifolds and give applications to a problem questioning that there exists a [lift]{} of a smooth map. More precisely, for a smooth map $f$ from an $m$-dimensional manifold into an $n$-dimensional manifold, does there exists an immersion, embedding or a map of an appropriate class $f_0$ or a [lift]{} into ${\mathbb{R}}^{n+k}$ such that for the canonical projection ${\pi}_{n+k,n}$, $f={\pi}_{n+k,n} \circ f_0$ holds? Note that such generalizations of maps have been done in several works. For exmaple, in 1996, Kobayashi and Saeki generalized stable maps into the plane of closed manifolds of dimensionl larger than $3$ to study geometry of such maps systematically and the author recently defined generalized maps of quotient maps induced from proper stable maps and more generally, proper triangulable smooth maps to show a theorem on the homology groups of Reeb spaces for more general spaces. As another example, a [shadow]{} [@turaev] is regarded as an extension of the quotient maps onto the Reeb spaces induced from stable fold maps on closed $3$-dimensional manifolds into the plane satisfying a condition on inverse images containing singular points and related studies are performed in [@costantinothurston] and [@ishikawakoda]. Last, studies of lifts of smooth maps are also fundametal and important in the studies of smooth maps and applications to algebraic and differential topology of differentaible manifolds. As a familiar example, plane curves can be lifted to classical knots in the $3$-dimensional space and there have been other various results on the problems. - Morse functions and stable maps of $2$ or $3$-dimensional manifolds into the plane are liftable to immersions or embeddings into appropriate dimensional spaces as shown in [@haefliger], [@levine], [@yamamoto] and [@yamamoto2] for example. - Generic maps between equi-dimensional manifolds are lifted to immersions or embeddings into one-dimensional higher spaces as done in [@saito] and see also [@blankcurley] for example. - Various special generic maps are lifted to immersions or embeddings as done in [@saekitakase] and [@nishioka] for example. - As a recent work [@kitazawa3], the author considered a problem questioning that a [normal spherical Morse function]{} can be lifted to a special generic map into an Euclidean space show several results. Here, we define a [normal spherical]{} Morse function, which is explained in [@kitazawa3], and see also [@saekisuzuoka] for example. \[def::1\] A stable fold map from a closed manifold of dimension $m$ into ${\mathbb{R}}^n$ satisfying $m \geq n$ is said to be [normal spherical]{} ([standard-spherical]{}) if the inverse image of each regular value is a disjoint union of (resp. standard) spheres (or points) and the connected component containing a singular point of the inverse image of a small interval intersecting with the singular value set at once in its interior is either of the following. 1. The ($m-n+1$)-dimensional standard closed disc. 2. A manifold PL homeomorphic to an ($m-n+1$)-dimensional compact manifold obtained by removing the interior of three disjoint ($m-n+1$)-dimensional smoothly embedded closed discs from the ($m-n+1$)-dimensional standard sphere. We review a fundamental property. \[prop:1\] For a normal spherical Morse function $f$ on a closed manifold $M$ of dimension $m>1$, the quotient map $q_f$ on $M$ to the Reeb space $W_f$ induces an isomorphism of homology, cohomology and homotopy groups whose degrees are smaller than $m-1$. In this paper, we study a problem related to the last one of the five kinds of results above and show an advanced result by make use of introduced generalized maps. The contents of the present paper is as the following. In the next section, we generalize the quotient maps to the Reeb spaces to smooth maps on closed manifolds onto compact smooth manifolds whose dimensions are lower than those of the source manifolds. In the last section, as a main work, we consider a spherical Morse function and represent this as a composition of two maps belonging to appropriate classes, which is regarded as a new answer for the problem studied in [@kitazawa3] and other related problems on lifts of smooth maps. Pseudo special generic maps. ============================ \[def:2\] Let $m>n$ be positive integers. A smooth map $f_p$ from a closed manifold $M$ of dimension $m$ onto a compact manifold $W_p$ of dimension $n$ satisfying $\partial W_p \neq \emptyset$ is said to be [pseudo special generic]{} if the following hold. 1. $f_p {\mid}_{{f_p}^{-1}(W_p-\partial W_p)}:{f_p}^{-1}(W_p-\partial W_p) \rightarrow W_p-\partial W_p$ gives a smooth $S^{m-n}$-bundle. 2. For a small collar neighborhood $N(\partial W_p)$ of $\partial W_p$ in $W_p$, regarded as a trivial bundle $\partial W_p \times [0,1]$ where $\partial W_p \times \{0\}$ corresponds to the boundary $\partial W_p$, for each point $(p,0) \in \partial W_p \times \{0\}$ and a small open neighborhood $U_p$, $f {\mid}_{{f_p}^{-1}(U_p \times [0,1])}:f^{-1}(U_p \times [0,1]) \rightarrow U_p \times [-1,1] $ has the same local form as a singular point of index $0$ of a fold map from an $m$-dimensional manifold into ${\mathbb{R}}^n$. From fundamental discussion of [@saeki] for example, ${f_p}^{-1}(N(\partial W_p))$ is a linear $D^{m-n+1}$-bundle over $\partial W_p$ given by the composition $f_p$ and the canonical projection and the bundle is seen as a normal bundle of the submanifold $f^{-1}(\partial W_p) \subset M$. All the quotient maps onto the Reeb spaces defined from special generic maps into Euclidean spaces are regarded as pseudo special generic. \[def:3\] A pseudo special generic map is said to be [trivial]{} if the bundles appearing in the two conditions of Definition \[def:2\] are trivial. 1. The Reeb space of a special generic map of a homotopy sphere into an Euclidean space is a contractible smooth manifold whose dimension is same as that of the target Euclidean space and the boundary is a homology sphere with coefficient ring $\mathbb{Z}$. The quotient map induced from a special generic map of a homotopy sphere into an Euclidean space of dimension smaller than $4$ is trivial. See [@saeki]. 2. Saeki, Takase [@saekitakase] and Nishioka [@nishioka] have shown that special generic maps satisfying appropriate differential topological conditions admit lifts which are immersions or embeddings and most of them are trivial. In addition, the author [@kitazawa] has given lifts of spherical Morse functions as special generic maps and seen that some of the resulting special generic maps are trivial and that some are not. Results. ======== \[thm:1\] 1. For a normal spherical Morse function $f:M \rightarrow \mathbb{R}$ on a closed manifold $M$ of dimension $m>2$ satisfying $m \neq 5$, there exist a compact smooth manifold $W_p$ of dimension $2$, a trivial pseudo special generic map $f_p:M \rightarrow W_p$ and a smooth map $g$ satisfying $f=g \circ f_p$. 2. Let $m>n$ be integers satisfying $m \neq 5$ and $n=3,4$ and let $m<6$ or $m \geq 6$ and the Gromoll filtration number of all the diffeomorphism on $D^m$ fixing all the boundary points is always larger than $n-1$ [(]{}for example $(m,n)=(13,3),(13,4)$; see [@crowleyschick] and also [@kitazawa][)]{}. For a normal standard-spherical Morse function $f:M \rightarrow \mathbb{R}$ on a closed manifold $M$ of dimension $m$, there exist a compact smooth manifold $W_p$ of dimension $n$, a trivial pseudo special generic map $f_p:M \rightarrow W_p$ and a smooth map $g$ satisfying $f=g \circ f_p$. For the proof, we need facts and technique precisely presented in [@cerf], [@saeki], [@wrazidlo] and [@kitazawa] and we review some of them. \[def:5\] Let $k$ be an integer larger than $6$. Let $l$ be a positive integer not larger than $k$ and let a diffeomorphism $\phi$ on the unit disc $D^{k-1} \subset {\mathbb{R}}^{k-1}$ fixing all the points in the boundary can be smoothly isotoped to ${\phi}_0$ so that ${{\pi}_{k-1,l-1}} \mid_{D^{k-1}} \circ {\phi}_0={\pi}_{k-1,l-1} {\mid}_{D^{k-1}}$. The [Gromoll filtration number]{} of the diffeomorphism $\phi$ is defined as the maximal number $l$. \[prop:2\] 1. \[prop:2.1\] Let $k \geq 6$ be an integer. We can obtain an orientation preserving diffeomorphism on $S^k$ by regarding $D^k$ as the hemisphere of the unit sphere of ${\mathbb{R}}^{k+1}$ by an injection $x \mapsto (x,\sqrt{1-{\parallel x \parallel}^2}) \in S^k$ and extending the diffeomorphism fixing all the points of the boundary on $D^k$ to the unit sphere $S^k$ by using the identity map. If the Gromoll filtration number is larger than $l$, then we can smoothly isotope the diffeomorphism to ${\phi}_0$ so that ${\pi}_{k+1,l} {\mid}_{S^k} \circ {\phi}_0 = {\pi}_{k+1,l} {\mid}_{S^k}$ holds. 2. \[prop:2.2\] For a positive integer $k$ smaller than $6$ and not $4$ and a positive integer $l \leq k$, then we can smoothly isotope an orientation diffeomorphism on $S^k$ to ${\phi}_0$ so that ${\pi}_{k+1,l} {\mid}_{S^k} \circ {\phi}_0 = {\pi}_{k+1,l} {\mid}_{S^k}$ holds. 3. \[prop:2.3\] Let $k \geq 6$ be an integer. Then the [Gromoll filtration number]{} of the diffeomorphism $\phi$ on the unit disc $D^{k-1} \subset {\mathbb{R}}^{k-1}$ fixing all the points in the boundary is larger than $1$. 4. \[prop:2.4\] Let $k$ be an integer larger than $6$, $\Sigma$ be a homotopy sphere of dimension $k$ and $f_{\Sigma}$ be a Morse function with just two singular points. then we can smoothly isotope any orientation preserving diffeomorphism on $\Sigma$ to ${\phi}_0$ so that $f_{\Sigma} \circ {\phi}_0 = f_{\Sigma}$ holds. We easily obtain the following, which is a fundamental proposition in the proof. \[prop:3\] 1. \[prop:3.1\] Let $k \geq 6$ be an integer and $l \leq k$. be a positive integer, then we have orientation reversing diffeomorphisms $r_{S^k}$ and $r_l$ on $S^k$ and ${\mathbb{R}}^l$, respectively so that ${\pi}_{k+1,l} {\mid}_{S^k} \circ r_{S^k} =r_l \circ {\pi}_{k+1,l} {\mid}_{S^k}$ holds. 2. \[prop:3.2\] For a positive integer $k$ smaller than $6$ and not $4$ and a positive integer $l \leq k$, then we have orientation reversing diffeomorphisms $r_{S^k}$ and $r_l$ on $S^k$ and ${\mathbb{R}}^l$, respectively so that ${\pi}_{k+1,l} {\mid}_{S^k} \circ r_{S^k} =r_l \circ {\pi}_{k+1,l} {\mid}_{S^k}$ holds. 3. \[prop:3.3\] Let $k$ be an integer larger than $6$ and $\Sigma$ be a homotopy sphere of dimension $k$ on which there exists an orientation reversing diffeomorphism and $f_{\Sigma}$ be a Morse function with just two singular points. then we have orientation reversing diffeomorphisms $r_{\Sigma}$ and $r$ on $S^k$ and ${\mathbb{R}}$, respectively and $f_{\Sigma} \circ r_{\Sigma} =r \circ f_{\Sigma}$ holds. We are enough to consider two appropriate antipodal poles and send each pole to another pole in the case where the source manifold is a unit sphere or each singular point to another singular point in the case where the source manifold is a general homotopy sphere with a Morse function with just two singular points. This gives us a desired orientation reversing diffeomorphism on the source homotopy sphere. A desired orientation reversing diffeomorphism on the target space is given by a reflection by a hyperplane containing the origin. See also FIGURE \[fig:1\]. ![Orientation reversing diffeomorphisms of a homotopy sphere and the image by a special generic map (the case of a canonical projection of a unit sphere; the arrows represent each orientation reversing diffeomorphism).[]{data-label="fig:1"}](sphereproj.eps){width="35mm"} We prove the first part. First, as done in [@kitazawa] for example, we locally lift Morse functions to smooth maps whose singular points are of the same form as that of a singular point of index $0$ of a fold map. First, for the connected component of the inverse image of a closed interval containing just one singular point, we lift as drawn in FIGURE \[fig:2\]. ![Local lifts of normal spherical Morse functions.[]{data-label="fig:2"}](locallift.eps){width="35mm"} For a connected component containing a singular point of index $0$, we lift as the left figure. The image is a $2$-dimensional closed disc with $2$ points in its corner. This is also regarded as a cobordism of special generic functions such that one of the function is null and the other is a function on a connected manifold or as a result a special generic function on a standard sphere. For a connected component containing a singular point of index $1$, we lift as the right figure. The image is a $2$-dimensional closed disc with $6$ points in its corner. This is also regarded as a cobordism of two special generic functions such that one of the function is a function on a connected manifold and that the other is a function on a manifold having two connected components. For cobordisms of special generic functions, see [@saeki3] for example. For the connected component of the inverse image of a closed interval containing no singular point, we lift the function as the product of a Morse function with just two singular points on the homotopy sphere appeared as the corresponding inverse image and the closed interval. Last, to glue together on each connected component of each point of the boudary of each closed interval in the target manifold, we apply Proposition \[prop:2\] and Proposition \[prop:3\]. By applying only Proposition \[prop:2\], we obtain a special generic map and this is also done in [@kitazawa]. By applying not only Proposition \[prop:2\], but also Proposition \[prop:3\], we can not always obtain a pseudo special generic map which can be realized as the quotient map of a special generic map; the Reeb space may be non-orientable. However, by the construction or the gluing, we can obtain the bundle of the first condition of Definition \[def:2\] as an orientable bundle and as a fundamental discussion on the topological properties of linear bundles, the bundle is trivial. We can construct the second bundle as a trivial bundle. Note that desired smooth map $g$ from the target space of the pseudo special generic map into $\mathbb{R}$ is also obtained as a global map; in the locall constructions, the map is of course constructed as a smooth map. The second part can be shown in a similar method. As a most important different point, we locally lift functions into a map of higher dimensional maps. For a connected component containing a singular point of index $0$, we lift so that the image is a $n$-dimensional closed disc whose corner is $S^{n-1}$. This is also regarded as a cobordism of special generic maps whose images are closed discs such that one of the map is null and the other is a map on a connected manifold or as a result regarded as the canonical projection of a unit sphere. For a connected component containing a singular point of index $1$, we lift so that image is a $n$-dimensional closed disc whose corner is three disjoint ($n-1$)-dimensional stadard spheres. This is also regarded as a cobordism of two special generic maps such that one of the map is regarded as the canonical projection of a unit sphere and that the other is a disjoint union of two canonical projections of unit spheres. For cobordisms of special generic functions and maps, see [@saeki3] and also [@sadykov] for example. For the connected component of the inverse image of a closed interval containing no singular point, we lift a function naturally as the higher dimensional version of the case above. We can discuss the last part similarly. Note that the 2nd homology and cohomology groups of the source manifold whose coefficient ring is $\mathbb{Z}$ vanish from Proposition \[prop:1\]. We can obtain the bundle of the first condition of Definition \[def:2\] as an orientable bundle over a $2$- or $3$-dimensional closed manifold and as a fundamental discussion on the topological properties of linear bundles, the bundle is trivial (since the second and third Stiefel Whitney classes vanish). We can construct the second bundle as a trivial bundle. Note that desired smooth map $g$ from the target space of the pseudo special generic map into $\mathbb{R}$ is also obtained as a global map. ![An example of lifts of normal spherical Morse functions.[]{data-label="fig:3"}](simplestlift.eps){width="35mm"} ![Local topology of the surfaces near the black intervals of FIGURE \[fig:3\]; the existence of the right case may make the target manifold of the pseudo special generic map non-orientable but in this case the bundles in the conditions of Definitions \[def:2\] can be orientable.[]{data-label="fig:4"}](localsurface.eps){width="35mm"} For examples, see FIGURE \[fig:3\] and FIGURE \[fig:4\] ($n=2$ case). Compare Theorem \[thm:1\] and Theorem 5 of [@kitazawa]; we cannot construct a lift as a special generic map so that the underlying pseudo special generic map is trivial when the source manifold is non-orientable and the dimension of the Euclidean space of the target is larger $2$. Last, we have the following. \[cor:1\] A spherical Morse function on a closed manifold of dimension larger than $2$ is represented as the composition of a smooth map into ${\mathbb{R}}^3$ regarded as the composition of a trivial pseudo special generic map and an appropriate smooth embedding of the target space of the map and the canonical projection ${\pi}_{3,1}$ defined by $(x_1,x_2,x_3) \in {\mathbb{R}}^3 \mapsto x_1 \in \mathbb{R}$. If the resulting pseudo special generic map is regarded as a map induced from a special generic map into the plane, then we can represent the Morse function as the composition of a special generic map into the plane obtained as a lift of the function and the canonical projection onto $\mathbb{R}$. We must sometimes push some local parts forward to realize the target space of the resulting pseudo special generic map in an appropriate Euclidean space or ${\mathbb{R}}^2 \times \mathbb{R}$. If in gluing local maps, we must apply the right case of FIGURE \[fig:4\] (\[fig:3\]), then we need to push this part forward to realize the target space of the resulting pseudo special generic map in an appropriate Euclidean space or ${\mathbb{R}}^2 \times \mathbb{R}$. From these discussions, we obtain the statement. ![The target manifold $W_p$ of a pseudo special generic map. $P^{\prime}$ and $P^{\prime \prime}$ are regions located in the left (right) side of the two thiin lines in the left (resp. right) and $P$ is the region of the center). Local surfaces in the black squares are like ones in FIGURE \[fig:4\].[]{data-label="fig:5"}](pseudospnew.eps){width="55mm"} \[thm:2\] A spherical Morse function $f$ on a closed manifold $M$ of dimension $m>2$ is represented as the composition of an embedding lift to ${\mathbb{R}}^n$ where $n \geq \max \{\frac{3m+3}{2},m+2+1\}+1$ holds and the canonical projection ${\pi}_{n,1}$ defined by $(x_1,\cdots,x_n) \in {\mathbb{R}}^n \mapsto x_1 \in \mathbb{R}$ and we can take the embedding lift so that the composition of the embedding and the canonical projection ${\pi}_{n,3}$ defined by $(x_1,\cdots,x_n) \in {\mathbb{R}}^n \mapsto (x_1,x_2,x_3) \in {\mathbb{R}}^3$ is a map as mentioned in Corollary \[cor:1\], represented as the composition of a trivial pseudo special generic map into the $2$-dimensional compact manifold and an embedding into ${\mathbb{R}}^3$. In the proof, we apply technique of constructions of embedding lifts of special generic maps such that the normal bundles of the singular sets are trivial used in [@nishioka] and for precise discussions, see also this.\ We represent the Morse function $f$ as a composition as Corollary \[cor:1\]; let $f_p$ be the pseudo special generic map into the $2$-dimensional compact manifold $W_p \subset {\mathbb{R}}^3$. We decompose this as presented in FIGURE \[fig:5\]. Let $C_p$ be a small collar neighborhood of $\partial W_p$. $C_p$ is, in the figure, the disjoint union of regions each of which is surrounded by a circle representing a connected component of the boundary $\partial W_p$ and a dotted circle. For the map ${f_p} {\mid}_{{f_p}^{-1}(C_p)}:{f_p}^{-1}(C_p) \rightarrow C_p \subset {\mathbb{R}}^3$, we can construct an embedding $e_C:{f_p}^{-1}(C_p) \rightarrow C_p \times {\mathbb{R}}^{n-3} \subset {\mathbb{R}}^{n}$ satisfying ${f_p} {\mid}_{{f_p}^{-1}(C_p)}={\pi}_{n,3} \circ e_C$. Since the pseudo special generic map is trivial, this lift can be extended to the whole space $W_p$ and to the outer space ${\mathbb{R}}^3$ by virtue of the fact that the space of all the smooth embeddings of $S^{m-2}$ into ${\mathbb{R}}^{n-3}$ with Whitney $C^{\infty}$ topology is simply connected or technique of [@nishioka]. From these arguments, we have the desired embedding. In [@kitazawa3], the author has shown a statement similar to Theorem \[thm:2\] for the case where we can lift the Morse function to a trivial pseudo special generic map represented also as a special generic map into ${\mathbb{R}}^2$ as Theorem 7 and in this, $n$ is assumed to be not smaller than $\max \{\frac{3m+3}{2},m+2+1\}$. Note also that in the case, the source manifold is assumed to be orientable by the reason that Nishioka’s technique [@nishioka] mentioned in the proof above is needed. [25]{} S. J. Blank and C. Curley, Desingularizing maps of corank one, Proc. Amer. Math. Soc. 80 (1980), 483–486. J. Cerf, Sur les diffeomorphismes de la sphère de dimension trois (${\Gamma}_4=0$), Lecture Notes in Math., Vol. 53, Springer-Verlag, 1968. F. Costantino and D. Thurston, $3$-manifolds efficiently bound $4$-manifolds, J. Topol. 1 (2008), 703–745. D. Crowley, T. Schick, The Gromoll filtration, KO-chracteristic classes and metrics of positive scalar curvature, Geometry Topoogy 17 (3) (2013), 1773–1789, arxiv:1204.6474v4. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Graduate Texts in Mathematics (14), Springer-Verlag(1974). A. Haefliger, Quelques remarques sur les applications différentiables d’une surface dans le plan, Ann. Inst Fourier, Grenoble 10 (1960), 47–60. J. T. Hiratuka and O. Saeki, Triangulating Stein factorizations of generic maps and Euler Characteristic formulas, RIMS Kokyuroku Bessatsu B38 (2013), 61–89. J. T. Hiratuka and O. Saeki, Connected components of regular fibers of differentiable maps, in “Topics on Real and Complex Singularities”, Proceedings of the 4th Japanese-Australian Workshop (JARCS4), Kobe 2011, World Scientific, 2014, 61–73. N. Kitazawa, On round fold maps (in Japanese), RIMS Kôkyûroku Bessatsu B38 (2013), 45–59. N. Kitazawa, Fold maps with singular value sets of concentric spheres, Hokkaido Mathematical Journal, Vol. 43, No. 3 (2014), 327–359. N. Kitazawa, Lifts of spherical Morse functions, arxiv:1805.05852v2. M. Ishikawa, Y. Koda, Stable maps and branched shadows of $3$-manifolds, Mathematische Annalen 367 (2017), no. 3, 1819–1863, arXiv:1403.0596. M. Kobayashi and O. Saeki, Simplifying stable mappings into the plane from a global viewpoint, Trans. Amer. Math. Soc. 348 (1996), 2607–2636. H. Levine, Classifying immersions into ${\mathbb{R}}^4$ over stable maps of $3$-manifolds into ${\mathbb{R}}^2$, Lecture Notes in Math., Vol. 1157, Springer-Verlag, Berlin, 1985. M. Nishioka, Desingularizing special generic maps into $3$-dimensional space, PhD thesis (Kyushu Univ.), arxiv:1603.04520v1. G. Reeb, Sur les points singuliers d'une forme de Pfaff complétement intègrable ou d'une fonction numérique, Comptes Rendus Hebdomadaires des Séances de I'Académie des Sciences 222 (1946), 847–849. R. Sadykov, Bordism groups of special generic mappings, Proc. Amer. Math. Soc., 133 (2005), 931–936. Y. Saito, On decomposable mappings of manifolds, J. Math. Kyoto Univ. 1 (1961/1962), 425–455. O. Saeki, Notes on the topology of folds, J.Math.Soc.Japan Volume 44, Number 3 (1992), 551–566. O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl. 49 (1993), 265–293. O. Saeki, Cobordism groups of special generic functions and groups of homotopy spheres, Japan. J. Math. (N. S. ) 28 (2002), 287–297. O. Saeki, K. Sakuma, On special generic maps into ${\mathbb{R}}^3$, Pacific J. Math. 184 (1998), 175–193. O. Saeki and K. Suzuoka, Generic smooth maps with sphere fibers J. Math. Soc. Japan Volume 57, Number 3 (2005), 881–902. O. Saeki and M. Takase, Desingularizing special generic maps, Journal of Gokova Geometry Topology 7 (2013), 1–24. M. Shiota, Thom’s conjecture on triangulations of maps, Topology 39 (2000), 383–399. V. G. Turaev, Shadow links and face models of statistical mechanics, J. Differential Geom. 36 (1992), 35–74. D. J. Wrazidlo, Standard special generic maps of homotopy spheres into Eucidean spaces, to appear: Topology and its Applications. arxiv:1707.08646. M. Yamamoto, Lifting a generic map of a surface into the plane to an embedding into $4$-space, Illinois J. Math. 51 (2007)., 705–721. M. Yamamoto, On embedding lifts over a Morse function on a circle, Singularity theory, geometry and topology, 31-43, RIMS Kokyuroku Bessatsu B38, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013.
--- abstract: 'Using a Bayesian methodology, we introduce the maximum a posteriori probability (MAP) estimator for quantum state and process tomography. We show that the maximum likelihood, the hedged maximum likelihood, and the maximum likelihood-maximum entropy estimator, and estimators of this general type, can be viewed as special cases of the MAP estimator. The MAP, like the Bayes’ mean estimator includes prior knowledge. For cases of interest to tomography MAP can take advantage of convex optimization tools making it numerically tractable. We show how the MAP and other Bayesian quantum state estimators can be corrected for noise produced if the quantum state passes through a noisy quantum channel prior to measurement. Numerical simulations on a single qubit indicate that on average, including these corrections significantly improve the estimate even when the measurement data are modestly large.' --- [Maximum a posteriori probability estimates for quantum tomography]{}\ Vikesh Siddhu\ Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.\ Date: 28$^{\text{th}}$ Jan’19 Introduction {#sct1} ============ The task of quantum state tomography is to estimate a quantum state or density operator by performing measurements. Its classical analogue is to estimate the parameters of a probability distribution by sampling from it several times. Quantum process tomography deals with the estimation of a noisy quantum channel or completely positive trace preserving map; its classical analogue is the estimation of a conditional probability distribution. An estimator is a procedure which uses the data from measurements to construct an estimate of the object of interest called the estimand. A point estimator provides a single best guess of the estimand, for example guessing the bias of a coin by flipping it several times, or locating a point in the qubit Bloch ball by measuring many identically prepared qubits. An interval estimator, more generally a set estimator, provides a set of plausible values for the estimand, for example a confidence interval for the bias of a coin, or a confidence region in the Bloch ball for a qubit state. In general, interval estimates provide more information, such as error bars, but are harder to construct than point estimates. For the latter error bars must be constructed independently. A lot of effort has been devoted to constructing good estimators for quantum states. Various point estimators [@PhysRevA.55.R1561; @PhysRevLett.105.200504; @PhysRevLett.107.020404; @PhysRevLett.105.150401; @teoBook] and interval estimators [@Blume-Kohout2012; @Ferrie2016; @Christandl2012; @PhysRevLett.117.010404] have been proposed. Maximum A Posteriori (MAP) point estimators are widely used in statistics, and have been applied in various fields of physics [@doi:10.1117/12.2212329; @Gursoy2015]. In this work we introduce the MAP estimator for quantum state and channel tomography. One begins with a prior probability density on the set of quantum states or channels, and using the measurement data the prior is updated to obtain a posterior density. The maximum of the posterior density gives the MAP estimate. The mean of the posterior density gives what is called the Bayes’ mean estimator (BME) [@Blume-Kohout2010]. We show that in many cases of interest in quantum tomography the MAP estimator can be computed efficiently using convex optimization tools [@boydVan04], some of these tools have been tailored for quantum information [@PhysRevA.95.062336; @Bolduc2017; @arXiv:1803.10062]. Obtaining the MAP estimate may be computationally simpler than computing the BME, which is evaluated by numerically integrating over the set of density operators. Several well known estimators, in particular the maximum likelihood estimator (MLE) [@PhysRevA.55.R1561], the hedged maximum likelihood estimator (HMLE) [@PhysRevLett.105.200504], and the maximum likelihood-maximum entropy (MLME) estimator [@PhysRevLett.107.020404], can be viewed as special cases of the MAP estimator corresponding to particular choices of the prior probability density. Thus, the MAP estimator provides a systematic framework for discussing estimators of this general type, and casts them in a new light. In addition we show how MAP estimators can be applied to quantum process tomography. Experimental setups are noisy, and it is useful to be able to correct the experimental data for noise. The MAP approach provides an easy way to do this if the noise can be represented by a noisy quantum channel with known parameters (that may have been determined in a separate experiment), through which the system of interest passes on its way to the measurement device. The rest of this paper is organised as follows. Section 2 is devoted to a general discussion of quantum state tomography and the linear inversion estimator; the material here is not new, but helps understand the later material. Section 3 introduces Bayesian estimators and the MAP estimator for quantum state and process tomography. In Sec. 4 we discuss how the simple noise model mentioned above can be incorporated in the MAP estimation framework, and present results from simulations that evaluate the effect of incorporating the noise model. Section 5 is a brief summary, and is followed by an appendix illustrating the use of convex optimization tools for minimizing convex functions over qubit density operators. Quantum State Tomography {#sct2} ======================== Let ${{\mathcal H}}$ be a $d$-dimensional Hilbert space, and ${{\mathcal L}}({{\mathcal H}})$ be the space of operators on ${{\mathcal H}}$. The set ${{\mathcal S}}$ of quantum states i.e. density operators forms a convex subset of ${{\mathcal L}}({{\mathcal H}})$. Measurements on quantum systems can be described using a POVM (positive operator-valued measure), a collection of positive operators that sum to the identity in ${{\mathcal H}}$. Let $\{ {\Lambda }_i \}$ be a POVM, and $\rho$ be a density operator. The probability $p_i$ of observing an outcome corresponding to the operator ${\Lambda }_i$ is $$p_i = {{\rm Tr}}(\rho {\Lambda }_i). \label{POVMProb}$$ One simple measurement scheme for doing quantum state tomography is to prepare $N$ quantum systems, each corresponding to density operator $\rho$, and independently measure each using the same POVM $\{{\Lambda }_i\}_{i=1}^k$. The measurements yield a data set ${\delta }= \{n_1, n_2,\dots ,n_k \}$, where $n_i$ is the number of measurement outcomes corresponding to ${\Lambda }_i$, and $\sum_i n_i = N$. The probability $\Pr({\delta }\|\rho)$ of observing the data set ${\delta }$ given the density operator $\rho$ is $$\Pr({\delta }\|\rho) = C_{{\delta }} \overset{k}{\underset{i = 1}{\prod}} p_i^{n_i}, \label{PdataRho}$$ where $C_{{\delta }} = N/ (n_1!) \dots (n_k!)$ is a normalization constant, which depends only on ${\delta }$. The linear inversion estimator $\hat{\rho}_{\text{inv}}$ is a simple method for estimating a system’s density operator $\rho$ when the measurement data are related to $\rho$ by a set of invertible linear equations. An example of this general strategy is the measurement scheme discussed above when the POVM $\{{\Lambda }_i\}_{i=1}^{d^2}$ forms a basis of ${{\mathcal L}}({{\mathcal H}})$. The dual basis $\{\bar {\Lambda }_i\}$ is defined by $$\lgl \bar {\Lambda }_i, {\Lambda }_j {\rangle }= {\delta }_{ij}, \quad i,j \in \{1, \dots, d^2\}, \label{dual}$$ where $\lgl \rho,\sigma {\rangle }= {{\rm Tr}}(\rho^{\dag} \sigma)$ is the Frobenius inner product. The set of invertible linear equations, $${{\rm Tr}}(\hat{\rho}_{\text{inv}} {\Lambda }_i) = \hat p_i := n_i/N, \quad i = 1, \dots, d^2, \label{linInvEq}$$ where $\hat p_i$ is an estimate of $p_i$, can be solved to obtain the linear inversion estimate $$\hat{\rho}_{\text{inv}} = \sum_{i=1}^{d^2} \hat p_i \bar {\Lambda }_i. \label{linINV}$$ The estimate $\hat p_i$ has a variance of $p_i(1-p_i)/N$, so one expects this strategy to work well when $N$ is large. The linear inversion estimator generalises in an obvious way when $\{{\Lambda }_i\}$ is a Hermitian basis of the operator space but not a POVM. The linear inversion estimator is quite special as it requires a set of invertible equations of the form . Note that estimates constructed using may not be valid quantum states. While they have unit trace, they may have negative eigenvalues. Bayesian Estimators {#sct3.1} =================== Bayes’ Rule ----------- For Bayesian estimators one chooses a prior probability for the estimand, and a model that relates observed data to the estimand. Using Bayes’ rule (discussed below), the prior is updated to obtain a posterior probability, and the latter is used to construct point or set estimates. We will be focusing on point estimates. Let $\Pr(\rho)$ be a prior probability measure on ${{\mathcal S}}$. It represents the belief or uncertainty about the quantum system prior to the measurement. For quantum state tomography, measurements are performed on many copies of a quantum system to generate a discrete data set ${\delta }$. A model $\Pr({\delta }\|\rho)$ is chosen, it relates ${\delta }$ to $\rho$ (see Eq. for an example) and represents the probability of obtaining the data given the quantum state. In the literature, $\Pr({\delta }\|\rho)$ for a fixed ${\delta }$ is often viewed as a non-negative function of $\rho$ called the likelihood function (for more examples see [@Smolin2012; @Singh2016]). The posterior probability measure $\Pr(\rho\|{\delta })$ on ${{\mathcal S}}$ given the data are obtained from Bayes’ rule $$\Pr(\rho\|{\delta }) = k \Pr({\delta }\| \rho) \Pr(\rho), \label{bayes}$$ where $k$, which depends on ${\delta }$, not $\rho$, is a normalization constant. As $\rho$ varies continuously, a useful way to express $\Pr(\rho)$ is by making $\rho$ a smooth one to one function $\rho({\textbf{x}})$ of a collection of real variables $X$, and introducing a non-negative prior probability density $p(\rho)$ such that $$\Pr(\rho \in {{\mathcal A}}) = \int_{\mathit{A}} p(\rho(\mathbf{x})) d\mathbf{x}, \label{PDPrior}$$ where ${{\mathcal A}}\subseteq {{\mathcal S}}$ is some subset of density operators, and $\mathit{A} \subseteq X$ the corresponding subset of parameters. The posterior probability density $$p(\rho\|{\delta }) = k \Pr({\delta }\|\rho)p(\rho), \label{postDen}$$ uses the same parametrization $X$ as before, and is related to the measure $\Pr(\rho\|{\delta })$ in a manner similar to . Note that for a given probability measure $\Pr(\rho)$ the density $p(\rho)$ depends on the choice of parametrization $\rho({\textbf{x}})$. Conversely, if $p(\rho)$ is held fixed, a different parametrization will lead to a different measure $\Pr(\rho)$. The same is true for the relationship between $\Pr(\rho\|{\delta })$ and $p(\rho\|{\delta })$. The Bayes’ mean estimator $\hat\rho_{\text{BME}}$ [@Blume-Kohout2010] is the expectation of $\rho$ in the posterior probability measure $\Pr(\rho\|{\delta })$, and can be written using the density $p(\rho\|{\delta })$ as $$\hat\rho_{\text{BME}} = \int \rho(\mathbf{x}) p(\rho(\mathbf{x})\|{\delta }) d\mathbf{x}. \label{BME}$$ While the terms in the integrand depend on the parametrization $X$ used for $\rho({\textbf{x}})$, the integral itself is independent of the parametrization, if $\Pr(\rho\|{\delta })$ is held fixed. If instead $p(\rho\|{\delta })$ is fixed, changing the parametrization may change $\Pr(\rho\|{\delta })$ (see comments following ) and alter $\hat\rho_{\text{BME}}$. Evaluating the integral numerically can be cumbersome as for $n$ qubits $X$ consists of $2^{2n}-1$ variables. MAP estimate ------------ \[sct3.2\] The maximum a posteriori probability (MAP) estimate $\hat \rho_{\text{MAP}}$, is the density operator $\rho$ for which the posterior probability density is maximum. Its advantage is that in many cases it can be easily computed. When the data set is large, one expects for a suitable prior that $\hat \rho_{\text{BME}}$ and $\hat \rho_{\text{MAP}}$ are close. Maximizing $p(\rho\|{\delta })$ is equivalent to maximizing $\log p(\rho\|{\delta })$, and since $k$ is independent of $\rho$ it follows from that $$\hat{\rho}_{\text{MAP}} = \underset{\rho \in {{\mathcal S}}}{\text{argmax}} \; [ \log \Pr({\delta }\|\rho) + \log p(\rho)]. \label{MAPForm}$$ Notice that for a given $\Pr(\rho)$, and thus $\Pr(\rho\|{\delta })$ as given by , $\hat{\rho}_{\text{MAP}}$ will depend on the parametrization $X$ for $\rho({\textbf{x}})$. See the comments above in connection with . Conversely, if $p(\rho)$, and thus $p(\rho\|{\delta })$ as given by , is held fixed, changing the parametrization may change $\Pr(\rho)$ and $\Pr(\rho\|{\delta })$ without altering $\hat{\rho}_{\text{MAP}}$. It is often the case in quantum tomography that $\log \Pr({\delta }\| \rho)$ is concave in $\rho$ (see Eq. for an example). If in addition, as is the case for a number of priors (see below), $\log p(\rho)$ is a concave function of $\rho$ then the same is true for the objective function on the right hand side of , and $\hat{\rho}_{\text{MAP}}$ can be efficiently computed using tools of convex optimization. If one knows that $\rho$ belongs to a discrete set of possibilities, the above discussion is modified in an obvious way; $\hat \rho_{\text{BME}}$ is the weighted average of finitely many density operators computed with respect to the left side of , and $\hat \rho_{\text{MAP}}$ is the density operator for which the left side of is maximum. The MLE, HMLE, and MLME estimators can be viewed as MAP estimators using suitable prior probability densities. The maximum likelihood estimate (MLE) $$\hat{\rho}_{\text{MLE}} = \underset{\rho \in {{\mathcal S}}}{\text{argmax}} \; \log \Pr({\delta }\|\rho), \label{MLE}$$ coincides with $\hat{\rho}_{\text{MAP}}$ in when the prior probability density $p(\rho)$ is independent of $\rho$. While MLE and the MAP estimate are the same for this special choice of prior, note that: the former is the density operator for which the data are most likely, and the latter is the most probable density operator given the data and the prior probability density. The hedged maximum likelihood estimate (HMLE) $$\hat{\rho}_{\text{HMLE}} = \underset{\rho \in {{\mathcal S}}}{\text{argmax}} \; \{ \log \Pr({\delta }\|\rho) + \log [\det(\rho)]^{{\beta }} \}, \quad {\beta }> 0, \label{HMLE}$$ is of the MAP form, where the prior $p(\rho) \propto [\det(\rho)]^{{\beta }}$ is called the hedging function; it guarantees a full rank estimate. When ${\beta }$ is an integer, this prior probability density can be viewed as a special case of an induced measure (see Eq. (3.5) in [@KZHS01]) obtained by choosing an ancillary system of dimension $k = {\beta }+ d$, defining the Haar measure on a $d \times k$ dimensional Hilbert space, then tracing out the ancillary system to induce a distribution on the space of $d$ dimensional density operators. The function $\log [\det(\rho)]^{{\beta }}$ is concave in $\rho$ for ${\beta }> 0$, and when $\log \Pr({\delta }\|\rho)$ is concave the HMLE can be efficiently computed. The maximum likelihood maximum entropy (MLME) estimate $$\hat{\rho}_{\text{MLME}} = \underset{\rho \in \mathbf {{\mathcal S}}}{\text{argmax}} \; [ \log \Pr({\delta }\|\rho) + {\lambda }S(\rho) ], \quad {\lambda }\geq 0, \label{MLME}$$ is a MAP estimate with a prior which is exponential in the von-Neumann entropy $S(\rho)=-{{\rm Tr}}(\rho \log \rho)$. Since $S(\rho)$ is concave in $\rho$, when $\Pr({\delta }\|\rho)$ is concave the MLME can be efficiently computed. Other possible advantages of the MLME estimator have been discussed in [@PhysRevLett.107.020404]. MAP and quantum process tomography {#sct3.3} ---------------------------------- Let ${{\mathcal H}}_{a}, {{\mathcal H}}_{a'}$ and ${{\mathcal H}}_b$ be finite dimensional Hilbert spaces with dimensions $d_{a}=d_{a'}=d$ and $d_b$, respectively. Let ${{\mathcal N}}:{{\mathcal L}}({{\mathcal H}}_{a'}) \mapsto {{\mathcal L}}({{\mathcal H}}_b)$ be a quantum channel, and ${{\mathcal I}}_a:{{\mathcal L}}({{\mathcal H}}_a) \mapsto {{\mathcal L}}({{\mathcal H}}_a)$ be the identity map on operators. Let $\{{\|a_i\rangle }\}$ and $\{{\|a'_i\rangle }\}$ be orthonormal basis of ${{\mathcal H}}_{a}$ and ${{\mathcal H}}_{a'}$ respectively, and ${\|\phi\rangle } = \sum_{i} {\|a_i\rangle }{\|a'_i\rangle }/\sqrt{d}$ be a maximally entangled bipartite state. The channel ${{\mathcal N}}$ can be completely characterised by a bipartite quantum state, sometimes called the Choi matrix or the dynamical operator: $${\Upsilon }= ({{\mathcal I}}\ot {{\mathcal N}}){\|\phi\rangle }{\langle\phi\|}, \quad {\Upsilon }\in {{\mathcal L}}({{\mathcal H}}_{ab}). \label{Choi}$$ The channel ${{\mathcal N}}$ is completely positive if and only if the operator ${\Upsilon }$ is positive semi-definite ([@Choi1975], see [@PatGrif11] for a diagrammatic proof), and ${{\mathcal N}}$ is trace preserving if $${{\rm Tr}}_{b}({\Upsilon })= \mathbb{I}_a/d, \label{partTr}$$ where ${{\rm Tr}}_{b}$ is the partial trace over ${{\mathcal H}}_b$, and $\mathbb{I}_a$ is the identity operator on ${{\mathcal H}}_a$. For any $A \in {{\mathcal L}}({{\mathcal H}}_{a'})$, one can show that $${{\mathcal N}}(A) = d \, {{\rm Tr}}_{a}[ (A^T \ot \mathbb{I}_b ){\Upsilon }], \label{chanOut}$$ where $A^T$ denotes the transpose of $A$ in the $\{{\|a_i\rangle }\}$ basis. So ${{\mathcal N}}$ is determined by the Choi matrix ${\Upsilon }$. Equation gives a one to one correspondence between ${{\mathcal M}}_{ab}$: the convex set of quantum channels mapping ${{\mathcal L}}({{\mathcal H}}_a)$ to ${{\mathcal L}}({{\mathcal H}}_b)$, and ${{\mathcal T}}_{ab}$: the convex set of density operators in ${{\mathcal L}}({{\mathcal H}}_{ab})$ with partial trace on ${{\mathcal H}}_b$ equaling $\mathbb{I}_a/d$. This correspondence can be used to construct a MAP estimator for a quantum channel as follows. Suppose measurements are performed with the aim of characterising the quantum channel ${{\mathcal N}}$ (see [@DLeu03; @NC97Tomo] for examples) and data ${\delta }$ are collected. As in Sec. \[sct3.2\], let $p({\Upsilon })$ be a prior probability density on ${{\mathcal T}}_{ab}$, and $\Pr({\delta }\|{\Upsilon })$ the probability of obtaining ${\delta }$ given ${\Upsilon }$. The MAP estimator for the Choi matrix $$\hat{{\Upsilon }}_{\text{MAP}} = \underset{{\Upsilon }\in {{\mathcal T}}_{ab}}{\text{argmax}} \; [ \log \Pr({\delta }\|{\Upsilon }) + \log p({\Upsilon }) ], \label{mapEst}$$ becomes a MAP estimate $\hat{{\mathcal N}}_{\text{MAP}}$ for the quantum channel when $\hat{{\Upsilon }}_{\text{MAP}}$ is inserted in Eq. . When the objective function on the right hand side of Eq. is concave in ${\Upsilon }$, $\hat{{\Upsilon }}_{\text{MAP}}$ can be efficiently computed using tools of convex optimization. Modelling Noise {#sct4} =============== Noise is present in any experimental setup. If its effect upon a tomographic measurement can be modeled by assuming a known noisy channel ${{\mathcal N}}$ (whose parameters have been determined by previous calibration measurements) preceding the final measurement as in Fig. 1, then ![Schematic diagram of a quantum state $\rho$ passing through a quantum channel ${{\mathcal N}}$ prior to being measured.](noisyMess_B.pdf) Bayesian estimators for $\rho$ can be obtained by replacing $\Pr({\delta }\|\rho)$ with $\Pr({\delta }\|{{\mathcal N}}(\rho))$ in and . The MAP estimate in becomes $$\hat{\rho}_{\text{MAP}} = \underset{\rho \in {{\mathcal S}}}{\text{argmax}} \; [ \log p({\delta }\|{{\mathcal N}}(\rho)) + \log p(\rho)], \label{MAPFormNoise}$$ In addition the MAP estimate of the state coming out of the channel can be shown to be, $$\hat{\sigma}_{\text{MAP}} ={{\mathcal N}}(\hat\rho_{\text{MAP}}). \label{MAPAfter}$$ The above construction is quite general. There is no restriction on ${{\mathcal N}}$, the form of $\Pr({\delta }\|\rho)$, $p(\rho)$, or the size of the quantum system $d$. Because ${{\mathcal N}}$ is a linear map, if $\log \Pr({\delta }\|\rho)$ is concave in $\rho$ so is $\log \Pr({\delta }\|{{\mathcal N}}(\rho))$. Thus when the tools of convex optimization allow an efficient calculation of the MAP estimate in the same will be true of . Since the MLE, MLME, and the HMLE are special cases of MAP, they can likewise be adapted to the the noisy setting. Note that the Gaussian noise models considered in [@Smolin2012; @Singh2016] are quite different: they are not based on a noisy channel, but instead on a special form of $\Pr({\delta }\|\rho)$. Example {#sct4.1} ------- Let $\{{\sigma }_s \}_{s \in \{x,y,z\}}$ be the Pauli matrices. A qubit density operator can be expressed in the Bloch parametrization, $$\rho({\textbf{r}}) = \frac{1}{2}(\mathbb{I} + {\textbf{r}}.\vec{{\sigma }}) := \frac{1}{2}(\mathbb{I} + r_x {\sigma }_x + r_y {\sigma }_y + r_z {\sigma }_z), \label{qBitBloch}$$ where the Bloch vector ${\textbf{r}}= (r_x,r_y,r_z)$, with norm $\|{\textbf{r}}\|$, belongs to the set of real variables $R=\{{\textbf{r}}\; \| \|{\textbf{r}}\| \leq 1\}$ called the Bloch ball. For doing tomography suppose $3N$ measurements, $N$ each in the eigenbasis $\{{\|+\rangle }_s, {\|-\rangle }_s\}$ of $\{{\sigma }_s\}_{s \in \{x,y,z\}}$, are performed on identically prepared quantum systems, each described by the density operator $\rho$. The probability of observing ${\|+\rangle }_s$, $$p_s = {{\rm Tr}}( \rho [+]_s) = (1 + r_s)/2, \quad s \in \{x,y,z\}, \label{pDef}$$ where $[+]_s$ denotes the projector on ${\|+\rangle }_s$. The measurement data set ${\delta }= \{n_s, N-n_s\}_{s \in \{x,y,z\}}$, where $n_s$ denotes the number of times ${\|+\rangle }_s$ is observed, has a probability $$\Pr({\delta }\|\rho) = C_{{\delta }} \underset{s \in \{x,y,z\}}{\prod} p_s^{n_s}(1-p_s)^{N-n_s}, \label{modDen}$$ where $C_{{\delta }} = (3N)!/(\prod_s n_s!(N-n_s)!)$. Using and , a simple MLE estimate $$\hat{\rho}_{\text{MLE}} = \underset{{\textbf{r}}\in R}{\text{argmax}} \; \log \Pr({\delta }\|\rho({\textbf{r}})) \label{MLECon}$$ can be obtained. However most tomography setups have noise. For example, when discussing noise in a nuclear magnetic resonance (NMR) experiment [@Vandersypen2001; @PhysRevA.70.032324; @PhysRevA.97.022302] one may use the model described in Fig. 1, where the channel ${{\mathcal N}}$, acting over time $t$ represents the combined action of two channels, the generalized amplitude damping ($T_1$) channel, $${{\mathcal A}}(\rho) = \sum_i A_i \rho A_i^{\dag}, \label{AD}$$ where, $$A_0 = \sqrt{p}\begin{pmatrix} 1&0\\ 0&\sqrt{1-\gamma} \end{pmatrix}, A_1 = \sqrt{p}\begin{pmatrix} 0&\sqrt{\gamma}\\ 0&0 \end{pmatrix}, A_2 = \sqrt{1-p}\begin{pmatrix} \sqrt{1-\gamma}&0\\ 0&1 \end{pmatrix}, A_3 = \sqrt{1-p}\begin{pmatrix} 0&0\\ \sqrt{\gamma}&0 \end{pmatrix},$$ $p \in [0,1]$, is the probability of finding the qubit in the ${\|0\rangle }$ state as $t \mapsto \infty$, $\gamma = 1 - e^{-t/T_1}$, $T_1$ is a time constant, and the phase damping ($T_2$) channel, $${{\mathcal B}}(\rho) = \sum_j B_j \rho B_j^\dag, \label{PD}$$ where, $$B_0 = \begin{pmatrix} 1&0\\ 0&\sqrt{1-{\lambda }} \end{pmatrix}, B_1 = \begin{pmatrix} 0&0\\ 0&\sqrt{{\lambda }} \end{pmatrix},$$ ${\lambda }= 1 - e^{-t/T_2}$, and $T_2$ is a time constant, and ${{\mathcal N}}= {{\mathcal A}}\circ {{\mathcal B}}= {{\mathcal B}}\circ {{\mathcal A}}$ [^1]. A MAP estimate which accounts for the noise, assuming a prior probability density independent of ${\textbf{r}}$ (under this probability density function, equal volumes of the Bloch ball have equal probability), can be obtained by using and , $$\hat{\rho}_{\text{MAP}} = \underset{{\textbf{r}}\in R}{\text{argmax}} \; \log \Pr({\delta }\|{{\mathcal N}}(\rho({\textbf{r}}))). \label{MAPCon}$$ We perform numerical simulations to test performance of MLE and MAP in and respectively. There is no unique metric to assess how close an estimate $\hat{\rho}$ is to the actual $\rho$. The fidelity ${{\mathcal F}}(\rho,\hat{\rho}) \equiv {{\rm Tr}}( \sqrt{\sqrt{\rho}\hat{\rho}\sqrt{\rho}} )$ and trace distance $ {{\mathcal D}}(\rho,\hat{\rho}) \equiv \frac{1}{2}\|\|\rho - \hat{\rho}\|\|_1$ are, however popular choices. Various parameter choices for ${{\mathcal A}}$ and ${{\mathcal B}}$ are used in the simulation by fixing $p = 1/2$, $T_1/T_2 = 10$ [@PhysRevA.97.022302], $t = k T_2$ and varying $k \in \{0.25,.5,.75,1.0,1.5,2.0,2.5 \}$. For any fixed $k$, we choose $2.5 \times 10^3$ qubit states uniformly in the Bloch ball. For each state we simulate the construction of the MLE and MAP estimates for various $N$ values. By averaging over all qubit states we arrive at the average log-infidelity ($\log_{10}[1 - {{\mathcal F}}(\rho,\hat{\rho})]$) and log-trace distance ($\log_{10}[{{\mathcal D}}(\rho,\hat{\rho})]$) of the MAP and MLE estimators (see Appendix for numerical techniques). For a fixed $k$, under both log-infidelity and log-trace distance, the behaviour of MAP and MLE estimators with the number of measurements ($3N$) is shown in Fig. 2. ![(Color online) Plot of the average log-infidelity and log-trace distance between the true and estimated state, against the number of measurements at $k=0.5$. The error bars represent one standard deviation. If the quantum channel ${{\mathcal N}}$ was perfect and the number of measurements was infinite, the log infidelity and log trace distance would be negative infinity for MAP and MLE. As the ordinate of the graph increases the performance of the estimator becomes worse. []{data-label="fig:pdfig"}](fig22.pdf) Under both metrics, MAP and MLE show qualitatively different behaviour. As the number of measurements increases from a small value, for both metrics, the average MAP value decreases and the average MLE value decreases but settles to a fixed number. Thus, on average the MAP estimate always improves with the number of measurements while the MLE improves up to a point and then ceases to change. ${{\mathcal N}}$ becomes more noisy as $k$ increases and this causes both the MAP and MLE curves in each of the plots in Fig. 2 to shift upwards. The upward shift implies that for a fixed $N$ with increasing $k$, on average the MAP and MLE estimates become worse under both metrics. In the case of MAP this effect of increasing $k$ can be mitigated by increasing the number of measurements which on average improves the estimate, however this is not always possible for MLE whose average value, under both metrics becomes fixed beyond a certain number of measurements and this fixed value also increases with $k$. Under both metrics and for all $k$ values tested, when the number of measurements is modestly large (roughly greater than $300$ in our case), on average MAP outperforms MLE. As the number of measurements increases further the average MAP value becomes an order of magnitude better than average MLE value, and eventually the error bars (one standard deviation about the average value) on the MAP and MLE curves cease to overlap. Thus numerical evidence on qubits shows, that except when the number of measurements are low, on average there is an advantage of using a MAP estimator which accounts for noise over a standard MLE. Conclusion {#sct5} ========== The maximum a posteriori probability (MAP) estimation framework for quantum state and process tomography introduced here combines a number of previous quantum state estimators, in particular the maximum likelihood, hedged maximum likelihood, and maximum likelihood-maximum entropy estimator, in a single framework using Bayesian methodology. In several cases of interest to quantum state tomography the MAP estimator becomes a convex optimization problem which should be numerically more tractable than the Bayes’ Mean Estimator. Using the Choi-Jamioł[l]{}kowski isomorphism, MAP estimation of quantum states can be extended to quantum channels, and the extension is expected to have similar advantages as the MAP estimate for quantum states. When the experimental noise can be represented by a known noisy channel preceding the measurement, the MAP estimator can be modified to take it into account and can be computed efficiently as long as the posterior probability density is log concave. Numerical results on qubits indicate that on average, such modifications can vastly improve estimates. Having a measure of reliability for any estimate is of significant value, and it would be interesting to construct such measures for the MAP estimate. Acknowledgments {#acknowledgments .unnumbered} =============== I am indebted to Robert B. Griffiths for valuable discussions, and thank Renato Renner, Yong Siah Teo, Carlton Caves, Ezad Shojaee and Simon Samuroff for their comments. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [@XSEDE], which is supported by National Science Foundation grant number ACI-1548562. Specifically, it used the Bridges system [@bridges], which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC). Appendix. Convex Optimization over qubit density operators ========================================================== Projected gradient descent is a very general iterative algorithm, used extensively for minimizing functions defined over convex sets. When the function is convex, successive function values obtained during the algorithm approach the global minimum. For some special convex sets, tools from convex analysis can be used to compute a bound on how close a given function value is to the global minimum. We provide an exposition of the projected gradient descent algorithm for minimizing any differentiable convex function over the set of qubit density operators, and illustrate a technique for checking how far a given function value is from the global minimum. Let $f:R \mapsto \mathbb{R}$ be a differentiable convex function, then $$f({\textbf{x}}^) = f^ = \underset{{\textbf{x}}\in R}{\min} \; f({\textbf{x}}),$$ is called the optimization problem, with objective function $f$, optimal ${\textbf{x}}^$ and optimum value $f^$. Projected gradient descent begins with some point inside $R$, then iteratively takes steps to move to a new point with a lower function value. The algorithm halts when some stopping criterion is met. New points are chosen by moving along the negative gradient direction by an amount called the step size, such movements may take one outside the set $R$, in which case we project onto the boundary of the set. If ${\textbf{x}}$ is a vector in $\mathbb{R}^3$ then its projection onto $R$, $$P_{R}({\textbf{x}}) = \begin{cases} {\textbf{x}}/\|{\textbf{x}}\| \quad \text{if} \quad \|{\textbf{x}}\| > 1 \\ {\textbf{x}}\quad \text{otherwise} \end{cases}.$$ A pseudo code for projected gradient descent is given in algorithm 1. There are several different ways of choosing a step size and a stopping criterion. We select the step size by using a method called backtracking line search. Let $$G_t({\textbf{x}}) \equiv ({\textbf{x}}- {\textbf{x}}^+(t))/t \label{genGrad}$$ be the generalized gradient at ${\textbf{x}}$ for step size $t$. A pseudo code for backtracking, to be used as a subroutine in algorithm 1 is provided in algorithm 2. Let $t = 1$ Backtracking line search ensures that the function decreases by at least $t^\|G_{t^}({\textbf{x}})\|^2/2$ in each iteration [@armijo1966]. One possible stopping criterion is to check whether the function hasn’t decreased appreciably over the past few iterations or $\|G_{t^}({\textbf{x}})\|$ is greater than a small constant. At any point ${\textbf{x}}\in R$, the surrogate duality gap [@jaggi13] $$g({\textbf{x}}) = \underset{{\textbf{r}}\in R}{\max} \; \langle \nabla f ({\textbf{x}}), {\textbf{x}}- {\textbf{r}}\rangle$$ upper bounds $\|f({\textbf{x}}) - f^\|$ and provides a measure of how close a function value at ${\textbf{x}}$ is to the global minimum value. Due to the simple structure of the set of qubit density operators, the surrogate duality gap can be easily computed. Let $\nabla F({\textbf{x}}) = \nabla f ({\textbf{x}}).\vec{{\sigma }}$ be a matrix, ${\lambda }_{\min}(\nabla F({\textbf{x}}))$ be its smallest eigenvalue, then $$g({\textbf{x}}) = \langle \nabla f ({\textbf{x}}), {\textbf{x}}\rangle - {\lambda }_{\min}(\nabla F({\textbf{x}})).$$ Given the ease of computing $g({\textbf{x}})$, checking whether its value is greater that a small constant also serves as a good stopping criterion. Once algorithm 1 converges and returns some $(\tilde{{\textbf{x}}}, f(\tilde{{\textbf{x}}}))$, computing $g(\tilde{{\textbf{x}}})$ gives an estimate of how far $f(\tilde{{\textbf{x}}})$ is from $f^$. The projected gradient algorithm discussed above can be used to solve optimization problems in Eq. and to obtain $\hat{\rho}_{\text{MLE}}$ and $\hat{\rho}_{\text{MAP}}$ respectively. The gradient of the objective function in these optimization problems can be computed analytically. Let ${\textbf{r}}= (r_x, r_y, r_z)$ be a Bloch vector for a qubit density operator $\rho$, then upto local unitaries at the input and output of a qubit channel ${{\mathcal N}}$, the Bloch vector ${\textbf{r}}'$ for ${{\mathcal N}}(\rho)$ can always be written as ${\textbf{r}}' = (l_x r_x + t_x, l_y r_y + t_y, l_z r_z + t_z)$ [@GriffithsNotesChan]. When ${{\mathcal N}}= {{\mathcal I}}$ $$l_s = 1, t_s = 0, \; s \in \{x,y,z\},$$ when ${{\mathcal N}}= {{\mathcal A}}\circ {{\mathcal B}}$ (see Eq. and ) $$l_x = l_y = \sqrt{(1-{\lambda })(1-\gamma)}, \; l_z = 1-\gamma, t_x=t_y=0, \;\text{and}\;t_z = \gamma(2p-1).$$ The gradient of the function on the right hand side in Eq. can be obtained using $$\frac{\partial }{ \partial r_s}\log \Pr({\delta }\|{{\mathcal N}}(\rho({\textbf{r}}))) = \frac{l_s}{2} \big(\frac{n_s}{p_s} - \frac{N-n_s}{1-p_s} \big), \quad s \in \{x,y,z\}.$$ Choosing ${{\mathcal N}}={{\mathcal I}}$ gives the gradient of the function in the right side of Eq. . A python implementation of projected gradient descent and surrogate duality gap computation over qubits, including a code that simulates the tomography reconstruction and generates plots for Fig. 2 is publicly available [@MAPQubit]. [10]{} Z. Hradil. Quantum-state estimation. , 55:R1561–R1564, Mar 1997. Robin Blume-Kohout. Hedged maximum likelihood quantum state estimation. , 105:200504, Nov 2010. Yong Siah Teo, Huangjun Zhu, Berthold-Georg Englert, Jaroslav Řehá ček, and Zden ěk Hradil. Quantum-state reconstruction by maximizing likelihood and entropy. , 107:020404, Jul 2011. David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. Quantum state tomography via compressed sensing. , 105:150401, Oct 2010. Yong Siah Teo. . World Scientific, 2015. Robin Blume-Kohout. . arXiv:1202.5270, 2012 Christopher Ferrie and Robin Blume-Kohout. . , 116(9):090407, mar 2016. Matthias Christandl and Renato Renner. . , 109(12), 2012. Philippe Faist and Renato Renner. Practical and reliable error bars in quantum tomography. , 117:010404, Jul 2016. Aaron C. Chan, Kazuhiro Kurokawa, Shuichi Makita, Young-Joo Hong, Arata Miyazawa, Masahiro Miura, and Yoshiaki Yasuno, Quantitative optical coherence tomography by maximum a-posteriori estimation of signal intensity , 96971W (8 March 2016) Doga G[ü]{}rsoy, Tekin Bi[ç]{}er, Jonathan D. Almer, Raj Kettimuthu, Stuart R. Stock, and Francesco De Carlo. Maximum a posteriori estimation of crystallographic phases in x-ray diffraction tomography. , 373(2043):20140392, Jun 2015. Robin Blume-Kohout. . , 12(4):043034, Apr 2010. Stephen Boyd and Lieven Vandenberghe. . Cambridge University Press, 2004. Jiangwei Shang, Zhengyun Zhang, and Hui Khoon Ng. Superfast maximum-likelihood reconstruction for quantum tomography. , 95:062336, Jun 2017. Eliot Bolduc, George C. Knee, Erik M. Gauger, and Jonathan Leach. Projected gradient descent algorithms for quantum state tomography. , 3(1):44, 2017. George C. Knee, Eliot Bolduc, Jonathan Leach, and Erik M. Gauger. Quantum process tomography via completely positive and trace-preserving projection. , 98:062336, Dec 2018. John A. Smolin, Jay M. Gambetta, and Graeme Smith. . , 108(7):070502, Feb 2012. Harpreet Singh, Arvind, and Kavita Dorai. . , 380(38):3051–3056, Sep 2016. Karol Zyczkowski and Hans-Jürgen Sommers. Induced measures in the space of mixed quantum states. , 34(35):7111, 2001. Man-Duen Choi. Completely positive linear maps on complex matrices. , 10(3):285 – 290, 1975. Patrick J. Coles, Li Yu, Vlad Gheorghiu, and Robert B. Griffiths. Information-theoretic treatment of tripartite systems and quantum channels. , 83:062338, Jun 2011. Debbie W. Leung. Choi’s proof as a recipe for quantum process tomography. , 44(2):528–533, 2003. Isaac L.Chuang and M. A. Nielsen. Prescription for experimental determination of the dynamics of a quantum black box. , 44(11-12):2455–2467, 1997. Lieven M. K. Vandersypen, Matthias Steffen, Gregory Breyta, Costantino S. Yannoni, Mark H. Sherwood, and Isaac L. Chuang. Experimental realization of shor’s quantum factoring algorithm using nuclear magnetic resonance. , 414:883, Dec 2001. M. S. Anwar, J. A. Jones, D. Blazina, S. B. Duckett, and H. A. Carteret. Implementation of nmr quantum computation with parahydrogen-derived high-purity quantum states. , 70:032324, Sep 2004. Harpreet Singh, Arvind, and Kavita Dorai. Evolution of tripartite entangled states in a decohering environment and their experimental protection using dynamical decoupling. , 97:022302, Feb 2018. J. Towns, T. Cockerill, M. Dahan, I. Foster, K. Gaither, A. Grimshaw, V. Hazlewood, S. Lathrop, D. Lifka, G. D. Peterson, R. Roskies, J. R. Scott, and N. Wilkins-Diehr. Xsede: Accelerating scientific discovery. , 16(5):62–74, Sept.-Oct. 2014. Nicholas A. Nystrom, Michael J. Levine, Ralph Z. Roskies, and J. Ray Scott. Bridges: A uniquely flexible hpc resource for new communities and data analytics. In [Proceedings of the 2015 XSEDE Conference: Scientific Advancements Enabled by Enhanced Cyberinfrastructure]{}, XSEDE ’15, pages 30:1–30:8, New York, NY, USA, 2015. ACM. Larry Armijo. Minimization of functions having lipschitz continuous first partial derivatives. , 16(1):1–3, 1966. Martin Jaggi. Revisiting frank-wolfe: Projection-free sparse convex optimization. In Sanjoy Dasgupta and David McAllester, editors, [Proceedings of the 30th International Conference on Machine Learning]{}, volume 28 of [ Proceedings of Machine Learning Research*]{}, pages 427–435, Atlanta, Georgia, USA, 17–19 Jun 2013. PMLR. Robert B. Griffiths. Quantum computation and quantum information theory course notes: Quantum channels, kraus operators, povms. <http://quantum.phys.cmu.edu/QCQI/index.html>, Oct 2013. Vikesh Siddhu. Python implementation of map estimation over qubits density operators. <https://github.com/vsiddhu/MAPqubit.git>, January 2018. [^1]: In general, a quantum channel obtained by first applying some quantum channel ${{\mathcal A}}$ and then some channel ${{\mathcal B}}$ is different from the one obtained by applying ${{\mathcal B}}$ first and then ${{\mathcal A}}$. However, in this special case where ${{\mathcal A}}$ is the qubit amplitude damping and ${{\mathcal B}}$ is the qubit phase damping channel, changing the order has no effect.
--- abstract: 'A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we are interested in the function sat$_{t}(n,p)$, defined to be the minimum number of edges that a $K_{p}$-saturated graph on $n$ vertices can have if it has minimum degree at least $t$. We prove that sat$_{t}(n,p) = tn - O(1)$, where the limit is taken as $n$ tends to infinity. This confirms a conjecture of Bollobás when $p = 3$. We also present constructions for graphs that give new upper bounds for sat$_{t}(n,p)$ and discuss an analogous problem for saturated hypergraphs.' author: - 'A. Nicholas Day [^1]' date: - - title: Saturated Graphs of Prescribed Minimum Degree --- Introduction ============ We say a graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we will be particularly interested in the case where $H$ is the complete graph on $p$ vertices, denoted $K_{p}$. For further results on saturated graphs see surveys by either Faudree, Faudree and Schmitt [@Faudree] or Pikhurko [@Oleg]. Erdős, Hajnal and Moon [@EHM] showed that if $G$ is a $K_{p}$-saturated graph on $n$ vertices then $e(G) \geqslant n(p-2) - \binom{p-1}{2}$ and that the unique graph achieving equality is formed by taking a clique on $p-2$ vertices and fully connecting it to an independent set of size $n - (p - 2)$. The above extremal graphs have minimum degree $p-2$ and no $K_{p}$-saturated graph on at least $p$ vertices can have smaller minimum degree. Thus it is natural to ask: how few edges can a $K_{p}$-saturated graph have if it has minimum degree at least $t$ for $t \geqslant p-2$? Duffus and Hanson [@DuffusHanson] considered the function $$\text{sat}'_{t}(n,p) = \min \{ e(G): \|V(G)\| = n, G \text{ is } K_{p} \text{-saturated} , \delta(G) = t \} \nonumber$$ where $\delta(G)$ is the minimum degree of $G$. The graph obtained by fully connecting a single vertex to all other vertices is $K_{3}$-saturated and shows sat$'_{1}(n,3) \leqslant n - 1$. However, any $K_{3}$-saturated graph must be connected and so in fact sat$'_{1}(n,3) = n - 1$. Duffus and Hanson showed that for $n \geqslant 5$, sat$'_{2}(n,3) = 2n - 5$ and that the unique graphs achieving this are obtained by taking a $5$-cycle and repeatedly duplicating vertices of degree $2$, that is, picking a vertex of degree $2$ and adding a new vertex to the graph with the same neighbourhood as your chosen vertex. They also showed that for $n \geqslant 10$, sat$'_{3}(n,3) = 3n - 15$ and that any graph achieving this contains the Petersen graph as a subgraph. In this paper we will consider the function $$\text{sat}_{t}(n,p) = \min \{ e(G): \|V(G)\| = n, G \text{ is } K_{p} \text{-saturated} , \delta(G) \geqslant t \},\nonumber$$ and we define the set Sat$_{t}(n,p)$ to be $$\{ G : \|V(G)\| = n, G \text{ is } K_{p} \text{-saturated} , \delta(G) \geqslant t, e(G) = \text{sat}_{t}(n,p) \}.\nonumber$$ The distinction between sat$_{t}(n,p)$ and sat$'_{t}(n,p)$ is important as it is not clear that we always have equality between the two. However we will see from Theorem \[Thm1\] below that, for fixed $t$ and $p$, equality does hold for $n$ sufficiently large. The complete bipartite graph $K_{t,n-t}$ shows that for $n \geqslant 2t$, we have sat$_{t}(n,3) \leqslant tn - t^{2}$. This upper bound and Duffus and Hanson’s results led Bollobás [@Boll] to conjecture that for fixed $t$ we have sat$_{t}(n,3) = tn - O(1)$. For more general values of $p$, Duffus and Hanson [@DuffusHanson] also showed that sat$_{t}(n,p) \geqslant \frac{t + p -2}{2}n - O(1)$. Writing $\alpha(G)$ for the size of the largest independent set in $G$, Alon, Erdős, Holzman and Krivelevich showed in [@AEHK] that any $K_{p}$-saturated graph on $n$ vertices with fewer than $O(n)$ edges has $\alpha(G) \geqslant n - O(\frac{n}{\log \log n})$. This shows that sat$_{t}(n,p) \geqslant tn - O(\frac{n}{\log \log n})$ as $e(G) \geqslant \alpha(G)\delta(G)$. Pikhurko [@Oleg] improved this result to show that sat$_{t}(n,p) \geqslant tn - O(\frac{n \log \log n}{\log n})$. Our main result in this paper improves these results by confirming and generalising Bollobás’s conjecture: \[Thm1\] There exists a constant $c = c(t)$ such that if $G$ is a $K_{p}$-saturated graph of order $n$ and minimum degree at least $t$, then $e(G) \geqslant tn - c$. The proof of the above theorem is presented in Section 2. To see that, up to the value of the constant, this result is best possible consider the graph obtained from fully connecting a clique of size $p-3$ to the complete bipartite graph $K_{t-(p-3),n-t}$. This graph is $K_{p}$-saturated and has minimum degree $t$ and so shows that $$\begin{aligned} \text{sat}_{t}(n,p) \leqslant tn - t^{2} + t(p-3)- \binom{p-2}{2} \nonumber\end{aligned}$$ for $n \geqslant 2t - (p-3)$ and $ t \geqslant p-2$. From Theorem \[Thm1\] we obtain the following corollary: \[CoroThm1\] There exists $n_{0}(t,p)$ and $c(t,p)$ such that if $n \geqslant n_{0}(t,p)$ then $$\text{sat}_{t}(n,p) = \text{sat}'_{t}(n,p) = tn - c(t,p).\nonumber$$ The proof of this corollary can also be found in Section 2. In Section 3 we discuss constructing $K_{p}$-saturated graphs and prove an upper bound on sat$_{t}(n,3)$: \[Thm2\] For all $t \geqslant 4$ and all $n \geqslant t + \lfloor t/2 \rfloor \binom{t}{\lfloor t/2 \rfloor}$ there exists a $K_{3}$-saturated graph on $n$ vertices with minimum degree $t$ and $$\begin{cases} tn - t^{2} (1 + \binom{t}{t/2}/8) \text{ \phantom{abcdefgi}for t even} \\ tn - t^{2} -\frac{(t-1)(t^{2}-1)}{8t}\binom{t}{(t-1)/2} \text{ for t odd} \end{cases}\nonumber$$ edges. We will see that we can use this theorem to construct $K_{p}$-saturated graphs that show $$\begin{aligned} c(t,p) \geqslant C2^{t'}t'^{3/2}+\binom{t+1}{2} - \binom{t'+1}{2} \nonumber\end{aligned}$$ for some constant $C$, where $t' = t - (p-3)$. We also discuss $K_{p}$-semi-saturated graphs, (graphs that have the property that any new edge added to them creates a new copy of $K_{p}$), and show that their behaviour is very different from $K_{p}$-saturated graphs. Finally, in Section 4 we discuss a conjecture for a generalisation of Theorem \[Thm1\] to hypergraphs and look at constructing saturated hypergraphs. We remark that one may also ask how few edges a $K_{p}$-saturated graph can have if restrictions are placed on its maximum degree rather than its minimum degree. Results on this problem for $p = 3$ can be found in Füredi and Seress’s paper [@FurSer] and also in Erdős and Holzman’s paper [@EHol]. Results for the case $p = 4$ can also be found in Alon, Erdős, Holzman and Krivelevich’s paper [@AEHK]. There are currently no known results for $p \geqslant 5$. Proof of Theorem \[Thm1\] ========================= Before proceeding, we present a number of required preliminaries. Let $[m]$ denote the set of integers $\{1,2,\ldots,m\}$. If $\mathcal{A}$ is a family of subsets of $[m]$ let $\mathcal{A}_{k} = \{ A \in \mathcal{A}: \|A\| = k \}$. We say $\mathcal{A}$ is an antichain if it has the property that for all distinct pairs $A,B \in \mathcal{A}$ we have $A \nsubseteq B$. The LYM-inequality, due to Lubell [@LYM-L], Yamamoto [@LYM-Y] and Meshalkin [@LYM-M], tells us that if $\mathcal{A}$ is an antichain then $$\sum_{k = 0}^{m}\frac{\|\mathcal{A}_{k}\|}{\binom{m}{k}} \leqslant 1. \nonumber$$ We will use the following consequence of the LYM-inequality: \[PropLYM\] If $\mathcal{A}$ is an antichain of subsets of $[m]$ such that $\|A\| \leqslant s$ for all $A \in \mathcal{A}$ then $\|\mathcal{A}\| \leqslant m^{s}$. The LYM-inequality tells us that if $m \leqslant 2s$ then $ \|\mathcal{A}\| \leqslant \binom{m}{\lfloor{\frac{m}{2}}\rfloor}$, while if $m \geqslant 2s$ then $\|\mathcal{A}\| \leqslant \binom{m}{s}$. In both cases, $\|\mathcal{A}\| \leqslant m^{s}$. We will also use the following notation: For a graph $G$ and a vertex $v \in V(G)$, let $N(v)$ be the set of vertices in $G$ that are adjacent to $v$. For $X \subseteq V(G)$ let $N_{X}(v) = N(v) \cap X$, let $d_{X}(v) = \|N_{X}(v)\|$ and let $e(X)$ be the number of edges in the graph $G[X]$. For another set $Y \subseteq V(G)$ that is disjoint from $X$, let $e(X,Y)$ be the number of edges with one vertex in $X$ and one vertex in $Y$. Let $G$ be a $K_{p}$-saturated graph on vertex set $V$ with $\|V\| = n$ and $\delta(G) \geqslant t$. Let $R$ be a subset of $V$, set $R_{0} = R$ and for $i \geqslant 1$ let $$\begin{aligned} R_{i} &=& R_{i-1} \cup \{ v \in V : d_{R_{i-1}}(v) \geqslant t \}.\nonumber\end{aligned}$$ As $G$ is finite, there exists $m$ such that $R_{m} = R_{m+1}$. Let $\overline{R} = R_{m}$. Any vertex $x \in R_{i} \setminus R_{i-1}$ sends at least $t$ edges to $R_{i-1}$ and so $e(\overline{R}) \geqslant t(\|\overline{R}\| - \|R\|)$. Let $Y = V \setminus \overline{R}$, and for a vertex $v \in V$ define $$\begin{aligned} w(v) &=& d_{\overline{R}}(v) + \frac{1}{2}d_{Y}(v). \nonumber\end{aligned}$$ We call $w(v)$ the weight of $v$. We split $Y$ into good vertices and bad vertices - a vertex $v \in Y$ is good if it has weight at least $t$, otherwise it is bad. Our aim will be to show there is some constant $c_{1} = c_{1}(t)$ such that we can pick some $R$ with $\|R\| \leqslant c_{1}$ that leads to $Y$ having no bad vertices. If so, then $$\begin{aligned} e(G) &=& e(\overline{R}) + e(\overline{R},Y) + e(Y) \nonumber \\ & \geqslant & t(\|\overline{R}\|-\|R\|) + \sum_{y \in Y} w(y) \nonumber \\ & \geqslant & t(\|\overline{R}\| - c_{1}) + t\|Y\| \nonumber \\ & = & t(n - c_{1}),\nonumber\end{aligned}$$ as required. To achieve this we would like to show that if any set $R \subseteq V$ does lead to $Y$ having bad vertices then we can move a small number of vertices into $R$ so that the remaining bad vertices in $Y$ have strictly larger weight. If so, we can start with some small initial set of vertices $R$ and keep moving small numbers of vertices into it until $Y$ has no bad vertices. This idea of moving vertices into R fits naturally with our set up so far. Indeed, suppose that $R'$ is a set of vertices with $R \subseteq R'$. If we define $\overline{R'}$, $Y'$ and $w'$ in the same way we defined $\overline{R}$, $Y$ and $w$ for our original set $R$ then $\overline{R} \subseteq \overline{R'}$ and $Y \supseteq Y'$. As a result, each $x \in N(v)$ contributes at least as much to $w'(v)$ as it contributes to $w(v)$ and so $w'(v) \geqslant w(v)$ for all $v \in V$. Therefore the set of bad vertices in $Y'$ is a subset of the set of bad vertices in $Y$. It turns out that dealing with $w(v)$ directly is difficult and so we introduce a control function $$\begin{aligned} l(v) &=& d_{R}(v) + \frac{1}{2}d_{\overline{R}\setminus R}(v)+ \frac{1}{2t}\sum_{y \in N_{Y}(v)} d_{R}(y). \nonumber\end{aligned}$$ for all $v \in V$. Note that $ d_{R}(y) \leqslant t-1$ for all $y \in Y$ and so $l(v) \leqslant w(v)$ for all $v \in V$. As before, for a new set of vertices $R'$ we define $l'(v)$ in the same way we defined $l(v)$ for $R$. Moreover, if $R \subseteq R'$ we have that each $x \in N(v)$ contributes at least as much to $l'(v)$ as it contributes to $l(v)$ and so $l'(v) \geqslant l(v)$. We will use our control function to show that if some set $R \subseteq V$ does lead to $Y$ having some bad vertices then we can replace $R$ by some set $R'$ such that - $R \subseteq R'$, - $\|R'\| \leqslant \|R\| + t\|R\|^{t-1}$, - $l'(v) \geqslant l(v) + \frac{1}{2t}$ for all bad $v \in Y'$. This will be to sufficient to prove our theorem as we can start by taking $R = \{v\}$ for any vertex $v \in V$ and then repeatedly replace $R$ with $R'$ until the resulting $Y$ has no bad vertices. This will require at most $2t^{2}$ replacements of $R$ with $R'$ because after $2t^{2}$ replacements we have that any bad vertex $v \in Y$ would have $w(v) \geqslant l(v) \geqslant t$ which is not possible. Each time we replace $R$ with $R'$ we have $\|R'\| \leqslant \|R\| + t\|R\|^{t-1}$ and so, calling the final set $R^{}$, we have $\|R^{}\| \leqslant c_{1}(t)$ for some function $c_{1}(t)$. We now describe how to find a suitable set $R'$ given $R$. Suppose for some $R$ that the set of bad vertices in $Y$ is non-empty. Let $\mathcal{C} = \{ C_{1}, \ldots, C_{k} \}$ be the set of maximal elements (with respect to set inclusion) of $$\begin{aligned} \{ N_{R}(y): y \in Y \text{ is a bad vertex} \}. \nonumber\end{aligned}$$ Note that $\mathcal{C}$ is an antichain and each $C_{i}$ has at most $t-1$ elements. Thus, by Proposition \[PropLYM\], $k \leqslant \|R\|^{t-1}$. For each $C_{i}$ pick some representative $y_{i} \in Y$ such that $ C_{i} = N_{R}(y_{i}) $. As $y_{i} \in Y$, we have that $d_{\overline{R}}(y_{i}) < t$. Therefore, as $d(y_{i}) \geqslant t$, we can pick some $x_{i} \in Y$ such that $y_{i}$ and $x_{i}$ are adjacent.\ Let $$\begin{aligned} R' = R \cup \bigcup \limits_{i = 1}^{k} ( x_{i} \cup N_{\overline{R}}(x_{i})). \nonumber\end{aligned}$$ Clearly $R \subseteq R'$. As each $x_{i} \in Y$ we have $d_{\overline{R}}(x_{i}) \leqslant t-1$ and so $\|R'\| \leqslant \|R\| + tk \leqslant \|R\| + t\|R\|^{t-1}$. It remains to check that $l'(y) \geqslant l(y) + 1/2t$ for all bad $y \in Y' \subseteq Y$. For a bad vertex $y \in Y' \subseteq Y$ let $C_{i} \in \mathcal{C}$ be such that $N_{R}(y) \subseteq C_{i}$. Recall that every $v \in N_{Y}(y)$ contributes at least as much to $l'(y)$ as it did to $l(y)$. We have two cases to deal with depending on whether or not $y$ is adjacent to $x_{i}$. If $y$ is not adjacent to $x_{i}$ then there are a few further sub cases to deal with. If $y$ is adjacent to $x_{i}$ then $l'(y) > l(y) + \frac{1}{2}$ as $x_{i}$ now contributes 1 to $l'(y)$ (due to the fact that $x_{i} \in R'$) while it contributed at most $(t-1)/(2t)$ to $l(y)$. If $y$ is not adjacent to $x_{i}$ then there exists some clique $Z \subseteq V$ of order $p-2$ such that adding an edge between $y$ and $x_{i}$ turns $Z \cup \{ x_{i}, y \}$ into a copy of $K_{p}$. Recalling that $N_{R}(y)\subseteq N_{R}(y_{i})$, we note that $Z \nsubseteq R$ as otherwise $Z \cup \{ x_{i}, y_{i} \}$ would be an example of a copy of $K_{p}$ in $G$. Thus there exists some $z \in Z \setminus R$ such that $z$ is adjacent to $x_{i}$ and $y$. If $z \in \overline{R} \setminus R$ then $z \in R'$ and so $l'(y) \geqslant l(y) + \frac{1}{2}$ as $z$ contributed $\frac{1}{2}$ to $l(y)$ while it contributes 1 to $l'(y)$. If $z \in Y$ then its contribution to $l(y)$ is $d_{R}(z)/2t \leqslant (t-1)/2t$ and we have either $z \in \overline{R'}$ or $z \in Y'$. If $z \in \overline{R'}$ then its contribution to $l'(y)$ is at least $\frac{1}{2}$, while if $z \in Y'$ then its contribution is $d_{R'}(z)/(2t) \geqslant (d_{R}(z) + 1)/(2t)$ to $l'(y)$. In either case, $l'(y) \geqslant l(y) + \frac{1}{2t}$. From a more quantitative perspective, the proof of Theorem \[Thm1\] shows that $$\text{sat}_{t}(n,p) \geqslant tn - t(t+1)^{(t^{(2t^{2})})}. \nonumber$$ Indeed, take an initial set $R$ to be any $t+1$ vertices in $V$ and repeatedly replace $R$ with $R'$ until $Y$ has no bad vertices, as described above. Note that $\|R\| \geqslant t + 1$ implies that $\|R\| + t\|R\|^{t-1} \leqslant \|R\|^{t}$ and so for each replacement of $R$ with $R'$ we have $\|R'\| \leqslant \|R\|^{t}$. Thus after at most $2t^{2}$ replacements we are left with a set $R^{}$ with $\|R^{}\| \leqslant (t+1)^{(t^{(2t^{2})})}$ that leads to $Y$ having no bad vertices. It is possible to improve this lower bound on sat$_{t}(n,p)$ by choosing a more careful initial set $R$ and performing more attentive analysis. However the improvements are minimal with respect to the tower of exponentials in $t$. We do not believe that this lower bound for sat$_{t}(n,p)$ is close to the true value of sat$_{t}(n,p)$ and in Section 3 we give a conjecture giving what we believe to be its asymptotic behaviour. Let $n_{1}(t,p) = \max \{c(t+1),2t - (p-3) \}$ where $c(t)$ is as given in the statement of Theorem \[Thm1\]. For $t \geqslant p-2$, the graph obtained from fully connecting a clique of size $p-3$ to the complete bipartite graph $K_{t-(p-3),n-t}$ shows that sat$_{t}(n,p) < tn$. Theorem \[Thm1\] shows that if $n \geqslant c(t+1)$ then sat$_{t+1}(n,p) \geqslant tn$. Thus for $n \geqslant n_{1}(t,p)$ all graphs $G \in$ Sat$_{t}(n,p)$ have $\delta(G)=t$ and so sat$_{t}(n,p) = $ sat$'_{t}(n,p)$. Duplicating a vertex of degree $t$ in such a graph gives a $K_{p}$-saturated graphs on $n+1$ vertices with minimum degree $t$ and $sat_{t}(n,p) + t$ edges. Thus for $n \geqslant n_{1}(t,p)$ the sequence $tn - sat_{t}(n,p)$ is increasing in $n$ but bounded above by $c(t)$ and so is eventually constant. Therefore, for some $n_{0}(t,p)$, we may define $c(t,p)$ to be the constant such that sat$_{t}(n,p) = tn - c(t,p)$ for $n \geqslant n_{0}(t,p)$. Constructing $K_{p}$-saturated graphs ===================================== Let $\{ X_{1}, \ldots, X_{r} \} = \{ X \subseteq [t] : 1 \in X, \|X\| = \lfloor t/2 \rfloor \}$ and for each $i$ let $Y_{i} = \{k \in [t] : k \notin X_{i} \}$. We construct a graph $G$ on $n$ vertices as follows: $G$ has vertex set $V$ comprising of vertex classes $H, V_{1},\ldots,V_{r}, W_{1},\ldots,W_{r}, C$ where - $H = \{h_{1}, \ldots, h_{t} \}$, - each $V_{i}$ has $\lfloor t/2 \rfloor$ vertices, - each $W_{i}$ has $\lceil t/2 \rceil$ vertices, - $C$ has the remaining $n - t - \lfloor t/2 \rfloor \binom{t}{\lfloor t/2 \rfloor}$ vertices. The edges of $G$ are as follows: - $C$ is fully connected to $H$, - each $V_{i}$ is fully connected to the set $\{ h_{k} : k \in X_{i} \}$, - each $W_{i}$ is fully connected to the set $\{ h_{k} : k \in Y_{i} \}$, - each $V_{i}$ is fully connected to $W_{i}$. We claim that this graph has minimum degree $t$ and is $K_{3}$-saturated. First we check that $\delta(G) = t$. Each vertex $c \in C$, $v \in V_{i}$ and $w \in W_{i}$ has degree $t$. As $t \geqslant 4$, each vertex $h_{k}$ (other than $h_{1}$) is fully connected to at least one $V_{i}$ and at least one $W_{i}$ and so has degree at least $t$. We also have that $h_{1}$ meets at least 3 of the $V_{i}$ and so has degree at least $3 \lfloor t/2 \rfloor > t$. Therefore $\delta(G) = t$. Suppose there is a triangle in $G$. No vertex class $C$, $H$, $V_{i}$ or $W_{i}$ contains an edge and so all vertices of the triangle must lie in different vertex classes. This means that at least one vertex of the triangle is in some $V_{i}$ or some $W_{i}$. Each $V_{i}$ only sends edges to $H$ and $W_{i}$, while each $W_{i}$ only sends edges to $H$ and $V_{i}$. Thus, for some $i$ and some $k$, one vertex of the triangle is in $V_{i}$, another vertex of the triangle is in $W_{i}$ and the final vertex is $h_{k}$. However, as $X_{i} \cap Y_{i} = \varnothing$, no $h_{k}$ is adjacent to both $V_{i}$ and $W_{i}$. Thus no such triangle can exist. Finally we need to check that adding any new edge to $G$ creates a triangle. We first check the case of adding an edge inside any vertex class. $C$ is fully connected to $H$ and so adding any edge to $C$ creates a triangle. As $t \geqslant 4$, we have that for each pair $k,l \in [t]$, there is some $Y_{i}$ such that $k,l \in Y_{i}$ unless $k$ or $l$ is $1$, in which case there is some $X_{i}$ such that $k,l \in X_{i}$. Thus adding an edge in $H$ from $h_{k}$ to $h_{l}$ creates a triangle with either a vertex in some $V_{i}$ or a vertex in some $W_{i}$. For each $i$, adding an edge in $V_{i}$ or $W_{i}$ creates a triangle with some vertex in $H$. Now we check that adding an edge between vertex classes create a triangle. $C$ is fully connected to $H$ so we cannot add any edges between these two classes. Adding an edge between $C$ and some $V_{i}$ or $W_{i}$ creates a triangle with some vertex in $H$. If we add an edge between some $h_{k}$ and some vertex in $V_{i}$ then we must have that $k \notin X_{i}$ and so $k \in Y_{i}$. This means we create a triangle with any vertex in $W_{i}$. By the same reasoning, adding any edge between $H$ and any $W_{i}$ creates a triangle with any vertex in $X_{i}$. As each $V_{i}$ is fully connected to $W_{i}$, we cannot add any edges between them. For each pair $i,j \in [r]$ with $i \neq j$ we have that $X_{i} \cap X_{j} \neq \varnothing$, $X_{i} \cap Y_{j} \neq \varnothing$ and $Y_{i} \cap Y_{j} \neq \varnothing$. Thus, for example, adding an edge from $V_{i}$ to $V_{j}$ creates a triangle with $h_{k}$ for some $k \in X_{i} \cap X_{j}$. Similarly adding an edge between $V_{i}$ and $W_{j}$ for $i \neq j$ or adding an edge between $W_{i}$ and $W_{j}$ creates a triangle with some vertex in $H$. Therefore $G$ is $K_{3}$-saturated. To count the edges of $G$ we note that each vertex of $C$ sends $t$ edges to $H$ and for each $i \in [r]$ the pair $V_{i}$ and $W_{i}$ sends $\lfloor t/2 \rfloor ^{2} + \lceil t/2 \rceil ^{2} = \lceil t^{2}/2 \rceil$ edges to $H$. Moreover, each pair $V_{i}$ and $W_{i}$ has $\lfloor t/2 \rfloor \lceil t/2 \rceil = \lfloor t^{2}/4 \rfloor$ edges between them. Thus, recalling that $\|C\| = n - t - \lfloor t/2 \rfloor \binom{t}{\lfloor t/2 \rfloor}$ and $r = \binom{t-1}{\lfloor t/2 \rfloor-1}$, we have $$\begin{aligned} e(G) &=& t\|C\| + r( \lceil t^{2}/2 \rceil + \lfloor t^{2}/4 \rfloor) \nonumber \\ &=& \begin{cases} tn - t^{2} (1 + \binom{t}{t/2}/8) \text{ \phantom{abcdefgi}for t even} \\ tn - t^{2} -\frac{(t-1)(t^{2}-1)}{8t}\binom{t}{(t-1)/2} \text{ for t odd} \end{cases} \nonumber\end{aligned}$$ as required. Recall that we defined $c(t,p)$ to be the constant such that sat$_{t}(n,p) = tn - c(t,p)$ for $n$ sufficiently large. Theorem \[Thm2\] shows that $c(t,3) \geqslant C2^{t}t^{3/2}$ for some constant $C$. We conjecture that this upper bound is asymptotically best, that is, $c(t,3) = O(2^{t}t^{3/2})$. We can use the construction in Theorem \[Thm2\] to construct $K_{p}$-saturated graphs with minimum degree $t$. For a graph $G$, let $G^{}$ be the graph obtained by adding a new vertex to $G$ and fully connecting it to all other vertices of $G$. If $G$ is a $K_{p}-$saturated graph with minimum degree at least $t$, then $G^{}$ is a $K_{p+1}-$saturated graph with minimum degree at least $t+1$. This shows that $c(t+1,p+1) \geqslant c(t,p) + t + 1$. As Theorem \[Thm2\] shows that $c(t,3) \geqslant C2^{t}t^{3/2}$ for some constant $C$, we have that $c(t,p) \geqslant C2^{t'}t'^{3/2}+\binom{t+1}{2} - \binom{t'+1}{2} $ where $t' = t - (p-3)$. We can also consider the above construction from the other direction. Suppose $G$ is a $K_{p}-$saturated graph with minimum degree at least $t$. If $G$ has a conical vertex, a vertex connected to all other vertices, then removing this vertex from $G$ leaves a $K_{p-1}-$saturated graph with minimum degree at least $t-1$. Hajnal [@Haj] showed that if $G$ is a $K_{p}$-saturated graph without a conical vertex then $\delta(G) \geqslant 2(p-2)$. Corollary \[CoroThm1\] shows that for any $t$ and $p$ and for $n$ sufficiently large, if $G \in$ Sat$_{t}(n,p)$ then $\delta(G) = t$. Thus, if $t < 2(p-2)$, these graphs must have a conical vertex and so are of the form $G^{}$ where $G \in$ Sat$_{t-1}(n-1,p-1)$. This leads us to the question: For which values of $n,t$ and $p$ do we have that all graphs in Sat$_{t}(n,p)$ are of the form $G^{}$ for some $G \in$ Sat$_{t-1}(n-1,p-1)$? Recall that the two properties that make a graph $H$-saturated are that the graph is $H$-free and that adding a new edge to the graph creates a copy of $H$. Suppose we are interested in graphs that only have the second property of $H$-saturation, that is, adding a new edge to the graph creates a new copy of $H$. We call such a graph $H$-semi-saturated and define $$\begin{aligned} \text{s-sat}_{t}(n,p) &=& \min \{ e(G): \|V(G)\| = n, G \text{ is } K_{p} \text{-semi-saturated}, \delta(G) \geqslant t \}. \nonumber\end{aligned}$$ It turns out that constructing semi-saturated graphs with few edges is much easier than constructing saturated graphs. Any graph that is $K_{p}$-saturated is also $K_{p}$-semi-saturated and so s-sat$_{t}(n,p) \leqslant$ sat$_{t}(n,p)$. Erdős, Hajnal and Moon’s proof [@EHM] that sat$_{p-2}(n,p) = n(p-2) - \binom{p-1}{2}$ began by first showing that s-sat$_{p-2}(n,p) = n(p-2) - \binom{p-1}{2}$ and so we have s-sat$_{p-2}(n,p) =$ sat$_{p-2}(n,p)$. Naturally one may ask if s-sat$_{t}(n,p) =$ sat$_{t}(n,p)$ for all $t,n$ and $p$, however it turns out that for $t > p-2$ this is not the case. Theorem \[Thm3\] shows that s-sat$_{t}(n,p) = \frac{t+p-2}{2}n - O(1)$ for fixed $t$ and $p$: \[Thm3\] For $n\geqslant 4t$ and $t \geqslant p-2$, we have $$\begin{aligned} \frac{t''(n-t-1)}{2} +t - \binom{p-2}{2}\leqslant \text{s-sat}_{t}(n,p) \leqslant \Big\lceil \frac{t''(n-(p-2))}{2} \Big\rceil + \binom{p-2}{2} \nonumber\end{aligned}$$ where $t'' = t + p -2$. For each pair of positive integers $m, s$ with $m>s$, let $F(m,s)$ be a graph on $m$ vertices with $\big\lceil \frac{ms}{2} \big\rceil$ edges and minimum degree $s$; it is easy to check that such graphs exist. The graph formed by taking a clique of size $p-2$ and fully connecting it to the graph $F(n-(p-2),t-(p-2))$ is a $K_{p}$-semi-saturated graph on $n$ vertices with minimum degree $t$, and thus proves the upper bound. Duffus and Hanson [@DuffusHanson] proved that if $G$ is a $K_{p}$-saturated graph on $n$ vertices then $$\label{eq1} e(G) \geqslant \frac{(n-\delta-1)(\delta+p-2)}{2} +\delta - \binom{p-2}{2},$$ where $\delta = \delta(G)$. However their proof in fact holds true for $K_{p}$-semi-saturated graphs, and so we use their result to prove ours. Let $G$ be a $K_{p}$-semi-saturated graph on $n$ vertices with $\delta(G) = t + c$ and s-sat$_{t}(n,p)$ edges, where $c$ is a non-negative integer. Our upper bound shows s-sat$_{t}(n,p) < \frac{n(t+p-2)}{2}$ and so $c < p - 2$. Thus, as $n \geqslant 4t > c + 2t + p -3$, we have that (\[eq1\]) is minimised with respect to $c$ when $c = 0$. Substituting $t$ for $\delta$ in (\[eq1\]) gives the required result. Saturated Hypergraphs ===================== We now turn our attention to $r$-uniform hypergraphs, which we also refer to as $r$-graphs. For a set $S$ of distinct vertices of an $r$-graph $G$, we define its degree, $d(S)$, to be the number of edges of $G$ that contain $S$. We define the minimum $s$-degree of $G$ to be $$\begin{aligned} \delta_{s}(G) = \min \{ d(S) : S \subseteq V(G), \|S\| = s \}. \nonumber\end{aligned}$$ We say $G$ is $K_{p}^{r}$-saturated if it contains no copy of $K_{p}^{r}$ as a subgraph, but the addition of any new edge to $G$ creates one. Bollobás [@Boll2] proved that any $K_{p}^{r}$-saturated $r$-graph has at least $\binom{n}{r} - \binom{n-p + r}{r}$ edges. Moreover, he proved that the unique $r$-graph achieving equality is formed by picking a set of $p - r$ vertices and having the edges of $G$ consist of all edges that contain at least one of these $p - r$ points. These extremal $r$-graphs all have $\delta_{r-1}(G) = p - r$, and so it is natural to ask, how few edges can a $K_{p}^{r}$-saturated $r$-graph on $n$ vertices with $\delta_{r}(G) \geqslant t$ have for $t \geqslant p - r$? We make the following conjecture on the behaviour of such $r$-graphs: \[Conj2\] Let $t,p$ and $r$ be integers with $t \geqslant p - r \geqslant 1$. If $G$ is a $K_{p}^{r}$-saturated $r$-graph on $n$ vertices with $\delta_{r-1}(G) \geqslant t$ then $$\begin{aligned} e(G) \geqslant \frac{tn^{r-1}}{(r-1)!} + O(n^{r-2}). \nonumber\end{aligned}$$ When $r = 2$ the conjecture is given by Theorem \[Thm1\]. The following theorem shows that if Conjecture \[Conj2\] is true then it is the asymptotically best result that one could hope for: \[Thm4\] Let $t,p$ and $r$ be integers with $t \geqslant p - r \geqslant 1$. For all $n \geqslant rt - (r-1)(p - (r + 1))$, there exists a $K_{p}^{r}$-saturated $r$-graph $G$ on $n$ vertices with $\delta_{r-1}(G) = t$ and $$\begin{aligned} e(G) = \frac{tn^{r-1}}{(r-1)!} + O(n^{r-2}). \nonumber\end{aligned}$$ We remark that looking at minimum $(r-1)$-degrees of $r$-graphs seems to be the natural choice to work with, due to the fact that the optimal $K_{r+1}^{r}$-saturated $r$-graphs have $\delta_{r-1}(G) = 1$. However, one could just as easily consider minimum $s$-degrees for any value of $s$. The following construction is based on a construction of Sidorenko [@Sido] which is used to prove lower bounds in the hypergraph Turán problem. Our dialogue follows that of Section 9 from Keevash’s survery [@Keevash] on hypergraph Turán problems. We begin by dealing with the case $p = r+1$ and will use this as the starting point for all larger values of $p$. We construct an $r$-graph $G$ with vertex set $V$, where $\|V\| = n$. We divide $V$ into vertex classes $A_{1}, \ldots, A_{r}$ where $\|A_{1}\| = n - t(r-1)$ and $\|A_{i}\| = t$ for $i \geqslant 2$. The edges of $G$ are the subsets $B \subseteq V$ of size $r$ that do not have the following property: there exists $j$ such that $$\label{eq2} \sum_{i =0 }^{s-1} \|B \cap A_{j+i}\| \geqslant s+1$$ for $s = 1, \ldots, r-1$. (We set $A_{i} = A_{i - r}$ if $i > r$ and $A_{i} = A_{i + r}$ if $i < 1$). We claim that this graph is $K_{r}^{r+1}$-free and has $\delta_{r-1}(G) = t$. To help illustrate why both of these are true, we describe the lorry driver puzzle: Suppose there is a circular road with $r$ cities on it and $r+1$ units of fuel distributed between the cities in integer amounts. A lorry driver travels clockwise around the road, collecting fuel at each city they visit but using $\frac{r+1}{r}$ units of fuel to travel from one city to the next. Show that it is always possible for the lorry driver to start at one of the cities with an empty tank and make a complete journey around the road, before returning to their starting point. The solution to the puzzle is as follows; suppose instead a new lorry driver starts at any city with enough units of fuel to make a complete circuit. If we monitor their fuel levels as they travel, the point at which their fuel levels are lowest is where the original lorry driver should start their journey. Their fuel levels will never drop below $0$, and so they can make a complete circuit. In the context of our hypergraph $G$, we look at the vertex classes $A_{i}$ as the cities arranged clockwise in the order $A_{1}, A_{2}, \ldots, A_{r}$. Certain vertices will represent units of fuel. For example, as $\big\lceil s \frac{r+1}{r} \big\rceil = s+1$ for $s \leqslant r-1$, property (\[eq2\]) can now be read as: $B$ is not an edge of $G$ if there exists $j$ such that if one unit of fuel is placed at each $v \in B$, then the lorry driver can complete a successful journey that starts at $A_{j}$ and finishes at $A_{j-1}$. It is now easy to see that our first claim, that $G$ is $K_{r}^{r+1}$-free, is true. Indeed, if $R$ is a set of $r+1$ vertices of $G$ and a unit of fuel is placed at each $v \in R$ then, by the solution to the lorry driver puzzle, we know there is some $j$ such that the lorry driver can start at $A_{j}$ and make a complete journey around the road, returning back to $A_{j}$. As $r \geqslant (r-1) \frac{r+1}{r}$, the lorry driver can advance distance $r-1$ with the first $r$ units of fuel they collect. Therefore the first $r$ vertices of $R$ that the lorry driver meets have property (\[eq2\]) and so are not an edge in $G$. Thus no such $R$ forms a $K_{r}^{r+1}$ and so $G$ is $K_{r}^{r+1}$-free. To prove that $\delta_{r-1}(G) = t$ we will need the following lemma: \[Lemma2\] If $R$ is a set of $r-1$ vertices of $G$, there is some $i$ such that $R \cap A_{i} = \varnothing$ and $R \cup a$ is an edge of $G$ for all $a \in A_{i}$. The proof of Lemma \[Lemma2\] can be found at the end of this section. Lemma \[Lemma2\] shows that $\delta_{r-1}(G) \geqslant t$. On the other hand, if $R$ is a set of $r-1$ vertices in $A_{1}$, the only edges that contain $R$ are precisely those of the form $\{R,v \}$ where $v$ is any vertex in $A_{r}$. Therefore $\delta_{r-1}(G) \leqslant t$ and so in fact $\delta_{r-1}(G) = t$. While our graph $G$ is $K_{r}^{r+1}$-free, it may not be $K_{r}^{r+1}$-saturated. We form a new graph $G'$ by adding as many edges to $G$ as we can that do not result in a $K_{r}^{r+1}$ being formed. $G'$ will be a $K_{r}^{r+1}$-saturated $r$-graph with $\delta_{r-1}(G') \geqslant \delta_{r-1}(G) = t$. Note that in the process of adding edges to $G$ to form $G'$, no edge of the form $\{R,v\}$, where $R \subseteq A_{1}$ and $v \in V \setminus (A_{r} \cup R)$, can be added to $G$. Indeed, for any such $R$ and $v$, if the edge $\{R,v\}$ is added to $G$ then the set $\{R,v,w\}$ forms a $K_{r}^{r+1}$, where $w$ is any vertex in $A_{r}$. Thus $\delta_{r-1}(G') = t$ and all but $O(n^{r-2})$ of the edges in $G'$ have $r-1$ vertices in $A_{1}$ and a single vertex in $A_{r}$. Therefore $$\begin{aligned} e(G') &=& t\binom{n-t(r-1)}{r-1} + O(n^{r-2}) \nonumber \\ &=& \frac{tn^{r-1}}{(r-1)!} + O(n^{r-2}). \nonumber\end{aligned}$$ To construct a $K_{p}^{r}$-saturated $r$-graph with $\delta_{r-1}(G) = t$ for any $p \geqslant r+1$, we begin by constructing a $K_{r+1}^{r}$-saturated $r$-graph $G$ on $n - p + r + 1$ vertices with $\delta_{r-1}(G) = t - p + r + 1$ as described above. Then add $p - (r+1)$ new vertices to $G$ as well as every edge that contains at least one of these new vertices. This gives an $r$-graph $G^{}$ that is $K_{p}^{r}$-saturated with $\delta_{r-1}(G^{}) = t$ and still has $$e(G^{}) = \frac{tn^{r-1}}{(r-1)!} + O(n^{r-2}). \nonumber$$ In the context of the lorry driver puzzle, our aim is to show that for any set $R$ of $r-1$ vertices, there exists some $i$ such that $A_{i} \cap R = \varnothing$ and if we place a unit of fuel at each vertex of $R$ and one more at $A_{i}$, then there is no $j$ such that a journey can be made from $A_{j}$ to $A_{j-1}$. Suppose that for some set $R$ the lemma does not hold. That means for every $i$ such that $R \cap A_{i} = \varnothing$, there exists some $j$ such that if we place a unit of fuel at every vertex $v \in R$ and then one additional unit of fuel at $A_{i}$, the lorry driver can travel from $A_{j}$ to $A_{j-1}$. Thus, for each $i$ such that $R \cap A_{i} = \varnothing$, we may define $j(i)$ to be such that $A_{j(i)}$ is the furthest city from $A_{i}$ (in the clockwise direction) that such a journey could be made. Note that in order for the lorry driver to complete such a journey starting from $A_{j(i)}$, we must have that $R \cap A_{j(i)-1} = \varnothing$ and $i \neq j(i)-1$. Moreover, the cities $A_{j(i)}, A_{j(i) + 1}, \ldots, A_{i}$ contain at least $\|j(i), j(i) + 1, \ldots, i\|$ of the units of fuel of $R$ between them as the lorry driver can certainly make it to $A_{i}$ starting from $A_{j(i)}$ using just the fuel collected from $R$ along the way. We define a sequence $(i_{m},j_{m})_{m \geqslant 0}$ such that $R \cap A_{i_{m}} = \varnothing$ and $j_{m} = j(i_{m})$ as follows: Let $(i_{0},j_{0})$ be such that the distance travelled from $A_{j_{0}}$ to $A_{i_{0}}$ is maximised among all pairs $A_{j(i)}$ and $A_{i}$ with $R \cap A_{i} = \varnothing$. For $m \geqslant 0$, set $i_{m+1} = j_{m} - 1$ and $j_{m+1} = j(i_{m+1})$. Let $l$ be the smallest integer such that when travelling from $A_{i_{l+1}}$ to $A_{i_{l}-1}$, the lorry driver passes over $i_{0}$. In other words, $l$ is the smallest integer such that $i_{0} \in \{ i_{l+1}, i_{l+1} + 1, \ldots, i_{l}-1 \}$. We claim that $i_{l+1} = i_{0}$. Suppose that $i_{l+1} \neq i_{0}$. If so, then when travelling from $A_{j_{l}}$ to $A_{i_{l}}$ the lorry driver starts at or passes over $A_{i_{0}}$. As the lorry driver can travel from $A_{j_{l}}$ to $A_{j_{l}-1} = A_{i_{l+1}}$ using the fuel collected from $R$ and one additional unit of fuel at $A_{i_{l}}$, they could certainly complete the same journey if the additional unit of fuel was placed at $A_{i_{0}}$ instead of $A_{i_{l}}$. Therefore, by the definition of $j_{0} = j(i_{0})$, we must have that $A_{j_{l}}$ is no further from $A_{i_{0}}$ than $A_{j_{0}}$ is, i.e., $j_{0} \in \{j_{l}, j_{l} + 1, \ldots, i_{0} - 1\}$. However, this contradicts our choice of $i_{0}$ and $j_{0}$ as the distance travelled from $A_{j_{l}}$ to $A_{i_{l}}$ is strictly greater than the distance travelled from $A_{j_{0}}$ to $A_{i_{0}}$. Therefore we did have $i_{l+1} = i_{0}$. For each $m \in \{0,1, \ldots, l\}$, let $\mathcal{P}_{m} = \{ A_{j_{m}},A_{j_{m} + 1}, \ldots, A_{i_{m}}\}$. As $i_{l+1} = i_{0}$, we have that the sets $\mathcal{P}_{0}, \mathcal{P}_{1}, \ldots, \mathcal{P}_{l}$ partition the circle of cities into disjoint intervals. We also have that each $\mathcal{P}_{m}$ contains at least $\|\mathcal{P}_{m}\|$ units of fuel of $R$ in it. However, summing this over $m = 0, 1, \ldots, l$ tells us that $\|R\| \geqslant r$, which contradicts the fact that $\|R\| = r-1$. Thus the lemma holds for all sets $R$ of $r-1$ vertices of $G$. Acknowledgements ================ I would like to thank Robert Johnson for his valuable advice and for always pointing me in the right direction. This research was funded by an EPSRC doctoral studentship. [1]{} N. Alon, P. Erdős, R. Holzman, M. Krivelevich. (1996) On k-saturated graphs with restrictions on the degrees. J. Graph Theory, 23(1), $1-20$. B. Bollobás. (1965) On generalized graphs. Acta Math. Acad. Sci. Hungary, 16, $447-452$. D. A. Duffus, D. Hanson. (1986) Minimal $k$-saturated and color critical graphs of prescribed minimum degree. J. Graph Theory, 10(1), $55 - 67$. P. Erdős, A. Hajnal, J. W. Moon. (1964) A Problem In Graph Theory. Amer. Math. Monthly, 71, $1107 - 1110$. P. Erdős, R. Holzman. (1994) On maximal triangle-free graphs. J. Graph Theory 18(6)* $585-594$. J. Faudree, R. Faudree, J. Schmitt. (2011) A survey of minimum saturated graphs and hypergraphs. Electronic J. Combinatorics 18 DS19. Z. Füredi, Á. Seress. (1994) Maximal triangle-free graphs with restrictions on the degrees. J. Graph Theory, 18(1), $11-24$. R. L. Graham, M. Grőtschel, L. Lovász, editors. (1995) Handbook of combinatorics. Vol. 2. Elsevier Science B.V., Amsterdam, $1271$. A. Hajnal.(1965) A theorem on k-saturated graphs. Canad. J. Math. 17, $720-724$. P. Keevash. (2011) Hypergraph Turán problems. Surveys in Combinatorics 2011, $83-140$. D. Lubell. (1966) A short proof of Sperner’s theorem J. Combin. Theory, 1, $299$ L.D. Meshalkin. (1963) A generalization of Sperner’s theorem on the number of subsets of a finite set Theor. Probability Appl., 8, $203-204$. O. Pikhurko. (2004) Results and Open Problems on Minimum Saturated Graphs. Ars Combinatoria, 72, $111-127$. A. Sidorenko. (1981) Systems of sets that have the T-property. Moscow University Mathematics Bulletin, 36, $22-26$. E. Sperner. (1928) Ein Satz über Untermenger einer endlichen Menge. Mathematische Zeitschrift, 27, $544-548$. K. Yamamoto. (1954), Logarithmic order of free distributive lattices. J. Math. Soc. Japan, 6, $343-353$. [^1]: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
--- author: - 'Doron Gepner[^1][Research supported in part by NSF grant PHY-80-19754.]{}' date: December 1987 title: 'String Theory on Calabi–Yau Manifolds:' --- phyzzx mydef \#1[\#1\^]{} Recently, string theory on Calabi–Yau manifolds was constructed and was shown to be a fully consistent, space–time supersymmetric string theory. The physically interesting case is the case of three generations. Intriguingly, it appears at the present that there is a unique manifold which gives rise to three generations. We describe in this paper a full fledged string theory on this manifold in which the complete spectrum and all the Yukawa couplings can be computed exactly. String theory is a rarity among physical theories. For twenty years it has been developed without experimental input. Certainly, it is a beautifully consistent theory. Yet, it remained the theoretician playground. The purpose of this note is to make a step towards the confrontation of string theory with experimental physics as a candidate theory for the unification of all natural forces. We do this by exploring the physically relevant case, the case of three generations, in the vast framework of the recently discovered string theory on Calabi–Yau manifolds. String theories incorporate an elaborate and tightly woven structure with many consistency requirements. In particular, a superstring propagating in flat Minkowski space is a consistent theory only if this space is ten dimensional. The real world is however four dimensional. A possible solution to this problem is to consider a world which is a manifold $M\times K$, where $M$ is the usual four dimensional Minkowski space and $K$ is some tiny, ‘invisible’, manifold, an idea first suggested in the late twenties by Kaluza and Klein . The internal manifold $K$ would then give rise to the observable forces and the particular choice for the manifold has a profound influence on the physical predictions of the theory. To implement the Kaluza–Klein idea in string theory, one needs to study string propagation on the manifold $M\times K$, where $K$ is some curved internal manifold. The first study of string propagation in curved space was described in , where a closed bosonic string theory on a manifold $K$, which is a Lie group manifold, was considered. As a result, string theory on a group manifold was shown to be a fully consistent, full fledged string theory, obeying all the severe constrains that string theory should obey. The constraints of the existence of sensible vertex operators, which are in one to one correspondence with the physical spectrum, unitarity at the tree level and at the one loop level, were all shown to be obeyed. Moreover, as a result of this work, it became clear that a full fledged string theory can be constructed, along the same lines, from any conformal field theory in two dimensions. For phenomenological reasons, one would actually like to study the case where the string theory has space–time supersymmetry. This symmetry between fermions and bosons enables the elimination of tachyons and the potential resolution of questions like the vanishing of the cosmological constant and the hierarchy problem. The problem of the existence of supersymmetric string theory in curved space was open for a long time. Initial interest in the question came from the work of , in which the equations of classical ten dimensional supergravity were studied and were shown to have a solution, provided the manifold $K$ is a complex manifold of vanishing first Chern class (Calabi–Yau manifold). This gives initial indication that string theory on such a manifold might exist, since the same classical equations expresses the lowest order contribution to the conformal anomaly of the sigma model on the manifold , which describes string propagation on it. However, the tiniest conformal anomaly renders such a string theory inconsistent so one needs to study higher order contributions. Indeed, such a contribution in the four loop level was reported in , implying that the naive string theory on a Calabi Yau (CY) manifold is inconsistent. Arguments were suggested that it may be possible to modify the metric and many workers in the field believed that, at least for large radius, a nearby string theory exists. This is not the only problem. The existence of metrics of $SU(3)$ holonomy on Calabi Yau manifolds was conjectured by Calabi and proved by Yau . However, no metric of a compact Calabi Yau manifold is known explicitly. The writing of such a metric is a hard mathematical problem, which even for the simplest $K3$ surface has been open for decades. Consequently, the propagation of a classical Newtonian particle on such a manifold is untractable, since one cannot compute distances. Thus, it appears to be entirely hopeless to compute the physical predictions from the seemingly much harder problem of the propagation of a quantized string on such a manifold. In a recent series of papers the author has put forward the construction of string theory on Calabi Yau manifolds and have shown them to be fully consistent, space–time supersymmetric string theories, where all the physical predictions can be computed exactly. The new idea, which avoids all the aforementioned difficulties, is to proceed in two stages. At the first stage any possible geometrical interpretation was ignored and new space–time supersymmetric string theories were constructed from scratch , solving the stringent constraints that string theory must obey. At the second stage, by studying the massless spectrum of these new string theories and comparing it with the results of equivariant index theorems in particular Calabi Yau geometries, it was shown that these theories correspond to string propagation on Calabi–Yau manifolds. The logic behind this procedure is that since string theory is a rarely constrained system, it should be possible to recover any such theory by simply considering string theory and its constraints, per se. This proves the consistency and existence of CY string theories, as well as giving their actual construction. To carry out the first stage it was needed to understand how to construct space–time supersymmetric string theories. Previously, the only known method to get space–time supersymmetry in string theory was in the context of theories made entirely out of free fermions, along the lines of the GSO construction . In order to get supersymmetry one needs some projection. The problem is that modular invariance almost always prevents one from projecting out any fields in a general conformal field theory. It is a kind of completeness condition. Thus it appears that supersymmetry cannot be achieved in general conformal field theory. Surprisingly, I was able to show that there is a very general supersymmetry projection, which works in consistency with modular invariance and can be implemented in any theory with $N=2$ superconformal invariance, leading to space–time supersymmetry. The supersymmetry charge is given by $Q=\exp(i\phi)$, where $J=\partial_z\phi$ is the $U(1)$ current algebra part of the $N=2$ superconformal algebra. In addition, one demands that the total $U(1)$ charge, in both the compactified and uncompactified dimensions, should be an integer. The crucial point is that under the modular transformation $\tau\rarrow -{1\over\tau}$, these two conditions are exchanged and thus can be implemented simultaneously in any $N=2$ superconformal field theory, without ruining modular invariance[^2][The old GSO projection is actually a particular case of the new G projection. The fact that the GSO projection can be written in terms of an $N=2$ algebra is known for some time, and was used in orbifold calculations . Preliminary observations that the $G$ projection might be possible were reported in .]{}. This new projection, the $G$ projection, then leads to space–time supersymmetry in any $N=2$ superconformal field theory. The next issue which needed to be addressed is how the left and right movers are correlated in the string theory. Again, the constraint of modular invariance is exceedingly restrictive. In general conformal field theory, in order to be able to achieve modular invariance, it is almost always required that the left movers and the right movers to be identical ‘half theories’. In a heterotic–like string theory, where the left movers are fermionic and the right movers are bosonic, this presents a formidable problem, since the left and right movers, by definition, are completely different. This problem is solved by a simple and completely general map which takes any superstring theory into a heterotic–like string theory . A general superstring theory in $d+2$ dimensions contains a $d$ dimensional flat superstring, described in the light–cone gauge by $d$ world sheet free fermions and $d$ world–sheet free bosons. The world–sheet fermions realize a level one $SO(d)$ current algebra, both in the right and in the left moving sectors. The map that takes a superstring into a heterotic string is then simply to replace the $SO(d)$ representations by $SO(24+d)$ or $E_8\times SO(8+d)$ ones in the right moving sector, where one exchanges the vector by the singlet and changes the sign of the two spinors. The effect of this is to exchange the fermions that carry a space–time index with fermions that carry an internal index. This map preserves modular invariance and spin–statistics and thus sends any consistent superstring theory in $d+2$ dimensions into a consistent heterotic string in $d+2$ dimensions. In the case of space–time supersymmetric superstring compactification to four dimensions the resulting gauge groups are either $SO(26)$ or $E_8\times E_6$. The $E_6$ is obtained by combining $SO(10)$ with the superconformal $U(1)$. These gauge groups are the same as the ones obtained in the supergravity models on CY manifolds . In the following was demonstrated: [String theory on a Calabi–Yau manifold exists, it is a full fledged, fully consistent string theory, which is space–time supersymmetric. All string theories on a CY manifold have the structure described above. In addition, any string theory which has this structure is a string theory on some Calabi–Yau manifold.]{} More generally, this structure corresponds to a string compactification to $10-2k$ dimensions, with propagation on a manifold of $SU(k)$ holonomy for the cases $k=1,2,3$. Presently, the proof of the above statement is incomplete. A partial proof and additional conclusive evidence are presented in . Our main tool in exploring this structure are the minimal $N=2$ superconformal field theories. The reason is that these are the only presently known $N=2$ superconformal field theories. [^3][Except of course for the trivial realization in terms of free fermions. This realization corresponds to the case of flat tori and their discrete quotients (orbifolds) .]{} The minimal models have the trace anomaly $$c={3k\over k+2}\qquad{\rm for\ }k=1,2,\ldots,\infty.\e$$ The primary fields in the minimal models are labeled by three integers for the left movers, $l$,$q$, and $s$, which obeys $0\leq l\leq k$, $q$ which is defined modulo $2(k+2)$ and $s$ which labels the sector and is defined modulo $4$. In addition, the right movers carry another set of such quantum numbers. We denote such a primary field by $\Phi_{l,q,s,\bar l,\bar q,\bar s}$. The dimension and charge of this field are then $$\Delta={l(l+2)-q^2\over 4(k+2)}+{s^2\over8}+{\rm integer},\qquad Q=-{q\over k+2}+{s\over2}+2({\rm integer}).\e$$ One can get the correct total trace anomaly by an arbitrary tensoring of these models. The number of possibilities for consistent string theories is enormous, at least several millions in the case of $c=9$ which corresponds to a four dimensional string theory. It appears that by these possibilities alone one can get all Calabi–Yau manifolds up to diffeomorphisms . The rule for computing the spectrum is simple: anything that can appear should appear. One starts from any modular invariant $N=2$ theory, and implements the $G$ projection, along with the map into heterotic–like string theory. The massless states in a tensor product of $c<3$ minimal theories can then be described as all the states obeying the following conditions, (C1) The left and right states have a total $U(1)$ charge which is odd integral. (C2) The states are either in the Ramond sector of all the sub-theories or all in the Neveu–Schwarz sector. (C3) In each of the discrete models, the $l$ and $\bar l$ quantum numbers are correlated according to any of the $A^1_1$ invariants, which were classified in . The left and right $q$ and $s$ quantum numbers are equal. These conditions guarantee modular invariance. (C4) In addition, we add to the spectrum states which can be obtained by the action of $Q$, the supersymmetry charge. This condition implements space–time supersymmetry. We also add to the spectrum states related by $G_iG_j$ where the $G_i$ are the superconformal stress tensors in any of the sub-theories In order to identify these string theories as string propagation on Calabi–Yau manifolds we explored the massless spectrum. In the supergravity models the gauge group is $E_8\times E_6$, the number of generations ($27$ of $E_6$) is equal to $h^{2,1}$ and the number of anti–generation is $h^{1,1}$ . If, in addition, the manifold has some automorphism group (this is the physically interesting case) then, by equivariant index theorems, the generations and anti–generations must transform in as the forms. By comparing the automorphisms with the discrete symmetries of the string theory and the way the massless spectra transform, we then obtain a highly unambiguous, model by model, identification of string theories with the spectrum expected for particular manifolds. An example of this procedure will be described later in the context of the three generations case. As a result we find that the massless spectrum of a string theory on a Calabi–Yau manifold is as following, 1\) The gauge symmetry is $E_8\times E_6$ or $SO(26)$ times a possible extra gauge symmetry. 2\) The theories have $N=1$ space–time supersymmetry. 3\) The number of generations is $h^{2,1}$ and the number of anti–generations is $h^{1,1}$. 4\) The $E_6$ singlets are divided according to: singlets that perturb the complex structure (their number is $h^{2,1}$), singlets that change the radii (their number is $h^{1,1}$), singlets coming from $H^1({\rm End}\,T)$ and a number of Higgs singlets equal in number to the dimension of the extra gauge symmetry. 5\) The automorphisms of the surface appear as discrete symmetries of the string theories and the massless spectra transform as their corresponding forms. The physically interesting case is when the number of generations is three, arising when the manifold has the Euler number $\|\chi\|=6$. Models with more than three generations tend to have problems with fast proton decay, as well as the flow of coupling constants , and thus can probably be ruled out. The first examples of CY manifolds with $\chi=6$ were described by Tian and Yau . In refs. , a comprehensive computer search for all complete intersection manifolds with $\|\chi\|=6$ was carried out, and it was shown that no additional such manifold exists. It is also known that there are no orbifolds , which corresponds to propagation on a flat singular limit of some CY manifold (e.g. the Z manifold ), that have three generations. In addition, all the known manifolds with $\chi=-6$ are either diffeomorphic to one another, or ill defined . Thus, strikingly, there appears to be a [unique ]{}manifold with three generations, the Tian–Yau manifold. In this paper we describe a string theory on this manifold. Our starting point is a CY manifold with Euler number $\chi=-54$. It can be described as the hypersurface, $S$, in $CP^2\times CP^3$, described as the manifold of solutions of the polynomial equations $$\eqalign{P_1&=z_0^3+z_1^3+z_2^3+z_3^3=0,\cr P_2&=z_1x_1^3+z_2x_2^3+z_3x_3^3=0,\cr}\e$$ where $[z_0,z_1,z_2,z_3]\in CP^3$ and $[x_1,x_2,x_3]\in CP^2$. This manifold has a vanishing first Chern class as follows from the existence of a holomorphic three form, which can be written as $$\mu=\oint \oint{\epsilon_{ijkl}z_i\d z_j \wedge\d z_k \wedge\d z_l\,\epsilon_{ijk} x_i\d x_j\wedge\d x_k\over P_1P_2},\e$$ where the integrals are taken around close contours surrounding the surfaces $P_1=0$ and $P_2=0$. In order to compute the Hodge numbers for this manifold it is enough to find $h^{2,1}$, since $\chi=2(h^{1,1}-h^{2,1})$. Now, the Hodge number $h^{2,1}$ counts the number of deformations of the complex structure in manifolds which admit a metric of $SU(3)$ holonomy. The reason is that these deformations are given, in general, by $(1,0)$ forms with values in the tangent bundle, $H^1(T)$ (e.g. see ), which can in turn be converted to $(1,2)$ forms using the holomorphic three form. The deformations of the complex structure for the particular surface $S$ may all be described as perturbations of the defining equation (3). We may perturb $P_1$ by adding any of the $20$ polynomials which are cubic in $z$ and of zero order in $x$, or perturb $P_2$ by any of the $40$ polynomials which are linear in $z$ and cubic in $x$. In total there are $60$ possible polynomials. However, polynomials related by a linear redefinition of $z$ or $x$ correspond to the same complex structure. There are $25$ such redefinitions and thus the net number of perturbations is $h^{2,1}=35$. Since $\chi=-54$ we also find $h^{1,1}=8$. The surface $S$ enjoys a large global automorphism group. First, we can permute the indices $1,2,3$ by an arbitrary permutation, $p\in S_3$, of these indices: $z_i\rarrow z_{p(i)}$ simultaneously with $x_i\rarrow x_{p(i)}$. Next, we have a $Z_3\times Z_9^3$ automorphism group given by different phases. Denoting by $\{r_0,r_1,r_2,r_3\}$ an element of $Z_3\times Z_9^3$, where $r_0$ is defined modulo $3$ and the other $r$ modulo $9$, its action is given by $$z_i\rarrow e^{2\pi ir_i/3} \qquad {\rm for\ }i=0,1,2,3 \e$$ $$x_i\rarrow e^{-2\pi i r_i/9} x_i \qquad {\rm for\ } i=1,2,3.\e$$ Since an overall phase is irrelevant in $CP^n$, the group element $g=\{1,1,1,1\}$ acts trivially. To summarize, the global automorphism group is $G=S_3\semidirect(Z_3\times Z_9^3)/(g)$. It is of order $1458$. Under the automorphism group $G$ the deformations of the complex structure transform like their corresponding polynomial perturbations. We denote by a column vector the perturbations, where the up (down) component perturb $P_1$ ($P_2$). Due to the freedom to linearly redefine $z$, the perturbations of $P_1$ may all be assumed to be linear in any of the $z_i$. Similarly, redefinitions of $x$ allow us to write the perturbations of $P_2$ as $z_i x_j^2 x_k$, $z_i x_j^3$, or $z_i x_1x_2x_3$, where $i\neq j$. The possible perturbations then come in the patterns, $$\eqalignno{\pmatrix{z_0z_1z_2\cr0\cr}&\qquad (1,3,3,0)&{(3)\qquad}\cr \pmatrix{z_1z_2z_3\cr0\cr}&\qquad (0,3,3,3)&{(1)\qquad}\cr \pmatrix{0\cr z_0x_1^3\cr}&\qquad (1,6,0,0)&{(3)\qquad}\cr \pmatrix{0\cr z_0x_1^2x_2\cr}&\qquad (1,-2,-1,0)&{(6)\qquad}\cr \pmatrix{0\cr z_0x_1x_2x_3\cr}&\qquad (1,-1,-1,-1)&{(1)\qquad}\cr \pmatrix{0\cr z_1x_2^3\cr}&\qquad (0,3,6,0)&{(6)\qquad}\cr \pmatrix{0\cr z_1x_2^2x_3\cr}&\qquad (0,3,-2,-1)&{(6)\qquad}\cr \pmatrix{0\cr z_1 x_2^2 x_1\cr}&\qquad (0,2,-2,0)&{(6)\qquad}\cr \pmatrix{0\cr z_1 x_1 x_2 x_3\cr}&\qquad (0,2,-1,-1)&{(3)\qquad}\cr}$$ where we denote by $(m_0,m_1,m_2,m_3)$ the charge of a vector, $v$, in any of the one dimensional irreducible representation of $Z_3\times Z_9^3$. A vector in this representation transforms as $$v\rarrow e^{2\pi i(r_0m_0/3+r_1m_1/9+r_2m_2/9+r_3m_3/9)} v\e$$ under $\{r_0,r_1,r_2,r_3\}\in G$. Since $g=(1,1,1,1)\in G$ is equivalent to $0\in G$, it must act trivially in all the representations of $G$ and thus $$3m_0+m_1+m_2+m_3=0\mod 9,\e$$ for all the representations $(m_0,m_1,m_2,m_3)$. In addition, the holomorphic three form $\mu$ transforms in the representation $(1,2,2,2)$ of $G$. The subgroup of $G$ which commutes with supersymmetry, $H$, is given by the elements that act trivially on the holomorphic three form, $$H=\big\{\{r_0,r_1,r_2,r_3\}\in G\,\big\vert\,\, 3r_0+2r_1+2r_2+2r_3=0 \mod 9 \big\}.\e$$ All the permutations commute with supersymmetry. The other elements of $G$, which are not in $H$, are $R$ symmetries. The $(2,1)$ forms are obtained from the deformations of the complex structure, which are elements of $H^1(T)$, by multiplying with holomorphic three form. Thus, the $(2,1)$ forms transform like the deformations times the holomorphic three form, i.e. they differ by the charge $(1,2,2,2)$. The transformation properties of the $(1,1)$ forms under $G$ may be computed using Lefshets fixed point theorem. Let $f$ be some element of the automorphism group $G$. Then $f$ acts on the cohomology group as some matrix, $f^$. Lefshets fixed point theorem tells us that $$\sum_{p,q} (-1)^{p+q}\Tr_{H^{(p,q)}} f^=\chi(M_f),\e$$ where $\chi(M_f)$ is the Euler character of the submanifold which is fixed by $f$, $M=\{x\in M\vert f(x)=x\}$. By calculating all the Euler numbers in eq. (10), we find that the eight $(1,1)$ forms transform as $$(0,0,0,0) \times 2,\qquad (1,3,6,6),\qquad (2,3,3,6).\e$$ Let us turn now to one forms with values in the endomorphism of the tangent bundle, $H^1({\rm End\,} T)$. In the field theory, such forms give rise to massless $E_6$ singlets . These forms are in correspondence with deformations of the tangent bundle. Denote a tangent vector by $(U_a,V_b)$, where $U_a$ is a tangent vector of $CP^3$ (a=0,1,2,3), and $V_b$ is a tangent vector in $CP^2$, $b=1,2,3$. The tangent vectors $U_a$ and $V_b$ are defined modulo the equivalence relation $$U_a\sim U_a+\lambda z_a,\qquad V_b\sim V_b+\rho x_b,\e$$ for any $\lambda$ and $\rho$. In addition, the vector $(U_a,V_b)$ must be tangent to the two surfaces $P_1$ and $P_2$. This implies, \#1\#2[[\#1\#2]{}]{} $$\eqalign{&\parr{P_1}{z_a}U_a=\parr{P_1}{x_b}V_b=0\cr &\parr{P_2}{z_a}U_a=\parr{P_2}{x_b}V_b=0\cr}\e$$ A simple method to deform the tangent bundle is to change equation (13) by adding to it some small perturbation. We can perturb any of the partial derivatives in (13) by adding to it an arbitrary polynomial of the same bi-degree as the corresponding partial derivative. Denote a perturbation by the matrix $$M=\pmatrix{P_a&Q_b\cr L_a&R_b}.\e$$ Eq (13) is then perturbed by $(\parr{P_1}{z_a}+P_a)U_a=0$, etc. The bi-degrees of the polynomials $P$, $Q$, $L$ and $R$ are $(2,0)$, $(0,0)$, $(0,3)$ and $(1,2)$, respectively. In addition, the equivalence relation eq. (12) implies $$P^az_a=Q^bx_b=L^az_a=R^bx_b=0.\e$$ Any such set of polynomials defines a perturbation of the tangent bundle. All the perturbations come either from $P$ or from $R$. To perturb $P$ we may take $P_a=C_a P/z_a$, for an arbitrary $P$ which is of bi-degree $(3,0)$ and where the $C_a$ are some constants which obey $\sum C_a=0$. The possible $P$ come in the patterns $$\eqalignno{z_0^2z_1&\qquad (2,3,0,0)& {(3)\qquad}\cr z_0z_1^2& \qquad (1,6,0,0)& {(3)\qquad}\cr z_1^2z_2& \qquad (0,6,3,0)& {(6)\qquad}\cr z_0z_1z_2& \qquad (1,3,3,0)& {(2\times 3)\qquad}\cr z_1z_2z_3& \qquad (0,3,3,3)& {(2\times 1)\qquad}\cr}$$ where the numbers above denote the $Z_3 Z_9^3$ charges and multiplicities. Similarly, the perturbations of $R$ can be written as, $R_b=C_b R/x_b$, where the constants $C_b$ obey, $\sum C_b=0$, and $R$ is any polynomial of bi-degree $(1,3)$. The possible $R$’s fall into the patterns $$\eqalignno{z_0x_1^2x_2& \qquad (1,-2,-2,0) & {(6)\qquad}\cr z_1x_1^2x_2& \qquad (0,1,-1,0) & {(6)\qquad}\cr z_2x_1^2x_2& \qquad (0,-2,2,0) & {(6)\qquad}\cr z_3x_1^2x_2& \qquad (0,-2,-1,3) & {(6)\qquad}\cr z_0x_1x_2x_3& \qquad (1,-1,-1,-1) & {(2\times 1) \qquad}\cr z_1x_1x_2x_3& \qquad (0,2,-1,-1) & {(2\times 3)\qquad }\cr}$$ In total we find $52$ elements of $H^1({\rm End\,}T)$. There can be more deformations which cannot be obtained in this way. Using spectral sequences it should be possible to compute the entire cohomology. In the field theory limit the number of generations ($27$ of $E_6$) is $35$, corresponding to the $35$ harmonic $(2,1)$ forms, and the number of anti–generations ($\bar{27}$ of $E_6$) is $8$, corresponding to the harmonic $(1,1)$ forms. The net number of generations is $\half\vert\chi \vert=27$. Consider now the theory made by gluing one copy of the $k=1$ model with three copies of the $k=16$ model. In addition, in condition (C3) we use the sporadic modular invariant at level $16$ . The resulting spectrum may be easily computed from (C1–C4). The theory, denoted by $1^116^3$, contains $35$ generations ($27$ of $E_6$), $8$ anti–generations ($\bar{27}$ of $E_6$) and $197$ massless $E_6$ singlets. As will be seen the theory $1^116^3$ corresponds to a string theory on the manifold $S$. The number of generations and anti–generations are indeed the same as $h^{2,1}$ and $h^{1,1}$ for this manifold. What are the discrete symmetries of the theory $1^116^3$? The $k$’th minimal model has a $Z_{k+2}$ discrete symmetry. Thus, the theory $1^116^3$ has a $Z_3\times Z_{18}^3$ symmetry. In addition, we can permute the three identical $k=16$ sub–theories. However, by examining the massless spectrum, it becomes clear that each of the $Z_2$ subgroups of $Z_{18}$ acts trivially on the spectrum and thus may be ignored. Hence the symmetry group of the theory $1^116^3$ is $G=(Z_3\times S_3 \semidirect Z_9^3)/Z_9$. Denote an element of $Z_3\times Z_9^3$ by $\{r_0,r_1,r_2,r_3\}$. The quotient by $Z_9$ corresponds to the fact that the total superconformal $U(1)$ charge of all the fields is an odd integer, implying that the element $\{1,1,1,1\}$ acts trivially on all the fields in the theory, so the actual symmetry group is a quotient by the $Z_9$ subgroup generated by this element. We see that the theory $1^116^3$ has a symmetry group which is isomorphic to the automorphism group of the hypersurface $S$. Under the $Z_{k+2}$ charge of the $k$ minimal model a field in the theory, $\Phi_{l,q,s,\bar l, \bar q,\bar s}$, has a charge which is $$Q=(q+\bar q)/2 \mod (k+2).\e$$ We assume in this definition that $q=\bar q\mod2$. This does not create any problem for the non R symmetries. For some of the $R$ symmetries, however, eq. (16) may imply that the charges are ill defined, suggesting that these group elements are bad symmetries that should be ignored. In the case at hand, though, no such problem arises. The R symmetries, since they do not commute with space–time supersymmetry, are very tricky. Different supersymmetry partners transform differently under them, and similarly, the different representations of $SO(10)$, which make the $27$ or $\bar{27}$ of $E_6$, transform differently. We would like to compare the transformation properties of the various massless fields in the spectrum of the $1^116^3$ theory, with those that are predicted in the field theory. The first thing we note is that the plus and minus chirality components of the $E_6$ gluino field come from the $H^{0,0}$ and $H^{0,3}$ Dolbeault cohomology groups or the positive chirality gluino corresponds to the unique constant $(0,0)$ form and the negative chirality gluino corresponds to the antiholomorphic $(0,3)$ form. The adjoint representation of $E_6$ decomposes into SO(10) as $78=1+16+\bar{16}+45$. The different $SO(10)$ representations always transform differently under the R symmetries. Now, for a fixed $SO(10)$ representation, say the singlet, the different $Z_{k+2}$ charges of the positive and negative chirality gluinos will always differ by $1$. This is simply because these two modes are related by the square of the supersymmetry charge , $Q^2$, which, in turn, carries the $Z_{k+2}$ charge which is $1$ for all the sub–theories. On the other hand, this ratio corresponds to the holomorphic $(3,0)$ form. Thus, the holomorphic $(3,0)$ form always carry the $Z_{k+2}$ charge $1$, when this charge is defined as in eq. (16). From eq. (4) we see that the $(3,0)$ form has the $Z_3\times Z_9^3$ charge which is $(1,2,2,2)$. On the other hand, the positive and negative chirality gluinos differ by a $Z_3\times Z_{18}^3$ charge which is $(1,1,1,1)$. Using this correspondence we can ‘fix the normalization’ of the discrete charges. The discrete charges of the automorphisms of the manifold $(m_0,m_1,m_2,m_3)$, which are elements of $Z_3\times Z_9^3$, and the conformal field theory charges $(Q_0,Q_1,Q_2,Q_3)$, which are defined according to (16) and are elements of $Z_3\times Z_{18}^3$, are then seen to be related as $$m_0=Q_0\mod 3,\qquad m_i=2Q_i\mod 9,\quad {\rm for\ }i=1,2,3.\e$$ Next, we can check whether the generations and anti–generations transform as they are supposed to, in the field theory limit. Under the non R symmetries (i.e. the ones which commute with SUSY) the generations and anti–generations must transform like their corresponding $(2,1)$ and $(1,1)$ forms. The R symmetries are trickier since, as discussed earlier, different supersymmetry components transform under them differently. Which component, then, should we compare with the forms? The answer is that the correct component for supersymmetry multiplets in the $27$ or the $\bar{27}$ of $E_6$ is the scalar (helicity zero) which is a vector of $SO(10)$. The reason is that such scalars are related to the $E_6$ singlets which perturb the radius (in the $\bar{27}$ case) or deform the complex structure (in the $27$ case) and thus must transform in precisely the same way as the forms of $H^1(T)$ (for $27$) or the $(1,1)$ forms (for $\bar{27}$) do. The following is an enumeration of the $35$ generations in the $1^116^3$ theory, \#1[\_[\#1]{}]{}\#1[\_[\#1]{}]{} \#1[\|\_[\#1]{}]{} $$\eqalignno{ % 201 (4) % ( 1, 3,2) ( 4,32,0) (16,20,0) (16,20,0) (3)&\quad\p{1,2,1,1,3,2}\q{12,13,1,4,32,0}\q{0,1,1,16,20,0}^2 & { ( 1, 6,0,0) }\quad\cr % ( 1, 2,1) (12,13,1) ( 0, 1,1) ( 0, 1,1) % % 288 (3) % ( 1, 3,2) ( 8,28,0) (12,24,0) (16,20,0) (6)&\quad\p{1,2,1,1,3,2}\q{8,9,1,8,28,0}\q{4,5,1,12,24,0}\q{0,1,1,16,20,0} & { (1,-2,-1,0) }\quad\cr % ( 1, 2,1) ( 8, 9,1) ( 4, 5,1) ( 0, 1,1) % % 302 (2) %( 1, 3,2) (10,26,0) (10,26,0) (16,20,0) (3)&\quad\p{1,2,1,1,3,2}\q{6,7,1,10,26,0}^2\q{0,1,1,16,20,0} & { (1,3,3,0) }\quad\cr %( 1, 2,1) ( 6, 7,1) ( 6, 7,1) ( 0, 1,1) % % 314 (1) %( 1,3,2) (12,24,0) (12,24,0) (12,24,0) (1)&\quad\p{1,2,1,1,3,2}\q{4,5,1,12,24,0}^3 & { (1,-1,-1,-1)}\quad\cr %( 1, 2,1) ( 4, 5,1) ( 4, 5,1) ( 4, 5,1) % % 431 (9) %( 0, 2,2) ( 4,32,0) (10,26,0) (16,20,0) (6)&\quad\p{0,1,1,0,2,2}\q{12,13,1,4,32,0}\q{6,7,1,10,26,0}\q{0,1,1,16,20,0} & { (0,6,3,0) }\quad\cr %( 0, 1,1) (12,13,1) ( 6, 7,1) ( 0, 1,1) ( % % 458 (8) %( 0,2,2) ( 6,30,0) ( 8,28,0) (16,20,0) (6)&\quad\p{0,1,1,0,2,2}\q{10,11,1,6,30,0}\q{8,9,1,8,28,0}\q{0,1,1,16,20,0} & { (0,2,-2,0) }\quad\cr %( 0, 1,1) (10,11,1) ( 8, 9,1) ( 0, 1,1) ( % % 462 (7) %( 0,2,2) ( 6,30,0) (12,24,0) (12,24,0) (3)&\quad\p{0,1,1,0,2,2}\q{10,11,1,6,30,0}\q{4,5,1,12,24,0}^2 & { (0,2,-1,-1) }\quad\cr %( 0, 1,1) (10,11,1) ( 4, 5,1) ( 4, 5,1) ( % % 472 (6) %( 0,2,2) ( 8,28,0) (10,26,0) (12,24,0) (6)&\quad\p{0,1,1,0,2,2}\q{8,9,1,8,28,0}\q{6,7,1,10,26,0}\q{4,5,1,12,24,0} & { (0,-2,3,-1) }\quad\cr %( 0, 1,1) ( 8, 9,1) ( 6, 7,1) ( 4, 5,1) % %478 (5) %(0,2,2) (10,26,0) (10,26,0) (10,26,0) (1)&\quad\p{0,1,1,0,2,2}\q{6,7,1,10,26,0}^3 & { (0,3,3,3) }\quad\cr %( 0, 1,1) ( 6, 7,1) ( 6, 7,1) ( 6, 7,1) % }$$ The fields in the list correspond to anti–spinors which are singlets of $SO(10)$. We denoted by $\Phi$ and $\Theta$ the fields from the $k=1$ and $k=16$ theories. The six indices on each field correspond to the three left quantum numbers $(l,q,s)$ and the three right quantum numbers. The numbers on the right are the $Z_3 Z_9^3$ charges of each of the fields. The $Z_9$ charges are computed according to $m_i=q_i+\bar q_i-3\mod9$, for $i=1,2,3$; the $Z_3$ charge is $m_0=-q_0-\bar q_0\mod3$. We see that the $35$ generations transform in precisely the same representation of $G$ as the deformations of the complex structure do (p. 10). We can now check the anti–generations. The $8$ anti–generations, along with their corresponding $Z_3\times Z_9^3$ charges, are enumerated below, $$\eqalignno{ % %* 295 %( 0, 0,0) ( 8, 8,0) (14,14,0) (14,14,0) (3)\quad&\p{0,1,1,0,0,0}\q{2,3,1,8,8,0}\p{8,9,1,14,14,0}^2 & { (1,3,6,6)}\quad\cr %( 0, 1,1) ( 2, 3,1) ( 8, 9,1) ( 8, 9,1) % % 315 %( 1,3,2) (12,12,0) (12,12,0) (12,12,0) (1)\quad&\p{1,2,1,1,3,2}\q{4,5,1,12,12,0}^3 & { (0,0,0,0)}\quad\cr %( 1, 2,1) ( 4, 5,1) ( 4, 5,1) ( 4, 5,1) % %* 615 %( 1, 1,0) ( 8, 8,0) ( 8, 8,0) (14,14,0) (3)\quad&\p{1,2,1,1,1,0}\q{2,3,1,8,8,0}\q{2,3,1,8,8,0}\q{8,9,1,14,14,0} & { (2,3,3,6)}\quad\cr %( 1, 2,1) ( 2, 3,1) ( 2, 3,1) ( 8, 9,1) % %*** 622 %( 0,4,2) (10,10,0) (10,10,0) (10,10,0) (1)\quad&\p{0,1,1,0,4,2}\q{6,7,1,10,10,0}^3 & { (0,0,0,0)}\quad\cr %( 0, 1,1) ( 6, 7,1) ( 6, 7,1) ( 6, 7,1) % }$$ The fields above are also anti–spinors which are $SO(10)$ singlets. The $Z_3\times Z_9$ charge, $(m_0,m_1,m_2,m_3)$ is computed in this case by $m_0=-q_0-\bar q_0-1\mod3$, $m_i=q_i+\bar q_i+1\mod9$. Again, we see that the anti–generations transform in precisely the same way as the $(1,1)$ forms do, eq. (11). This completes the identification of the $1^116^3$ theory as a string theory on the hypersurface $S$. We can further compare the $E_6$ singlets. The $197$ singlets can be seen to contain, in a completely unambiguous way, $35$ modes which transform like $H^1(T)$, these are the singlets related to deformations of the complex structure, $8$ modes transforming like $(1,1)$ forms, these are singlets related to change of radii and $52$ singlets transforming like the modes of $H^1({\rm End}\, T)$ described earlier (p. 13). The remaining $102$ singlets may correspond to additional perturbations of the tangent bundle that we have not computed, or less likely, to accidental massless particles. The resolution of this question must await the complete calculation of $H^1({\rm End}\,T)$ for this manifold. The automorphism group of $S$ has a certain $Z_3\times Z_3$ subgroup, $H$, which will be very important for us. The first $Z_3$ is generated by the permutation: $z_1\rarrow z_2\rarrow z_3\rarrow z_1$ along with $x_1\rarrow x_2\rarrow x_3\rarrow x_1$, denoted by $h$. This can be seen to be a freely acting automorphism. The second $Z_3$ is generated by the $Z_3\times Z_9^3$ group element $g=\{0,3,6,0\}$. This $Z_3$ is not freely acting. Thus, the quotient manifold $S/H$ is a singular manifold. However, these singularities can be resolved, as discussed in ref. and the resulting manifold is a CY manifold with Euler number $\chi=-6$. Thus, a string theory on $S/H$ should have three generations. The spectrum of a heterotic string theory propagating on the manifold $S/H$ can be computed as a quotient of the theory $1^116^3$. The partition function of the $k$’th minimal model, twisted in the space and time directions by the $Z_{k+2}$ elements $x$ and $y$, respectively, is given by $$Z(x,y)={1\over2}e^{2\pi ixy/(k+2)}\sum_{l,q,s} e^{2\pi ixq/(k+2)} \chi^l_{q+2y}\chi^{l*}_{q,s},\e$$ where $\chi_{q,s}^l$ is the partition function of the $N=2$ conformal block with the quantum numbers $(l,q,s)$. Implementing eq. (18) in the string theory amounts to a simple modification of the conditions (C1–C4) and enables us to compute the spectrum of string propagation on the manifold $S/H$. Consider first string theory on the quotient manifold $S/(g)$. By a straightforward enumeration of states we find that this theory has $23$ generations, $14$ anti–generations and $173$ singlets. Of these, $17$ generations, $8$ anti–generations and $85$ singlets come from the untwisted sector. The Hodge numbers $h^{2,1}=23$ and $h^{1,1}=14$ are the same as those of a well known CY manifold, namely the one constructed by Tian and Yau . This manifold can be described as the intersection of three hypersurfaces of bi-degrees $(3,0)$, $(0,3)$ and $(1,1)$ in the product space $CP^3\times CP^3$. Its most symmetric shape is $$\sum_{i=0}^3 x_i^3=0,\qquad\sum_{i=0}^3 y_i^3=0,\qquad \sum_{i=0}^3 x_i y_i=0.\e$$ This manifold has the Hodge numbers $h^{2,1}=23$ and $h^{1,1}=14$. Thus, taking a quotient of it by the freely acting $Z_3$ automorphism group, which is generated by $$(x_0,x_1,x_2,x_3)\times (y_0,y_1,y_2,y_3) \rarrow (x_0,\alpha^2x_1,\alpha x_2,\alpha x_3)\times (y_0,\alpha y_1, \alpha^2y_2,\alpha^2 y_3),\e$$ where $\alpha=\exp(2\pi i/3)$, we get to a three generation manifold. The manifold $S/H$ is indeed diffeomorphic to the Tian–Yau manifold . The supergravity model on the Tian–Yau manifold was studied by a number of authors and the indications are that the discrete symmetries of the manifold may well be rich enough to prevent fast proton decay. Returning to the string theory on $S$, the next step after twisting the theory by $g$, is to further twist it by the permutation $h$. By projecting the spectrum of the string theory on $Q/(g)$ onto the $h$ invariant subspace, it is easy to write the spectrum of the closed string (untwisted) sector. We find in this sector $9$ generations, $6$ anti–generations and $62$ singlets. In addition, one needs to take into account the winding sectors. Since $h$ acts freely, these sectors do not contribute any generations or anti–generations. Thus, all together, in this string theory we find $9$ generations and $6$ anti–generations, or a net number of three generations. The Yukawa couplings in this three generation string theory can all be computed exactly since they are given as products of the structure constants of the $N=2$ minimal models. These, in turn, are related to the structure constants of the $SU(2)$ WZW models which have been studied by several authors . Using the isomorphism of states and vertex operators that create them out of the vacuum , one can express the vertex operators in this string theory in terms of WZW fields and free bosons, and thus to calculate the Yukawa couplings exactly (see for more explanation). In conclusion, we have presented a new string theory which corresponds to string propagation on a three generation Calabi–Yau manifold. It is a full fledged string theory in which all the physical predictions can be computed exactly. This string theory appears to be the unique viable candidate in its class. In addition, it comes intriguingly close to a realistic description of nature, which is moreover a consistent unification of gravity. I thank G. Faltings and D. Gross for interesting discussions. [^1]: [^2]: [^3]:
--- bibliography: - 'content/bib\_new.bib' ---
--- abstract: 'We make publicly available a catalog of calibrated environmental measures for galaxies in the five 3D-HST/CANDELS deep fields. Leveraging the spectroscopic and grism redshifts from the 3D-HST survey, multi wavelength photometry from CANDELS, and wider field public data for edge corrections, we derive densities in fixed apertures to characterize the environment of galaxies brighter than $JH_{140} < 24$ mag in the redshift range $0.5<z<3.0$. By linking observed galaxies to a mock sample, selected to reproduce the 3D-HST sample selection and redshift accuracy, each 3D-HST galaxy is assigned a probability density function of the host halo mass, and a probability that is a central or a satellite galaxy. The same procedure is applied to a $z=0$ sample selected from SDSS. We compute the fraction of passive central and satellite galaxies as a function of stellar and halo mass, and redshift, and then derive the fraction of galaxies that were quenched by environment specific processes. Using the mock sample, we estimate that the timescale for satellite quenching is $t_{\rm quench} \sim 2-5$ Gyr; longer at lower stellar mass or lower redshift, but remarkably independent of halo mass. This indicates that, in the range of environments commonly found within the 3D-HST sample ($M_h \lesssim 10^{14} M_\odot$), satellites are quenched by exhaustion of their gas reservoir in absence of cosmological accretion. We find that the quenching times can be separated into a delay phase during which satellite galaxies behave similarly to centrals at fixed stellar mass, and a phase where the star formation rate drops rapidly ($\tau_f \sim 0.4-0.6$ Gyr), as shown previously at $z=0$. We conclude that this scenario requires satellite galaxies to retain a large reservoir of multi-phase gas upon accretion, even at high redshift, and that this gas sustains star formation for the long quenching times observed.' author: - 'M. Fossati$^{,}$, D.J. Wilman$^{,}$, J.T. Mendel$^{,}$, R.P. Saglia$^{,}$, A. Galametz$^{,}$, A. Beifiori$^{,}$, R.Bender$^{,}$, J.C.C. Chan$^{,}$, M. Fabricius, K. Bandara, G.B. Brammer, R. Davies, N.M. F[ö]{}rster Schreiber, R. Genzel$^{,}$$^{,}$, W. Hartley, S.K. Kulkarni, P. Lang, I.G. Momcheva, E.J. Nelson$^{,}$, R. Skelton, L.J. Tacconi, K. Tadaki, H.[Ü]{}bler, P.G. van Dokkum, E. Wisnioski, K.E. Whitaker$^{,}$, E. Wuyts, S. Wuyts' title: \| Galaxy environment in the 3D-HST fields.\ Witnessing the onset of satellite quenching at $z\sim 1-2$. --- Introduction ============ It has long been known that galaxies are shaped by the environment in which they reside. Works by e.g., @Oemler74, @Dressler80, and @Balogh97 showed that galaxies in high-density environments are preferentially red and early-type compared to those in lower density regions. The more recent advent of large scale photometric and spectroscopic surveys confirmed with large statistics those early findings [@Balogh04; @Kauffmann04; @Baldry06]. Meanwhile, space and ground based missions have probed the geometry of our Universe. Those observations coupled to cosmological models have built the solid Lambda cold dark matter ($\Lambda$CDM) framework [@White78; @Perlmutter99], in which lower mass haloes are the building blocks of more massive structures. One of the major tasks for modern studies of galaxy formation is therefore to understand how and when galaxy evolution is driven by internal processes or the evolving environment that each galaxy experiences during its lifetime. While internal mechanisms, including ejective feedback from supernovae or active galactic nuclei, are deemed responsible for suppressing star formation in all galaxies [@Silk98; @Hopkins08], a galaxy can also directly interact with its environment when falling into a massive, gas- and galaxy-rich structure such as a galaxy cluster. At low redshift detailed studies of poster child objects [@Yagi10; @Fossati12; @Fossati16; @Merluzzi13; @Fumagalli14; @Boselli16] coupled with state-of-the-art models and simulations [@Mastropietro05; @Kapferer09; @Tonnesen10] have started to explore the rich physics governing those processes [e.g., @Boselli06; @Boselli14c; @Blanton09 for reviews]. Broadly speaking, they can be grouped into two classes. The first of them includes gravitational interactions between cluster or group members [@Merritt83] or with the potential well of the halo as a whole [@Byrd90], or their combined effect known as “galaxy harrassment” [@Moore98]. The second class includes hydrodynamical interactions between galaxies and the hot and dense gas that permeates massive haloes. This class includes the rapid stripping of the cold gas via ram pressure as the galaxy passes through the hot gas medium [@Gunn72]. Ram-pressure stripping is known to effectively and rapidly suppress star formation in cluster galaxies in the local Universe [@Solanes01; @Vollmer01; @Gavazzi10; @Gavazzi13a; @Boselli08; @Boselli14b]. Less directly influencing the galaxy’s current star formation, the multi-phase medium (e.g. warm, hot gas) associated to the galaxy (known as the “reservoir”) should be easier to strip than the cold gas. Even easier, the filamentary accretion onto the galaxy from the surrounding cosmic web will be truncated as the galaxy is enveloped within the hot gas of a more massive halo [@White91]. Both of these processes will suppress ongoing accretion onto the cold gas disk of the galaxy and lead to a more gradual suppression of star formation, variously labelled “strangulation” or “starvation” [e.g. @Larson80; @Balogh97] These processes are complicated in nature and the exact details of their efficiency and dynamics are still poorly understood. The situation is even more complicated when several of those processes are found to act together [@Gavazzi01; @Vollmer05]. A different approach to disentangle the role of environment from the secular evolution is to study large samples of galaxies and correlate their properties (e.g. star formation activity) to internal properties (e.g. stellar mass) and environment. In the local Universe, the advent of the Sloan Digital Sky Survey (SDSS) has revolutionized the field of large statistical studies and allowed for the effects of the environment on the galaxy population as a whole to be studied [@Kauffmann04; @Baldry06; @Peng10; @Peng12; @Wetzel12; @Wetzel13; @Hirschmann14]. One of the main results is that environmental quenching is a separable process that acts on top of the internal processes that regulate the star formation activity of galaxies. A crucial parameter to understand the collective effect of the several environmental processes is the timescale over which the star formation activity is quenched. Several authors took advantage of excellent statistics to estimate the average timescale for environmental quenching, accounting for internal quenching processes, and found that in the low redshift Universe this is generally long [$\sim 5-7$ Gyr; @McGee09; @DeLucia12; @Wetzel13; @Hirschmann14], while possibly shorter in clusters of galaxies [$\sim 2-5$ Gyr; @Haines15; @Paccagnella16] . At higher redshift, the situation is made more complex due to the more limited availability of spectroscopic redshifts which are paramount to depict an accurate picture of the environment. In the last decade, several ground based redshift surveys started to address this issue [@Wilman05; @Cooper06]. By exploiting the multiplexing of spectroscopic instruments at 8-10 meter class telescopes [e.g. VIMOS and GMOS, @Lilly07; @Kurk13; @Balogh14], these works showed that the environment plays a role in quenching the star formation activity of galaxies accreted onto massive haloes (satellite galaxies) up to $z\sim 1$ [@Muzzin12; @Quadri12; @Knobel13; @Kovac14; @Balogh16], although the samples are limited to massive galaxies or a small number of objects. Low-resolution space-based slitless spectroscopy is revolutionising this field providing deep and highly complete spectroscopic samples. The largest of those efforts is the 3D-HST survey [@Brammer12] which, by combining a large area, deep grism observations and a wealth of ancillary photometric data, provides accurate redshifts to $\Delta z/(1+z) \sim 0.003$ [@Bezanson16] for a large sample of objects down to low stellar masses ($\sim 10^{9} M_\odot$, and $\sim 10^{10} M_\odot$ at $z\sim1$ and $z\sim2$ respectively). The public release of their spectroscopic observations [@Momcheva16], in synergy with deep photometric observations [@Skelton14] has opened the way to an accurate quantification and calibration of the environment over the redshift range $z\sim 0.5-3$. Another source of uncertainty in the interpretation of correlations of galaxy properties with environment is the inhomogeneity of methods used for different surveys [e.g., @Muldrew12; @Haas12; @Etherington15] and the lack of calibration of important parameters such as halo mass. In @Fossati15, we studied how to link a purely observational parameter space to physical quantities (e.g., halo mass, central/satellite status) by analysing a stellar mass limited sample extracted from semi-analytic models of galaxy formation. To do so, we computed a projected density field in the simulation box and we tested different definitions of density at different redshift accuracy. Our method is Bayesian in nature (galaxies have well-defined observational parameters, while the calibration into physical parameters is probabilistic). This approach is best suited to statistical studies where the application of selection functions and observational uncertainties can be fully taken into account. In this paper, we extend this method to the 3D-HST survey by building up an environment catalogue which we make available to the community with this work. We then explore the efficiency and timescales for quenching of satellite galaxies over cosmic time ($z\sim 0-2$) by combining the 3D-HST data at high redshift with SDSS data in the local Universe in a homogeneous way. We also address the long standing issue of impurity and contamination of the calibrated parameters (the fact that the observations do not perfectly constrain the halo mass of the parent halo for each galaxy or its central/satellite status) by recovering the “pure” trends using the mock sample as a benchmark. The paper is structured as follows. In Section \[obssample\], we introduce the 3D-HST dataset. In Section \[sec\_environment\], we derive the local density for 3D-HST galaxies including accurate edge corrrections. Section \[sec\_overdensities\] presents the range of environments in the 3D-HST area and how they compare to known galaxy structures from the literature. In Section \[sec\_models\], we introduce the mock galaxy sample and how we calibrate it to match the 3D-HST sample. We then link models and observations in Section \[sec\_halomass\], and assign physical quantities to observed galaxies. In Section \[sec\_quenching\] we study the quenching of satellite galaxies at $0 < z < 2.5$, and derive quenching efficiency and timescales. Lastly, we discuss the physical implications of our findings in Section \[sec\_discussion\] and summarize our work in Section \[sec\_conclusions\]. All magnitudes are given in the AB system [@Oke74] and we assume a flat $\Lambda$CDM Universe with $\Omega_M = 0.3$, $\Omega_\Lambda = 0.7$, and $H_0 = 70~\rm{km~s^{-1}~Mpc^{-1}}$ unless otherwise specified. Throughout the paper, we use the notation $\log(x)$ for the base 10 logarithm of $x$. The observational sample {#obssample} ======================== In this work we aim at a quantification and calibration of the local environment for galaxies in the five CANDELS/3D-HST fields [@Grogin11; @Koekemoer11; @Brammer12] namely COSMOS, GOODS-S, GOODS-N, AEGIS and UDS. The synergy of these two surveys represents the largest effort to obtain deep space-based near-infrared photometry and spectroscopy in those fields. For a description of the observations and reduction techniques, we refer the reader to @Skelton14 and @Momcheva16 for the photometry and spectroscopy respectively. The CANDELS observations provide HST/WFC3 near infrared imaging in the F125W and F160W filters ([$J_{\rm 125}$]{} and [$H_{\rm 160}$]{} hereafter) for all the fields, while 3D-HST followed-up a large fraction of this area with the F140W filter ([$JH_{\rm 140}$]{} hereafter) and the WFC3/G141 grism for slitless spectroscopy. The novelty of this approach is to obtain low resolution ($R\sim100$) spectroscopy for all the objects in the field. Taking advantage of the low background of the [HST]{} telescope, it is possible to reach a depth similar to traditional slit spectroscopy from 10m class telescopes on Earth. Hereafter, we use the term “3D-HST” sample to refer to the combination of CANDELS and all the other space- and ground-based imaging datasets presented in @Skelton14, plus the grism spectroscopy of the 3D-HST program. The 3D-HST photometric catalog [@Skelton14] used [$H_{\rm 160}$]{} or [$JH_{\rm 140}$]{} as detection bands and its depth varies from field to field and across the same field due to the observing strategy of CANDELS. However, even in the shallowest portions of each field, the 90$\%$ depth confidence level is $H_{\rm 160} \sim 25$ mag. Beyond this magnitude limit, the star/galaxy classification (which is a key parameter for the environment quantification) becomes uncertain. The 3D-HST spectroscopic release [@Momcheva16] provides reduced and extracted spectra down to $JH_{\rm IR} = 26$ mag. The spectra are passed through the EAZY template fitting code [@Brammer08] along with the extensive ground- and space-based multiwavelength photometry. This results in “grism” redshifts, which are more accurate than photometric redshifts thanks to the wealth of stellar continuum and emission line features present in the spectra. However, only objects brighter than $JH_{\rm IR} = 24$ mag have been visually inspected, and have a `use_grism` flag that describes if the grism spectrum is used to compute the redshift. Incomplete masking of contaminating flux from nearby sources in the direction of the grism dispersion, residuals from spectra of bright stars, and corrupted photometric measurements can lead to this flag being set to 0 (“bad”). We include in the present analysis all galaxies brighter than $JH_{\rm 140} = 24$ mag, therefore limiting our footprint to the regions covered by grism and [$JH_{\rm 140}$]{} observations. We limit the redshift range to $0.5 < z < 3.0$. The lower limit roughly corresponds to the redshift where the $\rm H\alpha$ line enters the grism coverage and the upper limit is chosen such that the number density of objects above the magnitude cut allows a reliable estimate of the environment. It also allows follow-up studies targeting the rest-frame optical features from ground based facilities in the $J$, $H$, and $K$ bands (e.g. $\rm{KMOS}$, @Sharples13 and MOSFIRE, @McLean12) We exclude stars by requiring `star_flag` to be 0 or 2 (galaxies or uncertain classification). We do not use the `use_phot` flag because it is too conservative for our goals. Indeed this flag requires a minimum of 2 exposures in the F125W and F160W filters, and the object not being close to bright stars. The quantification of environment requires a catalog which is as complete as possible even at the expenses of more uncertain photometry (and photo-z) for the objects that do not meet those cuts. Nonetheless a $JH_{\rm 140} = 24$ mag cut allows a reliable star/galaxy separation for 99% of the objects and is at least 1 mag brighter than the minimum depth of the mosaics, thus alleviating the negative effects of nearby stars on faint sources. The final sample is made of 18745 galaxies. As a result of the analysis in @Momcheva16, each galaxy is assigned a “best” redshift. This is: 1. a spectroscopic redshift from a ladder of sources as described below. 2. a grism redshift if there is no spectroscopic redshift and `use_grism = 1` 3. a pure photometric redshift if there is no spectroscopic redshift and `use_grism = 0`. A `zbest_type` flag is assigned to each galaxy based on the conditions above. The best redshift is the quantity used to compute the environment for each galaxy in the 3D-HST fields. Spectroscopic redshifts are taken from the compilation of @Skelton14 which we complement with newer data. For the COSMOS field we include the final data release of the zCOSMOS bright survey [@Lilly07]. We find 253 new sources with reliable redshifts in the 3D-HST/COSMOS footprint mainly at $z<1$. In COSMOS and GOODS-S ,we include 95 objects from the DR1 [@Tasca16] of the VIMOS Ultra Deep Survey [@LeFevre15 VUDS, \[]. This survey mainly targets galaxies at $z>2$ therefore complementing zCOSMOS. We include 105 redshifts from the MOSFIRE Deep Evolution Field Survey [MOSDEF, @Kriek15] which provides deep rest frame optical spectra of galaxies selected from 3D-HST. For the UDS field, we also include 164 redshifts from VIMOS spectroscopy in a narrow slice of redshift ($0.6 < z < 0.7$, Galametz et al. in prep.) Lastly, we include 376 and 33 secure spectroscopic redshifts from $\rm{KMOS^{3D} }$ [@Wisnioski15] and VIRIAL [@Mendel15] respectively. Those large surveys use the multiplexing capability of the integral field spectrometer KMOS on the ESO Very Large Telescope to follow-up 3D-HST selected objects. The former is a mass selected survey of emission line galaxies at $0.7<z<2.7$, while the latter observed passive massive galaxies at $1.5<z<2.0$. In the selected sample, $20\%$ of the galaxies have a spectroscopic redshift, $64\%$ have a grism redshift, and only $16\%$ have a pure photometric redshift. In the next Section, we explore the accuracy of the grism and photometric redshifts as a function of the galaxy brightness and the $S/N$ of emission lines in the spectra. ![image](fig1.eps) Stellar masses and stellar population parameters are estimated using the FAST code [@Kriek09], coupled with @Bruzual03 stellar population synthesis models. Those models use a @Chabrier03 initial mass function (IMF) and solar metallicity. The best redshift is used for each galaxy together with the available space- and ground-based photometry. The star formation history is parametrized by an exponentially declining function and the @Calzetti00 dust attenuation law is adopted. Redshift accuracy {#3dhstzaccuracy} ----------------- A careful quantification of the grism and photometric redshift accuracy is paramount for a good calibration of the environmental statistics into physically motivated halo masses. In Section \[sec\_halomass\], we will show how these masses are obtained from mock catalogues selected to match the number density and redshift uncertainty of 3D-HST galaxies. The low-resolution spectra cover different spectral features as a function of galaxy properties and redshift. The most prominent features are emission lines, which are however limited to star forming objects. On the other hand, stellar continuum features (Balmer break, absorption lines) are present in the spectra of all galaxies with a $S/N$ that depends on the galaxy magnitude. Because all those features contribute to the redshift fitting procedure, we explore their impact on the redshift accuracy in bins of $S/N$ of the strongest emission line in the spectrum and [$JH_{\rm 140}$]{} total magnitude. Given the limited spectral coverage of the G141 grism, it is common to find only one prominent emission line feature in the spectrum [@Momcheva16]; this justifies our approach of using the $S/N$ of the strongest line. [We define the redshift accuracy ($\sigma_{v,{\rm acc}}$) as half the separation of the 16$^{\rm th}$ and 84$^{\rm th}$ confidence levels obtained from the probability density function (PDF) of grism redshifts as derived from the EAZY template fitting procedure. In the case of fits obtained without including the spectral information, it becomes a pure photometric redshift uncertainty.]{} A comparison of grism redshifts to spectroscopic redshifts shows that $\sim 800~\rm{km~s^{-1}}$ should be added to the formal uncertainty on the grism redshifts to obtain a scatter in $\Delta z/\sigma(z)$ with a $1\sigma$ width of unity. This “intrinsic grism” uncertainty can arise from morphological effects, i.e. the light-weighted centroid of the gas emission can be offset from that of the stars [see @Nelson16a; @Momcheva16]. In this analysis, we added the intrinsic uncertainty of the grism data in quadrature to the formal uncertainty from the fitting process. Figure \[fig1\] shows $\sigma_{v,{\rm acc}}$, for bins of emission line $S/N$ and [$JH_{\rm 140}$]{} magnitude. The bottom right panel (f) shows the accuracy of photometric redshifts for the same sources highlighting the significant improvement on the redshift quality when the spectra are included. From the top panels of Figure \[fig1\], it is clear how an emission line detection narrows the redshift PDF to the intrinsic uncertainty, irrespective of the stellar continuum features. At $S/N$ where the emission line becomes less dominant, we start to witness a magnitude dependence of the redshift accuracy. Brighter galaxies have better continuum detections and therefore a more accurate redshift. Even when there is no line detection (Panel e), the typical redshift uncertainties are a factor 2-3 lower than pure photometric redshifts. The inclusion of the spectra helps the determination of the redshifts even when the spectra are apparently featureless. [The grism redshift accuracy is comparable to the pure photometric redshift accuracy only for the faintest objects ([$JH_{\rm 140}$]{}$>$ 23 mag) with no emission line detection ($S/N<2$), a population which accounts for $\sim 10\%$ of our grism sample.]{} As a final note of caution, we highlight that whenever the information in the spectra is limited, the final grism redshift accuracy depends largely on the photometric data, whose availability depends on the field. Indeed, COSMOS and GOODS-S have been extensively observed with narrow or medium band filters [@Taniguchi07; @Cardamone10; @Whitaker11] resulting in better photometric redshifts compared to the other fields. However, as shown in Section \[sec\_halomass\] these field-to-field variations have negligible effects on our calibration of halo mass. Quantification of the environment {#sec_environment} ================================= There are many ways to describe the environment in which a galaxy lives [e.g., @Haas12; @Muldrew12; @Etherington15]. In this work, we apply to observational data the method we explored and calibrated in @Fossati15 and based on the work of @Wilman10. We use the number density of neighbouring galaxies within fixed cylindrical apertures because it is more sensitive to high overdensities, less biased by the viewing angle, more robust across cosmic times, and easier to physically interpret and calibrate than the N$^{\rm th}$ nearest neighbour methods [@Shattow13]. Density {#subsec_density} ------- We consider all 3D-HST galaxies selected in Section \[obssample\] to be part both of the primary (galaxies for which the density is computed) and neighbour samples. We calculate the projected density $\Sigma_{r_{\rm ap}}$ in a combination of circular apertures centered on the primary galaxies with radii $r_{\rm ap}$. The apertures range from 0.25 to 1.00 Mpc in order to cover from intra-halo to super-halo scales. For a given annulus defined by $r_{\rm ap}$, the projected density is given by $$\Sigma_{r_{\rm ap}} = \frac{w_{r_{\rm ap}}}{\pi \times r_{\rm ap}^2} \label{eqSigma}$$ where $w_{r_{\rm ap}}$ is the sum of the weights of galaxies in the neighbour sample falling at a projected distance on the sky $r < r_{\rm ap} $ from the primary galaxy and within a relative rest-frame velocity $\pm dv$. For the 3D-HST galaxies with a grism or spectroscopic redshift, the weights are set to unity (non weighted sum), while for galaxies with pure photometric redshifts, we apply a statistical correction for the less accurate redshifts as described in Section \[edgecor\]. The primary galaxy is not included in the sum therefore isolated galaxies have $\Sigma = 0$. We set the velocity cut at $dv = 1500 \rm{km~s^{-1}}$. This value is deemed appropriate for surveys with complete spectroscopic redshift coverage [@Muldrew12] and for 3D-HST given the quality of grism redshifts shown in Figure \[fig1\]. A small value of $dv$ avoids the peaks in the environmental density to be smoothed by interlopers in projection along the redshift axis. On the other hand, if only less accurate redshifts are available, a larger cut must be used to collect all the signal from overdense regions which is artificially dispersed along the redshift axis [see Figure 4 in @Fossati15; @Etherington15]. ![image](fig2.eps) Because the mean number density changes continuously with redshift, it is not possible to compare the local density ($\Sigma$) across time. Instead we define a relative overdensity $\delta$, which is given by: $$\delta{r_{\rm ap}} = \frac{\Sigma_{r_{\rm ap}} - \Sigma_{\rm mean}(z)}{\Sigma_{\rm mean}(z)} \label{eqoverdens}$$ where $\Sigma_{\rm mean}(z)$ is the average surface density of galaxies at a given redshift. This is obtained by computing the volume density of galaxies (per Mpc$^3$) in the whole survey and parametrising the redshift dependence with a third degree polynomial. This value is multiplied by the depth of the cylindrical aperture at redshift $z$ to obtain the surface density $\Sigma_{\rm mean}(z)$. Throughout the paper, we will mainly use the overdensity in terms of the logarithmic density contrast defined as $\log(1+\delta{r_{\rm ap}})$. Edge corrections {#edgecor} ---------------- The calculation of the environment of primary galaxies at the edges of the 3D-HST footprint (see Figure \[figfootprints\], red areas) suffers from incomplete coverage of neighbours that results into an underestimated density in the considered aperture. In large scale surveys [e.g. SDSS, @Wilman10], it is common practice to remove galaxies too close to the edges of the observed field. In the case of deep fields, however, the observed area is relatively small and the removal of such galaxies would reduce total number of objects significantly. One possible solution is to normalize the densities by the area of the circular aperture which is within the survey footprint in equation \[eqSigma\]. Although this is a simple choice, it assumes a constant density field and neglects possible overdense structures just beyond the observed field. A more accurate solution consists of building up galaxy catalogues for a more extended area than 3D-HST and then use galaxies within these areas as “pure neighbours” for the environment of the primary 3D-HST galaxies. Given the amount of publicly available data, this is possible in GOODS-S, COSMOS and UDS (see Figure \[figfootprints\], blue areas). In appendix \[edgecor\_cats\] we describe the data, depth and redshift quality of the catalogues we built in those fields. Here we present the edge correction method we developed and how it was tuned to perform the edge corrections in the other two fields GOODS-N and AEGIS. ![image](fig3.eps) ### Edge correction method for GOODS-S, COSMOS, and UDS {#Edgecormethod} The availability of spectroscopic redshifts in the extended area catalogs is limited (from $\sim5\%$ in COSMOS and UDS to $\sim 15\%$ in GOODS-S). We thus need to deal with the limited accuracy of photometric redshifts for the galaxies in those fields. The photo-z accuracy, which varies from field to field and depends on the redshift, brightness and color of the objects [@Bezanson16], is such that most of the sources which are part of the same halo in real space would not be counted as neighbours of a primary galaxy, simply due to the redshift uncertainty. @Fossati15 show that increasing the depth of the velocity window would recover most of the real neighbours but at the expense of a larger fraction of interlopers (galaxies which are not physically associated to the primary). Here, we thus exploit a different method. We assume that galaxies which are at small angular separation and whose redshifts are consistent within the uncertainties are, with a high probability, physically associated [e.g., @Kovac10; @Cucciati14]. If one of them has a secure spectroscopic redshift, we assign this to the others. Our method works as follows: - For each galaxy with a photometric redshift, we select all neighbours with a redshift within $dv_{\rm phot} = \pm10.000~\rm{km~s^{-1}}$. This value is chosen to recover most of the real neighbours given the average photo-z uncertainties. - Among those neighbours, we select the closest (in spatial coordinates) which has a secure spectroscopic (or grism) redshift. Here we assume grism redshifts to have a negligible uncertainty compared to photo-z. - We replace the photo-z of the galaxy of interest with this spec-z (or grism-z). Since the statistical validity of the assumption of physical association depends on the distance of the neighbour, for increasing distances we underestimate the true clustering. We correct for the bias by assigning a weight $w_{ph}$ to each galaxy. The weight is evaluated on a training sample made of galaxies in 3D-HST with $JH_{\rm 140} < 23$ mag. For each galaxy in the three fields, we compute the “real” density ($\Sigma_{\rm real}$ in a 0.75Mpc radius and $dv = \pm1500~\rm{km~s^{-1}}$) using spec-z or grism-z from 3D-HST. We then take for each galaxy its photometric redshift, and follow the procedure described above, but, instead of choosing the closest neighbour with a secure redshift, we select a random neighbour in different bins of projected sky distance (from 0 to 3 Mpc in bins of 0.5 Mpc width). Then we compute densities with each of those distance replacements separately and the fractional bias ($b_d$) as: $$b_d = \frac{\Sigma_{d}- \Sigma_{\rm real} }{\Sigma_{\rm real}} \label{eqedgecor}$$ where the $d$ subscript denotes the replacement with a spec-z of a galaxy found at distance $d$. By using the 3D-HST data, we make sure that there are always a large number of neighbours with a secure redshift, and we repeat this procedure 1000 times in order to uniformly sample the neighbours. Figure \[figedgecor\] left panel shows $b_d$ as a function of the real density in four bins of $d$. Clearly, the larger $d$ is, the more underestimated the real density will be, due to a decreasing fraction of correct redshift assignments. We then derive the median weight $w_{ph,d} = {\rm med}((b_d+1)^{-1})$ where the median is computed among all galaxies that have $\Sigma_{\rm real} > 9.5 {\rm Mpc^{-2}}$ (see the vertical dashed line in Figure \[figedgecor\] left panel). The density dependence of $b_d$ is negligible at these densities, therefore by avoiding underdense regions (where the uncertainty on $b_d$ is large) we obtain a robust determination of $w_{ph,d}$. Figure \[figedgecor\] middle panel shows $w_{ph,d}$ versus $d$, which we fit with a quadratic relation obtaining: $$w_{ph,d} = 9.66\times10^{-2} \times d^2 + 0.155 \times d + 0.946 \label{edgecorweights}$$ with the additional constraint that $w_{ph,d} \ge 1$ which corresponds to $w_{ph,d} = 1$ for $d<0.29$ Mpc. We tested that this relation, although obtained combining all fields, holds within the uncertainties when each field is considered separately. Lastly we show in Figure \[figedgecor\] right panel how the systematic bias is removed when the weight is applied to all neighbours when computing the density. This is consistent with no bias within the uncertainties for all the distance bins. ### Edge correction method for GOODS-N and AEGIS The GOODS-N and AEGIS fields do not have deep and extended near-infrared public catalogues that can be used to derive the edge corrections as presented above. As shown in Figure \[figfootprints\] (light blue shaded areas) the 3D-HST/CANDELS footprint slightly extends beyond the area covered by G141 grism observations (the main requirement for our primary sample). Therefore the 3D-HST/CANDELS catalogue itself can be used to perform edge corrections. We derive [$JH_{\rm 140}$]{} magnitudes from the [$J_{\rm 125}$]{} magnitudes using a linear function derived from the five 3D-HST fields ($JH_{140} = 1.000 \times J_{125} - 0.295$). We then use 3D-HST photometric redshifts (or spec-z where available) and apply the method described in Section \[Edgecormethod\]. However, the 3D-HST/CANDELS photometric catalogues do not extend enough beyond the primary sample area to ensure the apertures used to compute the density are entirely covered by the photometric catalog footprint. For this reason, we compute the densities using the area of the circular aperture within the photometric catalogue. We test this method by comparing the density ($\Sigma_{\rm real}$) in a 0.75Mpc aperture measured using the extended catalogues for COSMOS, GOODS-S, UDS and the density ($\Sigma$) measured correcting for the fraction of the aperture ($f_{\rm area,0.75}$) in the 3D-HST/CANDELS footprint. The result is shown in Figure \[edgecor\_frac\]. We note that although the median (red solid line) is consistent with no bias, the area correction introduces a scatter (dotted and dashed lines) which increases by decreasing the fraction of the aperture in the footprint. In conclusion, the environment catalogue released with this work includes all the primary galaxies in the five 3D-HST fields. The structure of the catalogue is described in Appendix \[envcatalogue\]. However, in the rest of this work we only include galaxies for which $f_{\rm area,0.75} > 0.9$ for the GOODS-N and AEGIS fields. The total number of objects in the primary 3D-HST sample with a robust determination of the environmental density is therefore reduced to 17397 ($93\%$ of the original sample). ![[Logarithmic offset between the density ($\Sigma_{\rm real}$) in a 0.75Mpc aperture measured using the extended catalogues for COSMOS, GOODS-S, UDS and the density ($\Sigma$) measured using the fraction of the aperture in the 3D-HST footprint ($f_{\rm area,0.75}$) as a function of the latter quantity]{}. The solid line is the median while dashed and dotted lines mark the 1$\sigma$ and 2$\sigma$ confidence intervals respectively. The offset between the two methods is zero, with a scatter which increases with decreasing fraction of the aperture in the 3D-HST footprint. []{data-label="edgecor_frac"}](fig4.eps){width="8cm"} Overdensities in the 3D-HST deep fields {#sec_overdensities} ======================================= ![image](fig5.eps){width="18cm"} ![image](fig6.eps){width="17cm"} ![image](fig7.eps){width="18cm"} ![image](fig8.eps){width="18cm"} ![image](fig9.eps){width="18cm"} In order to explore correlations of galaxy evolution with environment, we need to make sure the 3D-HST fields span a wide range of galaxy (over-)densities, and use known structures as a sanity check of our density estimates. Figures \[densityGOODSS\], \[densityCOSMOS\], \[densityUDS\], \[densityAEGIS\], and \[densityGN\] present the primary sample of 3D-HST galaxies in the five fields color coded by their overdensity in the 0.75 Mpc aperture in different redshift slices. This aperture corresponds to the typical virial radius of massive haloes ($M_h>10^{13.5} M_\odot$) in the redshift range under study. The range of density probed is wide and spans from isolated galaxies to objects for which the local number of neighbours is up to ten times larger than the mean at that redshift, reaching the regime of clusters or massive groups. In each Figure, we overplot the position and extent of X-Ray extended emission from the hot intragroup (and intracluster) medium that fills massive haloes. The exquisite depth of X-Ray data in the deep fields [@Finoguenov07; @Finoguenov10; @Finoguenov15; @Erfanianfar13] allows the detection of the hot gas from haloes down to $M_{\rm h} \sim 10^{13} M_\odot$. We find a very satisfactory agreement between our overdensities and the X-Ray emission position. Indeed, most of the X-Ray groups are coincident with large overdensities in our maps. On the other hand, not all the overdense structures identified in our work are detected in X-Ray. We speculate this is mainly due to the presence of more than one massive structure along the line of sight or that low mass groups may not yet be virialized. Lastly, we note that the redshift of the X-Ray emission is assigned based on the photometric or spectroscopic information available at the epoch of the publication of the catalogue; these data might not have been as accurate as the density field reconstruction performed in this work. Our analysis has therefore the potential to spectroscopically confirm more X-Ray groups and improve the quality of previous redshift assignments. Several other works have also analysed, with different techniques, the presence of overdense structures in the deep fields. We overplot on Figure \[densityGOODSS\] the position of overdensities found in the GOODS-S field by @Salimbeni09. These have been derived from the GOODS-MUSIC catalogue using a smoothed 3D density searching algorithm. These data have 15% spectroscopic redshifts and photometric redshifts for the remaining fraction. Because the smoothing technique is less able to constrain the size of the structure, we plot circles with an arbitrary radius. The structures within the 3D-HST footprint (except those at $z>2$) are confirmed with our data to be at least a factor of $2-3$ denser than the mean. The differences in samples and techniques hamper a more quantitative comparison. Our data confirm with a high degree of significance the detection of two well known super-structures, one at redshift $z=0.73$ [@Gilli03; @Adami05; @Trevese07] and one at redshift $z=1.61$ first detected by @Kurk09. The latter is made of 5 peaks in the photo-z map (which correspond to putative positions for the X-Ray emission, see Table 1 in @Finoguenov15). The main structure is robustly recovered by our analysis while the other sub-structures are only mild ($\log(1+\delta_{0.75}) \sim 0.5$) overdensities. In the COSMOS field (see Figure \[densityCOSMOS\]), @Scoville07 applied an adaptive smoothing technique [similar to @Salimbeni09] to find large scale structures at $z<1$. While their results do not constrain the size of the structure and are less sensitive to very compact overdensities, we do find that their detections in the 3D-HST footprint correspond to high overdensities in our work. Similarly in the UDS field, we do detect a very massive cluster surrounded by filaments and less massive groups (upper left panel of Figure \[densityUDS\]) at $z=0.65$ (Galametz et al. in prep.). Another well known structure in this field is located at $z=1.62$ [@Papovich10; @Tanaka10]. Despite being only partially covered by the 3D-HST grism observations (isolated pointing on the left of the contiguous field), we do find it corresponds to a large overdensity of galaxies thanks to our accurate edge corrections using UKIDSS-UDS photometric data. In summary, our reconstruction of the density field in the 3D-HST deep fields recovers the previously known massive structures across the full redshift range analysed in this work. The model galaxy sample {#sec_models} ======================= The goal of this work is to understand the environment of galaxies in the context of a hierarchical Universe. To reach this goal, we need to calibrate physically motivated quantities using observed metrics of environment by means of semi-analytic models (SAM) of galaxy formation. We make use of light cones from the latest release of the Munich model presented by @Henriques15. This model is based on the Millennium N-body simulation [@Springel05] which has a size of $500 h^{-1}$ Mpc. The simulation outputs are scaled to cosmological initial conditions from the Planck mission (Planck Collaboration XVI, [-@Planck14]): $\sigma_8 = 0.829, H_0 = 67.3 \rm{km~s^{-1}~Mpc^{-1}}, \Omega_\Lambda = 0.685, \Omega_M =0.315$. Although those values are slightly different from those used in our observational sample, the differences in cosmological parameters have a much smaller effect on mock galaxy properties than the uncertainties in galaxy formation physics [@Wang08; @Fontanot12; @Guo13a]. This model includes prescriptions for gas cooling, size evolution, star formation, stellar and active galactic nuclei feedback and metal enrichment as described by e.g. @Croton06 [@DeLucia07; @Guo11]. The most significant updates concern the reincorporation timescales of galactic wind ejecta that, together with other tweaks in the free parameters, reproduce observational data on the abundance and color distributions of galaxies from $z=0$ to $z=3$ [@Henriques15]. Our choice of this model is therefore driven by those new features which are critical for an accurate quantification of the environment. We make use of the model in the form of 24 light cones, which are constructed by replicating the simulation box evaluated at multiple redshift snapshots. Before deriving the density for the light cones, as described in Section \[subsec\_density\], we first match the magnitude selection and redshift accuracy of the 3D-HST survey. Sample selection ---------------- SAMs are based on N-body dark matter only simulations. Therefore (and opposite to observations), the galaxy stellar masses are accurate quantities, while observed magnitudes are uncertain and rely on radiative transfer and dust absorption recipes implemented in the models. On the other hand, magnitudes are direct observables in a survey (like 3D-HST) are therefore known with a high degree of accuracy. To overcome these limitations and the fact that [$JH_{\rm 140}$]{} magnitudes are not given in @Henriques15 cones, we employ a method that generates observed magnitudes for SAM galaxies by using observational constraints from 3D-HST. Each model galaxy is defined by its stellar mass ($M_{,\rm{mod}}$), $U-V$ rest frame color ($(U-V)_{\rm{mod}}$) and redshift ($z_{\rm{mod}}$). Similarly, 3D-HST galaxies are defined by stellar mass ($M_{,\rm{obs}}$), $U-V$ rest frame color ($(U-V)_{\rm{obs}}$), redshift ($z_{\rm{obs}}$), and magnitude ($JH_{\rm{obs}}$). The method works as follows: - For each bin of stellar mass (0.25 dex wide) and redshift (0.1 wide) we select all the model and 3D-HST galaxies. - For each model galaxy in this bin we rank the $(U-V)_{\rm{mod}}$ and we find the $(U-V)_{\rm{obs}}$ that corresponds to the same ranking. - We assign to the model galaxy a randomly selected stellar mass-to-light ratio $(M_/L_{JH})_{\rm obs}$ at $(U-V)_{\rm{obs}}\pm0.05$ drawn from the distribution of 3D-HST galaxies in the stellar mass and redshift bin of the mock galaxy of interest. - From $(M_/L_{JH})_{\rm obs}$, $M_{,\rm{mod}}$, and $z_{\rm{mod}}$, we compute $JH_{\rm{mod}}$ for the model galaxy. This method generates [$JH_{\rm 140}$]{} magnitudes for all the model galaxies down to $10^{8} M_\odot$. This is much deeper than the 3D-HST magnitude limit even at the lower end of our redshift range. We then select model galaxies down to a $JH_{\rm{mod,lim}}$ magnitude that matches the total number density of the primary targets ([$JH_{\rm 140}$]{}$<24$ mag) in the five 3D-HST fields to that in the 24 lightcones. This protects us from stellar mass function mismatches between the models and the observations (although those differences are very small in @Henriques15). We employ a $JH_{\rm{mod,lim}} = 23.85$ mag, which is very close to 24 mag further supporting the quality of the stellar mass functions in the models. Matching the redshift accuracy {#subsec_redshiftaccuracy} ------------------------------ After the model sample is selected, the next goal is to assign to each galaxy a redshift accuracy that matches as closely as possible to the one in 3D-HST. To do so, we should not only assign the correct fraction of spec-z, grism-z and photo-z as a function of observed magnitude but also assign an accuracy for the grism-z and photo-z as a function of physical properties such that the final distributions resemble those in Figure \[fig1\]. We showed in Section \[3dhstzaccuracy\] that the grism redshift accuracy depends on the signal-to-noise of the strongest emission line and the galaxy magnitude. For the latter, we use [$JH_{\rm 140}$]{} as derived above, while the former quantity needs to be parametrized in terms of other quantities available in the models. ![Left panel: emission line $S/N$ for 3D-HST galaxies as a function of the [$JH_{\rm 140}$]{} magnitude and line magnitude. Right panel: emission line $S/N$ obtained using the parametrization from equation \[eqlineSN\].[]{data-label="SNgrism"}](fig10.eps){width="9cm"} Figure \[SNgrism\] left panel shows how the emission line $S/N$ depends both on the line flux and the [$JH_{\rm 140}$]{} magnitude for galaxies with a measured line flux. For each galaxy, we take the flux (in units of erg cm$^{-2}$ s$^{-1}$) of the strongest line and we define the line magnitude as $m_{\rm{line}} = -2.5 \times \log(f_{\rm{line}})$. At fixed line flux, brighter galaxies have more continuum, thus decreasing the line $S/N$. This relation is well reproduced by the following parametrization: $$\log(S/N) = -0.33 \times(2\times m_{\rm{line}} - JH_{\rm 140}) + 19.85 \label{eqlineSN}$$ Figure \[SNgrism\] right panel shows the line $S/N$ obtained with this equation. The small differences between the two panels can be due to additional variables not taken into account (e.g. dust extinction or grism throughput). We tested (by perturbing the $S/N$ assigned to each model galaxy) that a more accurate parametrization of this relation is not required for the purpose of this paper. In order to obtain a synthetic line $(S/N)_{\rm mod}$ for the model galaxies, we first convert the star formation rate (SFR) of the model galaxies into an $\rm{H\alpha}$ flux (or $\rm{H\beta}$ flux where $\rm{H\alpha}$ is redshifted outside the grism wavelength range) by inverting the relation given in @Kennicutt98. We then obtain $S/N_{\rm mod}$ from $m_{\rm{line}}$, and $JH_{\rm{mod}}$ using equation \[eqlineSN\]. The rank in $(S/N)_{\rm mod}$ (and not the absolute value) is then matched to that in $S/N$ for the 3D-HST galaxies. Lastly, we assign to each mock galaxy a random grism redshift accuracy such that the observed distributions shown in Figure \[fig1\] are reproduced for the mock sample. A photometric redshift accuracy is also generated using the same distributions (as a sole function of $JH_{\rm{mod}}$). Each model galaxy is then defined by three redshifts: a spectroscopic redshift which is derived from the geometric redshift ($z_{\rm{GEO}}$) of the cones plus the peculiar velocity of the halo, a grism like redshift which is derived from the spec-z plus a random value drawn from a gaussian distribution with sigma equal to the grism redshift accuracy derived above, and a photometric redshift derived as the previous but using the photometric redshift accuracy. The last step in this procedure requires that for each galaxy only one of these three redshifts is selected to generate a “best” redshift. To do so, we work in bins of [$JH_{\rm 140}$]{} magnitude. For each bin of magnitude, the fraction of 3D-HST galaxies with spec-z, grism-z and photo-z is computed. Then in order of descending $(S/N)_{\rm mod}$, the spec-z is taken for a number of galaxies matching the fraction of galaxies with spec-z in the observational catalog, a grism-z is taken for an appropriate number of galaxies and lastly a photo-z is taken for the galaxies with the lowest $(S/N)_{\rm mod}$ which mimic line non-detections in the grism data. We stress that since the grism redshift accuracy is a function of $(S/N)_{\rm mod}$, the quality of grism redshifts for objects with marginal line detections is preserved by this method. Once a catalog of model galaxies is selected and their redshift accuracy matches the 3D-HST catalog, we compute the environment parameters as described in Section \[subsec\_density\]. The only minor difference is that, as the number of model galaxies is very large, we can remove objects closer than 1.0 Mpc from the edges of the cone, to avoid edge biases. Calibration of physical parameters {#sec_halomass} ================================== The local density of galaxies is not the only parameter that describes the environment in which a galaxy lives. Another important parameter is whether a galaxy is the dominant one within its dark matter halo (Central), or if it orbits within a deeper potential well (Satellite). [The definition of centrals and satellites in the mock sample is obtained from the hierarchy of subhaloes (the main units hosting a single galaxy). First, haloes are detected using a friends-of-friends (FOF) algorithm with a linking length $b=0.2$ [@Springel05]. Then each halo is decomposed into subhaloes running the algorithm [SUBFIND]{} [@Springel01], which determines the self-bound structures within the halo. As time goes by, the model follows subhaloes after they are accreted on to larger structures. When two haloes merge, the galaxy hosted in the more massive halo is considered the central, and the other becomes a satellite.]{} In this Section, we describe how we use the mock catalog to assign a halo mass probability density function (PDF) and a probability of being central or satellite to 3D-HST galaxies. The method builds on the idea of finding all the galaxies in the mock lightcones that match each 3D-HST galaxy in redshift, density, mass-rank (described below), and stellar mass (within the observational uncertainties). The main advantage of using multiple parameters is to break degeneracies which are otherwise dominant if only one parameter is used [e.g. to account for the role of stellar mass at low density, where halo mass depends more significantly on stellar mass than density; @Fossati15]. The stellar mass rank in fixed apertures {#sec_censat} ---------------------------------------- @Fossati15 explored how the rank in stellar mass of a galaxy in an appropriate aperture can be a good discriminator of the central/satellite status for a galaxy. This method, which complements the one usually used in local large scale surveys of galaxies based on halo finder algorithms, is more effective with the sparser sampling of high redshift surveys. We refer the reader to @Fossati15 for the details of how this method is calibrated. Here, we recall that we define a galaxy to be central if it is the most massive (mass-rank = 1) within an adaptive aperture that only depends on the stellar mass. Otherwise, if it is not the most massive (mass-rank $> 1$), it is classified as a satellite. The adaptive aperture is motivated by the fact that ideally, the aperture in which the mass-rank is computed should be as similar as possible to the halo virial radius to maximize the completeness of the central/satellite separation and reduce the fraction of spurious classifications. @Fossati15, defined this aperture as a cylinder with radius: $$r_0= 3\times 10^{(\alpha \log M_ + \beta)} ~\rm{[Mpc]}$$ where $M_$ is the stellar mass, $\alpha = 0.25$, and $\beta = -3.40$ are the parameters which describe the dependence of the virial radius with stellar mass. These values are calibrated using the models [see @Fossati15]. We also limit the aperture between 0.35 and 1.00 Mpc. The lower limit is set to avoid small apertures which would result in low mass galaxies being assigned mass-rank = 1 even if they are satellites of a large halo. The upper limit is approximately the radius of the largest haloes in the redshift range under study. The adaptive aperture radius (in Mpc) is therefore defined as: $$r = \begin{cases} 0.35 & \mbox{if } r_0 < 0.35 \\ r_0 & \mbox{if } 0.35 \le r_0 \le 1.00 \\ 1.00 & \mbox{if } r_0 > 1.00 \end{cases} \label{eqadaptiveap}$$ In this work, we have to consider the variable redshift accuracy of 3D-HST galaxies. Therefore fixing the depth of the cylinder to $\pm1500 \rm{km~s^{-1}}$ does not optimize the central versus satellite discrimination. We set the depth of the adaptive aperture cylinder (in $\rm{km~s^{-1}}$) to: $$dv= \begin{cases} 1500 & \mbox{if } \sigma_{v,{\rm acc}} < 1500 \\ \sigma_{v,{\rm acc}} & \mbox{if } 1500 \le \sigma_{v,{\rm acc}} \le 7500 \\ 7500 & \mbox{if } \sigma_{v,{\rm acc}} > 7500 \end{cases} \label{eqadaptivedepth}$$ where $\sigma_{v,{\rm acc}}$ is the redshift accuracy of the primary galaxy. By using the mock sample, we tested that this combination of upper and lower limits gives a pure yet sufficiently complete sample of central galaxies. The simple classification of centrals and satellites based on mass-rank only is subject to a variety of contaminating factors. For instance in galaxy pairs or small groups (where the mass of the real central and satellites are very close), it is difficult to use the stellar mass to robustly define which galaxy is the central. On the other hand, in the infalling regions beyond the virial radius of massive clusters, many central galaxies would be classified as satellites as analysed in detail in @Fossati15. In this work, we go beyond the simple dichotomic definition that each galaxy is either central or satellite using the mass-rank only. We combine multiple observables to derive a probability that each 3D-HST galaxy is central or satellite by matching observed galaxies to mock galaxies. This probabilistic approach naturally takes into account all sources of impurity and is of fundamental importance to separate the effects of mass and environment on the quenching of galaxies. Matching mock to real galaxies ------------------------------ In this Section, we describe how we match individual 3D-HST galaxies to the mock sample to access physical quantities unaccessible from observations only. Our method heavily relies on the fact that the distributions of stellar mass and density (and their bivariate distribution) are well matched between the mocks and the observations across the full redshift range. In the upper and right panels of Figure \[Mstar\_dens\_compare\], we show the distributions of density and stellar mass respectively, while the main panel shows the 2D histogram of both quantities. The overall agreement is very satisfactory and relates to the agreement of the observed stellar mass functions to that from @Henriques15, and to our careful selection of objects. The match of the density distributions also confirms that the redshift assignment for mock galaxies is accurate enough to reproduce the observed density distributions. A good match between models and observations is found for other apertures as well. In the future, it should be possible to improve our method by combining density information on several scales by means of machine learning algorithms. To match observed galaxies to mock galaxies, we also require an estimate of the uncertainty on both the density and the stellar mass. For the stellar mass, we use $\sigma({\log(M_)}) = 0.15$ dex [@Conroy09; @Gallazzi09; @Mendel14]. For the density, the error budget is dominated by the redshift uncertainty of each galaxy and the fact that for a sample of galaxies with given [$JH_{\rm 140}$]{} and emission line $S/N$, the redshift accuracy has a distribution with non zero width. This means that the redshift uncertainty of mock galaxies can only match the observational sample in a statistical sense. To test how the densities of individual galaxies are affected by the redshift uncertainty, we repeat 50 times the process of assigning a redshift to mock galaxies described in Section \[subsec\_redshiftaccuracy\]. We then compute the density for each of those samples independently and analyse the distribution of densities for each galaxy. We find that the distribution roughly follows a Poissonian distribution: $\sigma(\Sigma_{r_{\rm ap}}) = \sqrt{w_{r_{\rm ap}}}/(\pi \times r_{\rm ap}^2)$ Based on this evidence, we match each 3D-HST galaxy to the mock galaxies within $\pm 0.1$ in redshift space and within $\pm \sigma({\log(M_))}$ and $\pm \sigma(\Sigma_{0.75})$ for the stellar mass and density on the 0.75 Mpc scale respectively. The local density is a quantity that depends on the redshift accuracy both of the primary galaxy and of the neighbours, which in turn depends on the emission line strength in the grism data and the galaxy brightness (see Section \[3dhstzaccuracy\]). As a result the density peaks are subject to different degrees of smoothing if the neighbouring galaxies have a systematically poorer redshift accuracy in a given environment. Our mock catalogue is a good representation of the observational sample only if the SFR (from which the syntetic line $S/N$ is derived) and the stellar mass distributions as a function of environment are well reproduced by the SAM. @Henriques16 have shown that the H15 model is qualitatively able to recover the observed trends of passive fraction as a function of environment. By matching model galaxies with a redshift accuracy within $\pm 2000 ~\rm{km~s^{-1}}$ to that of the observed galaxy we introduce no bias in the halo mass distributions: for galaxies with less accurate redshifts, we simply obtain broader PDFs of halo mass. Lastly, we restrict the match for the most massive galaxies (mass-rank $=1$) to the most massive mock galaxies. The rest of the population (mass-rank $>1$) was matched to the same population in the mocks. ### A probabilistic determination of central versus satellite status {#pcensatcalib} [The central and satellite fractions of those matched mock galaxies are used to define a probability that the 3D-HST galaxy under consideration is central ($P_{\rm{cen}}$) or satellite ($P_{\rm{sat}}$): $$P_{\rm{cen}} = \frac{N_{\rm matched~cen}}{N_{\rm matched}},~~P_{\rm{sat}} = \frac{N_{\rm matched~sat}}{N_{\rm matched}} = 1-P_{\rm cen}$$]{} Figure \[PcenPsat\] shows the average values of those quantities in bins of logarithmic density contrast (see Section \[subsec\_density\]) in the 0.75 Mpc aperture and stellar mass for all the 3D-HST galaxies included in our sample. The average value of $P_{\rm{cen}}$ decreases with increasing density and decreasing stellar mass, and the opposite trend occurs for $P_{\rm{sat}}$. High mass haloes (high density regions) are indeed dominated by the satellite population, but objects with high stellar masses are more likely to be centrals. Galaxies in low density environments ($\log(1+\delta_{0.75})<0.2$) are almost entirely centrals. However, in the analysis performed in the next sections we use the values of $P_{\rm{cen}}$ and $P_{\rm{sat}}$ computed for each galaxy instead of the average values (@Kovac14 performs instead an average correction as a function of galaxy density). This takes into full account possible second order dependencies on mass-rank, redshift, or redshift accuracy. We also examine how $P_{\rm{sat}}$ varies as a function of the distance from the center of over-dense structures, such as massive groups or clusters of galaxies. To do so, we take the haloes more massive than $10^{13.5} M_\odot$ in the mock lightcones. We then select all galaxies in a redshift slice centered on the redshift of the central galaxy and within $\Delta z \leq \pm 0.01$ and compute their projected sky positions with respect to the central galaxy. We normalize their positions to the virial radius of the halo and remove the central galaxy. Figure \[Psat\_2Dstack\], top panel, shows the average value of $P_{\rm{sat}}$ as a function of normalized R.A. and Dec. offset from the center of the haloes. The black solid circles mark $r_{\rm vir}$ and $2\times r_{\rm vir}$. Figure \[Psat\_2Dstack\], bottom panel, shows the average value of $P_{\rm{sat}}$ (black solid line) as a function of radial distance from the center of the haloes. The red solid line shows the fraction of satellites in the same radial bins but using the mock definition of satellites. [Lastly, the red dashed line shows the value of $P_{\rm sat}$ including only SAM satellites living in the same halo of the central galaxy. ]{} Our Bayesian definition tracks well the SAM definition of satellites as a function of halo mass. However the real trend is smoothed due to both the transformation from real to redshift space, and the intrinsic uncertainty of our method to extract $P_{\rm{sat}}$ based on observational parameters. [Moreover, $P_{\rm{sat}}$ only drops to $40\%$ at $\sim5\times r_{\rm vir}$. This is caused by satellites from nearby haloes, while the contribution from satellites belonging to the same halo becomes negligible at $\sim3\times r_{\rm vir}$. Indeed, massive structures are embedded in filaments and surrounded by groups which will eventually merge with the cluster. Therefore, even at large distances from the center, the density is higher than the mean density (at $\sim5\times r_{\rm vir}$ the density is $\sim 4$ times higher than the average density)]{}. As a reference we show in Figure \[Psat\_2Dstack\], bottom panel, the value of $P_{\rm{sat}}$ for [a stellar mass and redshift matched sample of]{} galaxies living in average density environments ($0.8 < (1+\delta_{0.75}) < 1.2$, horizontal dashed line). ### The halo mass calibration {#halomasscalib} Similarly, we use the halo masses of matched central and satellite model galaxies to generate the halo mass PDFs given their type ($P_{M_h \| {\rm{cen}}}$ and $P_{M_h \| {\rm{sat}}}$ respectively). Figure \[halopdfexamples\] shows three examples of such PDFs for one object with high $P_{\rm{cen}}$, one with high $P_{\rm{sat}}$, and one object with an almost equal probability of being a central or a satellite. The vertical dashed lines mark the median halo mass for a given type. Although the total halo mass PDF can be double peaked (middle panel), the degeneracy between the two peaks is broken once the galaxy types are separated, making the median values well determined for each type independently. Testing calibrations -------------------- We test the halo mass calibration by comparing the halo mass distributions of the mock sample to the 3D-HST sample. In both panels of Figure \[Mhalodistr\], we plot the halo mass histograms for centrals and satellites of the entire mock sample. The number counts are scaled by the ratio of the volume between the 24 lightcones and the five 3D-HST fields. In the left panel of Figure \[Mhalodistr\] the dashed lines are the halo mass distributions of 3D-HST galaxies obtained by summing the full halo mass PDFs for centrals ($P_{M_h \| {\rm{cen}}}$, red dashed) and satellites ($P_{M_h \| {\rm{sat}}}$, blue dashed) weighted by $P_{\rm{cen}}$ and $P_{\rm{sat}}$ for each galaxy. The agreement with the mock sample distributions is remarkable. Although this is in principle expected because the halo mass PDFs for observed galaxies are generated from the mock sample, it should be noted that we perform the match in bins of redshift, redshift accuracy, stellar mass, density and mass-rank. The good agreement for the whole sample between the derived PDFs and the mock distributions (for centrals and satellites separately) should therefore be taken as an evidence that our method has not introduced any bias in the final PDFs. We take the median value of the halo mass PDFs given that each galaxy is a central ($M_{\rm h,50\|cen}$) or a satellite ($M_{\rm h,50\|sat}$) as an estimate of the “best” halo mass, weighted by $P_{\rm{cen}}$ and $P_{\rm{sat}}$. These values are shown in Figure \[Mhalodistr\], right panel. The agreement with the mock distributions is good. For central galaxies, the shape and extent of the distribution is well preserved. For satellite galaxies, the halo mass range is less extended than the one in the mocks; values above $10^{14.2} M_\odot$ and below $10^{12} M_\odot$ indeed only contribute through the tails of the PDFs, and therefore do not appear when the median of the PDFs are used. In the next Section, we make use of the full PDFs to derive constraints on the environmental quenching of satellite galaxies. However, the satisfactory agreement of single value estimates of halo mass with the mock distributions makes them a valuable and reliable estimate in science applications when the use of the full PDFs is not possible or feasible. Constraining environmental quenching processes at $z=0.5-2$ {#sec_quenching} =========================================================== In this Section, we explore the role of environment in quenching the star formation activity of galaxies over $0.5 < z < 2$ using 3D-HST data. It was first proposed by @Baldry06 that the fraction of passive galaxies depends both on stellar mass and environment in a separable manner. @Peng10, using the SDSS and zCOSMOS surveys, extended the independence of those processes to $z \sim1$. More recently, @Peng12 interpreted these trends in the local Universe by suggesting that central galaxies are only subject to “mass quenching” while satellites suffer from the former plus an “environmental quenching”. @Kovac14 similarly found that satellite galaxies are the main drivers of environmental quenching up to $z\sim0.7$ using zCOSMOS data. Here, we extend these analysis to higher redshift by exploring the dependence of the fraction of passive galaxies on stellar mass, halo mass and central/satellite status in order to derive the efficiency and timescale of environmental quenching. In Appendix \[app\_passfrac\_dens\], we show that we obtain consistent results using the observed galaxy density as opposed to calibrated halo mass. Passive fractions {#sec_passfrac_mhalo} ----------------- The populations of passive and star-forming galaxies are typically separated either by a specific star formation rate cut [e.g. @Franx08; @Hirschmann14; @Fossati15] or by a single color or color-color selection [e.g. @Bell04; @Weiner05; @Whitaker11; @Muzzin13; @Mok13]. In this work, we use the latter method and select passive and star forming galaxies based on their position in the rest-frame UVJ color-color diagram [@Williams09]. Following @Whitaker11, passive galaxies are selected to have: $$(U -V) > 0.88\times(V -J)+0.59$$ $$(U-V)>1.3,(V -J)<1.6~[0.5<z<1.5]$$ $$(U-V)>1.3,(V -J)<1.5~[1.5<z<2.0]$$ where the colors are rest-frame and are taken from @Momcheva16. Figure \[UVJdiagram\] shows the distribution of 3D-HST galaxies in the rest frame UVJ color-color plane. The red solid line shows the adopted division between passive and star forming galaxies. The fractions of passive centrals and satellites in bins of $M_{\rm{}}$ and $M_{\rm{h}}$ are computed as the fraction of passive objects in a given stellar mass bin where each galaxy is weighted by its probability of being central or satellite and the probability of being in a given halo mass bin for its type. Algebraically: $$f_{\rm{pass\|ty}} = \frac{\sum_i \left( \delta_{\rm pass,i} \times \delta_{M_* \rm{,i}} \times P_{\rm{ty,i}} \times \int_{M_{\rm h}} P_{M_{\rm h},i \| ty} dM_{\rm h}\right )}{\sum_i \left(\delta_{M_* \rm{,i}} \times P_{\rm{ty,i}} \times \int_{M_{\rm h}} P_{M_{\rm h},i \| ty} dM_{\rm h}\right)} \label{eqpassive_mhalo}$$ where ty refers to a given type (centrals or satellites), $\delta_{\rm pass,i}$ is 1 if a galaxy is UVJ passive and 0 otherwise, $\delta_{M_* \rm{,i}}$ is 1 if a galaxy is in the stellar mass bin and 0 otherwise, $P_{\rm{ty,i}}$ is the probability that a galaxy is of a given type [(see Section \[pcensatcalib\])]{} and $\int_{M_{\rm h}} P_{M_{\rm h},i \| ty} dM_{\rm h}$ is the halo mass PDF given the type integrated over the halo mass bin limits [(see Section \[halomasscalib\])]{}. The data points in Figure \[Passfrac\_Mhalo\] show the passive fractions in two bins of halo mass (above and below $10^{13} M_\odot$) and in three independent redshift bins. The median (log) halo masses for satellites are 12.36, 13.53 at $z=0.5-0.8$ for the lower and higher halo mass bin respectively; 12.41, 13.44 at $z=0.8-1.2$; and 12.43, 13.34 at $z=1.2-1.8$. The uncertainties on the data points cannot be easily evaluated assuming Binomial statistics because the number of galaxies contributing to each point is not a priori known. Indeed, $P_{\rm{ty,i}}$ and $\int_{M_{\rm h}} P_{M_{\rm h},i \| ty} dM_{\rm h}$ act as weights and all galaxies with a stellar mass within the mass bin do contribute to the passive fraction. To assess the uncertainties we use the mock lightcones (where each mock galaxy has been assigned a $P_{\rm{cen}}$ and $P_{\rm{sat}}$ and halo mass PDFs as if they were observed galaxies). In a given stellar mass bin we assign each model galaxy to be either passive or active such that the fraction of passive galaxies matches the observed one. Then we randomly select a number of model galaxies equal to the number of observed galaxies in that bin and we compute the passive fraction of this subsample using equation \[eqpassive\_mhalo\]. We repeat this procedure 50000 times to derive the $1\sigma$ errorbars shown in Figure \[Passfrac\_Mhalo\]. This method accounts for uncertainties in the estimate of $P_{\rm{ty,i}}$ and $\int_{M_{\rm h}} P_{M_{\rm h},i \| ty} dM_{\rm h}$ as well as cosmic variance. [The vertical dashed lines, in Figure \[Passfrac\_Mhalo\], mark the stellar mass completeness limit derived following @Marchesini09. In brief, we use the 3D-HST photometric catalog (down to [$JH_{\rm 140}$]{} = 25 mag) and we scale the stellar masses of the galaxies as if they were at the spectroscopic sample limit of [$JH_{\rm 140}$]{}= 24 mag (which defines the sample used in this work). The scatter of the points is indicative of the $M/L$ variations in the population at a given redshift. We then take the upper $95^{th}$ percentile of the distributions as a function of redshift as the stellar mass limit, which is approximately $\sim10^{9.5}$ and $\sim 10^{10.5}$ for old and red galaxies at $z=1$ and $z=2$ respectively. ]{} Below this mass we limit the upper edge of the redshift slice such that all galaxies in the stellar mass bin are included in a mass complete sample. A stellar mass bin is included only if the covered volume is greater than $1/3$ of the total volume of the redshift slice. This typically results in only one stellar mass bin below the completeness limit being included in the analysis. In the highest halo mass bin of Figure \[Passfrac\_Mhalo\] at $z=0.5-0.8$, the satellite passive fraction (integrated over all galaxies) is higher than the central passive fraction, with a marginal significance. The same trend can be observed in the other halo mass and redshift bins, although the separation of the observed satellite and central passive fractions becomes more marginal. In each redshift bin we also identify a sample of “pure” central galaxies ($P_{\rm cen} > 0.8$, irrespective of overdensity or halo mass), which provides a reference for the passive fraction of galaxies subject only to mass-quenching. The passive fraction of this sample $f_{\rm{pass\|cen,pure}}(M_)$ of centrals (which has an average $P_{\rm cen} = 0.95$) is shown as the thick red line in both halo mass bins. The separation of the observed satellite passive fraction from that of the pure sample of centrals is more significant (especially at $z<1.2$). Indeed, the passive fractions derived using equation \[eqpassive\_mhalo\] can be strongly affected by impurities in the central/satellite classification and by cross-talk between the two halo mass bins, given that each galaxy can contribute to both bins and types (see equation \[eqpassive\_mhalo\]). Any contribution of central galaxies to the satellite passive fraction, and vice versa, will reduce the observed difference between the two populations with respect to the “pure”, intrinsic difference. Recovering the “pure” passive fractions for satellite galaxies {#sec_decontamination} -------------------------------------------------------------- In order to recover the “pure” passive fraction for satellite galaxies as a function of halo mass, we perform a parametric model fitting to our dataset. We start by parametrizing the probability of a satellite galaxy being passive independently in each stellar mass bin as a function of log halo mass, using a broken function characterized by a constant value ($P_{\rm{pass,lo}}$) below the lower break ($M_{\rm br,lo}$) and another constant value ($P_{\rm{pass,hi}}$) above the upper break ($M_{\rm br,hi}$). In between the breaks, the passive fraction increases linearly. Algebraically, this 4-parameter function is defined as: $$P_{\rm{pass\|sat}}(M_{\rm h}) = \begin{cases} P_{\rm{pass,lo}} & \mbox{if } M_{\rm h} \leq M_{\rm br,lo} \\ m \times (\log \frac{M_{\rm h}}{M_{\rm br,lo}})+P_{\rm{pass,lo}} & \mbox{if } M_{\rm br,lo} < M_{\rm h} \leq M_{\rm br,hi} \\ P_{\rm{pass,hi}} & \mbox{if } M_{\rm h}>M_{\rm br,hi} \end{cases} \label{eqpassive_satmodel}$$ where $m = (P_{\rm{pass,hi}}-P_{\rm{pass,lo}})/(\log(M_{\rm br,hi})-\log(M_{\rm br,lo}))$. This function is chosen to allow for a great degree of flexibility. We make the assumption that satellite galaxies are not subject to environmental quenching below $M_{\rm br,lo}$, and therefore treat $P_{\rm{pass,lo}}$ as a nuisance parameter of the model with a Gaussian prior centered on the observed passive fraction of pure centrals $f_{\rm{pass\|cen,pure}}(M_)$ and a sigma equal to its uncertainty. For $P_{\rm{pass,hi}}$, instead we assume a semi-Gaussian prior with the same center and sigma as above, but only extending below the observed passive fraction of central galaxies (this implies that satellites are affected by the same mass-quenching as centrals). Above this value we assume a uniform prior. For the break masses we assume uniform priors. Table \[modelpars\] summarizes the model parameters, their allowed range, and the number of bins in which the range is divided to compute the posterior. Parameter Range Nbins Prior ---------------------- ------- ------- -------------------------------------------------------------- $\log M_{\rm br,lo}$ 11,15 80 Uniform $\log M_{\rm br,hi}$ 11,15 80 Uniform Gaussian (if $P_{\rm{pass,hi}} \leq f_{\rm{pass\|cen,pure}}$) Uniform (if $P_{\rm{pass,hi}} > f_{\rm{pass\|cen,pure}}$) : Table of the model parameters. []{data-label="modelpars"} The probability that each 3D-HST galaxy, $i$ is passive is: $$P_{\rm{pass,i}} = P_{\rm{cen,i}} \times P_{\rm{pass\|cen}}+P_{\rm{sat,i}} \times \int_{M_{\rm h}} P_{M_{\rm h}i \| {\rm sat}} \times P_{\rm{pass\|sat}} dM_{\rm h}$$ where $P_{\rm{pass\|sat}}$ is from equation \[eqpassive\_satmodel\] and $P_{\rm{pass\|cen}}=f_{\rm{pass\|cen,pure}}$. The likelihood space that the star forming or passive activity of 3D-HST galaxies in a stellar mass bin is reproduced by the model is computed as follows: $$\mathcal{L} = \prod_{i} \begin{cases} P_{\rm{pass,i}} & \mbox{if } i {\rm ~is~UVJ~passive} \\ 1-P_{\rm{pass,i}} & \mbox{if } i {\rm ~is~not~UVJ~passive} \end{cases}$$ We compute the posterior on a regular grid covering the parameter space. We then sample the posterior distribution and we apply the model described in equation \[eqpassive\_satmodel\] to obtain the median value of $P_{\rm{pass\|sat}}$ and its $1\sigma$ uncertainty as a function of halo mass. Lastly, we assign the probability of being passive to mock satellites in each stellar mass bin according to their model halo mass, and we compute the average passive fraction in the two halo mass bins (above and below $10^{13}M_\odot$). This results in the thick blue ($f_{\rm{pass\|sat,pure}}(M_)$) lines with $1\sigma$ confidence intervals plotted as shaded regions in Figure \[Passfrac\_Mhalo\]. We illustrate in Appendix \[app\_decontamination\] an example of this procedure applied to a single redshift bin. We verify that the separation seen in the pure passive fractions in Figure \[Passfrac\_Mhalo\] is real. To do so we randomly shuffle the position in the UVJ diagram for galaxies in each stellar mass bin (irrespective of environmental properties) to break any correlation between passive fraction and environment. Then we compute the observed passive fractions of centrals and satellites, and for the pure sample of centrals and we perform again the model fitting procedure. At $0.5 < z < 0.8$ we find that the pure satellite passive fraction is inconsistent with the null hypothesis (no satellite quenching) at a $\gtrsim 2 \sigma$ level in each stellar bin at high halo mass and 7 out of 8 stellar mass bins at low halo mass. The combined probability of the null hypothesis is $P <10^{-10}$ in either halo mass bin. The difference is smaller, but still very significant ($P \lesssim 10^{-5}$) at $0.8<z<1.2$. At $1.2<z<1.8$ the hypothesis of no satellite quenching is acceptable ($P \sim 0.4$) in the low halo mass bin, while it can be ruled out ($P \lesssim 10^{-5}$) at higher halo mass. [-@van-der-Burg13], @Kovac14, and @Balogh16 have found that the environment plays an important role in determining the star formation activity of satellites, at least up to $z \sim 1$. However these works have only probed relatively massive haloes ($M_{\rm h} \gtrsim 10^{13} M_\odot$). The depth of the 3D-HST sample allows us, for the first time, to extend these results to higher redshift, to lower mass galaxies and to lower mass haloes. Satellite quenching efficiency {#sec_convfrac_mhalo} ------------------------------ In order to further understand the increased passive fractions for satellite galaxies we compute the “conversion fractions” as first introduced by @van-den-Bosch08. This parameter, sometimes called the satellite quenching efficiency, quantifies the fraction of galaxies that had their star formation activity quenched by environment specific processes since they accreted as satellites into a more massive halo [see also @Kovac14; @Hirschmann14; @Balogh16]. It is defined as: $$f_{\rm conv} (M_, M_{\rm h}) = \frac{f_{\rm{pass\|sat,pure}}(M_, M_{\rm h})-f_{\rm{pass\|cen,pure}}(M_)}{1-f_{\rm{pass\|cen,pure}}(M_)} \label{eqConvfrac}$$ where $f_{\rm{pass\|sat,pure}}(M_, M_{\rm h})$ and $f_{\rm{pass\|cen,pure}}(M_)$ are the corrected fractions of quenched centrals and satellites in a given bin of $M_$ and $M_{\rm h}$ obtained as described above. In equation \[eqConvfrac\] we compare the sample of centrals at the same redshift as the satellites. This builds on the assumption that the passive fraction of central galaxies only depends on stellar mass and that the effects of mass and environment are independent and separable. The conversion fraction then represents the fraction of satellites which are quenched due to environmental processes above what would happen if those galaxies would have evolved as centrals of their haloes. A different approach would be to compare the passive fraction of satellites to that of centrals at the time of infall in order to measure the total fraction of satellites quenched since they were satellites [e.g., @Wetzel13; @Hirschmann14]. However this measurement includes the contribution of mass-quenched satellite galaxies, which we instead remove under the assumption that the physical processes driving mass quenching do not vary in efficiency when a galaxy becomes a satellite. We also caution the reader that equation \[eqConvfrac\] has to be taken as a simplification of reality as it does not take into account differential mass growth of centrals and satellites which can be caused by tidal phenomena in dense environments or different star formation histories. Figure \[Convfrac\_Mhalo\] shows the conversion fractions in the same bins of $M_{\rm{}}$, $M_{\rm{h}}$ and redshift as presented in Figure \[Passfrac\_Mhalo\]. Previous results from galaxy groups and clusters from @Knobel13, and @Balogh16 are plotted in our higher halo mass bin (colored points with errorbars). We also add the conversion fractions from @Kovac14 obtained from zCOSMOS data as a function of local galaxy overdensity. We plot their overdensity bins above the mean overdensity in our higher halo mass bin and the others in our lower halo mass bin following the overdensity to halo mass conversion given in @Kovac14. The agreement of our measurements with other works is remarkable considering that different techniques to define the environment (density and central/satellite status) and passiveness are used in different works. The satellite quenching efficiency tends to increase with increasing stellar mass and to decrease with increasing redshift at fixed stellar mass. In the lower halo mass bin, we note the presence of similar trends as at higher halo masses although the uncertainties are larger due to the smaller number of satellites. In our probabilistic approach this is due to the lower $P_{\rm sat}$ in low density environments as shown in Figure \[PcenPsat\]. Moreover, $f_{\rm conv}$ is poorly constrained at $M_ > 10^{11} M_\odot$ due to small number statistics of high mass satellites in the 3D-HST fields. Quenching timescales {#sec_tquench_mhalo} -------------------- A positive satellite conversion fraction can be interpreted in terms of a prematurely truncated star formation activity in satellite galaxies compared to field centrals of similar stellar mass. [We define the quenching timescale ($T_{\rm quench}$) as the average time elapsed from the first accretion of a galaxy as satellite to the epoch at which the galaxy becomes passive, and we estimate it by assuming that galaxies which have been satellites for longer times are more likely to be quenched [@Balogh00; @McGee09; @Mok14]. Indeed, the quenching can be interpreted to happen a certain amount of time after satellite galaxies cease to accrete material (including gas) from the cosmic web (see Section \[sec\_discussion\]).]{} In practice, we obtain quenching timescales from the distribution of $T_{\rm sat}$ for satellite galaxies, which we define as the time the galaxy has spent as a satellite of haloes of any mass since its first infall [e.g., @Hirschmann14]. For each bin of $M_{\rm{}}$, $M_{\rm{h}}$, and redshift we select all satellite galaxies in our mock lightcones which define the distribution of $T_{\rm sat}$. Then we select as the quenching timescale the percentile of this distribution which corresponds to $1-f_{\rm conv}(M_, M_{\rm h})$. This method builds on the assumption that the infall history of observed satellites is well reproduced by the SAM. Systematic uncertainties can arise in the analytic prescriptions used for the dynamical friction timescale of satellites whose parent halo has been tidally stripped in the N-body simulation below the minimum mass for its detection (the so-called “orphan galaxies”). When this time is too short, too many satellites merge with the central galaxy and are removed from the sample, and vice-versa when the time is too long. @Delucia10 explored the dynamical friction timescale in multiple SAMs, finding a wide range of timescales. However a dramatically wrong dynamical friction recipe impacts the fraction of satellites, the stellar mass functions, and the density-mass bivariate distribution, which we found to be well matched between the mocks and the observations. In principle low stellar mass galaxies ($M_* < 10^{10} M_\odot$) are more affected by the resolution limit of the simulation and their derived quenching timescales might be subject to a larger uncertainty compared to galaxies of higher stellar mass. We verified that this is not the case by comparing the distribution of $T_{\rm sat}$ in two redshift snapshots ($z=1.04$ and $z=2.07$) of @Henriques15 built on the Millennium-I and the Millennium-II simulations. The latter is an N-body simulation started from the same initial conditions of the original Millennium run but with a higher mass resolution at the expense of a smaller volume. The higher resolution means that the sub-haloes hosting low mass satellites galaxies, which can be tidally stripped, are explicitly tracked to lower mass and later times: while these are detected the recipe for dynamical friction is not invoked. We obtain consistent quenching timescales for Millennium-I and Millennium-II based mock catalogues, and therefore we conclude that the analytical treatment of orphan galaxies does not bias our results. Figure \[Tquench\_Mhalo\] shows our derived quenching timescales (black points) in the same bins of $M_{\rm{}}$ and $M_{\rm{h}}$ and redshift as presented in Figures \[Passfrac\_Mhalo\] and \[Convfrac\_Mhalo\]. The observed trend of $f_{\rm conv}$ with stellar mass that is found in both redshift bins turns into a trend of $T_{\rm quench}$. Quenching timescales increase to lower stellar mass in all redshift and halo mass bins, mainly a consequence of the decreasing conversion fraction. This parameter ranges from $\sim 4-5$ Gyr for low mass galaxies to $<2$ Gyr for the most massive ones, and is in agreement with that found by @Balogh16. Remarkably, the dependence of quenching timescale on halo mass is very weak. We overplot in each panel, as gray symbols, the quenching timescales obtained from our sample with the same procedure described above but without separating the data in two halo mass bins. In most of the stellar mass bins we find a good agreement, within the uncertainties, between the black and the gray points. The lack of a strong halo mass dependence is a consequence of the typically shorter time since infall for satellite galaxies in lower mass haloes which largely cancels the lower conversion fraction in low mass haloes, and suggests that the physical process responsible for the premature suppression of star formation in satellite galaxies (when the Universe was half of its present age) is largely independent of halo mass. A mild redshift evolution is also seen when comparing the redshift bins: passive satellites at higher redshift are quenched on a shorter timescale. In the next section we will further explore the redshift evolution of the quenching timescales from $0 < z < 2$ by combining the 3D-HST sample with a local galaxy sample from SDSS. Redshift evolution of the quenching timescales {#sec_redshiftevo} ---------------------------------------------- Figures \[Convfracz\] and \[Tquenchz\] show the evolution of the conversion fraction and the quenching timescale from redshift 0 to 2. We now concentrate on three bins of stellar mass, each of 0.5 dex in width, and ranging from $10^{9.5} M_\odot$ to $10^{11} M_\odot$. Given that $f_{\rm conv}$ (and consequently $T_{\rm quench}$) are poorly constrained at $M_ > 10^{11} M_\odot$ due to the low number statistics of massive satellites, we exclude more massive galaxies from these plots. Similarly, galaxies at $M_* < 10^{9.5} M_\odot$ are only included in the mass limited sample at the lowest end of the redshift range under study, therefore the redshift evolution of $f_{\rm conv}$, and $T_{\rm quench}$ cannot be derived for those low mass galaxies. A stellar mass bin appears in Figures \[Convfracz\] and \[Tquenchz\] only if the stellar mass range above the mass limit is more than half of the entire stellar mass extent of the bin. The values (solid lines) and their associated uncertainties (dashed lines) are obtained by performing the procedure described in the previous sections in overlapping redshift bins defined such that $\Delta z/(1+z) = 0.2$ where $\Delta z$ is the width of the redshift bin and $z$ its center. This means we span larger volumes at higher redshift, modulating the decrease in sample density (Malmquist bias) and retaining sufficient sample statistics. It is also close to a constant bin in cosmic time. The x-axis of both figures is scaled such that the width of the redshift bins is constant and is shown as the horizontal error bar. We include only galaxies in a stellar mass complete sample for each redshift bin. In addition to the 3D-HST based constraints, we add constraints at $z=0$, obtained using the same method to ensure homogeneity. The observational sample is drawn from SDSS and the mock sample from the redshift zero snapshot of the @Henriques15 model. We describe the details of how those datasets are processed in Appendix \[app\_SDSS\]. For this sample we restrict to stellar masses above $10^{9.5} M_\odot$ to avoid including low mass galaxies with large $V_{max}$ corrections. The evolution of $f_{\rm conv}$ as seen in Figure \[Convfrac\_Mhalo\] is now clearly visible over the large redshift range probed by 3D-HST. The fraction of environmentally quenched satellite galaxies is a function of $M_h$, $M_$ and redshift. At fixed redshift $f_{\rm conv}$ is higher for higher mass galaxies and at fixed stellar mass it is higher in more massive haloes. More notably, the redshift evolution follows a decreasing trend with increasing redshift such that at $z\sim 1.5$ the excess of quenching of satellite galaxies becomes more marginal (at least for massive galaxies) as first predicted by @McGee09 using halo accretion models. Several observational works reached a similar conclusion. @Kodama04, @DeLucia07a, and @Rudnick09 found a significant build-up of the faint end of the red sequence (of passive galaxies) in cluster environments from $z\sim1$ toward lower redshift. This implies an increase in the fraction of quenched satellites with decreasing redshift for low mass galaxies. Recently, @Darvish16 found that the environmental quenching efficiency tends to zero at $z>1$, although their analysis is only based on local overdensity and does not separate centrals and satellites. With the 3D-HST dataset we cannot rule out that satellite quenching is still efficient for lower mass satellites at $z>1.5$; deeper samples are required to robustly assess the satellite quenching efficiency at $z\sim 1.5-2.0$. Moving to the present day Universe (SDSS data) does not significantly affect the fraction of environmentally quenched satellites despite the age of the Universe nearly doubling compared to the lowest redshift probed by the 3D-HST sample. The redshift dependence of the quenching timescale originates from the combination of the evolution of $f_{\rm conv}$ and the distributions of infall times for satellite galaxies. The redshift evolution of $f_{\rm conv}$ in the high halo mass bin is well matched by the halo assembly history (at lower redshift they have been satellites on average for more time) and therefore $T_{\rm quench}$ is mostly independent of redshift. However, for lower mass galaxies a mild redshift evolution of $T_{\rm quench}$ might be present. However the slope is much shallower than the ageing of the Universe. For this reason, going to higher redshift, $T_{\rm quench}$ approaches the Hubble time and the satellite quenching efficiency decreases. Despite the large uncertainty on the quenching times at low halo mass, their redshift evolution appears to be largely independent of halo mass. This means that the halo mass dependence of the conversion fractions may be mostly driven by an increase in the time spent as satellites in more massive haloes. At $z=0$ a more significant difference is found between the quenching times in the two halo mass bins. In the next section we discuss which mechanism can produce these observational signatures. Discussion {#sec_discussion} ========== There is a growing consensus that the evolution of central galaxies is regulated by the balance between cosmological accretion, star formation and gas ejection processes in a so-called “equilibrium growth model” [e.g. @Lilly13]. The reservoir of cold gas in each galaxy is replenished by accretion, and will fuel star formation. As the rate of cosmological accretion is correlated with the mass of the halo, this regulates mass growth via star formation. As a result the eventual stellar mass is also tightly correlated with halo mass, driving a tight relation between star formation rate and stellar mass for normal star forming galaxies [the main sequence - MS - of star forming galaxies, e.g. @Noeske07]. When galaxies fall into a more massive halo the accretion of new gas from the cosmic web is expected to cease: such gas will instead be accreted (and shock heated) when it reaches the parent halo [@White91]. More recently @Dekel06 estimate that this process occurs at a minimum halo mass $M_h \sim 10^{12} M_\odot$, which is largely independent of redshift. This roughly corresponds to the minimum halo mass at which satellites are detected in the 3D-HST survey (see Figure \[Mhalodistr\]). Identification of the main mechanism ------------------------------------ There are several additional ways in which a satellite galaxy’s gas and stellar content can be modified through interaction with its environment, including stripping of the hot or cold gas, and tidal interactions among galaxies or with the halo potential itself. An important combined effect is to remove (partially or completely) the gas reservoir leading to the quenching of star formation. However, as pointed out by @McGee14, and @Balogh16, it might not be necessary to invoke these mechanisms of environmental quenching to be effective. The high SFR typical of galaxies at high redshift, combined with outflows, can lead to exhaustion of the gas reservoir in the absence of cosmological accretion. Our approach to link the conversion fractions to the distributions of time spent as satellite is based on the assumption that a galaxy starts to experience satellite specific processes at the time of its first infall into a larger halo and, in particular, that the cosmological accretion is shut off at that time. We now examine whether a pure exhaustion of the gas reservoir can explain the quenching times we observe, or whether additional gas-removal mechanisms are required. First we appeal to the similarity of quenching times in the two halo mass bins shown in Figure \[Tquenchz\] to support the pure gas exhaustion scenario. Other than at $z=0$, the derived quenching times are indeed consistent within the uncertainties, therefore the main quenching mechanism has to be largely independent of halo mass. Ram pressure stripping is often invoked as the main quenching mechanism for satellite galaxies in low redshift clusters [e.g., @Poggianti04; @Gavazzi13a; @Boselli14b]. Its efficiency is a function of the intracluster medium (ICM) density and the velocity of galaxies in the halo. More massive haloes have a denser ICM and satellites move faster through it which exerts a stronger dynamical pressure on the gas leading to faster stripping (and shorter quenching times) in more massive haloes [@Vollmer01; @Roediger05]. Our 3D-HST dataset does not extend to the extreme high mass end of the halo mass function in which ram pressure effects have been clearly observed [e.g., @Sun07; @Yagi10; @Merluzzi13; @Kenney15; @Fossati16], and so the lack of significantly shorter quenching times in the higher halo mass bin is consistent with the lack of stripping, and indeed of any strong halo-mass dependent gas-stripping process. However @Balogh16 find a small halo mass dependence of the quenching times comparing their GEEC2 group sample ($M_h \sim 10^{13.5} M_\odot$) to the GCLASS cluster sample ($M_h > 10^{14} M_\odot$). These evidences might indicate that dynamical stripping can play a minor role in more massive haloes even at $z \sim 1$. At $z=0$ instead, thanks to the large area covered by the SDSS dataset, a number of very massive haloes are included in the higher halo mass bin. This, combined with the presence of hot and dense ICM in massive haloes in the local Universe might be sufficient to explain the shorter quenching times in the high halo mass bin. @Haines15, and @Paccagnella16 found quenching timescales which are possibly shorter in massive clusters of galaxies ($\sim 2-5$ Gyr). However, a quantitative comparison is hampered by the different definitions of the quenching timescale. Delayed then Rapid or Continuous Slow quenching? {#sec_delayedorslow} ------------------------------------------------ Having ruled out gaseous stripping as the main driver of satellite quenching in the range of halo mass commonly probed by our samples ($M_h \lesssim 10^{14} M_\odot$), we now concentrate on how the gas exhaustion scenario can explain the observed values of $T_{\rm quench}$. To explain the quenching times at $z=0$, @Wetzel13 presented a model dubbed the “delayed then rapid” quenching scenario, shown in the top panel of Figure \[Quenchmodels\]. This model assumes that $T_{\rm quench}$ can be divided into two phases. During the first phase, usually called the “delay time” ($T_{d}$), the star formation activity of satellites on average follows the MS of central galaxies. After this phase the star formation rate drops rapidly and satellite galaxies become passive on a short timescale called the “fading time”. @Wetzel13 estimated an exponential fading with a characteristic timescale $\tau_f \sim 0.3-0.8$ Gyr which depends on stellar mass at $z=0$. At $z \sim 1 $, @Mok14 [@Muzzin14]; and @Balogh16 estimated the fading time to be $\tau_f \sim 0.4-0.9$ Gyr, by identifying a “transition” population of galaxies likely to be transitioning from a star forming to a passive phase. These values suggest little redshift evolution of the fading timescale with cosmic time. @McGee14 developed a physical interpretation of this model. These authors assumed that the long delay times are only possible if the satellite galaxy has maintained a multi-phase reservoir which can cool onto the galaxy and replenish the star forming gas (typically molecular) at roughly the same rate as the gas is lost to star formation (and potentially outflows). A constant molecular gas reservoir produces a nearly constant SFR according to the Kennicutt-Schmidt relation [@Schmidt59; @Kennicutt98a]. Then the eventual depletion of this cold gas results in the rapid fading phase. An alternative scenario would be that satellite galaxies retain only their molecular gas reservoirs after infall. In this case, if we assume a constant efficiency for star formation we should expect a star formation history which immediately departs from the MS, declining exponentially as the molecular gas is exhausted (“slow quenching” model shown in the bottom panel of Figure \[Quenchmodels\]). By using our data we directly test those two toy models. We use the star formation rates ($SFR(M_,z)$) for 3D-HST galaxies presented in the @Momcheva16 catalogue. By limiting to galaxies in the redshift range $0.5<z<1.5$, stellar mass range $9.5 < \log(M_) < 11$ and a maximum offset below the main sequence of 0.5 dex, we make sure that the SFR estimates are reliable and, for 91% of the objects, are obtained from [Spitzer]{} 24$\mu$m observations combined with a UV monochromatic luminosity to take into account both dust obscured and unobscured star formation. For the remaining 9% SFR estimates are from an SED fitting procedure [see @Whitaker14; @Momcheva16]. There is growing evidence of curvature in the MS, which becomes shallower at higher stellar mass. @Whitaker14, @Gavazzi15a, and @Erfanianfar16 interpreted this as a decline in star formation efficiency caused by the growth of bulges or bars in massive galaxies. To study the effects of environment above the internal processes driving the star formation efficiency at fixed stellar mass, we convert the SFR into an offset from this curved MS: $\Delta_{\rm MS,obs} = \log(SFR(M_,z)/SFR_{\rm MS}(M_,z)$ using the @Wisnioski15 parametrization of the MS from @Whitaker14. In order to test the two models we again resort to the mock sample. For each central galaxy in the mocks we assign a random offset from the main sequence obtained from a pure sample ($P_{\rm cen} > 0.8$) of observed centrals: $\Delta_{\rm MS,cen}$. For satellite galaxies, instead, their $\Delta_{\rm MS}$ is a function of their time spent as satellites ($T_{\rm sat}$) as follows: $$\Delta_{\rm MS}= \begin{cases} <-1 & \mbox{if } T_{\rm sat}>T_{\rm quench} \\ \Delta_{\rm MS,cen} & \mbox{if } T_{\rm sat}\le T_{\rm d} \\ \Delta_{\rm MS,cen}+\log(e^{-(T_{\rm sat}-T_{\rm d})/ \tau_f}) & \mbox{if } T_{\rm sat}>T_{\rm d} \end{cases} \label{eqMSoffset}$$ [where $T_{\rm quench}$ and $T_{\rm d}$ are the total quenching time and the delay time respectively, and $\tau_f$ is the characteristic timescale of the exponential fading phase.]{} The latter is computed for each galaxy independently such that the SFR drops 1 dex below the MS in ($T_{\rm quench}-T_{\rm d}$) Gyr. As we already computed $T_{\rm quench}$, the only free parameter remaining in this family of models is $T_{\rm d}$. We define the “slow quenching” model for $T_{\rm d}=0$, and “delayed then rapid” those where $ 0 < T_{\rm d} < T_{\rm quench}$. Figure \[MSoffset\] shows the distributions of $\Delta_{\rm MS}$ for 3D-HST satellites in two stellar mass bins, obtained as usual by weighting all galaxies by $P_{\rm sat}$, and for the two models obtained from the mock sample in the same way. The histograms are normalized to the total number of 3D-HST satellites in the same stellar mass bin (including UVJ passive galaxies). We stress that this comparison is meaningful because our models include the cross-talk between centrals and satellites. In the “delayed then rapid” scenario, the value of the delay time that best reproduces the observed data is $T_{\rm d}=T_{\rm quench}-1.4 (0.9)$ Gyr for the $10^{9.5}-10^{10.5}$ ($10^{10.5}-10^{11}$) stellar mass bins respectively. This means the average satellite fades with an $e$-folding timescale of $\tau_{\rm f} = 0.6 (0.4)$ Gyr. Our values are consistent with those from @Wetzel13 at $z=0$ and other independent estimates at high-z. [@Tal14 performed a statistical identification of central and satellite galaxies in the UltraVISTA and 3D-HST fields and found that the onset of satellite quenching occurs 1.5-2 Gyr later than that of central galaxies at fixed number density. These values are in good agreement with the delay times estimated in our work.]{} Conversely the “slow quenching” model predicts too many galaxies below the main sequence but which are not UVJ passive (“transition” galaxies). The fraction of 3D-HST satellites for which $\Delta_{\rm MS}>-0.5$ is 65% (46%), which compares to 67% (47%) for the “delayed then rapid” model; instead it drops to 52% (39%) for the “slow quenching” model. We tested that the distributions of $\Delta_{\rm MS}$ and the estimated fading times are not biased by inaccurate UV+IR SFR for AGN candidates in the sample. Because the CANDELS fields have uniform coverage of deep [Spitzer]{}/IRAC observations, we remove the sources selected by the IRAC color-color criteria presented in @Donley12. We find that neither the $\Delta_{\rm MS}$ distributions, nor the fading time estimates change appreciably. In conclusion the fading of the star formation activity must be a relatively rapid phenomenon which follows a long phase where satellite galaxies have a SFR which is indistinguishable from that of centrals. This is further supported by the evidence that the passive and star forming populations are well separated in color and SFR and that the “green valley” in between them is sparsely populated across different environments [@Gavazzi10; @Boselli14b; @Schawinski14; @Mok14]. The gas content of satellite galaxies ------------------------------------- Finally, we discuss the implications of the quenching times on the gas content of satellites at the time of infall. Because satellite galaxies are not thought to accrete gas after infall, their continued star formation occurs at the expense of gas previously bound to the galaxy. As previously discussed, @McGee14 explain the fading phase by the depletion of molecular gas. The depletion time of molecular gas ($T_{\rm depl, H_2}$) has been derived by several authors [@Saintonge11; @Tacconi13; @Boselli14; @Genzel15]. There is general consensus that this timescale (which is an $e$-folding time) is $\sim 1.5$ Gyr at $z=0$ and is $\sim 0.75$ Gyr at $z=1$. Moreover it is independent of stellar mass. In this framework we might expect fading times shorter than (or similar to) $T_{\rm depl, H_2}$, where shorter fading times are possible where a fraction of the gas is lost to outflows. Our fading times are indeed somewhat shorter than the molecular gas depletion times, consistent with this picture, but with a mass dependence which suggests a mass-dependent outflow rate. In Figure \[Tquench\_Tfade\] we show the ratio of the delay time to the fading time as a function of redshift in bins of stellar mass. Assuming that the fading phase is driven by depletion of molecular gas (in absence of further replenishment), this ratio informs us about the relative time spent refuelling the galaxy to keep it on the MS (from a gas reservoir initially in a warmer phase) to the time spent depleting the molecular gas. We note that the delay time is estimated via the quenching time (which is a function of stellar mass and redshift) while the fading time is computed only for 2 stellar mass bins at $0.5<z<1.5$, with $z=0$ fading timescales taken from @Wetzel13. Errors in Figure \[Tquench\_Tfade\] propagate only the errors on the total quenching time. For all stellar mass bins this ratio is above unity, which we interpret to mean that gas in a non-molecular phase is required to supply fuel for star formation, and this gas is likely to exceed the molecular gas in mass. The longer delay times at $z=0$ suggest that a smaller fraction of the gas mass is in molecular form. This model also implies that a significant fraction of the final stellar mass of satellite galaxies is built-up during the satellite phase. A multi-phase gas reservoir is observed in the local Universe in the form of ionized, atomic, and molecular hydrogen. Atomic hydrogen cools, replenishing the molecular gas reservoir which is depleted by star formation. In the local Universe, using the scaling relations derived from the [Herschel]{} Reference Sample [@Boselli14], the observed mass of atomic hydrogen is found to be 2-3 times larger than the amount of molecular hydrogen for our most massive stellar mass bin. This ratio increases to $\sim8$ for the lower mass objects, although with a large uncertainty. These numbers are consistent with the picture that much, if not all, of the reservoir required to maintain the satellite on the MS during the delay phase at $z\sim0$ can be (initially) in an atomic phase. It is also plausible that much of the gas reservoir bound to higher redshift galaxies is contained in a non-molecular form, and that this can be retained and used for star formation when the galaxies become satellites. Assuming that outflows are not only active during the fading phase but rather during the entire quenching time, the mass in the multi-phase gas reservoir needs to be even larger, although it is not straightforward to constrain by how much. In conclusion our work supports a “delayed then rapid” quenching scenario for satellite galaxies regulated by star formation, depletion and cooling of a multi-phase gas reservoir. Conclusions {#sec_conclusions} =========== In this work, we have characterized the environment of galaxies in the 3D-HST survey at $z=0.5-3.0$. We used the projected density within fixed apertures coupled with a newly developed method for edge corrections to obtain a definitive measurement of the environment in five well studied deep-fields: GOODS-S, COSMOS, UDS, AEGIS, GOODS-N. Using a recent semi-analytic model of galaxy formation, we have assigned physical quantities describing the properties of dark matter haloes to observed galaxies. Our results can be summarized as follows: 1. The 3D-HST deep fields host galaxies in a wide range of environments, from underdense regions to relatively massive clusters. This large variety is accurately quantified thanks to a homogeneous coverage of high quality redshifts provided by the 3D-HST grism observations. 2. As described in @Fossati15, a calibration of density into physically motivated quantities (e.g. halo mass, central/satellite status) requires a mock catalogue tailored to match the properties of the 3D-HST survey. We developed such a catalogue and performed a careful match to the observational sample. As a result, each 3D-HST galaxy is assigned a probability that it is a central or satellite galaxy with an associated probability distribution function of halo mass for each type. This Bayesian approach naturally takes into account sources of contamination in the matching process. We publicly release our calibrated environment catalogue to the community. 3. The 3D-HST sample provides us with a unique dataset to study the processes governing environmental quenching from $z\sim 2$ to the present day over a wide range of halo mass. As no galaxy has a perfectly defined environment, a Bayesian analysis including forward modelling of the mock catalogue allows us to recover “pure” passive fractions of central and satellite populations. We also estimated robust and realistic uncertainties through a Monte Carlo error propagation scheme that takes into account the use of probabilistic quantities. 4. By computing conversion fractions (i.e. the excess of quenched satellite galaxies compared to central galaxies at the same epoch and stellar mass) [@van-den-Bosch08], we find that satellite galaxies are efficiently environmentally quenched in haloes of any mass up to $z\sim 1.2-1.5$. Above these redshifts the fraction of passive satellites is roughly consistent with that of central galaxies. 5. Under the assumption that the earliest satellites to be accreted become passive first, we derive environmental quenching timescales. These are long ($\sim2-5$ Gyr at $z\sim0.7-1.5$; 5-7 Gyr at $z=0$) and longer at lower stellar mass. As they become comparable to the Hubble time by $z\sim1.5$, effective environmental quenching of satellites is not possible at earlier times. More remarkably, their halo mass dependence is negligible. By assuming that cosmological accretion stops when a galaxy becomes a satellite, we were able to interpret these evidences in a “gas exhaustion” scenario [i.e. the “overconsumption” model of @McGee14] where quenching happens because satellite galaxies eventually run out of their fuel which sustains further star formation. 6. We tested two toy models of satellite quenching: the “delayed then rapid” quenching scenario proposed by @Wetzel13 and a continuous “slow quenching” from the time of first infall. By comparing the observed SFR distribution for 3D-HST satellites to the predictions of these toy models we found that the scenario that best reproduces the data at $z\sim 0.5-1.5$ is “delayed then rapid”. Consistently with the results of @Wetzel13 at $z=0$, we find that the fading of the star formation activity is a relatively rapid phenomenon ($\tau_{\rm f} \sim 0.4-0.6$ Gyr, lower at higher mass) which follows a long phase where satellite galaxies have a SFR which is indistinguishable from that of centrals. 7. By linking the fading to the depletion of molecular gas we conclude that the “delayed then rapid” scenario is best explained, even at high redshift, by the presence of a significant multi-phase reservoir which can cool onto the galaxy and replenish the star forming gas at roughly the same rate as the gas is turned into stars. This analysis of satellite quenching is only one of many possible analyses that can be performed with the environmental catalogue built in this work. In the future, the advent of the [James Webb Space Telescope, WFIRST]{} and [Euclid]{} space missions, as well as highly multiplexed spectroscopic instruments from the ground (e.g., [MOONS]{} at VLT; [PFS]{} at Subaru), will provide excellent redshift estimates for fainter objects over a much larger area, to which similar techniques to calibrate environment can be applied. This, in combination with deeper scaling relations for the atomic and molecular gas components from the [Square Kilometer Array]{} and [ALMA]{} will revolutionize measurements to constrain how galaxies evolve and quench as a function of their environment. It is a pleasure to thank Nikhil Arora for his help in a preliminary examination of the local SDSS data, to Alessandro Boselli, Gabriella de Lucia, Sean McGee, Greg Rudnick, and Andrew Wetzel for useful discussion, to Michael Balogh and Alexis Finoguenov for providing catalogs in useful format, and to Stefano Zibetti and Tam[á]{}s Budav[á]{}ri for the work in helping deriving densities for the SDSS sample as presented in @Wilman10, and @Phleps14. We thank the anonymous referee for his/her comments that improved the quality of the manuscript. MFossati and DJW acknowledge the support of the Deutsche Forschungsgemeinschaft via Projects , and . JCCC acknowledges the support of the Deutsche Zentrum f[ü]{}r Luft- und Raumfahrt (DLR) via Project ID 50OR1513. KEW gratefully acknowledge support by NASA through Hubble Fellowship grant \#HF2-51368 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA. This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under programme ID(s) 175.A-0839(B), 175.A-0839(D), 175.A-0839(I), 175.A-0839(J), 175.A-0839(H), 175.A-0839(F), 092.A-0091, 093.A-0079, 093.A-0187, 094.A-0217, 094.A-0287, 095.A-0047, 095.A-0109, 096.A-0093, and 096.A-0025. The MOSDEF data were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. We recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. We acknowledge all the teams and observatories that provided datasets included in the photometric and spectroscopic catalogues used in this work. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. natexlab\#1[\#1]{} , C., [Mazure]{}, A., [Ilbert]{}, O., [et al.]{} 2005, , 443, 805 , H., [Allende Prieto]{}, C., [An]{}, D., [et al.]{} 2011, , 193, 29 , S., [Cristiani]{}, S., [Moscardini]{}, L., [et al.]{} 1999, , 310, 540 , S., [Vandame]{}, B., [Benoist]{}, C., [et al.]{} 2001, , 379, 740 , M. L. N., [Willner]{}, S. P., [Fazio]{}, G. G., [et al.]{} 2013, , 769, 80 , I. K., [Balogh]{}, M. L., [Bower]{}, R. G., [et al.]{} 2006, , 373, 469 , M., [Eke]{}, V., [Miller]{}, C., [et al.]{} 2004, , 348, 1355 , M. L., [Morris]{}, S. L., [Yee]{}, H. K. C., [Carlberg]{}, R. G., & [Ellingson]{}, E. 1997, , 488, L75 , M. L., [Navarro]{}, J. F., & [Morris]{}, S. L. 2000, , 540, 113 , M. L., [McGee]{}, S. L., [Mok]{}, A., [et al.]{} 2014, , 443, 2679 —. 2016, , 456, 4364 , E. F., [Wolf]{}, C., [Meisenheimer]{}, K., [et al.]{} 2004, , 608, 752 , R., [Wake]{}, D. A., [Brammer]{}, G. B., [et al.]{} 2016, , 822, 30 , R., [Hudelot]{}, P., [McCracken]{}, H. J., [et al.]{} 2012, , 545, A23 , M. R., & [Moustakas]{}, J. 2009, , 47, 159 , M. R., & [Roweis]{}, S. 2007, , 133, 734 , A., [Boissier]{}, S., [Cortese]{}, L., & [Gavazzi]{}, G. 2008, , 674, 742 , A., [Cortese]{}, L., [Boquien]{}, M., [et al.]{} 2014, , 564, A66 , A., & [Gavazzi]{}, G. 2006, , 118, 517 —. 2014, , 22, 74 , A., [Voyer]{}, E., [Boissier]{}, S., [et al.]{} 2014, , 570, A69 , A., [Cuillandre]{}, J. C., [Fossati]{}, M., [et al.]{} 2016, , 587, A68 , G. B., [van Dokkum]{}, P. G., & [Coppi]{}, P. 2008, , 686, 1503 , G. B., [van Dokkum]{}, P. G., [Franx]{}, M., [et al.]{} 2012, , 200, 13 , J., [Charlot]{}, S., [White]{}, S. D. M., [et al.]{} 2004, , 351, 1151 , G., & [Charlot]{}, S. 2003, , 344, 1000 , G., & [Valtonen]{}, M. 1990, , 350, 89 , D., [Armus]{}, L., [Bohlin]{}, R. C., [et al.]{} 2000, , 533, 682 , C. N., [van Dokkum]{}, P. G., [Urry]{}, C. M., [et al.]{} 2010, , 189, 270 , G. 2003, , 115, 763 , C., [Gunn]{}, J. E., & [White]{}, M. 2009, , 699, 486 , M. C., [Newman]{}, J. A., [Croton]{}, D. J., [et al.]{} 2006, , 370, 198 , D. J., [Springel]{}, V., [White]{}, S. D. M., [et al.]{} 2006, , 365, 11 , O., [Granett]{}, B. R., [Branchini]{}, E., [et al.]{} 2014, , 565, A67 , B., [Mobasher]{}, B., [Sobral]{}, D., [et al.]{} 2016, ArXiv e-prints, arXiv:1605.03182 , G., & [Blaizot]{}, J. 2007, , 375, 2 , G., [Boylan-Kolchin]{}, M., [Benson]{}, A. J., [Fontanot]{}, F., & [Monaco]{}, P. 2010, , 406, 1533 , G., [Weinmann]{}, S., [Poggianti]{}, B. M., [Arag[ó]{}n-Salamanca]{}, A., & [Zaritsky]{}, D. 2012, , 423, 1277 , G., [Poggianti]{}, B. M., [Arag[ó]{}n-Salamanca]{}, A., [et al.]{} 2007, , 374, 809 , A., & [Birnboim]{}, Y. 2006, , 368, 2 , J. L., [Koekemoer]{}, A. M., [Brusa]{}, M., [et al.]{} 2012, , 748, 142 , A. 1980, , 236, 351 , T., [Hildebrandt]{}, H., [Lerchster]{}, M., [et al.]{} 2009, , 493, 1197 , G., [Finoguenov]{}, A., [Tanaka]{}, M., [et al.]{} 2013, , 765, 117 , G., [Popesso]{}, P., [Finoguenov]{}, A., [et al.]{} 2016, , 455, 2839 , J., & [Thomas]{}, D. 2015, , 451, 660 , A., [Guzzo]{}, L., [Hasinger]{}, G., [et al.]{} 2007, , 172, 182 , A., [Watson]{}, M. G., [Tanaka]{}, M., [et al.]{} 2010, , 403, 2063 , A., [Tanaka]{}, M., [Cooper]{}, M., [et al.]{} 2015, , 576, A130 , F., [Springel]{}, V., [Angulo]{}, R. E., & [Henriques]{}, B. 2012, , 426, 2335 , M., [Fumagalli]{}, M., [Boselli]{}, A., [et al.]{} 2016, , 455, 2028 , M., [Gavazzi]{}, G., [Boselli]{}, A., & [Fumagalli]{}, M. 2012, , 544, A128 , M., [Wilman]{}, D. J., [Fontanot]{}, F., [et al.]{} 2015, , 446, 2582 , M., [van Dokkum]{}, P. G., [F[ö]{}rster Schreiber]{}, N. M., [et al.]{} 2008, , 688, 770 , M., [Fossati]{}, M., [Hau]{}, G. K. T., [et al.]{} 2014, , 445, 4335 , H., [Kosugi]{}, G., [Akiyama]{}, M., [et al.]{} 2008, , 176, 1 , A., & [Bell]{}, E. F. 2009, , 185, 253 , G., [Boselli]{}, A., [Mayer]{}, L., [et al.]{} 2001, , 563, L23 , G., [Fumagalli]{}, M., [Cucciati]{}, O., & [Boselli]{}, A. 2010, , 517, A73 , G., [Fumagalli]{}, M., [Fossati]{}, M., [et al.]{} 2013, , 553, A89 , G., [Consolandi]{}, G., [Dotti]{}, M., [et al.]{} 2015, , 580, A116 , E., [van Dokkum]{}, P. G., [Herrera]{}, D., [et al.]{} 2006, , 162, 1 , J. E., [Simpson]{}, C., [Rawlings]{}, S., [Read]{}, A. M., & [Watson]{}, M. 2007, , 381, 1369 , R., [Tacconi]{}, L. J., [Lutz]{}, D., [et al.]{} 2015, , 800, 20 , R., [Cimatti]{}, A., [Daddi]{}, E., [et al.]{} 2003, , 592, 721 , N. A., [Kocevski]{}, D. D., [Faber]{}, S. M., [et al.]{} 2011, , 197, 35 , J. E., & [Gott]{}, III, J. R. 1972, , 176, 1 , Q., [White]{}, S., [Angulo]{}, R. E., [et al.]{} 2013, , 428, 1351 , Q., [White]{}, S., [Boylan-Kolchin]{}, M., [et al.]{} 2011, , 413, 101 , Y., [Ferguson]{}, H. C., [Giavalisco]{}, M., [et al.]{} 2013, , 207, 24 , M. R., [Schaye]{}, J., & [Jeeson-Daniel]{}, A. 2012, , 419, 2133 , C. P., [Pereira]{}, M. J., [Smith]{}, G. P., [et al.]{} 2015, , 806, 101 , B. M. B., [White]{}, S. D. M., [Thomas]{}, P. A., [et al.]{} 2015, , 451, 2663 —. 2016, ArXiv e-prints, arXiv:1611.02286 , M., [De Lucia]{}, G., [Wilman]{}, D., [et al.]{} 2014, , 444, 2938 , P. F., [Cox]{}, T. J., [Kere[š]{}]{}, D., & [Hernquist]{}, L. 2008, , 175, 390 , B.-C., [Wang]{}, W.-H., [Hsieh]{}, C.-C., [et al.]{} 2012, , 203, 23 , L.-T., [Salvato]{}, M., [Nandra]{}, K., [et al.]{} 2014, , 796, 60 , O., [Arnouts]{}, S., [McCracken]{}, H. J., [et al.]{} 2006, , 457, 841 , O., [Capak]{}, P., [Salvato]{}, M., [et al.]{} 2009, , 690, 1236 , T., [Feigelson]{}, E. D., [Akritas]{}, M. G., & [Babu]{}, G. J. 1990, , 364, 104 , W., [Sluka]{}, C., [Schindler]{}, S., [Ferrari]{}, C., & [Ziegler]{}, B. 2009, , 499, 87 , G., [White]{}, S. D. M., [Heckman]{}, T. M., [et al.]{} 2004, , 353, 713 , G., [Heckman]{}, T. M., [Tremonti]{}, C., [et al.]{} 2003, , 346, 1055 , J. D. P., [Abramson]{}, A., & [Bravo-Alfaro]{}, H. 2015, , 150, 59 , Jr., R. C. 1998, , 36, 189 —. 1998, , 498, 541 , C., [Lilly]{}, S. J., [Kova[č]{}]{}, K., [et al.]{} 2013, , 769, 24 , T., [Yamada]{}, T., [Akiyama]{}, M., [et al.]{} 2004, , 350, 1005 , A. M., [Faber]{}, S. M., [Ferguson]{}, H. C., [et al.]{} 2011, , 197, 36 , K., [Lilly]{}, S. J., [Cucciati]{}, O., [et al.]{} 2010, , 708, 505 , K., [Lilly]{}, S. J., [Knobel]{}, C., [et al.]{} 2014, , 438, 717 , M., [van Dokkum]{}, P. G., [Labb[é]{}]{}, I., [et al.]{} 2009, , 700, 221 , M., [Shapley]{}, A. E., [Reddy]{}, N. A., [et al.]{} 2015, , 218, 15 , J., [Cimatti]{}, A., [Zamorani]{}, G., [et al.]{} 2009, , 504, 331 , J., [Cimatti]{}, A., [Daddi]{}, E., [et al.]{} 2013, , 549, A63 , R. B., [Tinsley]{}, B. M., & [Caldwell]{}, C. N. 1980, , 237, 692 , O., [Tasca]{}, L. A. M., [Cassata]{}, P., [et al.]{} 2015, , 576, A79 , B. D., [Brandt]{}, W. N., [Alexander]{}, D. M., [et al.]{} 2005, , 161, 21 , S. J., [Carollo]{}, C. M., [Pipino]{}, A., [Renzini]{}, A., & [Peng]{}, Y. 2013, , 772, 119 , S. J., [Le F[è]{}vre]{}, O., [Renzini]{}, A., [et al.]{} 2007, , 172, 70 , D., [van Dokkum]{}, P. G., [F[ö]{}rster Schreiber]{}, N. M., [et al.]{} 2009, , 701, 1765 , C., [Moore]{}, B., [Mayer]{}, L., [et al.]{} 2005, , 364, 607 , H. J., [Milvang-Jensen]{}, B., [Dunlop]{}, J., [et al.]{} 2012, , 544, A156 , S. L., [Balogh]{}, M. L., [Bower]{}, R. G., [Font]{}, A. S., & [McCarthy]{}, I. G. 2009, , 400, 937 , S. L., [Bower]{}, R. G., & [Balogh]{}, M. L. 2014, , 442, L105 , I. S., [Steidel]{}, C. C., [Epps]{}, H. W., [et al.]{} 2012, in , Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, 84460J , J. T., [Simard]{}, L., [Palmer]{}, M., [Ellison]{}, S. L., & [Patton]{}, D. R. 2014, , 210, 3 , J. T., [Saglia]{}, R. P., [Bender]{}, R., [et al.]{} 2015, , 804, L4 , P., [Busarello]{}, G., [Dopita]{}, M. A., [et al.]{} 2013, , 429, 1747 , D. 1983, , 264, 24 , A., [Balogh]{}, M. L., [McGee]{}, S. L., [et al.]{} 2013, , 431, 1090 —. 2014, , 438, 3070 , I. G., [Brammer]{}, G. B., [van Dokkum]{}, P. G., [et al.]{} 2016, , 225, 27 , B., [Lake]{}, G., & [Katz]{}, N. 1998, , 495, 139 , E., [Barmby]{}, P., [Rigopoulou]{}, D., [et al.]{} 2003, , 403, 493 , S. I., [Croton]{}, D. J., [Skibba]{}, R. A., [et al.]{} 2012, , 419, 2670 , A., [Wilson]{}, G., [Yee]{}, H. K. C., [et al.]{} 2012, , 746, 188 , A., [Marchesini]{}, D., [Stefanon]{}, M., [et al.]{} 2013, , 777, 18 , A., [van der Burg]{}, R. F. J., [McGee]{}, S. L., [et al.]{} 2014, , 796, 65 , E. J., [van Dokkum]{}, P. G., [F[ö]{}rster Schreiber]{}, N. M., [et al.]{} 2016, , 828, 27 , K. G., [Weiner]{}, B. J., [Faber]{}, S. M., [et al.]{} 2007, , 660, L43 , Jr., A. 1974, , 194, 1 , J. B. 1974, , 27, 21 , A., [Vulcani]{}, B., [Poggianti]{}, B. M., [et al.]{} 2016, , 816, L25 , C., [Momcheva]{}, I., [Willmer]{}, C. N. A., [et al.]{} 2010, , 716, 1503 , Y.-j., [Lilly]{}, S. J., [Renzini]{}, A., & [Carollo]{}, M. 2012, , 757, 4 , Y.-j., [Lilly]{}, S. J., [Kova[č]{}]{}, K., [et al.]{} 2010, , 721, 193 , S., [Aldering]{}, G., [Goldhaber]{}, G., [et al.]{} 1999, , 517, 565 , S., [Wilman]{}, D. J., [Zibetti]{}, S., & [Budav[á]{}ri]{}, T. 2014, , 438, 2233 , [Ade]{}, P. A. R., [Aghanim]{}, N., [et al.]{} 2014, , 571, A16 , B. M., [Bridges]{}, T. J., [Komiyama]{}, Y., [et al.]{} 2004, , 601, 197 , R. F., [Williams]{}, R. J., [Franx]{}, M., & [Hildebrandt]{}, H. 2012, , 744, 88 , E., & [Hensler]{}, G. 2005, , 433, 875 , G., [von der Linden]{}, A., [Pell[ó]{}]{}, R., [et al.]{} 2009, , 700, 1559 , A., [Kauffmann]{}, G., [Wang]{}, J., [et al.]{} 2011, , 415, 61 , S., [Castellano]{}, M., [Pentericci]{}, L., [et al.]{} 2009, , 501, 865 , D. B., [Salvato]{}, M., [Aussel]{}, H., [et al.]{} 2007, , 172, 86 , K., [Urry]{}, C. M., [Simmons]{}, B. D., [et al.]{} 2014, , 440, 889 , M. 1959, , 129, 243 , N., [Aussel]{}, H., [Benson]{}, A., [et al.]{} 2007, , 172, 150 , R., [Bender]{}, R., [Agudo Berbel]{}, A., [et al.]{} 2013, The Messenger, 151, 21 , G. M., [Croton]{}, D. J., [Skibba]{}, R. A., [et al.]{} 2013, , 433, 3314 , J., & [Rees]{}, M. J. 1998, , 331, L1 , R. E., [Whitaker]{}, K. E., [Momcheva]{}, I. G., [et al.]{} 2014, , 214, 24 , J. M., [Manrique]{}, A., [Garc[í]{}a-G[ó]{}mez]{}, C., [et al.]{} 2001, , 548, 97 , V., [White]{}, S. D. M., [Tormen]{}, G., & [Kauffmann]{}, G. 2001, , 328, 726 , V., [White]{}, S. D. M., [Jenkins]{}, A., [et al.]{} 2005, , 435, 629 , M., [Donahue]{}, M., & [Voit]{}, G. M. 2007, , 671, 190 , L. J., [Neri]{}, R., [Genzel]{}, R., [et al.]{} 2013, , 768, 74 , T., [Dekel]{}, A., [Oesch]{}, P., [et al.]{} 2014, , 789, 164 , M., [Finoguenov]{}, A., & [Ueda]{}, Y. 2010, , 716, L152 , Y., [Scoville]{}, N., [Murayama]{}, T., [et al.]{} 2007, , 172, 9 , L. A. M., [Le Fevre]{}, O., [Ribeiro]{}, B., [et al.]{} 2016, ArXiv e-prints, arXiv:1602.01842 , E. N., [Franx]{}, M., [van Dokkum]{}, P. G., [et al.]{} 2009, , 183, 295 , S., & [Bryan]{}, G. L. 2010, , 709, 1203 , D., [Castellano]{}, M., [Fontana]{}, A., & [Giallongo]{}, E. 2007, , 463, 853 , F. C., [Aquino]{}, D., [Yang]{}, X., [et al.]{} 2008, , 387, 79 , R. F. J., [Muzzin]{}, A., [Hoekstra]{}, H., [et al.]{} 2013, , 557, A15 , B., [Cayatte]{}, V., [Balkowski]{}, C., & [Duschl]{}, W. J. 2001, , 561, 708 , B., [Huchtmeier]{}, W., & [van Driel]{}, W. 2005, , 439, 921 , J., [De Lucia]{}, G., [Kitzbichler]{}, M. G., & [White]{}, S. D. M. 2008, , 384, 1301 , B. J., [Phillips]{}, A. C., [Faber]{}, S. M., [et al.]{} 2005, , 620, 595 , A. R., [Tinker]{}, J. L., & [Conroy]{}, C. 2012, , 424, 232 , A. R., [Tinker]{}, J. L., [Conroy]{}, C., & [van den Bosch]{}, F. C. 2013, , 432, 336 , K. E., [Labb[é]{}]{}, I., [van Dokkum]{}, P. G., [et al.]{} 2011, , 735, 86 , K. E., [Franx]{}, M., [Leja]{}, J., [et al.]{} 2014, , 795, 104 , S. D. M., & [Frenk]{}, C. S. 1991, , 379, 52 , S. D. M., & [Rees]{}, M. J. 1978, , 183, 341 , R. J., [Quadri]{}, R. F., [Franx]{}, M., [van Dokkum]{}, P., & [Labb[é]{}]{}, I. 2009, , 691, 1879 , D. J., [Balogh]{}, M. L., [Bower]{}, R. G., [et al.]{} 2005, , 358, 71 , D. J., [Zibetti]{}, S., & [Budav[á]{}ri]{}, T. 2010, , 406, 1701 , E., [F[ö]{}rster Schreiber]{}, N. M., [Wuyts]{}, S., [et al.]{} 2015, , 799, 209 , M., [Yoshida]{}, M., [Komiyama]{}, Y., [et al.]{} 2010, , 140, 1814 , D. G., [Adelman]{}, J., [Anderson]{}, Jr., J. E., [et al.]{} 2000, , 120, 1579 Extended catalogs for edge corrections in the GOODS-S, COSMOS, and UDS fields {#edgecor_cats} ============================================================================= GOODS-S ------- The GOODS-S field is part of a larger field known as the Extended Chandra Deep Field South [ECDFS, @Lehmer05]. This field has been covered by the Multiwavelength Survey by Yale-Chile [MUSYC, @Gawiser06] in 32 broad and medium bands from the optical to the medium infrared wavelengths. The broadband data originates from various sources [@Arnouts01; @Moy03; @Taylor09] and a consistent reduction and analysis is performed by the MUSYC team [@Cardamone10]. The source extraction is performed on a deep combined image of three optical filters ($BVR$) and reaches a depth of $\sim 25.5$ mag. Stars are removed from the catalogue by using the `star_flag` parameter. In order to select galaxies in a consistent way as for 3D-HST we need deep observations in a filter with a central wavelength as close as possible to that of WFC3/F140W ($1.4 \mu m$). However, the near infrared observations from MUSYC are shallow and only reach a depth of $J = 22.4$ mag. We therefore match the MUSYC catalogue with the Taiwan ECDFS Near-Infrared Survey [TENIS, @Hsieh12]. This survey provides deep $J$ and $Ks$ images of the ECDFS area with limiting magnitudes of 24.5 and 23.9 respectively. Hereafter, where sky coordinates matching between different catalogues is required we select the closest match within a 1 arcsec radius. The comparison of $J$ band magnitudes from the two surveys for sources above the sensitivity limit of the MUSYC data shows a remarkable agreement. We then match the MUSYC and 3D-HST/GOODS-S catalogue, again by sky coordinates. Using the galaxies that are present in both surveys we fit a linear function between [$JH_{\rm 140}$]{} and $J_{\rm TENIS}$ magnitudes. Given the significant overlap between the filters we neglect color terms in the fit. The best bisector fit [@Isobe90] is $JH_{\rm 140} = 0.99 \times J_{\rm TENIS} + 0.22$. Then we use this function to generate [$JH_{\rm 140}$]{} magnitudes for all objects in the MUSYC+TENIS catalogue. We evaluate the depth of the resulting catalogue by inspecting the histogram of the number counts in the [$JH_{\rm 140}$]{} band. Since this is obtained from deep $J_{\rm TENIS}$ data (24.5 mag), the limiting factor will be the depth of the $BVR$ selection band of MUSYC. The number counts increase linearly in log space up to $JH_{\rm 140} \sim 23.5$ and we therefore adopt this value for the selection. Since this limit is brighter than the one we set for the primary sample in 3D-HST, a fraction of the neighbours are missed. We correct for this by assigning to each MUSYC galaxy in eq. \[eqSigma\] a weight $w = 1.42$ that corresponds to the ratio of the cumulative luminosity function at $JH_{\rm 140} = 23.5$ and $JH_{\rm 140} = 24$ mag from the deeper 3D-HST catalogue. The most recent calculation of photometric redshifts in ECDFS is presented by @Hsu14. These authors combined the MUSYC photometry with TENIS and HST/CANDELS [@Guo13] in the GOODS-S area. We match our catalog with their catalog based on MUSYC ID and we find a match for each source. @Hsu14 also present a compilation of spectroscopic redshifts available in the literature which we use whenever available. Photo-zs are computed using LePhare* [@Arnouts99; @Ilbert06] and their accuracy depends primarily on the availability and depth of multiwavelength photometry. The GOODS-S area has deep HST coverage from CANDELS, but those galaxies are already present in our primary 3D-HST catalog. Therefore we are primarily interested in sources outside the CANDELS/3D-HST area. In the ECDFS footprint which is not covered by HST more than 30 photometric bands are available and photo-zs are quite accurate: $\sigma_{\rm NMAD} \sim 3000~(4000)~\rm{km~s^{-1}}$ for galaxies with $H<23 (H>23)$ respectively. Those values degrade where continuum spectral features (e.g. Balmer break) are redshifted outside the range observed with medium band filters ($z>1.5$), although low number statistics hampers a robust determination of the photometric redshift quality. Stellar masses are computed using the photometric data and the redshift information by choosing the same grid of templates used by @Skelton14 for the 3D-HST fields. We assess the quality of the stellar masses by comparing to those from @Skelton14, where MUSYC and 3D-HST overlap and we find a median offset of 0.01 dex and a scatter of 0.15 dex. The scatter is driven by the scatter in photometric redshifts in the two catalogs. As a last step we remove from this catalog galaxies in the 3D-HST footprint that satisfy the selection criteria for the primary environment sample, to obtain a pure catalogue that we use only for the edge corrections. COSMOS ------ The entire COSMOS 2deg$^2$ field has been observed in 30 photometric bands from UV to medium infrared (including several medium bands) [@Sanders07; @Taniguchi07; @Erben09; @Bielby12]. Photometric redshifts are computed using LePhare and are presented by @Ilbert09 for sources with $i^+ < 25$ mag. We include spectroscopic redshifts from zCOSMOS-bright [@Lilly07] where available. The photometric redshift uncertainty is evaluated by @Ilbert09 comparing photo-z to spec-z and is $\sigma_{\rm NMAD} \sim 2100~(9000)~\rm{km~s^{-1}}$ for galaxies with $i^+<22.5 (i^+>23)$ respectively. The latter value must be taken with caution as it is calibrated using a small number of objects. We remake this comparison by using 3D-HST spec-z and grism-z as a reference (restricting our analysis to the 3D-HST/COSMOS field). We divide the sample into bright ($i^+<22.5$ mag) and faint ($i^+ \ge 22.5$ mag) for $0.5 < z \le 1.5$ and irrespective of magnitude for $1.5 < z \le 3.0$. We note that for faint sources the effective magnitude limit is that of the grism redshift extraction $JH_{140} < 24$ mag and the comparison is limited by the degraded accuracy of grism redshifts for faint sources with no emission line detection (see Figure \[fig1\]). The redshift accuracy is: $\sigma_{\rm NMAD} \sim 3000~(7500)~\rm{km~s^{-1}}$ for the bright (faint) sample at low redshift and $\sigma_{\rm NMAD} \sim 8500~\rm{km~s^{-1}}$ for the high redshift sample. Those values are consistent with the determination by @Ilbert09 and the reduced accuracy at high redshift is due to the lack of narrow bands in the NIR. To overcome this limitation the Newfirm Medium Band Survey [NMBS, @Whitaker11] observed the COSMOS field with 5 medium band filters in the $J$ and $H$ bands and a broadband filter in $K$. As a result the accuracy of photometric redshifts is significantly improved [see Section 5 in @Whitaker11] and we use those photo-z where they are available. Deep $J$ band magnitudes are provided by the UltraVISTA survey [@Mccracken12]. After matching their catalog via sky coordinates we generate synthetic $JH_{\rm 140}$ magnitudes as described in the previous section and using the best fit: $JH_{\rm 140} = 0.98 \times J_{\rm UltraVISTA} + 0.31$. The depth of our catalog is limited by the depth of the $i^+$ selection band from @Ilbert09. The number counts increase linearly in log space until $JH_{140} \sim 23.0$ and we therefore adopt this value for the selection limit. As for the MUSYC catalog this limit is brighter than the one we set for the primary sample in 3D-HST therefore we assign to each galaxy in eq. \[eqSigma\] a weight ($w = 2.06$). Stars are removed from the catalog by using the `type` flag from @Ilbert09. We compute stellar masses as described in the previous section. The agreement with stellar masses from @Skelton14 is remarkable, with a median offset of 0.02 dex and a scatter of 0.20 dex. Lastly we remove the primary 3D-HST sources from this edge correction sample. UDS --- The 3D-HST UDS field is part of a larger field known as UKIDSS UDS. This field features deep near infrared $J$, $H$ , and $K$ observations with the UKIDSS telescope (Almaini et al. in prep) complemented by optical and medium infrared data [@Furusawa08; @Ashby13]. The UDS/DR8 catalog selection is performed in $K$ band and the completeness limit is $K\sim 24.6$ mag. As for the previous fields we exclude stars and we compute synthetic [$JH_{\rm 140}$]{} magnitudes using the best fit relation: $JH_{\rm 140} = 0.98 \times J_{\rm UKIDSS} + 0.19$. The depth of our catalog matches the limiting magnitude for the primary 3D-HST sample, thus we do not apply any statistical weight for the UKIDSS UDS galaxies when computing the density. Photometric redshifts (W. Hartley private comm.) have a typical accuracy of $\sigma \sim 9000~\rm{km~s^{-1}}$ due to the lack of narrow- or medium band photometry in this field. As for the other fields we compute stellar masses using the FAST code and we find a good agreement with the values from @Skelton14 for the 3D-HST/UDS field with an offset of -0.03 dex and a scatter of 0.22 dex. Again the last step is to remove the 3D-HST primary sources via positional matching with the @Skelton14 catalogue. Passive fraction as a function of density {#app_passfrac_dens} ========================================= Halo mass is the parameter which most easily allows the interpretation of environmental effects across cosmic time. It also allows for easier and less biased comparisons across different works. Moreover it can be directly linked to models (either semi-analytic or hydrodynamical) allowing a better understanding of which physical processes are most relevant at different halo masses. Density, on the other hand, depends on the depth (and to some extent the observing strategy) of each survey. Detailed and quantitative comparisons are also made difficult by different approaches to density [e.g., @Muldrew12; @Haas12; @Etherington15]. However it is a parameter directly obtained from the observed redshift space coordinates of the population of galaxies under investigation. In this respect it is less sensitive to the quality and uncertainties in the calibration of halo mass. In this Appendix we derive the passive fraction of galaxies in two bins of density and compare them to those obtained in Figure \[Passfrac\_Mhalo\] using halo mass. The observed fractions of passive centrals and satellites in bins of $M_{\rm{}}$ and density contrast $\log(1+\delta_{0.75})$ are given by $$f_{\rm{pass\|ty}} = \frac{\sum_i \left(\delta_{\rm pass,i} \times \delta_{M_ \rm{,i}} \times \delta_{\log(1+\delta_{0.75}),i} \times P_{\rm{ty,i}}\right)}{\sum_i \left(\delta_{M_* \rm{,i}} \times \delta_{\log(1+\delta_{0.75}),i} \times P_{\rm{ty,i}} \right)} \label{eqpassive_dens}$$ where ty refers to a given type (centrals or satellites), $\delta_{\rm pass,i}$ is 1 if a galaxy is UVJ passive and 0 otherwise, $\delta_{M_* \rm{,i}}$ is 1 of a galaxy is in the stellar mass bin and 0 otherwise, $\delta_{\log(1+\delta_{0.75}),i}$ is 1 if a galaxy is in the density bin and 0 otherwise, and $P_{\rm{ty,i}}$ is the probability that a galaxy is of a given type. In this equation the only uncertain property for each object is its central/satellite status, while the cross talk between multiple density bins is not present (as it was for halo mass). We therefore perform a simpler decontamination procedure. For each density, stellar mass and redshift bin, we assign to real centrals in the mocks a probability of being passive equal to the passive fraction of the pure sample of observed central galaxies $f_{\rm{pass\|cen,pure}}(M_,\delta_{0.75})$, while the passive fraction of satellites $f_{\rm{pass\|sat,pure}}(M_,\delta_{0.75})$ is a free parameter. Then we use equation \[eqpassive\_dens\] to compute the observed passive fractions for mock galaxies (therefore contaminating the “pure” values). We solve for $f_{\rm{pass\|sat,pure}}(M_*,\delta_{0.75})$ by maximising the likelihood that the contaminated passive fractions for mock galaxies match the observed passive fractions (jointly for centrals and satellites). This procedure is repeated 500 times in a Monte Carlo fashion in order to propagate the uncertainties on the datapoints to the “pure” (decontaminated) passive fractions. The decontaminated values of the passive fraction for centrals and satellites shown in Figure \[Passfrac\_Dens\] are qualitatively similar to those obtained in bins of halo mass in the same redshift slices (see Figure \[Passfrac\_Mhalo\]). We conclude that the dependence of environmental quenching when binned on local density is similar to that in bins of halo mass, where density is a more directly observed quantity. An example of the fitting procedure to recover the passive fraction of pure satellites {#app_decontamination} ====================================================================================== In this Appendix we illustrate the results of the fitting process described in Section \[sec\_decontamination\] for a single redshift bin. Figure \[Corner\_plot\_example\] presents the constraints on the model parameters (marginalised over the nuisance parameter $P_{\rm{pass\|cen}}$) for a single stellar mass and redshift bin. The panels along the diagonal show the marginalised posterior distributions for each for the three parameters $(M_{\rm br,lo},M_{\rm br,hi},P_{\rm pass,hi})$. The red solid lines show the median value of each parameter, and the black dashed lines show the $1\sigma$ confidence intervals. The off-diagonal panels show the marginalised posterior distributions for a pair of model parameters. The black contours show the $1\sigma$, $1.5\sigma$, and $2\sigma$ confidence intervals. The fits for the other stellar mass bins give qualitatively similar results. Figure \[Psat\_model\_example\] shows the median value (thick blue lines) of $P_{\rm{pass\|sat}}$ as a function of log halo mass and $1\sigma$ confidence intervals in each stellar mass bin. Despite the significant covariance of the model parameters, the shape of the passive fraction models for satellites is well determined. The horizontal black lines show the halo mass range that includes $90\%$ of the satellites in each stellar mass bin. The average passive fractions in the two halo mass bins above and below $10^{13}M_\odot$, presented as the thick blue lines in Figure \[Passfrac\_Mhalo\], are shown for each stellar mass bin in Figure \[Psat\_model\_example\] by the black points. We add in Figure \[Psat\_model\_example\] an additional test of the result presented in Section \[sec\_tquench\_mhalo\] that the quenching time is largely independent of halo mass. We compute a single quenching time per stellar mass bin without binning the data in halo mass. Then we compute which fraction of mock galaxies have $T_{\rm sat} > T_{\rm quench}$ as a function of halo mass. This is converted in a probability of being passive as a function of halo mass which we show as solid red lines (with $1\sigma$ confidence intervals as dashed lines) in Figure \[Psat\_model\_example\]. The agreement with the best fit values of $P_{\rm{pass\|sat}}$ is remarkable in most of the stellar mass bins, further supporting the result of a quenching timescale that is independent of halo mass. A z=0 sample from SDSS {#app_SDSS} ====================== Observational data ------------------ The $z=0$ points in Figures \[Convfracz\] and \[Tquenchz\] are obtained from a sample of galaxies in the local Universe selected from the SDSS [@York00] survey. We use the data from the SDSS DR8 database [@Aihara11] cross correlated with an updated version of the multi-scale density catalog from @Wilman10 [with densities computed according to equation \[eqSigma\]; updated DR8 catalog as used by @Phleps14; @Hirschmann14]. SDSS DR8 includes 5 color $ugriz$ imaging of 14555 square degrees. The spectroscopic part of the survey provides redshifts for 77% of objects brighter than a limit of $r=17.77$ across 8032 square degrees. Our sample is derived from the spectroscopic database. Luminosities are computed by k-correcting and adding the distance modulus to the Petrosian r-band magnitude. k-corrections are performed using the [k-correct idl]{} tool [@Blanton07]. We select as primary galaxies those with $M_r < -18$ mag and $0.015 < z < 0.08$. In contrast to the method we use at high redshift the sample of neighbours (galaxies used to calculate the density in equation \[eqSigma\]) is restricted to $M_r < -20$ mag. This ensures a volume limited sample for the neighbours in this redshift range, while for the primary galaxies we correct for volume incompleteness using $V_{\rm max}$ corrections. The primary sample numbers $\sim3\times10^5$ galaxies. Stellar masses and star formation rates are obtained from the JHU-MPA catalogues updated to DR7 [@Brinchmann04; @Kauffmann03]. For this work we use the density computed on a fixed scale of 1 Mpc, with a velocity cut of $dv = \pm1000 \rm{km~s^{-1}}$. This scale is larger than what we use in the 3D-HST sample in order to take into account the growth of structure with cosmic time. We stress that our results do not significantly depend on the scale chosen because the halo mass calibration is performed self consistently and we only compare calibrated quantities across the two samples. We have further computed stellar mass ranks for each primary galaxy in the adaptive aperture as described in Section \[sec\_censat\]. One limitation of the SDSS spectroscopic strategy is that not all the spectroscopic targets can be actually observed because two fibers cannot be placed closer than $55"$ on the sky and each patch of the sky is only observed once (although with small overlaps between adjacent spectroscopic plates). As a result the spectroscopic catalogue does not contain all the sources detected in the imaging. Spectroscopic incompleteness is taken into account in the computation of the densities as described by @Wilman10, and we further consider it when we match to the mock galaxy sample. Passive galaxies are selected using the specific star formation rate ($sSFR$) as a tracer. For consistency with previous studies [e.g. @Hirschmann14] we define passive galaxies those with $sSFR < 10^{-11} {\rm yr}^{-1}$. We note that this corresponds to a $\sim1$ dex offset from the main sequence of star forming galaxies at $z=0$, which is consistent with the division of UVJ star forming from UVJ passive galaxies adopted in Section \[sec\_passfrac\_mhalo\]. The model sample ---------------- We generate a model galaxy sample that matches the stellar mass and density distributions of the SDSS observational catalogue. To do so we take the SAM from @Henriques15 at the $z=0$ snapshot of the Millennium simulation. In this case we do not use lightcones but a three dimensional box because of the large area covered by SDSS and the single redshift bin. Densities are computed by projecting one of the axes of the box into a redshift axis as described in @Fossati15. We set an aperture size of 1 Mpc, a velocity cut $dv = \pm1000 \rm{km~s^{-1}}$, and we compute densities according to equation \[eqSigma\]. The model sample does not suffer from spectroscopic incompleteness; on the other hand the distribution of $r$-band magnitudes does not match perfectly the one obtained from the observations. To overcome both those issues at once we employ a method that iterates on the magnitude limits for the primary and the neighbours samples until the number density and the density distribution of the selected sample match the observational data. Before doing that we need to derive the total number of photometric galaxies in the SDSS DR8 footprint (more precisely in the area followed up by spectroscopy) that would have been observed if fiber collisions were not a limitation. We query the SDSS database for the number of galaxies in the spectroscopic database and the number of galaxies in the photometric database that would satisfy the criteria for spectroscopic follow-up. The ratio of those values is 0.769. Therefore the number density of mock galaxies needs to be $\rho_{\rm mod} = 1.3 \times \rho_{\rm SDSS,sp}$ where $ \rho_{\rm SDSS,sp}$ is the number density of primary galaxies in our observational catalogue once we account for $V_{\rm max}$ corrections. The absolute magnitude cuts we set in the models using this iterative method are $M_r < -17.6$ mag and $M_r < -19.0$ mag for the primary and the neighbour samples respectively. We note that these cuts are up to 1 mag deeper than those used in the SDSS sample. This difference arises in a non perfect match of the $r$-band luminosity function, while stellar mass functions are better matched between the SAM and the SDSS data. Figure \[Mstar\_dens\_compare\_sdss\] shows that, with this choice of magnitude limits, both the density and the stellar mass distributions are well matched. This is a critical step to trust our Bayesian approach to halo mass and central/satellite status. As a last step we assign to each SDSS galaxy (and to model galaxies) a probability that it is central ($P_{\rm{cen}}$) or satellite ($P_{\rm{sat}}$) and the halo mass PDFs $P_{M_h \| {\rm{cen}}}$ and $P_{M_h \| {\rm{sat}}}$ as described in Section \[sec\_halomass\]. Figure \[threeplots\_Mhalo\_SDSS\] shows the passive fraction for centrals and satellites, conversion fractions and satellite quenching timescales derived for the SDSS sample as described in Sections \[sec\_passfrac\_mhalo\], \[sec\_convfrac\_mhalo\], and \[sec\_tquench\_mhalo\]. Section \[sec\_redshiftevo\] contains the scientific discussion of these results in the context of the evolution of satellite quenching efficiency and timescales from $z=0$ to $z\sim 2$. Description of the environment catalogue for the 3D-HST sample {#envcatalogue} ============================================================== The environmental properties of 3D-HST galaxies are made available at <http://dx.doi.org/10.5281/zenodo.168056>. Conditional halo mass PDFs given that each galaxy is a central or a satellite and covering the range $10 < \log(M_h/M_\odot) < 15$ with 100 uniform bins are also available as separate tables in the same repository. Table \[envtableexample\] gives an example of the quantities provided in the catalog and the description of the columns follows: - \(1) 3D-HST field - \(2) 3D-HST photometric ID from @Skelton14 - \(3) 3D-HST spectroscopic (grism) ID from @Momcheva16 - \(4) fraction of the 0.75 Mpc aperture in the photometric catalogue - \(5) density of galaxies in an aperture of 0.75 Mpc radius (see eq. \[eqSigma\]) - \(6) overdensity of galaxies in an aperture of 0.75 Mpc radius (see eq. \[eqoverdens\]) - \(7) stellar mass rank in the adaptive aperture - \(8) and (9) probability that the galaxy is a central or a satellite - (10), (11), and (12) $16^{\rm th}$, $50^{\rm th}$, and $84^{\rm th}$ percentile of the log halo mass cumulative PDF given that the galaxy is a central - (13), (14), and (15) $16^{\rm th}$, $50^{\rm th}$, and $84^{\rm th}$ percentile of the log halo mass cumulative PDF given that the galaxy is a satellite
--- author: - 'E. J. Marchesini' - 'A. Paggi' - 'F. Massaro' - 'N. Masetti' - 'R. D’Abrusco' - 'I. Andruchow' - 'R. de Menezes' bibliography: - 'Biblio.bib' subtitle: 'Searching for the connection between X-rays and $\gamma$-rays in Fermi BL Lac objects' title: 'The $\gamma$-ray sky seen at X-ray energies I.' --- [BL Lac objects are an extreme type of active galactic nuclei (AGNs) that belong to the largest population of $\gamma$-ray sources: blazars. This class of AGNs shows a double-bumped spectral energy distribution that is commonly described in terms of a synchrotron self-Compton (SSC) emission process, whereas the low-energy component that dominates their emission between the infrared and the X-ray band is tightly connected to the high-energy component that peaks in the $\gamma$-rays. Two strong connections that link radio and mid-infrared emission of blazars to the emission in the $\gamma$-ray band are well established. They constitute the basis for associating $\gamma$-ray sources with their low-energy counterparts.]{} [We searched for a possible link between X-ray and $\gamma$-ray emissions for the subclass of BL Lacs using all archival Swift/XRT observations combined with Fermi data for a selected sample of 351 sources.]{} [ Analyzing $\sim$2400 ks of Swift/XRT observations that were carried out until December 2018, we discovered that above the $\gamma$-ray flux threshold $F_{\gamma}\approx3\times10^{-12}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$, 96% of all Fermi BL Lacs have an X-ray counterpart that is detected with signal-to-noise ratio higher than 3.]{} [ We did not find any correlation or clear trend between X-ray and $\gamma$-ray fluxes and/or spectral shapes, but we discovered a correlation between the X-ray flux and the mid-infrared color. Finally, we discuss on a possible interpretation of our results in the SSC framework.]{} Introduction {#sec:intro} ============ Blazars are a peculiar class of active galactic nuclei (AGNs) that is characterized by emission arising from a relativistic jet oriented at small angles [e.g., less than a few degrees, @Lister13] with respect to the line of sight. This jet emission overwhelms most of the radiation of their host galaxy [@BlandfordRees78]. Blazar emission is detected at all frequencies. It extends from radio [see, e.g., @Jorstad01; @Ciaramella04; @Ghirlanda10; @Lister19] and even low radio frequencies [see, e.g., @Nori14; @Giroletti16 for recent observational campaigns], infrared [IR, see, e.g., @Impey88; @Stevens94; @Massaro11; @DAbrusco12] and optical [see, e.g., @Carini92; @Marchesini16; @PenaHerazo17; @Marchesini19 for recent observational campaigns] to X-rays [see, e.g., @Singh85; @Giommi90; @Sambruna96; @Pian98; @Paggi13; @Landi15]. The emission is also detected in the $\gamma$-ray band [see, e.g., @Aharonian05; @Albert07; @Giannios09; @Tavecchio11; @Wehrle98; @Ackermann15] and shows strong variability. It also has flaring states in which spectral shape and/or luminosities change [see, e.g., @Hartman01; @Romero02; @Bottcher07; @Pandey17; @Kaur17; @Bruni18]. Blazars can be classified into two main categories: flat-spectrum radio quasars, and BL Lac objects. Empirically, the distinction between these two classes is based on emission features in their optical spectra [@Stickel91]. The first class shows strong and broad emission lines that are typical of normal quasars, while the spectra of the second class are almost featureless (i.e., emission lines with equivalent widths smaller than 5 Å). We here adopt the nomenclature established by Roma-BZCat [@Massaro15], where flat-spectrum radio quasars are labeled BZQs and BL Lac objects are BZBs. Blazars are the dominant class of active galaxies in the $\gamma$-ray sky [@Hartman99; @Mattox02; @Massaro15b]. In particular, $\sim$56% of all associated and classified $\gamma$-ray sources that have been detected by the Fermi Large Area Telescope (Fermi-LAT) four-year Point Source Catalog (3FGL) belong to this class [@Acero15]. In the past decade, follow-up spectroscopic campaigns [see, e.g., @Coso1; @Opt5; @Opt6; @Opt7; @Opt8] confirmed that most of the sources that were classified as “Blazar Candidates of Uncertain type” (BCUs), introduced in the 3FGL catalog, are indeed blazars of BL Lac type [@Massaro16b; @AlvarezCrespo16]. The same situation occurs in the preliminary version of the latest release of the Fermi catalog[^1] [@Opt9]. Moreover, follow-up campaigns of unassociated or unidentified $\gamma$-ray sources (UGSs) have also shown that a large portion of them appear to be associated with blazars [@Opt1; @Opt2; @Opt3; @Opt4]. The spectral energy distribution (SED) of blazars shows two components: the low-energy component peaks between infrared and X-rays, and the high-energy component peaks between hard X-rays and the $\gamma$-ray band. The low-energy component is attributed to synchrotron emission arising from electrons that are accelerated in the blazar jets, and the high-energy component is due to the inverse Compton (IC) process [see, e.g., @Ghisellini85; @Maraschi92; @Massaro06; @Tramacere07; @Finke08; @Paggi09b; @Tramacere11 for recent analyses]. For BZBs in particular, the two emission processes are strictly connected because seed photons for the IC emission are emitted by electrons through synchrotron radiation (i.e., the synchrotron self-Compton, or SSC, scenario). Two different subclasses were originally defined for BZBs to distinguish them on the basis of the ratio of their radio- to X-ray flux [@Maselli10a]: low-energy peaked BL Lacs (i.e., LBLs) and high-energy peaked BL Lacs (i.e., HBLs). This ratio is defined as $\Phi_{\rm{XR}}=10^{-3}\,\frac{\rm{F}_{\rm{X}}}{\rm{S}_{1.4}\Delta\nu}$, where $\rm{F}_{\rm{SX}}$ is the X-ray flux from the ROSAT (short forRöntgensatellit) survey [@Voges99] in the 0.1 to 2 keV band, $\rm{S}_{1.4}\Delta\nu$ is the radio flux density at 1.4 GHz multiplied by the band frequency width $\Delta\nu$. Thus, values of $\Phi_{\rm{XR}}$ greater than 1 point toward an HBL classification, while values lower than 1 indicate an LBL type of source. This corresponds to the limit $\frac{\rm{F}_{\rm{X}}}{\rm{S}_{1.4}}=1.0\times10^{-11}$. A strong link between $\gamma$-ray and radio emission in blazars, known as the $\gamma$- [@Stecker93; @Taylor07; @Bloom08; @Ghirlanda11], was discovered soon after the launch of the Energetic Gamma Ray Experiment Telescope (EGRET) on board the Compton Gamma-ray satellite [@Fichtel93]. This link was later confirmed through Fermi observations [@Mahony10; @Ackermann11; @Ghirlanda11; @Cutini14; @Lico14]. Moreover, combining $\gamma$-ray and mid-infrared observations, the latter collected with the Wide-field Infrared Survey Explorer [WISE; @Wright10], a tight connection between their emissions in these two bands has also been discovered [@Massaro11; @Massaro12b; @DAbrusco13; @Massaro16]. In particular, the $\gamma$-ray – infrared connection is strongly related not only to the blazar power, but also to their spectral shapes in the two different energy ranges; this is expected given the theoretical interpretation of their SED. These two connections strongly stimulated follow-up campaigns to search for blazar-like counterparts of UGSs, for example, in the radio band [@Massaro13b; @Nori14; @Lico14; @Giroletti16] and/or by applying statistical procedures to find them in the WISE catalogs [@DAbrusco13; @Coso2; @DAbrusco14; @DAbrusco19]. On the other hand, X-ray follow-up observations have also been carried out to search for blazar-like counterparts of UGSs [e.g., @Mirabal09; @Kataoka12; @Stroh13; @Paggi13; @Masetti13b; @Landi15; @Paiano17a], even if these counterparts not supported by any firmly established observational link or connection between the X-ray and $\gamma$-ray emission of blazars. Based on the SSC scenario that underlies the interpretation of BL Lac SEDs, a link between X-ray and $\gamma$-ray emission could be expected because emission of the low- and high-energy components is related to the same particle distribution. This is the main aim of the analysis presented here. We focus on BZBs, whose $\gamma$-ray emission is probably not significantly contaminated by inverse Compton radiation of seed photons that arises from regions outside the jet [@Sikora13]. We aim to determine the portion of $\gamma$-ray blazars that have an X-ray counterpart with respect to their $\gamma$-ray flux, and whether their $\gamma$-ray emission (i.e., flux and/or spectral shape) correlates with the X-ray emission, as occurs in the radio and in the mid-infrared bands. Our investigation will provide the necessary scientific background to support and justify ongoing [@Stroh13] and future X-ray follow-up campaigns of Fermi UGSs. A detailed study of the general behavior of BZBs in the X-ray band is crucial to improve the selection of BZB candidates within a UGS sample. Our search for a connection between X-rays and $\gamma$-rays is also supported by the fact that HBLs that are detected at TeV energies [e.g., @Piner14; @Piner18] are generally among the brightest X-ray sources in the extragalactic sky and their X-ray emission is also linked to the mid-infrared emission [@Massaro13]. To carry out our investigation, we analyzed observations from the X-ray Telescope (XRT) o board the Neil Gehrels Swift Observatory, performed before mid-December 2018, of $\gamma$-ray BZBs observed by Fermi. We made this choice because the ROSAT survey is relatively shallow. Only $\sim$60% percent of all known Fermi blazars are listed in the ROSAT survey catalog [@Voges99]. Nevertheless, Swift performs an X-ray follow-up campaign of Fermi UGSs[^2] [@Falcone13; @Stroh13; @Falcone14]. An extensive database is therefore available. This paper is organized as follows. In §2 we present the sample selection criteria, while in §3 we describe the Swift/XRT data reduction procedures. §4 is devoted to results of our analysis, and in §5 we describe a possible interpretation of our results within the SSC framework. Finally, §6 is dedicated to a brief summary and our main conclusions. Unless stated otherwise and throughout the whole paper, we adopted cgs units and a flat cosmology with $H_0=\,72\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$, and $\Omega_{\Lambda}=0.74$ [@Dunkley09]. Spectral indices $\alpha$ were defined so that the flux density $\rm{S}_{\nu}\propto\nu^{-\alpha}$, considering $\alpha<0.5$ as *. The AllWISE magnitudes in the \[3.4\]$\mu$m, \[4.6\]$\mu$m, and \[12\]$\mu$m nominal filters are in the Vega system, and are not corrected for Galactic extinction because this correction is negligible for Galactic latitudes $\|b\|>10^{\circ}$ [see, e.g., @DAbrusco13]. Sample selection {#sec:sample} ================ To assess whether the X-ray and $\gamma$-ray emission in BZBs is connected, we started by selecting all known Fermi* BZBs listed in the “clean” sample of the Third Catalog of Active Galactic Nuclei detected by theFermi-LAT [@Ackermann15 3LAC], and considered only those that belong to the fifth release of the Roma-BZCat [@Massaro15]. At this selection step our sample contained 580 of the original 1151 sources. All BZBs in BZCat have a counterpart in at least one of the main radio surveys [@White97; @Condon98; @Mauch03], and all selected BZBs are uniquely associated with $\gamma$-ray sources in the 3FGL catalog. We only included in our sample Swift/XRT observations that were performed in photon-counting (PC) mode that lay within a circular region of 6 arcmin angular separation around the BZB $\gamma$-ray positions (431 sources). Our choice of 6 arcmin corresponds to the average semimajor axis of the positional uncertainty ellipse of $\gamma$-ray sources listed in the 3FGL [@Acero15]. We chose only sources with total exposure times of between 1 and 20 ks because those with cumulative exposure times longer than 20 ks are generally pointed as follow-up observations of flaring states and are not snapshot observations. A similar criterion was adopted by @Mao16. The Swift X-ray campaign of UGSs is performed with a nominal 5 ks exposure time [@Stroh13], implying that the results achieved for our selected sample are suitable to carry out a future investigation of UGSs observed with Swift/XRT (Marchesini et al. 2019, in prep.). Most of the observed fields have a 5 ks exposure time; the average of our sample is 6.7 ks. Swift/XRT data reduction and analysis ===================================== Data processing {#sec:data} --------------- We adopted the same data reduction procedure as described in @Massaro08 [@Massaro08b; @Paggi13; @Massaro11b; @Massaro12c]. Here we report the basic details and highlight differences and improvements with respect to our previous analyses. Raw SWIFT/XRT data were download and reduced to obtain clean event files with the standard procedures using the <span style="font-variant:small-caps;">xrtpipeline</span> task, which is part of the Swift X-Ray Telescope Data Analysis Software [<span style="font-variant:small-caps;">XRTDAS</span>, @Capalbi05], and the High Energy Astrophysics Science Archive Research Center (HEASARC) calibration database (<span style="font-variant:small-caps;">CALDB</span>) version <span style="font-variant:small-caps;">x20180710</span>. In particular, using <span style="font-variant:small-caps;">xselect,</span> we excluded time intervals with count rates exceeding 40 counts per second, and time intervals where the CCD temperature exceeds -50$^{\circ}$C in regions located at the CCD edge [@DElia13]. Clean event files were merged using the <span style="font-variant:small-caps;">xselect</span> task, while corresponding exposure maps were merged with <span style="font-variant:small-caps;">ximage</span> software. Figure 1 shows a merged image obtained for the XRT field associated with 5BZBJ2005+7752 as an example of the final product obtained with our code. ![image](5BZBJ2005+7752.png) Source detection and photometry ------------------------------- A first detection run was performed over merged event files using the <span style="font-variant:small-caps;">DET</span> algorithm in <span style="font-variant:small-caps;">ximage</span>, to obtain pixel positions for every detection with signal-to-noise ratios (S/N) higher than 3. Because the exposure times chosen to carry out our investigation were relatively short, we focused on a photometric analysis. We ran the <span style="font-variant:small-caps;">SOSTA</span> task available within the <span style="font-variant:small-caps;">XIMAGE</span> package on the pixel positions obtained from <span style="font-variant:small-caps;">DET</span>. In particular, <span style="font-variant:small-caps;">SOSTA</span> takes into account the local background for each source to claim a detection, which achieves a more precise photometry than the <span style="font-variant:small-caps;">DET</span> algorithm. This was carried out on merged event files for the full 0.5-10 keV band, and also for the soft (0.5-2 keV) and hard (2-10 keV) bands. Sources detected with <span style="font-variant:small-caps;">SOSTA</span> all had an S/N between 3 and 25. We compared the resulting X-ray sources with those listed in the Swift-XRT point source (1SXPS) catalog [@Evans14]. Our procedure differs from that of 1SXPS in i) the choice of the S/N threshold applied to claim a detection, which is S/N$>$1.6, ii) the total number of Swift/XRT observations processed (1SXPS used those up to October 2012, while we reduced those up to December 2018), and iii) the choice of background regions (whose shape and size depend on the S/N of the source in 1SXPS). However, we found that our results agree with the 1SXPS catalog with differences of only a few percent (i.e., less than 5%) mainly due to the reasons highlighted above. Thus, we obtained positions, counts, and count rates for all sources. We flagged sources with full-band count rates higher than 0.5 photons per second, indicating the presence of pile-up. These are 13 sources, all of them with mild pile-up (i.e., with count rates lower than one photon per second). We then derived the hardness ratio ($\rm{HR}_{\rm{X}}$) for each source and full-band X-ray fluxes ($\rm{F}_{\rm{X}}$). The hardness ratio was computed as $(\rm{H}-\rm{S})/(\rm{H}+\rm{S})$, where $\rm{H}$ are counts in the band (2 to 10 keV) and $\rm{S}$ those in the band (0.5 to 2 keV). We verified that using counts or count rates to obtain $\rm{HR}_{\rm{X}}$ is equivalent because the exposure map is not energy dependent in the 0.5-10 keV band. Fluxes were obtained for each source using <span style="font-variant:small-caps;">PIMMS</span> [@Mukai93], assuming a power law with a photon index of 2.0 and values of the Galactic column density values obtained from the LAB Survey of Galactic HI [@Kalberla05]. The choice of the photon index affects the estimate of the X-ray flux by a factor of less than a few percent [@Massaro11c; @Massaro12d]. In Table 1 we report the X-ray data and information available from cross-matches that were performed with multifrequency archives (see Section 4.1 for details). In Col. 1 we report the BZCat name, in Cols. 2 to 4 the total counts and their error for the soft (0.5-2.0 keV), hard (2.0-10 keV), and total (0.5-10 keV) bands, in Cols. 5 to 10 the AllWISE magnitudes and their errors, and in Col. 11 the ratio of the X-ray to radio flux. Results ======= X-ray counterparts of $\gamma$-ray BL Lac objects ------------------------------------------------- In total, we found 1362 X-ray sources in the 351 BZB fields we reduced and analyzed. In Figure 2 we show the distribution of the total exposure times of our final sample, of all observed fields, considering those with at least one X-ray detection and those with no X-ray sources above an S/N of 3, separately. In 14 BZBs fields no XRT counterpart was found. All these sources were observed for a total exposure time shorter than 2.4 ks, with two exceptions: 5BZBJ1458+3720 (5.3 ks) and 5BZBJ0434-2342 (6.8 ks). The first does not show any detection in the XRT merged observation, and the second shows two X-ray sources at more than 6 arcmin from the BZB position. There is, however, a marginal detection with an S/N of 2.8 coincident with the BZB position. This is also detected in the 1SXPS catalog with an S/N of 1.9. Both were discarded because no detection above the S/N threshold of 3 was reported. Thus the lack of X-ray counterparts for these 14 cases is mainly imputable to our choice of the S/N threshold combined with short exposure time. ![Distribution of the total exposure time for all selected BZBs. The blue histogram indicates the X-ray merged event files with at least one X-ray source detected with an S/N greater than 3, while in red we show those without an X-ray detection.](Exptime.png){height="7.2cm" width="8.4cm"} The following cuts were also performed *: 1. Spurious X-ray detections due to artifacts and bad pixels or bad columns were discarded. 2. X-ray sources that were detected close ((i.e., closer than 8 arcsec) to a bright source were discarded as artifacts of the detection algorithm. The only exception was the source with the highest S/N. 3. Sources that were clearly extended with respect to the XRT 90% point spread function (PSF) [23 arcsec, @Moretti04] were discarded. 4. Sources not coincident with the BZB position (i.e., beyond 5.5 arcsec) and with negative counts on either soft and hard band (defined in Sec. 3.2) were discarded. 5. 5BZBJ1104+3812 (also known as Mrk 421), which presented severe pile-up contamination, was also discarded. Mrk 421 has been thoroughly studied in $\gamma$-rays and X-rays [e.g., @Brinkmann05; @Isobe10; @Banerjee19; @Hervet19 and references therein]. 6. Two BZBs that lie in the same field within the positional uncertainty region of the same Fermi* source, 3FGLJ0323.6-0109 (i.e., 5BZBJ0323-0111 and 5BZBJ0323-0108), were also discarded because of possible source confusion in $\gamma$-rays. In addition, we also cross-matched the X-ray positions with the AllWISE catalog with an uncertainty radius of 3.3 arcsec, following @DAbrusco13. All mid-infrared counterparts were correctly associated with the XRT detections, with two exceptions: 5BZBJ0335-4459 (at 3.8 arcsec) and 5BZBJ2131-2515 (at 3.5 arcsec)[^3]. Because they are the closest WISE sources and unique matches, we kept these two BZBs in our sample. Of the 351 Fermi BZBs observed by Swift/XRT for 1 to 20 ks, 337 were detected and 14 were not. In addition, 3 BZBs were discarded because of pile-up or source confusion. Thus, we built a clean sample of 334 BZBs with X-ray and mid-infrared counterparts, and a clean sample of 675 background and foreground X-ray sources lying within an angular separation of 6 arcmin from the $\gamma$-ray position. The flow chart shown in Figure 3 summarizes all our selection steps. ![Flow chart to highlight all steps we followed to build our final BZBs sample.](FlowChart1.png) X-ray properties of $\gamma$-ray BL Lac objects {#sec:results} ----------------------------------------------- We detected 337 BZBs using XRT data out of the original selected sample of 351 Fermi BZBs. This means that 96% of the BZB sources listed by Fermi show an X-ray counterpart when they are observed for more than 1ks. This strongly supports the existence of a connection between X-rays and $\gamma$-rays and certainly motivates X-ray follow-up observations at least to the level of the $\gamma$-ray flux of our current BZB sample to search for UGS counterparts [@Massaro13d]. In Figure 4 we show the distribution of the $\gamma$-ray energy flux in the 100 MeV to 100 GeV band for all Fermi BZBs in our sample, divided into those for which we found at least one X-ray counterpart and those for which no XRT counterpart was detected. We plot the $\gamma$-ray flux thresholds above which we found at least one X-ray counterpart for 100% ($\rm{F}_{\gamma}=1.7\times10^{-11}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}$), 98% ($\rm{F}_{\gamma}=7.0\times10^{-12}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}$), and 96% ($\rm{F}_{\gamma}=2.9\times10^{-12}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}$) of the sample. We expect that these thresholds can be used to find new BZBs among the Fermi UGSs. ![Distribution of $F_{\gamma}$ in the 100 MeV to 100 GeV band for all 351 BZBs in our selected sample. As in Fig. 2, the blue histogram indicates the X-ray merged event file with at least one X-ray source detected with S/N greater than 3, and in red we show those without an X-ray detection. The solid black line corresponds to the threshold above which 100% of the selected BZBs has an X-ray counterpart, while the dashed and dot-dashed black lines mark the 98% and 96% thresholds, respectively.](BZB_GFlux.png){height="7.2cm" width="8.4cm"} The nominal exposure time used for the UGS XRT follow-up campaign is 5 ks [@Stroh13]. Thus, we verified the fraction of BZBs that could be detected by rescaling the exposure time to this nominal value. We selected only sources for which the total (merged) exposure time was longer than 5 ks (222 sources), and then scaled their S/N assuming a Poisson distribution of the observed count rates. We confirm that 98% of our selected sample would still be detected at an S/N ratio greater than 3 with shorter (i.e., 5ks) exposure time. The HBL versus LBL classification was made following @Maselli10a, as mentioned in §1. The 1.4 GHz radio fluxes were taken from BZCat for each source, with the exception of 5BZBJ1326-5256 and 5BZBJ1604-4441, for which only the flux at 4.85 GHz was available. We assumed that the spectral shape of blazars is flat (radio spectral index equal to zero) at radio frequencies [see, e.g., @Healey07; @Massaro13b; @Massaro13c], therefore we considered the 4.85 GHz flux to be equal to that at 1.4 GHz. Because the XRT band (0.5-10 keV) differs from that of ROSAT (0.1-2.4 keV), which is the band that was originally used for this classification, we compared the use of full-band fluxes adjusted to the ROSAT band assuming a power law. The HBL and LBL classification is the same using either of the X-ray flux estimates. Then we adopted full-band XRT fluxes. In Table 1 we show the full-band XRT-to-radio flux ratio $\rm{F}_{\rm{X}}/\rm{S}_{1.4}$ instead of $\Phi_{\rm{XR}}$, so that the limit between classes is $\rm{F}_{\rm{X}}/\rm{S}_{1.4}>10^{-11}$. ---------------- ----------------- ---------------- ------------------------------------------------------- --------- --------------- --------- --------------- --------- --------------- -------------------------------- Source Soft Hard Flux W1 $\sigma_{W1}$ W2 $\sigma_{W2}$ W3 $\sigma_{W3}$ $\rm{F}_{\rm{X}}/\rm{S}_{1.4}$ \[photons\] \[photons\] $[\times10^{-12}\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}]$ \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] $[\times10^{-12}]$ 5BZBJ0001-0746 $22.3\pm5.5$ $11.4\pm3.9$ $0.3\pm0.05$ 12.68 0.01 11.76 0.01 9.17 0.04 1.4 5BZBJ0004-1148 $55.7\pm8.2$ $13.9\pm4.3$ $0.5\pm0.06$ 14.10 0.02 13.12 0.04 10.4 0.1 1.0 5BZBJ0008-2339 $533.7\pm26.0$ $122.2\pm13.0$ $5.5\pm0.2$ 13.75 0.02 13.29 0.03 11.3 0.2 150 5BZBJ0009+0628 $69.6\pm10.0$ $23.4\pm6.1$ $0.5\pm0.06$ 12.97 0.01 12.04 0.02 9.36 0.04 2.1 5BZBJ0009+5030 $44.6\pm7.5$ $9.1\pm3.4$ $0.4\pm0.05$ 12.97 0.02 12.44 0.02 10.37 0.08 29 5BZBJ0014-5022 $1051.0\pm37.0$ $293.4\pm19.0$ $7.6\pm0.2$ 14.74 0.03 14.6 0.1 11.9 (...) 590 5BZBJ0019+2021 $15.9\pm4.7$ $5.8\pm2.8$ $0.2\pm0.05$ 14.44 0.04 13.39 0.06 10.4 0.1 0.2 5BZBJ0019-8152 $91.3\pm11.0$ $9.8\pm4.0$ $0.6\pm0.06$ 12.209 0.007 11.331 0.007 8.98 0.02 6.8 5BZBJ0021-2550 $35.9\pm6.9$ $4.9\pm2.8$ $0.2\pm0.03$ 13.38 0.02 12.54 0.02 10.24 0.08 3.0 5BZBJ0022+0608 $67.3\pm8.9$ $19.8\pm5.2$ $0.4\pm0.05$ 12.92 0.02 11.88 0.01 9.08 0.04 1.2 ---------------- ----------------- ---------------- ------------------------------------------------------- --------- --------------- --------- --------------- --------- --------------- -------------------------------- In total, 175 sources out of the total 334 are HBLs, while the remaining 159 are classified as LBLs. As expected, errors in $\rm{HR}_{\rm{X}}$ and $\rm{F}_{\rm{X}}$ for HBL and LBL objects are smaller than for background or foreground objects because BZBs are generally one order of magnitude brighter (average flux $\rm{F}_{\rm{X}}=3.9\times10^{-12}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$) than background or foreground objects (average flux $\rm{F}_{\rm{X}}=2.7\times10^{-13}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$). On the other hand, HBLs and LBLs have softer X-ray spectra than background or foreground objects, which span all the possible $\rm{HR}_{\rm{X}}$ values. In Figure 5 we show $\rm{F}_{\rm{X}}$ versus $\rm{HR}_{\rm{X}}$ plotted for all detected sources (i.e., BZBs and background or foreground X-ray objects ). BZBs are separated into HBLs and LBLs with a color code indicating their $\Phi_{\rm{XR}}$ value, which extends from green (HBLs) to orange (LBLs), while sources with $\Phi_{\rm{XR}}\approx1$ are marked in yellow. The average value for the $\rm{HR}_{\rm{X}}$ is $\rm{HR}_{\rm{X}}=-0.63\pm0.09$ for HBLs and $\rm{HR}_{\rm{X}}=-0.46\pm0.16$ for LBLs. Sources with $\rm{F}_{\rm{X}}>2.0\times10^{-11}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$ (indicated with the black solid line) suffer from pile-up. The HBL and LBL subsamples lie in different regions, as do background or foreground objects. Intermediate sources (i.e., with $\Phi_{\rm{XR}}\approx1$) lie in between both regions, as expected. This is also true in all figures shown from this point on. To quantify the separation between each subsample in the $\rm{F}_{\rm{X}}$–$\rm{HR}_{\rm{X}}$ space, we performed a kernel density estimation (KDE) analysis, in order to estimate density thresholds for each subsample, following @DAbrusco09 [@Laurino11; @Massaro11; @Massaro13d; @DAbrusco19], and references therein. In Figure 6 we show $\rm{F}_{\rm{X}}$ versus $\rm{HR}_{\rm{X}}$ for background or foreground objects and for BZBs separated into HBLs and LBLs, with contours obtained from the KDE analysis for each subsample at 70%, 80%, and 90% isodensity levels drawn on the basis of the probability evaluated with the KDE. Although the subsamples overlap, the overlap is smallest when we consider a 90% density level. Moreover, HBLs lie in a very distinct region at all density levels. From Figures 5 and 6, it follows that the LBL and HBL classification following @Maselli10a coincides with our results obtained through a KDE analysis, with intermediate-type objects populating the overlapping isodensity area between both classes. In Figure 7 we also show the distribution of the angular separation between the X-ray and the $\gamma$-ray positions in the same field. X-ray detected BZBs on average appear to be closer to their $\gamma$-ray counterpart than background X-ray sources in the examined fields. This is confirmed through a Kolmogorov-Smirnov test, which yields a negligible chance coincidence probability, or p-chance $\rm{p}$. This implies a low contamination by background or foreground sources. ![$\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. $\rm{HR}_{\rm{X}}$ for all BZBs in the selected sample. Background or foreground X-ray sources are marked in gray, and BZB classified as HBL and as LBL are marked in green and orange, respectively. The solid black line separates BZBs with pile-up, which show $\rm{F}_{\rm{X}}>2.0\times10^{-11}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$.](FX_HRX.png){height="6.2cm" width="8.4cm"} ![$\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. $\rm{HR}_{\rm{X}}$ for all BZBs in the selected sample, same as in Figure 5. We show the 70%, 80%, and 90% isodensity contours as obtained from a KDE analysis for background or foreground sources in gray, HBLs in green, and LBLs in orange.](KDE_Ezequiel_70_80_90.png){height="7.2cm" width="8.4cm"} ![Angular separation between the X-ray and $\gamma$-ray position for all BZBs, in arcminutes. As in Fig. 5, BZBs classified as HBLs and LBLs are shown in green and orange, respectively, while background or foreground X-ray sources are shown in gray.](XG_Dist.png){height="7.6cm" width="8.4cm"} In Figure 8 we report $\rm{F}_{\rm{X}}$ versus the WISE mid-infrared magnitude at 12 $\mu$m (in the upper panel), and versus the $\gamma$-ray energy flux in the 100 MeV - 100 GeV band, taken from the 3FGL catalog (in the upper panel). The 12 $\mu$m magnitude is the least affected by Galactic extinction but is still more sensitive than the W4 magnitude at 22 $\mu$m [@DAbrusco12]. No clear trend is visible in either panel of Figure 8. This is expected because while $\gamma$-rays are dominated by the IC SED component, X-rays are due to synchrotron emission in HBLs but might be a combination of synchrotron and IC components in LBLs [@Bondi01; @Massaro08b]. No clear trend is visible between $\rm{F}_{\rm{X}}$ and the $\gamma$-ray energy flux. However, we highlight that because 96% $\gamma$-ray BZBs have a counterpart in the X-rays, this provides a direct link between the BZB emission in these two energy ranges. We recall that HBLs are on average an order of magnitude brighter in X-rays than LBLs. The average $\rm{F}_{\rm{X}}$ value is $(4.5\pm3.3)\times10^{-12}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$ for HBLs, and $(4.5\pm1.9)\times10^{-13}\,\rm{erg}\,\rm{cm}^{-2}\,\rm{s}^{-1}$ for LBLs. On the other hand, LBLs are on average 0.6 magnitudes brighter in mid-infrared than HBLs. ![Lower panel: $\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. the WISE magnitude at 12 $\mu$m for all selected BZBs. Upper panel: $\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. the $\rm{F}_{\gamma}$ in the 100 MeV - 100 GeV band, collected from the 3FGL. BL Lac objects classified as HBLs and LBLs are shown in green and orange, respectively.](FXFGW3.png){height="8.2cm" width="7.8cm"} ![Upper panel: $\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. the \[4.6\]-\[12\] $\mu$m mid-infrared color. Selected BZBs are classified as HBLs and LBLs and are marked in green and orange, respectively. Dashed lines indicate regression lines for LBLs (purple), HBLs (red), and the whole sample (black). Lower panel: $\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. the $\alpha_{\gamma}$ as reported in the 3FGL. Solid black lines mark the area within one standard deviation centered on the mean, and solid gray shows the area computed with the median absolute deviation centered on the median value.](FXW23GSIlog.png){height="10.2cm" width="7.8cm"} The distinction between HBLs and LBLs becomes clearer when we compare their X-ray, $\gamma$-ray, and mid-infrared spectral shapes. In Figure 9 we show $\rm{F}_{\rm{X}}$ versus the mid-infrared color computed with WISE magnitudes at 4.6$\mu$m and 12$\mu$m (upper panel) and versus the $\gamma$-ray spectral index reported in the 3FGL (lower panel). HBLs are brighter in X-rays and harder in $\gamma$-rays than LBLs [@Ghisellini10; @Ghisellini17]. In the lower panel of Figure 9 we also plot the average and median values of these parameters for both samples, together with their standard deviation and median absolute deviation, to highlight the clear distinction between the two subclasses. Although we were unable to establish a clear trend between fluxes or spectral shapes of BZBs in the X-ray and $\gamma$-ray band, we proved that there is a link between the emission in these two energy ranges given by the Fermi BZB high detection rate in the Swift/XRT observations analyzed here. This is different from what was discovered between radio, mid-infrared, and $\gamma$-ray emission for this class of AGNs, but it is crucial to support ongoing and future X-ray campaigns searching for blazars within sample of UGSs as well as for spectroscopic follow-up observations of X-ray sources that lie within the positional uncertainty region of unassociated Fermi sources. The previous distinction improves significantly when the mid-infrared color is used, mainly because it marks the difference in the synchrotron component between HBLs and LBLs better. We found a correlation between $\rm{F}_{\rm{X}}$ and the \[4.6\]-\[12\]$\mu$m WISE color index, with a slope of $\sim -0.32$ and a p-chance of $\rm{p}<1\times10^{-7}$ for the whole sample of BZBs (plotted as the black line in Figure 9). This is due to the shifting of the whole synchrotron component toward higher energies because X-rays and infrared trace two different sides of the synchrotron peak. The trend is clearer for HBLs than for LBLs possibly because the IC component in the SEDs of the latter subclass might contribute. The p-chance of both correlations is $\rm{p}\sim3.7\times10^{-7}$ for HBLs (plotted in red) and $\rm{p}\sim0.019$ for LBLs (plotted in purple), indicating that the second is not statistically significant. In almost all previous WISE and X-ray combined investigations [see, e.g., @Maselli13], the lack of uniform X-ray datasets did not allow a comparison of the behavior of HBLs and LBLs, as done here. In Figure 10 we show $\rm{HR}_{\rm{X}}$ versus the mid-infrared color and versus the $\gamma$-ray spectral index. We note that HBLs tend to cover a small range of X-ray hardness ratios; they appear to have softer X-ray spectra than LBLs. The average value for the $\rm{HR}_{\rm{X}}$ is $-0.63\pm0.09$ for HBLs and $-0.46\pm0.16$ for LBLs. The opposite occurs for the $\gamma$-ray spectral index and mid-infrared colors, where HBLs show a wider range of values than LBLs. ![Comparison of the spectral shape for all the BZBs in our sample. LBLs are shown in orange and HBLs in green. Upper panel: $\rm{HR}_{\rm{X}}$ in the 0.5-10 keV band vs. the \[4.6\]-\[12\] $\mu$m mid-infrared color. Lower panel: $\rm{HR}_{\rm{X}}$ in the 0.5-10 keV band vs. $\alpha_{\gamma}$. In both panels the average uncertainties on the parameters are shown as crosses in the top right corner.](HRXGSIW23.png){height="10.2cm" width="7.8cm"} Finally, for 77 BZBs a redshift estimate was available in Roma-BZCat, allowing us to compute their X-ray luminosities ($\rm{L}_{\rm{X}}$), as shown in Figure 11. HBLs tend to be more luminous than LBLs in the X-ray band. This might be due to the redshift difference, which on average is 0.283 for HBLs and 0.443 for LBLs. However, we also note that their corresponding $\gamma$-ray luminosities ($\rm{L}_{\gamma}$) are indeed quite similar for the same sample. ![Distribution of $\rm{L}_{\rm{X}}$ in the 0.5-10 keV band for the subsample of 77 BZBs with a well-determined redshift estimate reported in the Roma-BZCat [@Massaro15]. As in previous figures, LBLs are in orange and HBLs in green.](XLum.png) Theoretical interpretation ========================== The theoretical connection between synchrotron and IC processes in the SSC scenario has previosly been extensively investigated [see, e.g., @Dermer95; @Bloom96; @Mastichiadis97; @Dermer02; @Massaro06; @Tramacere11 and references therein]. However, possible observable connections between X-ray and $\gamma$-ray emissions in BZBs are not yet fully exploited, certainly not on a statistical sample as we analyzed here. As previously stated, while we carried out this investigation, we found that HBLs tend to be brighter in the X-rays than LBLs and cover a wider range of fluxes. On the other hand, LBLs cover a wide range of $\rm{HR}_{\rm{X}}$, being generally harder than HBLs in the X-rays. These results suggest that on average HBLs do not vary their X-ray spectral shape significanlty, but only their total integrated power, and the opposite holds for LBLs. According to the SSC scenario, we could interpret this as follows. For LBLs a decrease in synchrotron emission in the X-rays could be balanced by an increase in IC component. This implies a certain balance in the total X-ray flux, but a noticeable change in the spectral shape (i.e., X-ray hardness ratio $\rm{HR}_{\rm{X}}$). On the other hand, the fact that for HBLs, $\rm{HR}_{\rm{X}}$ is restricted to a narrow range of values implies that the position of the synchrotron peak remains constant on average, and that we only observe the high-energy tail of their synchrotron component in X-rays. We expect that the HBL behavior in X-rays is consistent with the theoretical scenario proposed in @Massaro11a on the basis of the acceleration mechanism proposed by @Cavaliere80. Thus, assuming that the beaming factor of HBL jets does not vary significantly, as the total number of emitting particles, and following @Paggi09a, we state that the frequency of the synchrotron peak scales as $\nu_{\rm{s}}\propto\gamma_{3\rm{p}}^2\rm{B}$, where $\rm{B}$ is the average magnetic field in the emission region and $\gamma_{3\rm{p}}$ is the peak of the $\gamma\,n(\gamma)$, with $n(\gamma)$ the particle energy distribution [see @Tramacere07; @Tramacere11 for details]. For HBLs, $\nu_{\rm{s}}$ remains constant on average, implying that $\gamma_{3\rm{p}}^2\rm{B}=\rm{const}$. Assuming that the IC emission occurs in the Thomson regime, we expect that the SED peak frequency of the high-energy component scales as $\nu_{\rm{ic}} \propto \gamma_{3\rm{p}}\,\nu_{\rm{s}}$, thus as $\gamma_{3\rm{p}}^2$ under the circumstances previously described. On the other hand, the peak height of the IC component being in general proportional to $\gamma_{3\rm{p}}^4\,\rm{B}^2$ does not show significant changes if particle energy distribution and magnetic field are the main driver of spectral variations [@Paggi11]. According to @Blandford77 and @Cavaliere02, the BZB luminosity should be limited by the Blandford-Znajek (BZ) limit [@Ghosh97; @Tchekhovskoy09], defined as $\rm{L}_{\rm{BZ}}\leq 8\times10^{45}\times\left(\frac{\rm{M_{\rm{BH}}}}{10^{9}\,\rm{M_{\rm{Sun}}}}\right)\,\rm{erg}\,\rm{s}^{-1}$, where $\rm{M}_{\rm{BH}}$ is the central black hole mass, and $\rm{M}_{\rm{Sun}}$ the mass of the Sun. This condition applies only if the output can be ascribed solely to the black hole (i.e., in “dry” sources), which is a useful benchmark to analyze whether accretion is negligible in BZBs [@Paggi09b]. There is evidence, however, that BZBs might be emitting more than what can be described within this benchmark alone [see, e.g., @Tavecchio16]. In presence of significant current accretion, we expect the jet launched from the central black hole to be more powerful [@Ghisellini14], and thus yielding higher observed luminosities that ultimately exceed the BZ limit, which instead applies to sources that are only powered by rotation. For our sample we found 27 black hole mass estimates, measured with different methods as reported by @Woo05 [@Plotkin11; @LeonTavares11; @Sbarrato12; @Shaw12; @Xiong14] and @Ghisellini15. We used them to compute the ratio of the sum of $\rm{L}_{\rm{X}}+\rm{L}_{\gamma}$ to the BZ luminosity limit. $\rm{L}_{\rm{X}}$ was computed from our data in the 0.5-10 keV band, and $\rm{L}_{\gamma}$ was obtained in the 100 MeV to 100 GeV band as listed in 3FGL. We adopted $\rm{L}_{\rm{X}}+\rm{L}_{\gamma}$ as an estimate of the bolometric luminosity because at least one of them is close to the SED peak. Thus, we expect the ratio to be below 1 if the BZ limit is applicable. In Figure 12 we plot this ratio versus the position of the rest-frame synchrotron peak taken from the 3FGL catalog. ![Synchrotron peak frequency vs. the ratio between the estimate of the bolometric luminosity and the BZ luminosity, computed as reported in Section 5. As in previous figures, LBLs are in orange and HBLs in green. The gray dashed line indicates a luminosity ratio of 1, below which “dry” BZBs should lie.](LBZ.png) It is clear that all HBLs are still below the BZ limit, which is expected because they are the less bolometrically luminous type of blazar [@Sambruna96]. In the LBL case, however, 8 out of 12 are above the BZ limit. Although the subsample is small, this is an indication that the BZ benchmark does not apply cleanly to the LBL scenario, as it does for HBLs. An explanation for this discrepancy between the BZ benchmark and the observed LBL luminosities might be that these eight sources are probably not ’dry’ sources as HBLs. Thus their broadband emission is contaminated by IC scattering of seed photons arising from surrounding gas and/or the accretion disk [@Abdo15; @Arsioli18]. This is also consistent with the fact that LBLs are more similar to BZQs, and in these cases, the accretion component is important. In particular, for all these cases, $\rm{L}_{\gamma}$ is more than one order of magnitude higher than the X-ray luminosity, which is a strong indication that emission in these sources is strongly affected by an external Compton process [@Arsioli18]. Summary and conclusions {#sec:conclusions} ======================= We characterized the X-ray properties of Fermi BZBs, searching for a possible connection between X-rays and $\gamma$-raysthat could motivate follow-up X-ray observations to discover new blazar-like counterparts of UGSs [@Stroh13; @Paggi13; @Massaro15c]. To achieve our goal, we built a sample of 351 BZBs listed by Fermi that were observed by the Swift/XRT telescope in photon-counting mode, with exposure times between 1 and 20 ks, and collected up to December 2018. 1. Of the 351 Fermi BZBs that were observed by XRT for more than 1ks, 96% (337) were detected with an S/N greater than 3 (mean S/N ratio of 25). We obtained accurate positions, counts, count rates, fluxes, and images for all of them in the soft X-ray band between 0.5 and 10 keV. 2. The BZBs are on average brighter in X-rays than other X-ray emitting sources that lie outside of the galactic plane ($\|b\|>10^{\circ}$). When they are classified as HBLs and LBLs, HBLs are generally brighter in X-ray flux and cover a wider range of flux values than LBLs. 3. On the other hand, the spectral shape of LBLs in X-rays is harder and less uniform than that of HBLs. This might be due to a more variable spectral shape in the X-ray band: for HBLs, the X-ray emission is dominated by the synchrotron process, while for LBLs the IC component is non-negligible. 4. We proved the direct link between the X-ray and $\gamma$-ray emission in BZBs, as supported by the high detection rate (96%) of their X-ray counterparts. This rate would remain unchanged even with shorter exposure times (5 ks). 5. The $\rm{F}_{\rm{X}}$ and mid-infrared color \[4.6\]-\[12\]$\mu$m ($p<1\times10^{-7}$) are correlated. X-ray brighter BZBs are bluer in the mid-infrared. 6. For a subsample of 77 sources we obtained redshift values that allowed us to compute distances and luminosities. We note that HBLs are bolometrically less luminous than LBLs. 7. For 27 sources out of these 77 with a determined redshift, we also searched for values of the central black hole mass in the literature. All HBLs are below their BZ limit. However, some LBLs exceed this limit. We speculate that this indicates that LBLs may be accreting more gas than HBLs, meaning that the ’dry’ hypothesis would be valid only in the latter, not in the former. Acknowledgements ================ E. J. Marchesini would like to thank Rocío I. Páez and M. Victoria Reynaldi for useful discussions on this work. This work is supported by the “Departments of Excellence 2018 - 2022’’ Grant awarded by the Italian Ministry of Education, University and Research (MIUR) (L. 232/2016). This research has made use of resources provided by the Compagnia di San Paolo for the grant awarded on the BLENV project (S1618\_L1\_MASF\_01) and by the Ministry of Education, Universities and Research for the grant MASF\_FFABR\_17\_01. F.M. acknowledges financial contribution from the agreement ASI-INAF n.2017-14-H.0 A.P. acknowledges financial support from the Consorzio Interuniversitario per la fisica Spaziale (CIFS) under the agreement related to the grant MASF\_CONTR\_FIN\_18\_02. This research has made use of data obtained from the high- energy Astrophysics Science Archive Research Center (HEASARC) provided by NASA’s Goddard Space Flight Center. This work is part of a project that has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Słodowska-Curie Grant Agreement No. 664931. Part of this work is based on the NVSS (NRAO VLA Sky Survey). The National Radio Astronomy Observatory is operated by Associated Universities, Inc., under contract with the National Science Foundation. The Molonglo Observatory site manager, Duncan Campbell-Wilson, and the staff, Jeff Webb, Michael White, and John Barry, are responsible for the smooth operation of the Molonglo Observatory Synthesis Telescope (MOST) and the day-to-day observing program of SUMSS. SUMSS is dedicated to Michael Large, whose expertise and vision made the project possible. The MOST is operated by the School of Physics with the support of the Australian Research Council and the Science Foundation for Physics within the University of Sydney. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. TOPCAT[^4] [@Taylor05] and STILTS [@Taylor06] were used for the preparation and manipulation of the images and the tabular data. Appendix ======== In this section we include complementary figures. In Figure 13 we show the X-ray full-band flux versus the magnitudes at 3.4 and 4.6 $\mu$m from the AllWISE catalog in the upper and lower panel, respectively. They do not differ from the results in the main text, as shown in Figure 10. In the same way, we show in Figure 14 the comparison between the X-ray full-band flux and the color index \[3.4\]-\[4.6\], which is comparable to Figure 11. Finally, in Figure 15 we show the X-ray hardness ratio versus the color index \[3.4\]-\[4.6\], which follows the same trend as discussed for Figure 10. ![$\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. the WISE magnitude estimate at a nominal wavelength of 3.4 $\mu$m (upper panel) and vs. that evaluated at 4.6 $\mu$m (lower panel). BZBs classified as LBL are shown in orange and HBLs are marked in green.](FXW1W2.png){height="10.2cm" width="7.8cm"} ![$\rm{F}_{\rm{X}}$ in the 0.5-10 keV band vs. the \[3.4\]-\[4.6\] $\mu$m mid-infrared color.](FX_W12.png){height="6.2cm" width="8.2cm"} ![$\rm{HR}_{\rm{X}}$ in the 0.5-10 keV band vs. the \[3.4\]-\[4.6\] $\mu$m. Average uncertainties on these two parameters are shown as crosses in the bottom right corner.](HRX_W12.png){height="6.2cm" width="8.2cm"} [^1]: https://arxiv.org/abs/1902.10045 [^2]: https://www.Swift.psu.edu/unassociated/ [^3]: 5BZBJ1046-2535 was not included because of light contamination from a nearby star. 5BZBJ2108-6637 is the only source for which no WISE counterpart was found. [^4]: http://www.star.bris.ac.uk/m bt/topcat/
--- abstract: \| The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $\delta(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\in\{1,2\}$, see [@PK-CA]. For $r\geq 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $\delta(\mathcal{P}(C_n))$ is equal to the degree of at least one of these vertices. If $r=3$ or if $n$ is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of $n$. [Key words.]{} Power graph, Cyclic group, Minimum degree, Edge connectivity, Euler’s totient function. [AMS subject classification.]{} 05C25, 05C07, 05C40 author: - Ramesh Prasad Panda - Kamal Lochan Patra - Binod Kumar Sahoo title: 'On the minimum degree of the power graph of a finite cyclic group' --- Introduction ============ Let $\Gamma$ be a simple graph with at least two vertices. The [edge connectivity]{} $\kappa'(\Gamma)$ of $\Gamma$ is the minimum number of edges whose deletion from $\Gamma$ gives a disconnected subgraph of $\Gamma$. The [vertex connectivity]{} $\kappa(\Gamma)$ of $\Gamma$ is the minimum number of vertices which need to be removed from $\Gamma$ so that the induced subgraph of $\Gamma$ on the remaining vertices is disconnected or has only one vertex. The latter case arises only when $\Gamma$ is a complete graph. The [minimum degree]{} of $\Gamma$, denoted by $\delta(\Gamma)$, is the minimum of the degrees of vertices of $\Gamma$. The study of vertex/edge connectivity is an interesting problem in graph theory. It is known that $\kappa(\Gamma)\leq \kappa'(\Gamma)\leq \delta(\Gamma)$, and $\kappa'(\Gamma)=\delta(\Gamma)$ if the diameter of $\Gamma$ is at most $2$, see Theorem 4.1.9 and Exercise 4.1.25 in [@west]. Power graph ----------- The notion of directed power graph of a group was introduced in [@kel-2000], which was further extended to semigroups in [@kel-2001; @kel-2002]. Then the undirected power graph of a semigroup, in particular, of a group was defined in [@CGS-2009]. Many researchers have investigated both directed and undirected power graphs of groups from different view points. More on these graphs can be found in the survey paper [@AKC-2013] and the references therein. Let $G$ be a finite group. The [power graph]{} of $G$, denoted by $\mathcal{P}(G)$, is the simple undirected graph with vertex set $G$, in which two distinct vertices are adjacent if one of them can be written as an integral power of the other. Since $G$ is finite, the identity element of $G$ is adjacent to all other vertices. So $\mathcal{P}(G)$ is a connected graph and its diameter is at most 2. By [@CGS-2009 Theorem 2.12], $\mathcal{P}(G)$ is a complete graph if and only if $G$ is a cyclic group of prime power order. It is proved in [@cur-2014 Theorem 1.3] and [@cur-2016 Corollary 3.4] respectively that, among all finite groups of a given order, the cyclic group of that order has the maximum number of edges and has the largest clique in its power graph. By [@dooser Theorem 5] and [@FMW Corollary 2.5], the power graph of a finite group is perfect, in particular, the clique number and the chromatic number coincide. Explicit formula for the clique number of the power graph of a finite cyclic group is given in [@mir Theorem 2] and [@dooser Theorem 7]. The full automorphism group of the power graph of a finite group is described in [@FMW-16 Theorem 2.2]. For a positive integer $n$, let $C_n$ denote the finite cyclic group of order $n$. The vertex connectivity of $\mathcal{P}(C_n)$ is studied in [@CP-ADM; @CPS; @CPS-1; @PK-JAA] and the exact value of $\kappa(\mathcal{P}(C_n))$ is obtained in the following cases: (i) $n$ is a product of distinct primes, (ii) $n$ is divisible by at most three distinct primes, (iii) $n$ is divisible by the square of its largest prime factor, and (iv) the smallest prime divisor of $n$ is greater than or equal to the number of distinct prime divisors of $n$. The above articles also provide some sharp upper bounds for $\kappa(\mathcal{P}(C_n))$. It is proved in [@PK-CA Theorem 6.7] that the vertex connectivity and the minimum degree of $\mathcal{P}(C_n)$ coincide if and only if either $n$ is a prime power or $n$ is twice of an odd prime power. For these values of $n$, the relation $\kappa(\mathcal{P}(C_n))\leq \kappa'(\mathcal{P}(C_n))\leq \delta(\mathcal{P}(C_n))$ implies that the vertex connectivity and the edge connectivity of $\mathcal{P}(C_n)$ are equal. Since the diameter of $\mathcal{P}(C_n)$ is at most $2$, the edge connectivity and the minimum degree of $\mathcal{P}(C_n)$ coincide for every $n$. Thus, in order to determine the edge connectivity of $\mathcal{P}(C_n)$, it is enough to find the minimum degree of $\mathcal{P}(C_n)$. Minimum degree of $\mathcal{P}(C_n)$ ------------------------------------ Throughout the paper, we shall identify $C_n$ with $\mathbb{Z}_n=\{0,1,2,\ldots,n-1\}$, the group of integers modulo $n$. The degree of a vertex $a\in C_n$ is denoted by $\deg(a)$. By [@MRS-JAA Lemma 3.4] (also see [@cur-2014 Lemma 2.7]), we have the following formula for $\deg(a)$: $$\label{eqn-2} \deg({a})=\frac{n}{b} +\underset{d\|b,\; d\neq b}\sum \phi \left( \frac{n}{d} \right) -1=\frac{n}{b} +\underset{d\|b}\sum \phi \left( \frac{n}{d} \right) - \phi \left( \frac{n}{b} \right) -1,$$ where $\phi$ is the Euler’s totient function and $b$ is the greatest common divisor of $a$ and $n$. If $a=0$ or $a$ is a generator of $C_n$, then $\deg(a)=n-1$. To determine $\delta(\mathcal{P}(C_n))$, our objective will be to identify a vertex of $\mathcal{P}(C_n)$ having minimum degree and then the degree of that vertex can be calculated using (\[eqn-2\]). The formula (\[eqn-2\]) implies that $\deg(a)=\deg(b)$. Thus the degree of a given non-zero vertex of $\mathcal{P}(C_n)$ is equal to the degree of some element of $C_n$ which is a divisor of $n$. Therefore, in order to identify a vertex of $\mathcal{P}(C_n)$ of minimum degree, we need to compare the degrees of all possible vertices which are divisors of $n$. If $n$ is a prime power, then $\mathcal{P}(C_n)$ is a complete graph and so $\delta(\mathcal{P}(C_n))=n-1=\deg(a)$ for every vertex $a\in C_n$. If $n>1$ is not a prime power, then $\mathcal{P}(C_n)$ is not a complete graph and so $\delta(\mathcal{P}(C_n))<n-1$. Hence the minimum degree of $\mathcal{P}(C_n)$ will be equal to the degree of a vertex which is a proper[^1] divisor of $n$. For certain values of $n$, a vertex of $\mathcal{P}(C_n)$ of minimum degree was obtained in [@PK-CA Theorem 4.6] which we mention below. [@PK-CA]\[mdv\] Let $p_1,p_2,p_3,p_4$ be prime numbers with $p_1<p_2<p_3<p_4$. Then the following hold: 1. If $n=p_1^{\alpha_1} p_2^{\alpha_2}$ for some positive integers $\alpha_1, \alpha_2$, then $\delta(\mathcal{P}(C_n)) = \deg\left({p_2^{\alpha_2}}\right)$. 2. If $n=p_1 p_2 p_3$, then $\delta(\mathcal{P}(C_n)) = \deg({p_3})$. 3. Let $n = p_1 p_2 p_3p_4$. If $n$ is odd or $p_4 \geq p_3 + \displaystyle \frac{2(p_3-1)}{p_2-1}$, then $\delta(\mathcal{P}(C_n)) = \deg({p_4})$, otherwise, $\delta(\mathcal{P}(C_n)) = \deg({p_3p_4})$. In this paper, we generalize the results stated in Proposition \[mdv\] to several other values of $n$. In view of Proposition \[mdv\](i), if necessary, we may assume that $n$ is divisible by at least three distinct prime numbers. The following theorem is proved in Section \[distinct-primes\] for the minimum degree of $\mathcal{P}(C_n)$ when $n$ is a product of distinct prime numbers. \[mindeg.main\] Let $n=p_1p_2\cdots p_r$, where $r \geq 3$ and $p_1,p_2,\ldots, p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. Then $$\delta(\mathcal{P}(C_n)) = \min\{\deg(p_{r-1}p_r), \deg(p_r)\}.$$ Further, $\delta(\mathcal{P}(C_n)) = \deg(p_r)$ if and only if $\phi(p_r) \geq \left( \dfrac{p_1p_2\cdots p_{r-2}}{\phi(p_1p_2\cdots p_{r-2})} - 1\right) \phi(p_{r-1})$. In particular, if $\phi(p_{r}) \geq (r-2) \phi(p_{r-1})$, then $\delta(\mathcal{P}(C_n)) = \deg({p_r})$. For general $n$, under certain conditions involving its prime divisors, the following theorem is proved in Section \[general-n\] on the minimum degree of $\mathcal{P}(C_n)$. \[thm.mindeg\] Let $n=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}$, where $r \geq 2$, $\alpha_1,\alpha_2,\ldots, \alpha_r$ are positive integers and $p_1,p_2,\ldots, p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. Suppose that any of the following two conditions holds: 1. $2\phi(p_{1}p_2 \cdots p_{r}) \geq p_1p_2\cdots p_r$, 2. $\phi(p_{i+1}) \geq r \phi(p_i)$ for each $i\in\{1,2,\ldots, r-1\}$. If $t\in\{2,3,\ldots, r\}$ is the largest integer such that $\alpha_t\geq \alpha_j$ for $2\leq j\leq r$, then $$\delta(\mathcal{P}(C_n)) = \min\{\deg\left(p_s^{\alpha_s}\right):t \leq s \leq r\}.$$ As an application of Theorem \[thm.mindeg\], we prove the following corollary in Section \[general-n\] which can be used to determine $\delta(\mathcal{P}(C_n))$ for many values of $n$. \[exact.mindeg\] Let $n=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}$, where $r \geq 2$, $\alpha_1,\alpha_2,\ldots, \alpha_r$ are positive integers and $p_1<p_2<\cdots <p_r$ are prime numbers. Suppose that any of the following two conditions holds: 1. $p_1\geq r+1$ and $p_r > rp_{r-1}$, 2. $p_{i+1} > rp_{i}$ for each $i\in\{1,2,\ldots, r-1\}$. Then $\delta(\mathcal{P}(C_n)) = \deg(p_r^{\alpha_r})$. For $r=3$, the following theorem is proved in Section \[three-primes\] which shows that the conclusion of Theorem \[thm.mindeg\] holds good without any condition involving the prime divisors of $n$. \[mindeg.3prime\] Let $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$, where $\alpha_1,\alpha_2,\alpha_3$ are positive integers and $p_1,p_2,p_3$ are prime numbers with $p_1<p_2<p_3$. Then $$\begin{aligned} \delta(\mathcal{P}(C_n)) = \min\{\deg\left({p_2^{\alpha_2}}\right),\;\deg\left({p_3^{\alpha_3}}\right)\}.\end{aligned}$$ Remark ------ We remark that Proposition \[mdv\] can be obtained from Theorems \[mindeg.main\] and \[thm.mindeg\]. 1. If $r=3$ and $n=p_1p_2p_3$, then $\phi(p_r)=\phi(p_3)> \phi(p_2)=(r-2)\phi(p_{r-1})$ and so Proposition \[mdv\](ii) follows from the last part of Theorem \[mindeg.main\]. 2. Suppose that $r=4$ and $n=p_1p_2p_3p_4$. If $n$ is odd, then $p_1\geq 3$ and so $2\phi(p_1p_2)> p_1p_2$ by Lemma \[prime.ineq\](ii). Then $\phi(p_4)>\phi(p_3) >\left( \dfrac{p_1p_2}{\phi(p_1p_2)} - 1\right) \phi(p_{3})$. If $n$ is even, then $p_1=2$ and so $1+\left( \dfrac{p_1p_2}{\phi(p_1p_2)} - 1 \right) \phi(p_3)=p_3 + \displaystyle \frac{2(p_3-1)}{p_2-1}$. In this case, $p_4\geq 1+ \left( \dfrac{p_1p_2}{\phi(p_1p_2)} - 1 \right) \phi(p_3)$ if and only if $p_4\geq p_3 + \displaystyle \frac{2(p_3-1)}{p_2-1}$. Then it follows that Proposition \[mdv\](iii) can be obtained from Theorem \[mindeg.main\]. 3. Finally, suppose that $n=p_1^{\alpha_1}p_2^{\alpha_2}$. If $p_1\geq 3$, then $2\phi(p_1p_2)> p_1p_2$ by Lemma \[prime.ineq\](ii). If $p_1=2$, then $\phi(p_{2}) \geq 2=2 \phi(p_1)$. Thus condition (i) or (ii) of Theorem \[thm.mindeg\] is satisfied and hence Proposition \[mdv\](i) follows from Theorem \[thm.mindeg\]. Preliminaries ============= Recall that $\phi$ is a multiplicative function, that is, $\phi(ab)=\phi(a)\phi(b)$ for any two positive integers $a,b$ which are relatively primes. We have $\phi(p^k)=p^{k-1}\phi(p)$ for any prime number $p$ and positive integer $k$. Also, $$\label{eqn-2-1} \underset{d\|m}\sum \phi(d) = m$$ for every positive integer $m$. We need the following two inequalities: the first one can be found in [@CPS-2 Lemma 3.1] and the second one was proved in [@CPS] while proving Corollary 1.4. [@CPS; @CPS-2]\[prime.ineq\] Let $p_1< p_2< \cdots < p_t$ be prime numbers. Then the following hold: 1. $(t+1)\phi(p_1p_2 \cdots p_t) \geq p_1p_2 \cdots p_t$, with equality if and only if $(t, p_1)=(1,2)$ or $(t, p_1,p_2)=(2,2,3)$. 2. If $p_1 \geq t+1$, then $2\phi(p_1 p_2 \cdots p_t) \geq p_1 p_2 \cdots p_t$, with equality if and only if $t=1$ and $p_1=2$. Certain inequalities involving degree of vertices of $\mathcal{P}(C_n)$ were proved in [@PK-CA Proposition 4.5]. From the proof of these inequalities, it can be seen that those inequalities are in fact strict and we have stated them accordingly in the following proposition. [@PK-CA]\[degcompare\] Let $n=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}$, where $r \geq 2$, $\alpha_1,\alpha_2,\ldots, \alpha_r$ are positive integers and $p_1,p_2,\ldots, p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. Then the following strict inequalities hold in $\mathcal{P}(C_n):$ 1. $\deg\left({p_1^{\alpha_1}}\right) > \deg\left({p_r^{\alpha_r}}\right)$. 2. $\deg\left({p_i^{\gamma}}\right) > \deg\left({p_i^{\beta}}\right)$ for $1 \leq i \leq r$ and $1 \leq \gamma < \beta \leq \alpha_i$. 3. $\deg\left({p_i^\beta}\right) > \deg\left({p_j^\beta}\right)$ for $1 \leq i < j \leq r$ and $1 \leq \beta \leq \min \{\alpha_i, \alpha_j\}$. 4. $\deg\left({p_1^{\beta_1}p_2^{\beta_2}\cdots p_r^{\beta_r}}\right) > \deg\left({p_2^{\beta_2}\cdots p_r^{\beta_r}}\right)$, where $1 \leq \beta_i \leq \alpha_i$ for each $i\in\{1,2,\ldots, r\}$. We need the strict inequality (\[eqn-5\]) stated in the following lemma while proving Corollaries \[exact.mindeg\] and \[coro\]. Let $n=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}$, where $r \geq 2$, $\alpha_1,\alpha_2,\ldots, \alpha_r$ are positive integers and $p_1,p_2,\ldots, p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. For $i\in\{1,2,\ldots, r-1\}$, the following strict inequality holds in $\mathcal{P}(C_n):$ $$\label{eqn-5} \deg\left({p_i^{\alpha_i}}\right) - \deg\left({p_r^{\alpha_r}}\right) > p_i^{\alpha_i-1} \left[(p_r-1)\phi\left(\frac{n}{p_i^{\alpha_i}p_r^{\alpha_r}}\right) - \frac{n}{p_i^{\alpha_i -1}p_r^{\alpha_r}}\right].$$ This follows from the proof of [@PK-CA Proposition 4.5(i)], in which we replace the subscript $1$ by $i$ and take $m=\dfrac{n}{p_i^{\alpha_i}p_r^{\alpha_r}}$. We note that the first inequality in the proof of [@PK-CA Proposition 4.5(i)] is strict. \[pp1\] Let $n=p_1p_2\cdots p_r$, where $p_1, p_2, \ldots, p_r$ are prime distinct numbers and let $a,b$ be two distinct proper divisors of $n$ such that $\dfrac{a}{b}=\dfrac{p_k}{p_l}$ for some $k,l\in\{1,2,\ldots ,r\}$. If $a<b$, then $\deg(a)>\deg(b)$. Note that both $a$ and $b$ have the same number of prime divisors, say $s$. Since $a<b$, we have $p_k<p_l$ and $1\leq s\leq r-1$. The lemma follows from Proposition \[degcompare\](iii) if $s=1$. Assume that $s\geq 2$. We have $$\begin{aligned} \underset{d \| a,\: d < a}\sum \phi \left(\frac{n}{d}\right)-\underset{d \| b,\: d < b}\sum \phi \left( \frac{n}{d}\right)& = \underset{d \| a,\: d < a, \: p_{k} \| d}\sum \phi \left( \frac{n}{d}\right)- \underset{d \| b,\: d <b, \: p_{l} \| d}\sum \phi \left( \frac{n}{d}\right)\\ & = \underset{u \vert \frac{a}{p_{k}},\: u < \frac{a}{p_{k}}}\sum \phi \left( \frac{n}{up_{k}}\right) - \underset{v \| \frac{b}{p_{l}},\: v < \frac{b}{p_{l}}}\sum \phi \left( \frac{n}{vp_{l}}\right)\\ & = \underset{u \| \frac{a}{p_{k}},\: u < \frac{a}{p_{k}}}\sum \left[\phi \left( \frac{n}{up_{k}}\right) - \phi \left( \frac{n}{up_{l}}\right) \right]\\ & = \left[\phi (p_{l}) - \phi (p_{k}) \right] \left[ \underset{u \| \frac{a}{p_{k}},\: u < \frac{a}{p_{k}}}\sum \phi \left( \frac{n}{up_{k}p_{l}}\right)\right].\end{aligned}$$ The second last equality in the above holds as $\dfrac{a}{p_k}=\dfrac{b}{p_l}$. Using the formula (\[eqn-2\]), we then get $$\begin{aligned} \deg(a) - \deg(b)& = \frac{n}{a}-\frac{n}{b} + \underset{d \| a,\: d < a}\sum \phi \left(\frac{n}{d}\right)-\underset{d \| b,\: d < b}\sum \phi \left( \frac{n}{d}\right)\\ & > \underset{d \| a,\: d < a}\sum \phi \left(\frac{n}{d}\right)-\underset{d \| b,\: d < b}\sum \phi \left( \frac{n}{d}\right)\\ & = \left[\phi (p_{l}) - \phi (p_{k}) \right] \left[ \underset{u \| \frac{a}{p_{k}},\: u < \frac{a}{p_{k}}}\sum \phi \left( \frac{n}{up_{k}p_{l}}\right)\right].\end{aligned}$$ Since $p_l >p_k$, we have $\phi (p_{l}) - \phi (p_{k})>0$ and it follows from the above that $\deg(a)>\deg(b)$. \[phi.sum\] Let $x=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}$, where $\alpha_1,\alpha_2,\ldots, \alpha_r$ are positive integers and $p_1,p_2,\ldots, p_r$ are prime numbers. If $y = p_1^{\beta_1}p_2^{\beta_2}\cdots p_r^{\beta_r}$, where $0\leq \beta_i \leq \alpha_i$ for $1 \leq i \leq r$, then $$\begin{aligned} \sum_{d \| y} \phi \left( \frac{x}{d}\right) = \left( \prod\limits_{\substack{1\leq i \leq r\\\alpha_i = \beta_i}} p_i^{\alpha_i} \right) \left(\prod\limits_{\substack{1\leq i \leq r\\\alpha_i > \beta_i}} \left(p_i^{\alpha_i} - p_i^{\alpha_i - \beta_i -1}\right) \right).\end{aligned}$$ If $\alpha_i=\beta_i$ for all $i\in\{1,2,\ldots, r\}$, then $y=x$ and the result follows from the fact that $\displaystyle\sum\limits_{d \|x} \phi \left( \frac{x}{d}\right) = \displaystyle\sum\limits_{d \|x} \phi \left( d\right)=x$. So assume that $\beta_i < \alpha_i$ for at least one $i\in\{1,2,\ldots, r\}$. For each $j\in\{1,2,\ldots, r\}$, observe that $$\label{eqn-3} \underset{0\leq \gamma_j \leq \beta_j}\sum \phi\left(p_j^{\alpha_j-\gamma_j}\right)= p_j^{\alpha_j}\; \mbox{ or }\; p_j^{\alpha_j} - p_j^{\alpha_j - \beta_j -1}$$ according as $\alpha_j =\beta_j$ or $\alpha_j >\beta_j$. We have $$\begin{aligned} \sum_{d \|y} \phi \left( \frac{x}{d}\right) = \sum_{\substack{1 \leq j \leq r \\ 0 \leq \gamma_j \leq \beta_j }} \phi \left( p_1^{\alpha_1 - \gamma_1} p_2^{\alpha_2 - \gamma_2}\cdots p_r^{\alpha_r - \gamma_r} \right) = \prod_{1\leq j \leq r} \left( \sum_{0 \leq \gamma_j \leq \beta_j} \phi\left(p_j^{\alpha_j-\gamma_j}\right)\right),\end{aligned}$$ and consequently the lemma follows using (\[eqn-3\]). Proof of Theorem \[mindeg.main\] {#distinct-primes} ================================ In this section, we take $n=p_1p_2\cdots p_r$, where $r \geq 3$ and $p_1,p_2,\ldots, p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. \[pp2\] $\delta(\mathcal{P}(C_n)) = \min \{\deg(p_sp_{s+1}\cdots p_r): 2 \leq s \leq r\}$. Let $\{k_1,k_2,\ldots,k_t\}$ be a proper subset of $\{1,2,\cdots,r\}$ with $k_1<k_2<\cdots <k_t$. Applying Lemma \[pp1\] repeatedly, we get $$\begin{aligned} \deg({p_{k_1}\cdots p_{k_{t-1}} p_{k_t}}) \geq \deg({p_{k_1}\cdots p_{k_{t-1}} p_r})\geq \cdots \geq \deg({p_{{r-t+1}}\cdots p_{r-1} p_r}).\end{aligned}$$ Since $\delta\left(\mathcal{P}(C_n)\right)$ is equal to the degree of a vertex which is a proper divisor of $n$, it follows from the above that $\delta(\mathcal{P}(C_n)) = \min \{\deg(p_sp_{s+1}\cdots p_r): 2 \leq s \leq r\}$. \[ineq.cond\] Let $3 \leq s \leq r$. Then $\deg({p_{s-1}p_{s}\cdots p_r}) \geq \deg({p_{s}p_{s+1}\cdots p_r})$ if and only if $$\begin{aligned} \label{prod.p.ineq} p_{s}p_{s+1}\cdots p_r \geq \left( \frac{p_1p_2\cdots p_{s-2}}{\phi(p_1p_2\cdots p_{s-2})} - 1 \right) \phi(p_{s-1}) +1. \end{aligned}$$ Further, $\deg({p_{s-1}p_{s}\cdots p_r}) =\deg({p_{s}p_{s+1}\cdots p_r})$ if and only if equality holds in (\[prod.p.ineq\]). We have $\dfrac{n}{p_{s-1}p_{s}\cdots p_r} - \dfrac{n}{p_{s}p_{s+1}\cdots p_r}=- p_1p_2\cdots p_{s-2} \phi(p_{s-1})$. Using (\[eqn-2-1\]), an easy calculation gives that $$\underset{d \| (p_{s-1}p_{s}\cdots p_r)}\sum \phi \left( \frac{n}{d}\right) - \underset{d \| (p_{s}p_{s+1}\cdots p_r)}\sum \phi \left( \frac{n}{d}\right)=\phi(p_1p_2\cdots p_{s-2}) p_{s}p_{s+1}\cdots p_r.$$ We also have $$\phi\left(\dfrac{n}{p_{s}p_{s+1}\cdots p_r} \right) - \phi\left(\dfrac{n}{p_{s-1}p_{s}\cdots p_r} \right)=\phi(p_1p_2\cdots p_{s-2} )(\phi(p_{s-1})-1).$$ Then, using the degree formula (\[eqn-2\]), it follows that $$\begin{aligned} & \deg ({p_{s-1}p_{s}\cdots p_r}) - \deg({p_{s}\cdots p_r})\\ & = - p_1p_2\cdots p_{s-2} \phi(p_{s-1})+ \phi(p_1p_2\cdots p_{s-2}) p_{s}p_{s+1}\cdots p_r + \phi(p_1p_2\cdots p_{s-2} )(\phi(p_{s-1})-1)\\ & = \phi(p_1p_2\cdots p_{s-2}) p_{s}p_{s+1}\cdots p_r - \left[p_1p_2\cdots p_{s-2} - \phi(p_1p_2\cdots p_{s-2} )\right] \phi(p_{s-1})- \phi(p_1p_2\cdots p_{s-2} ).\end{aligned}$$ Now it can be seen that $\deg({p_{s-1}p_{s}\cdots p_r}) \geq \deg({p_{s}p_{s+1}\cdots p_r})$ if and only if inequality (\[prod.p.ineq\]) holds. Also, $\deg({p_{s-1}p_{s}\cdots p_r}) =\deg({p_{s}p_{s+1}\cdots p_r})$ if and only if equality holds in (\[prod.p.ineq\]). \[higher-lower\] Let $3\leq s\leq r$. If $\deg (p_{s-1}p_{s}\cdots p_r) \geq \deg(p_{s}p_{s+1}\cdots p_r)$, then $$\deg (p_{s-2}p_{s-1}\cdots p_r) > \deg(p_{s-1}p_{s}\cdots p_r).$$ If $s=3$, then $\deg(p_1p_2\cdots p_r)=\deg(n)=\deg(0)=n-1>\dfrac{n\phi(p_1)}{p_1}=\deg(p_2p_3\cdots p_r)$ and the lemma follows. Assume that $4\leq s\leq r$. Observe that the inequality (\[prod.p.ineq\]) in the statement of Lemma \[ineq.cond\] is equivalent to the following inequality: $$\begin{aligned} \label{eq.mindeg2} \phi(p_1p_2\cdots p_{s-2}) (p_{s}p_{s+1}\cdots p_{r}-1) \geq \phi(p_{s-1}) \left[p_1p_2\cdots p_{s-2} - \phi(p_1p_2\cdots p_{s-2})\right]. \end{aligned}$$ Since $\deg (p_{s-1}p_{s}\cdots p_r) \geq \deg(p_{s}p_{s+1}\cdots p_r)$ by the given hypothesis, Lemma \[ineq.cond\] then implies that the inequality (\[eq.mindeg2\]) holds. We need to show that $$\deg (p_{s-2}p_{s-1}\cdots p_r) > \deg(p_{s-1}p_{s}\cdots p_r).$$ By Lemma \[ineq.cond\] again, it is enough to show that $$\begin{aligned} \label{eq.mindeg3} \phi(p_1p_2\cdots p_{s-3}) (p_{s-1}p_{s}\cdots p_{r}-1) > \phi(p_{s-2}) \left[p_1p_2\cdots p_{s-3} - \phi(p_1p_2\cdots p_{s-3})\right]. \end{aligned}$$ Since $\phi(p_{s-2})< p_{s-1}$, we have $$\begin{aligned} \label{eq.mindeg4} \phi(p_1p_2\cdots p_{s-2}) (p_{s}p_{s+1}\cdots p_{r}-1) & < \phi(p_1p_2\cdots p_{s-3})p_{s-1} (p_{s}p_{s+1}\cdots p_{r}-1) \nonumber\\ & < \phi(p_1p_2\cdots p_{s-3}) (p_{s-1}p_s\cdots p_r- 1). \end{aligned}$$ Moreover, $$\begin{aligned} \label{eq.mindeg5} \phi(p_{s-1}) \left[p_1p_2\cdots p_{s-2} - \phi(p_1p_2\cdots p_{s-2})\right] &> p_1p_2\cdots p_{s-3}p_{s-2} - \phi(p_1p_2\cdots p_{s-2}) \nonumber \\ & = p_1p_2\cdots p_{s-3} \left[\phi(p_{s-2})+1\right] - \phi(p_1p_2\cdots p_{s-2}) \nonumber \\ & > \phi(p_{s-2}) \left[p_1p_2\cdots p_{s-3} - \phi(p_1p_2\cdots p_{s-3})\right]. \end{aligned}$$ Now (\[eq.mindeg3\]) follows from the inequalities (\[eq.mindeg2\]), (\[eq.mindeg4\]) and (\[eq.mindeg5\]). If $r=3$, then $\delta(\mathcal{P}(C_n)) = \min \{\deg(p_2p_3),\deg({p_3})\}$ by Lemma \[pp2\]. Assume that $r\geq 4$. By Lemma \[prime.ineq\](i), we have $(r-2)\phi(p_1p_2\cdots p_{r-3})\geq p_1p_2\cdots p_{r-3}$, and this gives $$r-3\geq \frac{p_1p_2\cdots p_{r-3}}{\phi(p_1p_2\cdots p_{r-3})} - 1.$$ Then $$p_{r-1}p_r >(r-2)\phi(p_{r-2})\geq (r-3)\phi(p_{r-2}) +1\geq \left( \dfrac{p_1p_2\cdots p_{r-3}}{\phi(p_1p_2\cdots p_{r-3})} - 1\right) \phi(p_{r-2}) +1.$$\ So inequality (\[prod.p.ineq\]) is satisfied with $s=r-1$ and hence $\deg({p_{r-2}p_{r-1}p_r}) > \deg({p_{r-1}p_r})$ by Lemma \[ineq.cond\]. Then, using Lemma \[higher-lower\] repeatedly, we have $$\deg(p_2p_3\cdots p_r)>\deg(p_3p_4\cdots p_r)>\cdots >\deg({p_{r-2}p_{r-1}p_r}) > \deg({p_{r-1}p_r}).$$ Therefore, by Lemma \[pp2\], we get $$\delta(\mathcal{P}(C_n)) = \min\{\deg(p_{r-1}p_r), \deg(p_r)\}.$$ By Lemma \[ineq.cond\], $\delta(\mathcal{P}(C_n)) = \deg(p_r)$ if and only if $\phi(p_r) \geq \left( \dfrac{p_1p_2\cdots p_{r-2}}{\phi(p_1p_2\cdots p_{r-2})} - 1\right) \phi(p_{r-1})$. Now suppose that $\phi(p_{r}) \geq (r-2) \phi(p_{r-1})$. Since $(r-1)\phi(p_1p_2\cdots p_{r-2})\geq p_1p_2\cdots p_{r-2}$ by Lemma \[prime.ineq\](i), we have $$r-2\geq \frac{p_1p_2\cdots p_{r-2}}{\phi(p_1p_2\cdots p_{r-2})} - 1.$$ Therefore, $$p_r\geq (r-2) \phi(p_{r-1})+1\geq \left(\frac{p_1p_2\ldots p_{r-2}}{\phi(p_1p_2\ldots p_{r-2})}-1 \right) \phi(p_{r-1})+1.$$ Then $\deg(p_{r-1}p_r)\geq \deg (p_r)$ by Lemma \[ineq.cond\] and hence $\delta(\mathcal{P}(C_n)) = \deg(p_r)$. This completes the proof of Theorem \[mindeg.main\]. The following examples shows that all possibilities can occur in Theorem \[mindeg.main\] for the minimum degree of $\mathcal{P}(C_n)$. (i) If $n = 2 \cdot 13 \cdot 17 \cdot 19$, then $\delta(\mathcal{P}(C_n))=\deg(17 \cdot 19)< \deg(19)$. (ii) If $n = 2 \cdot 13 \cdot 17 \cdot 23$, then $\delta(\mathcal{P}(C_n))=\deg(23)<\deg(17\cdot 23)$. (iii) If $n=2\cdot 5\cdot 13\cdot 19$, then $\delta(\mathcal{P}(C_n))=\deg(13\cdot 19)=\deg(19)$. Note that if $n=2\cdot 3\cdot p_3\cdot p_4$ with $p_4=2p_3 -1$, then $\deg(p_{3}p_4)=\deg(p_4)$. Proof of Thereom \[thm.mindeg\] and Corolary \[exact.mindeg\] {#general-n} ============================================================= In this section, we take $n=p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}$, where $r \geq 2$, $\alpha_1,\alpha_2,\ldots, \alpha_r$ are positive integers and $p_1,p_2,\ldots, p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. \[prop1\] Let $m=p_{k_1}^{\beta_{k_1}}p_{k_2}^{\beta_{k_2}}\cdots p_{k_s}^{\beta_{k_s}}$, where $2 \leq s \leq r$, $k_1<k_2<\cdots < k_s$ and $1\leq \beta_{k_i}\leq \alpha_{k_i}$ for $1\leq i\leq s$. Suppose that one of the following two conditions holds: 1. $2\phi(p_{1} p_2\cdots p_{r}) \geq p_1p_2\cdots p_r$, 2. $\phi(p_{j+1}) \geq r \phi(p_j)$ for each $j\in \{1,2,\ldots, r-1\}$. Then $\deg(m) > \deg\left(\dfrac{m}{p_{k_i}} \right)$ for $i\in\{1,2,\ldots, s-1\}$ in $\mathcal{P}(C_n)$. Fix $i\in\{1,2,\ldots, s-1\}$. Using the degree formula (\[eqn-2\]), we have $$\xi:= \deg(m) - \deg\left(\frac{m}{p_{k_i}} \right) = \frac{n}{m} - \frac{np_{k_i}}{m} +\theta =\theta - \frac{n}{m} \phi(p_{k_i}),$$ where $$\theta:= \sum_{\substack{d \| m}} \phi \left( \frac{n}{d}\right) - \sum_{\substack{d \big\vert \frac{m}{p_{k_i}}}} \phi \left( \frac{n}{d}\right)+\phi\left(\dfrac{np_{k_i}}{m} \right) - \phi\left(\dfrac{n}{m}\right).$$ First calculate $\sum_{\substack{d \| m}} \phi \left( \frac{n}{d}\right) - \sum_{\substack{d \big\vert \frac{m}{p_{k_i}}}} \phi \left( \frac{n}{d}\right)$. Define $ n':= \dfrac{n}{p_{k_1}^{\alpha_{k_1}}p_{k_2}^{\alpha_{k_2}}\cdots p_{k_s}^{\alpha_{k_s}}} \times {p_{k_i}^{\alpha_{k_i} - \beta_{k_i}}}$. Then $$\sum_{d \| m} \phi \left( \frac{n}{d}\right) - \sum_{d \big\vert \frac{m}{p_{k_i}}} \phi \left( \frac{n}{d}\right) = \sum_{d \big\vert \frac{m}{p_{k_i}^{\beta_{k_i}}}} \phi \left( \frac{n}{dp_{k_i}^{\beta_{k_i}}}\right) = \phi \left( n'\right) \times \left(\sum_{d \big\vert \frac{m}{p_{k_i}^{\beta_{k_i}}}} \phi \left( \frac{p_{k_1}^{\alpha_{k_1}}p_{k_2}^{\alpha_{k_2}}\cdots p_{k_s}^{\alpha_{k_s}}}{dp_{k_i}^{\alpha_{k_i}}}\right)\right).$$ Taking $x=\dfrac{p_{k_1}^{\alpha_{k_1}}p_{k_2}^{\alpha_{k_2}}\cdots p_{k_s}^{\alpha_{k_s}}}{p_{k_i}^{\alpha_{k_i}}}$ and $y=\dfrac{p_{k_1}^{\beta_{k_1}}p_{k_2}^{\beta_{k_2}}\cdots p_{k_s}^{\beta_{k_s}}}{p_{k_i}^{\beta_{k_i}}}=\dfrac{m}{p_{k_i}^{\beta_{k_i}}}$ in Lemma \[phi.sum\], we get $$\begin{aligned} \label{align-num-1} \sum_{d \| m} \phi \left( \frac{n}{d}\right) - \sum_{d \big\vert \frac{m}{p_{k_i}}} \phi \left( \frac{n}{d}\right) & =\phi \left( n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} = \beta_{k_j}}} p_{k_j}^{\alpha_{k_j}} \right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \left( p_{k_j}^{\alpha_{k_j}} - p_{k_j}^{\alpha_{k_j} - \beta_{k_j} -1} \right) \right) \nonumber \\ & = \phi \left( n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} = \beta_{k_j}}} p_{k_j}^{\alpha_{k_j}} \right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} p_{k_j}^{\alpha_{k_j} - \beta_{k_j} -1} \left(p_{k_j}^{\beta_{k_j} +1} -1\right) \right).\end{aligned}$$ Next calculate $\phi \left( \dfrac{np_{k_i}}{m}\right) - \phi \left(\dfrac{n}{m}\right)$. We have $$\begin{aligned} \label{align-num-2} \phi \left( \frac{np_{k_i}}{m}\right) - \phi \left(\frac{n}{m}\right) & = \phi \left( \frac{n}{p_{k_1}^{\alpha_{k_1}}\cdots p_{k_s}^{\alpha_{k_s}}}\right) \phi \left( \frac{p_{k_1}^{\alpha_{k_1} - \beta_{k_1}} \cdots p_{k_s}^{\alpha_{k_s} - \beta_{k_s}}}{p_{k_i}^{\alpha_{k_i}-\beta_{k_i}}}\right)\nonumber \\ & \qquad \times \left[\phi\left(p_{k_i}^{\alpha_{k_i}-\beta_{k_i}+1}\right)-\phi\left(p_{k_i}^{\alpha_{k_i}-\beta_{k_i}}\right)\right] \nonumber \\ & \geq \phi \left( \frac{n}{p_{k_1}^{\alpha_{k_1}}\cdots p_{k_s}^{\alpha_{k_s}}}\right) \phi \left( \frac{p_{k_1}^{\alpha_{k_1} - \beta_{k_1}} \cdots p_{k_s}^{\alpha_{k_s} - \beta_{k_s}}}{p_{k_i}^{\alpha_{k_i}-\beta_{k_i}}}\right) \phi\left(p_{k_i}^{\alpha_{k_i}-\beta_{k_i}}\right) \left(p_{k_i}-2\right) \nonumber \\ & = \phi\left(n'\right) \phi \left( \frac{p_{k_1}^{\alpha_{k_1} - \beta_{k_1}} \cdots p_{k_s}^{\alpha_{k_s} - \beta_{k_s}}}{p_{k_i}^{\alpha_{k_i}-\beta_{k_i}}}\right) \left(p_{k_i}-2\right) \nonumber \\ & =\phi\left(n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} p_{k_j}^{\alpha_{k_j} - \beta_{k_j} -1}\phi\left(p_{k_j}\right)\right) \left(p_{k_i}-2\right),\end{aligned}$$ where equality in the above holds if and only if $\alpha_{k_i}=\beta_{k_i}$. Using (\[align-num-1\]) and (\[align-num-2\]), we get $$\begin{aligned} \theta & \geq \phi \left( n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} p_{k_j}^{\alpha_{k_j} - \beta_{k_j} -1}\right) \nonumber\\ & \qquad \times \left[\left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} = \beta_{k_j}}} p_{k_j}^{\alpha_{k_j}} \right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \left(p_{k_j}^{\beta_{k_j} +1} -1\right) \right) + \left( \prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \phi\left(p_{k_j}\right)\right) \left(p_{k_i}-2\right) \right] \label{align-num-3}\\ & = \phi \left( n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} p_{k_j}^{\alpha_{k_j} - \beta_{k_j} -1}\right)\times \left( \prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \phi\left(p_{k_j}\right)\right) \nonumber\\ & \qquad \times \left[\left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} = \beta_{k_j}}} p_{k_j}^{\alpha_{k_j}} \right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \left(p_{k_j}^{\beta_{k_j}} +\cdots + p_{k_j} + 1\right) \right) + \left(p_{k_i}-2\right) \right] \nonumber\\ & = \phi \left( n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \phi\left(p_{k_j}^{\alpha_{k_j} - \beta_{k_j}}\right)\right) \nonumber \\ & \qquad \times \left[ \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} = \beta_{k_j}}} p_{k_j}^{\alpha_{k_j}} \right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \left(p_{k_j}^{\beta_{k_j}} +\cdots + p_{k_j} + 1\right) \right) + \phi(p_{k_i})-1 \right] \nonumber \\ & \geq \phi \left( n'\right) \left(\prod^{s}_{\substack{j=1\\ j\neq i\\ \alpha_{k_j} > \beta_{k_j}}} \phi\left(p_{k_j}^{\alpha_{k_j} - \beta_{k_j}}\right)\right) \times \left[\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)\right] \nonumber\\ & \geq \frac{1}{m}\times \left[p_1^{\alpha_1-1}p_2^{\alpha_2-1}\cdots p_r^{\alpha_r-1} \phi(p_1p_2\cdots p_r) \times \left[\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)\right] \right] . \label{align-num-4}\end{aligned}$$ Note that equality holds in (\[align-num-3\]) if and only if $\alpha_{k_i}=\beta_{k_i}$, which follows from (\[align-num-2\]). It can be seen that equality holds in (\[align-num-4\]) if and only if $\alpha_{k_j}>\beta_{k_j}$ for all $j\in\{1,2,\ldots, s\}$. Combining these two facts, we thus have $$\begin{aligned} \theta & > \frac{1}{m}\times \left[p_1^{\alpha_1-1}p_2^{\alpha_2-1}\cdots p_r^{\alpha_r-1} \phi(p_1p_2\cdots p_r) \times \left[\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)\right] \right].\nonumber\end{aligned}$$ Finally, we get $$\begin{aligned} \label{align-num-6} \xi & > \frac{1}{m}\left[p_1^{\alpha_1-1}\cdots p_r^{\alpha_r-1} \phi(p_1p_2\cdots p_r) \times \left[\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)\right] \right] - \frac{n}{m} \phi(p_{k_i}) \nonumber \\ & = \frac{p_1^{\alpha_1-1}\cdots p_r^{\alpha_r-1}}{m} \left[\left[\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)\right] \phi(p_1p_2\ldots p_r) - \phi\left(p_{k_i}\right) p_1p_2\cdots p_r \right].\end{aligned}$$ Since $k_s >k_i$, we have $p_{k_s}>p_{k_i}$. So $\phi\left(p_{k_s}\right) > \phi\left(p_{k_i}\right)$ and hence $\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)>2\phi\left(p_{k_i}\right)$. If $2\phi(p_{1} p_2\cdots p_{r}) \geq p_1p_2\cdots p_r$, then it follows from (\[align-num-6\]) that $\deg(m) - \deg\left(\frac{m}{p_{k_i}} \right)>0$. If $\phi(p_{j+1}) \geq r \phi(p_j)$ for each $j\in \{1,2,\ldots, r-1\}$, then $\phi(p_{k_s})\geq r\phi(p_{k_i})$ and so $\phi\left(p_{k_s}\right) + \phi\left(p_{k_i}\right)\geq (r+1)\phi\left(p_{k_i}\right)$. Using Lemma \[prime.ineq\](i), it again follows from (\[align-num-6\]) that $\deg(m) - \deg\left(\frac{m}{p_{k_i}} \right)>0$. This completes the proof. Let $m$ be a proper divisor of $n$. We can write $m=p_{k_1}^{\beta_{k_1}}p_{k_2}^{\beta_{k_2}}\cdots p_{k_s}^{\beta_{k_s}}$ for some $s\in\{1,2,\ldots,r\}$, where $k_1<k_2<\cdots < k_s$ and $1\leq \beta_{k_i}\leq \alpha_{k_i}$ for $1\leq i\leq s$. If $s=1$, then $\deg(m)=\deg\left(p_{k_1}^{\beta_{k_1}}\right)\geq \deg\left(p_{k_1}^{\alpha_{k_1}}\right)$ by Proposition \[degcompare\](ii). So assume that $s\geq 2$. Then applying Proposition \[prop1\] repeatedly, we find that $\deg(m)>\deg\left(p_{k_s}^{\beta_{k_s}}\right)\geq \deg\left(p_{k_s}^{\alpha_{k_s}}\right)$. Here the last inequality holds again by Proposition \[degcompare\](ii). Thus $$\delta(\mathcal{P}(C_n))=\min\{\deg\left(p_{i}^{\alpha_{i}}\right): 1\leq i\leq r\}.$$ By Proposition \[degcompare\](i), we have $\deg\left(p_{1}^{\alpha_{1}}\right)> \deg\left(p_{r}^{\alpha_{r}}\right)$. Let $t\in\{2,3,\ldots, r\}$ be the largest integer such that $\alpha_t\geq \alpha_j$ for $2\leq j\leq r$. Then, by Proposition \[degcompare\](ii) and (iii), we have $\deg\left(p_{j}^{\alpha_{j}}\right)> \deg\left(p_{t}^{\alpha_{t}}\right)$ for $2\leq j\leq t-1$ (if $t\geq 3$). It now follows that $$\delta(\mathcal{P}(C_n)) = \min\{\deg\left(p_s^{\alpha_s}\right):t \leq s \leq r\}.$$ This completes the proof. Let $n=2\cdot 3 \cdot 5\cdot 11$. Then the minimum degree of $\mathcal{P}(C_n)$ is equal to $\deg(11)$ by Theorem \[mindeg.main\], but none of the two conditions mentioned in Theorem \[thm.mindeg\] is satisfied. Thus each of the two conditions stipulated in Theorem \[thm.mindeg\] is sufficient but not necessary. Let $n=2^2\cdot 7 \cdot 11 \cdot 13$. By Proposition \[degcompare\](iii), we have $\deg(13)<\deg(11)<\deg(7)$. Using the degree formula (\[eqn-2\]), it can be seen that $\deg(11\cdot 13)<\deg(13)$. This shows that if none of the two conditions stated in Theorem \[thm.mindeg\] is satisfied, then the minimum of degree of $\mathcal{P}(C_n)$ may not be equal to the degree of $p_i^{\alpha_i}$ for any $i\in\{2,3,\ldots,r\}$. If $p_1\geq r+1$, then $2\phi(p_1p_2\cdots p_r)\geq p_1p_2\cdots p_r$ by Lemma \[prime.ineq\](ii). If $p_{i+1} > rp_{i}$ for each $i\in\{1,2,\ldots,r-1\}$, then $\phi(p_{i+1})=p_{i+1} -1 > rp_{i} -1> rp_{i}-r= r \phi(p_i)$ for $1\leq i\leq r-1$. So, by Theorem \[thm.mindeg\], the minimum degree of $\mathcal{P}(C_n)$ is equal to $\deg\left({p_i^{\alpha_i}}\right)$ for some $i\in\{2,3,\ldots,r\}$ and the result follows for $r=2$. Assume that $r\geq 3$. For $2\leq i < r$, let $m=\dfrac{n}{p_i^{\alpha_i}p_r^{\alpha_r}}$. By Lemma \[prime.ineq\](ii), $\phi\left(\dfrac{p_1 p_2\cdots p_r}{p_i} \right)\geq \dfrac{p_1p_2 \cdots p_r}{rp_i}$ and so $$\begin{aligned} \label{align-num-7} p_i^{\alpha_i-1} \left[(p_r-1)\phi(m) - p_im\right] & = p_1^{\alpha_1-1} \cdots p_{r-1}^{\alpha_{r-1}-1} \left[\phi\left(\frac{p_1 p_2\cdots p_r}{p_i} \right) - {p_1 p_2\cdots p_{r-1}} \right] \nonumber\\ & \geq p_1^{\alpha_1-1} \cdots p_{r-1}^{\alpha_{r-1}-1} \left[\frac{p_1p_2 \cdots p_r}{rp_i} - {p_1p_2 \cdots p_{r-1}} \right] \nonumber \\ & = \frac{ p_1^{\alpha_1} \cdots p_{r-1}^{\alpha_{r-1}}}{rp_i} (p_r - rp_i).\end{aligned}$$ From the given conditions in both cases, we have $p_r> rp_{r-1}\geq rp_i$ and then the result follows from (\[eqn-5\]) and (\[align-num-7\]). Proof of Theorem \[mindeg.3prime\] {#three-primes} ================================== In this section, take $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$, where $\alpha_1,\alpha_2,\alpha_3$ are positive integers and $p_1,p_2,p_3$ are prime numbers with $p_1<p_2<p_3$. \[lem.beta2\] Let $i,j\in\{1,2,3\}$ with $i< j$. If $1\leq \beta_i\leq\alpha_i$, then $\deg\left(p_{i}^{\beta_i}p_{j}^{\beta_j}\right) > \deg\left( p_{i}^{\beta_i-1}p_{j}^{\beta_j} \right)$ for $\beta_j \geq 2$. Using the degree formula (\[eqn-2\]), we have $$\begin{aligned} \deg\left(p_{i}^{\beta_i}p_{j}^{\beta_j}\right) - \deg\left( p_{i}^{\beta_i-1}p_{j}^{\beta_j} \right) & = \frac{n}{p_i^{\beta_i}p_j^{\beta_j}}-\frac{n}{p_i^{\beta_i-1}p_j^{\beta_j}} + \sum_{\substack{d \big\vert p_i^{\beta_i}p_j^{\beta_j}}} \phi \left( \frac{n}{d}\right) - \sum_{\substack{d \big\vert p_i^{\beta_i-1}p_j^{\beta_j}}} \phi \left( \frac{n}{d}\right)\\ & \qquad + \phi\left(\frac{n}{p_i^{\beta_i-1}p_j^{\beta_j}} \right) - \phi\left(\frac{n}{p_i^{\beta_i}p_j^{\beta_j}} \right).\end{aligned}$$ Let $\{k\}=\{1,2,3\}\setminus \{i,j\}$. Then $$\label{eqn-6} \frac{n}{p_i^{\beta_i}p_j^{\beta_j}}-\frac{n}{p_i^{\beta_i-1}p_j^{\beta_j}}= - p_{i}^{\alpha_{i} - \beta_i} p_{j}^{\alpha_{j} - \beta_j} p_{k}^{\alpha_{k}} \phi(p_{i})$$ and $$\label{eqn-7} \sum_{\substack{d \big\vert p_i^{\beta_i}p_j^{\beta_j}}} \phi \left( \frac{n}{d}\right) - \sum_{\substack{d \big\vert p_i^{\beta_i-1}p_j^{\beta_j}}} \phi \left( \frac{n}{d}\right) \geq \phi \left({p_{i}^{\alpha_{i} - \beta_i}} p_{k}^{\alpha_{k}}\right) \left( p_{j}^{\alpha_j} - p_{j}^{\alpha_j-\beta_j-1} \right),$$ where equality holds if and only if $\alpha_j > \beta_j$. We also have $$\label{eqn-8} \phi\left(\frac{n}{p_i^{\beta_i-1}p_j^{\beta_j}} \right) - \phi\left(\frac{n}{p_i^{\beta_i}p_j^{\beta_j}} \right)\geq \phi \left({p_{i}^{\alpha_{i} - \beta_i}} p_{k}^{\alpha_{k}}\right) \phi\left(p_{j}^{\alpha_{j} - \beta_j}\right) ( \phi(p_{i}) -1 ),$$ where equality holds if and only if $\alpha_i = \beta_i$. From (\[eqn-6\]), (\[eqn-7\]) and (\[eqn-8\]), we get $$\begin{aligned} &\deg\left(p_{i}^{\beta_i}p_{j}^{\beta_j}\right) - \deg\left( p_{i}^{\beta_i-1}p_{j}^{\beta_j} \right)\\ & \geq \phi \left({p_{i}^{\alpha_{i} - \beta_i}} p_{k}^{\alpha_{k}}\right) \left[\left( p_{j}^{\alpha_j} - p_{j}^{\alpha_j-\beta_j-1} \right) + \phi\left(p_{j}^{\alpha_{j} - \beta_j}\right) ( \phi(p_{i}) -1 ) \right] - p_{i}^{\alpha_{i} - \beta_i} p_{j}^{\alpha_{j} - \beta_j} p_{k}^{\alpha_{k}} \phi(p_{i}) \\ & \geq p_{i}^{\alpha_{i} - \beta_i-1} p_{j}^{\alpha_{j} - \beta_j-1} p_{k}^{\alpha_{k}-1} \left[\phi\left( p_{i} p_{k} \right) \left[ \left( p_{j}^{\beta_j+1} -1 \right ) + \phi\left(p_{j}\right) ( \phi(p_{i}) -1 ) \right] - p_{i} p_{j} p_{k} \phi(p_{i}) \right] \\ & = p_{i}^{\alpha_{i} - \beta_i-1} p_{j}^{\alpha_{j} - \beta_j-1} p_{k}^{\alpha_{k}-1} \left[\phi\left( p_{i} p_{j} p_{k} \right) \left( p_{j}^{\beta_j} + \ldots + p_{j} + \phi(p_{i}) \right ) - p_{i} p_{j} p_{k} \phi(p_{i}) \right] \\ & = p_{i}^{\alpha_{i} - \beta_i-1} p_{j}^{\alpha_{j} - \beta_j-1} p_{k}^{\alpha_{k}-1} \phi(p_{i}) \left[ \phi\left(p_{j} p_{k} \right) \left( p_{j}^{\beta_j} + \ldots + p_{j} + \phi(p_{i}) \right ) - p_{i} p_{j} p_{k} \right].\end{aligned}$$ Since $\beta_j \geq 2$ and $j>i$ by the given hypotheses, we have $ p_{j}^{\beta_j}+\cdots + p_{j} + \phi(p_{i}) > 3 p_i$. So $\phi\left(p_{j} p_{k} \right) \left( p_{j}^{\beta_j} + \ldots + p_{j} + \phi(p_{i}) \right ) - p_{i} p_{j} p_{k}> 3p_i\phi\left(p_{j} p_{k} \right)- p_{i} p_{j} p_{k}\geq 0,$ where the last inequality follows using Lemma \[prime.ineq\](i). Hence the lemma follows. \[lem1\] Let $\{i,j,k\}=\{1,2,3\}$ with $i< j$. If $(p_{i}+ p_{j})\phi \left( p_{j}p_k \right) - p_ip_{j}p_k>0$, then $\deg\left(p_{i}^{\beta_i}p_{j}\right) > \deg\left(p_{j} \right)$ for $1\leq\beta_i\leq\alpha_i$. Using the degree formula (\[eqn-2\]), we get $$\begin{aligned} & \deg\left(p_{i}^{\beta_i}p_{j}\right) - \deg\left(p_{j} \right) = \frac{n}{p_i^{\beta_i}p_j}-\frac{n}{p_j} + \sum_{\substack{d \| p_i^{\beta_i}p_j}} \phi \left( \frac{n}{d}\right) - \sum_{\substack{d \| p_j}} \phi \left( \frac{n}{d}\right) + \phi\left(\frac{n}{p_j} \right) - \phi\left(\frac{n}{p_i^{\beta_i}p_j} \right).\end{aligned}$$ We have $$\label{eqn-9} \frac{n}{p_i^{\beta_i}p_j}-\frac{n}{p_j}= - p_{j}^{\alpha_{j} - 1}p_k^{\alpha_k} \left(p_{i}^{\alpha_i} -p_{i}^{\alpha_{i} - \beta_i}\right)$$ and $$\begin{aligned} \label{eqn-10} \sum_{\substack{d \| p_i^{\beta_i}p_j}} \phi \left( \frac{n}{d}\right) - \sum_{\substack{d \| p_j}} \phi \left( \frac{n}{d}\right)& =\phi \left(p_k^{\alpha_k}\right) \left(\sum_{l=1}^{\beta_i} \phi \left( {p_{i}^{\alpha_{i} - l}} \right) \right) \left(\sum_{l=0}^{1} \phi \left( {p_{j}^{\alpha_{j} - l}} \right) \right )\nonumber\\ & \geq \phi \left(p_k^{\alpha_k}\right) \left(\sum_{l=1}^{\beta_i} \phi \left( {p_{i}^{\alpha_{i} - l}} \right) \right) \left ( p_{j}^{\alpha_{j}} - p_{j}^{\alpha_{j} - 2} \right ),\end{aligned}$$ where equality holds if and only if $\alpha_j>1$. We also have $$\label{eqn-11} \phi\left(\frac{n}{p_j} \right) - \phi\left(\frac{n}{p_i^{\beta_i}p_j} \right)=\phi \left( p_{j}^{\alpha_{j} - 1}p_k^{\alpha_k} \right) \left(\phi\left(p_{i}^{\alpha_i}\right) - \phi\left(p_{i}^{\alpha_{i} - \beta_i}\right)\right).$$ From (\[eqn-9\]), (\[eqn-10\]) and (\[eqn-11\]), we get $$\begin{aligned} \deg\left(p_{i}^{\beta_i}p_{j}\right) - \deg\left(p_{j} \right) & \geq \phi \left(p_k^{\alpha_k}\right) \left(\sum_{l=1}^{\beta_i} \phi \left( {p_{i}^{\alpha_{i} - l}} \right) \right) \left ( p_{j}^{\alpha_{j}} - p_{j}^{\alpha_{j} - 2} \right )\\ & \quad + \phi \left( p_{j}^{\alpha_{j} - 1}p_k^{\alpha_k} \right) \left(\phi\left(p_{i}^{\alpha_i}\right) - \phi\left(p_{i}^{\alpha_{i} - \beta_i}\right)\right) - p_{j}^{\alpha_{j} - 1}p_k^{\alpha_k} \left(p_{i}^{\alpha_i} -p_{i}^{\alpha_{i} - \beta_i}\right) \\ & \geq p_{j}^{\alpha_{j} - 2} p_k^{\alpha_k-1} \Bigg[ \left ( p_{j}^{2} - 1 \right )\phi \left(p_k\right) \left(\sum_{l=1}^{\beta_i} \phi \left( {p_{i}^{\alpha_{i} - l}} \right) \right)\\ & \qquad\qquad + \phi \left( p_{j}p_k \right) \left(\phi\left(p_{i}^{\alpha_i}\right) - \phi\left(p_{i}^{\alpha_{i} - \beta_i}\right)\right) - p_{j}p_k \left(p_{i}^{\alpha_i} -p_{i}^{\alpha_{i} - \beta_i}\right) \Bigg]\\ & = p_{j}^{\alpha_{j} - 2} p_k^{\alpha_k-1} \Bigg[ \phi \left( p_{j}p_k \right) \Bigg[\left ( p_{j}+ 1 \right ) \left(\sum_{l=1}^{\beta_i} \phi \left( {p_{i}^{\alpha_{i} - l}} \right) \right) \\ & \qquad\qquad\qquad\qquad + \phi\left(p_{i}^{\alpha_i}\right) - \phi\left(p_{i}^{\alpha_{i} - \beta_i}\right) \Bigg] - p_{j}p_k \left(p_{i}^{\alpha_i} -p_{i}^{\alpha_{i} - \beta_i}\right)\Bigg]\\ & \geq p_{j}^{\alpha_{j} - 2} p_k^{\alpha_k-1} \bigg[ \phi \left( p_{j}p_k \right) \left[ p_{j} \left ( p_{i}^{\alpha_i-1} -p_{i}^{\alpha_{i} - \beta_i-1} \right ) + p_{i}^{\alpha_i} -p_{i}^{\alpha_{i} - \beta_i} \right]\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad - p_{j}p_k \left(p_{i}^{\alpha_i} -p_{i}^{\alpha_{i} - \beta_i}\right)\bigg]\\ & = p_{j}^{\alpha_{j} - 2} p_k^{\alpha_k-1} \left ( p_{i}^{\alpha_i-1} -p_{i}^{\alpha_{i} - \beta_i-1} \right ) \left[(p_{i} + p_{j}) \phi \left( p_{j}p_k \right) - p_ip_{j}p_k\right].\end{aligned}$$ Since $(p_{i}+ p_{j})\phi \left( p_{j}p_k \right) - p_ip_{j}p_k>0$, it follows from the above that $\deg\left(p_{i}^{\beta_i}p_{j}\right) > \deg\left(p_{j} \right)$. If $p_1 \geq 4$, then $2\phi(p_1 p_2 p_3) \geq p_1 p_2 p_3$ by Lemma \[prime.ineq\](ii). So $\delta(\mathcal{P}(C_n)) = \min\{\deg\left({p_2^{\alpha_2}}\right),\deg\left({p_3^{\alpha_3}}\right)\}$ by Theorem \[thm.mindeg\]. Now assume that $p_1 = 2$ or $3$. In view of Proposition \[degcompare\](i), (ii) and (iv), the minimum degree of $\mathcal{P}(C_n)$ can be attained at the vertex $p_{2}^{\alpha_2}$ or $p_{3}^{\alpha_3}$, or at a vertex of the form $p_{i}^{\beta_i}p_{j}^{\beta_j}$ for some $i,j\in\{1,2,3\}$ with $i<j$, where $1\leq \beta_i \leq \alpha_i$ and $1\leq \beta_j \leq \alpha_j$. Consider the vertices of the form $p_{i}^{\beta_i}p_{j}^{\beta_j}$ with $i<j$. We show that $\deg\left(p_{i}^{\beta_i}p_{j}^{\beta_j}\right) > \deg\left(p_{j}^{\beta_j} \right)$. Then Proposition \[degcompare\](ii) implies that $\deg\left(p_{j}^{\beta_j}\right)\geq \deg\left( p_j^{\alpha_j}\right)$ and this would complete the proof. If $\beta_j\geq 2$, then applying Lemma \[lem.beta2\] repeatedly we find that $\deg\left(p_{i}^{\beta_i}p_{j}^{\beta_j}\right)>\deg\left(p_j^{\beta_j}\right)$. Suppose that $\beta_j=1$. Let $\{k\}=\{1,2,3\}\setminus\{i,j\}$. We show that $$\label{eqn-12} (p_{i}+ p_{j})\phi \left( p_{j}p_k \right) - p_ip_{j}p_k>0.$$ Then Lemma \[lem1\] implies that $\deg\left(p_{i}^{\beta_i}p_{j}\right) > \deg\left(p_{j} \right)$. Clearly, (\[eqn-12\]) holds using Lemma \[prime.ineq\](i) if $p_{j} > 2p_i$. Since $p_j\neq 2p_i$, assume that $p_{j} < 2 p_i$. We have the following two cases. 1. $i=1$: Since $p_1\in\{2,3\}$ and $p_{j} < 2 p_1$, we have $j=2$ and $(p_1,p_2)=(2,3)$ or $(3,5)$. If $(p_1, p_2) = (2,3)$, then $\left(p_{1} + p_{2}\right)\phi\left(p_{2}p_3\right) - p_{1} p_{2} p_3 = 10 \phi \left(p_3\right)- 6 p_3 = 4 p_3 -10 > 0$ as $p_3\geq 5$. If $(p_1, p_2)=(3,5)$, then $p_3\geq 7$ and $\left(p_{1} + p_{2}\right)\phi\left(p_{2}p_3\right)- p_{1} p_{2} p_3 = 32 \phi\left(p_3\right)- 15 p_3 = 17p_3 -32 > 0$. 2. $(i,j)=(2,3)$: Here $k=1$ and $p_3\geq p_2 + 2$. If $p_k=p_1=2$, then $\left ( p_{2} + p_{3} \right)\phi \left(p_1p_{3}\right) - p_{1} p_{2} p_3 = p_3^2 - \left ( p_{2} + p_{3} \right ) - p_{2} p_3 =p_3( p_3- p_{2} -1)-p_2 > 0$. If $p_k=p_1=3$, then $\left(p_{2}+ p_{3}\right)\phi \left(p_1p_{3}\right) - p_{1} p_{2} p_3 = 2p_3^2 - 2\left ( p_{2} + p_{3} \right )- p_{2} p_3 =p_3( 2p_3- p_{2} -2)-2p_2 > 0.$ This completes the proof. \[coro\] If $p_3 \geq 2p_2+1$, then $\delta(\mathcal{P}(C_n)) = \deg\left(p_3^{\alpha_3}\right)$. By (\[eqn-5\]), we have $\deg\left({p_2^{\alpha_2}}\right) - \deg\left({p_3^{\alpha_3}}\right) > p_2^{\alpha_2-1} \left[(p_3-1)\phi \left(p_1^{\alpha_1}\right) - p_2p_1^{\alpha_1}\right]$. Since $p_3\geq 2p_2+1$, it follows that $\deg\left({p_2^{\alpha_2}}\right) - \deg\left({p_3^{\alpha_3}}\right)>p_1^{\alpha_1-1}p_2^{\alpha_2} \left (2\phi(p_1) - p_1\right) \geq 0$ and so $\delta(\mathcal{P}(C_n)) = \deg\left(p_3^{\alpha_3}\right)$ by Theorem \[mindeg.3prime\]. Take $n= 2 \cdot 3^3 \cdot 5$. Then $\delta(\mathcal{P}(C_n))=\deg(3^3)=113 < 125 = \deg(5)$. It follows that if $p_3 < 2p_2+1$, then $\delta(\mathcal{P}(C_n)) = \deg\left(p_2^{\alpha_2}\right)< \deg\left(p_3^{\alpha_3}\right)$ may occur. [10]{} J. Abawajy, A. Kelarev, and M. Chowdhury. Power graphs: a survey. , 1(2):125–147, 2013. I. Chakrabarty, S. Ghosh, and M. K. Sen. Undirected power graphs of semigroups. , 78(3):410–426, 2009. S. Chattopadhyay and P. Panigrahi. Connectivity and planarity of power graphs of finite cyclic, dihedral and dicyclic groups. , 18(1):42–49, 2014. S. Chattopadhyay, K. L. Patra, and B. K. Sahoo. Vertex connectivity of the power graph of a finite cyclic group. , to appear, DOI:10.1016/j.dam.2018.06.001. S. Chattopadhyay, K. L. Patra, and B. K. Sahoo. Vertex connectivity of the power graph of a finite cyclic group [II]{}. , to appear, DOI:10.1142/S0219498820500401. S. Chattopadhyay, K. L. Patra, and B. K. Sahoo. Minimal cut-sets in the power graph of certain finite non-cyclic groups, communicated. 2018. arXiv:1802.07646. B. Curtin and G. R. Pourgholi. Edge-maximality of power graphs of finite cyclic groups. , 40(2):313–330, 2014. B. Curtin and G. R. Pourgholi. An [E]{}uler totient sum inequality. , 163:101–113, 2016. A. Doostabadi, A. Erfanian, and A. Jafarzadeh. Some results on the power graphs of finite groups. , 41:73–78, 2015. M. Feng, X. Ma, and K. Wang. The full automorphism group of the power (di)graph of a finite group. , 52([P]{}art A):197–206, 2016. A. V. Kelarev and S. J. Quinn. A combinatorial property and power graphs of groups. In [Contributions to general algebra, 12 ([V]{}ienna, 1999)]{}, pages 229–235. Heyn, Klagenfurt, 2000. A. V. Kelarev and S. J. Quinn. Directed graphs and combinatorial properties of semigroups. , 251(1):16–26, 2002. A. V. Kelarev, S. J. Quinn, and R. Smolíková. Power graphs and semigroups of matrices. , 63(2):341–344, 2001. X. Ma, M. Feng, and K. Wang. The structure and metric dimension of the power graph of a finite group. , 43:82–97, 2015. M. Mirzargar, A. R. Ashrafi, and M. J. Nadjafi-Arani. On the power graph of a finite group. , 26(6):1201–1208, 2012. A. R. Moghaddamfar, S. Rahbariyan, and W. J. Shi. Certain properties of the power graph associated with a finite group. , 13(7):1450040, 18, 2014. R. P. Panda and K. V. Krishna. On connectedness of power graphs of finite groups. , 17(10):1850184, 20, 2018. R. P. Panda and K. V. Krishna. On minimum degree, edge-connectivity and connectivity of power graphs of finite groups. , 46(7):3182–3197, 2018. D. B. West. . Prentice Hall, Inc., Upper Saddle River, NJ, 1996. .3cm :\ [Ramesh Prasad Panda, Kamal Lochan Patra, Binod Kumar Sahoo]{} 1. School of Mathematical Sciences, National Institute of Science Education and Research (NISER), Bhubaneswar, P.O.- Jatni, District- Khurda, Odisha - 752050, India. 2. Homi Bhabha National Institute (HBNI), Training School Complex, Anushakti Nagar, Mumbai - 400094, India. [Emails]{}: rppanda@niser.ac.in, klpatra@niser.ac.in, bksahoo@niser.ac.in [^1]: A positive integer $a$ is called a [proper divisor]{} of $n$ if $a$ divides $n$ and $a\notin\{1,n\}$.
--- abstract: 'A simple cellular automata model for a two-group war over the same “territory” is presented. It is shown that a qualitative advantage is not enough for a minority to win. A spatial organization as well a definite degree of aggressiveness are instrumental to overcome a less fitted majority. The model applies to a large spectrum of competing groups: smoker-non smoker war, epidemic spreading, opinion formation, competition for industrial standards and species evolution. In the last case, it provides a new explanation for punctuated equilibria.' author: - \| [Serge Galam[^1]]{}\ Laboratoire des Milieux Désordonnés et Hétérogènes[^2],\ Tour 13 - Case 86, 4 place Jussieu,\ 75252 Paris Cedex 05, France\ \ [Bastien Chopard[^3],]{} [Alexander Masselot[^4]]{}\ Département d’Informatique, University of Geneva,\ 24 rue Général-Dufour, 1211 Genève 4, Switzerland,\ \ [Michel Droz[^5]]{}\ Département de Physique Théorique, University of Geneva,\ 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland, title: 'Competing Species Dynamics: Qualitative Advantage versus Geography' --- .5cm[PACS:]{} 01.75+m, 05.50+q, 89.90+n .5cm Physics has dealt with quite a success in describing and understanding collective behavior in matter. Very recently many physicists have used basic concepts and techniques from the physics of collective disorder to study a large spectrum of problems outside the usual field of physics such as social behavior [@kohring:96; @glance:93; @bonabeau:95], group decision making [@galam:97], financial systems [@levy:95] and multinational organizations [@galam:96]. See [@stauffer] for a review of these applications. A few years ago, Galam has developed a hierarchical voting model based on the democratic use of majority rule [@galam:90]. In the simplest case of two competing parties $A$ and $B$ with respective support of $a_0$ and $b_0=1-a_0$, it was shown that, for the $B$, winning the elections at the top of the hierarchy (i.e. after several tournaments) does not depend only on $b_0$ but also on the existence of some local biases. In particular, in the case of voting cells of four persons, a bias is introduced (usually in favor of the leading party, e.g. $B$) to solve the $2A$-$2B$ situations. Then, the critical threshold of support for the ruling party to win can be as low as $b_c=0.23$. The model showed how a majority up to $0.77 $ can self-eliminate while climbing up the hierarchy, using locally the democratic majority voting rule. This self-elimination occurs within only few hierarchical levels. Following this previous study, we address here the universal and generic problem of the competing fight between two different groups over a fixed area. We present a “voter model” which describes the dynamical behavior of a population with bimodal conflicting interests and study the conditions of extinction of one of the initial groups. This model can be thought of as describing the smoker - non smoker fight: in a small group of persons, a majority of smokers will usually convince the few others to smoke and vice versa. The point is really when an equal number of smokers and non-smokers meet. In that case, it may be assumed that a social trend will decide between the two attitudes. In the US, smoking is viewed as a disadvantage whereas, in France, it is rather well accepted. In other words, there is a bias that will select the winner party in an even situation. In our example, whether one studies the French or US case, the bias will be in favor of the smokers or the non-smokers, respectively. The same mechanism can be associated with the problem of competing standards (for instance PC versus Macintosh for computer systems or VHS versus Beta MAG for video systems). The choice of one or the other standard is often driven by the opinion of the majority of people one meets. But, when the two competing systems are equally represented, the intrinsic quality of the product will be decisive. Price and technological advance then play the role of a bias. Here we consider the case of four-person confrontations in a spatially extended system in which the actors (species $A$ or $B$) move randomly. The process of spatial contamination of opinion plays a crucial role in this dynamics. In the original Galam model [@galam:90], the density threshold for an invading emergence of $B$ is $b_c=0.23$ if the $B$ group has a qualitative bias over $A$. With a spatial distribution of the species, even if $b_0<b_c$, $B$ can still win over $A$ provided that it strives for confrontation. Therefore a qualitative advantage is found not to be enough to win. A geographic as well a definite degree of aggressiveness are instrumental to overcome the less fitted majority. The model we use to describe the two populations $A$ and $B$ influencing each other or competing for some unique resources, is based on the reaction-diffusion automata proposed by Chopard and Droz [@BC-EPL]. However, here, we consider only one type of particle with two possible internal states ($\pm1$), coding for the $A$ or $B$ species, respectively. The individuals move on a two-dimensional square lattice. At each site, there are always four individuals (any combination of $A$’s and $B$’s is possible). These four individuals all travel in a different lattice direction (north, east, south and west). The interaction takes place in the form of “fights” between the four individuals meeting on the same site. At each fight, the group nature ($A$ or $B$) is updated according to the majority rule, when possible, otherwise with a bias in favor of the best fitted group: - The local majority species (if any) wins: $$nA+mB\rightarrow \left\{ \begin{array}{ll} (n+m)A & \mbox{if $n>m$} \\ (n+m)B & \mbox{if $n<m$} \\ \end{array} \right.$$ where $n+m=4$. - When there is an equal number of $A$ and $B$ on a site, $B$ wins the confrontation with probability $1/2+\beta/2$. The quantity $\beta\in[0,1]$ is the bias accounting for some advantage (or extra fitness) of species $B$. The above rule is applied with probability $k$. Thus, with probability $1-k$ the group composition does not change because no fight occurs. Between fights both population agents perform a random walk on the lattice. This is achieved by shuffling randomly the directions of motion of the fours individuals present at each site and letting them move to the corresponding neighboring sites [@BC-EPL]. Initially, populations $A$ and $B$ are randomly distributed over the lattice, with respective concentrations $a_0$ and $b_0=1-a_0$. It is clear that the model richness comes from the even confrontations. If only odd fights would happen, the initial majority population would always win after some short time. The key parameters of this model are (i) $k$, the aggressiveness (probability of confrontation), (ii) $\beta$, the $B$’s bias of winning a tie and (iii) $b_0$, the initial density of $B$. The strategy according to which a minority of $B$’s (with yet a technical, genetic, persuasive advantage) can win against a large population of $A$’s is not obvious. Should they fight very often, try to spread or accept a peace agreement? We study the parameter space by running cellular automata implementing the above system. In the limit of low aggressiveness ($k\to 0$), the particles move a long time before fighting. Due to the diffusive motion, correlations between successive fights are destroyed and $B$ wins provided that $b_0>0.23$ and $\beta=1.$ This is the mean-field level of our dynamical model which corresponds to the theoretical calculations made by Galam in his election model [@galam:90]. More generally, and for $\beta={\rm const}$, we observe that $B$ can win even when $b_0<0.23$, provided it acts aggressively, i.e. by having a large enough $k$. Thus, there is a critical density $b_{death}(k)<0.23$ such that, when $b_0>b_{death}(k)$, all $A$ are eliminated in the final outcome. Below $b_{death}$, $B$ looses unless some specific spatial configurations of $B$’s are present. This is a general and important feature of our model: the growth of species $B$ at the expense of $A$ is obtained by a spatial organization. Small clusters that may accidentally form act as nucleus from which the $B$’s can develop. In other words, above the mean-field threshold $b_c=0.23$ there is no need to organize in order to win but, below this value only condensed regions will be able to grow. When $k$ is too small, such an organization is not possible (it is destroyed by diffusion) and the strength advantage of $B$ does not lead to success. Figure \[pbvsk0\] summarizes, as a function of $b_0$ and $k$, the regions where either $A$ or $B$ succeeds. It turns out that the separation curve satisfies the equation $(k+1)^7(b_0-0.077)=0.153$. It is also interesting to study the time needed to annihilate completely the looser. Here, time is measured as the number of fights per site (i.e. $kt$ where $t$ is the iteration time of the automaton). We observed that, in this case, the dynamics is quite fast and a few units of time are sufficient to yield a collective change of opinion. The previous results assume a contant bias. However, with the assumption that an individual surrounded by several of its congeners becomes more confident and thus less efficient in its fight, one may vary the bias $\beta$ as a function of the local density of $B$. For example, within a neighborhood of size $\ell^2$, the bias can decrease from 1 to 0 as follows : $\beta=1- b/(2\ell^2)$ if $0\le b\le 2\ell^2$ (local minority of $B$’s) and $\beta=0$ if $b>2\ell^2$ (local majority of $B$’s), where $b$ designates the number of $B$’s in the neighborhood. This rule produces an interesting and non-intuitive new behavior. Depending on the value of $\ell$, there is a region near $k=1$ such that the $A$ species can win by preventing the $B$’s from spreading in the environment. This is achieved by a very aggressive attitude of the $A$’s. Note that this effect is already present in the previous case ($\ell=1$ and $\beta={\rm const}$), but only on the line $k=1$ and for $b_0<0.2$. Figure \[pbvsk3\] summarizes the regions where either $A$ or $B$ succeeds when $\ell=7$. In addition to the separation line shown in light gray, the time needed to decimate the other opinion is indicated by the gray levels. We observe that this time may become large in the vicinity of the critical line. Depending on the time scale associated with the process, such a slow evolution may be interpreted as a coexistence of the two species (if a campaign lasts only a few days or a few weeks, the conflict will not be resolved within this period of time). We have shown that the correlations that may exist between successive fights may strongly affect the global behavior of the system and that an organization is the key feature to obtain a definite advantage over the other population. This observation is important. For instance, during a campaign against smoking or an attempt to impose a new system, it is much more efficient (and cheaper) to target the effort on small nuclei of persons rather than sending the information in an uncorrelated manner. Also, according to figure \[pbvsk3\], an hypothetical minority of smokers in France must harass non-smokers during social meetings (coffee break, lunch,...) rather often but not systematically, in order to reinforce their position. On the contrary, for an hypothetical majority of smokers in the US, either a smooth or a stiff harassment against the non-smokers is required to survive. Aggressiveness is the key to preserve the spatial organization. Refusing a fight is an effective way for the $A$ species to use its numerical superiority by allowing the $B$ individuals to spread. With this respect, a minority should not accept a peace agreement (which would results in a lower $k$) with the leading majority unless the strength equilibrium is modified (i.e. $B$ is better represented). Motion is also a crucial ingredient in the spreading process. There is a subtle tradeoff between moving and fighting. When little motion is allowed between fights ($k\to1$), the advantage is in favor of $A$ again. In an epidemic system, our model shows that two solutions are possible to avoid infestation: either one let the virus die of isolation (dilute state due to a small $k$) or one decimates it before it spreads (large $k$). Finally a simple variant of the above model provides a possible scenario to explain punctuated equilibria [@bak-sneppen:93] in the evolution of living organisms. It is well known that the transition between two forms of life may be quite abrupt. There is no trace of the intermediate evolutionary steps. To give some insights into this problem we modify our voter model by including a creation rate for the $B$ individuals ($A\to B$, with probability $p\ll 1$). In this context, the $B$ species is fitter than the $A$ species (the bias $\beta=1$) but the numerical advantage of $A$ is too strong for $B$ to survive. However, if the simulation is run for a long enough time, nucleation in this metastable state will happen, which will produce locally a very favorable spatial arrangement of $B$’s. These $B$’s will then develop and, very rapidly, eliminate all $A$’s. In other words, a very numerous species may live for a considerable amount of time without endangering competitors and suddenly, be decimated by a latent, fitter species. This scenario needs a strong statistical fluctuation but no additional external, global event. In conclusion, although the model we propose is very simple, it abstracts the complicated behavior of real life agents by capturing some essential ingredients. For this reason, the results we have presented may shed light on the generic mechanisms observed in a social system of opinion making. Acknowledgment {#acknowledgment .unnumbered} ============== We thank D. Stauffer for a careful reading of this manuscript. Part of this work was realized during the “Complexity and Chaos” workshop at ISI, under grant OFES 95.0046. [10]{} G.A. Kohring. Ising models of social impact: the role of cumulative advantage. , 6:301, 1996. N.S. Glance and B.A. Huberman. The outbreak of cooperation. , 17(4):281, 1993. E. Bonabeau, G. Theraulaz, and J.L. Deneubourg. Phase diagram of a model of self-organizing hierachies. , 217:373, 1995. S. Galam. Rational group decision making: A random field ising model at $t=0$. , 238:66–80, 1997. M. Levy, H. Levy, and S. Solomon. Microscopic simulation of the stock market. , 5:1087, 1995. S. Galam. Fragmentation versus stability in bimodal coalitions. , 230:174–188, 1996. S.M. de Oliveira, P.M.C. de Oliveira, and D. Stauffer. . Springer, in press. S. Galam. Social paradoxes of majority rule voting and renormalization group. , 61:943–951, 1990. B. Chopard and M. Droz. Microscopic study of the properties of the reaction front in an [$A+B\rightarrow C$]{} reaction-diffusion process. , 15:459–464, 1991. P. Bak and K. Sneppen. Punctuated equilibrium and criticality in a simple model of evolution. , 71:4083–4086, 1993. [^1]: galam@ccr.jussieu.fr [^2]: Laboratoire associé au CNRS (UMR n$^{\circ}$ 800) et à l’Université P. et M. Curie - Paris 6 [^3]: Bastien.Chopard@cui.unige.ch [^4]: Alexandre.Masselot@cui.unige.ch [^5]: Michel.Droz@physics.unige.ch
--- abstract: 'We present new radio continuum observations of NGC 253 from the Murchison Widefield Array at frequencies between 76 and 227 MHz. We model the broadband radio spectral energy distribution for the total flux density of NGC 253 between 76 MHz and 11 GHz. The spectrum is best described as a sum of central starburst and extended emission. The central component, corresponding to the inner 500 pc of the starburst region of the galaxy, is best modelled as an internally free-free absorbed synchrotron plasma, with a turnover frequency around 230 MHz. The extended emission component of the NGC 253 spectrum is best described as a synchrotron emission flattening at low radio frequencies. We find that 34% of the extended emission (outside the central starburst region) at 1 GHz becomes [ partially]{} absorbed at low radio frequencies. Most of this flattening occurs in the western region of the SE halo, and may be indicative of synchrotron self-absorption of shock re-accelerated electrons or an intrinsic low-energy cut off of the electron distribution. Furthermore, we detect the large-scale synchrotron radio halo of NGC 253 in our radio images. At 154–231 MHz the halo displays the well known X-shaped/horn-like structure, and extends out to $\sim 8$ kpc in $z$-direction (from major axis).' author: - 'A. D. Kapińska$^{1,2}$, L. Staveley-Smith$^{1,2}$, R. Crocker$^{3}$, G. R. Meurer$^1$, S. Bhandari$^{4,2}$, N. Hurley-Walker$^{5}$, A.R. Offringa$^{6}$, D.J. Hanish, N. Seymour$^5$, R. D. Ekers$^7$, M. E. Bell$^{7}$, J.R. Callingham$^{7,8,2}$, K. S. Dwarakanath$^{9}$, B.-Q. For$^{1}$, B. M. Gaensler$^{10,8,2}$, P. J. Hancock$^{5,2}$, L. Hindson$^{11,12}$, M. Johnston-Hollitt$^{12}$, E. Lenc$^{8,2}$, B. McKinley$^{13}$, J. Morgan$^{5}$, P. Procopio$^{13,2}$, R. B. Wayth$^{5,2}$, C. Wu$^1$, Q. Zheng$^{12}$, N. Barry$^{14}$, A. P. Beardsley$^{14}$, J. D. Bowman$^{15}$, F. Briggs$^3$, P. Carroll$^{15}$, J. S. Dillon$^{16}$, A. Ewall-Wice$^{16}$, L. Feng$^{16}$, L. J. Greenhill$^{17}$, B. J. Hazelton$^{14}$, J. N. Hewitt$^{16}$, D. J. Jacobs$^{15}$, H.-S. Kim$^{13,2}$, P. Kittiwisit$^{15}$, J. Line$^{13,2}$, A. Loeb$^{17}$, D. A. Mitchell$^{3,2}$, M. F. Morales$^{14}$, A. R. Neben$^{16}$, S. Paul$^{9}$, B. Pindor$^{13,2}$, J. C. Pober$^{18}$, J. Riding$^{13,2}$, S. K. Sethi$^{9}$, N. Udaya Shankar$^{9}$, R. Subrahmanyan$^{9,2}$, I. S. Sullivan$^{14}$, M. Tegmark$^{16}$, N. Thyagarajan$^{15}$, S. J. Tingay$^{5,19,2}$, C. M. Trott$^{5}$, R. L. Webster$^{13,2}$, S. B. Wyithe$^{13,2}$, R. J. Cappallo$^{20}$, A. A. Deshpande$^{9}$, D. L. Kaplan$^{21}$, C. J. Lonsdale$^{20}$, S. R. McWhirter$^{20}$, E. Morgan$^{16}$, D. Oberoi$^{22}$, S. M. Ord$^{5,7,2}$, T. Prabu$^{9}$, K. S. Srivani$^{9}$, A. Williams$^{5}$, C. L. Williams$^{16}$' title: Spectral energy distribution and radio halo of NGC 253 at low radio frequencies --- INTRODUCTION {#sec:intro} ============ Observing at low radio frequencies ($\lesssim0.5$ GHz) is of a particular value; low energy and old plasma can be revealed, tracing and constraining physical conditions in galaxies. In star forming galaxies the low surface brightness plasma forms e.g. extended halos associated with winds and large scale magnetic fields, or diffuse emission from galactic disks. Furthermore, measurements at low radio frequencies can help to distinguish for instance between thermal and non-thermal plasma, and their absorbing mechanisms, responsible for the level of observed radio emission. It is expected that the Square Kilometre Array (SKA) will unravel a large star-forming galaxy population [e.g. @2015aska.confE..70B; @2015aska.confE..68J], but before we can embark on a large scale study of star-forming and starburst galaxies and their evolution with continuum radio surveys, we need to understand the origin of the complex radio spectral energy distributions and morphologies of these galaxies. Nearby objects are ideal laboratories for this task. NGC 253 is the dominant galaxy in the nearby Sculptor Group, at a distance of 3.94 Mpc from the Local Group [@2003AA...404...93K] and velocity $cz=240$ km s$^{-1}$. It is an almost edge-on SBc type galaxy [@1991rc3..book.....D] observed at an inclination of $78.5^{\circ}$ [@1980ApJ...239...54P] and is considered a prototype of nuclear starburst galaxies [@1980ApJ...238...24R]. Its estimated stellar mass is $\sim4\times10^{10}$ M$_{\odot}$, with a prominent stellar halo of $2.5\times10^{9}$ M$_{\odot}$ extending up to 30 kpc above the disk [@2011ApJ...736...24B]. As one of the closest and most prominent galaxies, NGC 253 has been extensively studied in all wavelengths, including broadband radio continuum, polarization and [ H[i]{}]{} observations . Radio emission from starburst galaxies originates from two principal components: the non-thermal synchrotron emission from relativistic electrons spiralling in the interstellar magnetic field, and the thermal emission from electrons colliding with ions in the ionized interstellar medium (ISM) around hot stars. The sources of the non-thermal emission are predominantly cosmic rays accelerated by supernova remnants (SNR) that in NGC 253 ultimately create a prominent synchrotron radio halo [@1992ApJ...399L..59C]. [ Studies of the NGC 253 magnetic field suggest the disk wind model and large-scale dynamo action to be shaping the vertical structure of the field, which in turn enhances the cosmic ray transport through a collimation of strong, starburst driven superwind]{} . The starburst region of NGC 253 is violently active; the supernova rate of the inner 300 pc of the galaxy is estimated to be between $0.14$ and $2.4$ yr$^{-1}$, and the star formation rate is $\sim5$ M$_{\odot}$ yr$^{-1}$ [@2006AJ....132.1333L; @2014AJ....147....5R; @2015MNRAS.450L..80B; @2015MNRAS.450.3935L]. It has been suggested that up to half of the radio sources in the central starburst region are dominated by thermal emission: i.e. H[ii]{} regions characterized by a flat radio spectral index[^1] $\alpha \simeq 0.1$ and including at least one large supercluster of stars [@1997ApJ...488..621U; @1999ApJ...518..183K]. Outside the central starburst region the radio emission at GHz frequencies is dominated by steep spectrum diffuse emission and SNRs, but several strong thermal sources are detected [@2000AJ....120..278U]. Based on integrated radio continuum spectra, estimated 10% of the NGC 253 flux density at 1 GHz to be of thermal origin, increasing to 35% at 10 GHz. At low radio frequencies both of these principal components become pronounced. Synchrotron emission has steep spectrum, becoming dominant at sub-GHz frequencies due to the population of old, low energy electrons. However, such emission may be also subject to self-absorption in the case of compact objects. Thermal emission also becomes increasingly more absorbed with decreasing frequency. The free-free absorption in the central starburst of NGC 253 has previously been measured . Here, we present extensive low radio frequency ($<230$ MHz) imaging of NGC 253 obtained with the Murchison Widefield Array [MWA; @2013PASA...30...31B; @2013PASA...30....7T]. Our images are some of the deepest yet at these frequencies, and at low angular resolution they are especially sensitive to large-scale diffuse structure, allowing us to investigate the extent and frequency dependence of the radio halo. The paper is structured as follows. Our radio data and methods, including assumed models of radio spectra and model fitting, are described in §\[sec:data\] and §\[sec:methods\] respectively. Results are presented in §\[sec:results\]. The synchrotron radio halo of NGC 253 is discussed in §\[sec:halo\]. We discuss low frequency radio emission from NGC 253, its radio spectral energy distribution and radio spectral maps in §\[sec:spectral-props\]. Conclusions are given in §\[sec:conclude\]. Observations and data reduction {#sec:data} =============================== We use radio continuum data from the Galactic and Extragalactic All-Sky MWA Survey [GLEAM; @2015PASA...32...25W] and the MWA Epoch of Reionization experiment [MWA/EoR; @2013PASA...30...31B; @2016ApJ...819....8P]. The GLEAM survey provides unprecedented spectral coverage between 72 and 231 MHz, while the MWA/EoR image at 169 MHz is [ almost twice as deep as the most sensitive GLEAM image at 200 MHz (rms noise 4.1 mJy beam$^{-1}$ and 7.3 mJy beam$^{-1}$ respectively)]{}. In addition, the data have been observed and processed independently, providing a verification of our flux density calibration. The GaLactic and Extragalactic All-Sky MWA Survey (GLEAM) --------------------------------------------------------- The GLEAM survey observed the entire radio sky south of declination $+30^{\circ}$ at an angular resolution of approximately 1.7 arcmin (227 MHz) to 5 arcmin (76 MHz). At 154 MHz the GLEAM survey is sensitive to structures up to $10$ deg in angular scale, and has an instantaneous field of view of $25\times25$ deg$^{2}$. The observations were made in a meridian drift scan mode covering frequencies between 72 and 231 MHz with bandwidths of 7.68 MHz grouped in five 30.72 MHz-wide bands. These bands, centred on 87.7, 118.4, 154.2, 185.0 and 215.7 MHz (hereafter 88, 118, 154, 185 and 216 MHz), were observed sequentially as 112 sec snapshots; each frequency was observed every 10 min. During a night typically 8–10 h in hour angle were observed. Frequencies between 134 and 137 MHz were avoided due to satellite interference. For more details on the survey parameters and strategy see [@2015PASA...32...25W]. Here we use GLEAM data from the first year of observing [Data Release 1 from 2013 August – 2014 June; @2016MNRAS.GLEAM.subm]. The sky area covering NGC 253 was observed on 2013 August 10 and 2013 November 25. The full data reduction process is described in detail in [@2016MNRAS.GLEAM.subm]; here we summarize only the main calibration and imaging steps. The correlated data were first pre-processed with the [cotter]{} pipeline which performs flagging of data affected by radio frequency interference (RFI) and averaging of the data to 1s time and 40 kHz frequency resolution [@2010MNRAS.405..155O; @2015PASA...32....8O]. Standard calibration (phase and amplitude bandpass calibration) was done with [CASA]{} [Common Astronomy Software Applications package; @2007ASPC..376..127M]. Imaging and self-calibration were then performed using [WSClean]{} imager [@2014MNRAS.444..606O] that corrects for wide field $w$-term effects. Images of a 7.68 MHz bandwidth at 20 frequencies continuously distributed between 72 and 231 MHz (avoiding 134–137 MHz) and using a robust weighting $r=-1.0$ [@1995AAS...18711202B] were then created. Deconvolution has been performed at this stage, and details are provided in [@2016MNRAS.GLEAM.subm]. The primary beam correction of our GLEAM observations was done with the [@2015RaSc...50...52S] model down to the 10% level of the beam response. An additional calibration stage was necessary to correct for residual declination dependence of the flux density scale in the final mosaics arising from the limited accuracy of the adopted primary beam model. This was done by comparing flux density measurements of all unresolved sources extracted from GLEAM images above $8\sigma$ rms noise level to their radio spectra as predicted by three catalogues: VLA Low-Frequency Sky Survey redux [VLSSr; @2014MNRAS.440..327L], MRC and NRAO VLA Sky Survey [NVSS; @1998AJ....115.1693C]. The absolute flux density scale of the GLEAM images is accurate to 8%, which is included in the quoted uncertainties of the measurements [for details see @2016MNRAS.GLEAM.subm]. Images for each of five central frequencies centered on frequencies of 88, 118, 154, 185 and 216 MHz and of bandwidth 30.72 MHz were made. The two highest frequency images are further combined to create a ‘deep’ image at 200 MHz with a 61.4 MHz bandwidth. We also use the 7.68-MHz images for construction of the high resolution radio spectrum of NGC 253. The final synthesised beam sizes, rms and background noise levels in the deep 200 MHz image are $2.22\times2.12$ arcmin$^2$, ${\rm PA}=-78^\circ$, 11 mJy beam$^{-1}$ and 7.3 mJy beam$^{-1}$ respectively, and [ their range between the lowest and highest GLEAM frequencies is listed in Table \[tab:noisebm\]]{}. ------- ------------------ ----------- ------------------- ------------------- -- -- $\nu$ rms noise background (MHz) bmaj$\times$bmin PA (mJy beam$^{-1}$) noise (arcmin$^2$) (deg) (mJy beam$^{-1}$) 76 $5.03\times4.72$ $-18.8$ $107$ $-44$ 227 $1.73\times1.67$ $-26.0$ $12.8$ $-3.3$ ------- ------------------ ----------- ------------------- ------------------- -- -- : Range of angular resolution and noise values of the GLEAM data. []{data-label="tab:noisebm"} ![The 330 MHz image of NGC 253 from with overlaid contours from the TGSS ADR1 survey (white) and the MWA/EoR image (red). The TGSS contours start at $4\sigma$ local rms noise level ($\sigma=11.7$ mJy beam$^{-1}$) and increase as $\sigma2^i$ for $i>0$. The MWA/EoR0 contour marks the $4\sigma$ radio intensity at 169 MHz (16.4 mJy beam$^{-1}$). The sizes of the synthesised beams at 169 MHz (red), 330 MHz (green) and 150 MHz (black) are drawn in the top right corner. Background sources, not associated with the intrinsic emission of NGC 253, are labelled with numbers (see §\[sec:bkg-srcs\]). The color scale is in units of Jy beam$^{-1}$, and the pixel size is $5\times5$ arcsec$^2$.\ []{data-label="rys:tgss-carilli"}](carilli-tgss-eor.eps){width="86mm"} \[rys:carilli-halo\] \[rys:bkg-sources\] MWA Epoch of Reionization (EoR) data ------------------------------------ The observed MWA/EoR field that contains the Sculptor Group (EoR0 field) is centered on $\text{RA} = 0^{\rm h}$, $\text{Dec} = -27^{\circ}$, and was observed for a total of 30 hours between August and October 2013 in a combination of a tracking and drifting modes. In this hybrid mode the telescope tracks a set of discrete pointing centers through which the field of interest is drifting. The observations cover frequencies between $138.9-197.7$ MHz observed as two bands (low and high) each with an instantaneous bandwidth of $\Delta\nu=30.72$ MHz. The correlated MWA/EoR0 data were pre-processed with the [cotter]{} pipeline [@2015PASA...32....8O] and averaged to 4s time and 40 kHz frequency resolution. Calibration of the data was performed as a direction-independent self-calibration using the [mitchcal]{} tool [@2008ISTSP...2..707M] and was based on a bootstrapped sky model. The initial sky model was generated from the MWA Commissioning Survey [@2014PASA...31...45H], the MRC catalogue and the Sydney University Molonglo Sky Survey [SUMSS; @2003MNRAS.342.1117M]. Imaging was performed with the [WSClean]{} software that corrects for the non-zero $w$-term effects. During the imaging process 2,500 sources were peeled, and the images were created with a uniform weighting. The primary beam, and so the flux density scale, was corrected by applying the [@2015RaSc...50...52S] model. As shown by [@2014PASA...31...45H], this model is accurate to 10%, hence we add this error in quadrature to the quoted uncertainties of our measurements. The final image used in this paper is centred at 169.6 MHz (thereafter 169 MHz) with a total bandwidth of $\Delta\nu=58.8$ MHz, synthesised beam size $2.3\times2.3$ arcmin$^2$ and rms noise $4.1$ mJy beam$^{-1}$. The calibration and imaging process of the EoR0 data is presented and discussed in detail in [@2016MNRAS.458.1057O]. Other low frequency radio surveys --------------------------------- There are additional two all-sky low frequency radio surveys that include NGC 253: the 74 MHz VLSSr [@2014MNRAS.440..327L] and the 150 MHz Tata Institute of Fundamental Research (TIFR) Giant Metrewave Radio Telescope (GMRT) Sky Survey [ (TGSS) Alternative Data Release 1]{} [ADR1; @2016arXiv160304368I]. The TGSS survey observed the whole radio sky north of declination $-53^{\circ}$ at a frequency 150 MHz (bandwidth $\Delta\nu=16.7$ MHz) at an angular resolution $25\times 25/\text{cos}(\delta-19^{\circ})$ arcsec$^2$ at declinations south of $+19^{\circ}$. The instantenous field of view of the survey at half power at 150 MHz is $3.1\times3.1$ deg$^2$, with sensitivity to structures up to 68 arcmin in angular scale [@2016arXiv160304368I]. Since the absolute flux density calibration of the TGSS ADR1 may be uncertain up to 50% in some sky regions[^2], we independently verified the calibration in the area of NGC 253. We selected unresolved sources with flux density $>1$ Jy from the $5\times5$ deg$^2$ mosaic that included NGC 253 (R03\_D17). We compared the TGSS ADR1 flux densities of these sources with the predicted values based on the spectral modeling in which we used the VLSSr, GLEAM (deep 200 MHz), MRC and NVSS surveys. We found that the TGSS mosaic required scaling by a factor 1.02 in flux density, and the absolute flux density calibration was accurate to 7%; we further added this error in quadrature to the quoted uncertainties of our measurements. The VLSS survey [@2007AJ....134.1245C] observed the radio sky north of $-30^{\circ}$ at a frequency 74 MHz. Here we use the recent re-reduction of the survey data, the VLSSr [@2014MNRAS.440..327L]. VLSSr images have an angular resolution of $75\times75$ arcsec$^2$ and a theoretical sensitivity to structures of 13–37 arcmin in angular scale. We find that neither TGSS nor VLSSr are sensitive to the extended emission of NGC 253 (Figure \[rys:tgss-carilli\]). For this reason we use the TGSS data for the flux density measurement of the central starburst region, and both TGSS and VLSSr for measurements of the flux densities of background sources only. ![image](multiwave-v3.eps){width="180mm"} Methods {#sec:methods} ======= Flux density measurements {#sec:flux-dens} ------------------------- Measurements of the total flux density of NGC 253 were performed with [CASA]{} task [imstat]{} that provides a summed flux density within a specified regions of the image corrected for the synthesised beam. We masked all pixels below $2.6\sigma$ local rms noise level [ [@2012MNRAS.425..979H]]{}. For point sources the flux density was measured with the [AIPS]{} task [jmfit]{}; for each unresolved source we fit for two components, a Gaussian and a zero-level with a slope. The absolute flux density scale is set to the [@1977AA....61...99B] scale. Models of radio spectra {#sec:models-spectra} ----------------------- Radio sources often show simple spectra that can be approximated by a power-law. However, at low radio frequencies (a few hundred MHz and below) radio spectra are consistently more curved until a turnover frequency below which the spectra become inverted. The spectral turnover is typically caused by either synchrotron self-absorption and/or thermal free-free absorption [@2015ApJ...809..168C and references therein]. If there is no evidence for a spectral turnover in the radio spectra we construct here [ (see §\[sec:modselct\] for model selection method)]{}, we proceed with fitting a polynomial. The curved radio spectra are then modeled with an $n$th-order polynomial, that in the logarithmic scale takes a general form of $$\begin{aligned} \text{log}(S_{\nu}) & = \sum^n_{i=0} A_i \text{log}^i\left(\frac{\nu}{\nu_0}\right) \\ & = A_0 + A_1 \text{log}\left(\frac{\nu}{\nu_0}\right) + ... + A_n \text{log}^n\left(\frac{\nu}{\nu_0}\right), \end{aligned} \label{eqn:poly}$$ where $A_0$ is an offset parameter (equivalent to log($S_0$) in the simple power-law case), $A_1$ is the spectral index $-\alpha$, and $A_n$ are curvature parameters ($c_n$). In the linear space the model takes the following form $$S_\nu = \prod^n_{i=0} 10^{A_i\text{log}^i(\nu/\nu_0)},$$ which we use in our modeling to preserve Gaussian noise characteristics of the measurements. Where the data suggest or show a spectral turnover, the following models are tested: synchrotron self-absorption, free-free absorption or a combination of these and power-law components. ### Synchrotron self-absorption (SSA) At low radio frequencies the intensity of the synchrotron radiation may become sufficiently high (optically thick regime) for re-absorption, termed synchrotron self-absorption, to take over. The process may be important, or even dominant, for compact sources [@1990SvAL...16..339S; @1998ApJ...499..810C]. We model the synchrotron radio spectra that may turnover due to self-absorption at low radio frequencies as [e.g. @2003AJ....126..723T] $$S_\nu = S_{\tau=1} \left( \frac{\nu}{\nu_{\tau=1}} \right)^{-\alpha} \left ( \frac{1 - e^{-\tau(\nu)}}{\tau(\nu)}\right ), \label{eqn:SSA}$$ $$\tau(\nu) = (\nu/\nu_{\tau=1})^{-(\alpha+2.5)},$$ where $\nu_{\tau=1}$ is a frequency at which the optical depth ($\tau$) reaches unity. ----------- ----------------------- ----------------- -------------- ---------------- ------------ Frequency [Angular]{} References $[$MHz$]$ [resolution]{} (arcsec$^2$) 74 $263 \pm76^{\dagger}$ $75\times75$ a, 150 $212 \pm19$ $15.6^{\rm up}$ $162\pm17$ $36\times24$ b, 200 $233 \pm23$ $138\times126$ c 330 $190\pm10$ $72\times72$ d 610 $84 \pm15$ $114\times24$ e 843 $97 \pm10$ $47\times43$ f 1465 $63.0\pm2.5$ $66\times38$ g 1465 $53.0\pm3.0$ $17.2\pm1.0$ $26\pm2$ $30\times30$ d 1490 $5.6 \pm0.5$ $53.0\pm0.5$ $19.0\pm0.5$ h 4850 $19.0\pm1.0$ $6.6\pm0.3$ $7.9\pm0.4$ $30\times30$ d 8350 $9.5\pm0.5$ $84\times84$ d ----------- ----------------------- ----------------- -------------- ---------------- ------------ $\dagger$ Tentative detection ($2.5\sigma$). $^{\rm up}$ Upper limit, equal $3\times$ local rms noise level. Our measurement based on images from the quoted survey. [References.]{} (a) [@2014MNRAS.440..327L], (b) [@2016arXiv160304368I], (c) This publication, (d) , (e) , (f) [@1983PASAu...5..235R], (g) , (h) [@1987ApJS...65..485C]. ------------ ------------ ----------------- ---------------- ----------------- ----- ---------- Background polynomial $S_{1 \rm GHz}$ $\alpha$ $c_1$ dof $\chi^2$ source order (mJy) 1 2 $80.7 \pm1.9$ $0.79 \pm0.02$ $-0.25 \pm0.04$ 7 19.2 2 2 $21.9 \pm1.0$ $0.32 \pm0.07$ $-0.63 \pm0.11$ 1 3.5 3 1 $33.1 \pm1.4$ $0.88 \pm0.03$ – 1 2.8 ------------ ------------ ----------------- ---------------- ----------------- ----- ---------- ----------- ---------------- ---------------------------------------------- ------------------------------- ------------ Frequency [ Angular]{} [ Largest ]{} References $[$MHz$]$ [ resolution]{} [ angular scale]{} (arcmin$^2$) (deg) 76 $23.9\pm2.0$ $5.1\times4.7$ 29 a 80 $23.7\pm4.0$ $3.7\times3.7$ no info b,c 84 $22.6\pm1.9$ $5.1\times4.7$ 27 a 92 $20.9\pm1.7$ $5.1\times4.7$ 24 a 99 $20.0\pm1.6$ $5.1\times4.7$ 23 a 107 $20.3\pm1.7$ $5.1\times4.7$ 21 a 115 $20.0\pm1.6$ $5.1\times4.7$ 19 a 122 $18.8\pm1.6$ $5.1\times4.7$ 18 a 130 $17.8\pm1.5$ $5.1\times4.7$ 17 a 143 $17.9\pm1.5$ $5.1\times4.7$ 16 a 151 $16.6\pm1.4$ $5.1\times4.7$ 15 a 158 $16.1\pm1.3$ $5.1\times4.7$ 14 a 166 $15.7\pm1.3$ $5.1\times4.7$ 13 a 169 $15.0\pm1.2$ $5.1\times4.7$ 13 a 174 $16.4\pm1.3$ $5.1\times4.7$ 13 a 181 $15.7\pm1.3$ $5.1\times4.7$ 12 a 189 $15.3\pm1.2$ $5.1\times4.7$ 12 a 197 $15.4\pm1.2$ $5.1\times4.7$ 11 a 204 $15.8\pm1.3$ $5.1\times4.7$ 11 a 212 $15.1\pm1.2$ $5.1\times4.7$ 11 a 220 $14.7\pm1.2$ $5.1\times4.7$ 10 a 227 $15.0\pm1.2$ $5.1\times4.7$ 10 a 330 $16.5 \pm1.9$ $1.2\times 1.2$ 1.2 d, e 408 $15.7 \pm1.9$ $2.9\times2.86$ no info f 468 $15.1 \pm1.5$ $2.1\times2.1$, $5.2\times5.2 ^\diamondsuit$ extr. single dish$^\clubsuit$ c, g 610 $9.4 \pm0.6$ $1.9\times0.4$ no info h 843 $9.0 \pm0.9$ $0.72\times0.78$ no info (th: 1.1) i 960 $8.0 \pm0.12$ $20.2\times20.2$ single dish c, g 1100 $6.7 \pm0.08$ $4.2\times4.2$ no info, dense core j 1200 $6.68 \pm0.10$ $3.4\times3.4$ no info, dense core j 1300 $6.22 \pm0.07$ $3.2\times3.2$ no info, dense core j 1400 $5.89 \pm0.16$ $2.9\times2.9$ no info, dense core j 1410 $6.12 \pm0.12$ $15.5\times15.5$ single dish c, g 1430 $5.7 \pm0.5$ $0.91\times0.83$ no info (th: 0.8) h 1465 $5.9 \pm0.1$ $1.1\times0.63$ extr. single dish$^\clubsuit$ k 1465 $6.3 \pm1.1$ $0.5\times0.5$ 0.25 d, e 1490 $5.6 \pm0.5$ $0.9\times0.9$ 0.27 m 2650 $3.85 \pm0.12$ $8.3\times8.3$ single dish c, g 2695 $4.26 \pm0.14$ $4.9\times4.9$ single dish n 2700 $3.49 \pm0.12$ $8.0\times8.0$ single dish c, g 4850 $2.93 \pm0.13$ $2.7\times2.7$ single dish n 4850 $2.71 \pm0.14$ $0.5\times0.5$ single dish e 4850 $2.69 \pm0.10$ $4.2\times4.2$ single dish p 5009 $2.50 \pm0.23$ $4.0\times4.0$ single dish c, g 5009 $2.12 \pm0.09$ $4.0\times4.0$ single dish c, r 8350 $1.66 \pm0.08$ $1.4\times1.4$ single dish e 8700 $2.06 \pm0.12$ $1.5\times1.5$ single dish n 10550 $1.98 \pm0.18$ $1.2\times1.2$ single dish s 10700 $1.95 \pm0.15$ $1.2\times1.2$ single dish t ----------- ---------------- ---------------------------------------------- ------------------------------- ------------ $\diamondsuit$ Conflicting details given in the reference. $\clubsuit$ Corrected for zero-spacing missing flux density with extrapolation. [References.]{} (a) This publication, (b) [@1973AuJPA..27....1S], (c) [@1981AAS...45..367K], (d) [@1992ApJ...399L..59C], (e) , (f) [@1971MNRAS.152..403C], (g) [@1975AuJPA..38....1W], (h) , (i) [@1983PASAu...5..235R], (j) [@2010ApJ...710.1462W], (k) , (m) [@1987ApJS...65..485C], (n) [@1979AA....77...25B], (p) [@1994ApJS...90..179G], (r) [@1976AuJPA..39....1W], (s) , (t) [@1983AA...127..177K]. ----------- ----------------------- ---------------- ------------ Frequency [Angular]{} References $[$MHz$]$ [resolution]{} (arcsec$^2$) 150 $2.16\pm0.15$ $36\times24$ a, 330 $2.67 \pm0.16$ $33\times21$ b 610 $2.3 \pm0.2$ $114\times24$ c 1413 $2.33 \pm0.14\dagger$ $3\times1.8$ d 1450 $2.07 \pm0.04$ $33\times21$ b 1465 $2.04 \pm0.10$ $30\times30$ e 1660 $1.96 \pm0.04$ $33\times21$ b 4520 $1.36 \pm0.04$ $33\times21$ b 4850 $1.27 \pm0.06$ $30\times30$ e 4890 $1.30 \pm0.04$ $33\times21$ b 6700 $1.13 \pm0.04$ $37\times37$ f 7000 $1.04 \pm0.04$ $35\times35$ f 8090 $0.93 \pm0.03$ $33\times21$ b 8350 $0.98 \pm0.05$ $84\times84$ e 8470 $0.89 \pm0.03$ $33\times21$ b ----------- ----------------------- ---------------- ------------ : Flux density measurements of NGC 253 central starburst region. All measurements are in the same absolute flux density scale of [@1977AA....61...99B]. []{data-label="tab:spectra-core"} $\dagger$ Integrated. Our measurement based on images from the quoted survey. [References.]{} (a) [@2016arXiv160304368I], (b) , (c) , (d) [@1982ApJ...252..102C], (e) , (f) [@2010ApJ...710.1462W]. ![image](fig2-both-panels.eps){width="180mm"} ![image](NGC253-final-spectrum.eps){width="103mm"} ![image](NGC253-bkg-spectrum.eps){width="75mm"} ### Free-free absorption (FFA) The self-absorbed bremstrahlung (i.e. free-free absorbed) radio spectrum can be expressed as [e.g. @2002MNRAS.334..912M] $$S_\nu = S_{\tau=1} \left (\frac{\nu}{\nu_{\tau=1}}\right)^{2} (1 - e^{-\tau_{\rm ff}(\nu)}),$$ where the opacity coefficient is given by $$\tau_{\rm ff}(\nu) = (\nu/\nu_{\tau=1})^{-2.1}. \label{eqn:tau-FFA}$$ As discussed in §\[sec:intro\] radio emission from NGC 253 is a mixture of synchrotron emitting cosmic rays from SNR and thermal emission H[ii]{} regions. The free-free absorption is expected to start dominating at low radio frequencies, where the intensity of the electrons in the ionized gas becomes high (optically thick regime). For NGC 253 it is a natural assumption that the thermal plasma co-exists with the synchrotron emitting electrons, hence the radio spectrum can be modeled as a synchrotron power-law with an internal free-free absorbing screen [SFA; @2003AJ....126..723T], $$S_\nu = S_{0} \left(\frac{\nu}{\nu_0} \right)^{-\alpha} \left ( \frac{1 - e^{-\tau_{\rm ff}(\nu)}}{\tau_{\rm ff}(\nu)}\right ). \label{eqn:SFA}$$ Weighted non-linear least squares fitting {#sec:correlated-noise} ----------------------------------------- All measurements in this paper are considered independent of each other (in the GLEAM survey valid for flux densities $\gtrsim5$ Jy; see [@2016MNRAS.GLEAM.subm], thus a simple form of $\chi^2$ statistic is used for the fitting of the radio spectra, which at the same time is the goodness-of-fit of the fitted model ($M_i$), $$\chi^2 = \sum_{i}^n \left( \frac{\text{M}_i - \text{data}_i}{\text{error}_i} \right)^2 \label{eqn:chi2}$$ for $i=1,..,n$ data points. In minimization of Eqn. \[eqn:chi2\] we use the Levenberg-Marquardt algorithm [@1944Levenberg; @1963Marquardt] implemented in the Python[^3] module [lmfit]{} [@2014PythonLMFIT]. Model selection {#sec:modselct} --------------- For the formal model selection we use the Bayesian inference method. We follow the prescription outlined in [@2015ApJ...809..168C], with the log-likelihood function (the probability of observing the data given model parameters $\theta$) in the form of $$\text{ln}\mathcal{L}(\theta) = -\frac{1}{2}\sum_{i}^{n}\left[\frac{(\text{M}_i - \text{data}_i)^2}{\text{error}_i^2} + \text{ln}(2\pi\, \text{error}_i^2)\right].$$ Under the hypothesis that the models being compared ($M_1$, $M_2$) are equally likely, the model selection can be performed based solely on the Bayesian evidence ($Z$), where $$\Delta \text{ln}(Z) = \text{ln}(Z_2) - \text{ln}(Z_1)$$ and $$Z_{1,2} = \int \int ... \int \mathcal{L}(\theta) \Pi(\theta) d(\theta).$$ The dimensionality of the integration depends on the number of model parameters. If $\Delta\text{ln}(Z)\geqslant3$ model $M_2$ is strongly favoured over $M_1$. If $1<\Delta\text{ln}(Z)<3$ model $M_2$ is only moderately favoured over $M_1$, and if $\Delta\text{ln}(Z)<1$ the preference of one model over the other is inconclusive. For more discussion on the theoretical background of the method used see [@2015ApJ...809..168C]. We use the [MultiNest]{} tool for our calculations of the Bayesian evidence. [crrrrrrrr]{} Region & & $S_{\rm 169 MHz}$ & $S_{\rm 1.4GHz}^{\rm 169 res}$ & & $S_{\rm 200 MHz}$ & $S_{\rm 1.4GHz}^{\rm 200 res}$\ & & & & & &\ 1 & $0.54 \pm0.06$ & $242\pm26$ & $75\pm10$ & $0.57\pm0.06$ & $233\pm25$ & $75\pm10$\ 2 & $0.63 \pm0.04$ & $586\pm59$ & $149\pm13$ & $0.68\pm0.03$ & $581\pm51$ & $149\pm13$\ 3 & $0.57 \pm0.02$ & $883\pm89$ & $257\pm11$ & $0.62\pm0.02$ & $884\pm73$ & $257\pm11$\ 4 & $0.67 \pm0.04$ & $415\pm42$ & $97\pm9$ & $0.72\pm0.04$ & $412\pm36$ & $97\pm9$\ 5 & $0.31 \pm<0.01$ & $6558\pm656$& $3341\pm14$ & $0.34\pm<0.01$ & $6589\pm528$& $3341\pm14$\ 6 & $0.60 \pm0.07$ & $224\pm24$ & $61\pm10$ & $0.70\pm0.07$ & $246\pm25$ & $61\pm10$\ 7 & $0.58 \pm0.03$ & $488\pm49$ & $139\pm10$ & $0.63\pm0.03$ & $487\pm42$ & $139\pm10$\ 8 & $0.50 \pm0.02$ & $616\pm62$ & $211\pm9$ & $0.51\pm0.02$ & $578\pm49$ & $211\pm9$\ 9 & $0.53 \pm0.06$ & $216\pm23$ & $69\pm9$ & $0.51 \pm0.06$ & $192\pm21$ & $69\pm9$\ Results {#sec:results} ======= In what follows we refer to the ‘halo’ as the radio emission beyond the boundary of the optical disk of the galaxy. Given the low angular resolution of the MWA observations we distinguish only between the galaxy disk and the extended synchrotron halo. Background sources {#sec:bkg-srcs} ------------------ There are three discrete radio sources located within the extended emission of NGC 253; the sources are marked in Figure \[rys:bkg-sources\], and their positions are based on the NVSS and measurements. The discrete radio source no. 1 is located at RA(J2000)=$00^h47^m59^s.10$, Dec(J2000)=$-25^\circ18'22''.45$ at 200 MHz, and is most likely a background AGN [@1992ApJ...399L..59C]. The radio source is detected in the GLEAM images, but at low frequencies becomes increasingly confused with the NGC 253 halo emission. We measure the flux density of the source only in the deep GLEAM image (Table \[tab:spectra\]). The discrete radio source no. 2 is located at RA(J2000)=$00^h47^m12^s.01$, Dec(J2000)=$-25^\circ17'43''.9$ and is most likely a faint background AGN . This source is heavily embeded in the NGC 253 extended emission in the MWA images. The discrete radio source no. 3 is located at RA(J2000)=$00^h47^m44^s.91$, Dec(J2000)=$-25^\circ13'38''.4$. This source is embeded in the extended emission in our MWA images, but is clearly detected in the TGSS ADR1 image (Figure \[rys:bkg-sources\]). The flux density measurements of the background sources are listed in Table \[tab:spectra\], and the spectral modeling results are given in Table \[tab:bkgmodels\]. We subtract the estimated flux density contribution of these sources from the total flux density measurements of NGC 253. In addition, we model the background source no. 1 as a point source with a peak flux density $240$ mJy at 169 MHz and $223$ mJy at 200 MHz, and for pictorial purposes we subtract it directly from the radio image plane. The resulting radio contours are overlaid on H$\alpha$ and X-ray images in Figure \[rys:multi-images\] and discussed in §\[sec:halo\]. NGC 253 {#sec:ngc253} ------- Radio images of NGC 253 at six chosen radio frequencies are presented in Figure \[rys:radio-images\]. At 200 MHz, [ the deepest image from the presented here GLEAM observations,]{} the size of NGC 253 is 1310 arcsec (major axis) and 535 arcsec (minor axis) measured at a $\text{PA}=52^\circ$, with a total radio luminosity density $2.4(\pm0.1)\times10^{22}$ W Hz$^{-1}$. [ At 169 MHz (MWA/EoR0 image), the size increases by 3–6 percent, to 1440 arcsec (major axis) and 615 arcsec (minor axis), which may be a combination of the intrinsic increase in size and the uncertainty of the measurement]{}. ### Total radio emission Radio continuum spectra of NGC 253 between 76 MHz and 10.7 GHz are plotted in Figure \[rys:radio-spectra\] and the flux density measurements are tabulated in Table \[tab:spectra\]. Background radio sources (§\[sec:bkg-srcs\]) located within the diffuse emission of NGC 253 were subtracted from the total flux density measurements. In the construction of the radio spectrum we used archival data provided the measurements were of angular resolution comparable to GLEAM or were sensitive to low brightness emission on angular scales of at least 0.5 deg, the total flux density was integrated over the diffuse emission of NGC 253 and not fitted by Gaussian components, and the absolute flux density scale and the uncertainties of the measurements were quoted. We do not use measurements at angular resolution of $>20$ arcmin because of confusion of NGC 253 with nearby sources. We find the best fitting model to be a 2nd-order polynomial with $S_0=7.30\pm0.04$ Jy, $\alpha=0.56\pm0.01$ and a curvature $c_1=-0.12\pm0.01$ at a reference frequency of 1 GHz ($\chi^2=140$, with degrees of freedom: dof $=45$; Figure \[rys:radio-spectra\]), which is significantly preferred to a simple power-law ($\Delta \text{ln}(Z)=100.5\pm0.3$). ### Central starburst region {#sec:nucleus} The angular resolution of the MWA data is too low to resolve the central starburst region of NGC 253; the highest angular resolution achieved is 102 arcsec at 227 MHz (GLEAM) and 138 arcsec at 169 MHz (EoR), which is over three times the size of the NGC 253 starburst region . We construct radio spectra of the central starburst region using data from the literature and new measurements from the TGSS ADR1 survey (Figure \[rys:radio-spectra\], Table \[tab:spectra\]). We limit the measurements to those that are at an angular resolution comparable to the size of the central starburst region (approximately 20–30 arcsec). We find the best fitting model to be a 2nd order polynomial with $S_0=2.28\pm0.02$ Jy, $\alpha=0.20\pm0.01$ and a curvature $c_1=-0.24\pm0.01$ at a reference frequency of 1 GHz ($\chi^2=12.8$, dof $=13$; Figure \[rys:radio-spectra\]). We further attempt to model the spectral turnover, and we find SFA (Eqn. \[eqn:SFA\]) to be the best fitting model with $S_{\tau=1}=4.43\pm0.14$ Jy, $\alpha=0.43\pm0.01$ and $\nu_{\tau=1}=238\pm15$ MHz ($\chi^2=42.9$, dof $=13$). Based on the Bayesian evidence the SFA model (synchrotron plasma absorbed by an internal free-free absorbing screen) is preferred to the pure synchrotron self-absorption model, SSA ($\Delta \text{ln}(Z)=8.3\pm0.3$). ![image](spindx-all.eps){width="182mm"} ![image](NGC253-TTpltSS-gleam.eps){width="78mm"} ![image](NGC253-TTpltSS-eor.eps){width="78mm"} ### Spectral index maps {#sec:spindx-maps} Using the total flux density images at 200 MHz (GLEAM), 169 MHz (MWA/EoR) and 1.46 GHz [@1992ApJ...399L..59C], we created spectral index distribution maps and their corresponding uncertainty and signal-to-noise ratio maps (Figure \[rys:sp-idx\]). [ To create the spectral index maps we convolved the 1.46 GHz image to the resolution of the GLEAM image (for the 200 MHz–1.46 GHz spectral index map), and the MWA/EoR image (169 MHz–1.46 GHz spectral index map).]{} The spectral index maps suggest an apparent variation in $\alpha$ across the NGC 253 disc and halo. The central regions of the galaxy are dominated by a flat component ($\alpha=0.31-0.34$), coinciding with the central starburst. The gradual steepening along the minor axis seen by is seen only on the northern side of the galaxy in our maps. Further, the region extending SW from the central starburst region seems to be flatter ($\alpha \lesssim 0.53$) than in the other parts of the galaxy outside the central starburst ($\alpha \sim 0.60-0.65$). To verify the significance of the spectral index variation across the galaxy, we used the T-T method [@1962MNRAS.124..297T]. The method allows one to estimate a spectral index within defined regions of a source between two frequencies. We define nine regions within NGC 253 (Figure \[rys:sp-idx\]). Due to the low angular resolution of our observations and to avoid oversampling, only one data point is associated with each region. The results are shown in Table \[tab:TTplots\] and Figure \[rys:TTplots\]. We find that between 200 MHz and 1.465 GHz the apparently flat regions (Region 8 and 9) are statistically different from the other regions within NGC 253 apart from Region 1 (Figure \[rys:TTplots\], Table \[tab:TTplots\]). This spectral flattening is not present in the radio spectral index distribution map between 330 MHz and 1.46 GHz of , even though the same high frequency map is used. This clearly indicates that the flattening occurs at $<300$ MHz. There is also a slight flattening of the spectral index in the NE region perpendicular to the major axis (Region 1), although we find this flattening to be statistically different only from Region 2 (eastern NW halo), 4 (radio spur) and 5 (including central starburst) in the $\alpha^{\rm 200 MHz}_{\rm 1.4 GHz}$ map. All regions further flatten at 169 MHz, reducing the differences between spectral indices of the regions. Discussion ========== Low-frequency synchrotron radio halo {#sec:halo} ------------------------------------ A large-scale radio halo in NGC 253 was discovered and confirmed by [@1992ApJ...399L..59C] and extensively studied by , and . This synchrotron halo is most pronounced at low radio frequencies, with the estimated scale heights of $1.7\pm0.1$ kpc at 1.4 GHz and $2.5\pm0.2$ kpc at 330 MHz . Both the deep 200 MHz GLEAM image and the MWA/EoR image at 169 MHz reveal the extended synchrotron halo, which is at least as extensive as one detected in the [@1992ApJ...399L..59C] 330 MHz map (Figure \[rys:carilli-halo\]). ### Maximum vertical extent We measured the observed, projected maximum vertical extent of the NGC 253 disk and halo at 169 MHz as a function of the distance from the nuclear region along the major axis (Figure \[rys:scaleheights\], Table \[tab:scaleheights\]). The extent is measured perpendicular to the major axis (at $\text{PA} =-38^\circ$, [z]{}-direction) in steps of 132 arcsec (2.4 kpc) separately for the North (filled circles) and South (empty circles) side of the disk and halo as divided by the major axis ($\text{PA} = 52^\circ$). We find the projected radio halo to extend up to 4.75 kpc above the optical [$B$ band including 90 per cent total light, @1989spce.book.....L] and 6.3 kpc above the infrared [total $K_{\rm s}$ band, @2003AJ....125..525J] edge of the galaxy, reaching up to a total 7.9 kpc in $z$-direction. This is consistent with previous radio measurements at higher radio frequencies , as well as broadband X-ray observations of the galaxy’s extended extraplanar emission . attributed decrease of scale height they measured and modeled to the increased synchrotron losses in the central regions, where the magnetic field is highest. ### Halo morphology The shape of the radio halo in our MWA radio images (Figure \[rys:multi-images\]) resembles the ‘horn-like’ or ‘X-shaped’ structure seen at GHz radio frequencies , in H[i]{} , X-rays , H$\alpha$ (G. Meurer, priv.comm.; Figure \[rys:multi-images\]), UV [@2005ApJ...619L..99H] and far-IR [@2009ApJ...698L.125K]. The radio halo was investigated in detail by who, through modeling of the large-scale magnetic field, attributed its origin to disk wind, confirming previous suggestions . This is also in line with the H$\alpha$ and optical analyses of the inner starburst-driven superwind [@2011MNRAS.414.3719W]. In our MWA maps both the NE and NW halo regions are pronounced. The extended soft X-ray emission ($<1$ keV) of the halo, detected in the north-western direction from the NGC 253 disk, is interpreted as bubbles of hot low density gas . postulates that the large-scale magnetic field of the halo follows the walls of these bubbles, where it may be compressed, producing the X-shaped synchrotron radio halo as well as heating up pre-existing cold gas to X-ray energies. The northern halo can also be easily seen, in projection, in our Figure \[rys:multi-images\] where we overlay MWA/EoR intensity contours on the soft X-ray emission [XMM-Newton]{} image. ------ ------- ----- ----- ----- ----- -658 -11.8 113 2.0 100 1.8 -526 -9.5 213 3.8 211 3.8 -395 -7.1 262 4.7 327 5.9 -263 -4.7 396 7.1 440 7.9 -132 -2.4 416 7.5 367 6.6 0 0.0 311 5.6 291 5.2 132 2.4 270 4.9 291 5.2 263 4.7 297 5.5 323 5.8 395 7.1 279 5.0 260 4.7 526 9.5 201 3.6 171 3.1 658 11.8 56 1.0 100 1.8 ------ ------- ----- ----- ----- ----- : Projected maximum vertical extent of the NGC 253 disk and halo at 169 MHz ($h$) measured at a distance $x$ from the nuclear region along the major axis, where 0 is centred on the nucleus of the galaxy. The NE direction along the major axis is negative and SW is positive. The extent is measured perpendicular to the major axis (PA$ =-38^\circ$) separately for the North and South side of the disk and halo. The uncertainties on the measurements are 21.6 arcsec (equivalent to 0.4 kpc). See §\[sec:halo\] for discussion. []{data-label="tab:scaleheights"} ![ Projected maximum vertical extent of the NGC 253 disk and halo at 169 MHz ($h$) as a function of the distance from the nuclear region along the major axis ($x$), where 0 is centred on the nucleus of the galaxy. The NE direction along the major axis is negative and SW is positive. The extent is are measured perpendicular to the major axis (at $\text{PA} =-38^\circ$) at a step of 132 arcsec (2.4 kpc) separately for the North (filled circles) and South (empty circles) side of the disk and halo as divided by the major axis ($\text{PA} = 52^\circ$). The synthesised beam size of the radio image, $138\times138$ arcsec$^{2}$, is not included in the uncertainties and is drawn as solid horizontal line. Measured sizes of NGC 253 at optical $B$ band including 90 per cent of total light (dash-dotted line) and total infrared $K_{\rm s}$ band (dashed line) are drawn for reference. Plotted values are tabulated in Table \[tab:scaleheights\].[]{data-label="rys:scaleheights"}](NGC253-scaleheight.eps){width="88mm"} The SE region of the extended halo, the ‘spur’ [@1992ApJ...399L..59C], is contaminated by a background source. We modeled the background source as unresolved (at MWA angular resolution) and subtracted it from the 169 MHz EoR and deep 200 MHz GLEAM images as described in §\[sec:bkg-srcs\]. The residual emission, which we consider intrinsic to the ‘spur’ is shown in Figure \[rys:multi-images\] overplotted on the H$\alpha$ and X-ray images. Although slightly offset , the feature broadly coincides with the extended outflows at both frequencies as clearly seen in our figure. The spur has been previously interpreted as originating from the active star formation region in the NE end of the bar [@1988AJ.....95.1057W; @1992ApJ...399L..59C]. We confirm the association of the radio ‘spur’ with an extended H$\alpha$ outflow, which can be clearly seen in our Figure \[rys:multi-images\] [G. Meurer, priv.comm.; @2003PASP..115..928K]. In our radio spectral map between 200 MHz and 1.46 GHz the spur displays the steepest spectral index within the galaxy (Figure \[rys:sp-idx\]), gradually steepening from $\alpha\sim0.7$ to $\alpha\sim0.9$ outwards from the galaxy disk (though it still contains the background source no.1 with $\alpha=0.57$), which is indicative of ageing unabsorbed synchrotron plasma. Spectral properties of NGC 253 {#sec:spectral-props} ------------------------------ ### Broadband spectrum of total radio emission {#sec:nonthermal-emission} The broadband spectrum of the total radio emission from NGC 253 is steep, although flattening at MHz radio frequencies (Figure \[rys:radio-spectra\]). The total radio emission originates from SNRs, H[ii]{} regions (predominantly central starburst region; e.g. @1997ApJ...488..621U [-@1997ApJ...488..621U], @2000AJ....120..278U [-@2000AJ....120..278U], but cf. @1988AJ.....95.1057W [-@1988AJ.....95.1057W], @1996AJ....112.1429H [-@1996AJ....112.1429H]) and electrons (cosmic rays) freely spiralling in the large scale magnetic field . The steep spectrum is understood to be of a synchrotron origin. The flattening of the spectrum, however, may be due to a number of reasons, including some degree of absorption of the synchrotron emission (Eqn. \[eqn:SSA\]) and a low energy cut-off of the electron population, where in general it is assumed the electron energy ($E$) spectrum can be described as a power-law $N(E)\propto E^{-p}$ with the index $p$ related to the radio spectral index as $\alpha=(p-1)/2$. The low radio frequency flattening of the spectra of starburst galaxies is not unusual and it has been observed previously [e.g. @2015AJ....149...32M]. ![image](xray-lucero-spidx-v3.eps){width="170mm"} ![image](Halpha-spinx-therm-2-min0.001ctssec-nocolorbar.eps){width="87mm"} ![image](Halpha-spinx-therm-1-min0.001ctssec.eps){width="82.5mm"} We attempt to separate the contribution of the extended and central starburst emission to the total flux densities to see if the [ spectral]{} flattening can be attributed mostly to the absorption occurring in the central starburst region. To do this we simultaneously fitted the total flux density and the central starburst region spectra, assuming an underlying 2- or 3-component spectrum composed of the central starburst region (component C) and extended emission (component E, composed of one or two components). As an example of our fitting method, in the case where the component E is modeled as a power-law and component C as a free-free absorbed synchrotron emission, the model equation ($S_{\nu}^{\rm mod})$ takes the following form $$S_{\nu}^{\rm mod} = ( S_\nu^{\rm E} + S_\nu^{\rm C} )^{\rm tot} + ( S_\nu^{\rm C} )^{\rm cor},$$ where $$S_\nu^{\rm E} = S_0^{\rm ext} \left( \frac{\nu}{\nu_0}\right ) ^{-\alpha^{\rm ext}},$$ $$S_\nu^{\rm C} = S_0^{\rm cor} \left( \frac{\nu}{\nu_0}\right ) ^{-\alpha^{\rm cor}} \left (\frac{1-e^{-\tau_{\rm ff}(\nu)}}{\tau_{\rm ff}(\nu)} \right ).$$ $\tau_{\rm ff}(\nu)$ is given by Eqn. \[eqn:tau-FFA\] and the indices indicate: $\rm tot$ – total, $\rm cor$ – core, $\rm C$ – component C, $\rm E$ – component E, $\rm ext$ – extended. This model equation is then compared to our observed data ($S_{\nu}^{\rm obs}$), where $S_{\nu}^{\rm obs} = S_{\nu}^{\rm obs, tot} + S_{\nu}^{\rm obs, cor}$. [ In the 2-component model, we fit the component E with either simple power-law or a curved spectrum (2nd degree polynomial; Eqn. \[eqn:poly\]). In the 3-component model, we fit the component E with a combination of power-law and either a curved spectrum, synchrotron self-absorbed (Eqn. \[eqn:SSA\]) or synchrotron free-free absorbed component (Eqn. \[eqn:SFA\]). The component C is modeled with either 2nd-degree polynomial, self-absorbed synchrotron or synchrotron power-law emission with a free-free absorbing screen. ]{} We find the best fitting model to be the 3-component model, with (1) the component E modeled as a combination of a simple power-law and 2nd order polynomial, with a total flux density $S_0 = 5.08\pm0.50$ Jy and $\alpha = 0.71\pm0.01$ at reference frequency 1 GHz, and 34% of $S_0$ becoming absorbed at low frequencies as described by 2nd order polynomial with $c_1=-0.76\pm0.17$, and (2) the component C modeled as a synchrotron plasma with an internal free-free absorbing screen, with $S_{\tau=1} = 4.34\pm0.11$ Jy, $\nu_{\tau=1} = 231\pm14$ MHz, $\alpha_{\rm SSA} =0.41\pm0.01$. The 3-component model ($\chi^2 = 173$, dof $=58$) is favoured over any 2-component model, even if the component E is modeled as a 2nd order polynomial ($\Delta\text{ln}(Z)>4.3\pm0.3$). Preference of the 3-component model indicates that flattening of the component E spectrum at the lower frequencies is non-negligible. In principle, this flattening could be attributed to synchrotron self-absorption caused by shock re-acceleration of the halo/disk plasma, an external free-free absorbing screen, or an intrinsic low-energy cut-off of the electron distribution. Thermal free-free absorption can be [ largely excluded based on the limited]{} evidence for high thermal content in the NGC 253 halo, [ especially in the SW region (see §\[sec:spectral-maps\] for details)]{}. Although our Bayesian inference tests indicate that the flattening caused by the synchrotron self-absorption is moderately favoured over the low-energy cut-off in the electron distribution which could be inferred from the 2nd order polynomial fit ($\Delta\text{ln}(Z)=2.5\pm0.3$), we find that the former fit is associated with very high uncertainties. We also find that any model invoking multiple internal components that we tested is strongly favoured over an external free-free absorbing screen. Our results do not change in the absence of the data points between 300 and 600 MHz that may seem unusually high, which further strengthens our result that the radio emitting plasma in the disk and halo of NGC 253 is composed of at least two spectral components that behave differently. This result is also in line with our findings on the radio spectral index variation across the galaxy (discussed further in the next section). Furthermore, another important result of our spectral modeling is that the central starburst region is best modeled by the SFA model. Although, in principle, the 2nd order polynomial is statistically favoured, the SFA model is more realistic. Curved radio spectra can be explained by low energy cut-off of electron population, SSA, SFA or FFA models. As we have shown the SSA model is statistically ruled out (§\[sec:nucleus\]). Given the overwhelming evidence of significant thermal component in the central starburst [e.g. Figure \[rys:multi-images\]; @1997ApJ...488..621U; @1999ApJ...518..183K; @2011ApJ...739L..24K] coexisting with synchrotron plasma, the synchrotron free-free absorption model is more likely than the low energy cut-off in electron population. The plasma becomes optically thick around frequency $230$ MHz. Given this result, and under a simplified assumption that a uniform optical depth holds across the region, we estimate a typical emission measure [@1961RvMP...33..525O; @1978ppim.book.....S] of the absorbing gas towards the central starburst region to be very high, of the order $4\times10^5$ pc cm$^{-6}$ . ### Radio spectral index distribution maps {#sec:spectral-maps} We now consider the origin of the $\alpha$ variation across the NGC 253 disk and halo. The southern flattening occurs beyond the SW spiral arm, in the halo region. In Figure \[rys:lucero\] we overplot the $\alpha^{\rm 200 MHz}_{\rm 1.4 GHz}$ and extraplanar H[i]{} contours on the [XMM-Newton]{} soft X-ray image of NGC 253. The diffuse X-ray emission indicates ionized hot gas. As pointed out by [@2015MNRAS.450.3935L], the neutral cold H[i]{} gas seems to surround the X-ray emitting regions. The radio spectral index spatial variations seem to follow the distribution of the X-ray emission, with the $\alpha$ steepening occurring in the regions of intense soft X-ray emission (radio spur and NW halo) and the flattening around the voids of diffuse X-ray plasma (western SE halo, eastern NW halo). This distribution seems to also match the extraplanar H[i]{} emission, especially in the western SE halo region. [ In H$\alpha$ we detect faint diffuse emission in the NE halo and the southern ‘spur’ (Figure \[rys:multi-images\]), with the line fluxes measured down to $3\times 10^{-18}$ erg s$^{-2}$ s$^{-1}$ arcsec$^{-2}$. In these regions the spectral index seems to steepen (Figure \[rys:lucero\]B). There is, however, almost no H$\alpha$ emission present in the western SE halo, while most of the flattening of the component E in the modeling of §\[sec:nonthermal-emission\] is due to this region (Figure \[rys:lucero\]C, regions 8 and 9 in Figure \[rys:sp-idx\]).]{} It has previously been suggested that the halo gas originates from both galactic ‘fountains’ from the star-forming disk and a galactic superwind . This strong superwind may be pushing and collimating the neutral cold gas in the halo [@2015MNRAS.450.3935L]. In the case of strong collimation shocks one may observe flattening of radio spectra due to synchrotron self-absorption in transverse shocks. Our results seem to favour such a scenario. It is also worth noting that the spectrum flattening of the SW halo corresponds to an extended loop, or arch, seen in optical images . However, if the SW halo is predominantly diffuse, and of low density, the flattening may be rather due to an intrinsic low-energy cut off of the electron distribution. Another important note is that, based on our broadband radio spectrum modeling, the SW halo region cannot be fully responsible for the total spectrum flattening. Radio emission that becomes absorbed at lower frequencies constitutes more than 30% of the total extended radio emission at 1 GHz ($1.73\pm0.36$ Jy), while the SW region is only 0.77 Jy at that frequency, which means that the flattening must be also occurring, although in a smaller degree, in other regions across the disk and halo. Although the SW flattening of the radio spectral index is most likely of a synchrotron origin, an external free-free absorbing screen was also previously suggested. The foreground absorption model was favoured by [@2008AA...489.1029B] based on their X-ray data modeling and apparent differences of radio and X-ray halo morphologies. As proved later , deep continuum and polarization radio observations at both GHz and MHz frequencies reveal the horn-like structure of the radio halo, which directly resembles the X-ray diffuse emission. Based on the equipartition assumptions, find the magnetic field within the halo to be very high, $7-12 \mu$G, reaching as much as $160\pm20 \mu$G in the central regions and $46\pm10 \mu$G in the starburst outflow. The magnetic field in the central regions is strong enough for synchrotron emission to contribute a few per cent to the total X-ray emission [@2013ApJ...762...29L]. As discussed in the previous section, we also find that an external free-free absorbing screen is not a statistically preferred model. These new findings support models in which the total X-ray emission may indeed come from a combination of thermal and synchrotron plasma rather than multi-temperature pure thermal plasma with an externally caused absorption [cf. @2008AA...489.1029B]. Conclusions {#sec:conclude} =========== We present deep, low-frequency radio continuum images and flux density measurements of a nearby, archetypal starburst galaxy, NGC 253. Our data are part of the Galactic and Extragalactic All-Sky MWA Survey and the MWA EoR observations. The images span frequencies between 76 and 231 MHz at angular resolution of 1.7 – 5 arcmin and rms noise levels of 4 – 75 mJy (depending on frequency), and present the deepest measurements of NGC 253 at these low radio frequencies yet. Our main findings are summarized as follows. 1. We detect a large-scale synchrotron radio halo that at 154–231 MHz displays the X-shaped/horn-like structure seen at GHz radio frequencies, and is broadly consistent with other multiwavelength observations of NGC 253. 2. The projected maximum vertical extent of the synchrotron emission at 169 MHz extends up to 7.5 kpc NW (7.9 kpc SE) from the major axis of NGC 253, consistent with large-scale soft X-ray emission (extending 9 kpc NW) and X-ray outflow (6.3 kpc SE).\ 3. The radio spectrum of the central starburst region of NGC 253 is significantly curved at low radio frequencies, with the spectral turnover occurring around 230–240 MHz, which is for the first time statistically constrained. 4. The radio spectral index maps show significant spectral variations in the structure of NGC 253 between 200 MHz and 1.465 GHz. In particular, we isolate a region of statistically significant spectral flattening to the western side of the SE halo. However, as the SW region is rather faint at 1.46 GHz it cannot be fully responsible for the total spectrum flattening, which indicates that the flattening must be also occurring, likely in a smaller degree, in other regions across the disk and halo. 5. The broadband spectrum of integrated total radio emission of NGC 253 is best described as a sum of central starburst and extended emission, where the central starburst component is best modeled as an internally free-free absorbed synchrotron plasma, and the extended emission as synchrotron emission flattening at low radio frequencies. We also find that an external free-free absorbing screen is not a statistically preferred model when compared to models including multiple internal components. 6. We find that the extended emission of NGC 253 is best modeled by a combination of two synchrotron components, one of which becomes significantly absorbed at low radio frequencies. The flattening occurs at frequencies below $300$ MHz, and may be attributed to synchrotron self-absorption of shock re-accelerated electrons or an intrinsic low-energy cut off of the electron distribution. Acknowledgments {#acknowledgments .unnumbered} =============== ADK thanks P. A. Curran for valuable discussions on data modeling and constant encouragement in achieving the goals. The authors thank the anonymous referee for careful reading of the manuscript and suggestions that improved this paper. The authors thank W. Pietsch and D. Lucero for providing, respectively, X-ray and H[i]{} fits images of NGC 253, and O.I. Wong and X. Sun for helpful comments. The authors thank V. Heesen for 1.465 GHz image of NGC 253 and for helpful discussions. SB acknowledges funding for the ICRAR Summer Scholarship. This research was conducted under financial support of the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020. This scientific work makes use of the Murchison Radio-astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the operation of the MWA is provided by the Australian Government (NCRIS), under a contract to Curtin University administered by Astronomy Australia Limited. We acknowledge the Pawsey Supercomputing Centre which is supported by the Western Australian and Australian Governments. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication uses the following radio data reduction software: the Multichanel Image Reconstruction, Image Analysis and Display software [; @1995ASPC...77..433S], the Common Astronomy Software Applications package [; @2007ASPC..376..127M] and the Astronomical Image Processing System [AIPS]{}. [AIPS]{} is produced and maintained by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. natexlab\#1[\#1]{} , R. R. J., & [Ulvestad]{}, J. S. 1988, , 330, L97 , J. W. M., [Genzel]{}, R., [Pauliny-Toth]{}, I. I. K., & [Witzel]{}, A. 1977, , 61, 99 , J., [Bell]{}, E. F., [Chappell]{}, S. N., [Radburn-Smith]{}, D. J., & [de Jong]{}, R. S. 2011, , 736, 24 , M., [Pietsch]{}, W., [Trinchieri]{}, G., [et al.]{} 2008, , 489, 1029 , R., [Biermann]{}, P., [Emerson]{}, D. T., & [Wielebinski]{}, R. 1979, , 77, 25 , R., [Carilli]{}, C. L., [Holdaway]{}, M. A., & [Klein]{}, U. 1994, , 292, 409 , R., [Hutschenreiter]{}, G., & [Wielebinski]{}, R. 1982, , 106, 112 , G. J., [Beswick]{}, R. J., [D’Cruze]{}, M. J., [et al.]{} 2015, , 450, L80 , R., [Brinks]{}, E., [Perez-Torres]{}, M., [et al.]{} 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), 70 , K. M., [Rawlings]{}, S., & [Willott]{}, C. J. 1999, , 117, 677 , R., [Oosterloo]{}, T. A., [Fraternali]{}, F., [van der Hulst]{}, J. M., & [Sancisi]{}, R. 2005, , 431, 65 , A., [Goss]{}, W. M., & [Wellington]{}, K. J. 1983, , 54, 387 , J. D., [Cairns]{}, I., [Kaplan]{}, D. L., [et al.]{} 2013, , 30, 31 , D. S. 1995, in Bulletin of the American Astronomical Society, Vol. 27, American Astronomical Society Meeting Abstracts, 1444 , J., [Georgakakis]{}, A., [Nandra]{}, K., [et al.]{} 2014, , 564, A125 , J. R., [Gaensler]{}, B. M., [Ekers]{}, R. D., [et al.]{} 2015, , 809, 168 , M. J. 1971, , 152, 403 , C. L. 1996, , 305, 402 , C. L., [Holdaway]{}, M. A., [Ho]{}, P. T. P., & [de Pree]{}, C. G. 1992, , 399, L59 , R. A. 1998, , 499, 810 , A. S., [Lane]{}, W. M., [Cotton]{}, W. D., [et al.]{} 2007, , 134, 1245 , J. J. 1987, , 65, 485 , J. J., [Condon]{}, M. A., [Gisler]{}, G., & [Puschell]{}, J. J. 1982, , 252, 102 , J. J., [Cotton]{}, W. D., [Greisen]{}, E. W., [et al.]{} 1998, , 115, 1693 , G., [de Vaucouleurs]{}, A., [Corwin]{}, Jr., H. G., [et al.]{} 1991, [Third Reference Catalogue of Bright Galaxies. Springer, New York, NY (USA), 1991]{} , H.-J., [Brinks]{}, E., [Duric]{}, N., [Klein]{}, U., & [Skillman]{}, E. 1993, , 410, 626 , P., & [Blundell]{}, K. M. 2012, , 421, 108 , G. 1988, , 330, 672 , F., & [Hobson]{}, M. P. 2008, , 384, 449 , F., [Hobson]{}, M. P., & [Bridges]{}, M. 2009, , 398, 1601 , F., [Hobson]{}, M. P., [Cameron]{}, E., & [Pettitt]{}, A. N. 2013, ArXiv e-prints, arXiv:1306.2144 , M. R., [Wright]{}, A. E., [Burke]{}, B. F., & [Ekers]{}, R. D. 1994, , 90, 179 , C. A., [Murphy]{}, T., [Curran]{}, J. R., [Middelberg]{}, E., [Gaensler]{}, B. M. & [Norris]{}, R. P. 2012, , 425, 979 , V., [Beck]{}, R., [Krause]{}, M., & [Dettmar]{}, R.-J. 2009, , 494, 563 —. 2011, , 535, A79 , V., [Krause]{}, M., [Beck]{}, R., & [Dettmar]{}, R.-J. 2009, , 506, 1123 , C. G., [Walterbos]{}, R. A. M., & [Greenwalt]{}, B. E. 1996, , 112, 1429 , C. G., [Heckman]{}, T. M., [Strickland]{}, D. K., [et al.]{} 2005, , 619, L99 , E. 1991, , 251, 442 , E., [Smith]{}, P., & [van der Hulst]{}, J. M. 1984, , 137, 138 , N., [Morgan]{}, J., [Wayth]{}, R. B., [et al.]{} 2014, , 31, e045 , N., [Callingham]{}, J.R., [Hancock]{}, P. J., [et al.]{} 2017, , 464, 1146 , H. T., [Jagannathan]{}, P., [Mooley]{}, K. P., & [Frail]{}, D. A. 2016, ArXiv e-prints 1603.04368, arXiv:1603.04368 , T. H., [Chester]{}, T., [Cutri]{}, R., [Schneider]{}, S. E., & [Huchra]{}, J. P. 2003, , 125, 525 , M., [Seymour]{}, N., [Afonso]{}, J., [et al.]{} 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), 68 , H., [Yamagishi]{}, M., [Suzuki]{}, T., & [Onaka]{}, T. 2009, , 698, L125 , I. D., [Grebel]{}, E. K., [Sharina]{}, M. E., [et al.]{} 2003, , 404, 93 , Jr., R. C., [Armus]{}, L., [Bendo]{}, G., [et al.]{} 2003, , 115, 928 , A. A., [Chomiuk]{}, L., [Johnson]{}, K. E., [et al.]{} 2011, , 739, L24 , E., [Hora]{}, J. L., [Fazio]{}, G. G., [Hoffmann]{}, W., & [Deutsch]{}, L. 1999, , 518, 183 , U., [Urbanik]{}, M., [Beck]{}, R., & [Wielebinski]{}, R. 1983, , 127, 177 , H., [Witzel]{}, A., [Pauliny-Toth]{}, I. I. K., & [Nauber]{}, U. 1981, , 45, 367 , B. C., & [Thompson]{}, T. A. 2013, , 762, 29 , R. A., & [Peacock]{}, J. A. 1980, , 190, 903 , W. M., [Cotton]{}, W. D., [van Velzen]{}, S., [et al.]{} 2014, , 440, 327 , M. I., [Cram]{}, L. E., & [Burgess]{}, A. M. 1991, The Observatory, 111, 72 , M. I., [Mills]{}, B. Y., [Little]{}, A. G., [Crawford]{}, D. F., & [Sutton]{}, J. M. 1981, , 194, 693 , A., & [Valentijn]{}, E. A. 1989, [The surface photometry catalogue of the ESO-Uppsala galaxies]{} , E., & [Tingay]{}, S. J. 2006, , 132, 1333 , K. 1944, Quarterly of Applied Mathematics, 2, 164 , D. M., [Carignan]{}, C., [Elson]{}, E. C., [et al.]{} 2015, , 450, 3935 , D. 1963, Journal of the Society for Industrial and Applied Mathematics, 11, 431 , J., [Owen]{}, F., & [Eilek]{}, J. 2015, , 149, 32 , T., [Murphy]{}, T., [Buttery]{}, H. J., [et al.]{} 2003, , 342, 1117 , A. R., [Muxlow]{}, T. W. B., [Wills]{}, K. A., [Pedlar]{}, A., & [Beswick]{}, R. J. 2002, , 334, 912 , J. P., [Waters]{}, B., [Schiebel]{}, D., [Young]{}, W., & [Golap]{}, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. [Shaw]{}, F. [Hill]{}, & D. J. [Bell]{}, 127 , G. R., [Hanish]{}, D. J., [Ferguson]{}, H. C., [et al.]{} 2006, , 165, 307 , D. A., [Greenhill]{}, L. J., [Wayth]{}, R. B., [et al.]{} 2008, IEEE Journal of Selected Topics in Signal Processing, 2, 707 , M., [Stensitzki]{}, T., [Allen]{}, & D. B., [Ingargiola]{}, A. 2014, Zenodo, doi:10.5281/zenodo.11813 , S., [Klein]{}, U., & [Wielebinski]{}, R. 1997, , 322, 19 , A. R., [de Bruyn]{}, A. G., [Biehl]{}, M., [et al.]{} 2010, , 405, 155 , A. R., [McKinley]{}, B., [Hurley-Walker]{}, N., [et al.]{} 2014, , 444, 606 , A. R., [Wayth]{}, R. B., [Hurley-Walker]{}, N., [et al.]{} 2015, , 32, 8 , A. R., [Trott]{}, C. M., [Hurley-Walker]{}, N., [et al.]{} 2016, , 458, 1057 , L. 1961, Reviews of Modern Physics, 33, 525 , W. D. 1980, , 239, 54 , W., [Vogler]{}, A., [Klein]{}, U., & [Zinnecker]{}, H. 2000, , 360, 24 , J. C., [Hazelton]{}, B. J., [Beardsley]{}, A. P., [et al.]{} 2016, , 819, 8 , H., [Morgan]{}, J. S., [Lenc]{}, E., & [Tingay]{}, S. J. 2014, , 147, 5 , J. E., & [Harnett]{}, J. I. 1983, Proceedings of the Astronomical Society of Australia, 5, 235 , G. H., [Lebofsky]{}, M. J., [Thompson]{}, R. I., [Low]{}, F. J., & [Tokunaga]{}, A. T. 1980, , 238, 24 , R. J., [Teuben]{}, P. J., & [Wright]{}, M. C. H. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. [Shaw]{}, H. E. [Payne]{}, & J. J. E. [Hayes]{}, 433 , O. B., & [Higgins]{}, C. S. 1973, Australian Journal of Physics Astrophysical Supplement, 27 , V. I. 1990, Soviet Astronomy Letters, 16, 339 , L. 1978, [Physical processes in the interstellar medium]{}, doi:10.1002/9783527617722 , D. K., [Heckman]{}, T. M., [Weaver]{}, K. A., [Hoopes]{}, C. G., & [Dahlem]{}, M. 2002, , 568, 689 , A., [O’Sullivan]{}, J., [Lenc]{}, E., [et al.]{} 2015, Radio Science, 50, 52 , S. J. 2004, , 127, 10 , S. J., & [de Kool]{}, M. 2003, , 126, 723 , S. J., [Goeke]{}, R., [Bowman]{}, J. D., [et al.]{} 2013, , 30, 7 , A. J., [Pugh]{}, J. F., [Kenderdine]{}, S., & [Pauliny-Toth]{}, I. I. K. 1962, , 124, 297 , J. S. 2000, , 120, 278 , J. S., & [Antonucci]{}, R. R. J. 1997, , 488, 621 , J. V., [Wright]{}, A. E., & [Bolton]{}, J. G. 1976, Australian Journal of Physics Astrophysical Supplement, 39, 1 , W. H., [Kleinmann]{}, S. G., & [Ricker]{}, G. R. 1988, , 95, 1057 , R. B., [Lenc]{}, E., [Bell]{}, M. E., [et al.]{} 2015, , 32, 25 , M. S., [Smith]{}, L. J., & [Gallagher]{}, III, J. S. 2011, , 414, 3719 , P. K. G., & [Bower]{}, G. C. 2010, , 710, 1462 , B. J. 1975, Australian Journal of Physics Astrophysical Supplement, 38, 1 \[lastpage\] [^1]: Radio spectral index $\alpha$ is defined such that the flux density $S_\nu$ at a frequency $\nu$ is $S_\nu \propto \nu^{-\alpha}$. [^2]: http://tgssadr.strw.leidenuniv.nl/doku.php [^3]: http://www.python.org
=-5mm =-6mm =-1mm [SELF-DUAL SPIN-3 AND 4 THEORIES]{}\ \[7mm\] C. Aragone\ Departamento de Física, Universidad Simón Bolívar,\ Apartado 8900, Caracas 1080A, Venezuela\ \[4mm\] and\ \[4mm\] A. Khoudeir [^1]\ International Centre For Theoretical Physics, Trieste, Italy\ \[4.7cm\] ABSTRACT We present self-dual spin-3 and 4 actions using relevant Dreibein fields. Since these actions start with a Chern-Simons like kinetic term (and therefore) cannot be obtained through dimensional reduction) one might wonder whether they need the presence of auxiliary ghost-killings fields. It turns out that they must contain, also in this three dimensional case, auxiliary fields. Auxiliary scalars do not break self-duality: their free actions does not contain kinetic terms. Self-dual theories for odd dimensions were discovered time ago by Townsend, Pilch and van Nieuwenhuizen \[1\]. For abelian vector theories, they can be shown to be classically and quantum mechanically equivalent \[2\] to the Maxwell-Chern-Simons (MCS) \[3\] model, if one permits a non minimal coupling in the self-dual model while keeps the minimal one for the gauge invariant second order MCS theory. Or one can assume minimal coupling in both cases and then, although both models propagates one massive-spin 1 mode these theories will not be equivalent. Spin-2 presents a new feature: there are three topological spin-2 theories: linearized topological massive gravity \[4\], a second order Einstein-CS action \[5\] and the first order self-dual one \[6\]. In the vector case the topological massive action is second order, whereas the self-dual one is first order. Spin-two fields presents a new feature: exact topological massive gravity \[4\] is a third order action while self-dual gravity \[5\] is, by definition, first order. Self-dual gravity is a good example of the relevance of the Dreibein representation \[7\] for higher spin gauge fields: its more compact form is obtained when the spin two field is represented by the (linearized) unsymmetrized second rank tensor $w_{pa}$ where $p$ is the gauge index and $a$ is the flat remanent of a Lorentz index. Its gauge variation is given by $\delta w_{pa}=\partial_p\xi_a$. When dealing with higher spin particles ($s\geq 3$) one is always concerned with whether they can have consistent interactions with either other basic elementary systems or (at least) with themselves. Along this direction, recently it has been shown the existence of higher-spin self interacting bosonic theories \[14\]. These theories are third order in the basic fields, their structure is very similar to metric topological Chern-Simons gravity \[4\]. In $d=4$, bosons obey second order field equations. Precisely due to this fact, coupling them to abelian vectors (when charged) or to gravity (which is always mandatory because of the universality of gravity) leads to consider a wide variety of different types of non minimal coupling, once the canonical ones are shown not to work, as it is in general the case. The natural solution to this problem comes from charged string theory models which consistently contain in their spectrum all spins \[15\]. In dimension 3 we have the peculiarity of the existence of these first order, Dirac-like, bosonic self-dual theories for spin 1 and 2. It seems to us worthwhile to construct flat models for spin 3 and 4 in order to investigate whether they can be consistently coupled to abelian vectors or to gravity. Here we report about the precise, Dirac like, self-dual actions we found for spin 3 and 4. We want to mention an additional (more technical) problem. Massive spin-3 in dimensions $d\geq 4$ cannot avoid the presence of auxiliary fields as it is clearly shown by dimensional reduction from its massless, gauge invariant $d+1$ dimensional spin-3 ascendent action \[8\]. In $d=3$ it is hard to imagine what might be the 4-dimensional ascendent of a three dimensional self-dual action (whose kinetic term is essentially given by $\sim w_{(3)}\epsilon \partial w_{(3)}$). Therefore, one might ask again whether self-dual pure spin-3 (or higher) needs the presence of auxiliary fields. Even if self-dual spin-3 would not have needed auxiliary fields one should ask what is the fate of spin-4 since the real high spin field is spin-4. This is due to the fact, if one works in the symmetric representation where $w_{(4)}$ is the basic 4-index symmetric tensor which carries the physical massless excitations, $w_{(4)}$ has to be double traceless \[9\], i.e., $w\equiv w_{pprr}=0$. This condition is uniformly obeyed by any spins-s grater than 4, v.e. $w_{pprr\ell_1\cdots \ell_{s-4}}=0$. In the following we will show that both self-dual spin-3 and 4 actions require the presence of self-dual auxiliary fields of spin-1 and 0 for the former and spin-2 and 1 for the latter. The symmetric formulation of massless spin-3 in $d\geq 3$ was given in \[9\]. The first order Vierbein formulation was presented by Vasiliev \[7\] and a second order action was introduced in \[10\]. The associated massive spin-3 models are discussed in \[8\]. In three dimensions there exist additional possibilities, (at the abelian level) which perhaps, taking into account the analysis performed in \[5\] for the spin-2 case, will be 3: the topological massive third order formulation discovered by Damour and Deser \[10\], the first order self-dual action which is presented here and the intermediate second order action equivalent to these two similar to the spin-2 intermediate \[12\]. Since spin-3 is simpler we treat if first. Self-dual spin-3 action is the addition of three layers: $$S=S_3+S_{31}+S_{10}$$ were $$\begin{aligned} S_{3} &\equiv & 2^{-1}\mu <w_{p\bar{a}_1\bar{a}_2}\varepsilon^{pmn} \partial_m w_{n\bar{a}_1\bar{a}_2}>-6^{-1}\mu^2<\varepsilon^{pmn} \varepsilon^{abc}\eta_{pa}w_{m\bar{b}\bar{d}}w_{n\bar{c}\bar{d}}>,\\ S_{31} &\equiv & \mu^2<w_p u_p>+2^{-1}\alpha\mu <u_p\varepsilon^{pmn} \partial_m u_n>+2^{-1}\beta\mu^2<u_pu_p>,\\ S_{10} &\equiv & \mu<\phi\partial_pu_p>+2^{-1}\gamma <\phi \square \phi > +2^{-1}\delta\mu^2<\phi^2>.\end{aligned}$$ In three dimensions $[\phi ]=m^{1/2}=[w]=[u]$. The basic field $w_{p\bar{a}_1\bar{a}_2}$ is symmetric and traceless in its Dreibein Lorentz indices $w_{p\bar{a}_1\bar{a}_2}=w_{p\bar{a}_2\bar{a}_1},w_{p\bar{a}\bar{a}}=0$ while $p$ is a world index, unrelated to them. (In the following, a set of barred indices will indicate that the associated tensor is symmetric and traceless in this set.) The algebraically irreducible descomposition of $w_{p\bar{a_1}\bar{a_2}}$ is $$w_{p\bar{a}_1\bar{a}_2}=w_{\bar{p}\bar{a}_2\bar{a}_1}+ \varepsilon_{pa_1b}h_{\bar{b}\bar{a}_2}+\varepsilon_{pa_2b} h_{\bar{b}\bar{a}_1}+b(\eta_{pa_1}w_{a_2}\eta_{pa_2}w_{a_1}-2(3)^{-1} \eta_{a_1a_2}w_p). \eqno (5a)$$ The 15 independent components of $w_{p\bar{a}_1\bar{a}_2}$ are represented by the 7 components of $w_{p\bar{a}_1\bar{a}_2}$ plus the 5 needed to describe $h_{\bar{b}\bar{c}}$ plus the last 3 which determine $w_p\equiv w_{r\bar{r}\bar{p}}$, the unique nonvanishing trace of $w_{p\bar{a}_1\bar{a}_2}$. Taking the trace in Eq.(5a) one obtains $b=3/10$ and calculating the symmetric part of $\epsilon_b{}^{pa}w_{p\bar{a}\bar{a}}$ one is led to determine $h_{\bar{b}\bar{c}}$: $$h=h_{\bar{b}\bar{c}}=-6^{-1}(\varepsilon_b{}^{pa}w_{p\bar{a}\bar{c}}+ \varepsilon_c{}^{pa}w_{p\bar{a}\bar{b}}).\eqno (5b)$$ The first interesting fact is that $S_3$ has the good spin-3 and spin-2 behaviour. The associated field equations $E^{p\bar{a}_1\bar{a}_2}\equiv \delta S^3/\delta w_{p\bar{a}_1\bar{a}_2}=0$ propagate one parity sensitive spin-3 excitation, do not propagate neither the other possible spin-3 variable nor any spin-2 degree of freedom (those contained in $h^T_{\bar{a}\bar{b}}$, the transverse part of $h_{\bar{a}\bar{b}}:\partial_{\bar{a}}h^T_{\bar{a}\bar{b}}=0$. However, $S_3$ has spin-1 ghosts and this is the reason one has to add a second layer which will fix this situation. $S_{31}$ is a pure self-dual vector action for the auxiliary vector $u_p$ plus the simplest, contact term $\sim <w\ \ u_p>$. In general one might also consider terms $\sim \mu <w_p\epsilon^{pmn}\partial_mu_n>$ but we have been lucky and there is no need to include them. Addition of these two layers leads to $S_3+S_{31}$ whose field equations are $$\begin{aligned} E^{p\bar{a}_1\bar{a}_2} &\equiv & \varepsilon^{pmn}\partial_m w_{n\bar{a}_1\bar{a}_2}+6^{-1}\mu (\eta_{pa_1}w_{a_2}+\eta_{pa_2}w_{a_1}- w_{a_1\bar{p}\bar{a}_2}w_{a_2\bar{p}\bar{a}_1})\nonumber \\ & & +2^{-1}\mu (\eta_{pa_1}u_{a_2}+\eta_{pa_2}u_{a_1}-2(3)^{-1}\eta_{a_1a_2} u_p)=0,\\ F^p &\equiv & \alpha\varepsilon^{pmn}\partial_mu_n +\beta\mu u_p +\mu w_p=0.\end{aligned}$$ These two equations can be analyzed by further breaking of the algebraic decomposition (5a) in terms of its $SL(2,R)$ irreducible representations. We introduce the three dimensional covariant (and non local) $T$-projectors which, in the vector case, are $$u_p=u^T_p+\wh{\partial}_pu^L,\ \ \ \wh{\partial}_p\equiv \square^{-1/2} \partial_p,$$ $$\wh{\partial}_pu^T_p=0,\ \ \ \wh{\partial}_p\cdot \wh{\partial}_p=1. \eqno (8a)$$ For spin-2 and 3, similar decompositions for symmetric traceless second and third rank tensors have the form: $$h_{\bar{p}\bar{a}}=h^T_{\bar{p}\bar{a}}+ \wh{\partial}_{(\bar{p}}h^L_{\bar{a})},\ \ \ \wh{\partial}_ph^T_{\bar{p}\bar{a}}=0=h^T_{\bar{p}\bar{p}},\eqno (8b)$$ $$w_{\bar{p}\bar{a}\bar{b}}=w^T_{\bar{p}\bar{a}\bar{b}}+ \wh{\partial}_{(p}w^L_{\bar{a}\bar{b})},\ \ \ \wh{\partial}_pw^T_{\bar{p}\bar{a}\bar{b}}=0=w^T_{\bar{p}\bar{p}\bar{b}}. \eqno (8c)$$ Symmetric traceless transverse $3d$ tensors $(u_p^T,h^T_{\bar{p}\bar{a}},w^T_{\bar{p}\bar{a}\bar{b}}w^T_{\bar{p}\bar{a} \bar{b}\bar{c}})$ have two independent components corresponding to the two $P$-sensitive pseudospin-$j(j=1,2,3,4)$ excitation they can propagate. A final covariant spliting of these set (symmetric, traceless, transverse) tensors is obtained by means of the pure pseudospin-$j$ projectors $p_j^\pm w^T_{\bar{p}\bar{a}\bar{b}\cdots \bar{c}}$ \[6\] $$p^\pm_jw^T_{\bar{p}\bar{a}\bar{b}\cdots \bar{c}}\equiv w^{T^\pm}_{\bar{p}\bar{a}\bar{b}\cdots \bar{c}}=2^{-1} w^T_{\bar{p}\bar{a}\bar{b}\cdots \bar{c}}\pm\frac{1}{2j} \varepsilon_{(p}{}^{mn}\wh{\partial}_m w_{\bar{n}\bar{a}\bar{b}\cdots \bar{c})},$$ where the indicated symmetrization is the minimal one and does not carry a normalization coefficient. It is straightforward to check that $$p^+_j+p_j^-={\bf 1},\ \ p_j^+-p_j^-=\frac{1}{j} \varepsilon (.\ddot{\ }\wh{\partial}\cdots ).$$ Armed with these projectors one can analyse the behaviour of $E^{\bar{p}\bar{a}\bar{b}T}$, the spin-3 sector of Eq.(6). It turns out that $E^{\bar{p}\bar{a}\bar{b}T}$ propagates the spin-3$^+$ part of $w^T_{\bar{p}\bar{a}\bar{b}}$ and annihilates $w^{T-}_{\bar{p}\bar{a}\bar{b}}$. Then ones goes to the spin-2 sector and it is immediate to verify that $\partial_p E^{p\bar{a}b},\check{E}^{\bar{b}\bar{c}}\equiv \varepsilon_{(bpa}E^{p\bar{a}}{}_{\bar{c})}$ do not allow the propagation of $h^{T^\pm}_{\bar{a}\bar{b}}$. The spin-1 dynamical behaviour is determined by $\partial_{pa}E^{p\bar{a}\bar{b}}$, $\partial_b\check{E}^{\bar{b}\bar{a}},E^b\equiv E^{p\bar{p}\bar{b}}$ and $F^p$. In order not to have any spin-1 excitation alive we must choose $$\alpha = \beta =-18.$$ Unfortunately this is not the last step in order to get a pure pseudospin-3$^+$ propagation. $S_3+S_{31}$ has scalar ghosts and therefore they have to be destroyed by an auxiliary scalar $\phi$. This is the reason of having to add to the first two layers $S_3+S_{31}$ the last one, $S_{10}$ defined in Eq.(4). In principle one should have to consider the posibility of kinetic terms like $\sim \phi \square \phi$ which are the second order and therefore would break the full system self-duality. The fields equations derived from $S$ are $$\begin{aligned} \delta_w S &\sim & E^{p\bar{a}_1\bar{a}_2} = 0 \\ \delta_{u} S &\sim & 'F^p \equiv F^p -\partial_p \phi = 0 , \\ \delta_\phi S &\sim & G\equiv \gamma \square \phi +\delta \mu^2\phi + \mu\partial_p u_p = 0.\end{aligned}$$ There are five scalar excitations which the system might propagate $\wh{\partial}_{pab}w_{\bar{p}\bar{a}\bar{b}},\wh{\partial}_{ab} h_{\bar{a}\bar{b}}$, $\wh{\partial}_pw_p$, $\wh{\partial}_p u_p$, $\phi$. However, since $\partial_p E^{p\bar{a}\bar{b}}$ and $\check{E}^{\bar{b}\bar{c}}$ tells us that $$\mu h_{\bar{b}\bar{c}}=-3(\partial_bu_c+\partial_cu_b-2(3)^{-1}\eta_{ab} (\partial \cdot u)),\eqno (15a)$$ $$\partial_bw_c+\partial_cw_b-(\partial_pw_{b\bar{p}\bar{c}}+\partial_p w_{c\bar{p}\bar{b}})+3(\partial_bu_c+\partial_cu_b-2(3)^{-1}\eta_{bc} (\partial \cdot u))=0,\eqno (15b)$$ it is immediate that, if neither $\wh{\partial}_p u_p$ nor $\wh{\partial}_p w_p$ propagate (i.e., $\wh{\partial}_p u_p = 0 = \wh{\partial}_p w_p)\wh{\partial}_{pab}w_{\bar{p}\bar{a}\bar{b}}$ and $\wh{\partial}_{pa}h_{\bar{p}\bar{a}}$ will not propagate either. The key equations are the vanishing of $\partial_b E^{p\bar{p}\bar{b}}$, $\partial_p `F^p$ and $G$ where in the first one, makes use of Eqs.(5a) and (15). They can be written, respectively $$(12 \square +5(8)^{-1}\mu^2)\partial \cdot u +2^{-1}\mu^2\partial \cdot w=0, \eqno (16a)$$ $$\mu\beta\partial \cdot u+\mu\partial \cdot w -\square \phi =0, \eqno (16b)$$ $$\mu\partial \cdot u +(\gamma \square +\delta \mu^2)\phi=0. \eqno (16c)$$ Introducing the dimensionless operator $x\equiv \mu^{-1}\square^{1/2}$ it is straightforward to see that the inverse propagator of $\wh{\partial}\cdot w,\wh{\partial}\cdot u, \phi$ is $$\Delta (x)\equiv -(\gamma x^2 +\delta )(12x^2+5(8)^{-1})+ 2^{-1}x^2+2^{-1}\beta (\gamma x^2+\delta ).$$ These scalar variables (and consequently $\wh{\partial}_{pab}w_{\bar{p}\bar{a}\bar{b}},\wh{\partial}_{pa} h_{\bar{p}\bar{a}}$) do not propagate if the polynomial $\Delta (x)$ becomes zero order, i.e., $\Delta (x)\equiv \Delta_4\cdot x^0=\Delta_4\cdot 1$. This condition uniquely determines $\gamma ,\delta$ $$\gamma =0,\ \ \ \delta =(24)^{-1}.$$ Note that the vanishing of $\gamma$ makes action $S_{10}$ first order (scalars appear of the self-dual type too), leading to the final $S$ being fully first order. Observe that we do not claim mathematical uniqueness for a pure spin-3$^+$ (or 3$^+$) $3d$ action: in the scalar sector one could have consider coupling terms like $\sim\phi (\partial \cdot w)$. However, it seems to us that, if one starts with the right-spin Dreibein seed (in the case $S_3$), then $S_{31}$ is unique if we demand that it must be the vector self-dual action coupled in the softest possible ways to $S_3$ (the coupling term must be, at most, first order and if possible algebraic). The construction of the auxiliary scalar action $S_{10}$ again is unique: it contains the free self-dual scalar action ($\sim \mu^2\phi^2$, no Klein-Gordon kinetic term) and it is next-neighbour coupled to the auxiliary spin-1 field, discarding $\phi (\partial \cdot w)$ which is not of the next-neighbour type. All these results will be useful when dealing with the much complex case of spin-4. We start this analysis by introducing the spin-4 part of the final action $S_{42}$ with the right physical behaviour up to the spin-2 sector. It reads $$\begin{aligned} S_{42} &\equiv & (2)^{-1}\mu <w_{p\bar{a}\bar{b}\bar{c}}\varepsilon^{pmn} \partial_m w_{n\bar{a}\bar{b}\bar{c}}>-2^{-1}\mu^2 <\varepsilon^{pmn} \varepsilon^{abc}\eta_{pa}w_{m\bar{b}\bar{d}_1\bar{d}_2} w_{n\bar{c}\bar{d}_1\bar{d}_2}>\nonumber \\ [3mm] & & +\mu^2<w_{p\bar{p}\bar{a}\bar{b}}u_{ab}>+(2)^{-1}\alpha\mu <u_{pa} \varepsilon^{pmn}\partial_m u_{na}>+2^{-1}\beta\mu^2<\varepsilon^{pmn} \varepsilon^{abc}\eta_{pa}u_{mb}u_{nc}>,\end{aligned}$$ where $w_{p\bar{a}\bar{b}\bar{c}}$ is symmetric and traceless (ST) in its three last barred indices and $u_{pa}$ is an auxiliary self-dual second rank tensor, $[w]=[u]=m^{1/2}$. Their algebraically irreducible representations are, respectively $$w_{p\bar{a}\bar{b}\bar{c}}=w_{\bar{p}\bar{a}\bar{b}\bar{c}}+ \varepsilon_{p(ad}h_{d\bar{b}\bar{c})}+5(21)^{-1}\eta_{p(a}w_{\bar{b}\bar{c})} -2(21)^{-1}w_{p(\bar{a}}\eta_{bc)},$$ $$u_{pa}=u_{\bar{p}\bar{a}}+\varepsilon_{pad}h_d+3^{-1}\eta_{pa}u,\ \ h_d=-2^{-1}\varepsilon_d{}^{pa}u_{pa},\eqno (21a,b)$$ where $w_{p\bar{p}\bar{b}\bar{c}}\equiv w_{\bar{b}\bar{c}}$ and $u_{pp}$ are the unique non-vanishing contractions which can be made out of $w_{p\bar{a}\bar{b}\bar{c}}$ and $u_{pa}$, respectively. Symmetrizations are minimal with coefficient one in front and sets of barred indices continue to indicate ST tensors. Variations with respect the $w_{p\bar{a}\bar{b}\bar{c}}$ and $u_{pa}$ yield the initial set of field equations $$\begin{aligned} E_{p\bar{a}\bar{b}\bar{c}} & \equiv & \varepsilon_p{}^{mn}\partial_m w_{n\bar{a}\bar{b}\bar{c}}+\mu (3)^{-1}\{\eta_{p(a}w_{\bar{b}\bar{c})}- w_{(a\bar{b}\bar{c})\bar{p}}\}\nonumber\\ & & +\mu(3)^{-1}\{\eta_{p(a}u_{\bar{b}\bar{c})}-2(5)^{-1} \eta_{(ab}u_{\bar{c}\bar{p})}\}=0,\\ F_{pa}&\equiv &\mu w_{\bar{p}\bar{a}}+\alpha \varepsilon_p{}^{mn}\partial_m u_{na}+\mu\beta \varepsilon_p{}^{mn}\varepsilon_a{}^{bc}\eta_{nc}u_{mb}=0.\end{aligned}$$ The spin-4$^\pm$ excitations are carried on the transverse part of $w_{\bar{p}\bar{a}\bar{b}\bar{c}}:w^T_{\bar{p}\bar{a}\bar{b}\bar{c}}, \partial_p w^T_{\bar{p}\bar{a}\bar{b}\bar{c}}=0$ while there are two sets of spin-3 variables: those contained in $\wh{\partial}_p w_{\bar{p}\bar{a}\bar{b}\bar{c}}$ and those defined by $h^T_{\bar{a}\bar{b}\bar{c}}$. Use of the spin-4$^\pm$projectors defined in Eqs.(9) and (10) show that $E_{p\bar{a}\bar{b}\bar{c}}$ uniquely propagate spin-4$^+$ (make the spin-4$^-$ degree of freedom to cancel) and does not propagate neither ($\wh{\partial}_pw_{\bar{p}\bar{a}\bar{b}\bar{c}})^T$ nor $h^T_{\bar{a}\bar{b}\bar{c}}$. In fact, equations $\partial_pE_{p\bar{a}\bar{b}\bar{c}}=0=\varepsilon_{(a}{}^{pd} E_{p\bar{d}\bar{b}\bar{c})}$ are equivalent to $$\begin{aligned} 4\mu h_{\bar{a}\bar{b}\bar{c}} &=& 2(5)^{-1}\eta_{(ab}\partial_p u_{\bar{p}\bar{c})}-\partial_{(a}u_{\bar{b}\bar{c})},\\ \partial_{(a}w_{\bar{b}\bar{c})}-\partial_{p}w_{(a\ \bar{p}\bar{b}\bar{c})} &=& 2(5)^{-1}\eta_{(ab}\partial_pu_{\bar{p}\bar{c})}-\partial_{(a}u_{\bar{b}\bar{c})}.\end{aligned}$$ These equations say both $h_{\bar{a}\bar{b}\bar{c}}$ and $\partial_p$ $w_{\bar{p}\bar{a}\bar{b}\bar{c}}$ are curls of spin-2 objects and therefore their pure spin-3 parts have to vanish. Four variables describe the spin-2 sector of $S_{42}:(\wh{\partial}_{pa}w_{\bar{p}\bar{a}\bar{b}\bar{c}})^T$, $(\wh{\partial}_ph_{\bar{p}\bar{a}\bar{b}})^T$, $w^T_{\bar{p}\bar{a}}$, $u^T_{\bar{p}\bar{a}}$. The equations which determine their dynamical behaviour are $\partial_{pa}E^{p\bar{a}\bar{b}\bar{c}}=0$, $\check{E}^{\bar{a}\bar{b}\bar{c}}=0$, $E_{\bar{b}\bar{c}}\equiv E_{p\bar{p}\bar{b}\bar{c}}=0$ and $F_{pa}=0$. After some algebra one is led to a separated propagation eqaution for $u^T_{\bar{p}\bar{a}}\equiv \omega ,p^\pm \omega \equiv \omega^\pm$ $$\begin{aligned} (x^2 &+& 7(5)^{-1}-4(3)^{-1}\beta )(\omega^++\omega^-)+2(3)^{-1}x(\alpha x+\beta ) \omega^++2(3)^{-1}x(\alpha x -\beta )\omega ^-\nonumber \\ & & -4(3)^{-1}\alpha x(\omega^+-\omega^-)=0.\end{aligned}$$ Projecting on this spin-2$^+$ (2$^-$) subspaces we obtain the two uncoupled equations which determine their evolution $$\{ x^2(1+2(3)^{-1}\alpha )\mp 2(3)^{-1}(2\alpha -\beta )x+(7(5)^{-1}-4(3)^{-1} \beta )\}\omega^\pm =0,$$ (either all upper indices or all down right). Non-propagations of one of these two variables determines the values of $\alpha\beta$: $$\alpha = -3(2)^{-1},\ \ \beta =-3,$$ and, due to Eq.(27), entails the non-propagation of the other companion variable. $S_{42}$ (19) has been uniquely determined requesting its good physical behaviour in its highest spin sector ($s=4,3,2$). However, it contains vector and scalar ghosts. This is the reason why we have to add two additional layers. The most difficult of them is spin-1 fixing action. Its ambiguity stems in the wide range of mathematically consistent terms one might have to consider $ab$ initio. In principle $S_{21}$ may be $$\begin{aligned} S_{21} &\equiv & -2\lambda_1\mu <h_a\partial_bu_{\bar{a}\bar{c}}>+2\lambda_2 \mu <v_p\partial_ru_{\bar{r}\bar{p}}>\nonumber \\ & & \gamma_2\mu <h_a\varepsilon^{abc}\partial_bh_c>+\gamma_1 (2)^{-1}\mu <v_p \varepsilon^{pmn}\partial_mv_n>\nonumber \\ & & +\rho \mu^2<h^2_a>+\delta (2)^{-1}\mu^2<v_a^2>+2\varepsilon \mu^2 <h_pv_p> +2\kappa \mu <h_a \partial_b w_{\bar{b}\bar{a}}>\nonumber \\ & & +2\varphi \mu <v_p\partial_r w_{\bar{r}\bar{p}}>+2\sigma \mu <v_p \varepsilon^{pmn}\partial_m h_n>,\end{aligned}$$ which can be regarded as the addition of the self-dual action for the spin-1 variable $h_a$ contained in $u_{pa}$ plus the auxiliary self-dual action for the auxilary vector $u_p$ algebraically coupled through $\sim h\cdot v$ plus more bizarre terms like $\sim h_a \partial_b u_{\bar{b}\bar{a}}$, $h_a\partial_b w_{\bar{b}\bar{a}}$, $v_a\partial_b u_{\bar{b}\bar{a}}$, $v_a \varepsilon^{abc}\partial_b h_c$ and the exotic term $\sim v_a\partial_b w_{\bar{b}\bar{a}}$. We will not consider them, the first because we already have chosen a good kinetic term for $u_{pa}$ ($u_{pa} \varepsilon^{pmn}\partial_m u_{na}$ as in Eq.(19)), the last one because it is not of the next-neighbour type (it is spin-4$\cdot$spin-1) and second, third and fourth because we have decided to choose, whenever possible, algebraic couplings and we have already a spin-2$\cdot$spin-1 contact term $\sim h.v$. Therefore we rule out the present of terms $\sim v_a \partial_b u_{\bar{b}\bar{a}}v_a\varepsilon^{abc}\partial_bh_c$ as well as the need form a term $\sim h_a \partial_b w_{\bar{b}\bar{a}}$, a different coupling term linking spin-4 with spin-2 for the same reason. In other words we take $\lambda_1=\lambda_2=\kappa =\sigma =\varphi =0$ in $S_{21}$. Taking into account Eq.(21b) we write down in the modified spin-2 field equations which govern this system (note that $E^{\bar{p}\bar{a}\bar{b}\bar{c}}=0$ remains intact). They have the aspect $$`F_{pa}\equiv F_{pa}+\gamma_2(\partial_ph_a-\partial_a h_p)-\rho \varepsilon_{pab}h_b-\varepsilon \varepsilon_{pab} v_b =0.$$ An additional vector-like field equation appears after varying $v_p$, $$G_p\equiv \gamma_1 \varepsilon_p^{mn}\partial_mv_n +\delta\mu v_p+2 \varepsilon\mu h_p =0.$$ We want to determine $\gamma_1,\gamma_2,\rho ,\delta ,\varepsilon$ in such a way that none of the six spin-1 variables: $\omega_8\equiv (\wh{\partial}_{pab}w_{\bar{p}\bar{a}\bar{b}\bar{c}})^T$, $\omega_9\equiv (\wh{\partial}_{pa}h_{\bar{p}\bar{a}\bar{b}})^T$, $\omega_{11}\equiv (\wh{\partial}_p u_{\bar{p}\bar{a}})^T$, $\omega_{11}\equiv h^T_p$, $\omega_{12}\equiv h^T_p$, $\omega_{13}\equiv v^T_p$ can propagate. Since $\omega_8$ is given by $\partial_{pab}E_{p\bar{a}\bar{b}\bar{c}}$ in terms of the five remaining variables $\omega_9\cdots \omega_{13}$ we go after the non propagation of them. They are determined by $\partial_{ab}\check{E}_{\bar{b}\bar{a}\bar{c}}=0$, $\partial_bE_{\bar{b}\bar{c}}=0$, $\partial_p `F_{pa}=0$, $`\check{F}^b=0$ and $G^p=0$ After minor algebra and some use of Eq.(24) the five equations become $$\begin{aligned} & &4\mu\partial_{ab}h_{\bar{a}\bar{b}\bar{c}}+8(5)^{-1}\square \partial_a u_{\bar{a}\bar{c}}+5^{-1}\partial_c(\partial_{ab}u_{\bar{a}\bar{b}})=0,\\[3mm] & &-4\partial_{ab}h_{\bar{a}\bar{b}\bar{c}}-3^{-1}\varepsilon_c{}^{pr}\partial_p (\partial_bw_{\bar{b}\bar{r}}+4(3)^{-1}\mu\partial_pw_{\bar{p}\bar{c}}+ 7(5)^{-1}\mu\partial_p u_{\bar{p}\bar{c}}=0,\\[3mm] & &\mu\partial_pw_{\bar{p}\bar{a}}-3\mu\partial_pu_{\bar{p}\bar{a}}+(\rho -3) \mu\varepsilon_a{}^{pr}\partial_ph_r+2\mu\partial_au+\nonumber \\ & & \ \ \ \ \ +\gamma_2(\square h_a-\partial_a(\partial_ph_p))+\varepsilon\mu\varepsilon_a {}^{pr}\partial_pv_r=0,\\[3mm] & &3(2)^{-1}\partial_pu_{\bar{p}\bar{b}}+2(\rho -3)\mu h_b +2\varepsilon\mu v_b+ (2\gamma_2+2(3)^{-1})\varepsilon_b{}^{pr}\partial_ph_r- \partial_bu=0\end{aligned}$$ and Eq.(31) as it stands. Working in a similar way to what we did for the spin-3 case, the vanishing of $\omega_9\cdots \omega_{13}$ is equivalent to their non propagation and this is reached if $\Delta (x)=\Delta_0x^4+\cdots +\Delta_4\cdot 1$ becomes $\Delta_4 \cdot 1$. Straighforward calculations give $$\begin{aligned} \Delta (x)=&-&3(10)^{-1}\gamma_1(9\gamma_2+8)x^4+\{3(2)^{-1}\gamma_1(1-9(5)^{-1} \rho ')-3(5)^{-1}\delta (9(2)^{-1}\gamma_2+4)\}x^3\nonumber \\ &+&\{-27(5)^{-1}\gamma_1(2\gamma_2+3(2)^{-1})-27(10)^{-1}\delta\rho '+ 3(2)^{-1}\delta +27(5)^{-1}\varepsilon^2\}x^2\nonumber \\ &-&27(5)^{-1}\{\delta (2\gamma_2+3(2)^{-1})+2\gamma_1\rho '\}x +54(5)^{-1} \{2\varepsilon^2-\delta \rho '\}\cdot 1,\end{aligned}$$ where for convenience $\rho '\equiv \rho -3$. Requesting the vanishing of the coefficients $\Delta_{0,1,2,3}$ of the inverse propagator $\Delta_x$ one is lead to $$\begin{aligned} & &\gamma_2=-8(9)^{-1},\ \ \rho '=5(9)^{-1}=\rho -3,(\rho =-4\gamma_2),\nonumber \\ & &\gamma_1=-18(5)^{-1}\varepsilon^2,\ \ \delta =4\gamma_1=-72(5)^{-1} \varepsilon^2.\end{aligned}$$ Redefining 2 $\varepsilon v_p \to v_p$ the final unique form of $S_{21}$ becomes $$S_{21}=-8(9)^{-1}\mu<h_a\varepsilon^{abc}\partial_bh_c>-9(20)^{-1}\mu <v_p\varepsilon^{pmn}\partial_mv_n>$$ $$+32(9)^{-1}\mu^2<h_a^2>-9(5)^{-1}\mu^2<v^2_p>+<h_pv_p>\mu^2.\eqno (29b)$$ The action $S_{42}+S_{21}$ has the right physical properties up to spin-1. However, its scalar sector contains ghost which we have to exorcise by introducing an auxiliary self-dual scalar $\phi$. Its associated action $S_{10}$ constitutes the last layer we need to determine the final pure self-dual spin-4$^+$ action S. The most general scalar auxiliary action one can add to $S_{42}+S_{21}$ is $$\begin{aligned} S_{10}&\equiv &2a_1\mu <\phi\partial_pu_p>+2a_2\mu <\phi\partial_ph_p>+2a_7\mu <u\partial_ph_p>+2a_8\mu <u\partial_pv_p>\nonumber \\ & & a_5\mu^2 <\phi u>+2^{-1}a_3\mu^2<\phi^2>+2^{-1}a_4<\phi \square \phi>\nonumber \\ & & +2^{-1}a_6\mu^2<u^2>+2^{-1}a_9<u\square u>+a_{10}<u\square \phi >.\end{aligned}$$ Taking advantage of what we learned from the spin-3 case, we assume that there will be a final scalar auxiliary action fully self-dual, i.e., that there exists a non trivial $S_{10}$ with vanishing $a_4,a_9$ and $a_{10}$. We also assume a vanishing $a_7$, since this term can be seen as an unpleasant kinetic term to add to the self-dual actions $u_{\bar{p}\bar{a}}\varepsilon^{pmn}\partial_m u_{\bar{n}\bar{a}}$ and $h_p\varepsilon^{pmn}\partial_m h_n$. The final equations are $$E_{p\bar{a}\bar{b}\bar{c}}=0,\eqno (22)$$ $$\begin{aligned} ``F_{pa}&\equiv &`F_{pa}+\mu a_5\eta_{pa}\phi +a_2\varepsilon_{pa} {}^m\partial_m\phi +a_6\mu\eta_{pa}u +2a_8\eta_{pa}(\partial \cdot v)=0,\\[3mm] `G_p&\equiv &G_p-2\alpha_1\mu\partial_p\phi-2a_8\partial_p u=0,\\[3mm] H&\equiv &\delta S_{10}/\delta\phi =2a_1(\partial \cdot v)+2a_2(\partial \cdot h)+a_4\mu u+\mu a_3\phi=0.\end{aligned}$$ The scalar sector has eight independent variables: $$\begin{aligned} \omega_1&\equiv &\wh{\partial}_{pabc}w_{\bar{p}\bar{a}\bar{b}\bar{c}},\ \ \omega_2\equiv \wh{\partial}_{pab}h_{\bar{p}\bar{a}\bar{b}},\ \ \omega_3\equiv \wh{\partial}_{ab}w_{\bar{a}\bar{b}},\nonumber \\ \omega_4 &\equiv &\wh{\partial}_{ab}u_{\bar{a}\bar{b}},\ \ \omega_5\equiv \wh{\partial}_ah_a,\omega_6\equiv \mu u,\nonumber \\ \omega_7&\equiv &\wh{\partial}_av_a,\ \ \omega_8\equiv \mu \phi\end{aligned}$$ whose evolution is determined by $\partial_{pabc}E_{p\bar{a}\bar{b}\bar{c}},\partial_{abc} \check{E}_{\bar{a}\bar{b}\bar{c}},\partial_{bc}E_{\bar{b}\bar{c}}, \partial_{pa}``F_{pa},\partial_{b}``\check{F}_{b},\partial_{p}`G_{p}$ and $H$. The first set of 3 equations is derived from Eq.(22) taking into account the algebraic structure of $w_{p\bar{a}\bar{b}\bar{c}}$ as given in Eq.(20). It turns out to be $$\begin{aligned} & &-5\partial_{pabc}w_{\bar{p}\bar{a}\bar{b}\bar{c}}+5(21)^{-1}\square \partial_{pa}w_{\bar{p}\bar{a}}+3\square \partial_{\bar{p}\bar{a}} u_{\bar{p}\bar{a}}=0,\\ & &4\mu \partial_{pab}h_{\bar{p}\bar{a}\bar{b}}+9(5)^{-1}\mu \partial_{pa}u_{\bar{p}\bar{a}}=0,\\ & &-4\partial_{pab}h_{\bar{p}\bar{a}\bar{b}}+4(3)^{-1}\mu\partial_{pa} w_{\bar{p}\bar{a}}+7(5)^{-1}\mu\partial_{pa}u_{\bar{p}\bar{a}}=0.\end{aligned}$$ The second set comes from Eq.(39). It consists of $$\begin{aligned} & &\partial_{pa}``F_{pa}\equiv \mu\partial_{pa}w_{\bar{p}\bar{a}}-3\mu \partial_{pa}u_{\bar{p}\bar{a}}+\mu (2+a_6)\square u+\nonumber \\ & &+\mu a_5\square \phi +2a_8\square \partial_pv_p=0,\\[3mm] & &\partial_b``\check{F}_b\equiv 3(2)^{-1}\partial_{pa}w_{\bar{p}\bar{a}}+ 10(9)^{-1}\mu\partial_ph_p+\mu\partial_pv_p-\nonumber \\ & &-\square u-2a_2\mu \square \phi =0,\\[3mm] & & ``F_{pp}\equiv \partial_ph_p+(2+a_6)\mu u+a_5\mu\phi +2a_8\partial_p v_p=0.\end{aligned}$$ The last two equations are $$\partial_p `G_p\equiv \delta_\mu\partial_pv_p+\mu \partial_ph_p-2a_1\mu \square \phi-2a_8\square u=0,$$ and Eq.(41) $H=0$. In terms of the $\omega$-variables (42) Eqs.(43)-(45) allow to obtain $\omega_1,\omega_2,\omega_3$ as a function of $\omega_4$. In particular $$\omega_3=-3(20)^{-1}(9x^2+7)\omega_4.$$ Then it is immediate to realize that Eqs.(46)-(49), (41) become a decoupled subset of the full system. It can be written as $$\begin{aligned} & &-27(20)^{-1}(x^2+3)\omega_4+(2+a_6)\omega_6+2a_8x\omega_7+a_5\omega_8=0,\\ & &3(2)^{-1}x\omega_4+10(9)^{-1}\omega_5+\omega_7-x\omega_6-2a_2x\omega_8=0,\\ & &x\omega_5+(2+a_6)\omega_6+2a_8x\omega_7 +a_5\omega_8=0,\\ & &\omega_5-2a_8x\omega_6+\delta\omega_7-2a_1x\omega_8=0,\\ & &2a_2x\omega_5+a_5\omega_6+2a_1x\omega_7+a_3\omega_8=0.\end{aligned}$$ The inverse of this determinat $\Delta_{(a_1,a_2,a_3,a_5,a_6,a_8)}$ is the system’s propagator. We wish to determine the $a_1\cdots a_8$ coefficients in such a way that $\Delta (x)$ is a non-vanishing real number. First we investigate the possibility of having a solution with pure next-neighbours coupling terms, i.e., where $a_2=0=a_5$ (they are spin-2-spin-0 couplings). In this case $$\begin{aligned} & &\Delta (a_2=0=a_5)=-27(20)^{-1}x^2(x^2+3)(4a^2_1x^2+a_3(\delta -2a_8)) \nonumber \\ & &-18x^2(a^2_1a_6'+a_3a_8^2)-9(2)^{-1}\delta a_3a_6',\end{aligned}$$ where $a_6'\equiv 2+a_6$. Vanishing of its highest power coefficient leads to $$a_1=0,\eqno (57a)$$ and subsequent cancellation of quartic and quadratic terms impose $$a_3=0, \eqno (57b)$$ which seem an inconsistent possibility, since in this case $\Delta (56)$ becomes identically zero. However, since we are now thinking of not having $\phi$-dependent actions $(a_1=a_2=a_3=a_5=0)$ we have to consider the appropiate system of field equations which consists of Eqs.(22),(39) and (40) for these values of $a_{1,2,3,5}$ and does no longer contain Eq.(41). Its crucial decoupled part consists of Eqs.(51)-(54) $(a_1=a_2=a_3=a_5=0)$ and the non propagating character is determined by imposing to its associated (quartic) determinant to be a non zero real number. This leads us to determine $a_6$ and $a_8$ $$a_6=(5)^{-1}44,\ \ \ a_8=-9(10)^{-1}.$$ $S_{10}$ attains a very simple form $$S_{10}=-9(5)^{-1}\mu <u\partial_p v_p>+22(5)^{-1}\mu^2<u^2>,$$ where there is no auxiliary scalar field present This is the minimal solution. If one relaxes a little bit the assumption of considering only next-neighbours coupling and investigate the consequence of only imposing $a_2=0$ (leaving room for an algebraic non-next-neighbour spin-2-spin-0 coupling) we are led to $a_1=a_3=0,a_6,a_8$ arbitraries and $a_5$ arbitrary non-vanishing. Similarly, one might constraint $a_5$ to vanish and try to determine $a_2$. In this case one obtains (after redefining $\phi \to a_2\phi )$ $$\begin{aligned} & & a_1=2^{-1}\delta ,a_2=1, a_3=20\delta^2(2+a_6)(6a_6+12-5\delta^2)^{-1}, \nonumber \\ & &a_6\neq 44(5)^{-1},a_8=4^{-1}\delta ,\end{aligned}$$ and the corresponding full action is a pure spin-4$^+$ action too. It is worth observing that simplest, self-dual, next-neighbour coupled pure spin-4$^+$ is then given by: $$S=S_{42}(19)+S_{21}(29a)+S_{10}(59)$$ and contains only one auxiliary self-dual spin-2, $u_{ra}$, and one (self-dual) vector auxiliary field $v_r$, in addition to the fundamental physical spin-4 carrier $w_{r\bar{a}\bar{b}\bar{c}}$. In conclusion we have been able to uniquely construct self-dual spin-3 and 4 actions where auxiliary fields also appear in a self-dual form (including scalars) and where coupling terms are next-neighbours. In both cases we needed one self-dual auxiliary filed of spin s-2, s-3, up to spin-1. Since spin-4 clearly is the higher-spin case we may conjecture that this self-dual picture exists for arbitrary integer spin, where the unique non uniform structure is the final layer fixing the good spin-0 behaviour. An additional interesting question is what should be the higher spin structure of topologically massive theories. We are inclined to think that all of them will be of third-order, as it is the case for gravity and spin-3. It would also be interesting to see what is the connection between the present self-dual spin-3, and 4 formulations and the recently proposed \[13\] anyonic relativistic actions for spin-$j$ real, since this scheme consistently contains the self-dual abelian vector case. However, as we mentioned in the beginning, whether this Dirac-like bosonic structures can be consistently coupled either to abelian vectors or to gravity is a worthwhile question which deserves further analysis. [Acknowledgments]{} One of the authors (A.K.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the international Centre for Theoretical Physics, Trieste. REFERENCES 1. P.K. Townsend, K. Pilch and P. van Nieuwenhuizen, Phys. Lett. [136B]{} (1984) 38. 2. S. Deser and R. Jackiw, Phys. Lett. [139B]{} (1984) 371. 3. W. Siegel, Nucl. Phys. Lett. [B156]{} (1979) 135; R. Jackiw and S. Templeton, Phys. Rev. [D2]{} (1981) 2291; J. Schönfeld, Nucl. Phys. [B185]{} (1981) 157. 4. S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. [48]{} (1982) 975; Ann. of Phys. (New York) [140]{} (1982) 372. 5. C. Aragone, P.J. Arias and A. Khoudeir, Proceedings Silarg 7th, edited by M. Rosenbaum et al, World Scientific (1991) p.437. 6. C. Aragone and A. Khoudeir, Phys. Lett. [173B]{} (1986) 141; [Quantum Mechanics of Fundamental Systems]{}, ed. C. Teitlboin, (Plenum Press, New York, 1988). p.17. 7. C. Aragone and S. Deser, Nucl. Phys. [B270]{}, \[FS1\] (1980) 329; M.A. Vasiliev, Sov. J. Nucl. Phys. [32]{} (1980) 439. 8. C. Aragone, S. Deser and Z. Yang, Ann. of Phys. [179]{} (1987) 76. 9. C. Fronsdal, Phys. Rev. [D18]{} (1978) 3624. 10. C. Aragone and H. La Roche, Nuovo Cimento [A72]{} (1982) 149. 11. T. Damour and S. Deser, Ann. Inst. Henri Poicaré [47]{} (1987) 277. 12. C. Aragone and A. Khoudeir, in preparation. 13. R. Jackiw and V. Nair, Phys. Rev. [D43]{} (1991) 1942. 14. E.S. Fradkin and V. Y. Linetsky Ann. of Phys. [198]{} (1990) 293. 15. P.C. Argyres and C.R. Nappi, Phys. Lett. [B224]{} (1989) 89. [^1]: Permanent address: Departamento de Física, Facultad de Ciencias, Universidad de los Andes, Apartado 5100, Mérida, Venezuela
--- abstract: 'The Apache Point Observatory Galactic Evolution Experiment has measured the stellar velocities of red giant stars in the inner Milky Way. We confirm that the line of sight velocity distributions (LOSVDs) in the mid-plane exhibit a second peak at high velocities, whereas those at $\|b\| = 2{^\circ}$ do not. We use a high resolution simulation of a barred galaxy, which crucially includes gas and star formation, to guide our interpretation of the LOSVDs. We show that the data are fully consistent with the presence of a thin, rapidly rotating, nuclear disk extending to $\sim 1 {\mbox{$\>{\rm kpc}$}}$. This nuclear disk is orientated perpendicular to the bar and is likely to be composed of stars on x2 orbits. The gas in the simulation is able to fall onto such orbits, leading to stars populating an orthogonal disk.' author: - 'Victor P. Debattista, Melissa Ness, Samuel W.F. Earp, & David R.Cole' title: 'A Kiloparsec-Scale Nuclear Stellar Disk in the Milky Way as a Possible Explanation of the High Velocity Peaks in the Galactic Bulge' --- Introduction {#sec:intro} ============ Detections of high Galactic standard-of-rest velocity () peaks in the Apache Point Observatory Galactic Evolution Experiment (APOGEE) commissioning data were reported by [@nidever+12] across all fields at $4{^\circ}\leq l \leq 14{^\circ}$ and $-2{^\circ}\leq b \leq 2{^\circ}$. @nidever+12 interpreted the high- peaks as being due to stars in the Galactic bar. However, the peaks are not statistically significant in a number of fields [@li+14] and no high- peaks were found at negative longitudes in the Bulge Radial Velocity Assay (BRAVA), at $b$ $\sim = -4 {^\circ}$ [@kunder+12]. Additionally, no high- peaks can be found in pure $N$-body models [@li+14]. @molloy+15 demonstrated that resonant (2:1 and higher order) orbits, viewed on their own, were able to generate high- peaks. @aumerschoenrich15 proposed that such resonant orbits are populated by young stars recently trapped by the bar; they argued that the APOGEE selection function is biased toward such young stars. Bars have been implicated in building large gas reservoirs at the centers of galaxies, fuelling high star formation rates there. As in other barred galaxies, the Milky Way (MW)’s bar funnels gas inwards [@binney+91; @weiner_sellwood99; @fux99]. This gas gives rise to structures such as the Central Molecular Zone (CMZ), spanning $-1{^\circ}\la l \la 1.5{^\circ}$. The CMZ contains $5-10 \times 10^7 {\>{\rm M_{\odot}}}$ of molecular gas [@bally+87; @gusten89], driving a star formation rate of $\sim 0.14 {\>{\rm M_{\odot}}}$yr$^{-1}$ [@wardle_yusef-zadeh08]. A molecular gas disk extends across $\|l\| < 6{^\circ}$ and $\|b\| < 1.6{^\circ}$ [@boyce_cohen94; @dame_thaddeus94]. @liszt_burton80 and @ferriere+07 interpreted the observed molecular, atomic and ionized gas outside the CMZ to Galactic longitude $\|l\| \sim 10 {^\circ}$ as a (tilted) disk with semi-major axis of radius $\sim 1.4 {\mbox{$\>{\rm kpc}$}}$ with a hole at its center. In external galaxies, star formation in nuclear rings builds nuclear disks [@kormendy_kennicutt04]. In this Letter we demonstrate that the high- peaks in the line of sight velocity distributions (LOSVDs) are consistent with the presence of a nuclear disk in the MW. Simulation {#sec:simulation} ========== Here we use a high resolution simulation, with gas and star formation, which develops a bar, driving gas to the center and forming a stellar nuclear disk [@cole+14], to derive the kinematic signatures of such a disk. We use these to guide our interpretation of the APOGEE Data Release 12 [@alam+15] stellar velocity data for the inner MW. While the simulation was not designed to match the MW, @cole+14 showed that the nuclear disk that it forms is qualitatively similar to those in external galaxies. The simulation was evolved with the $N$-body$+$smoothed particle hydrodynamics code [gasoline]{} [@gasoline]. The galaxy forms out of gas cooling off a hot corona in pressure equilibrium within a dark matter halo of virial mass $M_{200} = 9 \times 10^{11} {\>{\rm M_{\odot}}}$. Both the dark matter halo and the initial gas corona are represented by $5 \times 10^6$ particles. As the gas cools and reaches high density, star formation is triggered. Star particles then provide feedback via winds from massive stars, and types Ia and II supernovae [@stinson+06]. Gas particles all have initial mass of $2.7 \times 10^4 {\>{\rm M_{\odot}}}$ and star particles are spawned from gas with $35\%$ of this mass. This high mass resolution allows us to use a high star formation threshold of 100 $\mathrm{cm}^{-3}$ for the gas [@governato+10]. By the end of the simulation the galaxy has a stellar mass of $6.5 \times 10^{10} {\>{\rm M_{\odot}}}$ in $\sim 1.1\times 10^7$ particles. This large number of star particles provides a very fine sampling of the mass distribution at the center of the model. Further details of the simulation are provided in @cole+14 The bar forms at around 4 Gyr. After 6 Gyr a prominent nuclear disk starts to form which, by 10 Gyr, has a semi-major axis of $1.5$ . The nuclear disk is perpendicular to the bar and its stellar streaming is perpendicular to the bar’s. At 10 Gyr the nuclear disk in the simulation is quite massive and is thus unlikely to match any nuclear disk in the MW. Therefore here we consider the model at two earlier times: at ${\mbox{$t_{1}$}}=6$ Gyr, before the nuclear disk forms, and at ${\mbox{$t_{2}$}}=7.5$ Gyr when a strong nuclear disk is established. Aside from the nuclear disk becoming more massive and the bar growing longer, the model at 10 Gyr is not qualitatively different from at . Scaling to the MW and Viewing Perspective ----------------------------------------- In order to compare to the MW, we rescale the model in both size and velocity. Size rescaling is accomplished by matching the size of the bar to that of the MW. Between and the average size of the bar in the simulation, as measured from the radius at which the phase of the $m=2$ Fourier moment deviates from a constant by more than $10{^\circ}$ [@aguerri+03], is 2.1 kpc. Assuming that the MW’s bar has a semi-major axis of $3.5 {\mbox{$\>{\rm kpc}$}}$ [@gerhard02], we scale all coordinates by a factor of 1.67. (Scaling to the more up-to-date bar size of @wegg+15, $5 {\mbox{$\>{\rm kpc}$}}$, leads to a nuclear disk which is much too large; because we seek a closer nuclear disk size match, we scale to the older bar size, but this is not to imply that the real MW bar semi-major axis is closer to $3.5{\mbox{$\>{\rm kpc}$}}$ than $5{\mbox{$\>{\rm kpc}$}}$.) The velocity scale factor is obtained by a least-squares fit to the line of sight velocity dispersion of the model to Abundances and Radial velocity Galactic Origins Survey [@ness+13b] data for all stars within Galactocentric radius $R_{\rm GC} < 3.5$ at $b$ = $5{^\circ}$, $7.5{^\circ}$ and $10{^\circ}$ across $\|l\| < 15{^\circ}$. We obtain a velocity scaling factor of 0.48. While these scalings lead to a model of roughly the right size and rotational velocity we stress that the model still does not match the MW and we only use it to qualitatively predict the expected trends in the MW, not their magnitude or precise location. We assume that the Sun is 8 from the Galactic Center, and place the observer at $y=-8$ . We orient the bar at $27{^\circ}$ to the line of sight [@wegg+gerhard13]. Since we compare our model with APOGEE [@alam+15] data, which targets bright red giant stars, we adopt a uniform selection function for star particles at $2 {\mbox{$\>{\rm kpc}$}}\leq R_s \leq 10 {\mbox{$\>{\rm kpc}$}}$, where $R_s$ is the distance from the Sun [@schultheis+14; @hayden+15]. Reducing the maximum $R_s$ to 8 kpc does not significantly alter our conclusions. We use an opening angle of $0.5{^\circ}$ for each LOSVD, to match the size of the smallest APOGEE bulge fields. The (off-plane) line of sight with the least particles contains over 2800 star particles while the best sampled (mid-plane) field has over 57,000 star particles; thus the shapes of the model LOSVDs are well determined. The top row of Figure \[fig:losvds\] shows the model’s surface density distribution. Line of Sight Velocity Distributions ------------------------------------ Viewing the model from the Solar perspective, we measure the distribution of line of sight velocities in the Galactocentric restframe, . Figure \[fig:losvds\] shows the LOSVDs for various lines-of-sight (indicated in the top row) in the mid-plane ($b=0{^\circ}$, second row) and off-plane ($b=2{^\circ}$, third row). At each LOSVD at $l\leq 12{^\circ}$ has a single peak, both in the mid-plane and off the plane. The LOSVDs have a shoulder to high , which @li+14 showed is produced by stars at large distances seen close to tangentially. The peak in moves to larger velocities with increasing $l$, but remains well below the Galaxy’s circular velocity. By the LOSVDs at $l = 8{^\circ}$ and $l = 10{^\circ}$ have developed a second, high- peak. This peak is more prominent than the low- peak, due to the model’s very vigorous star formation in the nuclear disk, roughly ten times higher than in the MW for the corresponding region. This very high star formation rate quickly leads to a relatively massive nuclear disk; thus the relative amplitudes of the low- and high- peaks are [not]{} predictions of the model. Indeed if we reduce the weight of star particles younger than 1 by a factor of 5, to compensate for the high star formation rate of the model, then the high- peaks become smaller than the main peaks, as seen in Figure \[fig:losvds\]. The distribution around the high- peak is narrower ( cooler) than that around the main peak and is skewed toward low . Interior to $l = 8{^\circ}$, the LOSVDs are broadened relative to those at , but no high- peak is evident. At $l \geq 14{^\circ}$ no high- peak is present in the mid-plane, indicating that the structure responsible for the feature does not extend this far. The off-plane and mid-plane LOSVDs are not substantially different at , aside from the mid-plane hosting more stars at ${\mbox{$V_{\rm GSR}$}}\geq 100~{\mbox{$\>{\rm km\, s^{-1}}$}}$. At the high- peaks, which dominate the mid-plane, are entirely absent in the $b = 2{^\circ}$ LOSVDs. Therefore the presence of a nuclear disk is only evident in the mid-plane. As in the MW, outside the nuclear disk, the off-plane LOSVDs at $(l,b) = (14{^\circ},2{^\circ})$ also contain a statistically significant high- peak/shoulder, but this is also present at , and is not related to the nuclear disk. Thus the kinematic signatures of a nuclear disk are (1) a second, high- peak at roughly the circular velocity, (2) which is absent a few degrees off the mid-plane, (3) is kinematically cooler than the low- peak, and (4) is skewed toward low . LOSVD Stacking -------------- The top row of Figure \[fig:losvds\] shows color-coded maps of the average , ; the peak velocities at orbit tangent points manifest as the characteristic “winged” pattern of the fields. Although the two maps show the model before and after the nuclear disk forms, they are not very different, indicating that the formation of the nuclear disk does not lead to a wholesale change of the galaxy as much as populating new parts of its phase space. At the low longitudes of the nuclear disk, large occurs only close to the galactic center while at other radii is smaller. Even with a survey the size of APOGEE, the number of stars in individual fields is still relatively small, giving a low signal-to-noise ratio for any second peak in any one field [@li+14]. In order to overcome this difficulty, we note that the of the second peak does not change significantly with longitude at $6{^\circ}\leq l \leq 10{^\circ}$. Therefore by stacking the LOSVDs we can enhance the signal-to-noise ratio of the high- peak. Because the main peak is dominated by stars streaming along the bar, and of these changes with $l$, the main peak in a stacked LOSVD will be quite broad. If we include $l < 4{^\circ}$, then the exponentially higher density of disk and bar stars near the center masks out any features at high . In the bottom panels of Figure \[fig:losvds\] we present a stack of the model’s LOSVDs at $l=6{^\circ}$, $8{^\circ}$ and $10{^\circ}$. As with the individual LOSVDs, a peak at high is evident at in the mid-plane but is absent at $b=2{^\circ}$. Moreover this second peak is still cooler than the low- peak, and remains skewed toward it. Thus stacking LOSVDs preserves the kinematic signatures of a nuclear disk, and provides a reliable method for searching for a nuclear disk in the APOGEE data. APOGEE data {#sec:apogee} =========== Data Selection -------------- We select APOGEE survey stars in the fields of interest, excluding stars with the STAR BAD flag (corresponding to poor stellar parameter fits) and those flagged as flux and telluric standards. Stars with a velocity scatter between different visits of more than $5 {\mbox{$\>{\rm km\, s^{-1}}$}}$ are also removed. (The same analysis including also stars flagged as STAR BAD, which leads to 763 in the plane and 1401 out of the plane, gives results in agreement with the more conservative cut.) The small numbers of stars in the APOGEE commissioning data resulted in peaks with low signal-to-noise ratio. We increase the statistical significance of a high- peak by stacking the APOGEE DR12 data in the longitude range $6{^\circ}\leq l \leq 8{^\circ}$ for fields in the mid-plane and off-plane at $\|b\| = 2{^\circ}$ (totalling 617 and 1114 stars, respectively). Table \[tab:fields\] lists the fields stacked together and the number of stars used from each field. ![The stacked APOGEE LOSVDs for the mid-plane (red histogram) and at $\|b\|=2{^\circ}$ (blue histogram). The fields used are listed in Table 1. The black lines show the two Gaussians fitted to the mid-plane LOSVD.[]{data-label="fig:apogeevgsrstack"}](histsplusfits.eps){width="\hsize"} Figure \[fig:apogeevgsrstack\] plots these two stacked APOGEE LOSVDs. The mid-plane stack has a clear second peak at ${\mbox{$V_{\rm GSR}$}}\sim 220 {\mbox{$\>{\rm km\, s^{-1}}$}}$, corresponding to roughly the circular velocity of the MW in the bulge region [@sofue+09]. No comparable second peak is visible in the off-plane stacked LOSVD, which is non-Gaussian and skewed toward high , it has a shoulder to high [@li+14]. A Kolmogorov-Smirnov test shows that the null hypothesis that the mid-plane and off-plane LOSVDs are drawn from the same distribution has a relatively low $p$-value of $0.04$. We fit two Gaussians to the mid-plane stacked LOSVD in the range $-300 {\mbox{$\>{\rm km\, s^{-1}}$}}\leq {\mbox{$V_{\rm GSR}$}}\ \leq 300 {\mbox{$\>{\rm km\, s^{-1}}$}}$, constrained such that the smaller Gaussian contains less than $25\%$ of the stars (to avoid fitting just the skewed low- distribution with two Gaussians). We obtain a low- component having mean velocity ${\mbox{$\left<{{\mbox{$V_{\rm GSR}$}}}\right>$}} = 24 {\mbox{$\>{\rm km\, s^{-1}}$}}$ and standard deviation ${\mbox{$\sigma_{\rm GSR}$}}= 57 {\mbox{$\>{\rm km\, s^{-1}}$}}$, while the high- component has ${\mbox{$\left<{{\mbox{$V_{\rm GSR}$}}}\right>$}} = 217 {\mbox{$\>{\rm km\, s^{-1}}$}}$ and ${\mbox{$\sigma_{\rm GSR}$}}= 44 {\mbox{$\>{\rm km\, s^{-1}}$}}$, making it cooler than the low- component. These two Gaussians are also shown in Figure \[fig:apogeevgsrstack\]. The velocity distribution at ${\mbox{$V_{\rm GSR}$}}\geq 200 {\mbox{$\>{\rm km\, s^{-1}}$}}$ hints at a skewness opposite to that of the main distribution, but the signal-to-noise ratio is still too low for a robust measurement. The high- Gaussian has a significant number of stars associated with it, and is significantly separated from the low- Gaussian. In order to test the likelihood of such a second peak arising purely from Poisson noise, we perform Monte-Carlo tests drawing 617 stars from the off-plane stacked LOSVD. Fitting two Gaussians as before to the resulting LOSVD, we label as $G_l$ and $G_h$ the low- and high- components, respectively. We repeat this procedure 100,000 times, and for each we compute $N_h/N_{\rm tot}$, the ratio of stars in the high- component to the total number of stars, and the overlap of the two components, defined as $$O = \int G_l G_h d{\mbox{$V_{\rm GSR}$}}.$$ The results are presented in Figure \[fig:statistics\]; the observed mid-plane stacked LOSVD has $N_h/N_{\rm tot} = 0.12$ and $O = 4.3$. Only $0.025\%$ of the Monte-Carlo samples have $N_h/N_{\rm tot} \leq 0.12$, while none of them have overlap $O \leq 8$, showing that the observed double-peaked mid-plane stacked LOSVD is highly unlikely to result from Poisson noise. The APOGEE data therefore show a statistically significant double-Gaussian LOSVD in the mid-plane, the properties of which agree with 3 of the 4 kinematic signatures of a nuclear disk from the simulation. While the signal-to-noise is too low to be sure if the high- peak is skewed to low , the data are suggestive that it is. Therefore a kiloparsec-scale nuclear disk can explain the high- peaks in the APOGEE data. ![Frequency distribution of properties of double-Gaussian fits for the off-plane APOGEE stacked LOSVD sub-sampled $N_s = 10^5$ times to 628 stars. The side-panels indicate the distributions over the individual variables, normalized to unit peak. The parameters for the fit to the mid-plane APOGEE stack are indicated by the filled red circle. In the side panels, the vertical dashed red lines indicate the values of $N_h/N_{\rm tot}$ and $O$ for the mid-plane stack. []{data-label="fig:statistics"}](statistics.eps){width="\hsize"} A simple estimate for the nuclear disk mass can be obtained from the fraction of stars in the high- component of the double-Gaussian fit to the mid-plane LOSVD. If we conservatively assume that the nuclear disk mass contained within $\|z\| \leq 150{\>{\rm pc}}$ and $4{^\circ}\leq \|l\| \leq 8{^\circ}$ is $12\%$ of the total mass of the Besançon Galaxy model [@robin+12] within this volume we obtain a lower limit to the mass of the nuclear disk $\sim 5.8 \times 10^7{\>{\rm M_{\odot}}}$. Discussion ========== Attempts to explain the high- peak directly via collisionless bar simulations fails [@nidever+12; @li+14]. However @molloy+15 demonstrated that resonant, bar-supporting 2:1 x1 (with some mixture of higher order resonance) orbits by themselves can produce second peaks. Subsequently @aumerschoenrich15 argued that the selection function of APOGEE favors young stars recently trapped into resonant orbits. Their interpretation requires that the stars in the high- peaks are younger. The other main 2:1 resonant orbit family of bars, the x2 family, is orientated perpendicular to the bar. This family is generally very poorly populated in the absence of gas [@sparke_sellwood87; @pfenniger_friedli91], but when gas is present it is driven inwards by the bar and settles into x2 orbits [@binney+91]. The gas can then form stars and produce nuclear rings and disks. We propose that the high- peak corresponds to a kiloparsec-scale disk composed of stars on x2 orbits. These orbits are stable and therefore our model does not require that the stars in the high- peak are young. Nuclear disks are known in many external galaxies [@scorza+vdbosch98; @zasov+moiseev99; @pizzella+02; @emsellem+04; @krajnovic+08; @ledo+10]; the presence of one in the MW is therefore not unusual. Nor is the kiloparsec scale unusual as a fraction of the bar size. For instance in NGC 3945 the ratio of semi-major axes of the nuclear disk to bar is $\sim 0.15-0.18$ [@erwin_sparke99; @cole+14], whereas for the MW this ratio is $\sim 0.2$, if we adopt @wegg+15’s $5 {\mbox{$\>{\rm kpc}$}}$ bar. The gas ring in the simulation is $\sim 5\times$ larger than the MW’s CMZ, which is coincident with a stellar disk [@launhardt+02; @schoenrich+15]. The large size of the gas ring in the model is a consequence of the still low resolution ($50 {\>{\rm pc}}$) of our simulation [@lizhi+15; @sormani+15]. This difference implies that the nuclear disk in the MW is not currently forming stars across its full extent. We anticipate that this proposal will inspire further detailed mapping of the central mid-plane of the MW. We will provide predictions from our model of a kiloparsec-scale nuclear disk elsewhere. , J. A. L., [Debattista]{}, V. P., & [Corsini]{}, E. M. 2003, , 338, 465 , S., et al. 2015, ArXiv e-prints , M., & [Sch[ö]{}nrich]{}, R. 2015, ArXiv e-prints , J., [Stark]{}, A. A., [Wilson]{}, R. W., & [Henkel]{}, C. 1987, , 65, 13 , J., [Gerhard]{}, O. E., [Stark]{}, A. A., [Bally]{}, J., & [Uchida]{}, K. I. 1991, , 252, 210 , P. J., & [Cohen]{}, R. J. 1994, , 107, 563 , D. R., [Debattista]{}, V. P., [Erwin]{}, P., [Earp]{}, S. W. F., & [Ro[v s]{}kar]{}, R. 2014, , 445, 3352 , T. M., & [Thaddeus]{}, P. 1994, , 436, L173 , E., et al. 2004, , 352, 721 , P., & [Sparke]{}, L. S. 1999, , 521, L37 , K., [Gillard]{}, W., & [Jean]{}, P. 2007, , 467, 611 , R. 1999, , 345, 787 , O. 2002, in ASP Conf. Ser. 273: The Dynamics, Structure & History of Galaxies: A Workshop in Honour of Professor Ken Freeman, 73 , F., et al. 2010, , 463, 203 , R. 1989, in IAU Symposium, Vol. 136, The Center of the Galaxy, ed. M. [Morris]{}, 89 , M. R., et al. 2015, , 808, 132 , J., & [Kennicutt]{}, R. C. 2004, , 42, 603 , D., et al. 2008, , 390, 93 , A., et al. 2012, , 143, 57 , R., [Zylka]{}, R., & [Mezger]{}, P. G. 2002, , 384, 112 , H. R., [Sarzi]{}, M., [Dotti]{}, M., [Khochfar]{}, S., & [Morelli]{}, L. 2010, , 407, 969 , Z., [Shen]{}, J., & [Kim]{}, W.-T. 2015, , 806, 150 , Z.-Y., [Shen]{}, J., [Rich]{}, R. M., [Kunder]{}, A., & [Mao]{}, S. 2014, , 785, L17 , H. S., & [Burton]{}, W. B. 1980, , 236, 779 , M., [Smith]{}, M. C., [Wyn Evans]{}, N., & [Shen]{}, J. 2015, ArXiv e-prints , M., et al. 2013, , 432, 2092 , D. L., et al. 2012, , 755, L25 , D., & [Friedli]{}, D. 1991, , 252, 75 , A., [Corsini]{}, E. M., [Morelli]{}, L., [Sarzi]{}, M., [Scarlata]{}, C., [Stiavelli]{}, M., & [Bertola]{}, F. 2002, , 573, 131 , A. C., [Marshall]{}, D. J., [Schultheis]{}, M., & [Reyl[é]{}]{}, C. 2012, , 538, A106 , R., [Aumer]{}, M., & [Sale]{}, S. E. 2015, ArXiv e-prints , M., et al. 2014, , 148, 24 , C., & [van den Bosch]{}, F. C. 1998, , 300, 469 , Y., [Honma]{}, M., & [Omodaka]{}, T. 2009, , 61, 227 , M. C., [Binney]{}, J., & [Magorrian]{}, J. 2015, , 449, 2421 , L. S., & [Sellwood]{}, J. A. 1987, , 225, 653 , G., [Seth]{}, A., [Katz]{}, N., [Wadsley]{}, J., [Governato]{}, F., & [Quinn]{}, T. 2006, , 373, 1074 , J. W., [Stadel]{}, J., & [Quinn]{}, T. 2004, New Astronomy, 9, 137 , M., & [Yusef-Zadeh]{}, F. 2008, , 683, L37 , C., & [Gerhard]{}, O. 2013, , 435, 1874 , C., [Gerhard]{}, O., & [Portail]{}, M. 2015, , 450, 4050 , B. J., & [Sellwood]{}, J. A. 1999, , 524, 112 , A. V., & [Moiseev]{}, A. V. 1999, in IAU Symposium, Vol. 194, Activity in Galaxies and Related Phenomena, ed. Y. [Terzian]{}, E. [Khachikian]{}, & D. [Weedman]{}, 279 V. P. D. is supported by STFC Consolidated grants \# ST/J001341/1 and \# ST/M000877/1, while D. R. C. is supported by STFC Consolidated grant \# ST/J001341/1. M. N. is funded by the European Research Council under the European Union’s Seventh Framework Programme (FP 7) ERC Grant Agreement \# 321035. The authors thank the ESF GREAT programme for funding which has supported this research. The simulation used in this paper was run at the High Performance Computing Facility of the University of Central Lancashire. We thank the anonymous referee for a very thoughtful report that helped substantially improve this paper. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
--- abstract: 'We propose an hybrid laser system consisting of a semiconductor external cavity laser associated to an intra-cavity diamond etalon doped with nitrogen-vacancy color centers. We consider laser emission tuned to the infrared absorption line that is enhanced under the magnetic field dependent nitrogen-vacancy electron spin resonance and show that this architecture leads to a compact solid-state magnetometer that can be operated at room-temperature. The sensitivity to the magnetic field limited by the photon shot-noise of the output laser beam is estimated to be around $250~\mathrm{fT/\sqrt{Hz}}$. Unlike usual NV center infrared magnetometry, this method would not require an external frequency stabilized laser. Since the proposed system relies on the competition between the laser threshold and an intracavity absorption, such laser-based optical sensor could be easily adapted to a broad variety of physical systems.' address: - '$^1$ Univ Rennes, CNRS, FOTON - UMR 6082, F-22305 Lannion, France' - '$^2$ Laboratoire Aimé Cotton, CNRS, Univ. Paris-Sud, ENS Cachan, Université Paris-Saclay, 91405 Orsay, France' - '$^3$ Light and Matter Physics Group, Raman Research Institute, Bangalore 560080, India' - '$^4$ Thales Research & Technology, 1 avenue Augustin Fresnel, Palaiseau, France' - '$^5$ University of New Mexico, Albuquerque, USA' - '$^6$ Universität des Saarlandes, Germany' - '$^7$ Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany' - '$^8$ Department of Physics, University of California, Berkeley, CA 94720-7300, USA' - '$^9$ Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA' author: - 'Yannick Dumeige$^1$, Jean-François Roch$^2$, Fabien Bretenaker$^{2,3}$, Thierry Debuisschert$^4$, Victor Acosta$^5$, Christoph Becher$^6$, Georgios Chatzidrosos$^7$, Arne Wickenbrock$^7$, Lykourgos Bougas$^7$, Alexander Wilzewski$^7$, and Dmitry Budker$^{7,8,9}$' title: Infrared laser magnetometry with a NV doped diamond intracavity etalon --- [Keywords]{}: diamond NV center, optical magnetometry, VCSEL Introduction ============ In recent years, the optical detection of the magnetic resonance between the electronic triplet $S=1$ spin states of the negatively charged nitrogen-vacancy (NV) color center in diamond and the measurements of the Zeeman shifts induced by an applied magnetic field has been used in a variety of solid-state magnetometers [@Rondin14]. Due to the special properties of the NV center, these systems can be operated in ambient conditions to detect a broad range of magnetic fields created by both physical and biological systems [@Schirhagl14; @CasolaYacobi18]. By raster scanning a single NV spin over a magnetized substrate and detecting the spin-dependent luminescence emitted by this atomic-like defect, the stray magnetic field created by the sample magnetization can be mapped with nanometer spatial resolution [@Balasubramanian08; @Maze08]. Compared to a single spin, the magnetic field sensitivity of an ensemble of NV centers contained in a macroscopic single-crystal diamond sample is increased by $\sqrt{N}$ where $N$ is the number of NV centers used as magnetic sensors [@Acosta09]. Due to this enhancement, continuous-wave magnetometry based on an NV ensemble has recently reached a sensitivity level of about ${15~\rm pT}/{\sqrt{\rm Hz}}$ [@Barry16; @Schloss18]. However this technique is constrained by the collection efficiency of the NV luminescence and by parasitic background light that spectrally overlaps the broad luminescence of the NV center with wavelength extending from 637 nm to about 800 nm. The spin state of the NV center can also be determined by the infrared (IR) optical transition associated with the singlet $S=0$ spin state [@Rogers08; @Acosta10b; @Kehayias13]. The corresponding scheme for detecting the perturbation by an applied magnetic field is based on the absorption of a signal beam tuned to this IR transition centered at $\lambda_s= 1042\,{\rm nm}$ wavelength [@Acosta10]. The magnetic-field dependent signal is then free from any background and the relevant photon detection efficiency can be almost ideal. Nevertheless, the low optical depth of the IR transition at room temperature, even for a dense ensemble of NV centers, needs to be compensated by a multi-pass configuration [@Clevenson15]. This enhancement scheme can be implemented by placing the NV doped diamond in an optical cavity resonant with the IR signal beam [@Dumeige13; @Jensen14]. A shot-noise limited sensitivity of $28~\mathrm{pT/\sqrt{Hz}}$ was recently achieved using a miniaturized Fabry-Perot cavity [@Chatzidrosos17] which could even be realized in integrated diamond photonics [@Gazzano17; @Bougas18]. In order to circumvent the previously mentioned drawbacks of luminescence based magnetometers, it was proposed to operate the NV center transition between the triplet spin states in the stimulated emission regime [@Faraon12; @Jeske16; @Savitski17] so that population inversion in the NV center levels provides the optical gain of a laser. By setting the laser at its threshold, sensitivities of about $\mathrm{fT/\sqrt{Hz}}$ are anticipated [@Jeske16]. Nevertheless the stimulated emission from the NV centers can be strongly affected by the excited state absorption (ESA) phenomena and by the photoconversion between the negatively-charged state $\mathrm{NV}^{-}$, with the previously described spin triplet structure, and the neutral charge state $\mathrm{NV}^{0}$ [@Subedi18]. These parasitic effects can make the implementation of NV$^-$ center magnetometry based on the visible optical laser amplification challenging [@Jeske17]. Here we propose to combine the IR absorption method and the laser threshold magnetometry method by considering a hybrid laser architecture which integrates the diamond sample containing the NV centers in an external-cavity laser. The optical gain in the laser is provided by an independent semiconductor material which is optically pumped. The laser threshold of the whole system is then sensitive to the applied magnetic field via the losses on the IR transition induced by the spin resonance of the NV centers. In this scheme, the ESA in the gain medium becomes irrelevant and has a marginally negative effect on the IR signal absorption efficiency. Using a rate equation model of the photodynamics of the NV center that takes into account its two charge states, we evaluate the magnetic field sensitivity of this hybrid laser system. Finally, we discuss the possible advantages of this sensor architecture for practical applications. Model of the spin-dependent NV center dynamics ============================================== The NV center consists of a nitrogen impurity linked to an adjacent vacant lattice site. In the negatively charged state NV$^-$ which consists of six electrons associated to the dangling bonds around the lattice vacancy, four of these electrons populate the lowest energy states [@Doherty13]. The remaining two electrons create both spin triplet $S=1$ states and spin singlet $S=0$ states that are associated to optical transitions within the $\rm 5.5~eV$ bandgap of diamond. In the spin triplet manifold of the ground electronic state $^{3}A_2$, the magnetic interaction between electron spins induces a zero-field splitting of $D\approx 2.87\,{\rm GHz}$ between the $m_S=0$ and $m_S=\pm 1$ spin projection sublevels along the intrinsic quantization axis that is defined by the N-to-V axis of the defect inside the crystal (Fig. \[fig\_energy\]a). According to selection rules determined from group-theory methods [@Doherty13; @Maze11], the optically electronic transitions between the triplet sublevels of the $^{3}A_2$ electronic ground state and the corresponding triplet sublevels of the excited electronic level $^{3}E$ are mainly spin-conserving. Due to a non-radiative decay path from the $m_S=\pm 1$ excited states through the metastable singlet states $^{1}A$ and $^{1}E$ and then preferentially back to the $m_S=0$ ground state (Fig. \[fig\_energy\]a), green laser optical excitation of the $^{3}A_2$ triplet sublevels polarizes the electron spin of the NV$^-$ center into the $m_S=0$ sublevel [@Thiering18]. The non-radiative leakage to the metastable $S=0$ state also induces a lower luminescence efficiency of the $m_S=\pm 1$ sublevels compared to $m_S=0$ so that the occupation probability in this ground state spin manifold of $m_S=\pm 1$ compared that of the $m_S=0$ can be determined by monitoring the photoluminescence (PL) intensity. These properties enable the optically detected magnetic resonance (ODMR) signal that can be induced by applying a microwave field resonant with the $m_S=0$ to $m_S=\pm 1$ transition. Since a magnetic field applied to the NV center induces Zeeman shifts that lift the degeneracy of the $m_S=\pm 1$ sublevels, the magnetic field amplitude can be determined by measuring these Zeeman shifts in the ODMR microwave frequency spectrum [@Taylor08]. The detection of spin polarization can also be realized by measuring the transmission of a signal IR beam that probes the absorption on the transition between the singlet metastable states $^{1}A$ and $^{1}E$ [@Acosta10]. Under green light continuous optical pumping and in the absence of resonant microwaves, the NV centers are pumped into the $m_S=0$ ground sublevel leading to a reduced occupation rate of the metastable singlet state $^{1}E$. In this off-resonance regime the IR signal transmission is maximal. For magnetic fields applied along one of the NV axis, when the microwave field frequency is resonant with frequency $ D \pm \gamma B_{\rm NV}/(2\pi)$, where $ \gamma=1.761 \times 10^{11} \; {\rm rad}\; {\rm s}^{-1} \; {\rm T}^{-1}$ is the NV gyromagnetic ratio and $B_{\rm NV}$ the projection on the NV axis of the applied magnetic field, the population is transferred into the $m_S = \pm 1$ ground state. The occupation rate of the $^{1}E$ state increases and the magnetic field dependent spin resonance can be detected as a lower transmission of the IR signal beam. A rate equation model is used to describe the photodynamics of the triplet and singlet states and to estimate the optical losses induced by the magnetic resonance between the sublevels of the $^{3}A_2$ ground state on the signal beam that propagates through the NV doped diamond sample [@Dumeige13; @Bougas18]. In order to take into account the photoionization process [@Dumeige04; @Manson06; @Aslam13; @Meirzada17] between $\mathrm{NV}^{-}$ and $\mathrm{NV}^{0}$ two supplementary levels associated to the $\mathrm{NV}^{0}$ neutral charge state [@Meirzada17] are added to this level scheme, as shown in Fig. \[fig\_energy\]b. In our configuration, the photoionization and the ESA only reduce the numerical value of the inferred IR absorption cross-section (see \[donnee\]) but are not an intrinsic limitation as it is the case for laser threshold magnetometry based on the visible transition. ![Energy diagram of the NV center. (a) Level scheme of the NV$^-$ center. As shown in the insert, the ground state $^3A_2$ is split into $m_S=0$ and $m_S=\pm 1$ sublevels due to spin-spin interaction and an external magnetic field applied to the NV center lifts the $m_S=\pm 1$ degeneracy. The NV$^-$ center is spin polarized into $m_S=0$ by optical pumping at $\lambda_g=532~\mathrm{nm}$. The resonance zero-phonon wavelength of the singlet transition is $\lambda_s=1042~\mathrm{nm}$. Typical lifetimes of the $^3E$ and $^1E$ levels are respectively $16~\mathrm{ns}$ and $600~\mathrm{ns}$. (b) Description of the photodynamics between the spin sublevels of the NV$^-$ and NV$^0$ ground and excited electronic states. $W_g$ is the pumping rate associated to the $\mathrm{NV}^{-}$ and $W_{g0}$ that of the $\mathrm{NV}^{0}$. $W_s$ is the transition rate of the IR resonance. $W_i$ and $W_r$ are respectively the ionization and recombination rates of the $\mathrm{NV}^{-} \rightleftarrows \mathrm{NV}^{0}$ transition. $W_{\rm MW}$ is the $m_S=0 \rightleftarrows m_S=\pm1$ transition rate induced by the resonant microwave field. The insert shows the pump (green) and signal (IR) configuration, with propagation through the diamond plate of thickness $e$.[]{data-label="fig_energy"}](Fig1.eps){width="12cm"} The spin sublevels $m_S=0$ and $m_S=\pm 1$ of the $^3 A_2$ state of the $\mathrm{NV}^{-}$ center are labelled $1$ and $2$ whereas $3$ and $4$ are the corresponding spin sublevels of the excited state $^3 E$. The ground and excited states of the singlet IR transition are respectively labelled $6$ and $5$. Finally, $7$ and $8$ are the $\mathrm{NV}^{0}$ ground and excited states. The radiative or non-radiative relaxation rate from $\alpha$ to $\beta$ levels is $k_{\alpha\beta}$; the values of these parameters are given in \[donnee\] and similar measurements can be found in [@Robledo11; @Kalb18]. The relaxation rate from $2$ to $1$ can be neglected since the associated spin-relaxation time, longer than $0.2~\mathrm{ms}$ at room temperature [@Mrozek15], is much longer than all other decay processes. When excited in the upper singlet state, the system can only decay to the lower singlet state and thus $k_{51}=k_{52}=0$. Finally, the optical transition are spin conserving and thus $k_{41}=k_{32}=0$. The optical depth of a diamond plate of thickness $e$ doped with the NV centers (see the insert of Fig. \[fig\_energy\]b) is obtained from the steady state solution of the following system calculated at each position indexed by $z$ in the diamond sample: $$\left\{ \begin{aligned} \frac{dN_1}{dt} & =-(W_g+W_{\rm MW})N_1+W_{\rm MW}N_2+k_{31}N_3+k_{61}N_6+\frac{W_r}{2}N_8\\ \frac{dN_2}{dt} & =W_{\rm MW}N_1-(W_g+W_{\rm MW})N_2+k_{42}N_4+k_{62}N_6+\frac{W_r}{2}N_8\\ \frac{dN_3}{dt} & =W_g N_1-(k_{31}+k_{35}+W_i)N_3\\ \frac{dN_4}{dt} & =W_gN_2-(k_{42}+k_{45}+W_i)N_4\\ \frac{dN_5}{dt} & =k_{35}N_3+k_{45}N_4-(k_{56}+W_s)N_5+W_sN_6\\ \frac{dN_6}{dt} & =(k_{56}+W_s)N_5-(W_s+k_{61}+k_{62})N_6\\ \frac{dN_7}{dt} & =W_iN_3+W_iN_4-W_{g0}N_7+k_{87}N_8\\ \frac{dN_8}{dt} & =W_{g0}N_7-(k_{87}+W_r)N_8, \end{aligned} \right.$$ where $N_{\alpha}(z)$ is the population density of state $\alpha$. The pumping rates are related to the pump $I_g$ and signal $I_s$ optical intensities through $W_g=\sigma_gI_g\lambda_g/(hc)$, $W_{g0}=\sigma_{g0}I_g\lambda_g/(hc)$, $W_i=\sigma_iI_g\lambda_g/(hc)$, $W_r=\sigma_rI_g\lambda_g/(hc)$ and $W_s=\sigma_sI_s\lambda_s/(hc)$, where the cross-sections $\sigma_{\beta}$ are given in \[donnee\]. The system is considered as closed and $\sum_{\alpha=1}^8 N_{\alpha}(z)=N_{\rm NV}$ where $N_{\rm NV}$ is the density of the NV centers contained in the diamond sample. The pump (green) and signal (IR) intensities at the output of the diamond sample are then obtained by: $$\left\{ \begin{aligned} \frac{dI_g}{dz} & =-\left[\sigma_g(N_1+N_2)+\sigma_{g0}N_7+\sigma_i(N_3+N_4)+\sigma_rN_8\right]I_g\\ \frac{dI_s}{dz} & =-\sigma_s(N_6-N_5)I_s.\\ \end{aligned} \right.$$ After integration along $z$, these equations determine the optical depth $\tau$ for the IR signal beam as a function of the green-light $W_g$ and microwave $W_{\rm MW}$ pumping rates: $$\tau=-\ln{\left[\frac{I_s(e)}{I_s(0)}\right]}.$$ Hybrid architecture for NV laser magnetometry ============================================= The proposed hybrid architecture shown in Fig. \[fig\_principle\]a is based on a vertical external cavity surface emitting laser (VECSEL). The gain medium is a half- vertical cavity surface emitting laser (VCSEL) consisting of semiconductor multiple InGaAs/GaAs quantum wells grown on a perfectly reflecting Bragg mirror both centered at $\lambda_s$ [@Baili07]. The output coupling mirror (M) of the laser cavity has a transmission coefficient $T$. A diamond thin plate containing a high concentration of NV centers is inserted inside the cavity. The semiconductor quantum wells are pumped using a laser at $\lambda_p=808~\mathrm{nm}$ and the NV centers are spin polarized by illuminating the diamond sample with a green laser at $\lambda_g=532\, {\rm nm}$ wavelength. The diamond plate operates as an intracavity etalon leading to single-mode operation of this external cavity semiconductor laser. The extra losses due to the NV absorption in the diamond plate, and thus the threshold and the efficiency of this hybrid laser depend on the spin state of the NV centers that are driven by the microwave field. Consequently, the output power $P_{\rm out}$ of the laser can be modified by the magnetic field $\mathbf{B}$ applied on the NV centers. As previously explained, the IR losses are increased when the microwave field is on-resonance, leading to a higher threshold and a lower efficiency compared to the off-resonance case as shown in Fig. \[fig\_principle\]b. Using these features the magnetic field dependent spin resonance can be detected by monitoring the IR laser output power. ![(a) Hybrid magnetometer architecture combining a half-VCSEL, a diamond thin plate (P) highly doped with NV centers, and an output coupling mirror (M). $\mathrm{L_1}$ and $\mathrm{L_2}$ are two focusing lenses. P is the diamond plate containing the NV centers. $P_g$, $P_p$ and $P_{\rm out}$ are respectively the power for NV polarization, the power for quantum well pumping, and the output power of the IR laser which is detected by a photodiode PD. The half-VCSEL represents the Bragg mirror and the semiconductor quantum wells which provide the optical gain in the laser cavity. (b) Operation principle of the magnetometer showing the threshold and the efficiency of the external cavity laser with the microwave (MW) field being either on-resonance or off-resonance.[]{data-label="fig_principle"}](Fig2.eps){width="12cm"} Parameters of the VECSEL {#VECSEL} ======================== The main requirements on the IR laser are (i) to operate in the regime of high-finesse cavity in order to increase the effective path of the IR signal in the diamond plate [@Dumeige13; @Jensen14; @Chatzidrosos17], (ii) to be low-noise since the magnetic field sensitivity is directly related to the IR optical signal noise and (iii) compactness. The VECSEL-based architecture is therefore a good candidate especially when the cavity length is chosen to reach the class A regime of laser operation (corresponding to a cavity lifetime longer than population inversion lifetime) enabling a photon shot-noise limited amplitude noise operation[@Baili09]. The parameters of the hybrid laser magnetometer are deduced from those given in Ref. [@Baili07] which describes a shot-noise limited semiconductor VECSEL emitting at a wavelength of $1~\mu\mathrm{m}$, close to $\lambda_s$. In the class A regime, the output power $P_{\rm out}$ of the IR laser is given by: $$\label{Output_power} P_{\rm out}=TP_{\rm sat}(r-1),$$ where $P_{\rm sat}$ is the pumping saturation power, and $r$ the rate of the pumping power $P_p$ above the laser threshold $P_{\rm th}$: $$\label{seuil1} r=\frac{P_p}{P_{\rm th}} = \frac{\eta P_p}{T+\epsilon} \,,$$ where $\epsilon$ are the losses introduced by the intracavity etalon for a round trip inside the cavity, and $\eta$ is the proportionality factor that relates the optical gain obtained after one round trip in the cavity to the pumping power $P_p$. With an intracavity etalon that ensures single-mode operation, the laser realized in Ref.[@Baili07] has a threshold power of $P_{\rm th}=700~\mathrm{mW}$ and provides an output power of $P_{\rm out}=50~\mathrm{mW}$ for $P_p=1~\mathrm{W}$ of pump power applied to the VECSEL. Considering an output coupling mirror with transmission $T=1~\%$, we then infer from Eq. (\[Output\_power\]) a saturation power of $P_{\rm sat}=11.7\,\mathrm{W}$. Without the intracavity etalon, the output power is $P_{\rm out}=140\,\mathrm{mW}$ for the same pump power. Since in this case $\epsilon=0$, we deduce $\eta=2.2\times 10^{-2}\,\mathrm{W}^{-1}$ by combining Eq. (\[Output\_power\]) and Eq. (\[seuil1\]). If we consider again the case of the etalon in the laser cavity, we have at the threshold $\eta P_{\rm th}=T+\epsilon$ so that $\epsilon=0.5~\%$. Intracavity diamond etalon and magnetic field sensitivity ========================================================= We now consider that the intracavity etalon consists of a diamond sample doped with NV centers, without any anti-reflection coating on the input and output facets. The etalon is illuminated using an additional green laser which polarizes the NV spins in the $m_S=0$ sublevel of the ground electronic state and also feeds the metastable singlet level (6) shown in Fig. \[fig\_energy\]b. Taking into account the optical thickness $\tau\ll 1$ of the diamond plate, the absorption of the IR beam due to the singlet transition of the NV centers then corresponds to additional intracavity optical losses $$\label{loss_diamond} \xi=2\chi\tau$$ where the factor 2 accounts for the round trip inside the cavity and $\chi=(n_d^2+1)/(2n_d)\approx 1.4$ is an enhancement factor of the losses which is induced by the high refractive index $n_d=2.4$ of the diamond plate (see \[enhancement\]). The laser pumping rate then becomes: $$\label{new_r} r=\frac{\eta P_p}{T+\epsilon+\xi}$$ where $\eta$ and $\epsilon$ have the values previously determined. The parameter $\xi$ corresponds to the useful losses of the diamond sample that determine the efficiency of the laser response to the applied magnetic field, as: $$\frac{\partial P_{\rm out}}{\partial B} =\frac{\partial P_{\rm out}}{\partial \tau}\cdot \frac{\partial \tau}{\partial \nu_{\rm ESR}}\cdot \frac{\partial \nu_{\rm ESR}}{\partial B}$$ where $\nu_{\rm ESR}$ is the resonance frequency of the microwave field with the dependence $\frac{\partial \nu_{\rm ESR}}{\partial B}= \frac{\gamma}{2\pi}$ to the applied magnetic field. If we assume that the spin resonance has a Lorentzian lineshape, the maximum of $\frac{\partial \tau}{\partial \nu_{\rm ESR}}$ is reached for $\tau_{\rm max}=(\tau_{\rm on}+3\tau_{\rm off})/4$ where $\tau_{\rm on}$ and $\tau_{\rm off}$ are the optical depths with respectively the microwave field being either on-resonance or off-resonance (see \[Odepth\]). This maximum value is then given by $$\label{max_sens} \left\|\frac{\partial \tau}{\partial \nu_{\rm ESR}}\right \|_{\rm max}=\frac{3\sqrt{3}}{4} \, \frac{\Delta\tau}{\Delta \nu_{\rm ESR}},$$ where $\Delta\tau=\left\|\tau_{\rm off}-\tau_{\rm on}\right\|$ and $\Delta \nu_{\rm ESR}$ is the full width at half maximum of the spin resonance. We assume here that the linewidth of the electronic spin resonance (ESR) is limited by the spin dephasing time $T_2^$ and by the spin polarization relaxation rate $\Gamma$ taking into account populations dynamics [@Dreau11] and related to $W_{\rm sat}$ the microwave saturation rate by $\Gamma=2W_{\rm sat}$, we thus can write $$\label{nuESR} \Delta \nu_{\rm ESR}=\frac{1}{\pi T_2^}\sqrt{1+\frac{\Omega_R^2T^_2}{\Gamma}},$$ where $\Omega_R$ the Rabi frequency is related to the microwave pumping rate by $W_{\rm MW}=\frac{\Omega_R^2T^_2}{2}$. Taking into account the pumping rate given by Eq. (\[new\_r\]), we can then determine the maximal response of the laser-based magnetometer: $$\label{sensibilite} \left\|\frac{\partial P_{\rm out}}{\partial B}\right\|_{\rm max}=\frac{3\sqrt{3}}{2} \chi\, \Delta\tau\, \, \frac{\gamma}{2\pi\Delta \nu_{\rm ESR}}\, T \, P_{\rm sat} \, \frac{\eta P_p}{(T+\epsilon+\xi_{\rm max})^2} \,$$ with $\xi_{\rm max} = 2 \chi \tau_{\rm max}$. Assuming that the laser output noise is at the limit of photon shot-noise we have $$\label{SN} \delta P_{\rm out}=\sqrt{\frac{P_{\rm out}hc \Delta f}{\lambda_s}},$$ where $\Delta f$ is the measurement bandwidth. The equivalent magnetic noise of the sensor $\delta B_{\rm min}=\frac{\delta P_{\rm out}}{\left\|\partial P_{\rm out}/\partial B\right\|_{\rm max}}$ can then be deduced from Eqs. (\[nuESR\]), (\[sensibilite\]) and (\[SN\]). Results ======= The simulations of the equivalent magnetic field noise are based on the laser parameters given in section \[VECSEL\] apart from the diamond etalon with thickness $e=100~\mu\mathrm{m}$ (note that we take into account parasitic losses due to diamond by taking $\epsilon=0.5~\%$). We also assume that the laser emission is tuned to the NV center IR transition. ![a) Normalized population $N_1$ versus microwave pumping rate $W_{\rm MW}$. b) Equivalent magnetic field noise for an IR NV center laser magnetometer versus Rabi frequency $\Omega_R$ of the microwave field. Config. 1: $N_{\rm NV}=4.4\times 10^{23}~\mathrm{m}^{-3}$ and $T_2^=390~\mathrm{ns}$. Config. 2: $N_{\rm NV}=2.8\times 10^{24}~\mathrm{m}^{-3}$ and $T_2^=150~\mathrm{ns}$. For both figures, calculations have been carried out for $I_g=40~{\rm kW}\, {\rm cm}^{-2}$. For the calculations of the equivalent magnetic field noise the on-resonance optical depth is calculated using the actual value of $\Omega_R$ whereas the off-resonance value is obtained for $\Omega_R=0$. Note that for the calculation of on-resonance optical depths, we consider that only 1/4 of the NV centers are aligned along the magnetic field. Furthermore in Fig. \[Rabi\]b) we used the following laser parameters: $T=0.03$ and $r=1.2$ which corresponds to an unoptimized value of the laser pumping rate.[]{data-label="Rabi"}](Fig3.eps){width="13.5cm"} We now consider two configurations with different realistic densities of NV centers. Config. 1 refers to $N_{\rm NV}=4.4\times 10^{23}~\mathrm{m}^{-3}$ and $T_2^=390~\mathrm{ns}$ [@Kubo11] whereas Config. 2 to $N_{\rm NV}=2.8\times 10^{24}~\mathrm{m}^{-3}$ and $T_2^=150~\mathrm{ns}$ [@Acosta09]. The length of the cavity and the curvature of the output mirror are such that the waist of the laser mode is $w_0=50~\mu\mathrm{m}$. The diamond etalon is located as close to the waist position as allowed by the pumping beam. We assume a green pump intensity $I_g=40~{\rm kW}\, {\rm cm}^{-2}$ corresponding to a mean power of $1.5~\mathrm{W}$. We first show in Fig. \[Rabi\]a) the population $N_1$ as a function of the microwave pumping rate for the two configurations. By fitting the results by $A+\frac{B}{1+W_{\rm MW}/W_{\rm sat}}$ with $A$, $B$ and $W_{\rm sat}$ as free parameters we can deduce the microwave saturation rate, for Config. 1: $W_{\rm sat}=4.9\times 10^5~\rm s^{-1}$ and for Config. 2: $W_{\rm sat}=2.1\times 10^5~\rm s^{-1}$. We then are able to plot the equivalent magnetic field noise $\delta B_{\rm min}$ versus the microwave Rabi frequency for the two studied configurations. In both cases, the equivalent magnetic field noise reaches an optimum coming from the trade-off between the increase of the contrast and the broadening of the ESR. The following optimal Rabi frequencies are used in the rest of the work: $\Omega_R=2\pi\times 3.4\times 10^5~\rm Hz$ for Config. 1 and $\Omega_R=2\pi\times 4.5\times 10^5~\rm Hz$ for Config. 2. Further optimization of the results are shown in Fig. \[fig\_plot\_results\] where the equivalent magnetic field noise $\delta B_{\rm min}$ is plotted as a function of the transmission of the output mirror for several pumping rates $r$. For both configurations an optimum output coupling is found depending on the IR absorption. Figure \[fig\_plot\_results\] also shows that by operating the laser close to its threshold (here $r=1.01$), the equivalent magnetic field noise can be strongly reduced, reaching for instance almost $700~\mathrm{fT}/\sqrt{\mathrm{Hz}}$ for the parameters of Config. 2. This value could be reduce to $250~\mathrm{fT}/\sqrt{\mathrm{Hz}}$ by using techniques to avoid ESR broadening due to microwave pumping [@Dreau11] and considering a higher Rabi frequency $\Omega_R=2\pi\times 1~\rm MHz$. ![Equivalent magnetic field noise optimization of the IR NV center laser magnetometer obtained for $I_g=40~\mathrm{kW/cm^2}$. (a) Config. 1, $\Omega_R=2\pi\times 3.4\times 10^5~\rm Hz$. (b) Config. 2, $\Omega_R=2\pi\times 4.5\times 10^5~\rm Hz$.[]{data-label="fig_plot_results"}](Fig4.eps){width="15cm"} Indeed, at its threshold, the laser becomes highly sensitive to the intracavity optical losses and thus to magnetic field fluctuations similarly to the behavior of visible laser threshold magnetometry [@Jeske16]. Finally comparison between the two configurations of Fig. \[fig\_plot\_results\] shows that the trade-off between the NV center density and the spin dephasing time associated with Config. 2 leads to an improved sensitivity. Note that once fundamental limits are achieved, the sensitivity scales as $1/\sqrt{N_{\rm NV}T_2^}$ [@Taylor08]. For the considered diamond thickness and waist size the spin projection noise determined by the total number of NV centers participating in the measurement is smaller than $30~\mathrm{fT}/\sqrt{\mathrm{Hz}}$ [@Taylor08]. This noise can therefore be neglected compared to the shot-noise limit set by the laser output photon flux. Note the spin projection noise limit could be reached by operating closer to the laser threshold. Conclusion ========== We have shown that magnetometry based on the IR absorption associated to the singlet states of the NV$^-$ center can be implemented by integrating a diamond sample containing the NV centers inside an external half-VCSEL cavity. This scheme does not require a narrow linewidth stabilized IR laser as in realizations based on multi-pass absorption in a resonant passive cavity [@Chatzidrosos17]. Compared to previous proposals consisting of a diamond laser using the NV$^-$ centers for optical amplification, the detrimental effects of both the parasitic ESA by the triplet excited state and the photoconversion to the NV$^0$ charge state are also circumvented since the optical gain is obtained from an independent system. Moreover, the use of a semiconductor material makes it possible to consider electric-current pumping which is of great interest for practical implementations avoiding the pump/signal configuration [@Chatzidrosos17; @Savitski17]. Our simulations show that a photon shot-noise limited sensitivity of about $700~\mathrm{fT}/\sqrt{\mathrm{Hz}}$ (and even $250~\mathrm{fT}/\sqrt{\mathrm{Hz}}$ if the ESR linewidth is limited by the spin dephasing time) can be reached for realistic parameters. Acknowledgement {#acknowledgement .unnumbered} =============== We acknowledge Isabelle Sagnes for fruitful discussions on the VCSEL fabrication. The work of JFR, FB, and TD is performed in the framework of the joint research lab between Laboratoire Aimé Cotton and Thales R&T. This project has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under the project DIADEMS (grant agreement No.611143), the German Federal Ministry of Education and Research (BMBF) within the Quantumtechnologien program (FKZ 13N14439) and from CNRS under the PICS project MOCASSIN. YD acknowledges the support of the Institut Universitaire de France and the Alexander von Humboldt Foundation. Photophysical parameters {#donnee} ======================== Table \[tableau\] gives the values of the photophysical parameters used in the simulations. Since we considered the transition between the two charge states NV$^-$ and NV$^0$, we updated the value of the IR absorption cross-section which was previously inferred from experimental data [@Dumeige13]. For this purpose, we used the same method consisting in adjusting the value of $\sigma_{s}$ to obtain the experimental value of the single-pass IR transmission reported in [@Acosta10]. \[tableau\] [@\[7]{}[l]{}]{} Parameter&Value&Reference $k_{31}=k_{32}$&$(66\pm5)~\mu\mathrm{s}^{-1}$&[@Tetienne12]$k_{35}$&$(7.9\pm4.1)~\mu\mathrm{s}^{-1}$&[@Tetienne12]$k_{45}$&$(53\pm7)~\mu\mathrm{s}^{-1}$&[@Tetienne12]$k_{61}$&$(1.0\pm0.8)~\mu\mathrm{s}^{-1}$&[@Tetienne12]$k_{62}$&$(0.7\pm0.5)~\mu\mathrm{s}^{-1}$&[@Tetienne12]$k_{56}$&$1.0~\mathrm{ns}^{-1}$&[@Acosta10]$k_{87}$&$(53\pm7)~\mu\mathrm{s}^{-1}$&[@Meirzada17]$\sigma_{g}$&$3.0\times 10^{-21}~\mathrm{m^2}$&[@Wee07]$\sigma_{g0}$&$1.8\sigma_{g}$&[@Meirzada17]$\sigma_{i}$&$(9.5\pm4.7)\times 10^{-21}~\mathrm{m^2}$&[@Meirzada17]$\sigma_{r}$&$(9.8\pm4.9)\times 10^{-21}~\mathrm{m^2}$&[@Meirzada17]$\sigma_{s}$&$(6.1\pm4.4)\times 10^{-23}~\mathrm{m^2}$& Effective optical depth of the diamond plate {#enhancement} ============================================ The maximum of transmission of the diamond plate is given by $$\mathcal{T}_{\mathrm{max}}=\frac{T_de^{-\tau}}{\left(1-R_de^{-\tau}\right)^2},$$ where $R_d=\left(\frac{n_d-1}{n_d+1}\right)^2$ and $T_d=1-R_d$ are the Fresnel coefficients associated to the index of refraction of diamond $n_d$. As $\tau\ll 1$, in the first-order of approximation, we have on one hand $e^{-\tau}\approx1-\tau$, on the other hand $\mathcal{T}_{\mathrm{max}}\approx1-\tau_{\mathrm{eff}}$ where $\tau_{\mathrm{eff}}\ll 1$ corresponds to an effective optical depth taking into account the multiple passes due to Fresnel reflections within the diamond plate. First-order calculations allow us to write $$\tau_{\mathrm{eff}}\approx\frac{1+R_d}{1-R_d}\tau,$$ which gives $\tau_{\mathrm{eff}}\approx\chi\tau$ with $$\chi=\frac{n^2_d+1}{2n_d},$$ representing the absorption enhancement factor due to Fresnel reflections. Spectral profile of the optical depth {#Odepth} ===================================== We assume a Lorentzian shape for the ESR, thus we can write $$\tau(x)=\tau_{\mathrm{off}}+\frac{\tau_{\rm on}-\tau_{\rm off}}{1+x^2},$$ where $x=\frac{2(\nu_{\mathrm{MW}}-\nu_{\rm ESR})}{\Delta\nu_{\mathrm{ESR}}}$ and $\nu_{\rm MW}$ is the frequency of the microwave. We have thus $\frac{\partial\tau}{\partial\nu_{\mathrm{ESR}}}=-\frac{2}{\Delta\nu_{\mathrm{ESR}}}\frac{\partial \tau}{\partial x}$. The maximum of sensitivity is obtained for $x=\frac{1}{\sqrt{3}}$ which gives $$\left.\frac{\partial \tau}{\partial x}\right\|_{x=\frac{1}{\sqrt{3}}}=\frac{3\sqrt{3}(\tau_{\rm off}-\tau_{\rm on})}{8},$$ and $$\tau\left(\frac{1}{\sqrt{3}}\right)=\frac{\tau_{\mathrm{off}}+3\tau_{\mathrm{on}}}{4}.$$ This maximal value of the optical depth is used to determine the optimal value of $\frac{\partial\tau}{\partial\nu_{\rm ESR}}$ given in Eq. (\[max\_sens\]). References {#references .unnumbered} ========== [10]{} url \#1[[\#1]{}]{}urlprefix\[2\]\[\][[\#2](#2)]{} Rondin L, Tetienne J P, Hingant T, Roch J F, Maletinsky P and Jacques V 2014 [Reports on Progress in Physics]{} [77]{} 056503 Schirhagl R, Chang K, Loretz M and Degen C L 2014 [Ann. Rev. Phys. Chem.]{} [65]{} 83–105 Casola F, Sar T and Yacoby A 2018 [Nat Rev Mater]{} [3]{} 17088 Balasubramanian G, Chan I, Kolesov R, Al-Hmoud M, Tisler J, Shin C, Kim C, Wojcik A, Hemmer P, Krueger A, Hanke T, Leitenstorfer A, Bratschitsch R, Jelezko F and Wrachtrup J 2008 [Nature]{} [455]{} 648 Maze J, Stanwix P, Hodges J, Hong S, Taylor J, Cappellaro P, Jiang L, Gurudev Dutt M, Togan E, Zibrov A, Yacoby A, Walworth R and MD L 2008 [ Nature]{} [455]{} 644 Acosta V M, Bauch E, Ledbetter M P, Santori C, Fu K M C, Barclay P E, Beausoleil R G, Linget H, Roch J F, Treussart F, Chemerisov S, Gawlik W and Budker D 2009 [Physical Review B]{} [80]{} 115202 Barry J F, Turner M J, Schloss J M, Glenn D R, Song Y, Lukin M D, Park H and Walsworth R L 2016 [Proceedings of the National Academy of Sciences]{} [113]{} 14133–14138 Schloss J M, Barry J F, Turner M J and Walsworth R L 2018 [Phys. Rev. Applied]{} [10]{}(3) 034044 Rogers L J, Armstrong S, Sellars M J and B M N 2008 [New J. Phys.]{} [ 10]{} 103024 Acosta V M, Jarmola A, Bauch E and Budker D 2010 [Phys. Rev. B]{} [ 82]{}(20) 201202 Kehayias P, Doherty M, English D, Fischer R, Jarmola A, Jensen K, Leefer N, Hemmer P, Manson N and Budker D 2013 [Phys. Rev. B]{} [88]{} 165202 ISSN 2469-9969 Acosta V M, Bauch E, Jarmola A, Zipp L J, Ledbetter M P and Budker D 2010 [ Appl. Phys. Lett.]{} [97]{} 174104 Clevenson H, Trusheim M E, Teale C, Schröder T, Braje D and Englund D 2015 [Nature Phys.]{} [11]{} 393–397 Dumeige Y, Chipaux M, Jacques V, Treussart F, Roch J, Debuisschert T, Acosta V, Jarmola A, Jensen K, Kehayias P and Budker D 2013 [Phys Rev B]{} [87]{} 155202 Jensen K, Leefer N, Jarmola A, Dumeige Y, Acosta V M, Kehayias P, Patton B and Budker D 2014 [Phys. Rev. Lett.]{} [112]{}(16) 160802 Chatzidrosos G, Wickenbrock A, Bougas L, Leefer N, Wu T, Jensen K, Dumeige Y and Budker D 2017 [Phys. Rev. Applied]{} [8]{}(4) 044019 Gazzano O and Becher C 2017 [Phys. Rev. B]{} [95]{}(11) 115312 Bougas L, Wilzewski A, Dumeige Y, Antypas D, Wu T, Wickenbrock A, Bourgeois E, Nesladek M, Clevenson H, Braje D, Englund D and Budker D 2018 [ Micromachines]{} [9]{} 276 Faraon A, Santori C M and Beausoleil R G 2012 [US Patent]{} US 2014/0072008A1 Jeske J, Cole J H and Greentree A D 2016 [New Journal of Physics]{} [ 18]{} 013015 Savitski V G 2017 [Journal of Physics D: Applied Physics]{} [50]{} 475602 Subedi S D, Fedorov V V, Peppers J, Martyshkin D V, Mirov S B, Shao L and Loncar M 2018 [Proc. SPIE]{} [10511]{} 105112D Jeske J, Lau D W M, Vidal X, McGuinness L P, Reineck P, Johnson B C, Doherty M W, McCallum J C, Onoda S, Jelezko F, Ohshima T, Volz T, Cole J H, Gibson B C and Greentree A D 2017 [Nat. Commun.]{} [8]{} 14000 Doherty M W, Manson N B, Delaney P, Jelezko F, Wrachtrup J and Hollenberg L C L 2013 [Phys. Rep.]{} [528]{} 1–45 Maze J, Gali A, Togan E, Chu Y, Trifonov A, Kaxiras E and Lukin M 2011 [New J Phys]{} [13]{} 025025 Thiering G and Gali A 2018 [arXiv]{} arXiv:1803.02561 Taylor J, Cappellaro P, Childress L, Jiang L, Budker D, Hemmer P, Yacobi A, Walsworth R and Lukin M 2008 [Nat. Phys.]{} [4]{} 810 Dumeige Y, Treussart F, Alléaume R, Gacoin T, Roch J F and Grangier P 2004 [Journal of Luminescence]{} [109]{} 61–67 Manson N B, Harrison J P and Sellars M J 2006 [Phys. Rev. B]{} [74]{}(10) 104303 Aslam N, Waldherr G, Neumann P, Jelezko F and Wrachtrup J 2013 [New Journal of Physics]{} [15]{} 013064 Meirzada I, Hovav Y, Wolf S and Bar-Gill N 2017 [arXiv]{} arXiv:1709.04776 Robledo L, Bernien H, van der Sar T and Hanson R 2011 [New Journal of Physics]{} [13]{} 025013 Kalb N, Humphreys P C, Slim J J and Hanson R 2018 [Phys. Rev. A]{} [ 97]{}(6) 062330 Mr[ó]{}zek M, Rudnicki D, Kehayias P, Jarmola A, Budker D and Gawlik W 2015 [EPJ Quantum Technology]{} [2]{} 22 Baili G, Alouini M, Dolfi D, Bretenaker F, Sagnes I and Garnache A 2007 [ Opt. Lett.]{} [32]{} 650–652 Baili G, Alouini M, Malherbe T, Dolfi D, Sagnes I and Bretenaker F 2009 [ EPL (Europhysics Letters)]{} [87]{} 44005 Dréau A, Lesik M, Rondin L, Spinicelli P, Arcizet O, Roch J F and Jacques V 2011 [Phys. Rev. B]{} [84]{}(19) 195204 Kubo Y, Grezes C, Dewes A, Umeda T, Isoya J, Sumiya H, Morishita N, Abe H, Onoda S, Ohshima T, Jacques V, Dréau A, Roch J F, Diniz I, Auffeves A, Vion D, Esteve D and Bertet P 2011 [Phys. Rev. Lett.]{} [107]{}(22) 220501 Tetienne J P, Rondin L, Spinicelli P, Chipaux M, Debuisschert T, Roch J F and Jacques V 2012 [New Journal of Physics]{} [14]{} 103033 Wee T L, Tzeng Y K, Han C C, Chang H C, Fann W, Hsu J H, Chen K M and Yu Y C 2007 [The Journal of Physical Chemistry A]{} [111]{} 9379–9386
--- abstract: 'We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.' address: - \| Dmitry Dolgopyat\ Department of Mathematics\ University of Maryland\ 4417 Mathematics Bldg, College Park, MD 20742, USA - \| Carlangelo Liverani\ Dipartimento di Matematica\ II Università di Roma (Tor Vergata)\ Via della Ricerca Scientifica, 00133 Roma, Italy. author: - Dmitry Dolgopyat - Carlangelo Liverani title: Random Walk in deterministically changing environment --- Introduction {#sec:intro} ============ The continuing interest in the limit properties of random walks has generated a remarkable amount of literature (see [@Sz; @Z] for a review of the field). In particular, many papers have addressed the case of random walks in dynamical environments. Apart from few papers in which special hypotheses are imposed on the form of the transition probabilities (implying that the process is reversible with respect to the stationary measure of the environment, see [@KV]) the authors have usually investigated the case in which the evolution of the environment is described by a Markov process with positive transition probabilities and the transition probabilities of the walk are close to constant (see, e.g., [@BMP; @BZ] and references therein). While such a situation has recently been settled in great generality ([@DKL]) the case of more complex local dynamics and/or far from constant transition probabilities is still wide open. In this paper we address the first issue establishing conditions under which the CLT holds for a deterministic local evolution with strong chaotic properties. Such a CLT is established for almost all the initial conditions of the environment with respect to a (natural) stationary measure (this is commonly called a [quenched CLT]{}). The deterministic dynamics is taken to be independent at each site (although our method can be easily extended to weakly interacting cases (cf. [@KL0; @DKL])). The single site dynamics is a piecewise expanding one dimensional map. While multidimensional expanding dynamics could be treated similarly the case of Anosov map poses a real problem. Indeed the technique used to control the environment dynamics is borrowed from the study of coupled map lattices (more precisely from [@KL]) and the extension of such a technique to coupled Anosov systems is still missing. In general, the extension to more general dynamics (with substantially weaker ergodic properties) would be of interest, but to obtain results in this direction new ideas seem to be needed. Note that the present strategy differs, in its probabilistic part, from the one used in [@DKL]. In particular, it is not necesary to prove absolutely continuity of the invariant measure of the environment as seen from the particle with respect to the invariant measure of the environment in a fixed reference frame. We hope that this simplification may be helpful in treating more general cases. The plan of the paper is the following. In section \[sec:model\] the system under investigation is explained in detail and the main result of the paper is precisely stated. This main result (Theorem \[thm:main\]) follows after estimating the asymptotic independence of two random walks in the same environment (Lemma \[lm:hot\]). In Section \[sec:proof\_env\] we study ergodic properties of the environment. Section \[sec:invariance\] contains the proof of the annealed (averaged) invariance principle. In section \[sec:clt-proof\] the proof of Lemma \[lm:hot\] is reduced to an estimate on the number of close encounters (Lemma \[lem:NL\]). Lemma \[lem:NL\] is proven in Section \[sec:twowalks\]. \[conv:contants\] In this paper we will use $C$ to designate a generic constant depending only on the quantities appearing in the Assumptions \[ass:assone\], \[ass:two\], \[ass:pert\], \[ass:ellipticity\] below. We will use instead $C_{a,b,c,\dots}$ for constants depending also on the parameters $a,b,c,\dots$. Consequently the actual numerical value of such constants may vary from one occurrence to the next. On the contrary we will use $C_1, C_2, \dots$, to designate constants whose value is fixed through the paper. Model and Results {#sec:model} ================= Let $I=[0,1]$ and $T:I\to I$ be a piecewise ${{\mathcal C}}^2$ topologically mixing map such that $\|D_xT\|\geq \lambda>2$ for each $x\in I$ for which the derivative is well defined. Then, $I^{{{\mathbb Z}}^d}=:\Theta$ is the space of environments (it is a measurable space with the product (Borel) $\sigma$-algebra ${{\mathcal T}}$) and $\theta\in\Theta$ is an environment on ${{\mathbb Z}}^d$. This environment evolves deterministically according to a map $F:\Theta\to\Theta$. \[ass:assone\] For each $\theta\in\Theta$ $$(F(\theta))_q:=T(\theta_q).$$ That is, the evolution is independent at each site.[^1] The evolution of the environment can be thus seen as a deterministic Markov process on the space $\Theta^{{{\mathbb N}}}=:\Omega$ such that, for all $(\theta^n)_{n\in{{\mathbb N}}}:={{\boldsymbol{\theta}}}\in\Omega$, $\theta^n:=F^n(\theta^0)$. If $\mu_0$ is the unique absolutely continuous invariant measure of $T$,[^2] then ${{\mu^e}}:=\otimes_{p\in{{\mathbb Z}}^d}\mu_0$ is the natural invariant measure for $F$ we are interested in. In fact, it is possible to show ([@KL]) that it is the only invariant measure in a reasonably large class of measures; see the precise statement below. We consider a bounded increment random walk $X_n$ in such an environment. More precisely, let $\Lambda:=\{ z\in{{\mathbb Z}}^d\;:\; \\|z\\|\leq {C_0}\}$ and $\Delta_n=X_{n+1}-X_n$, then the process is defined by the transition probabilities $$\label{eq:walk} {{\mathbb P}}(\{\Delta_n=z\}\;\|\;X_n,\theta^0)=\pi_z(\tau^{X_n}\theta^{n})$$ where $\pi_z\equiv 0$ for all $z\not\in\Lambda$, $\pi_z(\theta)$ depends only on $\{\theta_q\}_{q\in\Lambda}$, and, for each $z\in{{\mathbb Z}}^d$, $(\tau^z\theta)_i:=\theta_{i+z}$. We will be interested in the measure ${\bf P}_\nu$ on $\Omega\times ({{\mathbb Z}}^d)^{{\mathbb N}}$ determined by the above process when the environment is started with the measure $\nu$ and the walk starts from zero. We will use the notation ${\bf P}^e$ for ${\bf P}_{{{\mu^e}}}$. Finally, we will use ${{\mathbb E}}$ for the expectation with respect to the latter measure and ${{\mathbb E}}_\nu$ for the expectations with respect to the process ${\bf P}_\nu$. \[ass:two\] The functions $\{\pi_z\}_{z\in\Lambda}$ belong to ${{\mathcal C}}^1$. The next assumption depends on a parameter ${\varepsilon}>0$. \[ass:pert\] There exists $\{a_z\}_{z\in\Lambda}\subset {{\mathbb R}}_+$, $\sum_{z\in\Lambda}a_z=1$, such that $$\\|\pi_z-a_z\\|_{{{\mathcal C}}^1}\leq a_z{\varepsilon}.$$ In the following, when we will say “assumption \[ass:pert\] holds for ${\varepsilon}_i$" we will mean that it holds with ${\varepsilon}={\varepsilon}_i$. The values ${\varepsilon}_i$ will be taken small enough for Theorem \[thm:mixing\], Proposition \[lem:clt-averaged\], Lemma \[lem:up\] and Lemma \[lem:variation\] to hold. \[ass:ellipticity\] For each $l\in{{\mathbb Z}}^d\setminus\{0\}$, the function $\left\|\sum\limits_{z\in\Lambda}\pi_z e^{i\langle l,z\rangle}\right\|\in{{\mathcal C}}^0(I^\Lambda, {{\mathbb R}}_+)$ is not identically equal to $1$. It is well known that to study the properties of $X_n$ it is convenient to study the process of the environment as seen from the particle. In fact, such a process can be considered in several fashions of which the following will be relevant in the sequel. Process of the environment as seen from the particle ---------------------------------------------------- Consider the process ${{\boldsymbol{\omega}}}=:(\omega^n)_{n\in{{\mathbb N}}}\in\Omega$ described by the action of the Markov operator $S:L^\infty(\Theta)\to L^\infty(\Theta)$ defined by $$\label{eq:semigroup-env} Sf(\omega):=\sum_{z\in\Lambda}\pi_z(\omega)f\circ F(\tau^z\omega)=:\sum_{z\in\Lambda}S_z f.$$ It is easy to verify that the process ${{\boldsymbol{\omega}}}$, $\omega^0=\theta$, has the same distribution as the process $(\tau^{X_n}\theta^n)_{n\in{{\mathbb N}}}$, $\theta^0=\theta$. We can then consider the measure ${{\mathbb P}}_\nu$ on $\Omega$ of the associated Markov process started with a measure $\nu$. In analogy with the techniques used in the study of coupled map lattices [@KL] it is then natural to restrict the space of measures on which $S'$ acts.[^3] To this end we start by defining the following norms $$\label{eq:norms} \begin{split} &\|\mu\|:=\sup_{\|{\varphi}\|_{{{\mathcal C}}^0(\Theta,{{\mathbb R}})}\leq 1}\mu({\varphi})\\ &\\|\mu\\|:=\sup_{i\in{{\mathbb Z}}^d}\sup_{\|{\varphi}\|_{{{\mathcal C}}^0(\Theta,{{\mathbb R}})}\leq 1}\mu(\partial_{\theta_i}{\varphi}) \end{split}$$ We then consider the Banach space of complex valued measures[^4] $$\label{eq:banach} {{\mathcal B}}:=\{\mu\in{{\mathcal M}}(\Theta)\;:\; \\|\mu\\|<\infty\}.$$ It is easy to check that such measures have finite dimensional marginals absolutely continuous w.r.t. Lebesgue and the densities are functions of bounded variations with variations bounded by the norm of the measure. Moreover ${{\mu^e}}$ is the unique invariant measure for $F$ belonging to ${{\mathcal B}}$, [@KL]. \[thm:mixing\] For each dynamics $F$ satisfying assumption \[ass:assone\] and transition probabilities satisfying assumption \[ass:two\], the operator $S'$, is a bounded operator on ${{\mathcal B}}$. In addition, there exists ${\varepsilon}_0>0$, depending on $F$, such that if assumption \[ass:pert\] holds for ${\varepsilon}_0$, then there exists a unique invariant probability measure ${{\mu^w}}\in{{\mathcal B}}$ ($S'{{\mu^w}}={{\mu^w}}$). This measure enjoys the following properties: There exists $\eta\in(0,1)$ such that for each $\nu\in{{\mathcal B}}$ and local functions $\varphi,\phi\in{{\mathcal C}}^0$ each depending only on $L$ variables with the two sets of dependency having distance at least $M$ 1. $\|\nu(S^n\phi)-{{\mu^w}}(\phi)\nu(1)\|\leq CL\eta^n\|\phi\|_{\infty}\\|\nu\\|$ 2. $\|{{\mu^w}}(\varphi\phi)-{{\mu^w}}(\varphi){{\mu^w}}(\phi)\|\leq CL\eta^{\frac{M}{2{C_0}}}\|\varphi\|_{\infty}\|\phi\|_\infty$. The proof of the above theorem can be found in Section \[sec:mixone\]. \[rem:stationary\] Theorem \[thm:mixing\] implies that the process ${{\mathbb P}}_{{{\mu^w}}}$ is a stationary (and ergodic) process. Annealed statistical properties {#subsec:averaged} ------------------------------- \[lem:clt-averaged\] For each dynamics $F$ satisfying assumption \[ass:assone\] and transition probabilities satisfying assumption \[ass:two\], if assumption \[ass:pert\] is satisfied for ${\varepsilon}_0$ (where ${\varepsilon}_0>0$ is as in Theorem \[thm:mixing\]), then there exists a vector $v\in{{\mathbb R}}^d$ and a matrix ${{\Sigma^2}}\geq0$ such that, for each probability measure $\nu\in{{\mathcal B}}$ we have $$\begin{split} &\frac 1N {{\mathbb E}}_\nu(X_N)\to v \\ &\frac{X_N-vN}{\sqrt{N}}\Rightarrow {{\mathcal N}}\left(0, {{\Sigma^2}}\right)\quad under \ {\bf P}_\nu. \end{split}$$ Moreover, there exists ${C_1}>0$ such that, setting ${\tilde{X}}_N:=X_N-vN$, the following inequality holds for all $N\in{{\mathbb N}}$ and ${{t}}\in{{\mathbb R}}^d$: $$\left\|{{\mathbb E}}_\nu\left(e^{\frac i{\sqrt N}\langle {{t}},\tilde X_N\rangle}\right)-e^{-\frac12\langle{{t}},{{\Sigma^2}}{{t}}\rangle}\right\|\leq {C_1}(1+\\|{{t}}\\|^3)N^{-\frac 12}\\|\nu\\|.$$ Finally, if Assumption \[ass:ellipticity\] is also satisfied, then ${{\Sigma^2}}>0$. Let us start noticing that $$\frac 1N{{\mathbb E}}_\nu\left(X_N\right)=\frac 1N\sum_{k=0}^{N-1}{{\mathbb E}}_\nu(\Delta_k)= \frac 1N\sum_{k=0}^{N-1}{{\mathbb E}}_\nu\left({{\mathbb E}}_\nu(\Delta_k\;\|\;{{\mathcal F}}_k)\right),$$ where ${{\mathcal F}}_k:=\sigma\{\theta^0,X_1,\dots,X_k\}$. The relevance of the process as seen from the particle is due to the following fact: $$\label{eq:S} {{\mathbb E}}_\nu(\Delta_k\;\|\;{{\mathcal F}}_k)=\sum_{z\in\Lambda}z\,\pi_z(\tau^{X_{k}}\theta^k)=\sum_{z\in\Lambda}z\,\pi_z(\omega^k)=: g(\omega^k).$$ Thus, $$\frac 1N{{\mathbb E}}_\nu\left(X_N\right)=\frac 1N\sum_{k=0}^{N-1}\nu(g(\omega^k))=\frac 1N\sum_{k=0}^{N-1}[(S')^k\nu]( g).$$ Accordingly, Theorem \[thm:mixing\] implies $$\label{eq:average-one} \lim_{N\to\infty}\frac 1N{{\mathbb E}}_\nu\left(X_N\right)={{\mu^w}}(g)=:v.$$ To prove the CLT let ${\tilde{\Delta}}_n={\tilde{X}}_{n+1}-{\tilde{X}}_n,$ then $${{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\tilde X_N\rangle}\right)={{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\tilde X_{N-1}\rangle}{{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\Delta_{N-1}-v\rangle}\;\big\|\;{{\mathcal F}}_{N-1}\right)\right).$$ Since $${{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\Delta_{k}-v\rangle}\;\big\|\;{{\mathcal F}}_{k}\right)=\sum_{z\in\Lambda}\pi_z(\tau^{X_k}\theta^k)e^{\frac{i}{\sqrt N}\langle {{t}},z-v\rangle}$$ it is natural to introduce the operators, for all $t\in{{\mathbb C}}^d$, $$\label{eq:operator_clt} {{\mathcal M}}_{{{t}}} h(\theta):=\sum_{z\in\Lambda}\pi_z(\theta)e^{ \langle {{t}},z-v\rangle}h(\tau^zF(\theta))=\sum_{z\in\Lambda}e^{\langle {{t}},z-v\rangle}S_z h.$$ Then, $${{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\Delta_{k}-v\rangle}\;\big\|\;{{\mathcal F}}_{k}\right)=({{\mathcal M}}_{it/\sqrt{N}} 1)(\tau^{X_{k}}\theta^{k}),$$ and the reader can then check, by induction, the formula $$\label{eq:clt_formula} {{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\tilde X_N\rangle}\right)=\nu({{\mathcal M}}_{it/\sqrt{N}}^N 1).$$ The operator ${{\mathcal M}}_{{{t}}}'$ acting on the space ${{\mathcal B}}$ is an analytic perturbation of the operator $S'={{\mathcal M}}_0'$. Unfortunately, $S'$ does not have a nice spectrum on ${{\mathcal B}}$, so in order to apply usual perturbation theory, it is necessary to lift the dynamics to an appropriate space in the spirit of [@BGK]. We do so in section \[sec:pert\] where we prove the following result. \[lem:up\] Under the assumptions of Theorem \[thm:mixing\] there exists ${C_2}>0$ and a function $\alpha_{{t}}$ analytic near $\{\\|{{t}}\\|\leq {C_2}\}$ such that for each $n\in{{\mathbb N}}$, probability measure $\nu\in{{\mathcal B}}$ and local function $f$ depending on $L$ variables we have $$\begin{split} &\|\nu({{{\mathcal M}}}_{{{t}}}^nf)\|\leq CL\|\alpha_{{t}}^n\|\,\|f\|_\infty\\|\nu\\|\\ &\nu({{\mathcal M}}_{{{t}}}^n1)=\alpha_{{t}}^n(1+{{\mathcal O}}({{t}}\\|\nu\\|))+{{\mathcal O}}(\eta^n\\|\nu\\|). \end{split}$$ Moreover, $\alpha_0=1,$ $\dot{\alpha}_0=0$ and $\ddot{\alpha}_0\geq 0$ (the “dot" stands for the derivatives with respect to ${{t}}$). Finally, if Assumption \[ass:ellipticity\] is also satisfied, then $\ddot{\alpha}_0> 0$. Using Lemma \[lem:up\] and setting ${{\Sigma^2}}:= \ddot{\alpha}_0$, we have $$\nu({{\mathcal M}}_{i{{t}}/\sqrt N}^N 1)=\alpha_{i{{t}}/\sqrt N}^N(1+{{\mathcal O}}({{t}}N^{-\frac 12}\\|\nu\\|)+ {{\mathcal O}}(\eta^N\\|\nu\\|).$$ We can finally compute, for $\\|{{t}}\\|\leq CN^{\frac 16}$ and $N$ large enough,[^5] $$\begin{split} {{\mathbb E}}_\nu\left(e^{\frac{i}{\sqrt N}\langle {{t}},\tilde X_N\rangle}\right)&=\alpha_{i{{t}}/\sqrt N}^N+{{\mathcal O}}\left(\frac{1+\\|{{t}}\\|}{\sqrt N}\\|\nu\\|\right)\\ &=\left(1-\frac 1{2N}\langle{{t}},{{\Sigma^2}}{{t}}\rangle+{{\mathcal O}}(\\|{{t}}\\|^3N^{-\frac 32})\right)^N+{{\mathcal O}}\left(\frac {1+\\|{{t}}\\|}{\sqrt N}\\|\nu\\|\right)\\ &=e^{-\frac 1{2}\langle{{t}},{{\Sigma^2}}{{t}}\rangle+{{\mathcal O}}(\\|{{t}}\\|^3N^{-\frac 12})}+{{\mathcal O}}\left(\frac {1+\\|{{t}}\\|}{\sqrt N}\\|\nu\\|\right), \end{split}$$ from which Proposition \[lem:clt-averaged\] follows. Next, we need a large deviations estimate. \[lem:largedev\] Under the assumptions of Theorem \[thm:mixing\] and Assumption \[ass:ellipticity\] there exists $a_0>0$ such that for each $\nu\in {{\mathcal B}}$, $n, m\in{{\mathbb N}}$ and $a\in(0,a_0)$ the following holds true $${\bf P}_\nu\left(\left\{\left\|\frac 1m(\tilde X_{n+m}-\tilde X_n)\right\|\geq a\right\}\right)\leq Ce^{-Ca^2m}(\\|\nu\\|+1).$$ Again this large deviation result can be obtained by perturbation theory of the operator $S'$. Indeed, for each $w\in{{\mathbb R}}^d$, $\\|w\\|=1$ and $t\in{{\mathbb R}}$, $$\begin{split} {\bf P}_\nu&\left(\left\{\frac 1m\langle w,\tilde X_{n+m}-\tilde X_n\rangle\geq a\right\}\right)\leq {{\mathbb E}}_\nu\left(e^{{{t}}(\langle w,\tilde X_{n+m}-\tilde X_n\rangle-am)}\right)\\ &\quad\quad=e^{-tam}[(S')^n\nu]({{\mathcal M}}_{tw}^m1). \end{split}$$ Since by Theorem \[thm:mixing\] $\sup_{n\in{{\mathbb N}}}\\|(S')^n\nu\\|\leq C\\|\nu\\|$, we can apply Lemma \[lem:up\] and obtain, for $t\leq {C_2}$, $$\begin{split} {\bf P}_\nu&\left(\left\{\frac 1m\langle w,\tilde X_{n+m}-\tilde X_n\rangle\geq a\right\}\right)\leq Ce^{-tam}\alpha_{tw}^m(1+C\|t\|\\|\nu\\|)+C\eta^m\\|\nu\\|\\ &\leq Ce^{-tam}\left(1+\frac {t^2}2\langle w,{{\Sigma^2}}w\rangle+{{\mathcal O}}(\|t\|^3)\right)^m(1+\\|\nu\\|)+C\eta^m\\|\nu\\|\\ &\leq Ce^{-tam+\frac {mt^2}2\langle w,{{\Sigma^2}}w\rangle+{{\mathcal O}}(m\|t\|^3)}(1+\\|\nu\\|)+C\eta^m\\|\nu\\|. \end{split}$$ Finally, choosing ${{t}}=\frac {a}{\langle w,{{\Sigma^2}}w\rangle}$ and $a_0$ so small that the term ${{\mathcal O}}(t^3)$ is small with respect to $\frac {a_0^2}{2\langle w,{{\Sigma^2}}w\rangle}$, $\frac {a_0}{\langle w,{{\Sigma^2}}w\rangle}< {C_2}$ and $e^{-\frac {a_0^2}{2\langle w,{{\Sigma^2}}w\rangle}}\geq \eta$ $${\bf P}_\nu\left(\left\{\frac 1m\langle w,\tilde X_{n+m}-\tilde X_n\rangle\geq a\right\}\right)\leq Ce^{-Ca^2m}(1+\\|\nu\\|).$$ We then conclude by noticing that the above estimate for all the $w$ in the set $\{\pm e_i\}_{i=1}^d$, where $\{e_i\}_{i=1}^d$ is the standard base of ${{\mathbb R}}^d$, implies the Lemma. Main result: Quenched C.L.T {#subsec:general} --------------------------- Let ${\bf P}_\theta$ be the measure ${\bf P}^e$ conditioned to starting the environment in the configuration $\theta$. We will use ${{\mathbb E}}_\theta$ for the expectation with respect to ${\bf P}_\theta$. \[thm:main\] For each dynamics $F$ satisfying assumption \[ass:assone\] and transition probabilities satisfying assumptions \[ass:two\] and \[ass:ellipticity\], if assumption \[ass:pert\] is satisfied for ${\varepsilon}_1$ (where ${\varepsilon}_0\geq {\varepsilon}_1>0$ is as in Lemma \[lem:variation\]), then (using the same notations as in Proposition \[lem:clt-averaged\]), for ${{\mu^e}}$ almost all $\theta\in\Theta$ the following holds (a) $\frac {1}N X_N \to v$ $ {\bf P}_\theta\; a.s.;$ (b) (c) $\frac{X_N-vN}{\sqrt{N}}\Rightarrow {{\mathcal N}}\left(0, {{\Sigma^2}}\right)$ under ${\bf P}_\theta.$ Lemma \[lem:largedev\] implies the bound ${\bf P}^e(\{\|N^{-1} X_N-v\|\geq {\varepsilon}\})\leq Ce^{-C{\varepsilon}^{2}N}$; (a) follows then by applying Borel-Cantelli. To prove (b) let $\alpha\in (0,1)$ be a number to be specified later. Combining Lemma \[lem:largedev\] and Borel-Cantelli Lemma we see that for any $\delta>0,$ ${\bf P}^e$-almost surely for any $k$ and $0\leq j\leq 2^{(1-\alpha) k}$ we have[^6] $$\label{eq:smallosc} \max_{m\in [2^k+j 2^{\alpha k}, 2^k+(j+1) 2^{\alpha k}]} \left\|{\tilde{X}}_m-{\tilde{X}}_{2^k+j 2^{\alpha k}}\right\|\leq C_{\theta, X, \delta}\, 2^{(\frac{\alpha}{2}+\delta)k}$$ By Fubini Theorem for almost every $\theta$ holds ${\bf P}_\theta$ almost surely. Therefore it is enough to prove the convergence along the subsequence $n_{jk}=2^k +j 2^{\alpha k}$ where $0\leq j\leq 2^{(1-\alpha) k}.$ To conclude it suffices to prove that there exists $\beta>0$ and $b\in{{\mathbb N}}$ such that for each smooth function ${\varphi}:{{\mathbb R}}^d\to{{\mathbb R}}$ compactly supported in a box of size $L$ the following inequality holds $$\label{eq:hot0} {{\mathbb E}}\left(\left\|{{\mathbb E}}_\theta({\varphi}(N^{-\frac 12}\tilde X_N))-{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})\right\|^2\right)\leq C_L \|{\varphi}\|_{{{\mathcal C}}^b}N^{-\beta},$$ where ${{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}$ is the expectation with respect to the Gaussian measure ${{\mathcal N}}(0,{{\Sigma^2}})$. Indeed, denote $$\xi_{jk}={{\mathbb E}}_\theta\left({\varphi}\left(\frac{{\tilde{X}}_{n_{jk}}}{\sqrt{n_{jk}}}\right)\right).$$ Then and Chebyshev inequality imply $$\label{eq:hot2} {\bf P}^e\left(\left\{\left\|\xi_{jk}-{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})\right\|\geq {\varepsilon}\right\}\right)\leq C_L\|{\varphi}\|_{{{\mathcal C}}^{b}}{\varepsilon}^{-2}n_{jk}^{-\beta}.$$ Hence, by finally choosing $\alpha$ such that $\alpha+\beta>1$, $\xi_{jk}\to{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})$ almost surely. Next, choose a family ${\varphi}_m$ which is dense in ${{\mathcal C}}_0^0({{\mathbb R}}^d).$ Then, for almost every $\theta$, we have $${{\mathbb E}}_\theta\left({\varphi}_m\left(\frac{{\tilde{X}}_{n_{jk}}}{\sqrt{n_{jk}}}\right)\right)\to {{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi}_m)$$ for all $m.$ Then, for any such $\theta$ $${{\mathbb E}}_\theta\left({\varphi}\left(\frac{{\tilde{X}}_{n_{jk}}}{\sqrt{n_{jk}}}\right)\right)\to {{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})$$ for any continuous compactly supported ${\varphi}$ proving (b). The result is then proved provided is true. It turns out that can be conveniently interpreted in terms of two independent walks $X_N,Y_N$ in the same environment. In fact, calling ${{\mathbb E}}^2$ the expectation with respect to such a process it follows $$\begin{split} {{\mathbb E}}&\left(\left\|{{\mathbb E}}_\theta({\varphi}(N^{-\frac 12}\tilde X_N))-{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})\right\|^2\right)= {{\mathbb E}}^2({\varphi}(N^{-\frac 12}\tilde X_N){\varphi}(N^{-\frac 12}\tilde Y_N))\\ &\quad-2{{\mathbb E}}({\varphi}(N^{-\frac 12}\tilde X_N)){{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})+{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})^2\\ =&{{\mathbb E}}^2({\varphi}(N^{-\frac 12}\tilde X_N){\varphi}(N^{-\frac 12}\tilde Y_N))-{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})^2+{{\mathcal O}}(L^d\\|{\varphi}\\|_{{{\mathcal C}}^{d+1}}N^{-\beta}). \end{split}$$ where we have used the quantitative estimate in the Proposition \[lem:clt-averaged\].[^7] We have thus reduced the proof of the theorem to proving the following. \[lm:hot\] $$\left\|{{\mathbb E}}^2({\varphi}(N^{-\frac 12}\tilde X_N){\varphi}(N^{-\frac 12}\tilde Y_N))-{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})^2\right\|\leq C\\|{\varphi}\\|_{{{\mathcal C}}^b}N^{-\beta}.$$ Lemma \[lm:hot\] is proved in Section \[sec:clt-proof\]. Proofs: Environment {#sec:proof_env} =================== In this section we establish all the needed properties of the environment dynamics. The basic idea is to prove that the environment enjoys very strong mixing properties. Our first aim is to prove Theorem \[thm:mixing\], that is exponential decay of space-time correlations. Note that $S1=1$, hence $$\label{eq:contra} \|S'\mu\|\leq \|\mu\|,$$ that is $S$ is a contraction in the $\|\cdot\|$ norm. In addition, it is possible to prove (although we will not use it here) that there exists $A,B>0$ and $\sigma\in (2\lambda ^{-1},1)$ such that, for each $n\in{{\mathbb N}}$ and $\mu\in{{\mathcal B}}$,[^8] $$\\|(S')^n\mu\\|\leq A\sigma^{n}\\|\mu\\|+B\|\mu\|.$$ For finitely many sites the above estimate would suffice to prove that the operator $S'$ is quasi-compact and this, together with the topologically mixing assumption, would imply the existence of a spectral gap. Unfortunately, such a proof is based on the compactness of the unit ball $\{\mu\in {{\mathcal B}}\;:\;\\|\mu\\|\leq 1\}$ in the topology of the $\|\cdot\|$ norm which fails when one considers infinitely many sites. The obvious idea is to use explicitly the fact that the dynamics in different sites are independent, hence the system has a product structure, yet this is a subtle issue. To understand better the situation, let us recall few fact about the single site systems. At each site we have the dynamical system $(I=[0,1], T)$. Let us consider the norm in ${{\mathcal M}}(I)$ given by $$\\|\nu\\|_0=\sup_{\|{\varphi}\|_{{{\mathcal C}}^0}\leq 1}\mu({\varphi}'),$$ where ${\varphi}'$ is the derivative of ${\varphi}$. The Banach space $B=\{\nu\in{{\mathcal M}}\;:\;\\|\nu\\|_0<\infty\}$ consists of measures absolutely continuous with respect to the Lebesgue measure $m_{{{\mathcal L}}}$. In addition, if $d\nu=h\, dm_{{{\mathcal L}}}$, then the density $h$ is a function of bounded variation and $\|h\|_{BV}=\\|\nu\\|$.[^9] By a change of variable one can compute that, if $d\nu=h\, dm_{{{\mathcal L}}}$, then $d(T'\nu)=({{\mathcal L}}h) dm_{{{\mathcal L}}}$, where the operator ${{\mathcal L}}$ is defined as $${{\mathcal L}}h(x)=\sum_{y\in T^{-1}(x)}\|D_yT\|^{-1}h(y).$$ The operator ${{\mathcal L}}$ is often called the Ruelle-Perron-Frobenious transfer operator. It is well known that, if $T$ is topologically mixing, then the operator ${{\mathcal L}}$, acting on $BV$ has $1$ as a simple eigenvalue (corresponding to the unique invariant measure absolutely continuous with respect to Lebesgue) and enjoys a spectral gap, that is there exists $\eta_0\in (0,1)$ such that the rest of the spectrum is strictly contained in a disk of radius $\eta_0$ (see, e.g., [@Babook] for details). Clearly the above implies that $T'$ has a spectral gap when acting on $B$. Unfortunately, it turns out that the tensor products of $B$ is a too small a space to be really useful for our purposes.[^10] This is the reason why we have introduced the spaces ${{\mathcal B}}$ which is a generalization of measures with density of bounded variations to the infinite dimensional setting. Yet on such a space $S'$ does not behave very well and we will use an abstract covering space on which the dynamics will exhibit a spectral gap. More precisely, we would like to introduce a Banach space ${{\overline{{{\mathcal B}}}}}$ and two (possibly only partially defined) maps $\Psi:{{\mathcal B}}\to{{\overline{{{\mathcal B}}}}}$ and ${{\operatorname{Pr}\,}}:{{\overline{{{\mathcal B}}}}}\to{{\mathcal B}}$ and an operator $\bf S:{{\overline{{{\mathcal B}}}}}\to{{\overline{{{\mathcal B}}}}}$ such that the dynamics of the latter covers the dynamics of $S'$ as illustrated by the following commutative diagram $$\label{eq:commutation} \begin{CD} {{\overline{{{\mathcal B}}}}}@> {\bf S}^n >>{{\overline{{{\mathcal B}}}}}\\ @A{\Psi}AA @VV{{{\operatorname{Pr}\,}}}V\\ {{\mathcal B}}@>>{(S')^n}> {{\mathcal B}}\end{CD}$$ We will first define the space ${{\overline{{{\mathcal B}}}}}$ and the map $\bf S$. Then we will prove Theorem \[thm:mixing\] by proving that $\bf S$ has a spectral gap on ${{\overline{{{\mathcal B}}}}}$. Next we will obtain others, more refined, results by using the same strategy (albeit applied to different operators). Covering dynamics. {#sec:covering} ------------------ First we define the above mentioned abstract space. Let ${{\overline{{{\mathcal B}}}}}:={{\mathbb C}}\times\left[\times_{p\in{{\mathbb Z}}^d}{{\mathcal B}}_p\right]$ where[^11] $${{\mathcal B}}_p:=\{\mu\in{{\mathcal B}}:\mu({\varphi})=0\; \forall \,{\varphi}\in{{\mathcal C}}^0(\Theta)\text{ that do not depend on }\theta_p\}.$$ The vector space ${{\overline{{{\mathcal B}}}}}$ is a Banach space when equipped with the norm $$\\|{{\boldsymbol{\mu}}}\\|:=\sup\{\|c_\mu\|,\\|\mu_p\\|\;:\;p\in{{\mathbb Z}}^d\}\ .$$ Here we use the notational convention that an element ${{\boldsymbol{\mu}}}\in{{\overline{{{\mathcal B}}}}}$ has components $c_\mu\in{{\mathbb C}}$ and $\bar\mu:=(\mu_p)_p$ with $\mu_p\in{{\mathcal B}}_p$. Next we define a projection ${{\operatorname{Pr}\,}}:D\subset{{\overline{{{\mathcal B}}}}}\to{{\mathcal B}}$ and a map $\Psi:{{\mathcal B}}\to{{\overline{{{\mathcal B}}}}}$ allowing to transfer objects between the two spaces. Let $\mu_\in {{\mathcal B}}$ be a fixed probability measure on $I$ then $m=\otimes_{{{\mathbb Z}}^d}\mu_$ is a product probability measure on $\Theta$. For each ${{\boldsymbol{\mu}}}=(c_\mu,(\mu_p)_p)\in{{\overline{{{\mathcal B}}}}}$ and local function $f$ we define $$\label{eq:decomp} {{\operatorname{Pr}\,}}{{{\boldsymbol{\mu}}}}(f):=c_\mu\,m(f)+\sum_{p\in{{\mathbb Z}}^d}\mu_p(f),$$ which makes clear in which sense ${{\overline{{{\mathcal B}}}}}$ “covers" ${{\mathcal B}}$ ( or ${{\mathcal M}}(\Theta)$). Note that, although ${{\operatorname{Pr}\,}}{{{\boldsymbol{\mu}}}}(f)$ is well defined on each local function, ${{\operatorname{Pr}\,}}{{{\boldsymbol{\mu}}}}$ is not necessarily a measure. Let ${{\overline{{{\mathcal B}}}}}_M\subset {{\overline{{{\mathcal B}}}}}$ be such that the elements of ${{\operatorname{Pr}\,}}{{\overline{{{\mathcal B}}}}}_M$ give rise to bounded linear functionals on the space of local functions, and hence identify uniquely a measure.[^12] We will call such a measure ${{\operatorname{Pr}\,}}{{{\boldsymbol{\mu}}}}$. The choice of the map $\Psi$ is quite arbitrary, we will fix a convenient one. Consider a strict total ordering $\prec$ of ${{\mathbb Z}}^{d}$ such that $0\prec p$ for each $p\in{{\mathbb Z}}^{d}\setminus\{0\}$ and the sets $\{q\;:\;q\prec p\}$ are finite for each $p\in{{\mathbb Z}}^d$.[^13] Let $q_+$ be the successor of $q$ (that is, $q\prec q_+$ and there are no $q'\in{{\mathbb Z}}^d$ such that $q\prec q'\prec q_+$). For each $q\in{{\mathbb Z}}^d$ we can then consider the $\sigma$-algebra ${{\mathcal F}}_q^0$ determined by all the variables $\theta_{q'}$ with $q\preceq q'$, hence ${{\mathcal F}}_0^0$ is the complete $\sigma$-algebra. Next, for each $f\in{{\mathcal C}}^0(\Theta)$ and $q\in{{\mathbb Z}}^d$, define the operator $J_qf=m(f\;\|\;{{\mathcal F}}_q^0)-m(f\;\|\;{{\mathcal F}}_{q_+}^0)$. For each local function $f$ we can write[^14] $$f=m(f)+\sum_{q\in{{\mathbb Z}}^d}J_q(f)$$ Accordingly, for each $\mu\in{{\mathcal B}}$ we define $\mu_q(f):=J_q'\mu(f)\in{{\mathcal B}}_q$, and the lift $$\Psi(\mu):=(\mu(1),(J_q'\mu)_q)\ .$$ Note that $\Psi$ is a bounded operator. Indeed if $q\prec p$, then $$\|J_q'\mu (\partial_{\theta_p}{\varphi})\|\leq \|\mu\|\, \|J_q(\partial_{\theta_p}{\varphi})\|_\infty\leq 2 \|\mu\|\,\\|\mu_\\|\,\|{\varphi}\|_\infty.$$ If $q\succ p$, then $$\|J_q'\mu (\partial_{\theta_p}{\varphi})\|\leq \|\mu(\partial_{\theta_p}J_q{\varphi})\|\leq \\|\mu\\|\, \|J_q({\varphi})\|_\infty\leq 2 \\|\mu\\|\,\|{\varphi}\|_\infty.$$ Finally, for $q=p$, we have $\|J_q'\mu (\partial_{\theta_p}{\varphi})\|\leq (\|\mu\|\,\\|\mu_\\|+\\|\mu\\|)\|{\varphi}\|_\infty$. In other words there exists ${C_3}>0$, depending on the choice of $\mu_$, such that $$\label{eq:J-bound} \\|\Psi(\mu)\\|\leq {C_3}\\|\mu\\|.$$ Clearly, $\Psi({{\mathcal B}})\subset D$ and for each $\mu\in{{\mathcal B}}$ it holds true $${{\operatorname{Pr}\,}}(\Psi(\mu))=\mu.$$ Now that we know how to lift measures, we can address the dynamics. For all $z\in{{\mathbb Z}}^d$, $\tau^z\mu_q({\varphi}):=\mu_q({\varphi}\circ \tau^z)=0$ if ${\varphi}$ does not depend on $\theta_{q-z}$. Thus we can define the decomposition for $\tau^{z}\mu$ via the decomposition $\mu_q=J_q'\mu$ of $\mu$: $$\tau^{z}\mu=\sum_{q\in{{\mathbb Z}}^d}(\tau^{z}\mu)_q:=\sum_{q\in{{\mathbb Z}}^d}\tau^{z}\mu_{q-z}.$$ Setting $\Lambda_1=\cup_{z\in\Lambda}\tau^{-z}\Lambda$, we can define the covering dynamics ${\bf S}{{{\boldsymbol{\mu}}}}$ by $(c_\mu, \bar S\bar\mu+\bar \zeta c_\mu)$ where $\bar\zeta:=(\zeta_q)$ with $\zeta_q:=J_q'S'm=:J_q'\zeta$ and $$\label{eq:dynone} (\bar S\bar\mu)_p=\begin{cases}\sum_{z\in\Lambda}S_z'\mu_{p-z}+\sum_{q\in\Lambda_1}\sum_{z\in\Lambda}J_p'\hat S_z'\mu_{q-z}&\text{ for each }p\not\in \Lambda_1,\\ \sum_{z\in\Lambda}A_z'\mu_{p-z}+\sum_{q\in\Lambda_1}\sum_{z\in\Lambda}J_p'\hat S_z'\mu_{q-z}&\text{ for each }p\in \Lambda_1, \end{cases}$$ where $S_zf(\omega):=\pi_z(\omega)f\circ F\circ \tau^z(\omega)$, $A_zf(\omega):=a_z f\circ F\circ \tau^z(\omega)$ and $\hat S_z:=S_z-A_z$. It is easy to check the following. \[lem:covering\] The operator $\bf S$ is well defined as a bounded operator from ${{\overline{{{\mathcal B}}}}}$ to ${{\overline{{{\mathcal B}}}}}$. For each ${{\boldsymbol{\mu}}}\in{{\overline{{{\mathcal B}}}}}$ and continuous local function $f$ we have $\Pr({\bf S}{{\boldsymbol{\mu}}})(f)=\Pr({{\boldsymbol{\mu}}})(Sf)$ which implies that for each $n\in{{\mathbb N}}$ and $\mu\in{{\mathcal B}}$ we have $\Pr {\bf S}^n\Psi\mu={S^n}'\mu$ We have thus established a setting in which the commutative diagram holds true. Mixing properties of the environment {#sec:mixone} ------------------------------------ We have now the necessary machinery to deal with the statistical properties of the environment. Let us first discuss the environment dynamics $F$. Its basic properties are described by the so called Lasota-Yorke inequalities asserting that there exists $B>0$ such that, for all $n\in{{\mathbb N}}$,[^15] $$\label{eq:lasota_0} \begin{split} &\|F'\mu\|\leq \|\mu\|\\ &\\|(F')^n\mu\\|\leq (2\lambda^{-1})^n\\|\mu\\|+B\|\mu\|. \end{split}$$ Note that the above implies that $\{\\|(F')^n\mu\\|\}_{n\in{{\mathbb N}}}$ is bounded. \[lem:gap0\] There exists $\eta_\in (\eta_0,1)$ such that, for each $q\in{{\mathbb Z}}^d$, $\mu_q\in{{\mathcal B}}_q$, $$\\|(F')^n\mu_q\\|\leq C\eta_^n\\|\mu_q\\|.$$ For each local function ${\varphi}\in{{\mathcal C}}^0$, we can define ${\varphi}_{\theta_{\neq q}}^n(\xi):={\varphi}(\theta^{n,\xi})$, where $\theta^{n,\xi}_p=T^n\theta_p$ for each $p\neq q$ while $\theta^{n,\xi}_q=\xi$, $$\begin{split} \|(F')^n\mu_q({\varphi})\|&=\left\|\mu_q\left(\partial_{\theta_q}\int_0^1[\chi_{[0,\theta_q]}(\xi)-\theta_q] {\varphi}_{\theta_{\neq q}}^n(T^n\xi) d\xi\right)\right\|\\ &\leq \\|\mu_q\\|\cdot\left\|\int_0^1{{\mathcal L}}^n[\chi_{[0,\theta_q]}(\xi)-\theta_q] \cdot {\varphi}_{\theta_{\neq q}}^n(\xi) d\xi\right\|_\infty\\ &\leq C\\|\mu_q\\|\,\left\|{{\mathcal L}}^n[\chi_{[0,\theta_q]}-\theta_q]\right\|_{BV}\|{\varphi}\|_\infty\leq C\\|\mu_q\\|\eta_0^n\|{\varphi}\|_\infty, \end{split}$$ by the spectral gap of ${{\mathcal L}}$ and the fact that $m_{{{\mathcal L}}}(\chi_{[0,\theta_q]}-\theta_q)=0$, i.e. it is a zero average function. Then, by the Lasota-Yorke inequality, $$\begin{split} \\|(F')^{j+k}\mu_q\\|&\leq (2\lambda^{-1})^j\\|(F')^{k}\mu_q\\|+B\|(F')^{k}\mu_q\|\\ &\leq (2\lambda^{-1})^j\left[(2\lambda^{-1})^k\\|\mu_q\\|+B\|\mu\|\right]+B C\eta_0^k\\|\mu_q\\|\\ &\leq \left[(2\lambda^{-1})^{j+k}+ (2\lambda^{-1})^jB+BC\eta_0^k\right]\\|\mu_q\\|. \end{split}$$ The result follows by optimizing the choice of $j+k=n$. \[SLMult\] Multiplication by a ${{\mathcal C}}^1$ local function is a bounded operator on ${{\mathcal B}}.$ For any smooth local functions $\psi,\phi$, $\|\psi\|_\infty\leq 1$ we have $$\label{eq:times} \|\nu(\phi\cdot \partial_{\theta_i}\psi)\|=\left\|\nu\left(\partial_{\theta_i}\int_0^{\theta_i}(\phi\partial_{\theta_i}\psi)\right) \right \| \leq \\|\nu\\|\left\|\int_0^{\theta_i}\phi\partial_{\theta_i}\psi\right\|_\infty\leq 3\\|\nu\\|\, \|\phi\|_{{{\mathcal C}}^1}.$$ To use the above facts, it is convenient to introduce a more compact notation for the pieces that make up the operator $\bar S$. Let ${\mathds{1}}_A:{{\mathbb Z}}^d\to \{0,1\}$ be the characteristic function of the set $A\subset{{\mathbb Z}}^d$. Then define the operators $K_{z,p,q,\sigma}:{{\mathcal B}}\to{{\mathcal B}}$ by $K_{z,p,q,0}:={\mathds{1}}_{\{p\}}(q+z)A_z'$ and $K_{z,p,q,1}:={\mathds{1}}_{\Lambda_1^c}(p){\mathds{1}}_{\{p\}}(q+z) \hat S'_z+{\mathds{1}}_{\Lambda_1}(q+z)J_p'\hat S'_z$. With this notation can be rewritten as $$(\bar S\bar\mu)_p=\sum_{z\in\Lambda}\;\sum_{\sigma\in\{0,1\}}\;\sum_{q\in{{\mathbb Z}}^d}K_{z,p,q,\sigma}\mu_{q}.$$ Hence, iterating, $$\label{eq:Smu} (\bar S^n\bar\mu)_{q_0}=\!\!\!\!\sum_{z_1,\dots, z_n\in\Lambda}\;\sum_{\sigma_1,\dots,\sigma_n\in\{0,1\}}\;\sum_{q_1,\dots, q_n\in{{\mathbb Z}}^d}\!\!\!\! K_{z_1,q_0,q_1,\sigma_1}\cdots K_{z_n,q_{n-1},q_n,\sigma_n}\mu_{q_n}.$$ By Assumption \[ass:pert\], Lemma \[lem:gap0\], Lemma \[SLMult\] and the inequalities it follows that there exists a constant ${C_4}>0$, depending only on $F$ and $\pi_z$, such that $\sum_q\\|K_{z,p,q,1}\mu_q\\|\leq {C_4}\, {\varepsilon}a_z\\|\bar \mu\\|$ and $$\sum_{q_1,\dots, q_n\in{{\mathbb Z}}^d}\\|K_{z_1,q_0,q_1,0}\cdots K_{z_n,q_{n-1},q_n,0}\mu_{q_n}\\|\leq {C_4}\, a_{z_1}\cdots a_{z_n}\eta_^n\\|\bar\mu\\|$$ Accordingly, if ${C_4}^2{\varepsilon}+\eta_<1$, then there exists $n_\in{{\mathbb N}}$ and $\eta\in(\eta_,1)$ such that ${C_4}(\eta_+{C_4}^2{\varepsilon})^{n_}\leq \eta^{n_}<1$. This means that every $z_1,\dots,z_{n_}$ term in will be smaller than $\eta^{n_}a_{z_1}\cdots a_{z_{n_}}\\|\bar\mu\\|$, hence for all $n\in{{\mathbb N}}$, $$\label{eq:gap} \\| (\bar S^{n}\bar\mu)\\|\leq{C_4}\, \eta^{n-n_}\\|\bar\mu\\| \,.$$ Since ${\bf S}^n{{\boldsymbol{\mu}}}=(c_\mu, \bar S^n\bar \mu+c_\mu\sum_{k=1}^{n-1}\bar S^k\bar\zeta )$ and the series $\bar\zeta_=\sum_{k=1}^\infty \bar S^k\bar\zeta$ converges by , it follows that ${{\boldsymbol \mu}^w}:=(1,\bar\zeta_)$ is an invariant vector for ${\bf S}$. In addition $$\label{eq:gap2} \\|{\bf S}^n{{\boldsymbol{\mu}}}-c_\mu{{\boldsymbol \mu}^w}\\|\leq C\eta^{n}\\|{{\boldsymbol{\mu}}}\\|,$$ That is the operator $\bf S$ on ${{\overline{{{\mathcal B}}}}}$ has one as a simple maximal eigenvalue and a spectral gap. From this result we can obtain the decay of temporal correlation simply by projecting down to ${{\mathcal B}}$. Indeed, let $\mu$ be a probability measure and $\phi$ be a smooth local function depending only on the sites $A\subset {{\mathbb Z}}^d$ and let $L$ be the cardinality of $A$, then, by Lemma \[lem:covering\] and , , $$\label{eq:equi} \begin{split} \|\mu(\phi\circ S^n)&-{{\operatorname{Pr}\,}}({{\boldsymbol \mu}^w})(\phi)\|= \|{{\operatorname{Pr}\,}}({\bf S}^n(\Psi(\mu)-{{\boldsymbol \mu}^w}))(\phi)\|\\ &=\left\|\sum_{q\in A} ({\bf S}^n(\Psi(\mu)-{{\boldsymbol \mu}^w}))_q(\phi)\right\|\\ &\leq\sum_{q\in A} \\|({\bf S}^n(\Psi(\mu)-{{\boldsymbol \mu}^w}))_q\\| \,\|\phi\|_\infty\leq CL\eta^n(\\|\mu\\|+C)\|\phi\|_\infty. \end{split}$$ Thus, remembering , ${{\boldsymbol \mu}^w}\in{{\overline{{{\mathcal B}}}}}_M$, that is it gives rise to a bounded linear functional on local functions. Accordingly, we can define the measure ${{\mu^w}}={{\operatorname{Pr}\,}}({{\boldsymbol \mu}^w})$ which will be invariant by $S'$. Equation gives then the temporal correlation decay for such a measure. To have the spatial decay of correlations note that if $\varphi$ and $\phi$ are supported at a distance $M$, then their support, under the dynamics, grows at most linearly in time, thus it will take a time $\frac{M}{2{C_0}}$ before the supports have a common variable. Accordingly, since ${\varphi}\phi$ depends on $2L$ variables, (applied repeatedly to the product measure $m$) implies $$\begin{split} {{\mu^w}}(\varphi\phi)&=(S')^{M/2{C_0}}m(\varphi\phi)+{{\mathcal O}}(L\eta^{M/2{C_0}}\|\varphi\phi\|_\infty)\\ &=m(S^{M/2{C_0}}\varphi)\,m(S^{M/2{C_0}}\phi) +{{\mathcal O}}(L\eta^{M/2{C_0}}\|\varphi\phi\|_\infty)\\ &={{\mu^w}}(g){{\mu^w}}(\phi)+{{\mathcal O}}(L\eta^{M/2{C_0}}\|\varphi\phi\|_\infty). \end{split}$$ Perturbation Theory {#sec:pert} ------------------- In this section we prove Lemma \[lem:up\]. We deal with operators of the the type ${{\mathcal M}}_{{{t}}}f:=\sum_{z\in\Lambda}S_z(e^{\langle{{t}},z-v\rangle}f)$ where ${{t}}\in{{\mathbb C}}^d$. The problem is to study the spectrum for small ${{t}}$.[^16] First of all we need to lift the operator to our covering space. The obvious solution is to define ${\bf M}_{{{t}}}{{{\boldsymbol{\mu}}}}$ by $$\left(c_\mu m({{\mathcal M}}_{{t}}1)+\sum_{p\in\Lambda_1}\sum_{z\in\Lambda}e^{\langle {{t}}, z-v\rangle}\mu_{p-z}(\hat S_{z}1),\ \bar S_{{{t}}}\bar\mu+\bar\zeta c_\mu\right)$$ where $\bar\zeta=(\zeta_q):=(J_q' {{\mathcal M}}_{{t}}'m)$ and $$\label{eq:dytwo} (\bar S_{{{t}}}\bar\mu)_q=\begin{cases}\sum_{z\in\Lambda}e^{\langle{{t}}, z-v\rangle}S_{z}'\mu_{q-z}+\sum_{\substack{p\in\Lambda_1\\z\in\Lambda}}e^{\langle{{t}}, z-v\rangle}J_q'\hat S_{z}'\mu_{p-z}&\forall\;\;q\not\in \Lambda_1,\\ \sum_{z\in\Lambda}e^{\langle{{t}}, z-v\rangle}A_z'\mu_{q-z}+\sum_{\substack{p\in\Lambda_1\\z\in\Lambda}}e^{\langle{{t}}, z-v\rangle}J_q'\hat S_{z}'\mu_{p-z}&\forall\;\;q\in \Lambda_1. \end{cases}$$ A direct computation shows that, for each smooth local function ${\varphi}$, $\Pr({\bf M}_{{t}}{{\boldsymbol{\mu}}})({\varphi})=\Pr({{\boldsymbol{\mu}}})({{\mathcal M}}_{{t}}{\varphi})$, thus the lift covers the dynamics. In addition, one can easily check that ${\bf M}_0={\bf S}$ and that ${\bf M}_{{t}}$ is analytic in ${{t}}$.[^17] Accordingly, standard perturbation theory implies that there exists $\alpha_{{t}}$, ${{{\boldsymbol{\mu}}}}_{{t}}$, analytic in ${{t}}$, such that ${\bf M}_{{t}}{{{\boldsymbol{\mu}}}}_{{t}}=\alpha_{{t}}{{{\boldsymbol{\mu}}}}_{{t}}$ with $\alpha_0=1$, ${{\boldsymbol{\mu}}}_0={{\boldsymbol \mu}^w}$. We will normalize ${{\boldsymbol{\mu}}}_{{t}}$ so that ${{\boldsymbol{\mu}}}_{{t}}=(1,\bar\mu_{{t}})$. Setting $\mu_{{t}}:=\Pr ({{\boldsymbol{\mu}}}_{{t}})$, for each fixed local function $f$, $\mu_{{t}}(f)=\Pr{{\boldsymbol{\mu}}}_{{t}}(f)$ is analytic in ${{t}}$ since the sum implicit in the right hand side is just a finite sum.[^18] However Lemma \[lem:up\] requires a more quantitative information. By the arguments of section \[sec:mixone\] (see ) it follows that ${\bf M}_0={\bf S}=\Pi+R$ where $\Pi^2=\Pi$, $\Pi R=R\Pi=0$ and $\\|R^n\\|\leq C\eta^n$, for all $n\in{{\mathbb N}}$. Thus by standard perturbation theory (see [@Ka]), ${\bf M}_{{t}}=\alpha_{{t}}\Pi_{{t}}+R_{{t}}$ where $\|\alpha_0-\alpha_{{t}}\|\leq C\\|{{t}}\\|$, $\\|\Pi_t-\Pi\\|\leq C\\|t\\|$, $\\|R_{{t}}^n\\|\leq C\eta_{{t}}^n$, with $\eta_{{t}}\leq\eta+C\\|t\\|$. Hence, $\\|{\bf M}_{{t}}^n\\|\leq C\|\alpha_{{t}}\|^n+C\eta_{{t}}^n$ and, for each local function function $f$ depending only on $L$ variables $$\|{{\mathcal M}}_{{t}}^n\nu(f)\|=\|\Pr({\bf M}_{{t}}^n\Psi(\nu))(f)\|\leq L\|f\|_\infty\\|{\bf M}_{{t}}^n\Psi(\nu)\\|\leq C\|\alpha_{{t}}\|^n \\|\nu\\|L\|f\|_\infty$$ provided that $\|\alpha_{{t}}\|\geq \eta_{{t}}$ which holds for all $\\|t\\|<B$ for some $B>0$. This proves the first inequality of Lemma \[lem:up\]. To prove the second note that $\Pi_{{t}}{{\boldsymbol{\nu}}}=\ell_{{t}}({{\boldsymbol{\nu}}}){{\boldsymbol{\mu}}}_{{t}}$ with $\ell_0({{\boldsymbol{\nu}}})=[{{\boldsymbol{\nu}}}]_0$ hence[^19] $$\begin{split} {{\mathcal M}}_{{t}}^n\nu(1)&=\left[{\bf M}_{{t}}^n\Psi(\nu)\right]_0=\left[\alpha_{{t}}^n\Pi_{{t}}\Psi(\nu)+{{\mathcal O}}(\eta_{{t}}^n\\|\nu\\|)\right]_0\\ &=\left[\alpha_{{t}}^n\Pi_0\Psi(\nu)\right]_0+{{\mathcal O}}((\eta_{{t}}^n+\alpha_{{t}}^nC t)\\|\nu\\|) =\alpha_{{t}}^n(1+{{\mathcal O}}({{t}}\\|\nu\\|))+{{\mathcal O}}(\eta_{{t}}^n\\|\nu\\|). \end{split}$$ Finally, to study the derivatives of $\alpha$ we use the relation $\mu_t({{\mathcal M}}_t{\varphi})=\alpha_t \mu_t({\varphi})$ for any local smooth function ${\varphi}$. Differentiating with respect to ${{t}}$ yields $$\label{eq:pert-one} \begin{split} &\dot\mu_{{t}}({{\mathcal M}}_{{t}}{\varphi})+\mu_{{t}}(\dot {{{\mathcal M}}}_{{t}}{\varphi})=\dot\alpha_{{t}}\mu_{{t}}({\varphi})+\alpha_{{t}}\dot\mu_{{t}}({\varphi})\\ &\ddot\mu_{{t}}({{\mathcal M}}_{{t}}{\varphi})+2\dot\mu_{{t}}(\dot { {{\mathcal M}}}_{{t}}{\varphi})+\mu_{{t}}(\ddot {{\mathcal M}}_{{t}}{\varphi})= \ddot\alpha_{{t}}\mu_{{t}}({\varphi})+2\dot\alpha_{{t}}\dot\mu_{{t}}({\varphi})+\alpha_{{t}}\ddot\mu_{{t}}({\varphi})\\ &\dot\mu_{{t}}(1)=\ddot\mu_{{t}}(1)=0, \end{split}$$ Since $\dot{{\mathcal M}}_{{t}}=\sum_{z\in\Lambda}(z-v)e^{\langle t,z-v\rangle}S_z$ and $\ddot{{\mathcal M}}_{{t}}=\sum_{z\in\Lambda}(z-v)\otimes (z-v)e^{\langle t,z-v\rangle}S_z$ the above equations, for ${{t}}=0$ imply (substituting ${\varphi}=1$) $$\dot\alpha_0={{\mu^w}}(\dot{{\mathcal M}}_0 1)={{\mu^w}}(g-v)=0,$$ where $g$ is defined in and we have used . Next, substituting in the first of the , ${\varphi}=\sum_{k=0}^{n-1}S^k\phi$, for some local function $\phi$, we have $$\sum_{k=0}^{n-1}\dot{{\mathcal M}}_0'{{\mu^w}}(S^k\phi)=\dot\mu_0(({\mathds{1}}-S)\sum_{k=0}^{n-1}S^k\phi)=\dot\mu_0(\phi)- {{\operatorname{Pr}\,}}({\bf S}^{n-1}\dot{{\boldsymbol{\mu}}}_0)(\phi).$$ By , taking the limit for $n$ to infinity, we have $$\label{eq:mu-der} \dot\mu_0(\phi)=\sum_{k=0}^{\infty}\dot{{\mathcal M}}_0'{{\mu^w}}(S^k\phi).$$ Finally, the second of the , setting ${\varphi}=1$ and ${{t}}=0$, yields[^20] $$\label{eq:sec-der} \begin{split} \ddot\alpha_0&=2\sum_{n=0}^\infty{{\mu^w}}(\dot{{\mathcal M}}_0S^n\dot{{\mathcal M}}_01) +{{\mu^w}}(\ddot {{\mathcal M}}_0 1)\\ &=2\sum_{n=0}^\infty{{\mathbb E}}_{{{\mu^w}}}\left(\tilde \Delta_n\otimes\tilde \Delta_0\right)+{{\mathbb E}}_{{{\mu^w}}}\left(\tilde \Delta_0\otimes\tilde\Delta_0\right). \end{split}$$ Since $\dot {{\mathcal M}}1$ is a local function the sum is convergent. Hence $$\begin{split} \ddot{\alpha}_0&=\lim_{n\to\infty}\frac 1n\left[2\sum_{k,m=0}^n{{\mathbb E}}_{{{\mu^w}}}\left(\tilde \Delta_{m+k}\otimes\tilde \Delta_k\right)+{{\mathbb E}}_{{{\mu^w}}}\left(\tilde \Delta_k\otimes\tilde\Delta_k\right)\right]\\ &=\lim_{n\to\infty}\frac 1n{{\mathbb E}}_{{{\mu^w}}}\left({\tilde{X}}_n\otimes {\tilde{X}}_n\right)\geq 0. \end{split}$$ Finally, if there exists $w\in{{\mathbb R}}^d$ such that $\ddot{\alpha}_0w=0$, it means (from and Theorem \[thm:mixing\]) that there exists a constant ${C_5}>0$ such that, for all $n\in{{\mathbb N}}$,[^21] $${{\mathbb E}}_{{\mu^w}}\left(\|\langle w,{\tilde{X}}_n\rangle\|^2\right)\leq {C_5}\;; \quad {{\mathbb E}}\left(\|\langle w,{\tilde{X}}_n\rangle\|^2\right)\leq {C_5}$$ We can thus extract a subsequence $\{n_j\}$ such that $\langle w,{\tilde{X}}_{n_j}\rangle$ converges weakly almost surely to a random variable $Z$. Let $\psi={{\mathbb E}}_{{\mu^w}}(Z\;\|\; {{\mathcal F}}_0)$ and ${{\tilde{g}_w}}=\langle w,g-v\rangle$, then, for each ${{\mathcal F}}_0$ measurable smooth local function ${\varphi}$, $$\begin{split} {{\mathbb E}}_{{\mu^w}}({\varphi}(\psi-S\psi))&=\lim_{j\to\infty}{{\mathbb E}}_{{\mu^w}}({\varphi}({\tilde{X}}_{n_j}-{\tilde{X}}_{n_j+1}))=\lim_{j\to\infty}{{\mu^w}}({\varphi}({{\tilde{g}_w}}-S^{n_j+1}{{\tilde{g}_w}}))\\ &={{\mu^w}}({\varphi}{{\tilde{g}_w}}), \end{split}$$ where we have used Theorem \[thm:mixing\]. Thus ${{\tilde{g}_w}}=\psi-S\psi$, ${{\mu^e}}$-a.s.. This implies that, setting $M_0=0$, and $$\begin{split} M_{n+1}-M_n&=\langle w,{\tilde{\Delta}}_{n}\rangle-{{\mathbb E}}(\langle w,{\tilde{\Delta}}_n\rangle\;\|\;{{\mathcal F}}_n)+\psi(\omega^{n+1})-S\psi(\omega^n)\\ &=\langle w,{\tilde{\Delta}}_{n}\rangle+\psi(\omega^{n+1})-\psi(\omega^n), \end{split}$$ $M_n$ is a ${{\mathbb P}}_{{\mu^w}}$ stationary martingale. Moreover, $$\langle w,{\tilde{X}}_n\rangle=M_n-\psi(\omega^n)+\psi(\omega^0).$$ From this it follows that $$\begin{split} C\geq{{\mathbb E}}_{{\mu^w}}(\|M_n\|^2)&=\sum_{k=1}^{n-1}{{\mathbb E}}_{{\mu^w}}(\|\langle w,{\tilde{\Delta}}_{n}\rangle+\psi(\omega^{n+1})-\psi(\omega^n)\|^2)\\ &=\sum_{k=1}^{n-1}{{\mathbb E}}_{{\mu^w}}(\|\langle w,{\tilde{\Delta}}_{n}\rangle+\psi(\omega^{n+1})\|^2-\|\psi(\omega^n)\|^2)\\ &=(n-1)\left[{{\mathbb E}}_{{\mu^w}}(\|\langle w,{\tilde{\Delta}}_{1}\rangle+\psi(\omega^{1})\|^2-\|{{\tilde{g}_w}}(\omega^0)+S\psi(\omega^0)\|^2)\right]. \end{split}$$ Thus $\sum_z\pi_z\|\langle w,z-v\rangle+\psi\circ F\circ \tau^z\|^2=\left\|\sum_z\pi_z(\langle w,z-v\rangle+\psi\circ F\circ \tau^z)\right\|^2$, that is $\langle w,z-v\rangle+\psi\circ F\circ \tau^z={{\tilde{g}_w}}+S\psi=\psi$, ${{\mu^w}}$-a.s.. Next, let $\alpha_z={{\mu^w}}(\pi_z)$, then, $\sum_z\alpha_z=1$ and $\sum_z\langle w, z-v\rangle\alpha_z={{\mu^w}}({{\tilde{g}_w}})=0$. Hence, $$\sum_z\alpha_z \psi\circ F\circ \tau^z=\psi\quad {{\mu^w}}\text{-a.s.} .$$ Note that the operator $S_\alpha{\varphi}:=\sum_z\alpha_z {\varphi}\circ F\circ \tau^z$ defines a Markov process with invariant measure ${{\mu^e}}$ and satisfies the hypothesis of Theorem \[thm:mixing\]. Since $\psi=\sum_{k=0}^{n-1}S^k{{\tilde{g}_w}}+S^{n}\psi$, for each $\phi\in L^2({{\mu^w}})$, $$\lim_{j\to\infty}{{\mu^w}}(\phi \sum_{k=0}^{n_j-1}S^k{{\tilde{g}_w}})=\lim_{j\to\infty}{{\mathbb E}}_{{\mu^w}}(\phi(\omega^0)\langle w,X_{n_j}\rangle)={{\mu^w}}(\phi \,\psi).$$ In addition, assumption \[ass:pert\] implies that setting,[^22] for each smooth local function $\phi$, $\nu_{n,\phi}({\varphi})={{\mu^w}}(\phi S_\alpha^n{\varphi})$, $$\|\nu_{n,\phi}({\varphi})\|\leq \frac{(1+{\varepsilon})^n}{(1-{\varepsilon})^n}\|\phi\|_\infty{{\mu^w}}(S^n\|{\varphi}\|)= \frac{(1+{\varepsilon})^n}{(1-{\varepsilon})^n}\|\phi\|_\infty{{\mu^w}}(\|{\varphi}\|).$$ Thus $\nu_{n,\phi}$ is absolutely continuos with respect to ${{\mu^w}}$ with density $\rho_{n,\phi}\in L^\infty({{\mu^w}})$. Accordingly,[^23] $$\begin{split} {{\mu^w}}(\phi\psi)&={{\mu^w}}(\phi S_\alpha^n \psi)=\lim_{j\to\infty}\sum_{k=0}^{n_j-1}{{\mu^w}}(\phi S_\alpha^n S^k {{\tilde{g}_w}})\\ &=\sum_{k=0}^{n_l-1}\left[{{\mu^e}}( S^k {{\tilde{g}_w}}){{\mu^w}}(\phi)+{{\mathcal O}}(C_\phi\eta^{n}k^d)\right] +\lim_{j\to\infty}\sum_{k=n_l}^{n_j-1}{{\mathcal O}}(C_\phi\eta^k)\\ &=\sum_{k=0}^{n_l-1}{{\mu^e}}( S^k {{\tilde{g}_w}}){{\mu^w}}(\phi)+C_\phi{{\mathcal O}}(\eta^{n_l}+\eta^nn_l^d). \end{split}$$ Taking first the limit for $n\to\infty$ and the one $l\to\infty$ yields ${{\mu^w}}(\phi\psi)={{\mu^w}}(\phi){{\mu^e}}(\psi)$. That is $\psi$ is ${{\mu^w}}$ almost surely constant. This implies that ${{\tilde{g}_w}}=0$ and hence $\langle w,z-v\rangle\pi_z=0$, ${{\mu^w}}$ a.s.. This is equivalent to saying that the vectors in the set $\{(\langle e_1,z\rangle,\dots, \langle e_d,z\rangle)\}_{z\in\Lambda}\cup\{(1,\dots,1)\}$ are linearly dependent over ${{\mathbb R}}$, but this implies that they are linearly dependent over ${{\mathbb Z}}$. In other words we can assume that $w\in{{\mathbb Z}}^d$. Finally, since $\pi_z$ is smooth, we have $\langle w,z\rangle=\langle w,v\rangle$ unless $\pi_z\equiv 0$, which contradicts Assumption \[ass:ellipticity\]. Variation bounds for conditional measures {#sec:variation} ----------------------------------------- In the previous subsection we obtained several results for random walks provided that we start the environment in a measure with “density" of bounded variation. Here we show why such measures constitute a natural class for the problem at hand. More precisely we shall show that if we start with a nice measure and condition on a behavior of a walk during an initial time interval we still have a good control on the variation of densities. For future needs we consider two random walks $(X_t,Y_t)$ evolving in the same environment starting respectively at $a,b\in{{\mathbb Z}}^d$ and with the environment at time zero distributed according to the measure $\nu\in{{\mathcal B}}$. Let ${{\mathbb P}}_{a,b,\nu}^2$ be the measure on $(\Theta\times {{\mathbb Z}}^{2d})^{{\mathbb N}}$ associated to such a process and ${{\mathbb E}}_{a,b,\nu}^2$ the corresponding expectation ${{\mathbb P}}^2_\nu:={{\mathbb P}}^2_{0,0,\nu}$ and ${{\mathbb E}}^2_\nu:={{\mathbb E}}^2_{0,0,\nu}$. Let $m\in{{\mathbb N}}$ and consider the $\sigma$-algebra ${{\mathcal F}}_m^{XY}=\sigma\{X_1,Y_1,\dots,X_m,Y_m\}$. We are interested in computing ${{\mathbb E}}_{a,b,\nu}^2(f(X,Y,\theta^m)\;\|\; {{\mathcal F}}_m^{XY})$ for each local ${{\mathcal F}}_m^{XY}\otimes{{\mathcal T}}$-measurable function $f$ and probability measure $\nu\in{{\mathcal B}}$. Thus, we are interested in the measures $\nu^{XY}_{a,b,m}$ defined by $${{\mathbb E}}^2_{a,b,\nu}(f(X,Y,\theta^m)\;\|\; {{\mathcal F}}_m^{XY})=:\int_\Theta f(X,Y,\theta)\;\nu^{XY}_{a,b,m}(d\theta).$$ \[lem:variation\] There exists ${C_6}>0$ and $0<{\varepsilon}_1\leq{\varepsilon}_0$ such that, if assumption \[ass:pert\] is satisfied for ${\varepsilon}_1$, then for each $m\in{{\mathbb N}}$, $a,b\in {{\mathbb Z}}^d$ and probability measure $\nu\in{{\mathcal B}}$ the following holds $$\\|\nu^{XY}_{a,b,m}\\|\leq {C_6}\\|\nu\\|.$$ Given two random walks realizations $X,Y:{{\mathbb N}}\to{{\mathbb R}}^d$, let us define the operators $$S_{ X,Y,k}f(\theta):=\pi_{z_k}(\tau^{X_k}\theta)\pi_{w_k}(\tau^{Y_k}\theta) f\circ F(\theta),$$ where $z_k=X_{k+1}-X_k$ and $w_k=Y_{k+1}-Y_k$. With such a notation we can write $$\nu^{XY}_{a,b,m}=\frac{S_{X,Y, m-1}'\cdots S_{X,Y, 0}'\nu}{S_{X,Y, m-1}'\cdots S_{X,Y, 0}'\nu(1)}.$$ Recalling and the Lasota-Yorke inequality for the map $F$ (see ), and using Assumption \[ass:pert\] we have $$\label{eq:time-dep-lasota} \\|S_{X,Y, k}'\nu\\|\leq 2\lambda^{-1}(1+{\varepsilon}_1)^2\\|\nu\\|a_{z_k}a_{w_k}+BS_{X,Y, k}'\nu(1).$$ Hence, for ${\varepsilon}_1$ such that $2\lambda^{-1}(1+{\varepsilon}_1)^2\leq \eta(1-{\varepsilon}_1)^{-2}<1$ we can iterate the above inequality and obtain $$\begin{split} &\\| S_{X,Y, m-1}'\cdots S_{X,Y, 0}'\nu\\| \leq \eta^m(1-{\varepsilon}_1)^{2m}\\|\nu\\|\prod_{k=0}^{m-1}a_{z_k}a_{w_k}\\ &\quad+B\sum_{j=0}^{m-1}\eta^{j}(1-{\varepsilon}_1)^{2j}\nu\left( S_{X,Y, 0}\cdots S_{X,Y, m-1-j}1\right)\prod_{k=m-j}^{m-1}a_{z_k}a_{w_k}\\ &\leq\left[\eta^m\\|\nu\\|+(1-\eta)^{-1}B\right]\nu\left( S_{X,Y, 0}\cdots S_{X,Y, m-1}1\right), \end{split}$$ which proves the Lemma with ${C_6}=1+(1-\eta)^{-1}B$.[^24] Annealed Invariance Principle. {#sec:invariance} ============================== This section is devoted to proving an averaged invariance principle. This result is used in Section \[sec:clt-proof\] to prove Lemma \[lm:hot\]. Consider the process $$\label{eq:path} \hat X^N_t=\frac{1}{\sqrt{N}}\left\{\tilde X_{\lceil tN\rceil}+(tN-\lceil tN\rceil)\tilde\Delta_{\lceil tN\rceil}\right\}.$$ Note that $\hat X^N_t\in{{\mathcal C}}^0([0,1],{{\mathbb R}}^d)$, by construction. In fact, Lemma \[lem:largedev\] implies higher regularity. \[lem:holder\] The family of processes $\{\hat X^N\}\subset {{\mathcal C}}^0([0,1],{{\mathbb R}}^d)$ is tight. Let $\varsigma\in (0,1/2)$, $$L_\varsigma(f):=\sup_{t,s\in[0,1]}\frac{\\|f(t)-f(s)\\|}{\|t-s\|^\varsigma},$$ and $K_L^\varsigma:=\{f\in{{\mathcal C}}^0([0,1],{{\mathbb R}}^d)\;:\; f(0)=0,\; L_\varsigma(f)\leq L\}$. By Lemma \[lem:largedev\] it follows that, for each $N\in{{\mathbb N}}$, $t\in [0,1]$ and $h\in [-t,1-t]$, $$\label{eq:holder} {{\mathbb P}}\left(\left\{\\|\hat X^N_{t+h}-\hat X^N_t\\|\geq L h^{\varsigma}\right\}\right)\leq e^{-CL^2h^{2\varsigma-1}}.$$ In addition, if $\\|\hat X^N_t\\|+\\|\hat X^N_{t+h}\\|\leq L^{1-\varsigma}$, then the set in is empty for all $h>L^{-1}$. Now the result follows in complete analogy with the usual proof of the Hölder continuity of the Brownian motion, based on applying the above estimates to the dyadic rationals, yielding $${{\mathbb P}}\left(\left\{\hat X^N\not\in K_L^\varsigma\right\}\right)\leq e^{-CL}.$$ Since $K_L^\varsigma$ are compact in ${{\mathcal C}}^0([0,1],{{\mathbb R}}^d)$ the tightness follows. Lemma \[lem:holder\] also allows us to prove the invariance principle. \[lem:invariance\] For each probability measure $\nu\in{{\mathcal B}}$ the process $\{\hat X^N_t\}$ converges in law to the Brownian motion with diffusion matrix ${{\Sigma^2}}$. In view of Lemma \[lem:holder\] we only need to check the convergence of finite dimensional distributions. We consider two dimensional distributions, the general case being very similar. Accordingly, let $t_1<t_2$ and fix $\xi_1, \xi_2.$ We have $${{\mathbb E}}_\nu\left(\exp(i\langle \xi_1, {\widehat{X}}_{t_1}^N\rangle+i\langle\xi_2 ,{\widehat{X}}_{t_2}^N\rangle)\right)= {{\mathbb E}}_\nu\left(\exp(i\langle[\xi_1+\xi_2],{\widehat{X}}_{t_1}^N\rangle)\exp(i\langle\xi_2 ,[{\widehat{X}}_{t_2}^N-{\widehat{X}}_{t_1}^N]\rangle)\right)=$$ $${{\mathbb E}}_\nu\left({{\mathbb E}}\left(\exp(i\langle\xi_2, [{\widehat{X}}_{t_2}^N-{\widehat{X}}_{t_1}^N]\rangle)\|{{\mathcal F}}_{[t_1 N]}\right) \exp(i\langle[\xi_1+\xi_2], {\widehat{X}}_{t_1}^N\rangle)\right).$$ By Lemma \[lem:variation\] and Proposition \[lem:clt-averaged\] we have[^25] $${{\mathbb E}}\left(\exp(i\langle\xi_2 ,[{\widehat{X}}_{t_2}^N-{\widehat{X}}_{t_1}^N]\rangle)\| {{\mathcal F}}_{[t_1 N]}\right)= \exp\left(-\frac{1}{2} \la \xi_2,\Sigma^2 \xi_2\ra(t_2-t_1)(1+o(1))\right)$$ and so using Proposition \[lem:clt-averaged\] again we obtain $${{\mathbb E}}_\nu\left(\exp(i\langle\xi_1, {\widehat{X}}_{t_1}^N\rangle+i\langle\xi_2, {\widehat{X}}_{t_2}^N\rangle)\right)\sim e^{-\frac{1}{2} \left[\la \xi_2,\Sigma^2 \xi_2\ra(t_2-t_1))+ \la (\xi_1+\xi_2), \Sigma^2(\xi_1+\xi_2)\ra t_1\right]}$$ Thus, $({\widehat{X}}_{t_1}^N, {\widehat{X}}_{t_2}^N)$ is asymptotically Gaussian with zero mean and the variance predicted by the Brownian Motion. Proofs: Quenched CLT via the study of two random walks {#sec:clt-proof} ====================================================== The goal of this section is to establish Lemma \[lm:hot\]. Lemma \[lem:holder\] shows that the distributions of the processes $(\hat X^N_t, \hat Y^N_t)$ are tight, hence they have accumulation points. Our next task is to characterize such accumulation points. Let us consider any accumulation point $(\hat X^\infty_{t},\hat Y^\infty_{t})$. We will see that $(\hat X_t^\infty,\hat Y^\infty_t)$ is a centered Gaussian random variables with variance $$\label{eq:vartwo} {{\Sigma^2_2}}:=t\begin{pmatrix}{{\Sigma^2}}&0\\0&{{\Sigma^2}}\end{pmatrix}.$$ More precisely, if we define the second order differential operator $\Delta_{{{\Sigma^2_2}}}:=\sum_{i,j}{{\Sigma^2_2}}_{ij}\partial_i\partial_j$ we have the following. \[lem:heat\] For any $\psi\in{{\mathcal C}}^{3}({{\mathbb R}}\times {{\mathbb R}}^d\times {{\mathbb R}}^d, {{\mathbb R}})$ we have $$\frac d{dt}{{\mathbb E}}^2(\psi(t,\hat X^\infty_{t},\hat Y^\infty_{t}))={{\mathbb E}}^2(\partial_t\psi(t,\hat X^\infty_{t},\hat Y^\infty_{t})+ \frac 12\Delta_{{\Sigma^2}}\psi(t,\hat X^\infty_{t},\hat Y^\infty_{t}) ).$$ More precisely, there exists $\beta\in(0,\frac 16)$ and $\vartheta\in (0,1-2\beta)$ such that, for all $N\in{{\mathbb N}}$ and $t,h\in [0,1]$ such that $h>N^{\vartheta-1}$ we have $$\begin{split} &\left\|{{\mathbb E}}^2\left(\psi(t+h,\hat X^N_{t+h},\hat Y^N_{t+h})-\psi(t,\hat X^N_{t},\hat Y^N_{t}) -h\left[\partial_t\psi(t,\hat X^N_{t},\hat Y^N_{t})+ \frac{1}{2} {\Delta_{{\Sigma^2}}\psi}(t,\hat X^N_{t},\hat Y^N_{t})\right] \right)\right\|\\ &\quad \leq C\\|\psi\\|_{{{\mathcal C}}^3}(N^{-\beta}h+h^{\frac 32}+N^{-\frac 12}). \end{split}$$ Thanks to the above Proposition for each $\phi\in{{\mathcal C}}_0^3({{\mathbb R}}^d\times {{\mathbb R}}^d)$ we can define $\psi$ by $$\label{eq:backheat} \begin{split} &\partial_t\psi+\frac 12\Delta_{{\Sigma^2}}\psi=0\\ &\psi(1,x,y)=\phi(x,y) \end{split}$$ and, by applying Proposition \[lem:heat\] with the choice $h=\lceil N^{2\beta}\rceil^{-1}$, obtain the wanted result: $$\begin{split} {{\mathbb E}}^2&(\phi(\hat X^N_{1},\hat Y^N_{1}))={{\mathbb E}}^2(\psi(1,\hat X^N_{1},\hat Y^N_{1}))\\ &={{\mathbb E}}^2(\psi(0,\hat X^N_{0},\hat Y^N_{0}))+\sum_{i=0}^{h^{-1}-1}{{\mathbb E}}^2(\psi((i+1)h,\,\hat X^N_{(i+1)h},\hat Y^N_{(i+1)h})-\psi(ih,\,\hat X^N_{ih},\hat Y^N_{ih}))\\ &={{\mathbb E}}^2(\psi(0,0,0))+{{\mathcal O}}(\\|\psi\\|_{{{\mathcal C}}^3}N^{-\beta})={{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2_2}})}(\phi)+{{\mathcal O}}(\\|\phi\\|_{{{\mathcal C}}^3}(N^{-\beta}+N^{-\frac 12+2\beta})), \end{split}$$ where we have used the explicit solution of .[^26] Remembering the form of ${{\Sigma^2_2}}$ (see ), Lemma \[lm:hot\], and hence Theorem \[thm:main\], follow. We start by the following Taylor expansion $$\label{eq:hoihoi} \begin{split} {{\mathbb E}}^2&(\psi(t+h,\hat X^N_{t+h},\hat Y^N_{t+h}))-{{\mathbb E}}^2(\psi(t,\hat X^N_{t},\hat Y^N_{t}))={{\mathbb E}}^2(\partial_t\psi(t,\hat X^N_{t},\hat Y^N_{t}))h\\ &+{{\mathbb E}}^2(\partial_x\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat X^N_{t+h}-\hat X^N_t) +\partial_y\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat Y^N_{t+h}-\hat Y^N_t))\\ &+\frac 12{{\mathbb E}}^2((\hat X^N_{t+h}-\hat X^N_t)\cdot\partial_x^2\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat X^N_{t+h}-\hat X^N_t))\\ &+{{\mathbb E}}^2((\hat Y^N_{t+h}-\hat Y^N_t)\cdot\partial_{xy}\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat X^N_{t+h}-\hat X^N_t))\\ &+\frac 12{{\mathbb E}}^2((\hat Y^N_{t+h}-\hat Y^N_t)\cdot\partial_y^2\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat Y^N_{t+h}-\hat Y^N_t))\\ &+\frac 12{{\mathbb E}}^2(\partial_{tx}\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat X^N_{t+h}-\hat X^N_t)) h\\ &+\frac 12{{\mathbb E}}^2(\partial_{ty}\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat Y^N_{t+h}-\hat Y^N_t)) h\\ &+{{\mathcal O}}\left(\|\partial_t^2\psi\|_\infty h^2+ \\|\psi\\|_{{{\mathcal C}}^3}\left[{{\mathbb E}}(\\|\hat X^N_{t+h}-\hat X^N_t\\|^3)+{{\mathbb E}}(\\|\hat Y^N_{t+h}-\hat Y^N_t\\|^3)\right]\right). \end{split}$$ Next, we will analyze the terms in equation one by one. First of all note that the remainders are of order $h^{\frac 32}$.[^27] Let us start the estimates with the term ${{\mathbb E}}(\partial_x\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat X^N_{t+h}-\hat X^N_t))$. To this end we consider the $\sigma$-algebra ${{\mathcal F}}^{XY}_m$ generated by $\{X_1,\dots, X_m,Y_1,\dots,Y_m\}$. Setting ${\tilde{g}}=g-v$, $\ell_t:=\lceil tN\rceil+1$ we can write $$\begin{split} {{\mathbb E}}((\hat X^N_{t+h}-\hat X^N_t)\;\|\; {{\mathcal F}}^{XY}_{\ell_{t}})&=N^{-\frac 12}\sum_{k=\ell_t}^{\ell_{t+h}}{{\mathbb E}}(\tilde \Delta_k^X\;\|\;{{\mathcal F}}^{XY}_{\ell_{t}})+{{\mathcal O}}(N^{-\frac 12})\\ &=N^{-\frac 12}\sum_{k=\ell_t}^{\ell_{t+h}} {{\mathbb E}}( (S^{k-\ell_{t}}{\tilde{g}})\circ \tau^{X_{\ell_{t}}}\;\|\;{{\mathcal F}}^{XY}_{\ell_{t}})+{{\mathcal O}}(N^{-\frac 12})\\ &=N^{-\frac 12}\sum_{k=\ell_t}^{\ell_{t+h}} [(S^{k-\ell_{t}}\tau^{X_{\ell_{t}}})'({{\mu^e}})^{XY}_{\ell_t}]({\tilde{g}})+{{\mathcal O}}(N^{-\frac 12}). \end{split}$$ where, by Lemma \[lem:variation\], $\\|({{\mu^e}})^{XY}_{\ell_t} \\|\leq C$ and hence $\\|(\tau^{X_{\ell_{t}}})'({{\mu^e}})^{XY}_{\ell_t} \\|\leq C.$ From Theorem \[thm:mixing\] and the fact that ${{\mu^w}}({\tilde{g}})=0$ it follows that $$\label{eq:first_term} \|{{\mathbb E}}^2(\partial_x\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat X^N_{t+h}-\hat X^N_t))\|\leq CN^{-\frac 12}\\|\psi\\|_{{{\mathcal C}}^1}.$$ In complete analogy we have $$\label{eq:second_term} \|{{\mathbb E}}^2(\partial_y\psi(t,\hat X^N_{t},\hat Y^N_{t})\cdot(\hat Y^N_{t+h}-\hat Y^N_t))\|\leq CN^{-\frac 12}\\|\psi\\|_{{{\mathcal C}}^1}.$$ The quadratic terms involving $\partial_{tx}$ and $\partial_{ty}$ are estimated in the same manner yielding terms of order $N^{-\frac 12} h\\|\psi\\|_{{{\mathcal C}}^2}$. The quadratic terms involving only $X$ or only $Y$ yield the following $$\label{eq:third_term} \begin{split} &{{\mathbb E}}^2((\hat X^N_{t+h}-\hat X^N_t)_i (\hat X^N_{t+h}-\hat X^N_{t})_j\partial_{x_ix_j}\psi) =({{\Sigma^2}})_{ij}{{\mathbb E}}^2(\partial_{x_ix_j}\psi)h+{{\mathcal O}}\left(\frac{\\|\psi\\|_{{{\mathcal C}}^2}}{\sqrt N}\right)\\ &{{\mathbb E}}^2((\hat Y^N_{t+h}-\hat Y^N_t)_i (\hat Y^N_{t+h}-\hat Y^N_{t})_j\partial_{y_iy_j}\psi) =({{\Sigma^2}})_{ij}{{\mathbb E}}^2(\partial_{y_iy_j}\psi)h+{{\mathcal O}}\left(\frac{\\|\psi\\|_{{{\mathcal C}}^2}}{\sqrt N}\right). \end{split}$$ Indeed, in analogy with what we have done before, we can condition with respect to ${{\mathcal F}}^{XY}_{\ell_t}$ and $$\label{eq:details2} \begin{split} {{\mathbb E}}^2&((\hat X^N_{t+h}-\hat X^N_t)_i (\hat X^N_{t+h}-\hat X^N_{t})_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})=\frac 1N\left\{ \sum_{k=\ell_t}^{\ell_{t+h}-1}{{\mathbb E}}^2(({\tilde{\Delta}}^X_k)_i({\tilde{\Delta}}^X_k)_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})\right.\\ &\left. +\sum_{k=\ell_t}^{\ell_{t+h}-1}\sum_{l=\ell_t}^{k-1}{{\mathbb E}}^2(({\tilde{\Delta}}^X_k)_i({\tilde{\Delta}}^X_l)_j+({\tilde{\Delta}}^X_l)_i({\tilde{\Delta}}^X_k)_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})\right\}\\ &+N^{-1}{{\mathcal O}}\left(1+\sum_{l=\ell_t+1}^{\ell_{t+h}-1}{{\mathbb E}}^2(({\tilde{\Delta}}^X_{\ell_{t+h}-1})_i({\tilde{\Delta}}^X_l)_j+({\tilde{\Delta}}^X_{\ell_{t+h}-1})_j({\tilde{\Delta}}^X_l)_i\;\|\;{{\mathcal F}}^{XY}_{\ell_t})\right), \end{split}$$ where the boundary terms of the type ${{\mathbb E}}^2(({\tilde{\Delta}}^X_{\ell_{t}-1})_i({\tilde{\Delta}}^X_l)_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})$ have been estimated as in . To estimate such an expression note that $${{\mathbb E}}^2(({\tilde{\Delta}}^X_k)_i({\tilde{\Delta}}^X_l)_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})= (S^{j-\ell_t}\tau^{X_{\ell_t}})'\nu^{XY}_{\ell_t}\left({\tilde{g}}_jS^{k-j}{\tilde{g}}_k\right)$$ Since Lemma \[lem:variation\] and Theorem \[thm:mixing\] imply $\\|(S^{j-\ell_t}\tau^{X_{\ell_t}})'\nu^{XY}_{\ell_t}\\|\leq C\\|{{\mu^e}}\\|$ and also that $\\|{\tilde{g}}_j(S^{j-\ell_t}\tau^{X_{\ell_t}})'\nu^{XY}_{\ell_t}\\|\leq C\\|{{\mu^e}}\\|$ we can apply Theorem \[thm:mixing\] $$\begin{split} \left\|{{\mathbb E}}^2(({\tilde{\Delta}}^X_k)_i({\tilde{\Delta}}^X_l)_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})\right\|= &\nu^{XY}_{\ell_t}(S^{j-\ell_t}\tau^{X_{\ell_t}}{\tilde{g}}_j){{\mu^w}}({\tilde{g}}_k)+{{\mathcal O}}(\eta^{k-j})={{\mathcal O}}(\eta^{k-j})\\ \left\|{{\mathbb E}}^2(({\tilde{\Delta}}^X_k)_i({\tilde{\Delta}}^X_l)_j\;\|\;{{\mathcal F}}^{XY}_{\ell_t})\right\|=&\nu^{XY}_{\ell_t}(1){{\mu^w}}({\tilde{g}}_jS^{k-j}{\tilde{g}}_k)+{{\mathcal O}}(\eta^{j-\ell_t})\\ =&{{\mu^w}}({\tilde{g}}_jS^{k-j}{\tilde{g}}_k)+{{\mathcal O}}(\eta^{j-\ell_t}). \end{split}$$ Using such estimates in and remembering formula (for $\ddot\alpha_0={{\Sigma^2}}$) the first equation of follows. The second equation of is proven in complete analogy. Finally, we must deal with the mixed quadratic term. $${{\mathbb E}}^2((\hat X^N_{t+h}-\hat X^N_t)_i (\hat Y^N_{t+h}-\hat Y^N_{t})_j)\;\|\;{{\mathcal F}}_{\ell_t}^{XY}) =\!\!\!\!\sum_{k,m=\ell_t-1}^{\ell_{t+h}}\!\!\!\!\frac{{{\mathbb E}}^2((\tilde \Delta_k^X)_i (\tilde\Delta_m^Y)_j \;\|\;{{\mathcal F}}^{XY}_{\ell_{t}})}N+{{\mathcal O}}( \frac 1{\sqrt N}).$$ If $\|k-m\|\geq A\ln N$, then, for $A>\ln \eta^{-1}$, by Lemma \[lem:variation\] and Theorem \[thm:mixing\] it follows that $$\begin{split} {{\mathbb E}}^2((\hat X^N_{t+h}-\hat X^N_t)_i (\hat Y^N_{t+h}-\hat Y^N_{t})_j)\;\|\;{{\mathcal F}}_{\ell_t}^{XY}) =&\!\!\!\!\sum_{\substack{\ell_t+2A\ln N\leq k\leq\ell_{t+h}\\\|m-k\|\leq A\ln N}}\!\!\!\!\!\!\!\!\!\! \frac{{{\mathbb E}}^2((\tilde \Delta_k^X)_i (\tilde\Delta_m^Y)_j \;\|\;{{\mathcal F}}^{XY}_{\ell_{t}})}{N}\\ &\quad+{{\mathcal O}}( N^{-\frac 12}). \end{split}$$ Next, suppose that $\|m-k\|\leq A\ln N$ and $\|X_{k-A\ln N}- Y_{k-A\ln N}\|> 4{C_0}A\ln N$. Assume, to fix our ideas, that $k\leq m$. Then, for all times $l$ such that $\|l-k\|\leq A\ln N$, the two walks explore disjoint parts of the environment. Thus, we can consider the process started at time $k-A\ln N$ with the conditional measure $({{\mu^e}})^{XY}_{k-A\ln N}$ and with the walks starting from $a=X_{k-A\ln N}, b=Y_{k-A\ln N}$, $\\|a-b\\|>4{C_0}A\ln N$. If we set ${\bf f}=S^{A\ln N}\tau^{X_k}{\tilde{g}}_i$ and ${\bf h}=S^{m-k+A\ln N}\tau^{Y_k}{\tilde{g}}_j$ we have that the two functions depend on different sets of variables (let $B\subset {{\mathbb Z}}^d$ be the set of variables on which $\bf f$ depends and $B'$ the ones relative to $\bf h$) and $${{\mathbb E}}^2((\tilde \Delta_k^X)_i (\tilde\Delta_m^Y)_j\;\|\;{{\mathcal F}}^{XY}_{k-A\ln N})=({{\mu^e}})^{XY}_{k-A\ln N}({\bf f}\,{\bf h}).$$ We can then define its Newtonian potential of $\bf f$, $$\label{eq:newton} \begin{split} \Psi(\theta)&=\frac{1}{\|B\|(\|B\|-2)\alpha_{\|B\|}}\int_{I^{{{\mathbb Z}}^d}}\\|\theta^B-\vartheta^B\\|^{-\|B\|+2}{\bf f}(\vartheta)d\vartheta^B \otimes_{j\not\in B}\mu_0(d\vartheta_j)\\ &=\frac{1}{\|B\|(\|B\|-2)\alpha_{\|B\|}}\int_{I^{B}}\\|\theta^B-\vartheta^B\\|^{-\|B\|+2}{\bf f}(\vartheta^B)d\vartheta^B. \end{split}$$ where $\theta^B=(\theta_l)_{l\in B}$ and $\alpha_{l}$ is the volume of the unit ball in ${{\mathbb R}}^l$. It is well known that, for $\theta^B$ in the interior of $I^B$ $$\sum_{l\in B}\partial_{\theta_l\theta_l}\Psi=\bf f.$$ Thus, remembering Lemma \[lem:variation\], we can write[^28] $$\begin{split} &\left\|{{\mathbb E}}^2((\tilde \Delta_k^X)_i (\tilde\Delta_m^Y)_j\;\|\;{{\mathcal F}}^{XY}_{k-A\ln N})\right\|\leq\sum_{l\in B}\left\|({{\mu^e}})^{XY}_{k-A\ln N}(\partial_{\theta_l}(\partial_{\theta_l}\Psi\cdot {\bf h}))\right\|\\ &\quad\leq \|B\|\cdot \\|({{\mu^e}})^{XY}_{k-A\ln N}\\|\cdot \sup_{l\in B} \|\partial_{\theta_l}\Psi\cdot {\bf h}\|_\infty \leq CA^d{C_0}^d\|\ln N\|^d\cdot \|g\|_\infty\cdot\sup_{l\in B} \|\partial_{\theta_l}\Psi\|_\infty. \end{split}$$ By we have, for $l\in B$, $$\partial_{\theta_l}\Psi(\theta)=\int_{I^{{{\mathbb Z}}^d}}\frac{\theta_l-\vartheta_l}{\|B\|(\|B\|-2)\alpha_{\|B\|}\cdot\\|\theta^{B}-\vartheta^B\\|^{\|B\|}} \;{\bf f}(\vartheta)d\vartheta^B \otimes_{j\not\in B}\mu_0(d\vartheta_j)=:\nu^\theta_l({\bf f}).$$ Unfortunately, $\nu^\theta_l\not \in{{\mathcal B}}$ due to the singularity of the kernel. To take care of this problem we need to isolate the singularity. For each $r>0$ let $\chi_r\in{{\mathcal C}}^\infty({{\mathbb R}}^B,[0,1])$ such that $\chi_r(\theta^B)=0$ for all $\\|\theta^B\\|\leq r$ and $\chi_r(\theta^B)=1$ for all $\\|\theta^B\\|\geq2r$. Clearly $\chi_r$ can be chosen radial and so that $\sup_l\|\partial_{\theta_l}\chi_r\|_\infty\leq Cr^{-1}$. We then define $$\nu^\theta_{l,r}(\phi):=\int_{I^{{{\mathbb Z}}^d}}\frac{(\theta_l-\vartheta_l)\cdot\chi_r(\theta-\vartheta)}{\|B\|(\|B\|-2)\alpha_{\|B\|}\cdot\\|\theta^{B}-\vartheta^B\\|^{\|B\|}} \;\phi(\vartheta)\;\;d\vartheta^B \otimes_{j\not\in B}\mu_0(d\vartheta_j)$$ and $\mu^\theta_{{l,r}}(f):=\nu^\theta_l(\phi)-\nu^\theta_{l,r}(\phi)$. A direct computation shows that $\|\mu^\theta_{l,r}\|\leq C r(A{C_0}\ln N)^{-d}$ and $\\|\nu^\theta_{l,r}\\|\leq C(A{C_0}\ln N)^{-d}\ln r^{-1}$. Since $\|{\bf f}\|\leq 2\|g\|_\infty$ we can finally use Theorem \[thm:mixing\] to estimate $$\begin{split} &\left\|{{\mathbb E}}^2((\tilde \Delta_k^X)_i (\tilde\Delta_m^Y)_j\;\|\;{{\mathcal F}}^{XY}_{k-A\ln N})\right\| \leq C\left\{ r+\sup_{\substack{l\in B\\\theta\in I^B}}\|(\tau^{X_k})'\nu^\theta_{l,r}(S^{A\ln N}{\tilde{g}}_i)\|\right\}\\ &\quad \leq C\left\{ r+{{\mu^w}}({\tilde{g}}_i)+\sup_{\substack{l\in B\\\theta\in I^B}}\\| (\tau^{X_k})'\nu^\theta_{l,r}\\|\eta^{A\ln N}\right\}\\ &\quad\leq C\left\{ r+N^{-1}\ln r^{-1}\right\}\leq CN^{-1}\ln N \,, \end{split}$$ where we have chosen $r=N^{-1}$. In conclusion, $$\label{eq:fourth_term} \begin{split} \big\|{{\mathbb E}}^2(&(\hat X^N_{t+h}-\hat X^N_t)_i \partial_{x_iy_j}\psi(t,\hat X^N_t,\hat Y^N_t)(\hat Y^N_{t+h}-\hat Y^N_{t})_j)\big\|\leq CN^{-\frac 12}\\|\psi\\|_{{{\mathcal C}}^2}\\ &+ A\ln N\;N^{-1}{{\mathbb E}}^2({{\rm Card}}\{t\leq N: \\|X_t-Y_t\\|\leq 4{C_0}A\ln N\})\\|\psi\\|_{{{\mathcal C}}^2}. \end{split}$$ In Section \[sec:twowalks\] we prove the following bound. \[lem:NL\] Let $A$ be a large constant and set $L_N:=A\ln N$. There exists $\delta_0\in (0,1)$ such that $$\label{NL} {{\mathbb E}}^2({{\rm Card}}\{t\leq N: \\|X_t-Y_t\\|\leq L_N\})\leq C N^{\delta_0} \quad (N\in{{\mathbb N}}).$$ Lemma \[lem:NL\] allows to estimate the last term in the right hand side of by $$C N^{\delta_0-1}\ln N \\|\psi\\|_{{{\mathcal C}}^2}=C h (h^{-1} N^{\delta_0-1}) \ln N \\|\psi\\|_{{{\mathcal C}}^2},$$ proving the proposition by choosing $\beta=\frac{1-\delta_0}6$ and $\vartheta=1-3\beta$. Two walks estimates {#sec:twowalks} =================== In Section \[sec:clt-proof\] we proved that Lemma \[lm:hot\] (and hence Theorem \[thm:main\]) holds provided the average number of times two walks come closer than $A\ln N$ in time $N$ is smaller than $N^{\delta_0}$ for some $\delta_0\in(0,1)$. The purpose of this section is to prove such an estimate and therefore conclude the argument. On the number of close encounters {#sec:two-crossing} --------------------------------- The proof of inequality can be reduced to the following simpler inequality. \[lem:Ex\] There exist $\rho\in(0,1), {C_7}>0$ such that for any $m\in{{\mathbb N}}$ and for any $a, b$ such that $\\|a-b\\|> L_N$, we have $$\label{Ex} {{\mathbb P}}^2\left(\left\{\\|X_j-Y_j\\|>L_N \;\;\;\forall\, j\in\{m,\dots,m+N\}\right\}\;\|\; X_m=a, Y_m=b\right)\geq \frac{{C_7}}{N^{\rho}} ,$$ (Here ${{\mathbb P}}^2$ is the underlying probability for the process $(\theta_t,X_t,Y_t)$ started with $\theta_0$ distributed according to ${{\mu^e}}$). We postpone the proof of the above Lemma until finishing the proof of . Notice that Assumption \[ass:ellipticity\] implies that the walks can move in different directions with positive probability. In particular, there exists $\gamma>0$ such that for each $a,b\in{{\mathbb Z}}^d$, $m\in{{\mathbb N}}$ and $\delta>0$, $$\label{eq:fast-escape} {{\mathbb P}}^2\left(\left\{\\|X_{m+L_N^2}-Y_{m+L_N^2}\\|\geq L_N\right\}\;\bigg\|\; X_m=a,Y_m=b\right)\geq c(\delta)\gamma^{\delta L_N},$$ Indeed $${{\mathbb P}}^2\left(\left\{\\|X_{m+\delta L_N}-Y_{m+\delta L_N}\\|\geq \delta L_N\right\}\;\bigg\|\; X_m=a,Y_m=b\right)\geq \gamma^{\delta L_N}$$ the latter being the probability of one fixed path in which $X_i,Y_i$ get further and further apart at each step. On the other hand $$\label{LNCLT} {{\mathbb P}}^2\left(\left\{\\|X_{m+L_N^2-\delta L_N}-Y_{m+L_N^2-\delta L_N}\\|\geq L_N\right\}\;\bigg\|\; \\|X_m-Y_m\\|\geq \delta L_N \right)\geq c(\delta) ,$$ To verify let $W^{(1)}(t)$ and $W^{(2)}(t)$ be independent Brownian Motions such that $W^{(2)}(0)-W^{(1)}(0)=\bv,$ where $\\|\bv\\|=\delta.$ Let $\hat\bv:=\bv\,\\|\bv\\|^{-1}$, and $$c(\delta):=\frac{1}{2}P\left(\\|W^{(2)}(1)-W^{(1)}(1)\\|>1, \quad \text{and for all } t\in [0,1] \right.$$ $$\la W^{(2)}(t), \hat\bv \ra> \la W^{(2)}(0), \hat\bv \ra-\frac{\delta}{3} \text{ and } \left. \la W^{(1)}(t), \hat\bv \ra< \la W^{(1)}(0), \hat\bv \ra-\frac{\delta}{3}\right)$$ Observe that the invariance principle established in Lemma \[lem:invariance\], Lemma \[lem:variation\] and the fact that local dynamics are independent implies a two particle invariance principle as long as the walkers explore disjoint regions in the phase space. Therefore the probability that two walkers grow $L_N$ apart exploring disjoint regions of the phase space is at least $c(\delta)$ for large $N$ proving . Choose $\varrho<1-\rho$. Then, $$\label{eq:long-escape} \begin{split} {{\mathbb P}}^2&\left(\left\{\sup_{m\leq i\leq m+N^\varrho}\\|X_i-Y_i\\|\leq L_N\right\}\;\bigg\|\; X_m=a,Y_m=b\right)\\ &\leq \prod_{j=1}^{N^\varrho L_N^{-2}}(1-c(\delta)\gamma^{\delta L_N}) \leq e^{-c(\delta) \gamma^{\delta L_N}L_N^{-2}N^\varrho}\leq e^{-CN^{\varrho/2}}, \end{split}$$ provided that $\delta$ is sufficiently small. Next, consider the sets $B_R^-:=\{(x,y): \\|x-y\\|\leq R\}$, $B_R^+:=\{\\|x-y\\|> R\}$ and the stopping times, for $k>0$, $$\begin{split} s_0&:=\inf\left\{j\in {{\mathbb N}}\;:j>0,\;\;(X_{j-1},Y_{j-1})\in B^-_{L_N}, (X_{j},Y_{j})\in B^+_{L_N}\right\},\\ s_{2k}&:=\inf\left\{j\in {{\mathbb N}}\;:\; j> s_{2k-2},\; (X_{j-1},Y_{j-1})\in B^-_{L_N}, (X_{j},Y_{j})\in B^+_{L_N}\right\},\\ s_1&:=\inf\left\{j\in {{\mathbb N}}\;:\;j>s_0,\;(X_{j-1},Y_{j-1})\in B^+_{L_N},\; (X_{j},Y_{j})\in B^-_{L_N}\right\},\\ s_{2k+1}&:=\inf \left\{j\in{{\mathbb N}}\;:\; j> s_{2k-1},\, (X_{j-1},Y_{j-1})\in B^+_{L_N},\; (X_{j},Y_{j})\in B^-_{L_N}\right\}. \end{split}$$ Clearly, $s_{2k}<s_{2k+1}<s_{2k+2}$ and $s_k>k$. As $X_0=Y_0$, these stopping times are adapted to the filtration ${{\mathcal F}}^{XY}_t$. Note that the $s_{2k}$ are upcrossing times hence $\\|X_t-Y_t\\|\leq L_N$ for all the $t\in\{s_{2k-1}, \dots, s_{2k}-1\}$. With this notation, implies $${{\mathbb P}}^2\left(\left\{\sup_{i\leq N} (s_{2i}-s_{2i-1})>N^\varrho\right\}\right) \leq N\sup_{i\leq N}{{\mathbb P}}^2 \left(\left\{s_{2i}-s_{2i-1}>N^\varrho\right\}\right) \leq Ne^{-N^{\varrho/2}}.$$ Let us set $J:=\inf \{k\in{{\mathbb N}}\;:\;s_k\geq N\}$, clearly $J\leq N$, $$\label{eq:NL0}{{\mathbb E}}^2({{\rm Card}}\{n<N\;:\;\\|X_n-Y_n\\|\leq L_N\}) \leq N^2e^{-N^{\varrho/2}}+N^\varrho{{\mathbb E}}^2(J).$$ It remains to investigate the length of the intervals of time in which the two walks are further apart than $L_N$. Let $S_n:=\{\sup_{k\leq n}(s_{2k+1}-s_{2k})<N\}$, and denote by ${{\mathcal F}}_{s_{2k}}^{XY}$ the $\sigma$-algebra associated to the filtration ${{\mathcal F}}_t^{XY}$ and the stopping time $s_{2k}$. Then, by , $$\begin{split} {{\mathbb P}}^2(\{J>n\}) &\leq {{\mathbb E}}^2\left({\mathds{1}}_{S_{n}}\right) = {{\mathbb E}}^2({\mathds{1}}_{S_{n-1}}{{\mathbb P}}^2 (\{s_{2n+1}-s_{2n}<N\}\;\|\;S_{n-1}))\\ &= {{\mathbb E}}^2\left({\mathds{1}}_{S_{n-1}}{{\mathbb P}}^2 \left(\{s_{2n+1}-s_{2n}<N\}\;\|\; {{\mathcal F}}^{XY}_{s_{2n}}\right)\right)\\ &\leq \left(1-\frac {C_0}{N^\rho}\right){{\mathbb E}}^2({\mathds{1}}_{S_{n-1}}) \leq \dots \leq \left(1-\frac {C_0}{N^\rho}\right)^{n}. \end{split}$$ Thus, letting $1-\varrho>\alpha>\rho$, it follows that $${{\mathbb P}}^2(\{J>N^\alpha\})\leq Ce^{-C_0N^{\alpha-\rho}}.$$ which means that ${{\mathbb E}}^2(J)\leq N^\alpha+N\,{{\mathbb P}}^2(\{J>N^\alpha\})\leq CN^\alpha$. In view of this proves provided we have chosen $\delta_0$ so that $\varrho+\alpha<\delta_0$. Our program is thus completed once we prove . To this end an intermediate result is needed. \[LmFarInd\] Given $R>0$, take two points $a_R$ and $b_R$ such that $\\|a_R-b_R\\|=R.$ Consider two walks starting at $a_R$ and $b_R$ respectively with the environment given by the probability distribution $\nu\in{{\mathcal B}}$ and define the stopping time $\tau_{\delta,R}$ as the first time $n>0$ such that $$\\|X_n-Y_n\\|\leq\frac{R}{1+\delta} \text{ or } \\|X_n-Y_n\\|\geq(1+\delta)R.$$ For each ${C_8}>0$, if $\\|\nu\\|\leq {C_8}$, then there exist $R_\delta\in{{\mathbb R}}_+,$ $c_1, c_2>0$ such that for each $R\geq R_\delta$ $${{\mathbb P}}_\nu^2\left(\{\\|X_{\tau_{\delta,R}}-Y_{\tau_{\delta,R}}\\|\geq (1+\delta)R\}\right)\geq \frac{1}{2+\delta}-c_1 e^{-c_2/\delta} \,.$$ Let $T_R$ be the first time $n>0$ such that $$\max(\\|X_n-X_m\\|, \\|Y_n-Y_m\\|)\geq \frac{R}{2}.$$ Then, by Section \[sec:invariance\] the pair $$\left(\frac{X_{\min(T_R, t R^2)}-X_m}{R}, \frac{Y_{\min(T_R, t R^2)}-X_m}{R} \right)$$ is asymptotic, when $R\to\infty$, to a pair of independent Brownian Motions $$(W^{(1)}(t), W^{(2)}(t))\in{{\mathbb R}}^{d}\times {{\mathbb R}}^d\quad W^{(1)}(0)=0, \quad \\|W^{(2)}(0)\\|=1$$ stopped at time $T$ when one of them wanders more than $1/2$ from its starting position.[^29] Let $\btau$ be the first time $\\|W^{(1)}(t)-W^{(2)}(t)\\|=(1+\delta)$ or $\\|W^{(1)}(t)-W^{(2)}(t)\\|=(1+\delta)^{-1}.$ Recall that $$P(\{\\|W^{(1)}(\btau)-W^{(2)}(\btau)\\|=(1+\delta)\})\geq\frac{1}{2+\delta}$$ (the worst case is then $d=1$, see e.g. [@RY], Section XI.1). On the other hand $$\label{NearFar} P(T<\btau)\leq P(\btau>\delta)+P(T<\delta)$$ and both terms are $O(e^{-c/\delta}),$ the first one because $$P(\btau>(k+1)\delta^2\|\btau>k\delta^2)\leq\gamma<1$$ and the second one by Hoeffding’s inequality (see e.g. [@GS]). Now[^30] $$\begin{split} &{{\mathbb P}}(\{\\|X_{\tau_{\delta, R}}-Y_{\tau_{\delta, R}}=R(1+\delta)\})\\ &\quad\geq {{\mathbb P}}(\{\\|X_{\tau_{\delta, R}}-Y_{\tau_{\delta, R}}=R(1+\delta)\text{ and } \tau_{\delta, R}<T_R\})\geq \frac{1}{2+\delta}-c_1 e^{-c_2/\delta} . \end{split}$$ We now use the following comparison criterion (proved in section \[sec:comparison\]). \[PrComp\] Suppose $\xi_1, \xi_2\dots \xi_n\dots$ is a random process such that $\xi_n=\pm 1 $ and for all $n$ $$P(\xi_n=1 \| \xi_1\dots \xi_{n-1})\geq p .$$ Let ${{\tilde\xi}}_1, {{\tilde\xi}}_2 \dots {{\tilde\xi}}_n\dots $ be iid random variables such that ${{\tilde\xi}}_n=\pm 1,$ and $P({{\tilde\xi}}_n=1)=p.$ Let $${{\mathcal X}}_n=\sum_{j=1}^n \xi_n+ {{\mathcal X}}_0\quad {{\tilde{{{\mathcal X}}}}}_n=\sum_{j=1}^n {{\tilde\xi}}_n+ {{\tilde{{{\mathcal X}}}}}_0.$$ Then for any $ \alpha_1<\alpha< \alpha_2$ $$P({{\mathcal X}}_k\text{ reaches } \alpha_2 \text{ before } \alpha_1\|{{\mathcal X}}_0=\alpha)\geq P({{\tilde{{{\mathcal X}}}}}_k\text{ reaches } \alpha_2 \text{ before } \alpha_1\|{{\tilde{{{\mathcal X}}}}}_0=\alpha)$$ Recall that by Gambler’s Ruin Formula for $p\neq 1/2$ $$\label{GR} P({{\tilde{{{\mathcal X}}}}}_k\text{ reaches } \alpha_2 \text{ before } \alpha_1\|{{\tilde{{{\mathcal X}}}}}_0=\alpha)= \frac{\left(\frac{p}{1-p}\right)^{\alpha_2-\alpha}-\left(\frac{p}{1-p}\right)^{\alpha_2-\alpha_1}} {1-\left(\frac{p}{1-p}\right)^{\alpha_2-\alpha_1}}$$ Let $X_m=a$ and $Y_m=b$ with $\\|a-b\\|\geq L_N$ and $\kappa\in(\frac 12,1)$. Using ellipticity of Assumption \[ass:ellipticity\] for the first $\delta L_N$ steps we see that with probability greater than $N^{-c\delta}$ our walkers move distance $(1+\delta) L_N$ apart without getting within distance $L_N$ from each other. Let $\tau_1$ be the first time after $m$ when our walkers move distance $(1+\delta) L_N$ apart and let $\tau_{n+1}$ be the first time after $\tau_n$ when $$\\|X_j-Y_j\\|\geq (1+\delta) \\|X_{\tau_n}-Y_{\tau_n}\\| \text{ or } \\|X_j-Y_j\\|\leq (1+\delta)^{-1} \\|X_{\tau_n}-Y_{\tau_n}\\|.$$ Applying Lemma \[PrComp\] to ${{\mathcal X}}_n=\frac{\ln\\|X_{\tau_n}-Y_{\tau_n}\\|}{\ln(1+\delta)}$ with $\alpha_1=\frac{\ln L_N}{\ln(1+\delta)}$, $\alpha=\alpha_1+1$, $\alpha_2=\frac{\kappa\ln N}{\ln(1+\delta)}$ and using Lemma \[LmFarInd\], taking into account Lemma \[lem:variation\], to estimate the probability of moving apart we conclude from that for each $\epsilon>0$, by choosing $\delta$ small and $N$ large enough, the probability that the walkers move distance $N^\kappa L_N$ apart without getting within distance $L_N$ from each other is at least $c\delta N^{-\kappa-\epsilon}$. Hence there is a polynomially small probability of making an excursion of size $N^\kappa L_N$ before returning to a distance $L_N$. On the other hand once we have such a big excursion Lemma \[lem:largedev\] implies that it will take more than $N$ steps to come back, indeed $$\begin{split} {{\mathbb P}}^2&\left(\left\{\inf_{\ell+1\leq j\leq \ell+N} \\|X_j-Y_j\\|\leq L_N\right\}\;\bigg\| \;\\|X_\ell-Y_\ell\\|\geq N^\kappa L_N\right)\\ & \leq CNe^{-CN^{2\kappa-1}}. \end{split}$$ The last two estimates imply Lemma \[lem:Ex\] Comparison Lemma {#sec:comparison} ---------------- Let $U_1, U_2\dots U_n\dots $ be random variables which are independent and uniformly distributed on $[0,1].$ Define $$\xi_n^=-1 \text{ if } U_n<P(\xi_n=-1\|\xi_1=\xi^_1, \dots, \xi_{n-1}=\xi_{n-1}^) \text{ and } \xi_n=1 \text{ otherwise} .$$ Also let ${{\tilde\xi}}_n^=-1$ if $U_n<1-p$ and ${{\tilde\xi}}_n^=1$ otherwise. Let $${{\mathcal X}}_n^=\sum_{j=1}^n \xi_j^, \quad {{\tilde{{{\mathcal X}}}}}_n^=\sum_{j=1}^n {{\tilde\xi}}_j^.$$ Then $\{{{\mathcal X}}^_n\}$ has the same distribution as $\{{{\mathcal X}}_n\},$ $\{{{\tilde{{{\mathcal X}}}}}^_n\}$ has the same distribution as $\{{{\tilde{{{\mathcal X}}}}}_n\}$ and ${{\mathcal X}}^_n\geq {{\tilde{{{\mathcal X}}}}}^_n.$ [99]{} V. Baladi, , volume 16 of [Advanced Series in Nonlinear Dynamics]{}. World Scientific, Singapore, 2000. A. Bandyopadhyay and O. Zeitouni, [Random Walk in Dynamic Markovian Random Environment]{}, ALEA (Latin American Journal of Probability and Statistics) [1]{} (2006) 205–224. J.-B. Bardet, S. Gouëzel, G. Keller, [Limit theorems for coupled interval maps]{}, preprint (2006). C. Boldrighini, R. A. Minlos, A. Pellegrinotti, [Random Walk in a Fluctuating Random Environment with Markov Evolution,]{} in [On Dobrushin’s way. From probability theory to statistical physics,]{} (Ed. R.A.Minlos, Senya Shlosman, Yu.M. Suhov) AMS Transl. Ser. 2, [198]{} (2000) 13–35. AMS, Providence, RI. D. Dolgopyat, G.Keller, C.Liverani, [Random Walk in Markovian Environment]{}, to appear on Annals of Probability. G.Grimmett, D.Stirzaker, [Probability and Random Processes]{} $3^{rd}$ edition, Oxford Univ. Press (2001). T.Kato, [Perturbation theory for linear operators]{}, Springer (1966). G.Keller, C.Liverani,[A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps]{} , in [Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems]{}, J.Chazottes and B.Fernandez Eds., Lecture Notes in Physics, Vol. [671]{}, pp. 115–151, Springer (2005) G.Keller, C.Liverani,[Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension]{}, Communications in Mathematical Physics, [262]{}, 1, 33–50, (2006). C.Kipnis and S.R.S.Varadhan, [Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions]{}. Comm. Math. Phys. [104]{} (1986), pp. 1Ð19. D. Revuz and Yor M. [Continuous martingales and Brownian motion,]{} 3d edition. Grundlehren der Mathematischen Wissenschaften, [293]{} (1999) Springer-Verlag, Berlin. A.-S.Sznitman, [Topics in random walk in random environment]{}, in Notes of course at School and Conference on probability, ICPT Lecture Series, 203-266, Triestre (2004). O. Zeitouni, [Random walks in random environment]{}. Lectures on probability theory and statistics, 189–312, Lecture Notes in Math., [1837]{}, Springer, Berlin, 2004. [^1]: The case of weakly coupled maps can be treated similarly by using the techniques introduced in [@KL] and used here to study the present, simpler, case. [^2]: For the existence and uniqueness (among measures absolutely continuous w.r.t. Lebesgue) see [@Babook]. [^3]: As usual the dual operator $S'$ is defined as $S'\nu(f)=\nu(Sf)$ for all $f\in {{\mathcal C}}^0$. [^4]: By ${{\mathcal M}}(\Theta)$ we designate the set of complex valued finite Borel measures on $\Theta$. [^5]: If $\\|t\\|\geq CN^{\frac 16}$ the last statement of Proposition \[lem:clt-averaged\] is obvious: the left hand side is $\leq 2$. [^6]: The $X$ in $C_{\theta, X, \delta}$ stands for the dependence on the random walk $\{X_n\}_{n\in{{\mathbb N}}}$. [^7]: If $\hat{\varphi}$ is the Fourier transform of ${\varphi}$, then Proposition \[lem:clt-averaged\] yields, for each $\rho<\frac16$ and $b\in{{\mathbb N}}$, $$\begin{split} &\left\|{{\mathbb E}}({\varphi}(N^{-\frac 12}\tilde X_N))-{{\mathbb E}}_{{{\mathcal N}}(0,{{\Sigma^2}})}({\varphi})\right\|\leq C\int_{\\|t\\|\geq N^{\rho}} \|\hat {\varphi}(t)\|dt+C\int_{\\|t\\|\leq N^{\rho}}(1+\\|t\\|^3)N^{-\frac 12}\|\hat{\varphi}(t)\|dt\\ &\leq C_dL^d\\|{\varphi}\\|_{{{\mathcal C}}^b}\int_{N^\rho}^\infty x^{-b+d-1}dx+C_dL^d\\|{\varphi}\\|_{{{\mathcal C}}^0}N^{-\frac 12}\int_0^{N^{\rho}} (1+x^3)x^{d-1}dx, \end{split}$$ which gives the advertised result provided $b>d$ and $\rho$ is chosen small enough. [^8]: See for the definition of ${{\mathcal B}}$. [^9]: The equality of the norms follows from the usual weak definition of $BV$, the fact that the measures must be absolutely continuos can be easily proved by approximating a measure with finite norm by one with a smooth density (just use a mollifier) and remembering that the unit ball of $BV$ is compact in $L^1$. See [@KL0] for more details. [^10]: The problem is already present for two sites since $BV(I)\otimes BV(I)\neq BV(I^2)$. [^11]: For example, if $\nu_p,\nu'_p$ $(p\in{{\mathbb Z}}^d)$ are probability measures on $I$ such that $\nu_p=\nu_{p}'$ for all $p\neq q$, and we set $\nu:=\otimes_{p\in{{\mathbb Z}}^d}\nu_p$, $\nu':=\otimes_{p\in{{\mathbb Z}}^d}\nu_p'$, then $\nu-\nu'\in{{\mathcal B}}_q$. [^12]: Since the local functions are dense in the continuous ones by the Stone-Weierstrass theorem. [^13]: For example, one can start from zero and spiral out over larger and larger cubical shells. [^14]: As $f$ is local there exists a box $\Lambda_\subset \Theta$ such that $f$ depends only on the variables $\{\theta_q\;:\;q\in\Lambda_*\}$, but this means that the sum consists only of finitely many terms. [^15]: The first is trivial since $\|F'\mu({\varphi})\|=\|\mu({\varphi}\circ F)\|\leq\|\mu\|\,\|{\varphi}\|_\infty$. For the second, given any smooth local function ${\varphi}$, let ${\varphi}_{\theta_{\neq q}}(\xi):={\varphi}(\theta^{\xi})$, where $\theta^{\xi}_p=T\theta_p$ for each $p\neq q$ while $\theta^{\xi}_q=\xi$. Next introduce a function $\phi$, piecewise linear in the variable $\theta_q$ such that ${\varphi}_{\theta_{\neq q}}(\theta_p)-\phi(\theta)=0$ for each $\theta_p$ on the discontinuity values of $T$. By construction ${\varphi}_{\theta_{\neq q}}(T\theta_p)-\phi(F\theta)$ is then a Lipschitz function in $\theta_p$, thus $$\begin{split} \|F'\mu(\partial_{\theta_q}{\varphi}\|&=\|\mu((\partial_{\theta_q}({\varphi}-\phi))\circ F)\|+ \|\mu\|\,\|\partial_{\theta_q}\phi\|_\infty \leq \|\mu(\partial_{\theta_q}(\|D_{\theta_q}T\|^{-1}({\varphi}-\phi)\circ F)\|+ C\|\mu\|\,\|{\varphi}\|_\infty\\ &\leq \\|\mu\\|\, \|D_{\theta_q}T\|^{-1}({\varphi}-\phi)\circ F\|_\infty+C\|\mu\|\,\|{\varphi}\|_\infty \leq \left[2\lambda^{-1} \\|\mu\\|+C\|\mu\|\right]\|{\varphi}\|_\infty. \end{split}$$ The above yields $\\|F'\mu\\|\leq 2\lambda^{-1} \\|\mu\\|+C\|\mu\|$ which iterated yields the wanted result with $B=(1-2\lambda^{-1})^{-1} C$. See [@Babook; @KL0] if more details are needed. [^16]: This problem is already well investigated, see in particular [@BGK], here we treat it in detail only because we need some explicit estimates not readily available in the literature. [^17]: For the first assertion note that $$\sum_{p\in\Lambda_1}\sum_{z\in\Lambda}\mu_{p-z}(\hat S_{z}1)= \sum_{p\in{{\mathbb Z}}^d}\sum_{z\in\Lambda}\mu_{p-z}(\hat S_{z}1)=\sum_{p\in{{\mathbb Z}}^d}\sum_{z\in\Lambda}\mu_{p}(S_{z}1) =\sum_{p\in{{\mathbb Z}}^d}\mu_p(S1)=0.$$ For the latter just write it as power series of ${{t}}$. [^18]: Note that $\mu_{{t}}$ is not necessarily a measure and gives rise to an analytic object only when applied to a local function. We will abuse notations by writing $\dot\mu_{{t}}$ to mean the functional on local functions defined by ${{\operatorname{Pr}\,}}(\frac{d}{d{{t}}}{{\boldsymbol{\mu}}}_t)({\varphi})$. [^19]: Here we use the notation $[{{\boldsymbol{\nu}}}]_0$ to designate the components $c_\nu$ of the vector ${{\boldsymbol{\nu}}}=(c_\nu,\bar\nu)$. [^20]: Remember that $\ddot \alpha_{{t}}=(\partial_{{{t}}_i}\partial_{{{t}}_j}\alpha_{{t}})$ is a $d\times d$ matrix. [^21]: The latter follows by Theorem \[thm:mixing\]. Let $\beta=\sum_z\langle w,z-v\rangle^2\pi_z$, and $G_j=\sum_z \langle w, z-v\rangle S_z(S^j\langle w,g-v\rangle)$, then $${{\mathbb E}}\left(\|\langle w,{\tilde{X}}_n\rangle\|^2\right)=\sum_{k=0}^{n-1}{{\mu^e}}(S^k \beta)+\sum_{j=1}^{n-1}\sum_{k=0}^{n-j-1}{{\mu^e}}(S^kG_j)$$ and $\|{{\mu^e}}(S^k \beta)-{{\mu^w}}(\beta)\|\leq C\eta^{k}$, $\|{{\mu^e}}(S^kG_j)-{{\mu^w}}(G_j)\|\leq Cj^d\eta^k$. Moreover, setting $\nu_{k,z}({\varphi}):= \langle w, z-v\rangle{{\mu^e}}(S^k S_z{\varphi})$, for ${\varepsilon}$ in assumption \[ass:pert\] such that $(1+3{\varepsilon})2\lambda^{-1}<1$ we have (by equation and ) $\\|\nu_{k,z}\\|\leq C$. Hence Theorem \[thm:mixing\] yields $$\|{{\mu^e}}(S^kG_j)\|\leq \|\sum_z\|\nu_{k,z}(S^j(g-v))\|\leq C\eta^j$$ Thus we can write $$\left\|{{\mathbb E}}\left(\|\langle w,{\tilde{X}}_n\rangle\|^2\right)-{{\mathbb E}}_{{\mu^w}}\left(\|\langle w,{\tilde{X}}_n\rangle\|^2\right)\right\|\leq C\sum_{k=0}^n\eta^k+C\sum_{j=1}^{n-1}\left[\sum_{k=0}^j \eta^j+\sum_{k=j+1}^{n-j}j^d\eta^k\right]\leq C.$$ [^22]: Indeed, ${{\mu^w}}(\pi_z)\leq a_z (1+{\varepsilon})$, thus $\pi_z\geq (1-{\varepsilon})a_z\geq(1-{\varepsilon})(1+{\varepsilon})^{-1}\alpha_z$. [^23]: Note that imply $\\|S_\alpha'\mu\\|\leq\sum_z\alpha_z\\|F'\mu\\|\leq(2\lambda)^{-1}\\|\mu\\|+B\|\mu\|$ and $\|S_\alpha\mu\|\leq\|\mu\|$. Thus, iterating, for each $n\in{{\mathbb N}}$, $\\|(S_\alpha^n)'\mu\\|\leq C\\|\mu\\|$. [^24]: Note that, for a probability measure, $1=\nu(1)=\nu(\partial_{\theta_i}\theta_i)\leq\\|\nu\\|$. [^25]: In fact, Lemma \[lem:variation\] considers two walks, yet the corresponding result for one walk can be obtained by integrating over the second walk. Moreover, for each smooth function $f:{{\mathbb R}}^{2d}\to{{\mathbb R}}$, $$\begin{split} {{\mathbb E}}_\nu(&f(\hat X^N_{t_2},\hat X^N_{t_1})\;\|\;{{\mathcal F}}_{[t_1N]}) ={{\mathbb E}}_\nu\left({{\mathbb E}}\big(f(\hat X^N_{t_2},\hat X^N_{t_1})\;\|\; {\tilde{X}}_{[t_1N]},\theta^{[t_1N]}\big)\;\|\;{{\mathcal F}}_{[t_1N]}\right)\\ &=\nu^X_{{\tilde{X}}_{[t_1N]},[t_1N]}\left({{\mathbb E}}(f(\hat X^N_{t_2},\hat X^N_{t_1}))\;\|\; {\tilde{X}}_{[t_1N]},\theta^{[t_1N]})\right)\\ &={{\mathbb E}}_{X^N_{t_1},\nu^X_{{\tilde{X}}_{[t_1N]},[t_1N]}}\left({{\mathbb E}}(f(\hat X^N_{t_2-t_1},\hat X^N_{0})\;\|\; {\tilde{X}}_{0},\theta^{0})\right) ={{\mathbb E}}_{X^N_{t_1},\nu^X_{{\tilde{X}}_{[t_1N]},[t_1N]}}\left(f(\hat X^N_{t_2-t_1},\hat X^N_{0})\right). \end{split}$$ Proposition \[lem:clt-averaged\] can then be applied after translating $\nu^X_{{\tilde{X}}_{[t_1N]},[t_1N]}$ by ${\tilde{X}}_{[t_1N]}$. [^26]: Indeed, equation is just the backward heat equation, thus for $t\in (0,1)$ $$\psi(t,x,y)=\frac1{(4\pi)^{d}\det(\Sigma_2^{-1}) (1-t)^d}\int_{{{\mathbb R}}^{2d}} e^{-\frac{\langle(x-z,y-w),\Sigma_2^{-2}(x-z,y-w)\rangle}{2(1-t)}}\phi(z,w) \;dz\,dw.$$ [^27]: \[foot:momenta\] In fact Lemma \[lem:largedev\] implies, for each $p\in{{\mathbb N}}$, that $$\begin{split} {{\mathbb E}}(\\|\hat X^N_{t+h}-\hat X^N_t\\|^p)&\leq N^{\frac p2}h^p{{\mathbb E}}\left(\left\\|\frac{1}{\lceil hN\rceil}(\tilde X_{\lceil (t+h)N\rceil}-\tilde X_{\lceil tN\rceil})\right\\|^p\right)\\ &\leq Ch^{\frac p2}+C_pN^{\frac p2}h^p\int_{(Nh)^{-\frac 12}}^\infty x^{p-1}{{\mathbb P}}\left(\left\{ \left\\|\frac{1}{\lceil hN\rceil}(\tilde X_{\lceil (t+h)N\rceil}-\tilde X_{\lceil tN\rceil})\right\\|\geq x\right\}\right)dx\\ &\leq Ch^{\frac p2}+C_pN^{\frac p2}h^p\int_{(Nh)^{-\frac 12}}^\infty x^{p-1} e^{-Cx^2hN} dx\leq C_p h^{\frac p2}. \end{split}$$ [^28]: Remember that the marginal of $({{\mu^e}})^{XY}_\ell$ on $B\cup B'$ is absolutely continuous with respect to Lebesgue, hence the boundary of $I^B$ has zero measure, moreover $\partial_{\theta_l}\Psi$ is a continuous function on $I^{{{\mathbb Z}}^d}$. [^29]: More precisely, the fact that each component is approximately Brownian comes from Section \[sec:invariance\] while independence is due to the fact that the walkers explore non-intersecting regions in the phase space and can be proven by the same arguments use to estimate . [^30]: Here $c_1$ should be taken a little bit larger than the implied constants in to take into account that our process is only approximated by Brownian Motion.
--- author: - \| Irina Penner[^1]\ Humboldt-Universität zu Berlin\ Unter den Linden 6, 10099, Berlin, Germany - \| Anthony Réveillac[^2]\ Université Paris-Dauphine\ CEREMADE UMR CNRS 7534\ Place du Maréchal De Lattre De Tassigny\ 75775 Paris cedex 16 France\ \ title: Risk measures for processes and BSDEs --- [Abstract: The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded [càdlàg ]{}processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs. ]{} [Key words: Convex risk measures for processes, Discounting ambiguity, Model ambiguity, Cash subadditivity, Decomposition of optional measures, BSDEs. ]{} [AMS 2010 subject classification: Primary: 60G07; Secondary: 91B30, 91B16, 60H10, 60G40. JEL subject classification: D81. ]{} Introduction ============ Classical risk assessment methods in Mathematical Finance focus on uncertain payoffs, that are described by random variables on some probability space. In this context, the payments are usually assumed to be discounted, and their timing does not matter for the risk evaluation beyond that. However, the assumption that time value of money can be resolved by a simple discounting procedure is too restrictive in many situations. The purpose of the present paper is to provide a risk assessment method in continuous time, that accounts not only for model ambiguity, but also for uncertainty about time value of money. An axiomatic approach to assessing risks in Mathematical Finance was initiated in [@adeh97; @adeh99; @fs2; @fr2] by introducing the concepts of coherent and convex monetary risk measures. One of the main axioms of a monetary risk measure, which distinguishes it from a classical utility functional, is cash invariance. A cash invariant risk measure computes the minimal capital requirement, that has to be added to a position in order to make it acceptable. On the other hand, as argued in [@er08], cash invariance is a too stringent requirement, since it postulates that future payoffs and present capital reserves are expressed in terms of the same numéraire. Therefore, while monetary risk measures provide a robust method to deal with model ambiguity, they do not allow one to deal with the issue of discounting ambiguity. To remedy this drawback, a new type of risk measures was introduced in [@er08], where the axiom of cash invariance is replaced by cash subadditivity. It was noted in [@afp9], that risk measures for processes introduced in [@cdk4; @cdk6] provide an alternative approach to the problem of discounting uncertainty. The more flexible framework of stochastic processes allows one to relax the axiom of cash invariance without loosing the interpretation of a risk measure as a minimal capital requirement. Consequently, risk measures for processes provide a natural framework to deal with both model ambiguity, and uncertainty about time value of money. Moreover, uncertainty about time value of money has a rather general interpretation in this context: It includes interest rate ambiguity, but also robust optimal stopping problems for american type options as in [@rie9; @BayraktarYao11a; @BayraktarYao11b]. And restricted to random variables, risk measures for processes reduce to cash subadditive risk measures introduced in [@er08]. The general structure becomes visible through the robust representation of a convex risk measure for processes given in [@afp9 Theorem 3.8, Corollary 3.9] in discrete time framework. One of the main goals of the present paper is to extend this result of [@afp9] to continuous time framework. It requires two steps: a dual representation of a monetary convex risk measure on the set of bounded [càdlàg ]{}processes in terms of suitably penalized optional measures, and a decomposition of optional measures into the model and the discounting components. The latter decomposition result is of independent mathematical interest. It provides a Fubini-type disintegration of a positive finite measure on the optional $\sigma$-field into a randomized stopping time $D$, which defines a random measure on the time-axis, and into a local martingale $L$, which can be essentially seen as a model on the underlying probability space. In discrete time, such decomposition was proved in [@afp9 Theorem 3.4]; a continuous time version appeared independently in [@kard10 Theorem 2.1]. In Theorem \[decomposition\] we complement the result of [@kard10] by providing necessary and sufficient conditions for a couple $(L,D)$ to define an optional measure. We also give a more precise statement on the uniqueness of the decomposition, and, in difference to [@kard10], a direct proof of it. Omitting technical details, our discussion shows that taking expectation on the optional $\sigma$-field essentially amounts to computing expectation of a discounted process on the underlying probability space. A robust representation of a risk measure for processes in terms of optional measures as in [@afp9] seems therefore fairy natural. Mathematical precision of this idea is however technically demanding in continuous time framework, since there is no dominating measure on the optional $\sigma$-field, that would allow one to apply the usual $L^\infty$-$L^1$ duality as in the context of random variables. A general dual representation of a convex risk measure for bounded [càdlàg ]{}processes given in [@cdk4 Theorem 3.3] involves pairs of optional and predictable measures, respectively. However, all examples given in [@cdk4], and also examples of risk measures defined by BSDEs in the present paper can be represented in terms of ordinary optional measures only. We provide therefore conditions, under which the representation from [@cdk4] reduces to such a simplified form. One of the reasons for popularity of classical risk measures is their well established relation to the concepts of BSDEs and $g$-expectations in continuous time Brownian framework. The papers [@peng04; @ro6; @bek8] were among the first to identify a solution of a BSDE with a convex driver as a time consistent dynamic risk measure. The strong notion of cash invariance in this context is reflected by the condition that the driver of the BSDE does not depend on the current level of the risk $y$. If the driver does depend on $y$ and is monotone, the solution to the corresponding BSDE becomes cash subadditive; this was noted in [@er08]. In the present paper we aim to establish an analogous link between risk measures for processes and BSDEs. The results of [@er08] suggest to consider to this end BSDEs with monotone convex drivers, which in our case should depend on the whole path of the process. Indeed, we show that a BSDE with a convex monotone generator defines a time consistent dynamic convex risk measure for processes, if the generator depends on the sum $X+Y$ of the current levels of the capital requirement $Y$, and the cumulated cash flow $X$. Moreover, one may add a reflection term to such a BSDE, ensuring that the sum $Y+X$ stays above zero. The resulting reflected BSDE still fits into the format of risk measures for processes. Whereas dependence of the driver on $Y+X$ corresponds to interest rate ambiguity, the reflection term appears in case of uncertainty about stopping times; this becomes visible in the dual representations we provide for the corresponding BSDEs. The paper is organized as follows: After fixing setup and notation in Section \[prelim\], and recalling basic facts about risk measures for processes in Section \[sec:def\], we focus on the structure of optional measures in Section \[sec:dec\]. This section is presented in a self-contained way, and might be read independently of the rest of the paper. The main result here is Theorem \[decomposition\], which provides decomposition of optional measures. The predictable case is treated in Proposition \[prop:pr\]; the section ends with the discussion of how one may associate a probability measure to the local martingale appearing in the decomposition. Section \[sec:robrep\] deals with duality theory for bounded [càdlàg ]{}processes. Section \[sec:discamb\] combines the results of Sections \[sec:dec\] and \[sec:robrep\] by providing a general robust representation of a monetary convex risk measure for processes; Section \[BSDE\] is devoted to BSDEs. Some technical results used in Section \[BSDE\] are proved in the Appendix. Preliminaries and notation {#prelim} ========================== In this paper we consider a filtered probability space $(\Omega,\mathcal{F}_T,(\mathcal{F}_t)_{t\in [0,T]},\P)$ satisfying usual conditions. The time horizon $T$ is a fixed number in $[0,\infty]$. For $T=\infty$ we assume that ${\mathcal{F}}_T=\sigma (\cup_{t\in[0,\infty)}{\mathcal{F}}_t)$. We denote by $\mathcal{O}$ (respectively by $\mathcal{P}$) the optional (respectively predictable) $\sigma$-field with respect to $(\mathcal{F}_t)_{t\in[0,T]}$. For any ${\mathcal{F}}_T\times[0,T]$ measurable process $X$ we denote by $^{o}X$ (respectively $^{p}X$) its optional (respectively predictable) projection. We use [càdlàg ]{}versions of any (local) martingales. For an adapted càdlàg process $X$ we denote by $X^c$ the continuous part of $X$, and by $\Delta_\tau X$ the jump of $X$ at a stopping time $\tau$ with $0\le\tau\le T$, i.e., $\Delta X_\tau:=X_\tau-X_{\tau-}$. ${\rm Var}(X)$ denotes the variation of $X$, $[X]$ the quadratic variation, and $\langle X\rangle$ the continuous part of quadratic variation, as long as these processes are well defined. For any two adapted càdlàg processes $X$ and $Y$ we write $X\le Y$, if $X_t\le Y_t$ for all $t$ $\P$-a.s.. As usually, $\int_t^\cdot$ denotes the (stochastic) integral over $(t,\cdot]$. If the lower bound $t$ should be included into the integration area, we use the notation $\int_{[t,\cdot]}$. By ${\mathcal{R}^\infty}$ we denote the set of all adapted càdlàg processes $X$ that are essentially bounded, i.e., such that $$\\|X\\|_{{\mathcal{R}^\infty}}:=\\|X^\\|_{L^\infty}<\infty,\quad\text{where}\quad X^:=\sup_{0\le t\le T}\|X_t\|.$$ Convex risk measures for processes {#sec:def} ================================== The notion of monetary convex risk measures for processes, that we use in this paper, was introduced in [@cdk4]. It was also studied in [@cdk5], [@cdk6], [@afp9]. In this section we recall definitions and some basic results from these papers. A process $X\in{\mathcal{R}^\infty}$ should be understood in our framework as a value process, which models the evolution of some financial value. It can also be seen as a cumulated cash flow. For instance, the process $m{\textbf{1}}_{[t,T]}$ describes a single payment of $m$ amounts of cash at time $t\le T$. This interpretation is in line with the axiom of cash invariance in the next definition. \[def:rm\] A map $\rho:\,{\mathcal{R}^\infty}\to\real$ is called a monetary convex risk measure for processes if it satisfies the following properties: - Cash invariance: for all $m\in\real$, $$\rho(X+m{\textbf{1}}_{[0,T]})=\rho(X)-m;$$ - (Inverse) Monotonicity: $\rho(X)\ge\rho(Y)$ if $X\le Y$; - Convexity: for all $\lambda\in[0,1]$, $$\rho(\lambda X+(1-\lambda)Y)\le\lambda\rho(X)+(1-\lambda)\rho(Y);$$ - [Normalization]{}: $\rho(0)=0$. A convex risk measure is called a coherent risk measure for processes if it has in addition the following property for all $X\in{\mathcal{R}^\infty}$: - [Positive homogeneity]{}: for all $\lambda\in\real$ with $\lambda\ge0$, $$\rho(\lambda X)=\lambda\rho(X).$$ If $\rho$ is a monetary convex risk measure for processes, the functional $\phi:=-\rho$ defines a monetary or money based utility functional, which is sometimes alternatively used in the literature. \[rem:ac\] 1. The axioms of inverse monotonicity and convexity in Definition \[def:rm\] go back to the classical utility theory, and have obvious interpretations. Normalization is assumed merely for notational convenience, any convex risk measure $\tilde\rho$ with $\tilde\rho(0)\in\real$ can be normalized by passing to $\rho:=\tilde\rho-\tilde\rho(0)$. 2. Cash invariance gives rise to the monetary interpretation of a risk measure as follows: We define the acceptance set of a monetary convex risk measure as $${\mathcal{A}}:={\left\{\,} \newcommand{\rk}{\right\}}X\in{\mathcal{R}^\infty}{\;\big\|\;}\rho(X)\le0\rk.$$ By convexity and monotonicity the set ${\mathcal{A}}$ is convex and solid. Cash invariance yields the following representation of a risk measure: $$\label{defcr} \rho(X)=\inf{\left\{\,} \newcommand{\rk}{\right\}}m\in \real{\;\big\|\;}X+m{{\textbf{1}}_{[0, T]}}\in{\mathcal{A}}\rk.$$ In other words, $\rho(X)$ is the minimal capital requirement, that has to be added to the process $X$ at time $0$ in order to make it acceptable. Conversely, a functional defined by for a given convex solid set ${\mathcal{A}}$ is a (not necessarily normalized) monetary convex risk measure for processes. In difference to a monetary risk measure for random variables, cf., e.g., [@fs11 Definition 4.1], the axiom of cash invariance in Definition \[def:rm\] specifies the timing of the cash flow: Only payments made at the same time as the risk assessment shift it in a linear way. This makes risk measures for processes sensitive to the timing of the payment, and establishes a conceptional difference to the more common notion of risk measures for random variables. Even if restricted to random variables, i.e., to processes of the form $X{\textbf{1}}_{[T]}$ for some $X\in{L^{\infty}(\Omega, {\mathcal{F}}_T,\P)}$, a risk measure in the sense of Definition \[def:rm\] does not reduce to a risk measure in the sense of [@fs11 Definition 4.1]. This aspect was noted in [@afp9 Section 5], and it can be made precise using the notion of cash subadditivity. \[def:ca\] A convex risk measure for processes $\rho$ is called - cash subadditive, if for all $t\geq 0$ and $m\in \real$ $$\begin{aligned} \rho(X+m1_{[t,T]})&\geq \rho(X)-m\;\;\text{for}\;\; m\geq 0\\(\text{resp.}\; &\leq\;\;\text{for}\; m\leq 0);\end{aligned}$$ - cash additive at $t$ for some $t>0$, if $$\rho(X+m1_{[t,T]})=\rho(X)-m,\quad \forall\; m\in\real;$$ - cash additive, if it is cash additive at all $t\in[0,T]$. The notion of cash subadditivity was introduced by El Karoui and Ravanelli [@er08] in the context of risk measures for random variables. It appears naturally in the context of risk measures for processes, as noted in [@afp9 Proposition 5.2]. \[prop:ca\] Every convex risk measure for processes is cash subadditive. Follows directly from monotonicity and cash invariance. Due to cash subadditivity property, risk measures for processes provide a more flexible framework than risk measures for random variables. They allow to capture not only model uncertainty, but also uncertainty about the time value of money. This will be made precise in Section \[sec:discamb\], and requires two steps: The first step consists in providing a dual representation of a monetary convex risk measure on ${\mathcal{R}^\infty}$ in terms of suitably penalized optional measures. In the second step, optional measures will be decomposed into state price deflators, describing the model component, and randomized stopping times, describing the discounting component. We begin with the latter decomposition result for optional measures. Decomposition of optional measures {#sec:dec} ================================== In this section we analyze the structure of finite positive measures $\mu$ on the optional $\sigma$-field ${\mathcal{O}}$, that have no mass on $\P$-evanescent sets. Such measures are called optional $\P$-measures in [@dm2], here we simply call them optional measures. The set of optional measures will be denoted by ${\mathcal{M}}({\mathcal{O}})$, and the subset of optional measures $\mu$ with $\mu({\mathcal{O}})=1$ by ${\mathcal{M}}_1({\mathcal{O}})$. We also introduce the spaces $${\mathcal{B}}^1={\left\{\,} \newcommand{\rk}{\right\}}a=(a_t)_{t\in[0,T]} {\;\big\|\;}a\, \text{adapted, right-continuous, of finite variation},\,{{\rm Var}}(a)\in L^1(\P)\rk, $$ and the space of random measures $${\mathcal{B}}^1_+:={\left\{\,} \newcommand{\rk}{\right\}}a\in{\mathcal{B}}^1 {\;\big\|\;}a_{0-}:=0, a\: \text{non-decreasing}\rk.$$ Due to Dol[é]{}ans representation result, cf., e.g., [@dm2 Theorem VI 65], $\mu\in{\mathcal{M}}({\mathcal{O}})$ if and only if there exists a process $a\in{\mathcal{B}}^1_+$ such that $$\label{om} \E_{\mu}[X]=\E\left[\int_{[0, T]}X_sda_s\right]$$ for every bounded optional process $X$. So we can (and will) identify the space ${\mathcal{M}}({\mathcal{O}})$ with ${\mathcal{B}}^1_+$, and the space ${\mathcal{M}}_1({\mathcal{O}})$ with $$\label{eq:z1} {\mathcal{Z}}_{1}:={\left\{\,} \newcommand{\rk}{\right\}}a\in{\mathcal{B}}^1_+ {\;\big\|\;}\E[a_T]=1\rk.$$ Next we prove an auxiliary result on extension of local martingales; we apply here terminology and results from [@Jacod Chapter V]. For a given non-decreasing sequence of stopping times $(\tau_n)_{n\in\inte}$ such that $\tau:=\lim_n\tau_n$ is a predictable stopping time, we consider a stochastic interval of the form $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$. The interval can be either open or closed at the right boundary $\tau$: Defining $B:=\cap_n\{\tau_n<\tau\}$, we have that $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket=\llbracket 0,\tau\llbracket$ on $B$, and $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket=\llbracket 0,\tau\rrbracket$ on $B^{\rm c}$. We call a process $L$ a local martingale (resp. a semimartingale, a supermartingale) on $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$, if for any stopping time $\sigma$ such that $\llbracket 0,\sigma\rrbracket\subseteq\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$ the stopped process $L^\sigma$ is a local martingale (resp. a semimartingale, a supermartingale). The following lemma extends [@ctr12 Proposition 1], cf. also [@bkx12 Lemma 6.10] to non-continuous local martingales. \[lemmma:locmart\] Let $(\tau_n)_{n\in\inte}$ be an non-decreasing sequence of stopping times, such that $\tau=\lim_n\tau_n$ is a predictable stopping time. Assume further that $L$ is a nonnegative local martingale on the stochastic interval $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$. Then there exists a [càdlàg ]{}local martingale $\tilde{L}=(\tilde{L}_t)_{t\in[0,T]}$, such that $\tilde{L}=\tilde{L}^\tau$, and $L=\tilde{L}$ on $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$. We define the extension of $L$ as $$\label{L:extension}\tilde L_t:= \begin{cases} L_t & \text{on $\{t<\tau\}$},\\ L_\tau & \text{on $\{\tau\le t\le T\}\cap B^{\rm c}$},\\ {\lim_{s\uparrow\tau, s\in\llbracket 0,\tau\llbracket\cap\qu}L_s}& \text{on $\{\tau\le t\le T\}\cap B$}, \end{cases}$$ where $B=\cap_n\{\tau_n<\tau\}$. Since $L$ is a nonnegative supermartingale on $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$, the left limit $L_{\tau-}$ exists $\P$-a.s.. In particular, the process $\tilde L$ is well defined, $L=\tilde{L}$ on $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket$, and $\tilde L$ is a supermartingale on $[0,T]$ by [@Jacod Lemma 5.17, Proposition 5.8]. In fact, $\tilde L$ is a local martingale. To see this, we use the Doob-Meyer decomposition of the supermartingale $\tilde L=\tilde M-\tilde a$, where $\tilde M$ is a local martingale, and $\tilde a$ a predictable non-decreasing process. Since $\tilde L$ is a local martingale on $\llbracket 0,\tau\llbracket$ and constant on $\rrbracket \tau,T\rrbracket$, uniqueness of the Doob-Meyer decomposition implies $$\tilde a_t= \begin{cases} 0 & \text{on $t<\tau$},\\ \Delta\tilde a_\tau & \text{on $\tau\le t\le T$}. \end{cases}$$ We will show that $$\label{deltal} \E{\left[}\Delta\tilde a_\tau{\right]}=\E{\left[}\Delta\tilde M_\tau{\right]}-\E{\left[}\Delta\tilde L_\tau{\right]}=0,$$ which implies $\tilde a\equiv 0$, and proves that $\tilde L=\tilde M$ is a local martingale. In order to see , note that $\E[\Delta\tilde M_\tau]=0$, since $\tau$ is predictable, and $\tilde M$ a local martingale. Moreover, by [@Jacod Theorem 5.3], cf. also [@ctr12 Lemma 1], there exists a non-decreasing sequence of stopping times $(\sigma_n)_{n\in\inte}$, such that $\cup_{n\in\inte}\llbracket 0,\tau_n\rrbracket=\cup_{n\in\inte}\llbracket 0,\sigma_n\rrbracket$, and $L^{\sigma_n}$ is a uniformly integrable martingale for each $n$. We have $\Delta \tilde L_\tau=\lim_n\Delta L_\tau^{\sigma_n}$, since $\Delta \tilde L_\tau=0$ on $B$, and $\Delta \tilde L_\tau=\Delta L_\tau$ on $B^{\rm c}$. In addition, $\|\Delta L_\tau^{\sigma_n}\|\le\|\Delta \tilde L_\tau\|\in L^1(\P)$ for all $n\in\inte$, since $\tilde L$ is a nonnegative supermartingale. Hence, dominated convergence implies $$\E[\Delta\tilde L_\tau]=\lim_n\E{\left[}\Delta L_\tau^{\sigma_n}{\right]}=0,$$ where we have used that $\tau$ is predictable and $L^{\sigma_n}$ is a martingale for the second equality. This concludes the proof. We are now ready to state the main result of this section. \[decomposition\] A process $a:=(a_t)_{t\in [0,T]}$ is an non-decreasing, right-continuous, adapted process with $a_{0-}=0$ and $\E[a_T]=1$, if and only if there exists a pair of adapted [càdlàg ]{}processes $(L,D):=(L_t,D_t)_{t\in [0,T]}$, such that - $L$ is a non-negative local martingale with $L_0=1$ and $L_{T-}=\E{\left[}L_T\|{\mathcal{F}}_{T-}{\right]}$; - $D$ is a non-increasing process with $D_{0-}=1$ and $\{D_T>0\}\subseteq\{L_T=0\}$; - The non-negative supermartingale $(L_tD_t)_{t\in[0,T]}$ is of class (D); - $\displaystyle{a_t=-\int_{[0,t]} L_s dD_s} \quad \forall t \in [0,T]$, with the convention $a_0=-L_0\Delta D_0=1-D_0$. The processes $L$ and $D$ are unique up to undistinguishability on $\llbracket0,\tau\llbracket$, where $$\tau:=\inf{\left\{\,} \newcommand{\rk}{\right\}}t\in[0,T]{\;\big\|\;}a_t=a_T\rk.$$ Moreover, the pair $(L,D)$ can be chosen such that in addition - $L_t=L_0+ \int_0^t \textbf{1}_{\{D_{s-}>0\}} dL_s$, $ D_t=1+\int_0^t \textbf{1}_{\{L_s>0\}} \, dD_s \quad \forall\, t\in [0,T]$ holds. Under this condition $L$ and $D$ are essentially unique on $[0,T]$. Dolean’s representation result, cf., e.g., [@dm2 Theorem VI 65], implies immediately the following corollary. \[cor:dec\] We have $\mu\in{\mathcal{M}}({\mathcal{O}})$ if and only if there exists a pair of processes $(L,D)$, satisfying properties 1)-3) of Theorem \[decomposition\], where in 1) $L_0=\mu({\mathcal{O}})$, such that $$\label{eq:md} \E_{\mu}[X]=\E{\left[}-\int_{[0,T]}X_sL_sdD_s{\right]}$$ for every bounded optional process $X$. Before giving the proof of Theorem \[decomposition\], let us note that a discrete time version of it appeared in [@afp9 Theorem 3.4], and a continuous time version was proved in [@kard10 Theorem 2.1]. Here we complement the result of [@kard10] by providing necessary and sufficient conditions for a couple $(L,D)$ to define an optional measure. In particular, sufficiency requires property 3), that did not appear in [@kard10 Theorem 2.1]. We also provide a more precise statement on the uniqueness of the couple $(L,D)$, and, in difference to [@kard10], a direct proof of it. It involves only conditions 1), 2), and 4) of Theorem \[decomposition\], and hence applies also to [@kard10 Theorem 2.1]. In Theorem \[decomposition\] we choose the process $D$ to be non-increasing, i.e., the measure $-dD$ to be positive, since in our framework $D$ is interpreted as a discounting process. One can always switch to the non-decreasing process $K:=1-D$ as in [@kard10 Theorem 2.1], in order to have a positive measure in the representation . [Proof of Theorem \[decomposition\]]{}. The proof will be obtained in several steps. We begin with the “only if” part.\ Step 1\ We consider the non-negative supermartingale $U$ defined by $$\label{u} U_t:=\E[a_T \vert \mathcal{F}_t]-a_t=:M_t-a_t,\qquad t\in[0,T],$$ The process $U$ is of class (D), and it is a potential if and only if $\Delta a_T=0$. We define the stopping times $$\label{taun} \tau_n:=\inf{\left\{\,} \newcommand{\rk}{\right\}}t\in [0,T]{\;\big\|\;}U_{t}\le\frac{1}{n}\rk,\qquad n\in\inte,$$ and $$\begin{aligned} \label{tau} \tau:=\lim_{n\to\infty}\tau_n&=\inf{\left\{\,} \newcommand{\rk}{\right\}}t\in [0,T]{\;\big\|\;}U_{t-}=0 \textrm{ or } U_t=0\rk\\\nonumber&=\inf{\left\{\,} \newcommand{\rk}{\right\}}t\in[0,T]{\;\big\|\;}a_t=a_T\rk. \end{aligned}$$ We have $\tau\le T$ $\P$-a.s., and $U$ vanishes on $\llbracket\tau,T\rrbracket$ by [@dm2 Theorem VI.17].\ To determine the process $D$, we set $D_{0-}:=1$, and define $(D_t)_{t\in [0,T]}$ as the unique solution of the SDE $$\label{eq:DSDE} D_t=1-\int_0^{t} \frac{D_{s-}}{U_s+\Delta a_s} da_s, \qquad t\in [0,T],$$ i.e., $$\label{eq:D} D_t:=\exp\left(-\int_0^{t} \frac{1}{U_s} d a_s^c \right) \prod_{0\le s\leq t, \, \Delta a_s>0} \frac{U_s}{U_s+\Delta a_s}, \qquad t\in [0,T].$$ Note that $D$ is well-defined, right-continuous, and non-increasing on $[0,T]$, with $D_0=1-a_0$, $D=D^\tau$, $\{D_{\tau-}=0\}\subseteq\{U_{\tau-}=0\}$, and $\{D_\tau=0\}\subseteq\{\Delta a_\tau>0\}\cup\{D_{\tau-}=0\}$ $\P$-a.s..\ The process $L$ should be intuitively defined as the stochastic exponential of $\int_0^{\cdot}\textbf{1}_{\{U_{s-}>0\}} \frac{1}{U_{s-}} d M_s$. In order to make this definition rigorous, let $A:= \{U_{\tau-}=0\}$, and denote by $\tau_{A}$ the restriction of $\tau$ to $A$, i.e., $$\tau_A:= \begin{cases} \tau & \text{on $A$},\\ T & \text{otherwise}. \end{cases}$$ Note that $\tau_A$ is a predictable stopping time, since $\tau_A=\lim_n\tilde\tau_n$, where $$\tilde \tau_n:= \begin{cases} \tau_n & \text{on $\{\tau_n<\tau\}$},\\ T & \text{on $\{\tau_n=\tau\}$}. \end{cases}$$ Since $M=M^\tau$, and $\frac{1}{U_{s-}}$ is bounded on $\llbracket 0,\tilde \tau_n\rrbracket\cap\llbracket 0,\tau\rrbracket$, the stochastic integral $\int_0^{\cdot} \frac{1}{U_{s-}} d M_s$ is well defined on each $\llbracket 0,\tilde\tau_n\rrbracket$, and hence on $\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$. Thus we can define the process $L$ as the stochastic exponential of the local martingale $\int_0^{\cdot}\frac{1}{U_{s-}} d M_s$ on $\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$, i.e. $$\label{eq:Lloc1} L_t:= \exp \left(\int_0^{t} \frac{1}{U_{s-}} d M_s^c - \frac12 \int_0^{t} \left\|\frac{1}{U_{s-}}\right\vert^2 d \langle M \rangle_s \right)\times \prod_{\substack{0<s\leq t,\\ \Delta M_s\ne0}} \left(1+\frac{\Delta M_s}{U_{s-}}\right)$$ for $(\omega,t)\in\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$. Then $L$ solves $$\label{eq:Lloc} L_t = 1 +\int_0^{t} \frac{L_{s-}}{U_{s-}} d M_s,$$ and is a non-negative local martingale on $\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$. By Lemma \[lemmma:locmart\], $L$ can be extended to a local martingale on $[0,T]$, which we also denote by $L$. It follows from and from , that $L$ solves the SDE $$\label{eq:L} L_t = 1 +\int_0^{t} \textbf{1}_{\{U_{s-}>0\}} \frac{L_{s-}}{U_{s-}} d M_s, \qquad t \in [0,T],$$ and can be written as $$\begin{aligned} \label{eq:Lexp} \nonumber L_t:= \exp \left(\int_0^{t\wedge\tau} \right.& \left. \frac{1}{U_{s-}} d M_s^c - \frac12 \int_0^{t\wedge\tau} \left\|\frac{1}{U_{s-}}\right\vert^2 d \langle M \rangle_s \right)\\ &\times \prod_{\substack{0<s\leq t\wedge\tau,\\ \Delta M_s\ne0}} \left(1+\frac{\Delta M_s}{U_{s-}}\right),\qquad \qquad t\in[0,T]. $$ We slightly deviate here from the usual definition of a stochastic exponential by allowing the continuous part of $L$ to become zero. Indeed, the set $$\left\{L_{\tau}=0\right\}=\left\{\lim_{v\uparrow\tau_A}\int_0^{v\wedge\tau_A} \left\|\frac{1}{U_{s-}}\right\vert^2 d \langle M \rangle_s =\infty\right\}\subseteq A$$ might have positive probability, cf. [@kard10 Example 2.5]. We use in the convention $L_t(\omega):=0$ for $\omega\in\{L_\tau=0\}\cap\{\tau\le t\}$. Note that the jump part of $L$ is well defined at $\tau$, since $\Delta M_\tau=0$ on $\{U_{\tau-}=0\}$.\ It follows either from or from , that $$\label{ceL} \E{\left[}L_T{\;\big\|\;}{\mathcal{F}}_{T-}{\right]}=\E{\left[}L_{T-}+\textbf{1}_{\{U_{T-}>0\}} L_{T-}\frac{\Delta M_T}{U_{T-}}{\;\big\|\;}{\mathcal{F}}_{T-}{\right]}=L_{T-},$$ since $\E{\left[}\Delta M_T{\;\big\|\;}{\mathcal{F}}_{T-}{\right]}=0$ both for $T<\infty$ and $T=\infty$ due to the fact that $M$ is a uniformly integrable martingale. Step 2\ We show that $D$ and $L$ provide a multiplicative decomposition of $U$, i.e., $$\label{eq:da} U_t=L_t D_t \qquad \forall \,t\in [0,T].$$ First we prove this equality on $\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$. To this end, we note that by the same argumentation as in Step 1, the stochastic integral $\int_0^{\cdot} \frac{1}{U_{s-}} dU_s$ is well defined on $\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$. Thus $U$ can be written as the stochastic exponential of $\int_0^{\cdot} \frac{1}{U_{s-}} dU_s$, i.e., $$\label{eq:tempdec} U_t =U_0\exp\left(\int_0^{t} \frac{1}{U_{s-}} dU_s^c -\frac12 \int_0^{t} \left\vert\frac{1}{U_{s-}}\right\vert^2 d\langle M \rangle_s \right) \times \prod_{\substack{0<s\leq t,\\ \Delta U_s\ne0}} \left( \frac{U_s}{U_{s-}} \right)$$ on $\cup_{n\in\inte}\llbracket 0,\tilde\tau_n\rrbracket$. Plugging and into , and noting that $$\prod_{\substack{0<s\leq t,\\ \Delta U_s\ne0 }}\frac{U_s}{U_{s-}} =\prod_{\substack{0<s\leq t,\\ \Delta M_s\ne0 }}\left(1+\frac{\Delta M_s}{U_{s-}}\right) \times \prod_{\substack{0<s\leq t,\\ \Delta a_s\ne0}} \frac{U_s}{U_{s}+\Delta a_s},$$ we obtain . It remains to prove for $(\omega, t)\in\llbracket\tau_A, T\rrbracket$ and $\omega\in\{U_{\tau_A-}=0\}$. Thanks to the existence of the left limits, we obtain $$0=U_{\tau_A-}=L_{\tau_A-}D_{\tau_A-}=L_{\tau_A}D_{\tau_A},$$ where we have used that $\Delta D_{\tau_A}=\Delta L_{\tau_A}=0$ on $\{U_{\tau_A-}=0\}=\{U_{\tau-}=0\}$ by definitions of $D$ and $L$. Hence, $$U_{t}=U_{\tau_A}=0=L_{\tau_A}D_{\tau_A}=L_{t}D_{t}\quad\text{for $(\omega, t)\in\llbracket\tau_A, T\rrbracket$ and $\omega\in\{U_{\tau_A-}=0\}$.}$$ This concludes the proof of . Note that implies in particular property 3) of the theorem, and $\{D_T>0\}\subseteq\{L_T=0\}$, since $U_T=0$.\ Step 3:\ We now prove properties 4) and 5) of the theorem. First note that 4) holds at $0$ by definitions of $D$ and $L$. Hence it remains to prove $$\label{3a} a_t-a_0=-\int_0^t L_s dD_s\qquad \forall\,t\in[0,T].$$ The multiplicative decomposition , integration by parts, and the SDE yield for each $t\in[0,T]$ $$\begin{aligned} \label{3} \nonumber U_t= L_{t} D_{t} &= L_0D_0+\int_0^{t} L_s dD_s + \int_0^{t} D_{s-} dL_s\\ \nonumber &=U_0+\int_0^{t} L_s dD_s +\int_0^{t} \textbf{1}_{\{U_{s-}>0\}} dM_s\end{aligned}$$ Since $\{M_{s}>0\}\subseteq\{U_{s-}>0\}$, we have $$M_t-M_0=\int_0^{t} \textbf{1}_{\{M_{s}>0\}} dM_s=\int_0^{t} \textbf{1}_{\{U_{s-}>0\}} dM_s,$$ and thus $$\int_0^{t} L_s dD_s=U_t-U_0-(M_t-M_0)=-(a_t-a_0).$$ Concerning property 5), note that by definition of $D$ we have ${\{U_{t-}>0\}}\subseteq {\{D_{t-}>0\}}$ for all $t\in [0,T]$. Thus implies for each $t\in[0,T]$ $$L_t = L_0 +\int_0^{t } \textbf{1}_{\{U_{s-}>0\}} \frac{L_{s-}}{U_{s-}} d M_s = L_0 +\int_0^t \textbf{1}_{\{D_{s->0}\}} dL_s.$$ Similarly, by definition of $L$ we have ${\{U_t + \Delta a_t >0\}}\subseteq {\{L_t>0\}}$ for all $t\in [0,T]$, and hence yields for each $t\in[0,T]$ $$D_t = 1-\int_0^t \textbf{1}_{\{U_s+\Delta a_s>0\}} \frac{D_{s-}}{U_s+\Delta a_s} da_s = 1+\int_0^t \textbf{1}_{\{L_s>0\}} dD_s.$$\ \ Step 4:\ In order to prove uniqueness, we first show that every pair of processes $(\tilde{L},\tilde{D})$ satisfying properties 1), 2), and 4) of Theorem \[decomposition\] provides a multiplicative decomposition of the supermartingale $U$ defined in , that is, $$\label{eq:U=tLtD} U_t = \tilde{L}_t \tilde{D}_t \qquad \forall\, t \in [0,T].$$ This holds clearly at $T$, since $U_T=0$ and $\{\tilde D_T>0\}\subseteq\{\tilde L_T=0\}$. In order to prove on $[0,T)$, let $(\sigma_n)_{n\geq 1}$ be a localizing sequence for $\tL$, i.e. $\sigma_n\nearrow T$ $\P$-a.s., and $L^{\sigma_n}$ is a uniformly integrable martingale for each $n$. Let further $\sigma$ be any stopping time. Then property 4) yields $$\begin{aligned} 1 &=& \E\left[a_T\right] = \E[a_T - a_{\sigma\wedge \sigma_n} + a_{\sigma\wedge \sigma_n} - a_{0-}]\\ &=& \E[a_T - a_{\sigma\wedge \sigma_n}] - \E\left[ \int_0^{\sigma\wedge \sigma_n} \tL_s d\tD_s{\right]}-\E{\left[}L_0\Delta D_0\right]\\ &=&\E[U_{\sigma\wedge \sigma_n}] - \E\left[ \int_0^{\sigma\wedge \sigma_n} \tL_{\sigma\wedge \sigma_n} d\tD_s \right]-\E{\left[}L_0\Delta D_0\right]\\ &=& \E[U_{\sigma\wedge \sigma_n}] - \E\left[ \tL_{\sigma\wedge \sigma_n} \tD_{\sigma\wedge \sigma_n}\right] + 1,\end{aligned}$$ where we have used uniform integrability of the martingale $L^{\sigma_n}$, and [@dm2 VI.57]. Hence, by [@dm1 IV.87 b)] the processes $U^{\sigma_n}$ and $\tL^{\sigma_n} \tD^{\sigma_n}$ are indistinguishable for each $n$. Since $\sigma_n\to T$ $\P$-a.s., holds on $[0,T)$.\ In particular, since $\Delta a_t=-\tilde L_t\Delta \tilde D_t$ by 4), yields $U_t+\Delta a_t=\tilde{L}_t \tilde{D}_{t-}$ for all $t$. This implies on $\llbracket 0, \tau\llbracket$: $$\tilde{D}_t - \tilde{D}_0 = \int_0^t d\tilde{D}_s = \int_0^t \frac{\tilde{L}_s \tilde{D}_{s-}}{\tilde{L}_s \tilde{D}_{s-}} d\tilde{D}_s = -\int_0^t \frac{\tilde{D}_{s-}}{U_s+\Delta a_s} da_s.$$ So $\tilde{D}$ is a solution to the SDE on $\llbracket 0, \tau\llbracket$, and thus coincides with $D$ on this set. Since $U=LD=\tL\tD$, this implies further $L=\tilde L$ on $\llbracket0,\tau\llbracket$, and $L_{\tau-}=\tilde L_{\tau-}$, $D_{\tau-}=\tilde D_{\tau-}$. Moreover, since $L_\tau D_\tau=\tL_\tau\tD_\tau=U_\tau=0$, property 4) yields $$L_\tau D_{\tau-} =-L_\tau\Delta D_\tau=-\Delta a_\tau=-\tilde L_\tau\Delta \tilde D_\tau=\tilde L_\tau \tilde D_{\tau-}.$$ Thus $\tilde L_\tau= L_\tau>0$ on $\{\Delta a_\tau>0\}$, and hence $\tilde D_\tau= D_\tau=0$ on $\{\Delta a_\tau>0\}$ by and , which implies already $D=\tilde D$ on $\{\Delta a_\tau>0\}$. On $\{\Delta a_\tau=0\}$ we have $\tilde D_\tau=\tilde D_{\tau-}=D_{\tau-}= D_\tau$, and $\tilde L_{\tau-}= L_{\tau-}=0$ on $\{\Delta a_\tau=0\}\cap\{D_{\tau-}>0\}$, since $U_{\tau-}=L_{\tau-}D_{\tau-}=0$ on $\{\Delta a_\tau=0\}$. Non-negativity and local martingale property imply then $L=\tilde L$ on $\{\Delta a_\tau=0\}\cap\{D_{\tau-}>0\}$.\ If we assume in addition, that $(\tilde L, \tilde D)$ satisfies property 5) of the theorem, we obtain also $\tilde L=\tilde L^\tau=L^\tau=L$ on $\{\Delta a_\tau>0\}\cup\{D_{\tau-}=0\}$, and $\tilde D=\tilde D^\tau=D^\tau=D$ on $\{\Delta a_\tau=0\}\cap\{D_{\tau-}>0\}$, which proves equality (in the sense of undistinguishability) on $[0,T]$.\ Step 5:\ We now proof the “if” part of the theorem. Obviously, any two processes $L$ and $D$ satisfying properties 1) and 2) define a non-decreasing, right-continuous, adapted process $a$ via 4). It remains to prove that $\E{\left[}a_T{\right]}=1$. To this end let $(\sigma_n)_{n\in\inte}$ be a localizing sequence for $L$. Note that w.l.o.g. we can assume that $\sigma_n<T$ for all $n$, otherwise we switch to $\sigma_n\wedge(T-\frac{1}{n})$ in case $T<\infty$. Using 1), 2), 4), uniformly integrability of the martingale $L^{\sigma_n}$, and [@dm2 VI.57] we obtain for each $n\in\inte$: $$\begin{aligned} \label{uniq} \nonumber \E[a_{\sigma_n}]&=\E{\left[}- \int_{[0,\sigma_n]}L_{t\wedge\sigma_n}dD_t{\right]}=\E{\left[}- \int_{[0,\sigma_n]}L_{\sigma_n}dD_t{\right]}\\ &=\E{\left[}-L_{\sigma_n}D_{\sigma_n}+L_0D_{0-}{\right]}=\E{\left[}- L_{\sigma_n}D_{\sigma_n}{\right]}+1\end{aligned}$$ By monotone convergence, $\E[a_{\sigma_n}]\to\E[a_{T-}]$ with $n\to\infty$, and $\E{\left[}L_{\sigma_n}D_{\sigma_n}{\right]}\to\E{\left[}L_{T-}D_{T-}{\right]}$, since $LD$ is of class (D). Moreover, $$\E{\left[}L_{T-}D_{T-}{\right]}=\E{\left[}L_{T}D_{T-}{\right]}=\E{\left[}- L_{T}\Delta D_{T}{\right]}=\E{\left[}\Delta a_T{\right]},$$ where we have used $L_{T-}=\E{\left[}L_T\|{\mathcal{F}}_{T-}{\right]}$, $D_T=0$ on $\{L_T>0\}$ and 4). Hence implies $$\E{\left[}a_T{\right]}=\E{\left[}a_{T-}{\right]}+\E{\left[}\Delta a_T{\right]}=1.$$ $\square$\ \[rem:inter\] 1. Our proof of Theorem \[decomposition\] is based on the idea that any pair of processes $(L,D)$ satisfying conditions 1)-3) provides a multiplicative decomposition of the supermartingale $U$ defined in . The construction of $L$ and $D$ is inspired by the classical multiplicative decomposition results as in [@iw65], [@Jacod Theorem 6.17]. However, in difference to these results, the non-increasing process $D$ in our case is in general not predictable, even if the corresponding process $a$ is. As it can be seen from , $D$ is predictable, if $a$ is predictable, and it does not jump at the same time as the martingale $M$; cf. also Remark \[ctsfiltr\] later on in text. 2. In [@afp9 Theorem 3.4], which is a discrete time counterpart of Theorem \[decomposition\], the non-increasing process $D$ is predictable. However, this is just a matter of notation: The process $D$ appearing in [@afp9 Theorem 3.4] corresponds to the predictable process $D_-$ of Theorem \[decomposition\]. Indeed, if $(L,D)$ is a couple of processes as in Theorem \[decomposition\], and if we can associate a measure $Q$ on $(\Omega, {\mathcal{F}})$ to the local martingale $L$, as explained later on in text, representation takes the form $$\E_{\mu}[X]=\E_{Q}{\left[}\int_{[0,T]}D_{s-}dX_s{\right]}$$ for any bounded semimartingale $X$ with $X_{0-}:=0$. This representation corresponds to (3.8) of [@afp9 Theorem 3.4]. Clearly, Theorem \[decomposition\] provides for any predictable process $a$ a decomposition $(L,D)$, such that $\int_0^\cdot L_tdD_t$ is predictable. However, if one seeks to construct a predictable process $a$ starting with a couple $(L,D)$, it requires more conditions than 1)-3) of Theorem \[decomposition\] to ensure predictability. In this case, $D$ should “compensate” the non-predictable jumps of the local martingale $L$, i.e., the jump process $(\sum_{s\le t} L_s\Delta D_s)_t$ should be predictable. This additional assumption is not very handy. In the predictable case it seems more natural to use a different construction, namely $a=\int L_-dD$ with a predictable process $D$ and a local martingale $L$. This is done in the next proposition. \[prop:pr\] A process $a:=(a_t)_{t\in [0,T]}$ is an non-decreasing, right-continuous, predictable process with $a_{0-}=0$ and $\E[a_T]=1$, if and only if there exists a pair of adapted [càdlàg ]{}processes $(L,D)$, satisfying properties 1)-3) of Theorem \[decomposition\], such that in addition $D$ is predictable, and - $\displaystyle{a_t=-\int_{[0,t]} L_{s-} dD_s} \quad \forall t \in [0,T]$ with the convention $L_{0-}:=1$ holds. The processes $L$ and $D$ are unique up to undistinguishability on $\llbracket0,\tau\llbracket$, where $\tau$ is as in Theorem \[decomposition\]. Moreover, the pair $(L,D)$ can be chosen such that in addition - $L_t=L_0+ \int_0^t \textbf{1}_{\{D_{s}>0\}} dL_s$, $ D_t=1+\int_0^t \textbf{1}_{\{L_{s-}>0\}} \, dD_s \quad \forall\, t\in [0,T]$ holds. Under this condition $L$ and $D$ are essentially unique on $[0,T]$. The proof of the “if” part follows exactly as in Step 5 of the proof of Theorem \[decomposition\]: Obviously, the process $a$ defined by 4’) is predictable, and, since $D$ is predictable, the equality holds in the same way for $\E{\left[}- \int_{[0,\sigma_n]}L_{t\wedge\sigma_n-}dD_t{\right]}$. To prove “only if”, we use the classical multiplicative decomposition of the supermartingale $U$ defined in as $$U_t=\E[a_T \vert \mathcal{F}_t]-a_t=M_t-a_t,\qquad t\in[0,T].$$ The construction of $D$ and $L$ basically follows as in the proof on Theorem \[decomposition\], with the difference that $U_-$ has to be replaced by the predictable projection of $U$, denoted by $^pU$. The process $D$ is defined via $$\label{eq:DSDEpr} D_t=1-\int_0^{t} \frac{D_{s-}}{^pU_s} da_s, \qquad t\in [0,T],$$ i.e., $D_{0-}:=1$ and $$\label{eq:D:pr} D_t:=\exp\left(-\int_0^{t} \frac{1}{^pU_s} d a_s^c \right) \prod_{0\le s\leq t, \, \Delta a_s>0} \frac{^pU_s}{U_{s-}}, \quad t\in [0,T].$$ $D$ is well-defined, predictable, right-continuous, and non-increasing on $[0,T]$. We have also $D=D^\tau$, where $\tau$ is the stopping time defined in .\ To define the process $L$, let $B:= \{^pU_{\tau}=0\}$, and denote by $\tau_{B}$ the restriction of $\tau$ to $B$. Due to [@Jacod (6.23), (6.24), Corollary 6.28], there exists an non-decreasing sequence of stopping times $(\sigma_n)$, such that $\frac{1}{^pU}{\textbf{1}}_{\llbracket0,\sigma_n\rrbracket}\le n$ for all $n\in\inte$, $\tau=\lim_n\sigma_n$, and $$\cup_n\llbracket0,\sigma_n\rrbracket= \cup_n\llbracket0,\tau_n\rrbracket\cap\llbracket0,\tau_{B}\llbracket=\llbracket0,\tau\rrbracket\cap\llbracket0,\tau_{B}\llbracket,$$ where $\tau_n$ are stopping times defines in . Hence we have $\tau_B=\lim_n\tilde\sigma_n$, where $$\tilde \sigma_n:= \begin{cases} \sigma_n & \text{on $\{\sigma_n<\tau\}$},\\ T & \text{on $\{\sigma_n=\tau\}$}, \end{cases}$$ and $\tau_B$ is a predictable stopping time.\ Using the same argumentation as in Step 1 of the proof of Theorem \[decomposition\], we define the process $L$ as the stochastic exponential of the local martingale $\int_0^{\cdot}\frac{1}{^pU_{s}} d M_s$ on $\cup_{n\in\inte}\llbracket 0,\tilde\sigma_n\rrbracket$, and extend it to a local martingale on $[0,T]$ as in Lemma \[lemmma:locmart\]. This yields $$L_t = 1 +\int_0^{t} \textbf{1}_{\{^pU_{s}>0\}} \frac{L_{s-}}{^pU_{s}} d M_s, \qquad t \in [0,T],$$ and $$\begin{aligned} \label{eq:Lexp:pr} \nonumber L_t=\exp \left(\int_0^{t\wedge\tau} \frac{1}{^pU_{s}} d M_s^c - \right.& \left.\frac12 \int_0^{t\wedge\tau} \left\|\frac{1}{^pU_{s}}\right\vert^2 d \langle M \rangle_s\right)\\& \times \prod_{\substack{0<s\leq t\wedge\tau,\\ \Delta M_s\ne0 }} \left(\frac{U_s}{^pU_{s}}\right),\qquad t\in[0,T].\end{aligned}$$ $L$ is well defined at $\tau$, since $\Delta M_\tau=a_\tau-M_{\tau-}=-^pU_{\tau}$, and thus $\Delta M_\tau=0$ on $\{^pU_{\tau}=0\}$. We also have $$ \E{\left[}L_T{\;\big\|\;}{\mathcal{F}}_{T-}{\right]}=\E{\left[}L_{T-}+\textbf{1}_{\{^pU_{T}>0\}} L_{T-}\frac{\Delta M_T}{^pU_{T}}{\;\big\|\;}{\mathcal{F}}_{T-}{\right]}=L_{T-}.$$ Due to [@Jacod Theorem 6.31], $L$ and $D$ provide a multiplicative decomposition of $U$, i.e., $$\label{eq:da:pr} U_t=L_t D_t$$ holds on $\cup_n\llbracket0,\sigma_n\rrbracket=\llbracket0,\tau\rrbracket\cap\llbracket0,\tau_B\llbracket$. Since $U=U^\tau$, $L=L^\tau$, and $D=D^\tau$, holds also on $\cup_{n\in\inte}\llbracket 0,\tilde\sigma_n\rrbracket$. It remains to prove for $(\omega, t)\in\llbracket\tau_B, T\rrbracket$ and $\omega\in\{^pU_{\tau}=0\}$. To this end, note that $D_{\tau_B}=0$ on $\{^pU_{\tau}=0\}\cap\{\Delta a_{\tau_B}>0\}$ by , hence $0=U_{\tau_B}=L_{\tau_B}D_{\tau_B}$ on this set. On the set $\{^pU_{\tau}=0\}\cap\{\Delta a_{\tau_B}=0\}$ we have $^pU_{\tau}=U_{\tau-}$, thus $\tau_B=\tau_A$, and we can conclude as in Step 2 of the proof of Theorem \[decomposition\].\ Thanks to and integration by parts formula, we have $$U_t=\E{\left[}a_T\|{\mathcal{F}}_t{\right]}-a_t=L_tD_t=\int_0^tD_sdL_s+\int_{[0,t]}L_{s-}dD_s,\qquad t\in[0,T],$$ and thus property 4’) follows from the uniqueness of the Doob-Meyer decomposition. Concerning property 5’), note that by definition of $D$ we have ${\{^pU_{t}>0\}}\subseteq {\{D_{t}>0\}}$ for all $t\in [0,T]$, and hence $$L_t = L_0 +\int_0^{t } \textbf{1}_{\{^pU_{s}>0\}}dL_s= L_0 +\int_0^t \textbf{1}_{\{D_{s>0}\}} dL_s.$$ Similarly, by definition of $L$ we have ${\{^pU_t >0\}}\subseteq {\{L_{t-}>0\}}$ for all $t\in [0,T]$, thus $$\begin{aligned} D_t = 1-\int_0^t \textbf{1}_{\{^pU_s>0\}} \frac{D_{s-}}{^pU_s} da_s= 1+\int_0^t \textbf{1}_{\{L_{s-}>0\}} dD_s, \quad t\in[0,T].\end{aligned}$$\ In order to prove uniqueness, we can again apply the same argumentation as in Step 4 of the proof of Theorem \[decomposition\], to conclude that every pair of processes $(\tilde{L},\tilde{D})$ satisfying properties 1)-4’) of Proposition \[prop:pr\] provides a multiplicative decomposition of the supermartingale $U$. Hence uniqueness on $\cup_n\llbracket0,\sigma_n\rrbracket=\llbracket 0, \tau\rrbracket\cap\llbracket 0, \tau_B\llbracket$ follows from [@Jacod Corollary 6.28, Theorem 6.31]. In particular, we have $L_{\tau_-}=\tL_{\tau_-}$, $D_{\tau_-}=\tD_{\tau_-}$, and $\tilde L_\tau= L_\tau=0$ on $B\cap\{L_{\tau_B-}=0\}$ due to the local martingale property. Moreover, since $$0={}^pU_{\tau_B}=U_{\tau_B-}-\Delta a_{\tau_B}=\tL_{\tau_B-}\tD_{\tau_B-}+\tL_{\tau_B-}\Delta \tD_{\tau_B}= L_{\tau_B-}\tD_{\tau_B}$$ on $B$, we have $\tilde D_\tau= D_\tau=0$ on $B\cap\{L_{\tau_B-}>0\}$. Property 5’) implies further $\tilde D_\tau= D_\tau$ on $B\cap\{L_{\tau_B-}=0\}$, $\tilde L_\tau= L_\tau$ on $B\cap\{L_{\tau_B-}>0\}$, and also $\tilde D= D$, $\tL=L$ on $\rrbracket\tau, T\rrbracket$. This concludes the proof. \[ctsfiltr\] 1. If $(L, D)$ is the decomposition of a predictable process $a$ as in Theorem \[decomposition\], and $(\tL,\tD)$ its decomposition as in Proposition \[prop:pr\], then $a=\int LdD=\int \tL_-d\tD$, but in general we do not have $D=\tD$ and $L=\tL$. As it can be seen from , , , and , we have $D=\tD$ and $L=\tL$ if and only if the martingale $M=(\E[a_T\|{\mathcal{F}}_t])_{t\in[0,T]}$ and the process $a$ do not jump at the same time, i.e., iff the bracket process $[M,a]=(\sum_{s\le t}\Delta M_s\Delta a_s)_{t\in[0,T]}$ is undistinguishable from $0$. 2. In a view of the previous remark, the decompositions $(L, D)$ as in Theorem \[decomposition\], and $(\tL,\tD)$ as in Proposition \[prop:pr\] coincide if the filtration $({\mathcal{F}}_t)$ is continuous. 3. It follows directly from (resp. ) and property 4) (resp. 4’)), that the process $a$ is purely discontinuous if and only if the process $D$ is purely discontinuous. In the rest of this section we discuss how one can associate a measure $Q$ on $(\Omega, {\mathcal{F}}_T)$ to the local martingale $L$; in this case representation takes the form $$\label{eq:qu} \E_{\mu}[X]=\E_Q[-\int_{[0,T]}X_sdD_s].$$ We fix a process $a\in{\mathcal{Z}}_1$, or alternatively, a measure $\mu\in{\mathcal{M}}_1({\mathcal{O}})$, and denote by $(L,D)$ the corresponding decomposition satisfying conditions 1)-4) of Theorem \[decomposition\]. The three following cases can occur:\ Case 1: $L$ is a uniformly integrable martingale. Then we can define a probability measure $Q$ on the $\sigma$-field ${\mathcal{F}}_T$ in a straightforward way by $\frac{dQ}{dP}:=L_T$. We have $Q\ll P$, and $D_T=0$ $Q$-a.s. Since $L$ is uniformly integrable martingale, [@dm2 VI.57] yields for any bounded optional process $X$ $$\label{eq:measure} \E{\left[}\int_0^TX_tL_tdD_t{\right]}=\E{\left[}L_T\int_0^TX_tdD_t{\right]}=\E_Q{\left[}\int_0^TX_tdD_t{\right]}.$$ \[uimart\] Case 1 holds in particular, if the measure $\mu$ is concentrated on $\Omega\times\{T\}$, i.e., if $a_t=0$ for all $t\in[0,T)$. Then the supermartingale $U$ defined in coincides with the uniformly integrable martingale $(\E{\left[}a_T\|{\mathcal{F}}_t{\right]})$ on $[0,T)$, and it’s multiplicative decomposition is given by $L_t:=\E{\left[}a_T\|{\mathcal{F}}_t{\right]}$, $t\in[0,T]$, and $D_t:=1$, $t\in[0,T)$, $D_T:=0$. In this case takes the form $$\E_{\mu}[X]=\E_Q[X_T].$$ Case 2: $L$ is a true martingale on $[0,T)$, which is not uniformly integrable, i.e. $\E[L_T]<1$. Note that this case can occur also if $T<\infty$, cf. [@kkn12 Remark 1.3]. In this case one can associate a measure $Q$ to the process $L$, if the filtration satisfies some additional technical conditions: Assume that ${\mathcal{F}}_T={\mathcal{F}}_{T-}=\bigvee_{t\in[0,T)}{\mathcal{F}}_t$, and that $({\mathcal{F}}_t)_{t\in[0,T)}$ is the so called $N$-augmentation of some filtered probability space, as defined in [@NajNik09 Proposition 2.4], see also [@Bichteler]. Moreover, assume that the non-augmented filtered probability space satisfies condition (P) of [@NajNik09 Definition 4.1], cf. also [@par67]. The main idea in this case is to use Parthasarathy’s ([@par67]) measure extension result, as done in [@f72]; see also [@NajNik09 Corollary 4.10], [@afp9 Theorem 3.4], [@kkn12 Theorem 1.1]. We define a measure $Q_t$ locally on each ${\mathcal{F}}_t$ by $\frac{dQ_t}{d\P}:=L_t$. Under the assumptions above, the consistent family $(Q_t)_{t\in[0,T)}$ can be extended to a unique measure $Q$ on ${\mathcal{F}}_T$, such that $Q\|_{{\mathcal{F}}_t}=Q_t$ for all $t$. Note that $Q$ is locally absolutely continuous with respect to $\P$, i.e., $Q\ll \P$ on each ${\mathcal{F}}_t$, $t\in[0,T)$, but $Q$ is not absolutely continuous with respect to $\P$ on ${\mathcal{F}}_T$. For this reason the filtration $({\mathcal{F}}_t)$ cannot be completed with zero sets of ${\mathcal{F}}_T$. However, in this case the “usual conditions” can be replaced by $N$-usual conditions, cf. [@NajNik09] and [@Bichteler]. \[ashkan\] Assume that ${\mathcal{F}}_T={\mathcal{F}}_{T-}$, and that $(\Omega, ({\mathcal{F}}_t)_{t\in[0,T)}, \P)$ is the $N$-augmentation of a filtered probability space that satisfies the property (P). Let $a\in{\mathcal{Z}}_1$ with the decomposition $(L,D)$ as in Theorem \[decomposition\], such that the process $L$ is a martingale on $[0,T)$. Then there exists a probability measure $Q$ on ${\mathcal{F}}_T$, that is locally absolutely continuous with respect to $\P$, such that $D_T=0$ $Q$-a.s. and $$\label{eq:underQ} \E{\left[}\int_{[0,T]}X_tda_t{\right]}=\E_Q{\left[}-\int_{[0,T]}X_tdD_t{\right]}$$ for any bounded optional process $X$. We define the measure $Q$ as explained above. Let $\tau_n$ be any sequence of stopping times such that $\tau_n<T$, $\tau_n\nearrow T$ $P$-a.s.. Then $L^{\tau_n}$ is a uniformly integrable martingale for each $n$, and the same argumentation as in yields for any bounded optional process $X$ $$\E{\left[}\int_0^{\tau_n}X_tL_tdD_t{\right]}=\E_Q{\left[}\int_0^{\tau_n}X_tdD_t{\right]},\qquad n\in\inte.$$ By dominated convergence, $\E{\left[}\int_0^{T-}X_tL_tdD_t{\right]}=\E_Q{\left[}\int_0^{T-}X_tdD_t{\right]}$, and it remains to prove the equality at $T$. To this end we argue as in [@afp9]: By [@kls79 Lemma 2, Lemma 3], the limit $L_{T-}=\lim_{t\to T}L_t$ exists $\P$- and $Q$-a.s., and the measure $Q$ has Lebesgue decomposition on ${\mathcal{F}}_T$ with respect to $\P$ given by $$\label{lebesgue} Q[A]=\int_AL_{T-}d\P+Q[A\cap\{L_{T-}=\infty\}],\qquad A\in{\mathcal{F}}_T.$$ Moreover, since $(D_t)_{t\in[0,T)}$ is non-increasing under $Q$, the limit $D_{T-}$ exists also $Q$-a.s. By construction, the random variable $D_T$ is defined under $\P$, and hence under $Q$ only on the set $\{L_{T-}<\infty\}$. We define $D_T:=0$ on the set $\{L_{T-}=\infty\}$. Note further that $L_T=L_{T-}$ $\P$-a.s., since ${\mathcal{F}}_T={\mathcal{F}}_{T-}$ and $\E[L_T\|{\mathcal{F}}_{T-}]=L_{T-}$. This implies $$Q[\{D_T>0\}]=\E_\P{\left[}{\textbf{1}}_{\{D_T>0\}}L_{T-}{\right]}+Q[\{D_T>0\}\cap\{L_{T-}=\infty\}]\\ =\E_\P{\left[}{\textbf{1}}_{\{D_T>0\}}L_{T}{\right]}=0,$$ where we have used that $L_TD_T=0$ $\P$-a.s.. Moreover, we have $D_{T-}=0$ on $\{L_{T-}=\infty\}$ $Q$-a.s. thanks to and the fact that $LD$ is of class (D). Indeed, we have for any sequence of stopping times $(\tau_n)$ as above $$\begin{aligned} \label{clsD} \nonumber\E_Q{\left[}D_{T-}{\textbf{1}}_{\{L_{T-}=\infty\}}{\right]}&=\E_Q{\left[}D_{T-}{\right]}-\E_\P{\left[}L_{T-} D_{T-}{\right]}\\ \nonumber&=\E_Q{\left[}D_{T-}{\right]}-\lim_n\E_\P{\left[}L_{\tau_n} D_{\tau_n}{\right]}\\ &=\E_Q{\left[}D_{T-}{\right]}-\lim_n\E_Q{\left[}D_{\tau_n}{\right]}=0,\end{aligned}$$ where the the last equality holds due to monotone convergence. Hence we obtain $$\begin{aligned} \E_\P{\left[}X_TL_T\Delta D_T{\right]}&= \E_\P{\left[}- X_TL_{T-}D_{T-}{\right]}\\ &=\E_Q{\left[}- X_TD_{T-}{\right]}-\E_Q{\left[}- X_TD_{T-}{\textbf{1}}_{\{L_{T-}=\infty\}}{\right]}\\ &=\E_Q{\left[}X_T\Delta D_T{\right]},\end{aligned}$$ where we have used that $L_T=L_{T-}$, $L_TD_T=0$ $\P$-a.s., , $D_{T-}=0$ on $\{L_{T-}=\infty\}$, and $D_{T}=0$ $Q$-a.s.. This proves also at $T$ and completes the proof. Note that $LD$ is of class (D) under $\P$ if and only if $D_{T-}=0$ on $\{L_{T-}=\infty\}$ $Q$-a.s.. Indeed, the “only if” part was proved in . To see that also the converse is true, we define the stopping times $$\sigma_n:=\inf{\left\{\,} \newcommand{\rk}{\right\}}t{\;\big\|\;}L_t\ge n\rk,\qquad n\in\inte.$$ By monotone convergence $$0=\E_Q{\left[}D_{T-}{\textbf{1}}_{\{L_{T-}=\infty\}}{\right]}=\lim_n\E_Q{\left[}D_{\sigma_n}{\textbf{1}}_{\{\sigma_n<T\}}{\right]}=\lim_n\E_\P{\left[}L_{\sigma_n}D_{\sigma_n}{\textbf{1}}_{\{\sigma_n<T\}}{\right]}.$$ Since $0\le D\le 1$ $\P$-a.s., [@dm2 Theorem VI.25] implies that $LD$ is of class (D). Case 3: If $L$ is a strict local martingale, and the filtration $({\mathcal{F}}_t)_{t\in[0,T)}$ is a standard system (cf. [@par67], [@f72]), it is still possible to associate a measure $Q$ to $L$, as done in [@f72], cf. also [@kkn12 Theorem 1.8]. However, in this case not even the $N$-augmentation of the filtration can be used, and one would have to work with a non-completed filtration. This imposes many technical restrictions, and goes beyond the scope of the present paper. Robust representation of convex risk measures on ${\mathcal{R}^\infty}$ {#sec:robrep} ======================================================================= In this section we first recall some notation and the representation result for convex risk measures on ${\mathcal{R}^\infty}$ from [@cdk4]. We consider the space of pairs of finite variation processes $$\begin{aligned} {\mathcal{A}}^1:=\Big\{a:[0,T]\times\Omega\to\real^2 {\;\big\|\;}& a=(a^{\rm op}, {a^{\rm pr}})=(a^{\rm op}_t, {a^{\rm pr}}_t)_{t\in[0,T]},\\ &{a^{\rm op}},{a^{\rm pr}}\, \text{right continuous, of finite variation},\\ &{a^{\rm pr}}\, \text{predictable}, {a^{\rm pr}}_0=0,\\ &a^{\rm op}\, \text{optional, purely discontinuous},\\&{\rm Var}({a^{\rm pr}})+{{\rm Var}}({a^{\rm op}})\in L^1(\P)\Big\}.\end{aligned}$$ The space ${\mathcal{A}}^1$ is a Banach space with the norm $$\\|a\\|_{{\mathcal{A}}^1}:=\E{\left[}{\rm Var}({a^{\rm pr}})+{{\rm Var}}({a^{\rm op}}){\right]},$$ and any element of ${\mathcal{A}}^1$ defines a linear form on ${\mathcal{R}^\infty}$ via $$\label{linform2} a(X):=\E{\left[}\int_0^TX_{t-}d{a^{\rm pr}}_t+\int_{[0,T]}X_{t}d{a^{\rm op}}_t{\right]},\quad X\in{\mathcal{R}^\infty}.$$ Let further ${\mathcal{A}}^1_+$ denote the subset of all non-decreasing elements of ${\mathcal{A}}^1$, and $${\mathcal{Z}}_1^d:={\left\{\,} \newcommand{\rk}{\right\}}a=({a^{\rm pr}}, {a^{\rm op}})\in{\mathcal{A}}^1_+{\;\big\|\;}\\|a\\|_{{\mathcal{A}}^1}=1\rk.$$ Given a subset $\hat{\mathcal{Z}}$ of ${\mathcal{Z}}_1^d$, a function $\gamma\::\: {\mathcal{Z}}_1^d\to[0,\infty]$ is called a penalty function on $\hat{\mathcal{Z}}$, if $$\inf_{a\in\hat{\mathcal{Z}}}\gamma(a)=0.$$ For a monetary convex risk measure for processes $\rho$, a typical penalty function is the conjugate of $\rho$: $$\label{eq:alpha} \alpha(a):=\rho^(a):=\sup_{X\in{\mathcal{R}^\infty}} \left(a(-X)-\rho(X)\right)= \sup_{X\in{\mathcal{A}}} a(-X),\quad a\in{\mathcal{Z}}_1^d.$$ Here ${\mathcal{A}}$ denotes the acceptance set defined in Remark \[rem:ac\]. As usually, dual representation of a convex risk measure is closely related to its continuity properties. A monetary convex risk measure for processes $\rho$ is called - continuous from above with respect to sup-convergence in probability* (resp. with respect to pointwise convergence in probability), if $$\lim_{n\to\infty}\rho(X^n)=\rho(X)$$ for every non-increasing sequence $(X^n)\subset{\mathcal{R}^\infty}$ and $X\in{\mathcal{R}^\infty}$, such that $(X^n-X)^\to0$ in probability (resp. such that $X_t^n-X_t\to0$ $\P$-a.s. for all $t\in[0,T]$). - continuous from below with respect to sup-convergence in probability* (resp. with respect to pointwise convergence in probability), if $$\lim_{n\to\infty}\rho(X^n)=\rho(X)$$ for every non-decreasing sequence $(X^n)\subset{\mathcal{R}^\infty}$ and $X\in{\mathcal{R}^\infty}$, such that $(X_t^n-X_t)^\to0$ in probability (resp. such that $X_t^n-X_t\to0$ $\P$-a.s. for all $t\in[0,T]$). The following result was proved in [@cdk4 Theorem 3.3]. \[cdk\] For a functional $\rho$ on ${\mathcal{R}^\infty}$ the following conditions are equivalent: 1. $\rho$ can be represented as $$\label{robrep} \rho(X)=\sup_{a\in{\mathcal{Z}}_1^d}\left(a(-X)-\gamma(a)\right),\quad X\in{\mathcal{R}^\infty},$$ with a penalty function $\gamma$ on ${\mathcal{Z}}_1^d$. 2. $\rho$ is a monetary convex risk measure that is continuous from above with respect to sup-convergence in probability. Moreover, if (1)-(2) are satisfied, the function $\alpha$ defined in is a penalty function on ${\mathcal{Z}}_1^d$ such that $$\alpha(a)\le\gamma(a)\quad\text{for all}\quad a\in{\mathcal{Z}}_1^d,$$ and the representation holds also with $\gamma$ replaced by $\alpha$. For any $a=({a^{\rm pr}}, {a^{\rm op}})\in{\mathcal{A}}^1$, the linear form can be written as $$\begin{aligned} \label{linform3} \nonumber a(X)&=E\left[\int_{(0,T]}X_{t-}da_t^{\text{pr}}+\int_{[0,T]}X_{t}da_t^{\text{op}}\right]\\ &=E\left[\int_{[0,T]}X_{t}d(a_t^{\text{pr}}+a_t^{\text{op}})-\sum_{0< t\le T}{}^p(\Delta X)_{t}\Delta a_t^{\text{pr}}\right],\end{aligned}$$ where $^p(\Delta X)$ denotes the predictable projection of the purely discontinuous part of $X\in{\mathcal{R}^\infty}$. For $a\in{\mathcal{Z}}_1^d$, the process ${a^{\rm pr}}+{a^{\rm op}}$ defines a normalized optional measure as we have considered in Section \[sec:dec\]; cf. . However, the linear form in involves an additional singular term $\sum{}^p(\Delta X)\Delta a^{\text{pr}}$, depending on the nature of the jumps of $X$. Our main goal in the rest of this section will be finding conditions on the risk measure $\rho$, under which it can be represented in terms of ordinary optional measures, as defined in . This simplified form is particularly useful for construction of risk measures for processes, e.g., all examples in [@cdk4 Section 5], and also our examples in Section \[BSDE\] are of this form. We begin by noting that the space of optional measures ${\mathcal{B}}^1$ defined in Section \[sec:dec\] can be identified with a subspace of ${\mathcal{A}}^1$. \[BsubsetA\] To any $a\in{\mathcal{B}}^1$ we can associate a pair $\tilde{a}:=(a^{\rm c}, a-a^{\rm c})\in{\mathcal{A}}^1$, where $a^{\rm c}$ denotes the continuous part of $a$, and $a-a^{\rm c}$ its purely discontinuous part. Then $\\|\tilde{a}\\|_{{\mathcal{A}}^1}=\E{\left[}{{\rm Var}}(a){\right]}$, and $$\label{sameforms} \tilde{a}(X)=\E{\left[}\int_{[0,T]}X_{t}da_t{\right]}.$$ Conversely, any pair of processes $\tilde{a}=({a^{\rm pr}}, {a^{\rm op}})\in{\mathcal{A}}^1$ such that ${a^{\rm pr}}$ is continuous, defines an element $a:={a^{\rm pr}}+{a^{\rm op}}\in{\mathcal{B}}^1$ such that holds. Thus we can identify ${\mathcal{B}}^1$ with the subspace $$ {\left\{\,} \newcommand{\rk}{\right\}}\tilde{a}=({a^{\rm pr}}, {a^{\rm op}})\in{\mathcal{A}}^1{\;\big\|\;}{a^{\rm pr}}\:\text{continuous}\rk$$ of ${\mathcal{A}}^1$, and for any $a\in{\mathcal{B}}^1$ the linear form $a(X)$ takes the form on ${\mathcal{R}^\infty}$. The key to the dual representation of a convex risk measure is an appropriate continuity property. The reason why a pair* of processes appears in the robust representation is condition of continuity from above with respect to sup-convergence in probability. By [@dm2 Lemma VII 2], sup-convergence for [càdlàg ]{}functions amounts to pointwise convergence of the paths and of their left limits. Thus any positive linear functional on ${\mathcal{R}^\infty}$, that is continuous from above with respect to sup-convergence in probability, is of the form , and involves two processes of finite variation, cf. [@dm2 Theorem VII 2]. On the other hand, by Daniell-Stone Integration Theorem (cf., e.g., [@fs11 Theorem A.49]), any positive linear functional on ${\mathcal{R}^\infty}$, that is continuous from above with respect to pointwise convergence in probability, can be represented as in for some $a\in{\mathcal{B}}_+^1$. This suggests to make a stronger requirement of continuity from above with respect to pointwise convergence in probability, in order to obtain a representation of a risk measure in terms of ${\mathcal{Z}}_1$. The requirement is necessary: \[lem:suff\] Let $\rho$ be a functional on ${\mathcal{R}^\infty}$ such that 1. $\rho$ can be represented as $$\label{robrepbeta} \rho(X)=\sup_{a\in{\mathcal{Z}}_1}\left(a(-X)-\gamma(a)\right),\quad X\in{\mathcal{R}^\infty},$$ with a penalty function $\gamma$ on ${\mathcal{Z}}_1$. Then 1. $\rho$ is a monetary convex risk measure, that is continuous from above with respect to pointwise convergence in probability. It is easy to see that $\rho$ satisfies the axioms of Definition \[def:rm\]. Continuity from above follows by standard arguments as, e.g., in the proof [@fs11 Lemma 4.21]. We conjecture, that conditions 1) and 2) of Lemma \[lem:suff\] are in fact equivalent. Unfortunately, after spending quite some time thinking about it, we are neither able to prove that 2) implies 1), nor could we find a counterexample. We could prove representation under the assumption of continuity from below with respect to pointwise convergence in probability. This is a stronger requirement than continuity from above, as shown in the next lemma. The result of this lemma is well known in the context of convex risk measures for bounded random variables, cf., e.g., [@fs11 Remark 4.25]. However, the proof there relies on the particular representation of a risk measure for random variables, and cannot be applied in our present framework. The following general argument was communicated to us by Michael Kupper, and we thank him for allowing us to include it in this paper. \[michael\] Let ${\mathcal{X}}$ be a topological vector space, and $\rho:{\mathcal{X}}\to\real$ any convex functional such that $\rho(X)\le\rho(Y)$ for any $X,Y\in{\mathcal{X}}$ with $Y\le X$. Assume further that $\rho$ is continuous from below in the following sense: $$\rho(X_n)\searrow\rho(X)\quad\text{for any non-decreasing sequence}\quad (X_n)\subset {{\mathcal{X}}}, X_n\nearrow X.$$ Then $\rho$ is continuous from above, i.e., $$\rho(X_n)\nearrow\rho(X)\quad\text{for any non-increasing sequence}\quad (X_n)\subset {{\mathcal{X}}}, X_n\searrow X.$$ W.l.o.g. we can assume that $\rho(0)=0$, otherwise consider $\tilde{\rho}(\cdot):=\rho(\cdot)-\rho(0)$.\ First we show that continuity from below at $0$ implies continuity from above at $0$. Indeed, let $(X_n)\subset{\mathcal{X}}, X_n\searrow 0$. Then monotonicity, convexity, $\rho(0)=0$, and continuity from below at $0$ imply $$0\ge\rho(X_n)\ge-\rho(-X_n)\nearrow0.$$ For the general case, let $(X_n)\subset{\mathcal{X}}, X_n\searrow X_0$, and consider the functional $$\tilde{\rho}(X):=\rho(X+X_0)-\rho(X_0),\quad X\in{\mathcal{X}}.$$ It is easy to see that $\tilde{\rho}$ is a monotone convex functional with $\tilde{\rho}(0)=0$, continuous from below in $0$. By the previous argument $\tilde{\rho}$ is continuous from above at $0$, which implies $$\rho(X_n)\nearrow\rho(X_0),$$ i.e., $\rho$ is continuous from above. Continuity from below with respect to sup-convergence in probability for convex risk measures on ${\mathcal{R}^\infty}$ was characterized in [@assa11 Theorem 3.1]. The following theorem combines this result with the argumentation inspired by [@fs11 Theorem 4.22]. \[contfrombelow\] Let $\rho$ be a monetary convex risk measure on ${\mathcal{R}^\infty}$, that is continuous from below with respect to pointwise convergence in probability. Then $\rho$ has representation , where any penalty function $\gamma$ is concentrated on the set ${\mathcal{Z}}_1$ of normalized optional measures. In particular, $\rho$ has the representation , and the supremum is attained, i.e., we have $$\label{robrepmax} \rho(X)=\max_{a\in{\mathcal{Z}}_1}\left(a(-X)-\gamma(a)\right),\quad X\in{\mathcal{R}^\infty}.$$ Moreover, the level sets $$\label{ls} \Lambda_c:={\left\{\,} \newcommand{\rk}{\right\}}a\in{\mathcal{Z}}_1^d{\;\big\|\;}\alpha(a)\le c\rk,\quad\quad c>0,$$ are compact in $\sigma({\mathcal{B}}^1, {\mathcal{R}^\infty})$. For the proof we will use the following lemma, which is a reformulation of [@fs11 Lemma 4.23], and can be proved in completely analogous way in our present context. \[fs\_lemma\] Let $\rho$ be a monetary convex risk measure on ${\mathcal{R}^\infty}$ with the representation , and consider the level sets $\Lambda_c$ defined in . Then for any sequence $(X_n)$ in ${\mathcal{R}^\infty}$ such that $0\le X_n\le 1$, the following two conditions are equivalent: 1. $\rho(\lambda X_n)\to\rho(\lambda{\textbf{1}}_{[0,T]})$ for each $\lambda\ge1$. 2. $\inf_{a\in\Lambda_c}a(X_n)\to1$ for all $c>0$. Proof of Theorem \[contfrombelow\]. First we note that by Lemma \[michael\] $\rho$ is continuous from above with respect to pointwise convergence in probability, hence also with respect to sup-convergence in probability, and by Theorem \[cdk\] $\rho$ has representation with some penalty function $\gamma$ on ${\mathcal{Z}}_1^d$. We will show that $\gamma(a)<\infty$ implies $a\in{\mathcal{Z}}_1$. It suffices to prove this for the minimal penalty function $\alpha$. To this end let $(Y^n)_{n\in\inte}$ be a sequence in ${\mathcal{R}^\infty}$ such that $Y^n_t\searrow0$ $\P$-a.s. for all $t$, and consider $X^n:={\textbf{1}}_{[0,T]}-\delta Y^n$, where $\delta>0$ is chosen such that $X^n_t\ge0$ for all $t$ (e.g. $\delta:=\frac{1}{\\|Y^0\\|_{{\mathcal{R}^\infty}}+1}$ does the job). Then $0\le X^n\le1$, and $\lambda X^n\nearrow\lambda{\textbf{1}}_{[0,T]}$ $\P$-a.s. for all $t$ for any $\lambda>0$. Continuity from below implies $\rho(\lambda X^n)\searrow\rho(\lambda{\textbf{1}}_{[0,T]})$, and by Lemma \[fs\_lemma\] $$1-\delta a(Y^n)=a(X^n)\to1\quad\text{for all}\quad a\in\Lambda_c.$$ Hence $a(Y^n)\searrow0$ for all $a\in\Lambda_c$. i.e., $a$ is continuous from above with respect to pointwise convergence in probability. By Daniell-Stone Integration Theorem (cf., e.g., [@fs11 Theorem A.49], [@dm1 Theorem III 35]), there exists a positive measure $\mu$ on $(\Omega, {\mathcal{O}})$ such that $a(X)=\int Xd\mu$ for all $X\in{\mathcal{R}^\infty}$. As in the proof of [@dm2 Theorem VII 2], it can be seen that $\mu$ disappears on $\P$-evanescent sets. Then, due to Dolean’s representation result [@dm2 Theorem VI 65], and uniqueness of the linear form , we can identify $a$ with some $\tilde{a}\in{\mathcal{B}}_+^1$ as in Remark \[BsubsetA\]. This proves (with some abuse of notation) that $a\in{\mathcal{B}}^1\cap{\mathcal{Z}}_1^d={\mathcal{Z}}_1$ for any $a\in{\mathcal{Z}}_1^d$ such that $\alpha(a)<\infty$. In particular, representation holds. Moreover, since $\rho$ is continuous from below with respect to sup-convergence in probability, [@assa11 Theorem 3.1] implies that the supremum in is attained for each $X\in{\mathcal{R}^\infty}$ by some $\bar a\in{\mathcal{Z}}_1^d$. We must have $\gamma(\bar a)<\infty$ in this case, and thus $\bar a\in{\mathcal{Z}}_1$. Compactness of the sets $\Lambda_c$ for any $c>0$ in $\sigma({\mathcal{B}}^1, {\mathcal{R}^\infty})$ follows also from [@assa11 Theorem 3.1]. $\square$\ Model and discounting ambiguity {#sec:discamb} =============================== This section combines the results of Sections \[sec:dec\] and \[sec:robrep\]. We denote by ${\mathcal{L}_{+}}$ the set of all non-negative [càdlàg ]{}local martingales $L=(L_t)_{t\in[0,T]}$ such that $L_{T-}=\E{\left[}L_T\|{\mathcal{F}}_{T-}{\right]}$, and by ${\mathcal{L}_{+}}^1$ the set of all $L\in{\mathcal{L}_{+}}$ with $L_0=1$. For $L\in{\mathcal{L}_{+}}$, ${{\mathcal{D}}(L)}$ denotes the set of all processes $D$ satisfying conditions 2)-3) of Theorem \[decomposition\], i.e., $$\begin{aligned} {{\mathcal{D}}(L)}:=\Big\{D=(D_t)_{t\in[0,T]}{\;\big\|\;}& D\;\text{adapted, right-continuous, non-increasing, s.t.}\; D_{0-}=1,\\&\qquad \quad\quad\{D_T>0\}\subseteq\{L_T=0\},\,\text{and}\; LD\;\text{is of class (D)}\Big\}. \end{aligned}$$ Correspondingly, ${{\mathcal{D}}^{\rm pr}(L)}$ denotes the set of all predictable processes as in Proposition \[prop:pr\] without jump at $0$, i.e., $$ {{\mathcal{D}}^{\rm pr}(L)}:=\Big\{D\in {{\mathcal{D}}(L)}{\;\big\|\;}D\;\text{predictable}, D_0=1\Big\},$$ and ${\mathcal{D}}^{\rm d}(L)$ the set of all $D\in{{\mathcal{D}}(L)}$ such that $D$ is a purely discontinuous process. We also introduce the set $$\S^1_+:={\left\{\,} \newcommand{\rk}{\right\}}(L, D, L',D'){\;\big\|\;}L,L'\in{\mathcal{L}_{+}}, L_0+L'_0=1, D\in{{\mathcal{D}}^{\rm pr}(L)}, D'\in{\mathcal D}^{\rm d}(L')\rk.$$ By Theorem \[decomposition\], Proposition \[prop:pr\], and 3) of Remark \[ctsfiltr\], we can identify the sets ${\mathcal{Z}}_1^d$ and $\S^1_+$, i.e., a process $a=({a^{\rm pr}},{a^{\rm op}})\in{\mathcal{Z}}_1^d$, iff there exist $(L,D, L',D')\in\S^1_+$, such that ${a^{\rm pr}}=\int_0^\cdot L_{s-}dD_s$, ${a^{\rm op}}=\int_{[0,\cdot]}L'_{s}dD'_s$. We also deliberately identify penalty functions $\gamma$ on ${\mathcal{Z}}_1^d$ and on $\S^1_+$ via $$\gamma(L,D, L',D'):=\gamma\left(\int_0^\cdot L_{s-}dD_s, \int_{[0,\cdot]}L'_{s}dD'_s\right)\quad\text{for}\;(L,D, L',D')\in\S^1_+.$$ Combining Theorem \[cdk\] with Theorem \[decomposition\] and Proposition \[prop:pr\], we obtain the following corollary. \[cdk\_rev\] For a functional $\rho$ on ${\mathcal{R}^\infty}$ the following conditions are equivalent: 1. For each $X\in{\mathcal{R}^\infty}$ we have $$\label{eq:repcdk} \rho(X)=\sup_{(L,D, L', D')\in\S^1_+}\left(\E{\left[}-\int_0^TX_{t-}L_{t-}dD_t-\int_{[0,T]}X_{t}L'_{t}dD'_t{\right]}-\gamma(L,D, L', D')\right)$$ with a penalty function $\gamma$ on $\S^1_+$. 2. $\rho$ is a monetary convex risk measure that is continuous from above with respect to sup-convergence in probability. Thanks to Theorem \[contfrombelow\], dual representation takes a simpler form under the assumption of continuity from below with respect to pointwise convergence in probability: \[cor:contfrombelow\] If $\rho$ is a monetary convex risk measure on ${\mathcal{R}^\infty}$ that is continuous from below with respect to pointwise convergence in probability, it has the representation $$\label{robrepmax1} \rho(X)=\sup_{L\in{\mathcal{L}_{+}}^1}\sup_{D\in{{\mathcal{D}}(L)}}\left(\E{\left[}\int_{[0,T]}X_tL_tdD_t{\right]}-\gamma(L,D)\right),\quad X\in{\mathcal{R}^\infty},$$ where $$\gamma(L,D):=\gamma\left(\int_{[0,\cdot]} L_{s}dD_s\right)$$ is a penalty function on ${\mathcal{Z}}_1$. Moreover, the supremum in is attained by some $L\in{\mathcal{L}_{+}}^1$ and $D\in{{\mathcal{D}}(L)}$ for each $X\in{\mathcal{R}^\infty}$. The local martingales $L$ and $L'$ in the representation play the roles of state price deflators, whereas the predictable non-increasing processes $D_-$ and $D'_-$ define discounting processes for this deflators, cf. 2) of Remark \[rem:inter\]. In difference to and , representations and make visible the roles of model ambiguity, as described by local martingales, and of discounting ambiguity, as described by corresponding non-increasing processes. In addition, a risk measure with representation , that does not reduce to , distinguishes between inaccessible and predictable jumps of the cumulated cash flow. Appearance of discounting processes in the representations and reflects cash subadditivity of the risk measure, whereas cash additivity at time $s>t$ implies that there is no discounting between $t$ and $s$ in all relevant models. This was noted in [@afp9 Corollary 5.10, Proposition 5.11], and is extended to our present framework by the next proposition. \[propcash\] Let $\rho$ be a convex risk measure for processes with representation . Then it is cash additive at time $s\in(0,T]$ if and only if $$\label{ca} D_{s-}=1\;\text{on}\;\, \{L_s>0\},\quad\text{and}\qquad D'_{s-}=1\;\text{on}\;\, \{L'_s>0\}\quad \quad \P\text{-a.s.}$$ for all $(L,L', D, D')\in\S^1_+$ such that $\gamma(L,L', D, D')<\infty$. In this case $\rho$ admits the representation $$\label{eq:cdkca} \rho(X)=\sup_{(L,D, L', D')\in\S^1_+}\left(\E{\left[}-\int_{[s,T]}X_{t-}L_{t-}dD_t-\int_{[s,T]}X_{t}L'_{t}dD'_t{\right]}-\gamma(L,D, L', D')\right)$$ and $\rho$ is cash additive up to time $s$, i.e., at all times $t\in[0, s]$.\ In particular, $\rho$ is cash additive if and only if it reduces to a risk measure on ${L^{\infty}(\Omega, {\mathcal{F}}_T,\P)}$, i.e., $\rho$ is of the form $$\label{reprrv} \rho(X)=\sup_{Q\in{\mathcal{M}}(\P)}\left(\E_Q[-X_T]-\tilde\gamma(Q)\right),$$ where ${\mathcal{M}}(\P)$ denotes the set of all probability measures on $(\Omega, {\mathcal{F}}_T)$ that are absolutely continuous with respect to $\P$, and $\tilde\gamma$ is a penalty function on ${\mathcal{M}}(\P)$. Since $L$ and $L'$ are local martingales, and $D$ and $D'$ non-increasing processes with $D_{0-}=D'_{0-}=1$, condition is equivalent to $$\label{ca1} \int_{(0,s)}L_{t-}dD_t=\int_{[0,s)}L'_tdD'_t=0\qquad\P\text{-a.s.}.$$ Choose $(L,L', D, D')\in\S^1_+$ such that $\gamma(L,L', D, D')<\infty$, and assume that condition does not hold. Then $$\E{\left[}\int_{[s, T]}L_{t-}dD_t+\int_{[s,T]}L'_tdD'_t{\right]}>-1,$$ and we can find $m\in\real$ such that $$\E{\left[}\int_{[s, T]}L_{t-}dD_t+\int_{[s,T]}L'_tdD'_t{\right]}-\frac{\gamma(L,L', D, D')}{m}>-1.$$ This implies that $$\begin{aligned} \rho\left(m{\textbf{1}}_{[s,T]}\right)&=m\sup_{(L,D, L', D')\in\S^1_+}\left(\E{\left[}-\int_{[s,T]}L_{t-}dD_t-\int_{[s,T]}L'_{t}dD'_t{\right]}-\frac{\gamma(L,D, L', D')}{m}\right)\\&>-m,\end{aligned}$$ which contradicts the cash additivity property at time $s$. Hence holds, and representation reduces to . In particular, if $\rho$ is cash additive at $T$, amounts to ${a^{\rm op}}={a^{\rm pr}}=0$ on $[0,T)$ for all $({a^{\rm pr}}, {a^{\rm op}})\in{\mathcal{Z}}_1^d$ such that $\gamma({a^{\rm pr}}, {a^{\rm op}})<\infty$. Due to Remark \[uimart\], in this case $L+L'$ is a uniformly integrable martingale and defines a probability measure $Q\in{\mathcal{M}}(\P)$ via $\frac{dQ}{d\P}:=L_T+L'_T$. It follows as in Remark \[uimart\] $$\E{\left[}-\int_0^TX_{t-}L_{t-}dD_t-\int_{[0,T]}X_{t}L'_{t}dD'_t{\right]}=\E_Q{\left[}X_T{\right]}$$ for any $X\in{\mathcal{R}^\infty}$, and any $(L,L', D, D')\in\S^1_+$ such that $\gamma(L,L', D, D')<\infty$. This proves with $$\tilde\gamma(Q):=\gamma\left(\frac{1}{2}\frac{dQ}{d\P}, \frac{1}{2}\frac{dQ}{d\P}, 1-\delta_{\{T\}}, 1-\delta_{\{T\}}\right),\quad Q\in{\mathcal{M}}(\P),$$ where $\delta_{\{T\}}$ denotes the Dirac measure at $T$. Risk measures and BSDEs {#BSDE} ======================= This section links risk measures for processes to BSDEs. We consider here risk measures in the dynamic framework. For $0\leq t\leq s\leq T$, we define the projection $\pi_{t,s}:\mathcal{R}^{\infty}\to \mathcal{R}^{\infty}$ as $$\pi_{t,s}(X)_r={\textbf{1}}_{[t,T]}(r)X_{r\wedge s},\quad r\in[0,T],$$ and we use the notation $\mathcal{R}_{t,s}^{\infty}:=\pi_{t,s}(\mathcal{R}^{\infty})$, and $\mathcal{R}_t^{\infty}:=\pi_{t,T}(\mathcal{R}^{\infty})$. Risk assessment at time $t$ takes into account the available information, and is described by a conditional convex risk measure for processes $\rho_t$. \[def:cond\] A map ${\rho_t}\,:\,\mathcal{R}_t^{\infty}\,\rightarrow\,{L^{\infty}(\Omega, {\mathcal{F}}_t,\P)}$ for $t\in(0,T]$ is called a conditional convex risk measure for processes if it satisfies the following properties for all $X,Y\in\mathcal{R}_t^{\infty}$: - Conditional cash invariance: for all $m\in{L^{\infty}(\Omega, {\mathcal{F}}_t,\P)}$, $$\rho_t(X+m{\textbf{1}}_{[t, T]})=\rho_t(X)-m;$$ - Monotonicity: ${\rho_t}(X)\ge{\rho_t}(Y)$ if $X\le Y$; - Conditional convexity: for all $\lambda\in{L^{\infty}(\Omega, {\mathcal{F}}_t,\P)}$ with $0\le \lambda\le 1$, $${\rho_t}(\lambda X+(1-\lambda)Y)\le\lambda{\rho_t}(X)+(1-\lambda){\rho_t}(Y);$$ - [Normalization]{}: ${\rho_t}(0)=0$. A sequence $({\rho_t}){_{t\in[0,T]}}$ is called a dynamic convex risk measure for processes if, for each $t$, ${\rho_t}\,:\,\mathcal{R}_t^{\infty}\,\rightarrow\,{L^{\infty}(\Omega, {\mathcal{F}}_t,\P)}$ is a conditional convex risk measure for processes.\ For $X\in{\mathcal{R}^\infty}$ we use the notation $${\rho_t}(X):={\rho_t}(\pi_{t,T}(X)).$$ A dynamic convex risk measure for processes is called time consistent, if $$\rho_t(X)=\rho_t(X{\textbf{1}}_{[t,s)}-\rho_{s}{\textbf{1}}_{[s,T]}(X))$$ for all $X\in{\mathcal{R}^\infty}$, and all $t\in[0,T]$, $s\in[t,T]$. \[rem:condca\] Also Definition \[def:ca\] of cash subadditivity can be extended to the conditional case in a straightforward way. By the same argument as in Proposition \[prop:ca\] every conditional convex risk measure for processes is cash subadditive. From now on we shell assume that the time horizon $T$ is finite, and the filtration $({\mathcal{F}}_t)_{t\in[0,T]}$ is the augmentation of the filtration generated by a $d$-dimensional Brownian motion $(W_t)_{t\in[0,T]}$. In this context, it is well known that a solution to a BSDE $$\label{bsde} Y_t=-X_T+\int_t^T g(s,Y_s, Z_s)ds-\int_t^TZ_sdW_s,\qquad t\in[0,T],$$ for a Lipschitz or quadratic growth driver $g=g(s,y,z)$ defines a dynamic convex risk measure for random variables, if the driver is convex in $z$ and does not depend on $y$; cf. [@peng04], [@ro6], [@bek8], and the references therein. The latter requirement is due to the strong notion of cash additivity in the framework of random variables. As pointed out in [@er08], a solution to a BSDE becomes cash subadditive, if the driver is monotone in $y$ and convex in $(y,z)$. In the sequel we want to modify in a way that it would define a dynamic convex risk measure for processes. As we have seen in Proposition \[prop:ca\], every risk measure for processes is cash subadditive; and this suggests to consider BSDEs with monotone convex drivers as in [@er08]. However, in our framework the BSDE should depend on the whole path of the process $X$ rather then just on its terminal value $X_T$. So for a fixed $X$ in ${\mathcal{R}^\infty}$ we will consider a BSDE of the following form: $$\label{qbsde} Y_t=-X_T+\int_t^T g(s,Y_s+X_s,Z_s)ds-\int_t^T Z_s dW_s,\qquad t\in[0,T].$$ Another example of a BSDE depending on a process is given by reflected BSDE, where the solution $Y$ of is required to stay above an “obstacle” process $X$, cf. [@ekppq97]. Thus we may also add a reflection condition to the BSDE , and consider the RBSDE $$\begin{aligned} \label{bsdegen} \nonumber&Y_t=-X_T+\int_t^T g(s,Y_s+X_s,Z_s)ds-\int_t^T Z_s dW_s+K_T-K_t,\quad t\in[0,T], \\ &\text{with }\\ \nonumber &Y_t \ge -X_t \quad \forall t \in [0,T],\quad \mbox{ and }\quad \int_0^T (Y_{s-}+X_{s-}) dK_s=0.\end{aligned}$$ In the sequel we will make the following assumptions on the driver $g: \Omega \times [0,T]\times \real \times \real^d \to \real$:\ \ (H1)\[Lipschitz\] For any $(y,z)\in \real^{1+d}$, the stochastic process $(\omega,t)\mapsto g(\omega,t,y,z)$ is progressively measurable. In addition, there exists $C_{Lip}>0$, such that $$\|g(\omega,t,y_1,z_1)-g(\omega,t,y_2,z_2)\|\leq C_{Lip} (\|y_1-y_2\|+\|z_1-z_2\|) \quad \forall (y_1,y_2,z_1,z_2)\in \real^{2+2d} \;\; \P\otimes dt\text{-a.e.}.$$ (H1’)\[Quadratic growth\] For any $(y,z)\in \real^{1+d}$, the stochastic process $(\omega,t)\mapsto g(\omega,t,y,z)$ is progressively measurable. In addition, there exists $C>0$, such that $$\|g(\omega,t,y,z)\|\leq C (1+ \|y\|+\|z\|^2) \quad \forall (y,z)\in\real^{1+d} \;\; \P\otimes dt\text{-a.e.}$$ (H2)\[Convexity\] $g$ is convex in $(y,z)$, i.e., $\forall (y_1,y_2,z_1,z_2,\lambda) \in \real^{2+2d}\times [0,1]$, $$g(\omega,t,\lambda y_1 + (1-\lambda) y_2,\lambda z_1 + (1-\lambda) z_2) \leq \lambda g(\omega,t,y_1,z_1) +(1-\lambda) g(\omega,t,y_2,z_2) \quad \P\otimes dt\text{-a.e.}.$$ (H3)\[Monotonicity\] $g$ non-increasing in $y$.\ \ (H4)\[Normalization\] $g(\omega, t,0,0)=0\quad\;\,\P\otimes dt$-a.s..\ \ Before recalling existence result for the equations under interest, we point out that assumptions (H1) and (H1’) from one hand, and assumptions (H2)-(H4) on the other hand are not of the same nature. Indeed, as it will be seen in the sequel, (H1) (resp. (H1’)) guarantees existence and uniqueness of a (maximal) solution, whereas assumptions (H2)-(H4) ensure that the solution satisfies the basic axioms of a risk measure for processes. \[rem:h\] In BSDEs and a given process $X$ shifts the driver $g$. However, for each $X\in{\mathcal{R}^\infty}$ we can define a new driver $h^X:\Omega \times [0,T]\times \real \times \real^d \to \real$ as $$h^X(\omega,t, y,z):=g(\omega, t,y+X_t(\omega),z).$$ By definition, $h^X$ directly inherits properties (H1)-(H3) (or (H1’)-(H3)) from $g$ for each $X\in{\mathcal{R}^\infty}$, and the BSDEs and can be written in the more conventional form in terms of the driver $h^X$. \[prop:rmbsde\] Under assumption (H1) (resp. (H1’)), there exists for each $X\in{\mathcal{R}^\infty}$ a unique triple $(Y,Z,K)$ in $\S^2 \times \H^2_d \times \S^2_{\uparrow}$, that is a solution of the RBSDE (resp. a unique couple $(Y,Z)$ in $\S^2 \times \H^2_d$, that is a maximal solution of the BSDE ). Here $$\S^2:=\left\{X:=(X_t)_{t\in [0,T]}{\;\big\|\;}X\; \textrm{progressively measurable, {c\`adl\`ag}}, \; \E\left[\sup_{t\in [0,T]} \|X_t\|^2\right]<\infty \right\},$$ $$\H^2_d:=\left\{X:=(X_t)_{t\in [0,T]}{\;\big\|\;}X\; \textrm{progressively measurable, $d$-dim.}, \; \E\left[\int_0^T \|X_t\|^2 dt \right]<\infty \right\},$$ and $\S^2_{\uparrow}$ denotes the subset of elements in $\S^2$ which are non-decreasing. Using Remark \[rem:h\], existence and uniqueness follow from classical results such as [@h2; @LepXu5; @px5]) for the RBSDE under (H1), and [@Kobylanski] for the BSDE under (H1’). \[rk:indfu\] To stress the dependence on a given process $X\in{\mathcal{R}^\infty}$, we will sometimes denote the BSDEs and by BSDE$(X)$, and the solution $Y$ of the BSDE$(X)$ at time $t$ by $Y_t(X)$. Note that by uniqueness of the (maximal) solution on $[t,T]$, we have $Y_t(X)=Y_t(\pi_{t,T}(X))$, which is in line with our convention $\rho_t(X) = \rho_t(\pi_{t,T}(X))$. For $0\le s\le t\le T$, we will also write $Y_{s,t}(X)$ to denote the solution of BSDE(X) on $[0,t]$ at time $s$. Accordingly, $Y_{s,t}(X)= Y_{s,t}(\pi_{s,t}(X))$, and $Y_t=Y_{t,T}$. The next proposition identifies the (maximal) solution $Y=Y(X)$ of and as a dynamic risk measure for processes. \[prop:rm\]Under the assumptions (H1)-(H4) (resp. (H1’)-(H4)), the (maximal) solution $(Y_{t})_{t\in[0,T]}$ of the RBSDE (resp. of the BSDE ) defines a time consistent dynamic convex risk measure for processes via $$\rho_t(X):=Y_t(X),\quad t\in[0,T],\quad X\in{\mathcal{R}^\infty}.$$ We only deal with the reflected case here, and simply indicate the main arguments for the non-reflected quadratic growth case.\ (i) To prove convexity, let $X^1,X^2\in{\mathcal{R}^\infty}$ and $\lambda\in[0,1]$; we have to show that $$Y(\lambda X^1 + (1-\lambda) X^2) \leq \lambda Y(X^1) + (1-\lambda) Y(X^2).$$ To this end we denote by $(Y^i,Z^i, K^i)$ the solutions of the BSDE for $X=X^i$ ($i=1,2$), and set $\tilde{X}:=\lambda X^1 + (1-\lambda) X^2$, $\tilde{Y}:=\lambda Y(X^1) + (1-\lambda) Y(X^2)$, $\tZ:=\lambda Z^1+(1-\lambda) Z^2$, and $\tK:=\lambda K^1+(1-\lambda) K^2$. Convexity of $g$ in $(y,z)$ implies $$\lambda g(r,Y_r^1+X_r^1,Z_r^1)+(1-\lambda) g(r,Y_r^2+X_r^2,Z_r^2) \geq g(r, \tY_r+ \tilde{X}_r, \tZ_r), \quad \P\text{-a.s.}.$$ Thus we have for any $0\le t_1\leq t_2\le T$ $$\begin{aligned} \tY_{{t_1}}&=\tY_{{t_2}} + \int_{{t_1}}^{{t_2}} \left(\lambda g(r,Y_r^1+X_r^1,Z_r^1)+(1-\lambda) g(r,Y_r^2+X_r^2,Z_r^2)\right)dr -\int_{{t_1}}^{{t_2}} \tZ_r dW_r +\int_{{t_1}}^{{t_2}} d\tK_r\\ &\ge\tY_{{t_2}} + \int_{{t_1}}^{{t_2}} g(r, \tY_r+ \tilde{X}_r, \tZ_r) dr -\int_{{t_1}}^{{t_2}} \tZ_r dW_r. $$Hence $\tY$ is a supersolution of the classical BSDE with driver $g$ and terminal condition $\tX_T$, and $\tY\ge\tX$. As it is proved in [@px5 Theorem 2.1], $Y(\lambda X^1 +(1-\lambda) X^2)$ is the smallest supersolution of the (classical) BSDE with driver $g$ and terminal condition $\tilde{X}_T$ which dominates $\tilde{X}$. Thus $$\tY_t \geq Y_t(\lambda X^1 + (1-\lambda) X^2) \qquad \forall t\in [0,T] \quad \P\text{-a.s.}.$$ In the non-reflected case, comparison theorem for maximal solutions of BSDEs (c.f., e.g., [@er08 Theorem 7.1]) provides the result.\ (ii) To prove (inverse) monotonicity, note that for any $X^1, X^2\in{\mathcal{R}^\infty}$ such that $X^1 \leq X^2$ we have $Y_T(X^1) \geq Y_T(X^2)$. Moreover, since $g$ is non-increasing in $y$, we have $h^{X^1}(t,y,z) \geq h^{X^2}(t,y,z)$ for all $(t,y,z)$. Thus monotonicity follows form the classical comparison principle for (R)BSDEs, cf., e.g., [@er08 Theorem 7.1] and [@h2 Theorem 1.5].\ (iii) We prove cash additivity at time $t$, i.e., $$Y_t(X+m{\textbf{1}}_{[t,T]})= Y_t(X)-m\qquad\forall m\in{L^{\infty}(\Omega, {\mathcal{F}}_t,\P)}.$$ Let $(\tY,\tZ,\tK)$ denote the solution of RBSDE($X+m{\textbf{1}}_{[t,T]}$). By definition, it holds that $$\tY_s + m = -X_T +\int_s^T g(r,\tY_r+X_r+m,\tZ_r) dr -\int_s^T \tZ_r dW_r+\int_s^T d\tK_s, \quad s\in [t,T].$$ Thus $(\tY+m,\tZ,\tK)$ is the solution of on $[t,T]$, and by uniqueness $\tY_t+m=Y_t(X)$.\ (iv) Due to the requirement $g(t,0,0)=0$ $\P\otimes dt$-a.s., $(0,0,0)$ is the unique solution to the BSDE$(0)$; this proves normalization.\ (v) We prove time consistency: $$Y_{t}(X{\textbf{1}}_{[t,s)}-Y_{s}(X) {\textbf{1}}_{[s,T]}(X))=Y_{t}(X)\qquad\forall t\in[0,T], s\in[t,T].$$ To this end, we first show that for $s\in[t,T]$ $$\label{eq:cons} Y_{t,T}(X)=Y_{t,s}(X {\textbf{1}}_{[t,s)} - Y_{s,T}(X) {\textbf{1}}_{[s]})$$ Indeed, if $(Y,Z,K)$ denotes the solution of RBSDE($X$), we have $$\begin{aligned} Y_{t,T}(X)&=-X_T + \int_s^T g(r,Y_r+X_r,Z_r) dr -\int_s^T Z_r dW_r + \int_s^T dK_r \\ &\qquad\quad\;\;+ \int_t^s g(r,Y_r+X_r,Z_r) dr -\int_t^s Z_r dW_r + \int_t^s dK_r\\ &=Y_{s,T}(X) + \int_t^s g(r,Y_r+X_r,Z_r) dr -\int_t^s Z_r dW_r + \int_t^s dK_r\\ &=Y_{t,s}(X {\textbf{1}}_{[t,s)} - Y_{s,T}(X) {\textbf{1}}_{[s]})\end{aligned}$$ due to uniqueness of the solution. Now let $(\tY,\tZ,\tK)$ denote the solution of the RBSDE$(X{\textbf{1}}_{[0,s)}-Y_{s,T}(X){\textbf{1}}_{[s,T]})$. Then we have $$\begin{aligned} \label{line1}\tY_t= & Y_{s,T}(X) + \int_t^s g(r,\tY_r+X_r,\tZ_r) dr -\int_t^s \tZ_r dW_r +\int_t^s d\tK_r\\ \label{line2}&-Y_{s,T}(X)+ Y_{s,T}(X)+ \int_s^T g(r,\tY_r-Y_{s,T}(X),\tZ_r) dr -\int_s^T \tZ_r dW_r +\int_s^T d\tK_r.\end{aligned}$$ Note further that equals to $Y_{t, T}(X)$ by , and is $0$, since $$\begin{aligned} Y_{s,T}(X)+ \int_s^T g(r,\tY_r-Y_{s,T}(X),\tZ_r) dr -\int_s^T \tZ_r dW_r +\int_s^T d\tK_r&=Y_s(-Y_{s,T}(X){\textbf{1}}_{[s,T]})\\&=Y_{s,T}(X)\end{aligned}$$ due to cash invariance and normalization as proved in (iii) and (iv). In the following we will provide dual representations for the risk measures associated to the BSDEs and . To this end we define the Legendre-Fenchel conjugate $g^:\Omega\times [0,T]\times \real\times\real^d\to \real\cup\{\infty\}$ of the convex generator $g$ as in [@er08]: $$\begin{aligned} g^(\omega, t,\beta,\mu)&=\sup_{(y,z)\in\real\times\real^d}{\left\{\,} \newcommand{\rk}{\right\}}-\beta y-\mu\cdot z-g(\omega,t,y,z)\rk\\ &=\sup_{(y,z)\in\qu\times\qu^d}{\left\{\,} \newcommand{\rk}{\right\}}-\beta y-\mu\cdot z-g(\omega,t,y,z)\rk.\end{aligned}$$ Moreover, we introduce the sets $${\mathcal{R}}:={\left\{\,} \newcommand{\rk}{\right\}}\beta=(\beta_t)_{t\in[0,T]}{\;\big\|\;}\beta\;\text{progressively measurable,}\; 0\le\beta\le C\;\,\P\otimes dt\text{-a.s.}\rk,$$ and $${\rm BMO}(\P):={\left\{\,} \newcommand{\rk}{\right\}}\mu=(\mu_t)_{t\in[0,T]}{\;\big\|\;}\mu\in\H^2_d,\;\exists B:\sup_{\tau \, \textrm{stopping time}} \E{\left[}\int_\tau^T\|\mu_s\|^2ds\|{\mathcal{F}}_\tau{\right]}\le B\; \P\text{-a.s.}\rk.$$ \[lem:gstar\] Assume that $g$ satisfies conditions (H1)-(H4) (resp. (H1’)-(H4)), and let $(Y, Z, K)$ (resp. $(Y,Z)$) be a solution to the BSDE (resp. to ) for a process $X\in{\mathcal{R}^\infty}$. Then $$\label{gstar} g(t,Y_t+X_t, Z_t)=\max_{(\beta, \mu)\in{\mathcal{R}}\times{\rm BMO}(\P)}{\left\{\,} \newcommand{\rk}{\right\}}-\beta_t(Y_t+X_t)-\mu_t\cdot Z_t-g^(t,\beta_t,\mu_t)\rk\quad\P\otimes dt\text{-a.s.},$$ where the maximum is attained by some $(\bar\beta, \bar\mu)\in{\mathcal{R}}\times{\rm BMO}(\P)$. Note first that (H1) together with (H4) implies (H1’), so it is sufficient to argue for $g$ satisfying quadratic growth condition (H1’). By definition of $g^$, we have “$\ge$” in , and standard convex duality and measurable selection results (cf. [@bek8 Lemma 7.5]) imply $$g(t,Y_t+X_t, Z_t)=-\bar\beta_t(Y_t+X_t)-\bar\mu_t\cdot Z_t-g^(t,\bar\beta_t,\bar\mu_t)\quad\P\otimes dt\text{-a.s.}$$ for some progressively measurable processes $\bar\beta$ and $\bar\mu$. We have to show that $0\le\bar\beta\le C$ and $\bar\mu\in$BMO$(\P)$. The first estimate follows from [@er08 Lemma 7.4], since $g^(t,\beta,\mu)=\infty$ for $\beta\notin[0,C]$. Moreover, the same argument as in [@er08 Lemma 7.4] implies that there exists $B>0$ such that $$\|\bar\mu_t^2\|\le B\left(1+\|Y_t\|+\|X_t\|+\|Z_t\|^2\right)\quad\P\otimes dt\text{-a.s.}.$$ As proved in the appendix, $Y$ is bounded, and $Z\in$BMO$(\P)$ for each $X\in{\mathcal{R}^\infty}$ both in and in . This proves that $\bar\mu\in$BMO$(\P)$. By classical results of Kazamaki [@Kazamaki Section 3.3], cf. also [@bek8 Theorem 7.2], every $\mu\in{\rm BMO}(\P)$ defines a probability measure $Q^\mu\approx\P$ on ${\mathcal{F}}_T$ via the density process $$\Gamma^\mu_t=\exp\left(\int_0^t\mu_sdW_s-\frac{1}{2}\int_0^t\|\mu_s\|^2ds\right),\qquad t\in[0,T].$$ Moreover, $W^\mu:=W-\int_0^\cdot\mu_sds$ is a $Q^\mu$-Brownian motion, and $\int_0^\cdot Z_sdW^\mu_s$ is a BMO$(Q^\mu)$-martingale for any $Z\in$BMO$(\P)$. Probability measures $Q^\mu$ will describe models appearing in the dual representations of the risk measures associated to BSDEs and . We also define for each $t\in[0,T]$ a family of discounting process $$\begin{aligned} \nonumber {\mathcal{D}}_t:=\Big\{(D_{t,s})_{s\in[t,T]}{\;\big\|\;}(D_{t,s})\;\text{adapted,}& \text{ non-increasing, right-continuous},\\&\; D_{t,t-}:=1, D_{t,T}=0\;\P\text{-a.s.}\Big\}.\end{aligned}$$ Every $D\in{\mathcal{D}}_0$ and a density process $\Gamma^\mu$ as above define a normalized optional measure $\nu$ as in Corollary \[cor:dec\] and via $$\E_{\nu}{\left[}X{\right]}=\E{\left[}-\int_{[0,T]}X_s\Gamma^\mu_sdD_{0,s}{\right]}=\E_{Q^\mu}{\left[}- \int_{[0,T]}X_sdD_{0,s}{\right]},\quad X\in{\mathcal{R}^\infty}.$$ If we define $\bar{\mathcal{F}}_t:=\sigma\left(\pi_{0,t}(X){\;\big\|\;}X\in{\mathcal{R}^\infty}\right)$, and $(D_{t,s})\in{\mathcal{D}}_t$ via $D_{t,s}:=\frac{D_{0,s}}{D_{0,t-}}$, $s\in[t,T]$, $\bar{\mathcal{F}}_t$-conditional expectation with respect to $\nu$ can be written as $$\E_{\nu}{\left[}X\|\bar{\mathcal{F}}_t{\right]}=X{\textbf{1}}_{[0,t)}+\E_{Q^\mu}{\left[}- \int_{[t,T]}X_sdD_{t,s}\|{\mathcal{F}}_t{\right]}{\textbf{1}}_{[t,T]},\quad X\in{\mathcal{R}^\infty}.$$ For $X\in{\mathcal{R}^\infty}_t$, this conditional expectation reduces to $\E_{Q^\mu}[- \int_{[t,T]}X_sdD_{t,s}\|{\mathcal{F}}_t]$, and it will appear in the conditional dual representation of the dynamic risk measures induced by BSDEs and . To be more precise, we will show that the risk measures induced by BSDEs and are of the form $$\label{eq:rmbsde} \rho_t(X)=\operatorname{ess\,sup}_{(\mu,D)\in{\rm BMO}(\P)\times{\mathcal{D}}_t}\left(E_{Q^\mu}{\left[}\int_{[t,T]}X_sdD_{t,s}\|{\mathcal{F}}_t{\right]}-\gamma_t(\mu,D)\right),\quad X\in{\mathcal{R}^\infty},$$ where $\gamma_t(\mu,D)$ is a penalty function on ${\rm BMO}(\P)\times{\mathcal{D}}_t$, and $t\in[0,T]$. This representation can be seen as a conditional version of , where the penalty function is concentrated on the local martingales of the form $\Gamma^\mu$, i.e., on probability measures $Q^\mu$, that are equivalent to the Wiener measure $\P$. In order to prove , let $(Y,Z,K)$ be the (maximal) solution of the BSDE$(X)$, fix $\mu\in{\rm BMO}(\P)$ and $D\in{\mathcal{D}}_t$. Applying integration by parts, taking conditional expectation with respect to $Q^{\mu}$ on both sides, and using that $\int_0^\cdot Z_sdW^\mu_s$ is a BMO$(Q^\mu)$-martingale, we obtain $$\begin{aligned} \nonumber Y_t =\, & Y_tD_{t,t-} = -Y_t\Delta D_{t,t}+Y_tD_{t,t}\\ \nonumber =\, & \E_{Q^\mu}{\left[}- Y_t\Delta D_{t,t}+ D_{t,T}Y_T-\int_t^T Y_sdD_{t,s}-\int_t^T D_{t,s-}dY_s{\;\big\|\;}{\mathcal{F}}_t{\right]}\\\label{l1} =\, & \E_{Q^\mu}{\left[}\int_{[t,T]} X_sdD_{t,s}{\;\big\|\;}{\mathcal{F}}_t{\right]}\\\label{l2} & +\E_{Q^\mu}{\left[}\int_t^T D_{t,s-}\left(g(s,Y_s+X_s,Z_s)+\mu_s\cdot Z_s\right)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\\\label{l3} & +\E_{Q^\mu}{\left[}-\int_{[t,T]} (Y_s+X_s)dD_{t,s}+\int_t^T D_{t,s-}dK_s{\;\big\|\;}{\mathcal{F}}_t{\right]},\end{aligned}$$ where the $dK$ term in disappears for the non-reflected BSDE . These computations lead to the following examples. \[er\] We consider the BSDE $$Y_t=-X_T+\int_t^Tg(s,Y_s+X_s, Z_s)ds-\int_t^TZ_sdW_s\qquad t\in[0,T],$$ where the driver $g$ satisfies assumptions (H1’)-(H4). This is the same framework as in [@er08 Section 7], but in our case the BSDE depends on the whole path of the process $X\in{\mathcal{R}^\infty}$. The results from [@er08] follow from our considerations if applied to processes $X:=X_T{\textbf{1}}_{[T]}$ for $X_T\in{L^{\infty}(\Omega, {\mathcal{F}}_T,\P)}$. For $\beta\in{\mathcal{R}}$ and $t\in[0,T]$, we introduce the discounting factors $$\label{Der} D_{t,s}:=e^{-\int_t^s\beta_udu}, \quad s\in[t,T),\quad\text{ and}\quad D_{t,T}=0.$$ Note that $(D_{t,s})\in{\mathcal{D}}_t$ for all $t$. \[th:robdarer\] The BSDE induces under assumptions(H1’)-(H4) a dynamic convex risk measure for processes $(\rho_t)_{t\in[0,T]}$ with the robust representation $$\begin{aligned} \label{robdarer} \nonumber\rho_t(X)=Y_t=\operatorname{ess\,sup}_{(\mu,\beta)\in{\rm BMO}(\P)\times{\mathcal{R}}}&\left(\E_{Q^{\mu}}{\left[}e^{-\int_t^T\beta_udu}(-X_T)-\int_t^T\beta_sX_se^{-\int_t^s\beta_udu}ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\right.\\ &-\left.\E_{Q^\mu}{\left[}\int_t^Te^{-\int_t^s\beta_udu}g^(s,\beta_s,\mu_s)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\right),\end{aligned}$$ where the essential supremum is attained for each $X\in{\mathcal{R}^\infty}$ by some $(\bar \mu,\bar \beta)\in{\rm BMO}(\P)\times{\mathcal{R}}$. Applying , , and with $(D_{t,s})$ defined in , we obtain $$\begin{aligned} \nonumber Y_t=&\E_{Q^\mu}{\left[}e^{-\int_t^T\beta_udu}(-X_T)-\int_t^T\beta_sX_se^{-\int_t^s\beta_udu}ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\\\label{ll2} &+\E_{Q^\mu}{\left[}\int_t^Te^{-\int_t^s\beta_udu}\left(g(s,Y_s+X_s,Z_s)+\beta_s(Y_s+X_s)+\mu_s\cdot Z_s\right)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}.\end{aligned}$$ By Lemma \[lem:gstar\], $\ge$ for all $(\mu, \beta)\in{\rm BMO}(\P)\times{\mathcal{R}}$, with equality attained at some optimal $(\bar \mu, \bar \beta)$. \[rem:abs\] 1. Theorem \[th:robdarer\] follows also directly from [@er08 Theorem 7.5], applied to the driver $h^X(t,y,z)=g(t,y+X_t,z)$ defined in Remark \[rem:h\]. Indeed, we have for all $\omega$, $t$, $\beta$, and $\mu$ $$(h^X)^(\omega,t,\beta,\mu)=\beta X_t(\omega)+g^(\omega,t,\beta,\mu).$$ 2. Note that $\rho_t$ in Theorem \[th:robdarer\] is of the form , with penalty function $$\gamma_t(Q^\mu,D)=\gamma_t(\mu,D)= \gamma_t(\mu,\beta)=\E_{Q^\mu}{\left[}\int_t^Te^{-\int_t^s\beta_udu}g^(s,\beta_s,\mu_s)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}.$$ This penalty function is concentrated on discounting measures $dD$, that are absolutely continuous with respect to the Lebesgue measure $\lambda$. This is due to the fact that the process $X$ appears only in the driver of , i.e., in the $\lambda$-absolutely continuous part of the BSDE. \[reflgen\] In this example we consider the BSDE $$\begin{aligned} \nonumber&Y_t=-X_T+\int_t^T g(s,Y_s+X_s,Z_s)ds-\int_t^T Z_s dW_s+K_T-K_t,\quad t\in[0,T], \\ \nonumber &Y_t \ge -X_t \quad \forall t \in [0,T],\quad \mbox{ and }\quad \int_0^T (Y_{s-}+X_{s-}) dK_s=0. \end{aligned}$$ For each $t\in[0,T]$, we define the set of stopping times $$\Theta_t:={\left\{\,} \newcommand{\rk}{\right\}}\tau{\;\big\|\;}\tau\;\text{is a stopping time},\;t\le\tau\le T\; \P\text{-a.s.}\rk,$$ and for $\tau\in\Theta_t$ and $\beta \in{\mathcal{R}}$ the discounting factors $D\in{\mathcal{D}}_t$ via $$\label{Dref} D_{t,t-}:=1,\qquad D_{t,s}:=e^{-\int_t^s\beta_sds}{\textbf{1}}_{\{\tau>s\}},\quad s\in[t,T].$$ \[th:robdarref\] The BSDE induces under assumptions (H1)-(H4) a dynamic convex risk measure for processes $(\rho_t)_{t\in[0,T]}$ with the robust representation $$\begin{aligned} \nonumber\rho_t(X)=Y_t=\operatorname{ess\,sup}_{(\mu,\beta,\tau)\in{\rm BMO}(\P)\times{\mathcal{R}}\times\Theta_t}&\left(\E_{Q^{\mu}}{\left[}e^{-\int_t^\tau\beta_udu}(-X_\tau)-\int_t^\tau\beta_sX_se^{-\int_t^s\beta_udu}ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\right.\\\label{sl2} &-\left.\E_{Q^\mu}{\left[}\int_t^\tau e^{-\int_t^s\beta_udu}g^(s,\beta_s,\mu_s)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\right)\end{aligned}$$ for all $X\in{\mathcal{R}^\infty}$. Applying , , and with $(D_{t,s})$ defined in , and using $dD_{t,s}=-{\textbf{1}}_{\{s\le\tau\}}\beta_s e^{-\int_t^s\beta_udu}ds-e^{-\int_t^\tau\beta_udu}\delta_{\{\tau\}}(ds)$, and $D_{t,s-}=e^{-\int_t^s\beta_udu}{\textbf{1}}_{\{\tau\ge s\}}$, we obtain $$\begin{aligned} \nonumber Y_t=&\E_{Q^\mu}{\left[}e^{-\int_t^\tau\beta_udu}(-X_\tau)-\int_t^\tau\beta_sX_se^{-\int_t^s\beta_udu}ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\\\label{li2} &+\E_{Q^\mu}{\left[}\int_t^\tau e^{-\int_t^s\beta_udu}\left(g(s,Y_s+X_s,Z_s)+\beta_s(Y_s+X_s)+\mu_s\cdot Z_s\right)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\\\label{li3} &+\E_{Q^\mu}{\left[}e^{-\int_t^\tau\beta_udu}(Y_\tau+X_\tau)+\int_t^\tau e^{-\int_t^s\beta_udu}dKs{\;\big\|\;}{\mathcal{F}}_t{\right]}.\end{aligned}$$ By Lemma \[lem:gstar\], $\ge$ for all $(\mu,\beta, \tau)$, with equality attained independently of $\tau$ at some $(\bar\mu,\bar\beta)\in{\rm BMO}(\P)\times{\mathcal{R}}$. Moreover, since $Y_t+X_t\ge0$ for all $t$, and $K$ is non-decreasing, $\ge 0$ for all $\tau\in\Theta_t$; this proves “$\ge$” in the representation. On the other hand, for any $\varepsilon>0$ we can define the stopping time $$\tau^\varepsilon:=\inf{\left\{\,} \newcommand{\rk}{\right\}}s\ge t{\;\big\|\;}Y_s\le -X_s+\varepsilon\rk\in\Theta_t.$$ It follows as in the proof of [@LepXu5 Proposition 3.1] that $K_{\tau^\varepsilon}-K_t=0$, and hence $$\E_{Q^{\bar\mu}}{\left[}e^{-\int_t^{\tau^\varepsilon}\bar\beta_udu}(Y_{\tau^\varepsilon}+X_{\tau^\varepsilon})+\int_t^{\tau^\varepsilon} e^{-\int_t^s\bar\beta_udu}dKs{\;\big\|\;}{\mathcal{F}}_t{\right]}\le \varepsilon.$$ This shows that the right-hand-side of the representation is larger or equal than $Y_t-\varepsilon$ for any $\varepsilon>0$, and proves the equality. \[reflriedel\] If the generator $g$ in the previous example does not depend on $y$, the BSDE takes the form $$\begin{aligned} \label{bsderied}&Y_t=-X_T+\int_t^T g(s,Z_s)ds-\int_t^T Z_s dW_s+K_T-K_t,\quad t\in[0,T], \\ \nonumber &Y_t \ge -X_t \quad \forall t \in [0,T],\quad \mbox{ and }\quad \int_0^T (Y_{s-}+X_{s-}) dK_s=0. \end{aligned}$$ In this case the conjugate $g^(t,\beta,\mu)=\infty$ if $\beta\not\equiv0$, and thus the penalty function in is concentrated on the discounting factors $D\in{\mathcal{D}}_t$ such that $D_{t,t-}=1$, and $D_{t,s}={\textbf{1}}_{\{\tau>s\}}$ for $s\in[t,T]$ and $\tau\in\Theta_t$. We write $g^(t,\mu):=g^(t,0,\mu)$; then Theorem \[th:robdarref\] takes the following form. \[th:robdarried\] The BSDE induces under assumptions (H1)-(H4) a dynamic convex risk measure for processes $(\rho_t)_{t\in[0,T]}$ with the robust representation $$\rho_t(X)=Y_t=\operatorname{ess\,sup}_{(\mu,\tau)\in{\rm BMO}(\P)\times\Theta_t}\left(\E_{Q^{\mu}}{\left[}-X_{\tau}{\;\big\|\;}{\mathcal{F}}_t{\right]}-\E_{Q^\mu}{\left[}\int_t^{\tau}g^(s,\mu_s)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\right)$$ for all $X\in{\mathcal{R}^\infty}$. This example was studied in [@morlais13; @bky10; @ried10] in the context of optimal stopping of risk measures for random variables. In our framework it appears naturally as an example of a risk measure for processes. \[reflnot\] [In order to identify a BSDE as a risk measure for processes, it seems to be crucial that the driver $g$, as well as the reflection term $K$ depend on the sum* $X+Y$. For instance, it was shown in [@ekppq97 Section 7] for the classical RBSDE $$\begin{aligned} &Y_t=-X_T+\int_t^T g(s, Y_s, Z_s)ds-\int_t^T Z_s dW_s+K_T-K_t,\quad t\in[0,T], \\ \nonumber &Y_t \ge -X_t \quad \forall t \in [0,T],\quad \mbox{ and }\quad \int_0^T (Y_{s}+X_{s}) dK_s=0, $$ under the assumptions that $X$ is continuous and $g$ satisfies (H1)-(H3), that $Y$ has the dual representation $$\label{robdarnot} Y_t(X)=\operatorname{ess\,sup}_{(\mu,\beta,\tau)\in{\rm BMO}(\P)\times{\mathcal{R}}\times\Theta_t}\left(\E_{Q^{\mu}}{\left[}e^{-\int_t^\tau\beta_udu}(-X_\tau)-\int_t^\tau e^{-\int_t^s\beta_udu}g^(s,\beta_s,\mu_s)ds{\;\big\|\;}{\mathcal{F}}_t{\right]}\right).$$ If $g$ (resp. $g^$) does not depend on $Y+X$, and the right-hand-side of does not take the form as in Theorem \[th:robdarref\], $Y$ does not define a conditional risk measure for processes in the sense of Definition \[def:cond\]: It does not satisfy the axiom of cash additivity. ]{} In general, using Lebesgue decomposition, we can write every measure $dD$ induced by a discounting process $D\in{\mathcal{D}}_0$ as a sum $dD^{\ll}+dD^{\perp}$, where $dD^{\ll}$ denotes the absolutely continuous, and $dD^{\perp}$ the singular part of $dD$ with respect to the Lebesgue measure $\lambda$. For instance, for $D$ defined in we have $$dD_{t,s}=\underbrace{-{\textbf{1}}_{\{s\le\tau\}}\beta_s e^{-\int_t^s\beta_udu}ds}_{dD^{\ll}}-\underbrace{e^{-\int_t^\tau\beta_udu}\delta_{\{\tau\}}(ds)}_{dD^{\perp}}.$$ As we have noted in Remark \[rem:abs\], only absolutely continuous discounting factors $dD^{\ll}$ appear in the robust representation of the risk measure, if there is no reflection, and only the driver of the BSDE depends on the sum $Y+X$. On the other hand, as seen in Example \[reflriedel\], if there is reflection, and the driver does not depend on $Y+X$, absolutely continuous parts $dD^{\ll}$ disappear, and only singular parts $dD^{\perp}$ contribute to the robust representation. The study of general relation between BSDEs of type and risk measures of the form is subject of future research. Examples presented in this paper suggest that appearance of absolutely continuous discounting factors corresponds to the dependence of the driver $g$ on the sum $Y+X$, whereas appearance of the singular discounting terms is induced by the reflection term $K$ depending on $Y+X$. Also more general reflection terms, induced by more complex penalty function on $dD^{\perp}$, can be thought about. Appendix {#appendix .unnumbered} ======== We provide here estimates for the BSDEs and , that are used in the proof of Lemma \[lem:gstar\]. The results for quadratic BSDE follow basically from [@bek8; @er08]; the results for the reflected BSDE might be known, but since we did not find them explicitly written in the literature, we give the proofs here. Throughout this section we consider a BSDE under assumptions (H1)-(H4), and RBSDE under assumptions (H1’)-(H4). \[prop:appendix\] Let $(Y,Z,K)$ (resp. $(Y,Z)$) be the solution of (resp. the maximal solution of ) for $X\in{\mathcal{R}^\infty}$. Then $Y$ is bounded, and $Z\in{\rm BMO}(\P)$. To see that $Y$ is bounded, we use monotonicity, cash additivity, and normalization as proved in Proposition \[prop:rm\]. Let $\\|X\\|_{{\mathcal{R}^\infty}}=:B$, then $$Y_t(X)\le Y_t(0-B{\textbf{1}}_{[t,T]})=B\quad \P\text{-a.s.\ for all}\;\; t\in[0,T],$$ and the converse inequality follows in the same manner.\ The proof that $Z\in{\rm BMO}(\P)$ in the non-reflected quadratic case follows as in [@bek8 Proposition 7.3], using that $Y$ and $X$ are bounded. In the reflected case we use classical estimates, as for example in [@PossamaiZhou], where such technique is used in the context of second order BSDEs. Itô’s formula implies for any $\tau\in\Theta_0$ and any $\alpha>0$ that $$\begin{aligned} \label{eq:good} e^{-\alpha Y_\tau} &= e^{-\alpha Y_\tau} - \alpha \int_\tau^T e^{-\alpha Y_{s}} g(s,Y_s+X_s,Z_s) ds + \alpha \int_\tau^T e^{-\alpha Y_s} Z_s dW_s -\frac{\alpha^2}{2} \int_\tau^T e^{-\alpha Y_s} \|Z_s\|^2 ds \nonumber\\ & - \alpha \int_\tau^T e^{-\alpha Y_{s-}} dK_s -\sum_{\tau <s \leq T} [e^{-\alpha Y_s} - e^{-\alpha Y_{s-}} +\alpha e^{-\alpha Y_{s-}} \Delta_s Y ].\end{aligned}$$ Since $K$ is non-decreasing, and thus $\Delta_s Y =\Delta_s K \geq 0$, and since the mapping $x\mapsto e^{-x}-1+x$ is non-negative on $\real_+$, the last two terms are non-positive. Hence rewrites as: $$\begin{aligned} \frac{\alpha^2}{2} \int_\tau^T e^{-\alpha Y_s} \|Z_s\|^2 ds +e^{-\alpha Y_\tau} &\leq e^{-\alpha Y_T} - \alpha \int_\tau^T e^{-\alpha Y_{s}} g(s,Y_s+X_s,Z_s) ds + \alpha \int_\tau^T e^{-\alpha Y_s} Z_s dW_s. \end{aligned}$$ This implies, since $g$ has Lipschitz growth, and $X$ and $Y$ are bounded, that $$\begin{aligned} \frac{\alpha^2}{2} \int_\tau^T e^{-\alpha Y_s} \|Z_s\|^2 ds \leq e^{-\alpha Y_T} + C \alpha \int_\tau^T e^{-\alpha Y_{s}} (1+\|Z_s\|^2) ds + \alpha \int_\tau^T e^{-\alpha Y_s} Z_s dW_s, \end{aligned}$$ where we have used that $\|x\|\leq 1+\|x\|^2$. ($C$ in this proof denotes a generic constant, which can differ from line to line.) Using again the fact that $Y$ is bounded, we get that there exists a constant $\tilde{C}$ (which only depends on $T$ but not on $\tau$) such that $$\begin{aligned} (\frac{\alpha^2}{2} - C \alpha) \int_\tau^T e^{-\alpha Y_s} \|Z_s\|^2 ds &\leq \tilde{C} + \alpha \int_\tau^T e^{-\alpha Y_s} Z_s dW_s. \end{aligned}$$ Taking conditional expectations on both sides of this inequality leads to $$(\frac{\alpha^2}{2} - C \alpha) \E\left[\int_\tau^T e^{-\alpha Y_s} \|Z_s\|^2 ds \Big\vert \mathcal{F}_t \right] \leq \tilde{C} \quad \P\text{-a.s.},$$ which concludes the proof again by boundedness and $Y$ and by choosing $\alpha > 2C$. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Kostas Kardaras and Michael Kupper for helpful comments and discussions. The authors acknowledge support from the DFG Research Center <span style="font-variant:small-caps;">Matheon</span>. [10]{} B. Acciaio, H. F[[ö]{}]{}llmer, and I. Penner. Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles. , 16(4):669–709, 2012. P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Thinking coherently. , 10:68–71, 1997. P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. , 9(3):203–228, 1999. H. Assa. Lebesgue property of convex risk measures for bounded c[à]{}dl[à]{}g processes. , 18(3):335–349, 2011. P. Barrieu and N. El Karoui. Pricing, hedging and optimally designing derivatives via minimization of risk measures. In [Indifference pricing: Theory and applications]{}, Princeton Series in Financial Engineering, pages 77–146, Princeton University Press, Princeton, NJ, 2009. E. Bayraktar, I. Karatzas, and S. Yao. Optimal stopping for dynamic convex risk measures. , 54(3):1025–1067, 2012. E. Bayraktar, C. Kardaras, and H. Xing. Valuation equations for stochastic volatility models. , 3:351–373, 2012. E. Bayraktar and S. Yao. Optimal stopping for non-linear expectations—[P]{}art [I]{} , 121(2):185–211, 2011. E. Bayraktar and S. Yao. Optimal stopping for non-linear expectations—[P]{}art [II]{} , 121(2):212–264, 2011. K. Bichteler. , volume 89 of [ Encyclopedia of Mathematics and its Applications]{}. Cambridge University Press, Cambridge, 2002. P. Carr, T. Fisher, and J. Ruf. On the hedging of options on exploding exchange rates. , 2012. P. Cheridito, F. Delbaen, and M. Kupper. Coherent and convex monetary risk measures for bounded c[à]{}dl[à]{}g processes. , 112(1):1–22, 2004. P. Cheridito, F. Delbaen, and M. Kupper. Coherent and convex monetary risk measures for unbounded c[à]{}dl[à]{}g processes. , 9(3):369–387, 2005. P. Cheridito, F. Delbaen, and M. Kupper. Dynamic monetary risk measures for bounded discrete-time processes. , 11:no. 3, 57–106, 2006. C. Dellacherie and P.-A. Meyer. , volume 29 of [North-Holland Mathematics Studies]{}. North-Holland Publishing Co., Amsterdam, 1978. C. Dellacherie and P.-A. Meyer. , volume 72 of [ North-Holland Mathematics Studies]{}. North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales, Translated from the French by J. P. Wilson. N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez. Reflected solutions of backward [SDE]{}’s, and related obstacle problems for [PDE]{}’s. , 25(2):702–737, 1997. N. El Karoui and C. Ravanelli. Cash subadditive risk measures and interest rate ambiguity. , 19(4):561–590, 2009. H. F[[ö]{}]{}llmer. The exit measure of a supermartingale. , 21:154–166, 1972. H. F[[ö]{}]{}llmer and A. Schied. Convex measures of risk and trading constraints. , 6(4):429–447, 2002. H. F[[ö]{}]{}llmer and A. Schied. . Walter de Gruyter & Co., Berlin, extended edition, 2011. An introduction in discrete time. M. Frittelli and E. Rosazza Gianin. Putting order in risk measures. , 26(7):1473–1486, 2002. S. Hamad[[è]{}]{}ne. Reflected [BSDE]{}’s with discontinuous barrier and application. , 74(3-4):571–596, 2002. K. It[ô]{} and S. Watanabe. Transformation of [M]{}arkov processes by multiplicative functionals. , 15(fasc. 1):13–30, 1965. J. Jacod. , volume 714 of [Lecture Notes in Mathematics]{}. Springer, Berlin, 1979. Ju. M. Kabanov, R. [Š]{}. Lipcer, and A. N. [Š]{}irjaev. Absolute continuity and singularity of locally absolutely continuous probability distributions. [I]{}. , 107(149)(3):364–415, 463, 1978. C. Kardaras. Num[é]{}raire-invariant preferences in financial modeling. , 20(5):1697–1728, 2010. C. Kardaras, D. Kreher, and A. Nikeghbali. Strict local martingales and bubbles. , 2011. N. Kazamaki. , volume 1579 of [Lecture Notes in Mathematics]{}. Springer-Verlag, Berlin, 1994. M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. , 28(2):558–602, 2000. J.-P. Lepeltier and M. Xu. Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier. , 75(1):58–66, 2005. M.-A. Morlais. Reflected backward stochastic differential equations and a class of non linear dynamic pricing rule. , 85(1):1–26, 2013. J. Najnudel and A. Nikeghbali. A new kind of augmentation of filtrations. , 15(In honor of Marc Yor, suppl.):S39–S57, 2011. K. R. Parthasarathy. . Probability and Mathematical Statistics, No. 3. Academic Press Inc., New York, 1967. S. Peng. Nonlinear expectations, nonlinear evaluations and risk measures. In [Stochastic methods in finance]{}, volume 1856 of [Lecture Notes in Math.*]{}, pages 165–253. Springer, Berlin, 2004. S. Peng and M. Xu. The smallest [$g$]{}-supermartingale and reflected [BSDE]{} with single and double [$L^2$]{} obstacles. , 41(3):605–630, 2005. D. Possamai and C. Zhou. Second order backward stochastic differential equations with quadratic growth. , 2012. F. Riedel. Optimal stopping under ambiguity in continuous time. , Institute of Mathematical Economics, [B]{}ielefeld university. F. Riedel. Optimal stopping with multiple priors. , 77(3):857–908, 2009. E. Rosazza Gianin. Risk measures via [$g$]{}-expectations. , 39(1):19–34, 2006. [^1]: `penner@math.hu-berlin.de` [^2]: `anthony.reveillac@ceremade.dauphine.fr`
--- author: - 'J. Licandro' - 'T. Müller' - 'C. Alvarez' - 'V. Alí-Lagoa' - 'M. Delbo' date: 'Received September 15, 1996; accepted March 16, 1997' title: 'CanariCam/GTC observations of (99942) Apophis' --- [The potentially hazardous asteroid (PHA) (99942) Apophis is one of the most remarkable near-Earth asteroids (NEA) in terms of impact hazard. A good determination of its surface thermal inertia is very important in order to evaluate the Yarkovsky effect on its orbital evolution. ]{} [We present thermal infrared observations obtained on January 29, 2013, with CanariCam mid-infrared camera/spectrograph attached to the Gran Telescopio CANARIAS (GTC, Roque de los Muchachos Observatory, La Palma, Spain) using the Si2-8.7, Si6-12.5, and Q1-17.65 filters with the aim of deriving Apophis’ diameter ($D$), geometric albedo ($p_V$), and thermal inertia ($\Gamma$).]{} [ We performed a detailed thermophysical model analysis of the GTC data combined with previously published thermal data obtained using [Herschel]{} [Space]{} [Observatory]{} PACS instrument at 70, 100, and 160 $\mu$m.]{} [The thermophysical model fit of the data favors low surface roughness solutions (within a range of roughness slope angles $rms$ between 0.1 and 0.5), and constrains the effective diameter, visible geometric albedo, and thermal inertia of Apophis to be $D_{eff} =$ 380 – 393 m, $p_V = $ 0.24–0.33 (assuming absolute magnitude $H = 19.09 \pm 0.19$) and $\Gamma =$ 50 – 500 Jm$^{-2}$ s$^{-0.5}$ K$^{-1}$, respectively.]{} Introduction ============ The potentially hazardous asteroid (PHA) (99942) Apophis (hereafter Apophis) is a near-Earth asteroid (NEA) that has a small but non-zero chance of impacting the Earth and it is one of the most remarkable NEAs in terms of impact hazard. With the available data, it is known that Apophis will have an extremely close approach on April 13, 2029, at 5.7 Earth radii from the Earth’s center, just below the altitude of geosynchronous Earth satellites; by means of a statistical analysis, Farnocchia et al. [@Farnocchia2013] find an impact probability greater than $10^{-6}$ for an impact in 2068. The computation of the orbital evolution of this object is limited by the insufficient knowledge of the role played by the non-gravitational Yarkovsky effect, which produces a steady orbital drift as a consequence of the momentum carried away by the thermal emission of the object (see, e.g., Bottke et al., [@Botkke2002], Giorgini et al., [@Giorgini2002]). The Yarkovsky effect depends upon several poorly known parameters, such as the albedo, size, thermal inertia, and pole orientation of the object. Recently Vokrouhlický et al. [@Vokrouhlicky] evaluated the Yarkovsky effect on the orbital evolution of Apophis. In addition, improved knowledge of the physical properties of Apophis is desirable for other reasons: (1) the European Commission H2020-PROTEC-2014 funded project NEOShield-2 will base its study of NEO mitigation strategy on the case of Apophis and (2) the possible implications would need to be addressed if an impact were to occur. A first determination from polarimetric observations of the geometric albedo $p_V = 0.33 \pm 0.08$ is presented in Delbo et al. [@Delbo2007]. They also obtained an absolute magnitude of $H = 19.7 \pm 0.4$ mag, which led to an effective diameter $D_{eff} = 270 \pm 60$ m, slightly smaller than earlier estimates in the range of 320 to 970 m depending on the assumed albedo. Müller et al. [@Mueller2014] published the first far-infrared observations of Apophis using the [Herschel Space Observatory]{} PACS instrument. They obtained data at 70, 100, and 160 $\mu$m at two epochs and performed a detailed thermophysical model (TPM) analysis. They used the spin and shape model and absolute magnitude $H = 19.09 \pm 0.19$ by Pravec et al. [@Pravec2014] and obtained an effective diameter $D_{eff} = 375 ^{+14}_{-10}$ m, a geometric albedo in the $V$-band $p_V = 0.30 ^{+0.05}_{-0.06}$, and a thermal inertia of $\Gamma = 600 ^{+200}_{-350}$ Jm$^{-2}$s$^{-0.5}$K$^{-1}$. The albedo determinations agree very well; the difference between the $D_{eff}$ determined by Müller et al. and the value derived by Delbo et al. [@Delbo2007] from their $p_V$ determination is the result of a different value of $H$ ($H = 19.09$ and $H = 19.7,$ respectively). The Vokrouhlický et al. [@Vokrouhlicky] results use the Pravec et al. [@Pravec2014] and Müller et al. [@Mueller2014] results to evaluate the Yarkovsky effect on the orbital evolution of Apophis; in particular, they use the range of $\Gamma$ values provided by Müller et al. [@Mueller2014]. In this paper we present thermal infrared observations obtained on January 29, 2013, with the CanariCam mid-infrared instrument attached to the Gran Telescopio CANARIAS (GTC, Roque de los Muchachos Observatory, La Palma, Spain). Images of Apophis were obtained using three different filters (Si2-8.7, Si6-12.5, and Q1-17.65). These fluxes, obtained at wavelengths which are closer to the wavelengths in which the Apophis thermal emission peaks than the [Herschel]{} measurements, are used together with [Herschel]{} data to better constrain the thermophysical model presented in Müller et al. [@Mueller2014]. The paper is organized as follows. In Sect. \[sec:data\] we present the observations and describe the reduction process and photometry. The TPM is described and the results are presented in Sect. \[sec:TPM\]. Finally the discussion and conclusions are presented in Sect. \[sec:discussion\]. Observations and data reduction {#sec:data} =============================== The observations were performed on January 29, 2013, with CanariCam (see Telesco et al. [@Telesco]) in imaging mode at the 10.4 m Gran Telescopio CANARIAS. Non-sidereal guiding was not available at the time of the observations (it was fully implemented in 2013, but later on) and therefore Apophis had to be tracked by applying offsets to the telescope every time the target was about to move off the CanariCam field of view (FOV) ($25'' \times 19''$). The standard star HD59381 was also observed on the same night as a flux calibrator. The data were taken in the Si2-8.7, Si6-12.5, and Q1-17.65 filters whose central wavelengths are 8.7, 12.5, and 17.65 $\mu$m, respectively (see Table \[TableObs\]). The telescope’s secondary mirror was chopping at 2 Hz with a chop throw of $7''$ along the east-west direction. Nodding of the telescope axes was performed every 47 seconds with a nod throw of $7''$, also in the east-west direction, to minimize the radiative offset. Object Date UT start UT end Filter on-source (s) eff. on-source (s) --------- ------------- ------------ ------------ ---------- --------------- -------------------- Apophis 2013-Jan-29 23:09:22.1 23:47:29.7 Q1-17.65 908.316288 371.5839 Apophis 2013-Jan-29 23:52:38.6 23:56:13.0 Si2-8.7 80.739226 80.73923 Apophis 2013-Jan-29 22:04:37.9 22:11:33.6 Si6-12.5 165.148416 74.83288 HD59381 2013-Jan-30 00:23:27.3 00:26:57.0 Q1-17.65 82.574208 82.57421 HD59381 2013-Jan-30 00:13:12.1 00:16:45.5 Si2-8.7 80.739226 80.73923 HD59381 2013-Jan-30 00:17:33.4 00:21:03.6 Si6-12.5 82.574208 82.57421 Data were processed using a set of dedicated PyRAF [^1] scripts developed within our group. CanariCam raw images consist of a series of individual frames (savesets). The savesets are stored in multi-extension FITS files (MEF), which have the structure of \[320,240,2,M\]\[N\]. The first two numbers represent the detector’s X and Y dimensions in pixels ($320 \times 240$). The third dimension represents the number of chop positions, namely on-source and off-source positions. M represents the number of savesets in each nod position and N the number of nods (nod beams A and B), which follow the sequence A-BB-A. Off-source savesets were subtracted from the corresponding on-source savesets for each nod beam. For each individual saveset, we determined the source centroid using SExtractor (Bertin & Arnouts [@BertinArnouts]). Savesets were then geometrically aligned using the shifts calculated from the centroid positions with respect to centroid from the first saveset and stacked to produce the final net signal image. This shift-and-add technique improves the image quality, and therefore the sensitivity. Shift-and-add is particularly important for the Apophis data since the object was drifting accross the CanariCam detector by several pixels from one saveset to the next. Additionally, owing to the relatively short on-source time in each saveset (5.9 s) and the rapid movement of Apophis accross the FOV, it was not possible to calculate image centroid in all savesets. Hence, those savesets where the centroid was not obtained were discarded from the total signal, yielding an effective on-source time smaller than the observed on-source time (see Table \[TablePhot\]). The resulting images of Apophis and the standard star used are shown in Figure \[apo\_images\]. Aperture photometry was performed in the reduced images using PyRAF. An aperture of radius $0.6''$ was used in all filters in the Apophis as well as in the standard star HD59381 images. A sky annulus with a radius of $2.4''$ and width of $0.8''$ was used to determine the sky as the median of all pixel values within the annulus area. The in-band flux in Jy for each CanariCam filter was obtained by integrating the standard star template spectrum (Cohen et al. [@Cohen]) multiplied by the filter transmission curve. Finally, the in-band flux for HD59381 was divided by the measured ADU/s within the $0.6''$ radius aperture and then multiplied by the measured ADU/s within the same aperture in Apophis. No color correction is applied as it is much smaller ($< 1\%$) than the uncertainties. The final flux densities and the FWHM of the PSF in each image can be found in Table \[TablePhot\]. Object Filter Flux (Jy) FWHM ($''$) --------- ---------- ------------------ ----------------- Apophis Si2-8.7 $0.14 \pm 0.01 $ $0.25 \pm 0.01$ Apophis Si6-12.5 $0.24 \pm 0.02$ $0.32 \pm 0.04$ Apophis Q1-17.65 $0.31 \pm 0.07$ $0.43 \pm 0.01$ HD59381 Si2-8.7 $8.80 \pm 0.04$ $0.25 \pm 0.01$ HD59381 Si6-12.5 $4.93 \pm 0.04$ $0.32 \pm 0.04$ HD59381 Q1-17.65 $2.43 \pm 0.04$ $0.43 \pm 0.01$ : \[TablePhot\]Photometric data. Thermophysical modeling {#sec:TPM} ======================= Thermophysical models (TPM) are powerful tools used to derive asteroid sizes from their thermal infrared data. If the shape, the spin axis orientation, and the rotational period of the object are well characterized and if enough data are available, TPMs also allow the thermal inertia and the macroscopic roughness of the surface to be constrained. In brief, TPMs model the temperature on each surface element of the shape model –typically triangular facets– at every observation epoch by accounting for the corresponding energy budget, i.e., how much incident solar radiation is absorbed at and conducted onto the surface. This depends on the distance from the asteroid to the Sun and on the physical properties of the surface (albedo, emissivity, macroscopic roughness, conductivity, etc.), but it also depends critically on the shape of the object and its rotational phase at the moment of the observations since they determine the illumination geometry of each facet. Thus, the TPM requires any available convex shape model in combination with the spin axis orientation and rotational properties as input. The heat conduction onto the surface is controlled by the thermal inertia $\Gamma$, while the infrared beaming effects are calculated via a surface roughness model implemented as concave, spherical crater segments on the surface and parametrized by the root mean square ([rms]{}) slope angle. Once the temperatures are modeled, the model fluxes that the observer would measure can be computed given the particular observational geometry and they can be fit to the data. The observational geometry refers to the heliocentric and geocentric distances and the phase angle –the angle subtended by the observer and the Sun from the point of view of the asteroid. Müller et al. (2014) applied a TPM based on the work by Lagerros ([@Lagerros96], [@Lagerros97], [@Lagerros98]) and Müller & Lagerros ([@Mueller1998], [@Mueller2002]) to model the thermal data of Apophis obtained with the [Herschel Space Observatory]{} PACS instrument. Taking the tumbling rotational state and shape model given by Pravec et al. ([@Pravec2014]), Müller et al. obtained a thermal inertia of $\Gamma = 600 ^{+200}_{-350}$ Jm$^{-2}$s$^{-0.5}$K$^{-1}$, an effective diameter of $D_{eff} = 375 ^{+14}_{-10}$ m, and a corresponding visible ($V$-band) geometric albedo of $p_V = 0.30 ^{+0.05}_{-0.06}$ for Apophis. In this paper we use the same TPM and rotational state model to study a combination of GTC and [Herschel]{} data to better constrain the Müller et al. (2014) results. We combined all GTC and [Herschel]{} data using the results obtained in Müller et al. ([@Mueller2014]), with [Herschel]{} data alone as a starting guess. We assumed a constant emissivity of 0.9 at all wavelengths. We also used the mean absolute magnitude $H_V = 19.09 \pm 0.19$ mag derived by Pravec et al. ([@Pravec2014]) under the assumption of a slope parameter of $G = 0.24 \pm 0.11$. The observational circumstances of [Herschel Space Observatory]{} data are summarized in Table 1 of Müller et al. ([@Mueller2014]); GTC observations were obtained with the asteroid at heliocentric and geocentric distances r$_{helio} = 1.080$ AU and $\Delta_{obs} = 0.113,$ respectively, and a phase angle $\alpha = -31.7$ degrees. The illumination is similar to that shown in Müller et al. ([@Mueller2014]), Fig. 3, right panel, except that it is at a smaller phase angle (CanariCam: -31.7 deg, PACS: -61.4 deg) and a different orientation of the body (as seen from GTC). In Fig. \[chi2\] we plot the reduced $\chi^2$-values calculated for the radiometric analysis of the combined GTC and [Herschel]{} data for different roughness slope angles versus thermal inertia. The roughness slope angles ($rms$) range from 0.0 to 0.9. The minimum reduced $\chi^2$ would be $\sim$0.8 and the statistical error would be $\sigma\sim 0.6$ so we find acceptable fits (with $\chi^2$-values lower than 1.55) for all levels of roughness (see, e.g., Press et al. [@Press1986]), which means that we cannot unambiguously constrain roughness and thermal inertia: low-roughness combined with low $\Gamma$ fit similarly well to high-roughness and high-$\Gamma$ solutions. On the other hand, the minima of the reduced $\chi^2$-values are lower for low-roughness solutions. This is also illustrated in the ratios of observed-to-modeled fluxes presented in Fig. \[ajustes\], where we compare the ratios of the observed fluxes to our modeled fluxes for the extreme roughness cases. We note in particular that observed-to-modeled fluxes of the GTC/CanariCam data are particularly sensitive to roughness and are a slightly less sensitive for larger roughness. These considerations led us to favor the solutions with low surface roughness in our analysis. Thus, within a range of $rms$ between 0.1 and 0.5, we constrain the size, visible geometric albedo, and thermal inertia of Apophis to be $D_{eff} =$ 380 – 393 m, $p_V = $ 0.27–0.29, and $\Gamma =$ 50 – 500 Jm$^{-2}$s$^{-0.5}$K$^{-1}$. We note that the given range of $p_V$ is obtained assuming the absolute magnitude $H=19.09$ from Pravec et al. ([@Pravec2014]), considering the uncertainty in $H$ ($H = 19.09 \pm 0.19$) and so the range of possible albedo is wider ($p_V = $ 0.24–0.33). [ccccc]{} rms & D$_{eff}$ (m) & p$_V$ & $\Gamma$ & $\chi^2_{reduced}$\ 0.0 & 389 & 0.278 & 126 & 0.624\ 0.1 & 389 & 0.278 & 159 & 0.674\ 0.2 & 387 & 0.281 & 200 & 0.784\ 0.3 & 384 & 0.285 & 251 & 0.922\ 0.5 & 380 & 0.292 & 316 & 1.114\ 0.9 & 375 & 0.300 & 398 & 1.262\ ![Reduced $\chi^2$ values for models considering different roughness slope angles $rms =$ 0.0, 0.1, 0.2, 0.3, 0.5 and 0.9. Notice that the minima of the reduced $\chi^2$-values are lower for low-roughness solutions, but we find acceptable fits (with $\chi^2$-values lower than 1.55 shown as an horizontal line) for all levels of roughness. []{data-label="chi2"}](26888_hk_fig2.pdf){width="7cm"} ![Observed GTC/CanariCam (this paper) and Herschel/PACS (from Müller et al. [@Mueller2014]) divided by model fluxes for the two extreme cases with roughness slope angles $rms =$ 0.0 ($upper$) and 0.9 ($lower$). Notice that observed-to-modeled fluxes of the GTC/CanariCam data are particularly sensitive to roughness and worsen for larger roughness, which led us to favor the solutions with low surface roughness in our analysis[]{data-label="ajustes"}](26888_hk_fig3a.pdf "fig:"){width="6cm"} ![Observed GTC/CanariCam (this paper) and Herschel/PACS (from Müller et al. [@Mueller2014]) divided by model fluxes for the two extreme cases with roughness slope angles $rms =$ 0.0 ($upper$) and 0.9 ($lower$). Notice that observed-to-modeled fluxes of the GTC/CanariCam data are particularly sensitive to roughness and worsen for larger roughness, which led us to favor the solutions with low surface roughness in our analysis[]{data-label="ajustes"}](26888_hk_fig3b.pdf "fig:"){width="6cm"}\ Discussion and conclusions {#sec:discussion} ========================== Images of Apophis were obtained using three different filters (Si2-8.7, Si6-12.5, and Q1-17.65) with the CanariCam instrument in imaging mode at the 10.4 m Gran Telescopio CANARIAS (GTC) at El Roque de los Muchachos Observatory (La Palma, Canary Islands, Spain). The derived fluxes, which are closer to the wavelengths in which Apophis’ thermal emission peaks than those reported by Müller et al. ([@Mueller2014]) using [Herschel]{} space telescope data, are used together with Müller et al. reported fluxes to better constrain the thermophysical model of Apophis also presented in Müller et al. Our fitting of the TPM to the combined GTC/CanariCam and PACS/$Herschel$ favors the solutions with low surface roughness (within a range of $rms$ roughness slope angles between 0.1 and 0.5) and it constrains the size, visible geometric albedo, and thermal inertia of Apophis to be $D_{eff} =$ 380 – 393 m, $p_V = $ 0.27–0.29, and $\Gamma =$ 50 – 500 Jm$^{-2}$ s$^{-0.5}$ K$^{-1}$. These results agree very well within the uncertainties with those reported by Müller et al. ([@Mueller2014]) using only the $Herschel$ data ($D_{eff} = 375^{+14}_{-10}$ m, $p_V = 0.30^{+0.05}_{-0.06}$, a thermal inertia $\Gamma $ in the range 250-800 Jm$^{-2}$ s$^{-0.5}$ K$^{-1}$, with a best solution at $\Gamma =$ 600 Jm$^{-2}$ s$^{-0.5}$ K$^{-1}$ for $rms = 0.5$), but point to a somewhat lower value of thermal inertia, closer to the lower range given in Müller et al., and a slightly larger effective size (closer to the wider range given in Müller et al.). The albedo value also agrees with the value reported by Delbo et al. ([@Delbo2007]) ($p_V =0.33 \pm 0.08$) derived from polarimetric observations. The thermal inertia of Apophis also closely fits the trend of $\Gamma$ vs. $D_{eff}$ reported by Delbo & Tanga ([@Delbo2009]). Finally, the presented improvement in the determination of the thermal inertia is very important in order to evaluate the Yarkovsky effect on the orbital evolution of Apophis (Vokrouhlický et al., [@Vokrouhlicky]). Using the Müller et al. ([@Mueller2014]) range of possible $\Gamma$-values, they estimate the drift of the Apophis orbital semimajor axis $\langle da/dt \rangle$ to be in a range between $-11$ x $10^{-4}$ and $-15$ x $10^{-4}$ au/Myear. Using the range of thermal inertia obtained in this paper, the range of $\langle da/dt \rangle$ that can be derived from Fig. 1 in Vokrouhlický et al. ([@Vokrouhlicky]) is slightly smaller, between $-6$ x $10^{-4}$ and $-14$ x $10^{-4}$ au/Myear. We acknowledge Ben Rositis for his useful comments that helped to improve the manuscript. Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma. We acknowledge Petr Scheirich & Petr Pravec for the provision of shape orientation, spin-axis orientation, rotational properties, etc for the GTC observing epochs, and Ben Rositis for his useful comments that helped to improve the manuscript. JL acknowledges support from the project ESP2013-47816-C4-2-P (MINECO, Spanish Ministry of Economy and Competitiveness). MD and V. Ali-Lagoa acknowledge support from the NEOShield-2 project that has received funding from the European UnionÕs Horizon 2020 research and innovation program under grant agreement No 640351. The work of V. Ali-Lagoa and MDB was supported by the French Agence National de la Recherche (ANR) SHOCKS [99]{} Bertin, E. & Arnouts, S., 1996, A&AS, 117, 393 Bottke, W. F., Morbidelli, A., Jedicke, R. et al., 2002, Icarus, 156, 399 Cohen, M., Walker, R. G. Carter, B. et al. 1999, AJ, 117, 1864 Delbo, M.,& Tanga, P., 2009, P&SS, 57, 259 Delbo, M., Cellino, A. & Tedesco, E., 2007, Icarus, 188, 266 Farnocchia, D., Chesley, S. R., Chodas, P.W., et al. 2013, Icarus, 224, 192 Giorgini, J., Ostro, J., Benner, L. et al., 2002, Science, 296,132 Lagerros, J. S. V. 1996, A&A, 310, 1011 Lagerros, J. S. V. 1997, A&A, 325, 1226 Lagerros, J. S. V. 1998, A&A, 332, 1123 Müller, T. & Lagerros, J. S. V. 1998, A&A, 338, 340 Müller, T. & Lagerros, J. S. V. 2002, A&A, 381, 324 Müller, T., Kiss, C., Scheirich, P. et al., 2014, A&A, 566, 22M Pravec, P., Scheirich, P., Durech, J. et al. 2014, Icarus, 233, 48 Press, W. H., Flannery, B. P., & Teukolsky, S. A. 1986, Numerical recipes (Cambridge: University Press) Telesco C. M., Ciardi D., French J., et al., 2003, in SPIE Conference Series, Vol. 4841, 913 Vokrouhlický, D., Farnocchia, D., Capek, D. et al. 2015, Icarus, 252, 277 [^1]: PyRAF is a product of the Space Telescope Science Institute, which is operated by AURA for NASA.
--- abstract: 'The number and the location of monopoles in Lattice configurations depend on the choice of the gauge, in contrast to the obvious requirement that monopoles, as physical objects, have a gauge-invariant status. It is proved, starting from non-abelian Bianchi identities, that monopoles are indeed gauge-invariant: the technique used to detect them has instead an efficiency which depends on the choice of the abelian projection, in a known and well understood way.' author: - 'A. Di Giacomo' bibliography: - 'sample.bib' title: Gauge invariance and lattice monopoles --- [ address=[Dipartimento di Fisica and INFN, Pisa, Italy]{} ]{} Introduction ============ Monopoles are relevant excitations of $QCD$: in particular they can condense in the vacuum and produce dual superconductivity, which is a candidate mechanism for confinement [@'tHP][@m][@'tH2]. If dual superconductivity is the mechanism of confinement monopoles are expected to dominate the critical dynamics at the deconfinement phase transition. Evidence is claimed that they also dominate below it, down to zero temperature[@suz][@pol]. On the lattice monopoles have been studied from two complementary perspectives - The symmetry of the vacuum state: an order parameter is introduced, which is the vacuum expectation value ($vev$) of a magnetically charged operator, and can discriminate between dual superconducting and normal phase[@digz; @dp; @ddp; @ddpp; @dlmp; @ccos; @3; @vpc]. - The observation of monopoles in lattice configurations, trying to extract a monopole effective action, in particular to check monopole dominance[@sco],[@suz; @pol]. In this paper we shall concentrate on the second approach. The procedure to detect monopoles in lattice configurations was first developed in Ref.[@dgt] for $U(1)$ gauge theory. Any excess over $2\pi$ of the abelian phase of a plaquette is interpreted as existence of a Dirac string through it, and a monopole exists in an elementary cube whenever a net number of Dirac flux-lines crosses the plaquettes at its border. The phase of a $U(1)$ plaquette is gauge invariant, and hence the procedure is well defined and unambiguous. For a non abelian gauge theory one has first to fix a gauge [@'tH2] and then apply the same procedure to the components along an abelian subgroup of the gauge group (Abelian projection). The result is strongly gauge dependent and because of that the existence of a monopole in a given location of a configuration is a gauge dependent property. That is of course physically unacceptable. The prototype magnetically charged configuration is the soliton solution of Ref.s[@'tH; @Pol] of an $SU(2)$ gauge theory coupled to a Higgs field in the adjoint representation. In that solution the magnetic $U(1)$ which couples to the magnetic charge coincides with the little group of the $vev$ in the unitary representation of the Higgs field which produces the symmetry breaking. The magnetic charge corresponds to a non-trivial homotopy, the solution being a mapping of the sphere $S_2$ at spatial infinity on $SU(2)/U(1)$. This feature is gauge invariant. In $QCD$ there is no Higgs field, but the magnetic $U(1)$ has to be a suitable subgroup of the gauge group. It follows from general arguments that the magnetic-monopole term in the multipole expansion of any field configuration is intrinsically abelian and obeys abelian equations of motion [@coleman]. Since at a superficial sight there is no preferred gauge direction in $QCD$ it was proposed in Ref.[@'tH2] that any operator in the adjoint representation could be used as an effective Higgs field to identify the magnetic $U(1)$ subgroup, physics being for some reason independent on that choice (Abelian projection). It was soon realized that the location and the number of monopoles in a given configuration was projection dependent, and most people did choose the so called maximal-abelian gauge as the most suitable for monopole dominance. In fact the number and the location of monopoles strongly depends on the choice of the abelian projection. Can monopoles be defined in a gauge-invariant way? We will show that this is possible, and that the difficulties outlined above are only apparent. By use of the Non Abelian Bianchi Identities ($NABI$), we will show that for any magnetically charged gauge field configuration there exists a privileged direction in color space which identifies the magnetic $U(1)$. As a consequence monopoles have a gauge invariant physical status, as they should. If, however, one tries to detect them by fixing an abelian projection and by looking for Dirac strings with the technique of Ref.[@dgt] the result is projection dependent. In the maximal abelian projection that technique works and monopoles are detected. In other abelian projections a fraction of the monopoles miss detection, and in the Landau Gauge projection no monopole is observed. Dual superconductivity, instead, defined as Higgs breaking of the residual $U(1)$ symmetry is a gauge invariant feature of the system. The non abelian Bianchi identities. =================================== The abelian Bianchi Identities are the homogeneous Maxwell’s equations $$\partial_{\mu} F^_{\mu \nu} =0$$ or $\vec \nabla \cdot \vec B =0$ and $\vec \nabla \wedge \vec E + \partial _{t} \vec B=0$. A violation of the Bianchi identities $$\partial_{\mu} F^_{\mu \nu} = j_{\nu} \label{abi}$$ means existence of a non-zero magnetic current $j_{\nu}$, which is conserved because of the antisymmetry of the tensor $F^_{\mu \nu}$, $$\partial_{\nu} j_{\nu} =0$$ The non abelian counterpart of Eq.(\[abi\]) is $$D_{\mu} G^_{\mu \nu} = J_{\nu} \label{nabi}$$ where $D_{\mu}$ denotes covariant derivative, $G^_{\mu \nu}$ is the dual of the non abelian field strength $G_{\mu \nu}$, $G^_{\mu \nu}= \frac{1}{2} \epsilon_{\mu \nu \rho \sigma} G_{\rho \sigma}$. The non-abelian current is covariantly conserved $$D_{\mu} J_{\mu} =0$$ as a consequence of Eq.(\[nabi\]). The four components of the current $J_{\mu}$ commute with each other, as always when the algebra of the generators of the symmetry does not involve generators of the Poincare’ group [@cm] Eq.(\[nabi\]) is gauge covariant. To extract the gauge invariant information contained in it one can go to the representation in which the currents are diagonal and project on a complete set of diagonal matrices. There are $r$ of them, if $r$ is the rank of the gauge group, by definition of rank. One possible choice are the fundamental weights $\phi^a_{0}$ ($a = 1,..,r$). There exists one fundamental weight for each simple root of the group algebra $\vec \alpha^a$, which commutes with the Cartan elements of the algebra $H_{i}$, $(i=1,..r)$ , $[\phi^a_{0}, H_{i}] =0$ and has commutators with the operators $E_{\pm \vec \alpha}$ related to the roots $\vec \alpha$ $[\phi^a_{0}, E_{\pm \vec \alpha}] = \pm (\vec c^a\cdot \vec \alpha) E_{\pm \vec \alpha} $ . In particular for the simple roots $\vec c^a\cdot \vec \alpha^b = \delta_{ab}$, so that, if we define $T^a_{3} \equiv \frac{1}{2} [ E_{ \vec \alpha^a}, E_{- \vec \alpha^a}]$, $Tr (\phi^a_{0} T^b_{3}) = \delta _{ab}$. If we denote by $\phi^a_{I} $ the matrix transforming in the adjoint representation which coincides with $\phi^a_{0}$ in the representation in which $J_{\mu}$ is diagonal , the projection of Eq.(\[nabi\]) described above reads $$Tr\left ( \phi^a_{I} D_{\mu} G^_{\mu \nu}\right) = Tr\left ( \phi^a_{I}J_{\nu}\right) \equiv j_{\nu}(I)$$ For reasons which will be clear in the following we can also operate the projection on the matrix $ \phi^a_{V}$ which, in the representation in which $J_{\mu}$ is diagonal, is defined as $ \phi^a_{V} = V(x) \phi^a_{0} V(x)^{\dagger} $ with $V(x)$ an arbitrary gauge transformation, getting $$Tr\left ( \phi^a_{V} D_{\mu} G^_{\mu \nu}\right) = Tr\left ( \phi^a_{V}J_{\nu}\right) \equiv j_{\nu}(V) \label{pr}$$ We shall denote by $F^a_{\mu\nu}(V)$ the ’tHooft tensor in the abelian projection in which $\phi^a_{V} $ is diagonal, i.e. the tensor which coincides with the abelian field strength of the residual $U(1)$ symmetry in the gauge in which $\phi^a_{V} $ is diagonal. In Ref.[@bdlp] we have proved the following theorem which is valid for a generic compact gauge group. [*THEOREM. As a consequence of Eq.\[nabi\] for any compact gauge group Eq.\[pr\] is equivalent to $$\partial_{\mu} F^{a}_{\mu \nu}(V) = j_{\nu}(V) \label{th}$$ ]{} The breaking of the abelian Bianchi identities, i.e. the magnetic currents of the residual $U(1)$ gauge field in any abelian projection, are the projections on the corresponding fundamental weight of the currents which break the non abelian Bianchi identities. Eq.(\[th\]) will be our tool to solve the problem we are concerned with. It implies $\partial_{\nu} j_{\nu}(V) =0$ for arbitrary $V$. The ’t Hooft Polyakov monopole revisited ======================================== We shall now check our theorem on the soliton configuration of Ref.s [@'tH; @Pol]. The configuration is a static solution of the SU(2) Higgs model, with the Higgs field in the adjoint representation. The Lagrangean has the form $$L = -\frac{1}{4}\vec G_{\mu \nu}\vec G_{\mu \nu} + (D_{\mu} \phi)^{\dagger}(D_{\mu} \phi) -V(\phi^2)$$ The notation is standard. The potential $V(\phi^2)$ contains a term linear in $\phi^2$ which has a negative $(-m^2)$ coefficient in the Higgs broken phase where the soliton exists, and a quadratic term in $(\phi^2)^2$ with positive coefficient $\lambda$. The solution is worked out in the so called hedgehog gauge, in which the Higgs field at the position $\vec r$ in physical space is directed as $\hat r$ in color space. In formulae $$\vec \phi(\vec r) = H(r) \hat r \label{hedg}$$ with $H(r)_{r \to \infty} \to v$ the vacuum expectation value of $\phi$ Eq.(\[hedg\]) explicitly shows the non trivial mapping of the $S_{2}$ sphere at spatial infinity on $SU(2)/U(1)$. The solution reads [@'tH; @Pol] $$\begin{aligned} \vec A_{0}& =&0 \\ A^a_{i}& = &\epsilon_{iak}\frac{r^k}{gr^2}[1 - K(gvr)]\end{aligned}$$ The function $K(x)$ depends on the parameters of the potential $V(\phi^2)$ but generically, modulo possible logs decays exponentially at large $r$ $K(x)_{x\to \infty} \propto \exp(-x)$ and, at small distances behaves as $[1 - K(x)]_{x\to0} \propto x^2$. It is trivial to realize, by inspecting the solution, that this gauge is nothing but the Landau gauge $$\partial_{\mu} A_{\mu}=0$$ The ’tHooft tensor, or better the field strength along the diagonal component $\sigma_3$ can explicitly be computed getting for the abelian magnetic field $$\vec b \approx _{r\to \infty} \frac{2 \hat r}{gr^2} \frac{z}{r}$$ The magnetic charge $Q_{m}$ computed as flux of the magnetic field through the sphere at infinity is trivially zero in this gauge! $Q_{m} =0 $ in Landau gauge. Let us now go to the unitary gauge in which the Higgs field is rotated to a fixed direction in color space, say the $3-$axis. The configuration can easily be computed (see e.g. the appendix of Ref.[@shnir]) and it is trivial to verify that this gauge is nothing but the maximal abelian gauge defined by the condition[@'tH2] $$\partial_{\mu}A_{\mu}^{\pm} \pm ig \left[A_{\mu}^3, A_{\mu}^{\pm} \right] =0$$ Moreover in this gauge the non abelian magnetic current is diagonal, so that it coincides with the one identified by $\phi^a(I)$ of Sect.2. In this gauge the magnetic charge can be computed either directly, as in the Landau gauge, or by our theorem. The solution being static the space components $J_{i}$ of the non abelian magnetic current vanish, and the temporal component is directly computed to be $$J_{0} = D_{i}B_{i} = \frac{2\pi}{g} \delta^3(\vec r) \sigma_{3}$$ There is one fundamental weight ($r=1$ for $SU(2)$) namely $\phi_{0} = \sigma_3$ . Our theorem Eq.(\[th\]) gives $$\vec \nabla\cdot \vec b= Tr(\phi_{0} J_{0}) = \frac{4\pi}{g}\delta^3(\vec r)$$ or $Q_{m} =\frac{1}{g} $ in the maximal abelian gauge. If our soliton were a lattice configuration and we had measured the magnetic charge as the flux at infinity of the abelian field in the $3-$direction, as is usually done on the lattice, we would have found a monopole of charge 2 Dirac units in the maximal abelian gauge, and no monopole at all in the Landau gauge. In both cases the monopole would be there, and with it the magnetic charge which is determined by the homotopy of the solution. It is not true that all abelian projections are equivalent. The monopole has a preferred direction in color space, which is, in this case, the direction of the Higgs breaking, which coincides with the direction of the magnetic field at large distances in the unitary representation. General case ============ The argument can be extended to a generic static configuration by use of a theorem due to Coleman\[[@coleman] ,Sect. 3.3\] [The magnetic monopole term in the multipole expansion of a generic static field configuration is abelian : it obeys abelian equations of motion and can be gauged along one direction in color space, modulo a global transformation.*]{} If the configuration is not static one can apply the argument to the superposition of it to the time-inverted configuration to isolate the magnetic field due to monopoles at large distances. Going to the gauge in which the asymptotic magnetic field is directed along a given direction in color space, say the $3-$axis, the non abelian magnetic field at large $r$ is fixed by the total magnetic charge $m$, to be $$\vec B = \frac{m}{2} \frac{\vec r}{2gr^3}\sigma_3$$ Since non diagonal terms in $\sigma_{\pm}$ are non leading it can easily be proved that also the leading term of the abelian magnetic field at large distances is fixed. $$\vec b \approx_{r\to\infty} \frac{m}{2} \frac{\vec r}{2gr^3}$$ The gauge field at large distances obeys the maximal abelian gauge condition. In the maximal abelian gauge, or in any gauge which differs from it by an arbitrary gauge transformation $W(\vec r)$ which tends to the identity as $r \to \infty$, the flux of the abelian magnetic field at large distances directly gives the correct magnetic charge. Monopole condensation and confinement. ====================================== The magnetic charge density in the maximal abelian projection is given by $$j_0(x, I) = Tr(\phi_I J_{0}(x) )$$ with $J_{0}$ the violation of the NABI Eq.(\[nabi\]), and $\phi_{I}$ the fundamental weight diagonal with it. The equal-time commutator of $j_{0}(x, I)$ with any local operator $O(y)$ carrying magnetic charge $m$ is $$\left[ j_0(\vec x, x_0,I), O(\vec y, x_0)\right] = m \delta^3(\vec x -\vec y)O(\vec y,x_0) + S.T.$$ By $S.T.$ we mean Schwinger terms. After integration over $\vec x$ the Schwinger terms give zero contribution and $$\left[Q(I),O(y)\right] = m O(y)$$ If $m\neq 0$ and $\langle O \rangle \neq 0$ the magnetic $U(1)$ is Higgs-broken, and there is dual superconductivity. In a generic abelian projection the magnetic current is $$j_{0}(x, V) = Tr( V(x)\phi_{I} V^{\dagger}(x) J_{0})$$ $j_{0}(x, V)$ is gauge invariant, but we can compute it in the gauge in which $J_{0}$ is diagonal and $\phi_{I}$ with it. Since $V(x) \phi_{I} V^{\dagger}(x)$ belongs to the algebra it will be $$V(x) \phi_{I} V^{\dagger}(x) = C(x,V) \phi_{I} +\sum _{\vec \alpha} E_{\vec \alpha} D^{\vec \alpha}(x,V) \label{CCC}$$ We have written the expansion having in mind the gauge group $SU(2)$ for simplicity. In the generic case the diagonal term will be, instead of a coefficient times $\phi_{I}$ a combination of the fundamental weights $\phi_{I}^a$ $( a= 1,..r)$, and the argument would be analogous. In the gauge chosen only the first term contributes and $$\left[Q(V),O(y)\right] = m O(y)C(y,V)$$ Since $C(y,V)$ is generically non-vanishing the operator $O$ will have a non zero charge also in the new abelian projection, and if $\langle O \rangle \neq 0$ also the new $U(1)$ is Higgs broken. Dual superconductivity is a gauge invariant physical property. This is in agreement with the numerical results obtained in Ref.[@suzu]. Lattice Monopoles. ================== The recipe to detect monopoles [@dgt] on the lattice is based on the measurement of the abelian magnetic flux in a given abelian projection through the boundary surface of elementary cubes. The above analysis shows that the magnetic flux is equal to the “true” magnetic flux in the maximal abelian projection, or in any other projection which differs from it by a continuous gauge transfomation which tends to the identity at large spatial distances. In the procedure it is assumed that the border of the elementary cube is far enough to be at infinity. The choice of the gauge on the lattice is then limited to the maximal abelian gauge itself, modulo a check that, defining the flux by larger cubes leaves the flux unchanged. This proves to be the case in the maximal abelian gauge [@ddmo][@bdd] In other abelian projections the magnetic flux is expected to be smaller then the true one[@bdlp]. This can be directly checked as follows[@bdd]: - Fix the gauge to be the maximal abelian : monopoles appear as elementary cubes with a net number of Dirac strings crossing the border. In the weak coupling regime, near the continuum limit, almost all of them will have one single string attached. - Assume as a first approximation that the monopole is in the centre of the cube and that the Dirac string is perpendicular to the plaquette crossed. - Perform gauge transformations depending on one parameter ${\bf a}$ $(0 \le {\bf a} \le1)$ of the form $$U({\bf a}) = \exp(- i\phi\frac{\sigma_3}{2}) \exp(- i{\bf a}\theta \frac{\sigma_2}{2}) \exp( i\phi\frac{\sigma_3}{2})$$ Here $\theta$ and $\phi$ are the polar angles with respect to the direction of the Dirac string. For ${\bf a}=0$ $U({\bf 0})=1$ and one stays in the maximal abelian gauge. For ${\bf a}=1$ $U({\bf 1})$ is nothing but the unitary transformation bringing from the maximal abelian to the Landau gauge [@shnir]. One can compute analytically the magnetic charge (flux of the abelian magnetic flux at large $r$) as a function of ${\bf a}$ , getting $$\frac{Q({\bf a})}{Q({\bf 0})} = \frac{1 + \cos({\bf a}\pi)}{2}$$ to be compared with lattice measurements. The comparison was done in Ref.[@bdd] after the time when this talk was presented, and is surprisingly positive, in spite of the approximations quoted above and of the discretization errors. This confirms that the approach is correct and that the differences between abelian projections are well understood. Concluding remarks ================== - Monopoles are defined by their topological structure (homotopy), which is gauge invariant. They are intrinsically abelian entities. - Each field configuration with non zero magnetic charge has an intrinsically built in preferred direction in color space, related to the magnetic monopole term in the multipole expansion of the field at large distances, which is the natural $U(1)$ coupled to monopole charge. This direction is selected by the so called maximal abelian gauge. In this gauge the recipe of Ref.[@dgt] really detects monopoles, defined as configurations with non trivial homotopy. - The magnetic charge of the same configuration measured by different abelian projection, is projection(gauge) dependent. This explains why the location and the number of monopoles can be different in different gauges. The recipe of Ref.[@dgt] in gauges other than the maximal abelian can miss monopoles. - Non abelian Bianchi identities allow to relate magnetic charges in different abelian projections. Aknowledgements =============== The author is grateful to Claudio Bonati, Massimo D’Elia, Luca Lepori and Fabrizio Pucci,who collaborated to the works Ref.[@bdlp], [@bdd] where the physics presented this talk was developed. [9]{} G. ’t Hooft, in High Energy Physics, EPS International Conference, Palermo 1975, A. Zichichi ed. S. Mandelstam, Phys. Rep. [23C]{}, 245 (1976). G. ’t Hooft, Nucl. Phys. B [190]{}, 455 (1981). T. Suzuki, I. Yotsuyanagi, Phys. Rev. D [42]{}, 4257 (1990). M. I. Polikarpov, Nucl. Phys. B (Proc. Suppl.) [53]{}, 134 (1997) A. Di Giacomo, Acta Phys.Polon.B25:215-226,1994. A. Di Giacomo, G. Paffuti, Phys. Rev. D [56]{}, 6816 (1997). L. Del Debbio,A. Di Giacomo, G. Paffuti, Phys. Lett. B [349]{}, 513 (1995). L. Del Debbio, A. Di Giacomo, G. Paffuti, P. Pieri, Phys. Lett. B [355]{}, 255 (1995). A. Di Giacomo, B. Lucini, L. Montesi, G. Paffuti. Phys. Rev. D [61]{}, 034503 (2000). P. Cea, L. Cosmai, Phys. Rev. D [76]{}, 031501 (2000). J. M. Carmona, M. D’Elia, A. Di Giacomo, B. Lucini, G. Paffuti, Phys. Rev. D [64]{}, 114507 (2001). A. I. Veselov, M. I. Polikarpov, M. N. Chernodub, JETP Lett.63:411-416,1996, Pisma Zh.Eksp.Teor.Fiz.63:392-397,1996. T. Sekido, K. Ishiguro,Y. Koma, Y. Mori, T. Suzuki, Phys. Rev. D [76]{}, 031501 (2007). T. A. DeGrand, D. Toussaint, Phys. Rev. D [22]{}, 2478 (1980). Erice, Italy (1981). G. ’t Hooft, Nucl. Phys. B [79]{}, 276 (1974). A. M. Polyakov, JETP Lett. [20]{}, 194 (1974). A. S. Kronfeld, M. L. Laursen, G. Schierholz, U. J. Wiese, Phys. Lett. B [198]{}, 516 (1987). T. Suzuki, M. Hasegawa, K. Ishiguro, Y. Koma, T. Sekido, Phys. Rev. D [80]{}, 054504 (2009). C. Bonati, A. Di Giacomo, L. Lepori and F. Pucci, Phys.Rev.D[81]{} 085022 (2010). A. Di Giacomo, L. Lepori and F. Pucci, JHEP [0810]{}, 096 (2008). T. Suzuki, K. Ishiguro, Y. Mori, T. Sekido, Phys. Rev. Lett.[94]{},132001 (2005) Ya. Shnir, Magnetic Monopoles, Springer (2005). A. Di Giacomo, M. Maggiore, S. Olejnik, Nucl. Phys. B [347]{}, 441 (1990). L. Del Debbio, A. Di Giacomo, M. Maggiore, S. Olejnik, Phys. Lett. B [267]{}, 254 (1991) S. R. Coleman, J. Mandula, Phys. Rev. [159]{}, 1251(1967). Ya. Shnir, Magnetic Monopoles, Springer (2005). S. Coleman, The Magnetic Monopole Fifty Years Later, Lectures given at the International School of Subnuclear Physics,Erice , Italy (1981). L. Del Debbio, A. Di Giacomo, M. Maggiore, S. Olejnik, Phys. Lett. B [267*]{}, 254 (1991) C. Bonati, M. D’Elia, A. Di Giacomo, Detecting monopoles on the lattice. e-Print: arXiv:1009.2425 \[hep-lat\], submitted for publication.
--- abstract: \| We investigate the long distance asymptotics of various correlation functions for the one-dimensional spin-1/2 Fermi gas with attractive interactions using the dressed charge formalism. In the spin polarized phase, these correlation functions exhibit spatial oscillations with a power-law decay whereby their critical exponents are found through conformal field theory. We show that spatial oscillations of the leading terms in the pair correlation function and the spin correlation function solely depend on $\Delta k_F$ and $2\Delta k_F$, respectively. Here $\Delta k_F =\pi(n_{\uparrow}-n_{\downarrow})$ denotes the mismatch between the Fermi surfaces of spin-up and spin-down fermions. Such spatial modulations are characteristics of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. Our key observation is that backscattering among the Fermi points of bound pairs and unpaired fermions results in a one-dimensional analog of the FFLO state and displays a microscopic origin of the FFLO nature. Furthermore, we show that the pair correlation function in momentum space has a peak at the point of mismatch between both Fermi surfaces $k=\Delta k_F$, which has recently been observed in numerous numerical studies. author: - 'J. Y. Lee and X. W. Guan' title: 'Asymptotic correlation functions and FFLO signature for the one-dimensional attractive spin-1/2 Fermi gas' --- Introduction ============ Bardeen-Cooper-Schrieffer (BCS) theory was formulated over 50 years ago as a microscopic theory for superconductivity. One of the ingredients in BCS theory is pairing between electrons with opposite momenta and spins, i.e., matching between the Fermi energies of spin-up and spin-down electrons. In the phase where the system is partially polarized, Fermi energies of spin-up and spin-down electrons become unequal. This leads to a non-standard form of pairing which was predicted independently by Fulde and Ferrell [@Fulde1964], and Larkin and Ovchinnikov [@Larkin1965]. Fulde and Ferrell discovered that under a strong external field, superconducting electron pairs have nonzero pairing momentum and spin polarization. At about the same time, Larkin and Ovchinnikov suggested that the formation of pairs of electrons with different momenta, i.e., $\vec{k}$ and $-\vec{k}+\vec{q}$ where $\vec{q}\neq 0$, is energetically favored over pairs of electrons with opposite momenta, i.e., $\vec{k}$ and $-\vec{k}$, when the separation between Fermi surfaces is sufficiently large. Consequently, the density of spins and the superconducting order parameter become periodic functions of the spatial coordinates. This non-conventional superconducting state is known in literature as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. More recently, theoretical predictions of the existence of an FFLO state in one-dimensional (1D) interacting fermions [@Yang1967; @Gaudin1967] have emerged by employment of various methods, such as Bethe ansatz (BA) [@Orso; @HuHui], density-matrix renormalization group (DMRG) [@Feiguin2007; @Rizzi2008; @Tezuka2008; @Meisner; @Luscher], quantum Monte Carlo (QMC) [@Batrouni2008], mean field theory [@Kinnunen2006; @Liu2008; @Zhao2008; @Cooper] and bosonization [@Yang2001]. At finite magnetization, it was found by Feiguin and Heidrich-Meisner [@Feiguin2007] that pair correlations for the attractive Hubbard model in a parabolic trapping potential has a power-law decay of the form $n^{\mathrm{pair}}\propto\cos(k_{\mathrm{FFLO}}\|x\|)/\|x\|^{\alpha}$ and the momentum pair distribution has peaks at the mismatch of the Fermi surfaces $k_{\mathrm{FFLO}}=\pi(n_{\uparrow}-n_{\downarrow})$. Wave numbers for the oscillations were numerically found as $\pi(n_{\uparrow}-n_{\downarrow})$ for the pair correlation function and as $2\pi(n_{\uparrow}-n_{\downarrow})$ for the density difference $\langle n_{\uparrow}-n_{\downarrow}\rangle$ [@Tezuka2008]. The FFLO pairing wave number was also confirmed by the occurrence of a peak in the pair momentum distribution corresponding to the difference between the Fermi momenta of individual species [@Rizzi2008; @Batrouni2008]. From mean field theory, it was demonstrated that the FFLO phase exists in the large-scale response of the Fermi gas [@Cooper] and even for temperatures up to $0.1T_{F}$ [@Liu2008]. On the other hand, critical behavior of 1D many-body systems with linear dispersion in the vicinities of their Fermi points can be described by conformal field theory. Some time ago, the critical behavior of the Hubbard model with attractive interaction was investigated by Bogoliubov and Korepin [@Bogoliubov1988; @Bogoliubov1989; @Bogolyubov1990; @Bogolyubov1992]. They showed that 1D superconductivity occurs when the average distance between electron pairs is larger than the average distance between individual electrons of these pairs. This means that the correlation function for the single particle Green’s function decays exponentially, i.e., $\langle\psi_{n,s}^{\dagger}\psi_{1,s}\rangle\rightarrow e^{- n/\xi}$ with $\xi=v_F/\Delta$ and $s=\uparrow,\, \downarrow$, whereas the singlet pair correlation function decays as a power of distance, i.e., $\langle\psi_{n,\uparrow}^{\dagger}\psi_{n,\downarrow}^{\dagger}\psi_{1,\uparrow}\psi_{1,\downarrow}\rangle\rightarrow n^{-\theta}$. Here $\Delta$ is the energy gap, and the critical exponents $\xi$ and $\theta$ are both greater than zero. This criterion is met when the external magnetic field is small, i.e., $H<H_{c}$. Once the external field exceeds the critical value, i.e., $H>H_{c}$, Cooper pairs are destroyed. Thus both of these correlation functions decay as a power of distance and the pairs lose their dominance, i.e., electrons become more or less independent of each other. So far, theoretical confirmation of the FFLO state in 1D still relies on numerical evidence of spatial oscillations in the pair correlations. Despite key features of the $T=0$ phase diagram [@Orso; @HuHui; @Guan2007; @Mueller; @Kakashvili; @Wadati] for the attractive Fermi gas were experimentally confirmed using finite temperature density profiles of trapped fermionic ${}^6$Li atoms [@Liao2009], the unambiguous theoretical confirmation and experimental observation of FFLO pairing is still an open problem. As remarked in Ref. [@Rizzi2008] that the 1D FFLO scenario proposed in Ref. [@Yang2001] does not apply to 1D attractive fermions where quantum phase transition from the fully-paired phase into the spin polarized phase does not belong to commensurate-incommensurate university class, also see Refs. [@Penc; @Guan2007]. For 1D attractive spin-1/2 fermions with polarization [@Yang1967; @Gaudin1967], the low-energy physics of the homogeneous system is described by a two-component Tomonaga-Luttinger liquid (TLL) of bound pairs and excess unpaired fermions in the charge sector and ferromagnetic spin-spin interactions in the spin sector [@Erhai]. In this paper, we determine the critical behavior of the single particle Green’s function, pair correlation function and spin correlation function within the context of a TLL. We show that the long distance asymptotics of various correlation functions provide a microscopic origin of FFLO pairing for 1D attractive fermions. This paper is organized as follows. We derive finite-size corrections for the ground state energy of the system in Section \[sec:Finite-size\_GS\]. In Section \[sec:Finite-size\_Ex\], we derive finite-size corrections for low-lying excitations and introduce the dressed charge formalism. Integral equations for each component of the dressed charge matrix is solved analytically in the strong coupling limit $\|c\|\gg 1$. In Section \[sec:Corr-func\], we derive correlation functions for different operators and discuss the signature of FFLO pairing. Finally, conclusions and remarks are made in Section \[sec:Conclusion\]. Ground state and finite-size corrections {#sec:Finite-size_GS} ======================================== We consider $N_{f}$ fermions with $SU(2)$ spin symmetry in a 1D system of length $L$ with periodic boundary conditions. The Hamiltonian for the spin-1/2 Fermi gas [@Yang1967; @Gaudin1967] is given by $$H=-\sum_{j=1}^{N_{f}}\frac{\partial^{2}}{\partial x_{j}^{2}}+2c\sum_{1\leq j< k\leq N_{f}}\delta(x_{j}-x_{k}),$$ where $c<0$ is the attractive interaction strength. This model is one of the most important exactly solvable quantum many-body systems. In recent years, it has attracted considerable attention from theory [@Orso; @HuHui; @Guan2007; @Mueller; @Kakashvili; @Wadati] and experiment [@Liao2009] due to evidence of the FFLO state. Systems exhibiting novel phase transitions at $T=0$ are particularly useful in studying TLL physics [@Erhai] and the nature of the FFLO state. The quasimomenta for unpaired fermions and bound pairs are given by $k_{j}$ and $\Lambda_{\alpha}\pm \mathrm{i}c'$ which satisfy the BA equations $$\begin{aligned} k_{j}L &=& 2\pi I_{j}+\sum_{\alpha=1}^{N_{b}}2\tan^{-1}\left(\frac{k_{j}-\Lambda_{\alpha}}{\|c'\|}\right), \\ 2\Lambda_{\alpha}L &=& 2\pi J_{\alpha}+\sum_{j=1}^{N_{u}}2\tan^{-1}\left(\frac{\Lambda_{\alpha}-k_{j}}{\|c'\|}\right) +\sum_{\beta=1}^{N_{b}}2\tan^{-1}\left(\frac{\Lambda_{\alpha}-\Lambda_{\beta}}{2\|c'\|}\right),\end{aligned}$$ where quantum numbers $I_{j}$ and $J_{\alpha}$ are given by $$I_{j}\equiv\frac{N_{b}}{2} \quad(\mathrm{mod}\phantom{a}1), \qquad J_{\alpha}\equiv\frac{N_{u}-N_{b}+1}{2}\quad(\mathrm{mod}\phantom{a}1). \label{eq:IandJ}$$ Here $c'=c/2$, and $N_{u}$ and $N_{b}$ denote the number of unpaired fermions and bound pairs, respectively. The energy and momentum for this system reads $$E=\sum_{j=1}^{N_{u}}k_{j}^{2}+\sum_{\alpha=1}^{N_{b}}2(\Lambda_{\alpha}^{2}-\|c'\|^{2}), \qquad P=\sum_{j=1}^{N_{u}}k_{j}+2\sum_{\alpha=1}^{N_{b}}\Lambda_{\alpha}.$$ We define monotonic increasing counting functions $z_{u}^{L}(k_{j}):=I_{j}/L$ and $z_{b}^{L}(\Lambda_{\alpha}):=J_{\alpha}/L$ and re-label the variables $k\rightarrow k_{u}$, $\lambda\rightarrow k_{b}$, $I_{j}\rightarrow I_{u,j}$ and $J_{\alpha}\rightarrow I_{b,\alpha}$ so that we can express the root densities in a general form as $$\begin{aligned} \rho_{u}^{L}(k_{u}) &:=& \frac{d}{dk_{u}}z_{u}^{L}(k_{u}) =\frac{1}{2\pi}-\frac{1}{L}\sum_{\alpha=1}^{N_{b}}a_{1}(k_{u}-k_{b,\alpha}),\label{eq:rho_u} \\ \rho_{b}^{L}(k_{b}) &:=& \frac{d}{dk_{b}}z_{b}^{L}(k_{b}) =\frac{1}{\pi}-\frac{1}{L}\sum_{j=1}^{N_{u}}a_{1}(k_{b}-k_{u,j})-\frac{1}{L}\sum_{\beta=1}^{N_{b}}a_{2}(k_{b}-k_{b,\beta}),\label{eq:rho_b}\end{aligned}$$ where $a_{n}(k)$ is defined by $$a_{n}(k)=\frac{1}{\pi}\frac{n\|c'\|}{(nc')^{2}+k^{2}}.$$ Here $k_{\alpha,j}$ (for $j=1,2,\ldots,N_{\alpha}$ and $\alpha=u,b$) denote the BA roots for unpaired fermions and bound pairs in the ground state. Using the Euler-Maclaurin formula for contributions up to $O(1/L^{2})$ when $L\gg 1$, the finite-size corrections to the root densities can be written in the generic form as $$\begin{aligned} \nonumber \rho_{\alpha}^{L}(k_{\alpha}) &=& \rho_{\alpha}^{(0)}(k_{\alpha})+\sum_{\beta=u,b}\int_{-Q_{\beta}}^{Q_{\beta}} K_{\alpha\beta}(k_{\alpha}-k_{\beta})\rho_{\beta}^{L}(k_{\beta})dk_{\beta} \\ && +\frac{1}{24L^{2}}\sum_{\beta=u,b}\left[\frac{K'_{\alpha\beta}(k_{\alpha}-Q_{\beta})}{\rho_{\beta}^{L}(Q_{\beta})} -\frac{K'_{\alpha\beta}(k_{\alpha}+Q_{\beta})}{\rho_{\beta}^{L}(-Q_{\beta})}\right], \qquad (\alpha=u,b) \label{eq:rho_gen}\end{aligned}$$ where $$\left( \begin{array}{c} \rho_{u}^{(0)}(k_{u}) \\ \rho_{b}^{(0)}(k_{b}) \\ \end{array} \right)=\left( \begin{array}{c} 1/2\pi \\ 1/\pi \\ \end{array} \right), \qquad \mathbf{K}(k)=\left( \begin{array}{cc} K_{uu}(k) & K_{ub}(k) \\ K_{bu}(k) & K_{bb}(k) \\ \end{array} \right)=\left( \begin{array}{cc} 0 & -a_{1}(k) \\ -a_{1}(k) & -a_{2}(k) \\ \end{array} \right).\label{eq:K_matrix}$$ Here, the Fermi points are denoted by $\pm Q_{\alpha}$. Notice that $\mathbf{K}(k)$ is a symmetric matrix. In order to calculate finite-size corrections for the ground state and low energy excitations, we introduce the thermodynamic Bethe ansatz (TBA) [@Y-Y; @Takahashi], which provides a powerful and elegant way to study the thermodynamics of 1D integrable systems. It becomes convenient to analyze phase transitions and low-lying excitations in the presence of external fields at zero temperature. In the thermodynamic limit, the grand partition function is $Z=tr(\mathrm{e}^{-\cal{H}/T})=\mathrm{e}^{-G/T}$, where the Gibbs free energy is given by $G = E - HM^z - \mu n - TS$, and is written in terms of the magnetization $H$, the chemical potential $\mu$ and the entropy $S$ [@Takahashi]. Equilibrium states satisfy the condition of minimizing the Gibbs free energy with respect to particle and hole densities for the charge and spin degrees of freedom (more details are given in Refs. [@Lai1971; @Lai1973; @Takahashi; @Schlottmann1993; @Guan2007]). At zero temperature, the ground state properties are determined by the dressed energy equations $$\varepsilon_{\alpha}(k_{\alpha})=\varepsilon_{\alpha}^{(0)}(k_{\alpha})+\sum_{\beta=u,b}\int_{-Q_{\beta}}^{Q_{\beta}} K_{\alpha\beta}(k_{\alpha}-k_{\beta})\varepsilon_{\beta}(k_{\beta})dk_{\beta}, \qquad (\alpha=u,b), \label{eq:dressedEnergy}$$ where $\varepsilon_{\alpha}^{(0)}(k_{\alpha})$ are given by $$\left( \begin{array}{c} \varepsilon_{u}^{(0)}(k_{u}) \\ \varepsilon_{b}^{(0)}(k_{b}) \\ \end{array} \right)=\left( \begin{array}{c} k_{u}^{2} \\ 2k_{b}^{2}-\|c\|^{2}/2 \\ \end{array} \right).$$ 1D many-body systems are critical at $T=0$ and exhibit not only global scale invariance but local scale invariance too, i.e., conformal invariance. The conformal group is infinite dimensional and completely determines the conformal dimensions and correlation functions when the excitations are gapless [@Belavin1984]. Conformal invariance predicts that the energy per unit length has a universal finite-size scaling form that is characterized by the dimensionless number $C$, which is the central charge of the underlying Virasoro algebra [@Blote1986; @Affleck1986]. From the density distributions (\[eq:rho\_gen\]) and dressed energy equations (\[eq:dressedEnergy\]), the finite-size corrections to the ground state energy is given by $$\varepsilon_{0}=\varepsilon_{0}^{\infty}-\frac{C\pi}{6L^{2}}\sum_{\alpha=u,b}v_{\alpha}, \label{eq:energy_Lfinalground}$$ where $C=1$, and $v_{u}$ and $v_{b}$ are the velocities of unpaired fermions and bound pairs, respectively. They are defined as $$v_{\alpha}:=\pm\left.\frac{d\varepsilon_{\alpha}(k_{\alpha})}{dp_{\alpha}(k_{\alpha})}\right\|_{k_{\alpha}=\pm Q_{\alpha}}=\pm\frac{\varepsilon'_{\alpha}(\pm Q_{\alpha})}{p'_{\alpha}(Q_{\alpha})}=\pm\frac{\varepsilon'_{\alpha}(\pm Q_{\alpha})}{2\pi\rho_{\alpha}(\pm Q_{\alpha})}, \qquad (\alpha=u,b),$$ where prime denotes the derivative with respect to $k_{\alpha}$ and $p_{\alpha}(k_{\alpha})=\lim_{L\rightarrow\infty}2\pi z_{\alpha}^{L}(k_{\alpha})$. The term $\varepsilon_{0}^{\infty}$ represents the ground state energy in the thermodynamic limit, i.e., $N,L\rightarrow\infty$. In the strong coupling limit, exact expressions for the velocities can be found in Refs. [@Guan2007; @Guan2010]. Low-lying excitations and dressed charge equations {#sec:Finite-size_Ex} ================================================== Critical phenomena of critical systems are described by finite-size corrections for their low-lying excitations. The method we use to study correlation functions of the spin-1/2 Fermi gas with attractive interaction follows closely the method set out in Refs. [@Woynarovich1989; @Kawakami1991; @Frahm1991; @Hubbardbook]. The conformal dimensions of two-point correlation functions can be calculated from the elements of the dressed charge matrix $\mathbf{Z}$. Long distance asymptotics of various correlation functions are then examined through the dressed charge formalism at the $T=0$. Three types of low-lying excitations are considered in the calculations of finite-size corrections. Type 1 excitation is characterized by moving a particle close to the right or left Fermi points outside the Fermi sea. It is equivalent to changing the quantum numbers $I_{\alpha,j}$ close to $I_{\alpha}^{\pm}$ for unpaired fermions ($\alpha=u$) and bound pairs ($\alpha=b$). $I_{\alpha}^{\pm}$ characterize the Fermi points of each Fermi sea and are given by $I_{\alpha}^{+}=I_{\alpha}^{\mathrm{max}}+1/2$ and $I_{\alpha}^{-}=I_{\alpha}^{\mathrm{min}}-1/2$. The change in total momentum from Type 1 excitations is $$\Delta P=\frac{2\pi}{L}\sum_{\alpha=u,b}(N_{\alpha}^{+}-N_{\alpha}^{-}),$$ and the change in energy is $$\begin{aligned} \nonumber \Delta E &=& \frac{2\pi}{L}\sum_{\alpha=u,b}\frac{\varepsilon'_{\alpha}(Q_{\alpha}\|Q^{\pm})} {p'_{\alpha}(Q_{\alpha}\|Q^{\pm})}(N_{\alpha}^{+}+N_{\alpha}^{-}) \\ &=& \frac{2\pi}{L}\sum_{\alpha=u,b}v_{\alpha}(N_{\alpha}^{+}+N_{\alpha}^{-}).\end{aligned}$$ Here $N_{\alpha}^{+}\geq 0$ ($N_{\alpha}^{-}\geq 0$) stems from the change in distribution of quantum numbers close to the right (left) Fermi points. This type of excitation is commonly known as particle-hole excitation. Type 2 excitation arises from the change in total number of unpaired fermions or bound pairs. It is characterized by the change in quantum numbers $$N_{\alpha}=I_{\alpha}^{+}-I_{\alpha}^{-}, \qquad (\alpha=u,b),$$ i.e., $\Delta N_{\alpha}=N_{\alpha}^{\mathrm{excited}}-N_{\alpha}^{\mathrm{ground}}$. On the other hand, Type 3 excitation is caused by moving a particle from the left Fermi point to the right Fermi point and vice versa. This type of excitation is also known as backscattering. It is characterized by the quantum numbers $$\Delta D_{\alpha}=\frac{I_{\alpha}^{+}+I_{\alpha}^{-}}{2}, \qquad (\alpha=u,b), \label{eq:Dalpha}$$ while leaving $\Delta N_{\alpha}$ unchanged. All three types of excitations can be unified in the following form of the finite-size corrections for the energy and total momentum of the system $$\begin{aligned} \Delta E &=& \frac{2\pi}{L}\left(\frac{1}{4}\phantom{.}^{t}(\Delta N)^{t}(\mathbf{Z}^{-1})\mathbf{VZ}^{-1}\Delta N+ \phantom{.}^{t}(\Delta D)\mathbf{ZV}^{t}\mathbf{Z}\Delta D+\sum_{\alpha=u,b}v_{\alpha}(N_{\alpha}^{+}+N_{\alpha}^{-})\right), \label{eq:energy_Lfinal} \\ \Delta P &=& \frac{2\pi}{L}\left(\phantom{.}^{t}\Delta N\Delta D+N_{u}\Delta D_{u}+N_{b}\Delta D_{b}+\sum_{\alpha=u,b}v_{\alpha}(N_{\alpha}^{+}-N_{\alpha}^{-})\right).\end{aligned}$$ Here we use the notations $$\begin{aligned} && \nonumber \Delta N=\left( \begin{array}{c} \Delta N_{u} \\ \Delta N_{b} \\ \end{array} \right),\qquad \Delta D=\left( \begin{array}{c} \Delta D_{u} \\ \Delta D_{b} \\ \end{array} \right),\\ &&\mathbf{V}=\left( \begin{array}{cc} v_{u} & 0 \\ 0 & v_{b} \\ \end{array} \right),\qquad \mathbf{Z}=\left( \begin{array}{cc} Z_{uu}(Q_{u}) & Z_{ub}(Q_{b}) \\ Z_{bu}(Q_{u}) & Z_{bb}(Q_{b}) \\ \end{array} \right).\end{aligned}$$ The dressed charge equations are a set of four coupled integral equations that read $$\begin{aligned} Z_{uu}(k) &=& 1-\int_{-Q_{b}}^{Q_{b}}a_{1}(k-\lambda)Z_{ub}(\lambda)d\lambda, \label{eq:dressed_uu} \\ Z_{ub}(k) &=& -\int_{-Q_{u}}^{Q_{u}}a_{1}(k-\lambda)Z_{uu}(\lambda)d\lambda-\int_{-Q_{b}}^{Q_{b}}a_{2}(k-\lambda)Z_{ub}(\lambda)d\lambda, \label{eq:dressed_ub} \\ Z_{bu}(k) &=& -\int_{-Q_{b}}^{Q_{b}}a_{1}(k-\lambda)Z_{bb}(\lambda)d\lambda, \label{eq:dressed_bu} \\ Z_{bb}(k) &=& 1-\int_{-Q_{u}}^{Q_{u}}a_{1}(k-\lambda)Z_{bu}(\lambda)d\lambda-\int_{-Q_{b}}^{Q_{b}}a_{2}(k-\lambda)Z_{bb}(\lambda)d\lambda. \label{eq:dressed_bb}\end{aligned}$$ Quantum numbers $\Delta D_{u}$ and $\Delta D_{b}$ (\[eq:Dalpha\]) are chosen based on the conditions given in Eq. and also on the conditions that $\Delta D_{u}\equiv\Delta N_{u}/2$ (mod 1) and $\Delta D_{b}\equiv\Delta N_{b}/2$ (mod 1). Combining both conditions together with the definition given in Eq. yields $$\Delta D_{u}\equiv\frac{\Delta N_{u}+\Delta N_{b}}{2}\quad\mathrm{(mod\phantom{a}1)},\qquad \Delta D_{b}\equiv\frac{\Delta N_{u}}{2}\quad\mathrm{(mod\phantom{a}1)}.$$ When the external magnetic field $H$ is smaller than the critical field, spin excitations for this model are gapped. Once $H$ exceeds this critical field, spin excitations become gapless and the system becomes conformally invariant. In this spin polarized phase, spin degrees of freedom are suppressed due to the ferromagnetic nature of excess unpaired fermions under a magnetic field. Therefore, bound pairs and excess unpaired fermions form two Fermi seas which can be described by a two-component TLL at low temperatures. Hence conformal invariance results in a universal finite-size scaling form of the energy shown in Eqs. and , and a universal form of the critical exponents of two-point correlation functions between primary fields $\langle O^{\dagger}(x,t)O(x',t')\rangle$ which are determined by the finite-size corrections of the model. Multi-point correlation functions can be derived by taking the product of two-point correlation functions. When $T=0$, the correlation functions of 1D systems decay as the power of distance, but when $T>0$ they decay exponentially. Following the standard calculations in Ref. [@Hubbardbook], the conformal dimensions are given by $$\begin{aligned} 2\Delta_{u}^{\pm} &=& \left(Z_{uu}\Delta D_{u}+Z_{bu}\Delta D_{b}\pm \frac{Z_{bb}\Delta N_{u}-Z_{ub}\Delta N_{b}}{2\det Z}\right)^{2}+2N_{u}^{\pm}, \\ 2\Delta_{b}^{\pm} &=& \left(Z_{ub}\Delta D_{u}+Z_{bb}\Delta D_{b}\pm \frac{Z_{uu}\Delta N_{b}-Z_{bu}\Delta N_{u}}{2\det Z}\right)^{2}+2N_{b}^{\pm},\end{aligned}$$ where $N_{\alpha}^{\pm}$ ($\alpha=u,b$) characterize the descendent fields from the primary fields. General two-point correlation functions at $T=0$ take the form $$\langle O(x,t)O(0,0)\rangle=\frac{\exp(-2\mathrm{i}(N_{u}\Delta D_{u}+N_{b}\Delta D_{b})x)} {(x-\mathrm{i}v_{u}t)^{2\Delta^{+}_{u}}(x+\mathrm{i}v_{u}t)^{2\Delta^{-}_{u}} (x-\mathrm{i}v_{b}t)^{2\Delta^{+}_{b}}(x+\mathrm{i}v_{b}t)^{2\Delta^{-}_{b}}}. \label{eq:corrfunc_gen}$$ The exponential oscillating term in the asymptotic behavior comes from Type 3 excitations, i.e., backscattering. Quantum numbers for the low-lying excitations completely determine the nature of the asymptotic behavior of these correlations. Here we are only concerned with the $T=0$ case. The four dressed charge equations can be broken up into sets of two pairs. Eqs. and constitute one pair, whilst Eqs. and make up the other. Since we are interested in the strong coupling limit $\|c\|\gg 1$, both sets of equations can be solved iteratively up to accuracy $1/\|c\|$. Let us consider the first set. Substituting Eq. into Eq. and iterating the terms give $$\begin{aligned} \nonumber Z_{ub}(k) &=& -\int_{-Q_{u}}^{Q_{u}}d\lambda a_{1}(k-\lambda)+\int_{-Q_{b}}^{Q_{b}}d\lambda\int_{-Q_{u}}^{Q_{u}}d\lambda' a_{2}(k-\lambda)a_{2}(\lambda-\lambda') \nonumber\\ && -\int_{-Q_{u}}^{Q_{u}}d\lambda\int_{-Q_{b}}^{Q_{b}}d\lambda'\int_{-Q_{u}}^{Q_{u}}d\lambda^{\prime\prime} a_{1}(k-\lambda)a_{1}(\lambda-\lambda')a_{1}(\lambda'-\lambda^{\prime\prime})+\ldots \label{eq:dressed_ub_iter}\end{aligned}$$ The functions $a_{n}(k)$ have leading order $1/\|c\|$, hence we can ignore all terms that have two or more multiples of $a_{n}(k)$. This procedure yields $$\begin{aligned} \nonumber Z_{ub}(Q_{b}) &\approx& -\int_{-Q_{u}}^{Q_{u}}d\lambda a_{1}(Q_{b}-\lambda) \approx -\frac{4Q_{u}}{\pi \|c\|}.\end{aligned}$$ Substituting Eq. into Eq. , we obtain $$\begin{aligned} Z_{uu}(Q_{u}) &=& 1+\int_{-Q_{b}}^{Q_{b}}d\lambda\int_{-Q_{u}}^{Q_{u}}d\lambda' a_{1}(Q_{u}-\lambda)a_{1}(\lambda-\lambda')+\ldots \\ &\approx& 1\end{aligned}$$ -- -- -- -- Next, we consider the second set of equations. Repeating the same arguments as before, Eq. at the Fermi point $Q_{b}$ becomes $$\begin{aligned} \nonumber Z_{bb}(Q_{b}) &=& 1-\int_{-Q_{b}}^{Q_{b}}d\lambda a_{2}(Q_{b}-\lambda)+\int_{-Q_{u}}^{Q_{u}}d\lambda\int_{-Q_{b}}^{Q_{b}}d\lambda' a_{1}(Q_{b}-\lambda)a_{1}(\lambda-\lambda') \\ && \nonumber +\int_{-Q_{b}}^{Q_{b}}d\lambda\int_{-Q_{b}}^{Q_{b}}d\lambda' a_{2}(Q_{b}-\lambda)a_{2}(\lambda-\lambda')+\ldots \\ &\approx& 1-\frac{2Q_{b}}{\pi \|c\|}.\end{aligned}$$ Eq. at the Fermi point $Q_{u}$ then reads $$\begin{aligned} \nonumber Z_{bu}(Q_{u}) &=& -\int_{-Q_{b}}^{Q_{b}}d\lambda a_{1}(Q_{u}-\lambda)+\int_{-Q_{b}}^{Q_{b}}d\lambda\int_{-Q_{b}}^{Q_{b}}d\lambda' a_{1}(Q_{u}-\lambda)a_{2}(\lambda-\lambda')+\ldots \\ &\approx& -\frac{4Q_{b}}{\pi \|c\|}.\end{aligned}$$ From Ref. [@Guan2007], the Fermi points in the strongly attractive limit are given by $$\begin{aligned} Q_{u} &\approx& \pi n_{f}P\left(1+\frac{2(1-P)}{\|\gamma\|}\right), \\ Q_{b} &\approx& \frac{\pi n_{f}(1-P)}{4}\left(1+\frac{(1-P)}{2\|\gamma\|}+\frac{2P}{\|\gamma\|}\right),\end{aligned}$$ where $n_{f}=N_{f}/L$ is the density of fermions per unit length, $\gamma=c/n_{f}$ is the dimensionless interaction parameter and $P=(N_{\uparrow}-N_{\downarrow})/N_{f}=N_{u}/N_{f}$ is the polarization. Inserting these relations into the expressions for dressed charges, we obtain $$\begin{aligned} Z_{uu}(Q_{u}) &\approx& 1, \qquad Z_{ub}(Q_{b}) \approx -\frac{4P}{\|\gamma\|}, \nonumber \\ Z_{bu}(Q_{u}) &\approx& -\frac{(1-P)}{\|\gamma\|}, \qquad Z_{bb}(Q_{b}) \approx 1-\frac{(1-P)}{2\|\gamma\|}.\label{eq:dressedCharges}\end{aligned}$$ In FIG. \[fig:Z\], the dressed charges are numerically calculated and plotted against polarization for different values of interaction strength $\|\gamma\|$. In the strong coupling limit, the external magnetic field $H$ is related to the polarization as $$H\approx\frac{n^{2}\|\gamma\|^{2}}{2}+2\pi^{2}n^{2}P^{2}\left(1+\frac{4(1-P)}{\|\gamma\|}-\frac{4P}{3\|\gamma\|}\right) -\frac{\pi^{2}n^{2}(1-P)^{2}}{8}\left(1+\frac{4P}{\|\gamma\|}\right).$$ With this relation, we can evaluate the dressed charges for different values of $H$. From the expressions for the dressed charges in Eq. , the conformal dimensions $\Delta_{\alpha}^{\pm}$ in terms of polarization are given by $$\begin{aligned} 2\Delta_{u}^{\pm} &\approx& \left(\Delta D_{u}\pm\frac{\Delta N_{u}}{2}\right)^{2}-\frac{8P}{\|\gamma\|}\left(\Delta D_{u}\pm\frac{\Delta N_{u}}{2}\right)\left(\Delta D_{b}\mp\frac{\Delta N_{b}}{2}\right)+2N_{u}^{\pm}, \\ \nonumber 2\Delta_{b}^{\pm} &\approx& \left(1-\frac{(1-P)}{\|\gamma\|}\right)\left(\Delta D_{b}\pm\frac{\Delta N_{b}}{2}\right)^{2} \\ && -\left(\frac{8P}{\|\gamma\|}\Delta D_{u}\mp\frac{(1-P)}{\|\gamma\|}\Delta N_{u}\right)\left(\Delta D_{b}\pm\frac{\Delta N_{b}}{2}\right)+2N_{b}^{\pm}.\end{aligned}$$ Correlation functions at zero temperature {#sec:Corr-func} ========================================= Here we consider 4 types of correlation functions, namely the single particle Green’s function $G_{\uparrow}(x,t)$, charge density correlation function $G_{nn}(x,t)$, spin correlation function $G^{z}(x,t)$, and pair correlation function $G_{p}(x,t)$. Each correlation function is derived based on the choice of $\Delta N_{u}$ and $\Delta N_{b}$. The one particle Green’s function, which is also called the Fermi-field (FF) correlation function in some literature, decays exponentially when the external magnetic field is not strong enough to overcome the gap associated with the breaking of bound states [@Bogoliubov1988; @Bogoliubov1989; @Bogolyubov1990; @Bogolyubov1992]. Once in the gapless phase, i.e., when $H_{c1}<H<H_{c2}$ where $H_{c1}$ and $H_{c2}$ are the critical fields mentioned in Ref. [@Guan2007], every correlation function at zero temperature decays spatially as some form of power law [@Belavin1984; @Blote1986; @Affleck1986; @Cardy1986; @Izergin1989]. $G_{\uparrow}(x,t)$ is characterized by $(\Delta N_{u},\Delta N_{b})=(1,0)$ which in turn allows quantum numbers $\Delta D_{u}\in\mathbb{Z}+1/2$ and $\Delta D_{b}\in\mathbb{Z}+1/2$. The leading terms are then given by $$\begin{aligned} \nonumber G_{\uparrow}(x,t) &=& \langle\psi_{\uparrow}^{\dagger}(x,t)\psi_{\uparrow}(0,0)\rangle \\ &\approx& \frac{A_{\uparrow,1}\cos\left(\pi(n_{\uparrow}-2n_{\downarrow})x\right)} {\|x+\mathrm{i}v_{u}t\|^{\theta_{1}}\|x+\mathrm{i}v_{b}t\|^{\theta_{2}}} +\frac{A_{\uparrow,2}\cos\left(\pi n_{\downarrow}x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{3}}\|x+\mathrm{i}v_{b}t\|^{\theta_{4}}},\end{aligned}$$ where the critical exponents are given by $$\begin{aligned} \theta_{1} &\approx& 1+\frac{4P}{\|\gamma\|}, \qquad \theta_{2} \approx \frac{1}{2}-\frac{(1-P)}{2\|\gamma\|}+\frac{4P}{\|\gamma\|}, \nonumber \\ \theta_{3} &\approx& 1-\frac{4P}{\|\gamma\|}, \qquad \theta_{4} \approx \frac{1}{2}-\frac{(1-P)}{2\|\gamma\|}-\frac{4P}{\|\gamma\|}.\end{aligned}$$ The first term in $G_{\uparrow}(x,t)$ comes from $(\Delta D_{u},\Delta D_{b})=(1/2,-1/2)$ and the second term comes from $(\Delta D_{u},\Delta D_{b})=(1/2,1/2)$. The constants $A_{\uparrow,1}$ and $A_{\uparrow,2}$ cannot be derived from the finite-size corrections for low-lying excitations. Here we only aim to evaluate the long distance asymptotics of these correlation functions. Instead of using $N_{u}$ and $N_{b}$ in the oscillation term, we choose to use $n_{\uparrow}=N_{\uparrow}/L$ and $n_{\downarrow}=N_{\downarrow}/L$ to elucidate the imbalance in the densities of spin-up and spin-down fermions. Both sets of variables are related by the relations $N_{u}=N_{\uparrow}-N_{\downarrow}$ and $N_{s}=N_{\downarrow}$. Next we consider the charge density correlation function $G_{nn}(x,t)$ together with the spin correlation function $G^{z}(x,t)$. Both of these correlation functions are characterized by the set of quantum numbers $(\Delta N_{u},\Delta N_{b})=(0,0)$ which allows quantum numbers $\Delta D_{u}\in\mathbb{Z}$ and $\Delta D_{b}\in\mathbb{Z}$. The leading terms are given by $$\begin{aligned} \nonumber G_{nn}(x,t) &=& \langle n(x,t)n(0,0)\rangle \\ &\approx& \nonumber n^{2}+\frac{A_{nn,1}\cos\left(2\pi(n_{\uparrow}-n_{\downarrow})x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{1}}} +\frac{A_{nn,2}\cos\left(2\pi n_{\downarrow}x\right)}{\|x+\mathrm{i}v_{b}t\|^{\theta_{2}}} \\ && +\frac{A_{nn,3}\cos\left(2\pi(n_{\uparrow}-2n_{\downarrow})x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{3}}\|x+\mathrm{i}v_{b}t\|^{\theta_{4}}}, \\ \nonumber G^{z}(x,t) &=& \langle S^{z}(x,t)S^{z}(0,0)\rangle \\ &\approx& \nonumber (m^{z})^{2} +\frac{A_{z,1}\cos\left(2\pi(n_{\uparrow}-n_{\downarrow})x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{1}}} +\frac{A_{z,2}\cos\left(2\pi n_{\downarrow}x\right)}{\|x+\mathrm{i}v_{b}t\|^{\theta_{2}}} \\ && +\frac{A_{z,3}\cos\left(2\pi(n_{\uparrow}-2n_{\downarrow})x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{3}}\|x+\mathrm{i}v_{b}t\|^{\theta_{4}}},\end{aligned}$$ where the operators $n(x,t)$ and $S^{z}(x,t)$ are given in terms of the fields as $$\begin{aligned} n(x,t) &=& \psi^{\dagger}_{\uparrow}(x,t)\psi_{\uparrow}(x,t)+\psi^{\dagger}_{\downarrow}(x,t)\psi_{\downarrow}(x,t), \\ S^{z}(x,t) &=& \frac{1}{2}\left(\psi^{\dagger}_{\uparrow}(x,t)\psi_{\uparrow}(x,t)-\psi^{\dagger}_{\downarrow}(x,t)\psi_{\downarrow}(x,t)\right).\end{aligned}$$ The critical exponents for asymptotic expressions of $G_{nn}(x,t)$ and $G^{z}(x,t)$ are $$\begin{aligned} \theta_{1} &\approx& 2, \qquad \theta_{2} \approx 2-\frac{2(1-P)}{\|\gamma\|}, \nonumber \\ \theta_{3} &\approx& 2+\frac{16P}{\|\gamma\|}, \qquad \theta_{4} \approx 2-\frac{2(1-P)}{\|\gamma\|}+\frac{16P}{\|\gamma\|}.\end{aligned}$$ The constant terms for $G_{nn}(x,t)$ and $G^{z}(x,t)$ come from the choice of quantum numbers $(\Delta D_{u},\Delta D_{b})=(0,0)$. The second, third and fourth terms arise from the choices $(1,0)$, $(0,1)$ and $(-1,1)$, respectively. Finally we consider the pair correlation function $G_{p}(x,t)$. This correlation function is characterized by the set of quantum numbers $(\Delta N_{u},\Delta N_{b})=(0,1)$ which allows quantum numbers $\Delta D_{u}\in\mathbb{Z}+1/2$ and $\Delta D_{b}\in\mathbb{Z}$. The leading terms are $$\begin{aligned} \nonumber G_{p}(x,t) &=& \langle\psi_{\uparrow}^{\dagger}(x,t)\psi_{\downarrow}^{\dagger}(x,t)\psi_{\uparrow}(0,0)\psi_{\downarrow}(0,0)\rangle \\ &\approx& \frac{A_{p,1}\cos\left(\pi(n_{\uparrow}-n_{\downarrow})x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{1}}\|x+\mathrm{i}v_{b}t\|^{\theta_{2}}} +\frac{A_{p,2}\cos\left(\pi(n_{\uparrow}-3n_{\downarrow})x\right)}{\|x+\mathrm{i}v_{u}t\|^{\theta_{3}}\|x+\mathrm{i}v_{b}t\|^{\theta_{4}}},\end{aligned}$$ where the critical exponents are given by $$\begin{aligned} \theta_{1} &\approx& \frac{1}{2}, \qquad \theta_{2} \approx \frac{1}{2}-\frac{(1-P)}{2\|\gamma\|}, \nonumber \\ \theta_{3} &\approx& \frac{1}{2}+\frac{8P}{\|\gamma\|}, \qquad \theta_{4} \approx \frac{5}{2}-\frac{5(1-P)}{2\|\gamma\|}+\frac{8P}{\|\gamma\|}.\end{aligned}$$ The first term in $G_{p}(x,t)$ arises from the choice of quantum numbers $(\Delta D_{u},\Delta D_{b})=(1/2,0)$, whilst the second term arises from the choice $(\Delta D_{u},\Delta D_{b})=(1/2,-1)$. The leading order for the long distance asymptotics of the pair correlation function $G_{p}(x,t)$ oscillates with wave number $\Delta k_F$, where $\Delta k_F =\pi(n_{\uparrow}-n_{\downarrow})$. Meanwhile, the leading order for the spin correlation function $G^{z}(x,t)$, which can also be thought of as the correlation of the density difference between spin-up and spin-down fermions, oscillates twice as fast with wave number $2\Delta k_F$. The oscillations in $G_{p}(x,t)$ and $G^{z}(x,t)$ are caused by an imbalance in the densities of spin-up and spin-down fermions, i.e., $n_{\uparrow}-n_{\downarrow}$, which gives rise to a mismatch in Fermi surfaces between both species of fermions. These spatial oscillations share a similar signature as the Larkin-Ovchinikov (LO) pairing phase [@Larkin1965]. Our findings of the wave numbers agree with those discovered through DMRG [@Feiguin2007; @Tezuka2008; @Rizzi2008], QMC [@Batrouni2008] and mean field theory [@Liu2008]. Though from conformal field theory, we see clearly that the spatial oscillation terms in the pair and spin correlations are a consequence of Type 3 excitations, i.e., backscattering for bound pairs and unpaired fermions. A comparison between our results and the results from numerical methods in Refs. [@Feiguin2007; @Tezuka2008; @Rizzi2008; @Batrouni2008] suggest that the coefficient $A_{p,1}$ is very much larger than the coefficient $A_{p,2}$ because the frequency of the oscillations in numerical studies of $G_{p}(x,t)$ is almost identical to $\pi(n_{\uparrow}-n_{\downarrow})$. This observation also applies to $G^{z}(x,t)$, where $A_{z,2}$ and $A_{z,3}$ are much smaller when compared with $A_{z,1}$. [c]{} The correlation functions in momentum space can be derived by taking the Fourier transform of their counterparts in position space. From Refs. [@Frahm1991; @Hubbardbook], the Fourier transform of equal-time correlation functions of the form $$g(x,t=0^{+}) = \frac{\exp(ik_{0}x)}{(x-\mathrm{i}0)^{2\Delta^{+}}(x+\mathrm{i}0)^{2\Delta^{-}}},$$ where $\Delta^{\pm}=\Delta_{u}^{\pm}+\Delta_{b}^{\pm}$ is given by $$\widetilde{g}(k\approx k_{0})\sim[\mathrm{sign}(k-k_{0})]^{2s}\|k-k_{0}\|^{\nu}.$$ The conformal spin of the operator is $s=\Delta^{+}-\Delta^{-}$ and the exponent $\nu$ is expressed in terms of the conformal dimensions as $\nu=2(\Delta^{+}+\Delta^{-})-1$. Hence the equal time correlation functions near the singularities $k_{0}$ for the one particle Green’s function, charge density, spin and bound pairs are $$\begin{aligned} \widetilde{G}_{\uparrow}(k) &\sim& [\mathrm{sign}(k-\pi(n_{\uparrow}-2n_{\downarrow}))]^{2s_{\uparrow}}\|k-\pi(n_{\uparrow}-2n_{\downarrow})\|^{\nu_{\uparrow}}, \label{eq:corr_mom1} \\ \widetilde{G}_{nn}(k) &\sim& [\mathrm{sign}(k-2\pi(n_{\uparrow}-n_{\downarrow}))]^{2s_{nn}}\|k-2\pi(n_{\uparrow}-n_{\downarrow})\|^{\nu_{nn}}, \\ \widetilde{G}^{z}(k) &\sim& [\mathrm{sign}(k-2\pi(n_{\uparrow}-n_{\downarrow}))]^{2s_{z}}\|k-2\pi(n_{\uparrow}-n_{\downarrow})\|^{\nu_{z}}, \\ \widetilde{G}_{p}(k) &\sim& [\mathrm{sign}(k-\pi(n_{\uparrow}-n_{\downarrow}))]^{2s_{p}}\|k-\pi(n_{\uparrow}-n_{\downarrow})\|^{\nu_{p}}, \label{eq:corr_mom4}\end{aligned}$$ where the exponents are given by $$\begin{aligned} && 2s_{\uparrow} \approx 1+\frac{4P}{\|\gamma\|}-\frac{(1-P)}{\|\gamma\|},\qquad \nu_{\uparrow} \approx \frac{1}{2}+\frac{8P}{\|\gamma\|}-\frac{(1-P)}{2\|\gamma\|}, \\ && 2s_{nn} = 2s_{z} \approx 0,\qquad \nu_{nn} = \nu_{z} \approx 1, \\ && 2s_{p} \approx 0,\qquad \nu_{p} \approx -\frac{(1-P)}{2\|\gamma\|}.\end{aligned}$$ We would like to stress that the momentum space correlation functions derived in Eqs. – are only accurate when the momenta $k$ are within the proximity of the wave numbers $k_{0}$, i.e., when $k\approx k_{0}$. FIG. \[fig:Gp\] plots $\widetilde{G}_{p}(k)$ against $k$ as polarization $P$ varies between 0 to 0.8. This figure is in qualitative agreement with the ones given in Refs. [@Feiguin2007; @Batrouni2008; @Rizzi2008]. We stress again that our plot is accurate only within the vicinity of the singularity, i.e., when $k$ approaches $\pi(n_{\uparrow}-n_{\downarrow})$. We plotted $\widetilde{G}_{p}(k)$ for the entire domain $k\in(0,\pi)$ so that readers can visualize the curves more easily. Conclusion {#sec:Conclusion} ========== In conclusion, we investigated various zero-temperature correlation functions for the spin-1/2 Fermi gas with attractive interaction. We derived the finite-size corrections for ground state and low-lying excitations of the model. Using conformal field theory, critical exponents of the correlation functions were given in terms of polarization and interaction strength. We found that the leading terms of the pair correlation function and the spin correlation function oscillate with frequencies $\pi(n_{\uparrow}-n_{\downarrow})$ and $2\pi(n_{\uparrow}-n_{\downarrow})$, respectively. We also found that backscattering between the Fermi points of bound pairs and unpaired fermions results in a 1D analog of the FFLO state and displays a microscopic origin of the FFLO nature. Furthermore, we showed that there is a peak in the pair correlation function in momentum space at $k=\pi(n_{\uparrow}-n_{\downarrow})$ which confirms the oscillation frequency. In the spin polarized phase, these correlation functions exhibit spatial oscillations with a power-law decay. This critical behaviour can be viewed as an analogy to long range order in 1D, i.e., the power law decay of the pair correlation function which is regarded as evidence of a superconducting/superfluid state. We also like to mention that from the dressed charge formalism, the asymptotic behavior of the correlation functions derived in this paper can be numerically obtained with high accuracy for arbitrary interaction strength. Additionally, by considering weakly perturbed inter-tube interactions or inter-lattice interactions (1D fermionic Hubbard model), quasi-1D correlations in the spin polarized phase can be calculated from perturbation theory [@Bogoliubov1989]. This provides a promising opportunity to estimate the critical temperature for high-Tc superconductors/superfluids by studying 1D to 3D trapped cold atoms. This work is supported by the Australian Research Council. We thank M. T. Batchelor, F. H. L. Essler, F. Heidrich-Meisner and K. Sakai for helpful discussions. XWG thanks T.-L. Ho for stimulating discussions to initiate this topic and acknowledges the Ohio State University for their kind hospitality. [99]{} P. Fulde and R. A. Ferrell, Phys. Rev. [135]{}, A550 (1964) A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP [20]{}, 762 (1965) C. N. Yang, Phys. Rev. Lett. [19]{}, 1312 (1967) M. Gaudin, Phys. Lett. A [24]{}, 55 (1967) G. Orso, Phys. Rev. Lett. [98]{}, 070402 (2007) H. Hu, X.-J. Liu and P. D. Drummond, Phys. Rev. Lett. 98, 070403 (2007) A. E. Feiguin and F. Heidrich-Meisner, Phys. Rev. B [76]{}, 220508(R) (2007) M. Tezuka and M. Ueda, Phys. Rev. Lett. [100]{}, 110403 (2008) M. Rizzi, M. Polini, M. A. Cazalilla, M. R. Bakhtiari, M. P. Tosi and R. Fazio, Phys. Rev. B [77]{}, 245105 (2008) F. Heidrich-Meisner, A. E. Feiguin, U. Schollwöck and W. Zwerger, Phys. Rev. A [81]{}, 023629 (2010) A. Lüscher, R. M. Noack and A. M. Läuchli, Phys. Rev. A [78]{}, 013637 (2008) G. G. Batrouni, M. H. Huntley, V. G. Rousseau and R. T. Scalettar, Phys. Rev. Lett. [100]{}, 116405 (2008) J. Kinunnen, L. M. Jensen and P. Törmä, Phys. Rev. Lett. [96]{}, 110403 (2006) X.-J. Liu, H. Hu and P. D. Drummond, Phys. Rev. A [78]{}, 023601 (2008) J. M. Edge and N. R. Cooper, Phys. Rev. Lett. [103]{}, 065301 (2009) E. Zhao and W. V. Liu, Phys. Rev. A [78]{}, 063605 (2008) K. Yang, Phys. Rev. B [63]{}, 140511(R) (2001) N. M. Bogoliubov and V. E. Korepin, Mod. Phys. Lett. B [1]{}, 349 (1988) N. M. Bogoliubov and V. E. Korepin, Int. J. Mod. Phys. B [3]{}, 427 (1989) N. M. Bogolyubov and V. E. Korepin, Theor. Math. Phys. [82]{}, 231 (1990) N. M. Bogolyubov and V. E. Korepin, Proceedings of the Steklov Institute of Mathematics [2]{}, 47 (1992) X. W. Guan, M. T. Batchelor, C. Lee and M. Bortz, Phys. Rev. B [76]{}, 085120 (2007) M. Casula, D M. Ceperley and E. J. Mueller, Phys. Rev. A [78]{}, 033607 (2008) P. Kakashvili and C. J. Bolech, Phys. Rev. A, 79, 041603(R) (2009) T. Iida and M. Wadati, J. Phys. Soc. Jpn, [77]{}, 024006 (2008) F. Woynarovich and K. Penc, Z. Phys. B [85]{}, 269 (1991) E. Zhao, X. W. Guan, W. V. Liu, M. T. Batchelor and M. Oshikawa, Phys. Rev. Lett. 103, 140404 (2009). Y. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur and E. J. Mueller, Nature [467]{}, 567 (2010) C. N. Yang and C. P. Yang, J. Math. Phys. [10]{}, 1115 (1969) M. Takahashi, [Thermodynamics of One-Dimensional Solvable Models]{}, Cambridge University Press (1999) C. K. Lai, Phys. Rev. Lett. [26]{}, 1472 (1971) C. K. Lai, Phys. Rev. A [8]{}, 2567 (1973) P. Schlottmann, J. Phys.: Condens. Matter [5]{} 5869 (1993) A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B [241]{}, 333 (1984) H. W. Blöte, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. [56]{}, 742 (1986) I. Affleck, Phys. Rev. Lett. [56]{}, 746 (1986) X. W. Guan, J.-Y. Lee, M. T. Batchelor, X.-G. Yin and S. Chen, Phys. Rev. A [82]{}, 021606(R) (2010) J. L. Cardy, Nucl. Phys. B [270]{} \[FS16\], 186 (1986) A. G. Izergin, V. E. Korepin and N. Yu Reshetikhin, J. Phys. A: Math. Gen. [22]{}, 2615 (1989) F. Woynarovich, J. Phys. A [22]{}, 4243 (1989) N. Kawakami and S. K. Yang, J. Phys. C [3]{}, 5983 (1991) H. Frahm and V. E. Korepin, Phys. Rev. B [43]{}, 5653 (1991) F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper and V. E. Korepin, [The One-Dimensional Hubbard Model]{}, Cambridge University Press (2005)
--- abstract: 'We study the computational complexity of reachability, coverability and inclusion for extensions of context-free commutative grammars with integer counters and reset operations on them. Those grammars can alternatively be viewed as an extension of communication-free Petri nets. Our main results are that reachability and coverability are inter-reducible and both NP-complete. In particular, this class of commutative grammars enjoys semi-linear reachability sets. We also show that the inclusion problem is, in general, coNEXP-complete and already $\Pi_2^\text{P}$-complete for grammars with only one non-terminal symbol. Showing the lower bound for the latter result requires us to develop a novel $\Pi_2^\text{P}$-complete variant of the classic subset sum problem.' address: - 'Max Planck Institute for Software Systems (MPI-SWS), Germany' - 'LSV, CNRS & ENS Cachan, Université Paris-Saclay, France' author: - Dmitry Chistikov - Christoph Haase - Simon Halfon bibliography: - 'bibliography.bib' title: 'Context-Free Commutative Grammars with Integer Counters and Resets' --- context-free commutative grammars ,communication-free Petri nets ,reset nets ,vector addition systems with states ,Presburger arithmetic ,subset sum
--- bibliography: - '../../Focusing\_and\_Channeling\_in\_Crystals/chan02.bib' ---
--- abstract: 'In this paper non-transversal intersection of the free and fixed boundary is shown to hold in any dimension for obstacle problems generated by fully nonlinear uniformly elliptic operators. Moreover, $C^1$ regularity results of the free boundary are obtained and a classification of blow-up solutions is given.' author: - Emanuel Indrei bibliography: - 'ngonref2.bib' title: 'Boundary regularity and non-transversal intersection for the fully nonlinear obstacle problem' --- Introduction ============ The dynamics of the free boundary are considered for strong $L^n$-solutions of the following PDE $$\label{eqF} \begin{cases} F(D^{2}u)=\chi_\Omega & \text{a.e. in }B_{1}^{+}\\ u=0 & \text{on }B'_{1} \end{cases}$$ .2in where $u \in W^{2,n}(B_{1}^{+})$, $F$ is a convex $C^1$ fully nonlinear uniformly elliptic operator, $\Omega$ is an open set and the free boundary is $\Gamma=\partial \Omega \cap B_1^+$. Under the structural assumptions on $F$, $u \in W^{2,p}(B_{1}^{+})$ for all $p<\infty$, and $u$ satisfies in the viscosity sense [@MR1376656]. It was recently shown in [@MR3513142] that $u \in C^{1,1}(B_{1/2}^{+})$, see also [@MR3542613] for the interior case. The class of bounded solutions is denoted by $P_1^+(0,M, \Omega),$ where $\|\|u\|\|_{L^\infty(B_1^+)} \le M$. In what follows, the tangential touch problem is considered for $$\Omega = \big(\{u \ne 0\} \cup \{\nabla u \neq 0\} \big) \cap \{x_n>0\}\subset \mathbb{R}_+^n.$$ It has been conjectured that the free boundary intersects the fixed boundary non-transversally and in two dimensions this was proved in [@MR3513142] (partial results have also been obtained in [@MR2065018]). The case of the Laplacian was treated in [@MR1950478; @MR1359745]. In this paper, the following theorem is established. (Non-transversal intersection, §\[blowup\]) \[tt\] There exists $r_0>0$ and a modulus of continuity $\omega$ such that $$\Gamma(u) \cap B_{r_0}^+ \subset \{x: x_n \le \omega(\|x'\|)\|x'\|\}$$ for all $u \in P_1^+(0,M, \Omega)$ provided $0 \in \overline{\Gamma(u)}$. As a consequence, if the solution is non-negative, then the free boundary is $C^1$ in a neighborhood of the origin. (Regularity, §\[regularity\]) \[c1r\] Let $u \in P_1^+(0, M, \Omega)$ be non-negative and $0 \in \overline{\Gamma(u)}$. There exists $r_0>0$ such that $\Gamma$ is the graph of a $C^1$ function in $B_{r_0}^+$. Moreover, the methods developed to prove the theorems lead to a classification of blow-up limits $$\lim_{r\rightarrow 0^+} \frac{u(rx)}{r^2}$$ which in the interior case was carried out in [@MR1745013]. (Uniqueness of Blow-Ups, §\[blowup\]) \[cul\] Suppose $u \in P_1^+(0,M, \Omega)$. If $0 \in \overline{\{u \neq 0\}}$ and $\nabla u(0)=0$, then the blow-up limit of $u$ at the origin has the form $$u_0(x)=ax_1x_n+bx_n^2$$ for $a, b \in \mathbb{R}$. A similar regularity result holds also in the two-phase case provided that the coincidence set satisfies a certain density assumption: given a set $E$, let $MD(E)$ denote the smallest distance between two parallel hyperplanes containing $E$. To measure thickness of the coincidence set corresponding to a solution $u \in P_1^+(0,M, \Omega)$, let $$\delta_r(u,x)=\frac{MD\big((B_1^+ \setminus \Omega)\cap B_r(x)\big)}{r}.$$ If there exists $\epsilon_0>0$ such that $$\delta_r(u,x^0) \ge \epsilon_0$$ for all $r>0$ and $x^0 \in \Gamma \cap B_1^+$, then the free boundary is $C^1$, see Corollary \[reg\]. In particular, Lipschitz free boundaries are $C^1$ and this is optimal in the sense that in general the free boundary is not $C^{1, \text{Dini}}$ [@MR1392033; @MR1698524; @MR1950478]. The regularity of the free boundary in the classical obstacle problem has been an area of intense research. If the solution is non-negative, it was shown in [@MR0454350] that under a density condition on the coincidence set, the free boundary is $C^1$ and higher regularity follows from [@MR0440187]; in general, there exist singularities [@MR0516201] and the structure of the singular set appeared in [@MR1658612]. Note that there is no density assumption in Theorem \[c1r\]. It is of interest to understand the most general conditions on the operator, $\Omega$, and boundary data to which this extends. In [@MR3357696; @MR3198649], the authors employed a novel method to handle the fully nonlinear uniformly elliptic case based on a harmonic analysis technique which appeared in [@MR2999297] where the authors proved optimal regularity for the solution of the no-sign obstacle problem under the weakest possible assumptions. This was further developed in [@MR3542613; @MR3625855] to obtain sharp regularity results for more general equations and also improve on existing results for the classical semilinear equation. The analysis is developed in such a way as to consider the most general configurations. In the above description, solutions with zero Dirichlet boundary data on the hyperplane were considered. If this is not the case, the free boundary may approach the fixed boundary at an angle [@MR2281197]. Several variations of the classical tangential touch problem in recent years have appeared in [@MR2874960; @MR2727672; @MR2267752; @MR2237208; @MR2142064]. It is of interest to study the largest function space for which uniform results hold. The $C^1$ regularity proved in this paper for the case when $u \ge 0$ is natural when considering the historical aspect of the problem where the solution represents the pressure in a liquid: consider water which penetrates a porous medium and divides it into a wet and dry part separated by an interface. The geometry of this interface subject to various boundary conditions was studied in [@MR693780]. However, mathematically, the no-sign case is more delicate since one has to understand different phases. For instance, blow-up sequences producing limits of the form $ax_1x_n+bx_n^2$ can be excluded in the one phase case. The idea of the proof for tangential touch is to understand the configuration of a blow-up solution which is not a half-space solution and connect it with the interior of the $0$-level set of $u$: if $\partial (\text{int} \{u=0\})$ intersects any ball around the origin, then all blow-up solutions must be half-space solutions. In [@MR3513142], the dimensional constraint is a necessary component of the proof since it relies on the fact that if $$u_j \rightarrow u_0=ax_1x_2+bx_2^2,$$ as $j \rightarrow \infty$, then $$\|\nabla u_j\|>0$$ in $B_r^+ \setminus B_\delta^+$ for $\delta<r$ and $j$ large; in higher dimensions this is not the case: e.g. consider for $t>0$, $z_t=(0,t,0)$ so that $$\nabla u_0(z_t)=0.$$ In the subsequent pages, the following idea is developed to circumvent this difficulty: given any $r>0$, take any cylinder oriented in the $x_1$-direction in $B_r^+$. Then there exist points in the cylinder whose $x_1$-coordinate is less than $-r/2$, such that $$\|\nabla u_j(x)\| \ge c$$ for $j$ large, where the constant is independent of $j$ and of the cylinder. As a result, if there exist elements of $\partial (\text{int} \{u=0\})$ inside $B_r^+$, then one can prove a monotonicity property to obtain information about the growth of the function in the $x_1$-direction inside this cylinder: suppose there exist non-negative constants $\epsilon, C$ such that $$C \partial_e u -u \ge -\epsilon$$ in $B_r^+$, then $$C \partial_e u - u \ge 0$$ in $B_{\frac{r}{2}}^+$, provided $\epsilon$ is small enough (see Lemma \[m\]). The analysis results in the following statement: either re-scalings converge to half-space solutions, or a certain regularity property holds, and this leads to the classification of blow-up limits. For the regularity, the idea is to show that if the solution is non-negative and there is contact between the fixed and free boundary, then the solution is close to a half-space solution. Thereafter, it follows that it is Lipschitz away from the origin, and hence $C^1$ by interior results and since tangential touch holds, the free normal converges to $e_n$. Preliminaries {#pre} ============= In what follows, $F$ is assumed to satisfy the following structural conditions. - $F(0)=0$. - $F$ is uniformly elliptic with ellipticity constants $\lambda_{0}$, $\lambda_{1}>0$ such that $$\mathcal{P}^{-}(M-N)\le F(M)-F(N)\le\mathcal{P}^{+}(M-N),$$ where $M$ and $N$ are symmetric matrices and $\mathcal{P}^{\pm}$ are the Pucci operators $$\mathcal{P}^{-}(M):=\inf_{\lambda_{0} \le N\le\lambda_{1}} \text{tr}(NM),\qquad\mathcal{P}^{+}(M):=\sup_{\lambda_{0}\le N\le\lambda_{1}}\text{tr} (NM).$$ - $F$ is convex and $C^1$. Let $\Omega$ be an open set. A continuous function $u$ belongs to $P_r^+(0,M, \Omega)$ if $u$ satisfies in the viscosity sense:\ 1. $F(D^2 u)=\chi_\Omega$ a.e. in $B_r^+$;\ 2. $\|\|u\|\|_{L^\infty(B_r^+)} \le M$;\ 3. $u=0$ on $\{x_n=0\} \cap \overline{B_1^+}=:B'_{1}$.\ In [@MR3513142] it was shown that $W^{2,p}$ solutions are $C^{1,1}$. Furthermore, given $u \in P_r^+(0,M, \Omega)$, the free boundary is denoted by $ \Gamma=\partial \Omega \cap B_r^+ $ and $ \Gamma_i=\partial (\text{int} \{u=0\}) \cap B_r^+. $ A cylinder with respect to the $e_1$-axis is denoted by $$S_{(\alpha, \beta)}(e_1)=\{(x_1,x'',x_n): (x_n-\beta)^2+\|x''\|^2<\alpha^2\}.$$ Non-transversal intersection and blow-ups ========================================= In this section, minimal assumptions are made on the set $\Omega$ in order to allow for general configurations. Technical tools --------------- The following lemma is similar to the interior case [@MR1745013; @MR3198649] and provides an improvement of monotonicity. \[m\] Let $u \in P_r^+(0,M, \Omega)$ where $\{u \neq 0\} \subset \Omega$. Let $e \in \mathbb{S}^{n-2} \cap e_n^{\perp}$ and suppose there exist non-negative constants $\epsilon_0, C_0$ such that $C_0 \partial_e u -u \ge -\epsilon_0$ in $B_r^+$. Then there exists $c=c(n, \Lambda, r)>0$ such that if $\epsilon_0 \le c$, then $C_0 \partial_e u - u \ge 0$ in $B_{\frac{r}{2}}^+$. By convexity of $F$, there exist measurable uniformly elliptic coefficients $a_{ij}$ such that $$F(D^2 u(x+he))-F(D^2 u(x)) \ge a_{ij}(\partial_{ij} u(x+he)-\partial_{ij} u(x))$$ if $x \in \Omega$ provided $h$ is small enough. Therefore, $$0 \ge a_{ij} \partial_{ij} \partial_e u \hskip .1in \text{in $\Omega$}.$$ Convexity also yields $$a_{ij} \partial_{ij} u \ge F(D^2 u(x))-F(0) = 1 \hskip .1in \text{in $\Omega$}.$$ Suppose now that there exists $y \in B_{\frac{r}{2}}^+$ for which $C_0 \partial_e u(y) - u(y) < 0.$ Let $w(x)=C_0 \partial_e u(x)-u(x)+\frac{\|x-y\|^2}{2n\Lambda}$. Since $\lambda Id \le (a_{ij}) \le \Lambda Id$, it follows by the above that $L w \le 0$ in $\Omega$ where $L=a_{ij}\partial_{ij}$. The maximum principle implies $\min_{\partial(\Omega \cap B_r^+)} w=\min_{\Omega \cap B_r^+} w < 0$. Note that $w\ge 0$ on $\partial \Omega$ and likewise on $\{x_n=0\}$. Therefore, the minimum occurs on $\partial B_r$ and thus $0 > -\epsilon_0+\frac{1}{8n \Lambda}r^2$, a contradiction if $\epsilon_0$ is small enough. \[rem1\] One may take $\epsilon_0=c r^2$, where $c>0$ depends only on the dimension and ellipticity constants of $F$. If $u \ge 0$, then $\partial_{e_n} u \ge 0$ on $\{x_n=0\} \cap B_r$ and Lemma \[m\] holds therefore in this case for all $e \in \mathbb{S}^{n-1}$ such that $e \cdot e_n \ge 0$. The next two lemmas highlight properties of the blow-up candidates. \[c\] Let $u_0(x)=ax_1x_n+bx_n^2$ with $a \neq 0$ and $R\ge1$. Then there exists $c=c(a,b)>0$ such that $$\inf_D \|\nabla u_0(x)\| \ge c,$$ where $D=\{x=(x_1,x'',x_n): R > \|x\| > R/2, \|x''\| \le \delta(R)\}$ for some $\delta(R)>0$. Note $\|\nabla u_0(x)\|^2=a^2x_n^2+a^2x_1^2+2abx_1x_n+4b^2x_n^2$ so that if $\|x_n\|>\frac{1}{3}$, then $\|\nabla u_0(x)\|^2 \ge \frac{a^2}{9}$. If $\|x_n\| \le \frac{1}{3}$, then for points that satisfy $\|x''\| \le \sqrt{\frac{5}{72}R}$, where $x''=(x_2,x_3,\ldots,x_{n-1})$, it follows that $$x_1^2 > \frac{23}{72} R^2.$$ If $b \neq 0$, let $\epsilon^2 \in (\frac{1}{a^2+4b^2}, \frac{1}{b^2}).$ Then $$\begin{aligned} \|\nabla u_0(x)\|^2&\ge(a^2+4b^2-\frac{1}{\epsilon^2})x_n^2+(a^2-\epsilon^2a^2b^2)x_1^2\\ &> (a^2-\epsilon^2a^2b^2)(\frac{23}{72} R^2).\end{aligned}$$ \[d\] Let $u_0(x)=ax_1x_n+bx_n^2$ with $a > 0$ and $R\ge 1$. Then there exists $C_0=C_0(a,b,R)>0$ such that $$C_0\partial_{x_1} u_0(x)-u_0(x) \ge 0$$ in $B_R^+$. The condition is equivalent to $ax_n(C_0-x_1) \ge b x_n^2$. Since $x_1 \le R$ and $0\le x_n \le R$, it follows that any $C_0 \ge \frac{b}{a}R+R$ satisfies the condition. A non-uniform version of the next result was shown in [@MR3513142]. It consists of the following alternative, either all re-scalings yield half space solutions or one may construct a specific sequence which produces a limit having the form in Lemma \[c\]. The main point here is that this procedure can be applied to blow-up limits of $\{u_j\}$ $\subset P_1^+(0,M,\Omega)$, i.e. limits of the form $$\lim_{k \rightarrow \infty} \frac{u_{j_k}(s_kx)}{s_k^2},$$ where $\{j_k\}$ is a subsequence of $\{j\}$ and $s_k \rightarrow 0^+$. \[th1\] Let $\{u_j\} \subset P_1^+(0,M, \Omega)$ and suppose $\{\nabla u_j \neq 0\} \cap \{x_n>0\} \subset \Omega$, $\nabla u_j(0)=0$. Then one of the following is true:\ (i) all blow-up limits of $\{u_j\}$ at the origin are of the form $u_0(x)=b x_n^2$ for some $b >0$;\ (ii) there exists a blow-up limit of $\{u_j\}$ of the form $ax_1x_n+bx_n^2$ for $a \neq 0$, $b \in \mathbb{R}$. Let $$N:= \limsup_{\|x\|\rightarrow 0, x_n>0} \frac{1}{x_n} \sup_{u \in \{u_j\}} \sup_{e \in \mathbb{S}^{n-2} \cap e_n^{\perp}} \partial_e u(x)$$ and consider a sequence $\{x^k\}_{k \in \mathbb{N}}$ with $x_n^k>0$, $u_{j_k} \in \{u_j\}$, and $e^k \in \mathbb{S}^{n-2} \cap e_n^{\perp}$ such that the previous limit is given by $$\lim_{k \rightarrow \infty} \frac{1}{x_n^k} \partial_{e^k} u_{j_k}(x^k).$$ Note that $N<\infty$ by $C^{1,1}$ regularity for the class $P_1^+(0,M, \Omega)$ and the boundary condition (see [@MR3513142]). By compactness, $e^k \rightarrow e_1 \in \mathbb{S}^{n-2}$ (along a subsequence) so that up to a rotation, $$N= \lim_{k \rightarrow \infty} \frac{1}{x_n^k} \partial_{x_1} u_{j_k}(x^k).$$ Next, if $$\tilde u_j(x):= \frac{u_{k_j}(s_jx)}{s_j^2} \rightarrow u_0(x)$$ for some sequence $s_j \rightarrow 0^+$, where the convergence is in $C_{loc}^{1,\alpha}(\mathbb{R}_+^n)$ for any $\alpha \in [0,1)$, $u_0 \in C^{1,1}(\mathbb{R}_+^n)$ satisfies the following PDE in the viscosity sense $$\label{m2} \begin{cases} F(D^{2}u_0)=1 & \text{a.e. in }\mathbb{R}_+^n\cap\Omega_0\\ \|\nabla u_0\|=0 & \text{in }\mathbb{R}_+^n\backslash\Omega_0\\ u=0 & \text{on }\mathbb{R}_+^{n-1}, \end{cases}$$ where $\Omega_0 =\{\nabla u_0 \neq 0 \} \cap \{x_n>0\}$. Note that $$\label{part} N \ge \lim_j \bigg\| \frac{\partial_{x_i} u_{k_j}(s_j x)}{s_j x_n}\bigg\| = \lim_j \bigg\| \frac{\partial_{x_i} \tilde u_j(x)}{x_n} \bigg\| =\bigg\|\frac{\partial_{x_i} u_0(x)}{x_n}\bigg\|$$ for all $i \in \{1,\ldots, n-1\}$. If $N=0$, then $\partial_{x_i} u_0=0$ for all $i \in \{1,\ldots,n-1\}$ so that $u_0(x)=u_0(x_n)$ and the conditions readily imply $u_0(x_n)=bx_n^2$. Since $N$ does not depend on the sequence $\{s_j\}$ it follows that in this case all blow-up limits have the previously stated form. Suppose that $N>0$, let $r_k=\|x^k\|$, and consider the re-scaling of $u_{j_k}$ with respect to $r_k$. Note that along a subsequence, $y^k:=\frac{x^k}{r_k} \rightarrow y \in \mathbb{S}^{n-1}$. By the choice of $r_k$, $$\lim_{k \rightarrow \infty} \frac{v(y^k)}{y_n^k}=\lim_{k \rightarrow \infty} \frac{\partial_{x_1} \tilde u_{k}(y^k)}{y_n^k}=\lim_{k \rightarrow \infty} \frac{\partial_{x_1} u_{j_k}(r_ky^k)}{r_ky_n^k} =N,$$ where $v= \partial_{x_1} u_0$. In particular, $$v(y)=Ny_n$$ and by the argument in [@MR3513142], $u_0(x)=ax_1x_n+bx_n^2$ with $a \neq 0$. Theorems \[tt\] and \[cul\] {#blowup} --------------------------- In what follows, the technical tools are utilized to prove that either all re-scalings yield half-space solutions or there exists a subsequence of the re-scalings which are classical solutions in a small half-ball around the origin. \[ke\] Suppose $\{u_j\} \subset P_1^+(0,M, \Omega)$. If $0 \in \overline{\{u_j \neq 0\}}$ and $\nabla u_j(0)=0$, then one of the following is true:\ (i) all blow-up limits of $\{u_j\}$ at the origin are of the form $u_0(x)=bx_n^2$ for $b>0$;\ (ii) there exists $\{u_{k_j}\} \subset \{u_j\}$ such that for all $R \ge 1$, there exists $j_R \in \mathbb{N}$ such that for all $j \ge j_R$, $$u_{k_j} \in C^{2,\alpha}(B_{\frac{Rr_{j}}{4}}^+),$$ where the sequence $\{r_{j}\}$ depends on $\{u_j\}$. Either all blow-up limits are of the form $u_0(x)=bx_n^2$ or there exists a subsequence, say $$\tilde u_j(x)=\frac{u_{k_j}(r_jx)}{r_j^2},$$ producing a limit of the form $u_0(x)=ax_1x_n+bx_n^2$ for $a>0$ (up to a rotation). Let $c=c(a,b)$ be the constant from Lemma \[c\] and note that since $\tilde u_j \rightarrow u_0$ in $C_{loc}^{1,\alpha}$, there exists $j_0=j_0(a, R) \in \mathbb{N}$ such that for every cylinder $S_{(\alpha, \beta)}(e_1)$ there exists $x \in S_{(\alpha, \beta)}(e_1) \cap B_R^+$ such that $\|\nabla \tilde u_j(x)\| \ge \frac{c}{2}$ for all $j \ge j_0$, where $R \ge 1$. Now choose a constant $C_0=C_0(a,b, R)>0$ (in fact, one may select any constant $C_0 \ge R(\frac{b}{a}+1)$) such that $$C_0 \partial_{x_1} u_0 -u_0 \ge 0$$ in $B_R^+$ and $j_0' \ge j_0$ for which $$\label{mon} C_0 \partial_{x_1} \tilde u_j - \tilde u_j \ge 0 \hskip .2 in \text{in $B_{\frac{R}{2}}^+$}$$ whenever $j \ge j_0'$ by Lemma \[m\]. Now fix $j \ge j_0'$ and suppose $z \in \Gamma_i(\tilde u_j) \cap B_{\frac{R}{2}}^+$. Then there exists a ball $B \subset \text{int} \{\tilde u_j=0\} \cap B_{\frac{R}{2}}^+$. Note that this ball generates a cylinder $S$ in the $e_1$- direction. Now select $x \in S \cap B_R^+$ for which $\|\nabla \tilde u_j(x)\|>0$ and $-R<x_1<-R/2$. In particular, there exists a small ball around $x$, say $\tilde B$ such that $F(D^2 \tilde u_j)=1$ in $\tilde B$ and one may assume $\tilde B \subset \{\tilde u_j \neq 0\}$. Note that $\tilde B$ is contained in the cylinder $S$ and let $E_t=\tilde B+te_1$ for $t \in \mathbb{R}$. If $t>0$ is such that $\overline{E_t} \cap \{\tilde u_j=0\} \neq \emptyset$, and for all $0\le s<t$, $E_s \cap \{\tilde u_j=0\} = \emptyset$, choose $y \in \overline{E_t} \cap \{\tilde u_j=0\}.$ Moreover, note that if $\tilde u_j > 0$ in $\tilde B$, then by it follows that $\tilde u_j$ is strictly positive at a point in $\{\tilde u_j=0\}$, a contradiction. Thus $\tilde u_j < 0$ in $\tilde B$. Next, by convexity of $F$ $$a_{kl} \partial_{kl} \tilde u_j \ge 0 \hskip .1in \text{in $E_t$}.$$ Since $0=\tilde u_j(y)>\tilde u_j(x)$ for $x \in E_t$ and $y$ satisfies an interior ball condition, then Hopf’s lemma implies that $\frac{\partial}{\partial n} \tilde u_j(y)>0$, where $n$ is the outer normal to the ball at $y$. Now, if there exists $z \in B_\delta(y)$ such that $\tilde u_j(z)>0$, then this contradicts the monotonicity, if $\delta>0$ is sufficiently small: $\overline{E_{\eta}} \subset B \subset int \{\tilde u_j=0\}$ for $\eta>0$ large enough and since $\tilde u_j(z)>0$, the monotonicity implies that $\tilde u_j(z+e_1s)>0$, for some $s>0$ such that $z+e_1s \in \{\tilde u_j=0\}$. Hence, $\tilde u_j\le 0$ on $B_\delta(y)$ and thus $\nabla \tilde u_j(y)=0$, a contradiction. The conclusion is that for $j \ge j_0'$, $$\Gamma_i(\tilde u_j) \cap B_{\frac{R}{2}}^+=\emptyset.$$ In particular, $(B_{\frac{R}{2}}^+ \setminus \Omega_j)^o=\emptyset$ and non-degeneracy implies that $\|B_{\frac{R}{2}}^+\setminus \Omega_j\|=0$. Thus the $C^{1,1}$ function $\tilde u_j$ satisfies $F(D^2 \tilde u_j)=1$ in $B_{\frac{R}{2}}^+$ in the viscosity sense and the up to the boundary Evans-Krylov theorem implies that $\tilde u_j \in C^{2,\alpha}(B_{\frac{R}{4}}^+).$ In particular, $u_{k_j} \in C^{2,\alpha}(B_{\frac{Rr_j}{4}}^+).$ If not, then there exists $\epsilon>0$ such that for all $k \in \mathbb{N}$ there exists $u_k \in P_1^+(0,M, \Omega)$ with $$\label{cont} \Gamma(u_k) \cap B_{1/k}^+ \cap \mathcal{C}_\epsilon \neq \emptyset,$$ where $0 \in \overline{\Gamma(u_k)}.$ Now we consider two cases. First, suppose all blow-ups of $\{u_k\}$ are half-space solutions. Let $x_k \in \Gamma(u_k) \cap B_{1/k}^+ \cap \mathcal{C}_\epsilon$ and set $y_k=\frac{x_k}{r_k}$ with $r_k=\|x_k\|$. Consider $\tilde u_k(x)=\frac{u_k(r_kx)}{r_k^2}$ so that $y_k \in \Gamma(\tilde u_k)$, $\tilde u_k \rightarrow bx_n^2$, $y_{k} \rightarrow y \in \partial B_1 \cap C_\epsilon$ (up to a subsequence), and $y \in \Gamma(u_0)$, a contradiction. Second, select a subsequence $\{u_{k_j}\}$ of $\{u_k\}$ such that for all $j \ge j_2$, $u_{k_j} \in C^{2,\alpha}(B_{\frac{r_{j}}{2}}^+)$, where $j_2 \in \mathbb{N}$ and the sequence $\{r_{j}\}$ depends on $\{u_k\}$. Since $0 \in \overline{\Gamma(u_{k_j})},$ there exists $$x_j \in \Gamma(u_{k_j}) \cap B_{\frac{r_{j}}{2}}^+$$ which contradicts the continuity of $F$. By Proposition \[ke\], either $u_0(x)=bx_n^2$ or $D^2u(0)$ exists and the rescaling of $u$ is given by $$u_j(x)=\frac{u(r_j x)}{r_j^2}=\langle x, D^2u(0)x\rangle + o(1).$$ Since $u_0(x',0)=0$ for $x' \in \mathbb{R}^{n-1}$, it follows that $u_0$ has the claimed form (up to a rotation). Combining the non-transversal intersection with [@MR3198649 Theorem 1.3], the following result holds. \[reg\] Let $u \in P_1^+(0,M, \Omega)$ and $0 \in \overline{\Gamma(u)}$. Suppose that for some $\epsilon_0>0$, $$\delta_r(u,x^0) \ge \epsilon_0$$ for all $r>0$ and $x^0 \in \Gamma \cap B_1^+$. Then there exists $r_0>0$ such that $$\Gamma \cap \overline{B_{r_0}^+}=\{x:x_n=\phi(x')\} \cap \overline{B_{r_0}^+},$$ where $\phi$ is $C^1$. Let $x^0 \in \Gamma \cap B_1^+$, then there exists a neighborhood of $x^0$ such that $\Gamma$ is represented as the graph of a $C^1$ function with respect to some coordinate system [@MR3198649 Theorem 1.3]. By Theorem \[tt\] it follows that the normal to the free boundary converges to $e_n$ so that this function can be taken with respect to the $\{x_n=0\}$ hyperplane. \[rema\] If the free boundary can be represented as the graph of a Lipschitz function close to a contact point, then the thickness condition is satisfied. In general, the free boundary is not more regular than $C^{1, \text{Dini}}.$ This is in sharp contradistinction to the interior case. The existence of non-tangential second derivatives follows in a standard way. Let $u \in P_1^+(0,M, \Omega)$ and $0 \in \overline{\Gamma(u)}$. Then $$\displaystyle\lim_{\|x\|\rightarrow 0}\partial_{ij} u(x)$$ exists non-tangentially for all $i,j \in \{1,\ldots, n\}$. Suppose $\{x^j\}$ is such that $x_n^j \ge \kappa \|(x')^j\|$ for some $\kappa>0$. Then, for all $j$ large, $x^j \in \Omega$. Let $$u_j(x)=\frac{u(r_jx)}{r_j^2}$$ where $r_j=\|x^j\|$ and $y^j=\frac{x^j}{r_j}$ so that along a subsequence, $y^j \rightarrow y \in \mathbb{S}^{n-1}$. There exists $\mu>0$ such that $B_\mu(y) \subset \Omega_j$ for all $j$ large so that $u_j \in C^{2,\alpha}(B_\mu(y))$. Therefore, $$D^2 u_j(y^j)=D^2 u(x^j) \rightarrow D^2 u_0(y).$$ Since $D^2 u_0$ is a constant matrix, it is independent of $y$, and therefore independent of the subsequence. In general, one cannot expect the tangential second derivatives to match the non-tangential derivatives. Consider the case when $0 \in \overline{\Gamma_i(u)}$ so that there exists a collection of balls $\{B_\alpha\}$ where $B_\alpha \subset int\{u=0\}$ and on $B_\alpha$, $D^2 u =0$. Regularity ========== Corollary \[reg\] follows in a standard way once non-transversal intersection is established (via interior regularity). In the physical case when $u \ge 0$, it turns out that one may dispense with density conditions. The key is to exploit the boundary condition and estimate a maximal mixed partial derivative. \[rl\] Suppose $u \in P_1^+(0,M, \Omega)$ and $u\ge 0$. Then for any $\epsilon>0$ there exists $r(\epsilon, M)>0$ such that if $x^0 \in \Gamma(u) \cap B_{1/2}^+$ and $d=x_n^0<r,$ then $$\sup_{B_{2d}^+(x^0)} \|u-h\| \le \epsilon d^2, \hskip .2in \sup_{B_{2d}^+(x^0)} \| \nabla u - \nabla h\| \le \epsilon d,$$ where $$h(x)= b[(x_n-d)^+]^2,$$ and $b>0$ depends on the ellipticity constants of $F$. Suppose not, then there exists $\epsilon>0$, non-negative $u_j \in P_1^+(0,M, \Omega)$, and $x^j \in \Gamma(u_j) \cap B_{1/2}^+$ with $d_j=x_n^j \rightarrow 0$, for which $$\sup_{B_{2d_j}(x^j)^+} \|u_j- b[(x_n-d_j)^+]^2\|>\epsilon d_j^2,$$ or $$\sup_{B_{2d_j}(x^j)^+} \|\nabla u_j- 2b(x_n-d_j)^+\|>\epsilon d_j.$$ Let $\tilde u_j(x)=\frac{u_j((x^j)'+d_jx)}{d_j^2}$ so that in particular $$\|\|\tilde u_j - h\|\|_{C^1(B_2^+(e_n))} \ge \epsilon,$$ where $h(x)=b[(x_n-1)^+]^2$. Since $\tilde u_j(e_n)=\|\nabla \tilde u_j(e_n)\|=0,$ the $C^{1,1}$ regularity of $\tilde u_j$ implies that $\|\tilde u_j(x)\| \le C\|x-e_n\|^2$. By passing to a subsequence, if necessary, $$\tilde u_j \rightarrow u_0$$ where $u_0 \in C^{1,1}(\mathbb{R}_+^n)$ satisfies the following PDE in the viscosity sense $$\label{m3} \begin{cases} F(D^{2}u_0)=1 & \text{a.e. in }\mathbb{R}_+^n\cap\Omega_0,\\ \|\nabla u_0\|=0=u_0 & \text{in }\mathbb{R}_+^n\backslash\Omega_0,\\ u_0=0 & \text{on }\mathbb{R}_+^{n-1}. \end{cases}$$ Now let $$N=\limsup_{\|x\|\rightarrow 0, x_n>0} \frac{1}{x_n} \sup_{u\in P_1^+\cap \{u \ge 0\}} \sup_{e \in \mathbb{S}^{n-2} \cap e_n^{\perp}} \sup_{y \in \overline{B_{1/2}^+} \cap \{x_n=0\}} \partial_e u(x+y)$$ and note that $N<\infty$ by $C^{1,1}$ regularity and the boundary condition: for any $e \in \mathbb{S}^{n-2} \cap e_n^{\perp}$ and $y \in \overline{B_{1/2}^+} \cap \{x_n=0\}$, it follows that $\partial_{e} u(x'+y)=0$. Furthermore, $$\label{part} N \ge \lim_j \bigg\| \frac{\partial_{x_i} u_j(d_j x+(x^j)')}{d_j x_n}\bigg\| = \lim_j \bigg\| \frac{\partial_{x_i} \tilde u_j(x)}{x_n} \bigg\| =\bigg\|\frac{\partial_{x_i} u_0(x)}{x_n}\bigg\|$$ for all $i \in \{1,\ldots, n-1\}$. In particular, let $v= \partial_{x_1} u_0$ so that in $\mathbb{R}_+^n$, $$\label{ine} \|v(x)\| \le Nx_n.$$ If $N=0$, then $\partial_{x_i} u_0=0$ for all $i \in \{1,\ldots,n-1\}$ and therefore $u_0(x)=u_0(x_n)$. Since $e_n$ is a free boundary point, it follows that $u_0=h$, a contradiction. Thus $N>0$ and there is a sequence $\{x^k\}_{k \in \mathbb{N}}$ with $x_n^k>0$, $u_k \in P_1^+(0,M, \Omega)$, $u_k \ge 0$, $y^k \in \overline{B_{1/2}^+} \cap \{x_n=0\}$, and $e^k \in \mathbb{S}^{n-2} \cap e_n^{\perp}$ such that $$N=\lim_{k \rightarrow \infty} \frac{1}{x_n^k} \partial_{e^{k}} u_k(x^k+y^k).$$ By compactness, $e^k \rightarrow e_1 \in \mathbb{S}^{n-2}$ (along a subsequence) so that up to a rotation, $$N= \lim_{k \rightarrow \infty} \frac{1}{x_n^k} \partial_{x_1} u_k(x^k+y^k).$$ Let $$\tilde u_k(x)=\frac{u_k(y^k+r_kx)}{r_k^2},$$ where $r_k=\|x^k\|$, $z^k=\frac{x^k}{r_k}$, and note that along a subsequence $z^k \rightarrow z \in \mathbb{S}^{n-1}$ and $\tilde u_k \rightarrow u_0$. It follows that $\partial_{x_1}u_0(z)=Nz_n$ and proceeding as in [@MR3513142] one deduces that $u_0(x)=ax_1x_n+cx_n+\tilde b x_n^2$ for $a \neq 0$ and $c, \tilde b \in \mathbb{R}$, contradicting that $u \ge 0$. By Lemma \[rl\] it follows that in a neighborhood of the origin, there is a cone of fixed opening that can be placed below and above each free boundary point. This implies that the free boundary is Lipschitz continuous. Away from the origin, it is therefore $C^1$ by interior results. Moreover, since the intersection of $\Gamma$ and the origin occurs non-transversally, the aperture of the cones can be taken arbitrarily close to $\pi$, and this implies that the free boundary is $C^1$ at the origin.
--- abstract: 'The Wilkinson Microwave Anisotropy Probe ([[WMAP]{}]{}) science team has released results from the first year of operation at the Earth-Sun L$_2$ Lagrange point. The maps are consistent with previous observations but have much better sensitivity and angular resolution than the [COBE]{} DMR maps, and much better calibration accuracy and sky coverage than ground-based and balloon-borne experiments. The angular power spectra from these ground-based and balloon-borne experiments are consistent within their systematic and statistical uncertainties with the [[WMAP]{}]{} results. [[WMAP]{}]{} detected the large angular-scale correlation between the temperature and polarization anisotropies of the CMB caused by electron scattering since the Universe became reionized after the “Dark Ages”, giving a value for the electron scattering optical depth of $0.17 \pm 0.04$. The simplest $\Lambda$CDM model with $n=1$ and $\Omega_{tot}=1$ fixed provides an adequate fit to the [[WMAP]{}]{} data and gives parameters which are consistent with determinations of the Hubble constant and observations of the accelerating Universe using supernovae. The time-ordered data, maps, and power spectra from [[WMAP]{}]{} can be found at `http://lambda.gsfc.nasa.gov` along with 13 papers by the [[WMAP]{}]{} science team describing the results in detail.' author: - '[[E. L. Wright]{}]{},' title: The WMAP Data and Results --- INTRODUCTION {#intro} ============ The cosmic microwave background (CMB) radiation was discovered by @penzias/wilson:1965. After its discovery, a small number of experimentalists worked for years to better characterize the spectrum of the CMB and to search for anisotropy in the CMB temperature. A leader of this effort, and of the [[WMAP]{}]{} effort, was our recently deceased colleague, Professor David T. Wilkinson of Princeton University. He was the supervisor of the doctoral theses that led to the second [@henry:1971] and third [@corey/wilkinson:1976] measurements of the dipole anisotropy of the CMB caused by the Solar System’s motion relative to the Universe. He was also a leading member of the [Cosmic Background Explorer (COBE)]{} mission team, which accurately characterized the spectrum of the CMB [@mather/etal:1990; @mather/etal:1999] and first discovered the intrinsic (non-dipole) anisotropy [@smoot/etal:1992; @bennett/etal:1992b; @kogut/etal:1992; @wright/etal:1992] of the CMB. The [[WMAP]{}]{} science working group, led by Principal Investigator Charles L. Bennett of Goddard Space Flight Center, was happy to have the opportunity to honor David T. Wilkinson by renaming the Microwave Anisotropy Probe as the Wilkinson Microwave Anisotropy Probe. @bennett/etal:2003 gives a description of the [[WMAP]{}]{} mission. @bennett/etal:2003b summarizes the results from first year of [[WMAP]{}]{} observations. @bennett/etal:2003c describes the observations of galactic and extragalactic foreground sources. @hinshaw/etal:2003 gives the angular power spectrum derived from the the [[WMAP]{}]{} maps. @hinshaw/etal:2003b describes the [[WMAP]{}]{} data processing and systematic error limits. @page/etal:2003b discusses the beam sizes and window functions for the [[WMAP]{}]{} experiment. @page/etal:2003c discusses results that can be derived simply from the positions and heights of the peaks and valleys in the angular power spectrum. @spergel/etal:2003 describes the cosmological parameters derived by fitting the [[WMAP]{}]{} data and other datasets. @verde/etal:2003 describes the fitting methods used. @peiris/etal:2003 describes the consequences of the [[WMAP]{}]{} results for inflationary models. @jarosik/etal:2003b describes the on-orbit performance of the [[WMAP]{}]{}radiometers. @kogut/etal:2003 describes the [[WMAP]{}]{} observations of polarization in the CMB. @barnes/etal:2003 describes the large angle sidelobes of the [[WMAP]{}]{} telescopes. @komatsu/etal:2003 addresses the limits on non-Gaussianity that can be derived from the [[WMAP]{}]{} data. OBSERVATIONS \[sec:obs\] ======================== [[WMAP]{}]{} Observatory was launched on 30 June 2001 at 19:46:46.183 UTC from Cape Canaveral by a Delta expendable launch vehicle. After three phasing loops in the Earth-Moon system [[WMAP]{}]{} executed a lunar-gravity-assist swingby one month after launch which catapulted [[WMAP]{}]{} to an orbit about the second Lagrange point of the Sun-Earth system, L$_2$. L$_2$ is only metastable so about 4 station-keeping maneuvers per year are required to keep [[WMAP]{}]{} on station. The spacecraft is actually in a “halo” orbit around L$_2$ and thus avoids the deep partial eclipse of the Sun that exists at L$_2$. [[WMAP]{}]{} observes at 5 different frequencies and can thus yield internal linear combination “no galaxy maps”. Figure \[fig:ilc\] shows such a map in the “polar eyeball” projection. This equal-area projection maps the North and South galactic hemispheres into circles with the Galactic center in the middle, the North Galactic Pole (NGP) in the center of the left circle, and the SGP in the center of the right circle. Figure \[fig:QVW\] shows the 41, 61 & 94 GHz maps as red, green & blue on the same temperature scale in the same projection as Figure \[fig:ilc\]. Analysis\[sec:analysis\] ======================== Given the extensive analysis of the [[WMAP]{}]{} data already posted on the astro-ph preprint server, or at the Legacy Archive for Microwave Background Data Analysis (LAMBDA at http://lambda.gsfc.nasa.gov), I see little point in repeating this lengthy discussion here. I have instead included Figure \[fig:fits\] which shows a $\Lambda$CDM model with a power law primordial power spectrum and zero spatial curvature. This particular model was the best fit found in a fairly small Monte Carlo Markov Chain computation that I did as an independent check of the main [[WMAP]{}]{} analysis papers. One easily sees that the [[WMAP]{}]{} data are quite consistent with the previous experiments \[ARCHEOPS [@benoit/etal:2003], BOOMERanG [@debernardis/etal:2000], DASI [@halverson/etal:2002], MAXIMA [@hanany/etal:2000], VSA [@grainge/etal:2003] & CBI [@padin/etal:2001]\], some of which overlap with [[WMAP]{}]{} in $\ell$-space, and that the $\Lambda$CDM model is consistent with both the [[WMAP]{}]{} data and the higher angular resolution interferometric data from CBI. The largest discrepancies among various CMB anisotropy experiments are due to systematic calibration uncertainties. Therefore, in doing these fits, a calibration correction for each experiment other than WMAP has been introduced as a new parameter in the fits, and the sum of the squares of calibration corrections divided by their stated uncertainties has been added to the $\chi^2$ of the model fit. None of the calibration corrections is inconsistent with its stated uncertainty. The data are plotted with the calibration corrections applied, which emphasizes the concordance between recent measurments of $C_\ell$. This plot is a good way to verify that [[WMAP]{}]{} is consistent with earlier experiments, but when setting limits on cosmological parameters one should not combine [[WMAP]{}]{} with experiments like ARCHEOPS that cover the same $\ell$ range since the cosmic variance is correlated between the two datasets. Combining [[WMAP]{}]{} only with high angular resolution datasets like ACBAR [@kuo/etal:2002] and CBI is the correct procedure. I then used the MCMC code to optimize the parameters for two alternative cosmological models that have published claimed fits to the CMB angular power spectrum. The “no CDM” model is an update of the @mcgaugh:2000 fit to the BOOMERanG data using CMBfast [@seljak/zaldarriaga:1996] with a zero CDM density. It is clear that this model is a terrible fit, and it was a terrible fit to the combined CMB dataset including COBE which existed in 2000. If CMBfast with no CDM were a good predictor of the anisotropy expected from the Modification Of Newtonian Dynamics then MOND would be killed by this bad fit. But CMBfast assumes that general relativity is a good description of gravity and thus @mcgaugh:2000 is not a definitive prediction of the $C_\ell$ expected under MOND. Thus the failure of the @mcgaugh:2000 model to fit the data only rules out this specific attempt to extend MOND to the early Universe. More recently @narlikar/etal:2003 claim to have calculated the CMB anisotropy expected in the Quasi-Steady-State Cosmology (QSSC), and also claim that the QSSC gives a better fit to the [binned]{} data than $\Lambda$CDM. This model is an [ad hoc]{} superposition of two populations of Gaussian blobs with two different sizes and a population of hard-edged circular spots all of a single size. These hard-edged circular disks have an oscillatory Fourier transform which gives a series of peaks in the angular power spectrum. But the low-$\ell$ behavior of this model is $C_\ell =$ const, which corresponds to a primordial power spectrum $P(k) \propto k^n$ with a power law index of $n=3$ which is $6\sigma$ away from the COBE value of $n = 1.2 \pm 0.3$ [@bennett/etal:1996]. @narlikar/etal:2003 hide this failing of their model by using a binning which puts all the COBE data into one bin. I have re-optimized the six arbitrary parameters in the @narlikar/etal:2003 model to give the best possible fit to the data in Figure \[fig:fits\], but even the best fit is totally unacceptable. As with MOND, if @narlikar/etal:2003 had presented a valid theory for the anisotropy predicted by the QSSC, then this bad fit would have killed the QSSC. But unfortunately neither the theory nor the fit is acceptable. Summary and Conclusions ======================= [[WMAP]{}]{} has presented the results from its first year of observation at L$_2$, and these data agree with the concordance $\Lambda$CDM model and yield dramatic improvements in the accuracy of the cosmological parameters. [[WMAP]{}]{} is funded for 3 more years of operation, and the results from a 4 year dataset will have a much improved signal-to-noise ratio for $\ell > 400$ in the temperature anisotropy angular power spectrum, and a much improved SNR for all $\ell$’s in the polarization measurements. The [[WMAP]{}]{} mission is made possible by the support of the Office of Space Sciences at NASA Headquarters and by the hard and capable work of scores of scientists, engineers, technicians, machinists, data analysts, budget analysts, managers, administrative staff, and reviewers. [33]{} natexlab\#1[\#1]{} Barnes, C. [et al.]{} 2003, , in press , C. L., [Banday]{}, A. J., [G[ó]{}rski]{}, K. M., [Hinshaw]{}, G., [Jackson]{}, P., [Keegstra]{}, P., [Kogut]{}, A., [Smoot]{}, G. F., [Wilkinson]{}, D. T., & [Wright]{}, E. L. 1996, , 464, L1 , C. L., [Bay]{}, M., [Halpern]{}, M., [Hinshaw]{}, G., [Jackson]{}, C., [Jarosik]{}, N., [Kogut]{}, A., [Limon]{}, M., [Meyer]{}, S. S., [Page]{}, L., [Spergel]{}, D. N., [Tucker]{}, G. S., [Wilkinson]{}, D. T., [Wollack]{}, E., & [Wright]{}, E. L. 2003, , 583, 1 , C. L., [Halpern]{}, M., [Hinshaw]{}, G., [Jarosik]{}, N., [Kogut]{}, A., [Limon]{}, M., [Meyer]{}, S. S., [Page]{}, L., [Spergel]{}, D. N., [Tucker]{}, G. S., [Wollack]{}, E., [Wright]{}, E. L., [Barnes]{}, C., [Greason]{}, M., [Hill]{}, R., [Komatsu]{}, E., [Nolta]{}, M., [Odegard]{}, N., [Peiris]{}, H., [Verde]{}, L., & [Weiland]{}, J. 2003, , in press , C. L., [Smoot]{}, G. F., [Hinshaw]{}, G., [Wright]{}, E. L., [Kogut]{}, A., [de Amici]{}, G., [Meyer]{}, S. S., [Weiss]{}, R., [Wilkinson]{}, D. T., [Gulkis]{}, S., [Janssen]{}, M., [Boggess]{}, N. W., [Cheng]{}, E. S., [Hauser]{}, M. G., [Kelsall]{}, T., [Mather]{}, J. C., [Moseley]{}, S. H., [Murdock]{}, T. L., & [Silverberg]{}, R. F. 1992, , 396, L7 , C. L. [et al.]{} 2003, , in press Benoit, A. [et al.]{} 2003, , submitted (astro-ph/0210305) , B. E. & [Wilkinson]{}, D. T. 1976, , 8, 351 , P., [Ade]{}, P. A. R., [Bock]{}, J. J., [Bond]{}, J. R., [Borrill]{}, J., [Boscaleri]{}, A., [Coble]{}, K., [Crill]{}, B. P., [De Gasperis]{}, G., [Farese]{}, P. C., [Ferreira]{}, P. G., [Ganga]{}, K., [Giacometti]{}, M., [Hivon]{}, E., [Hristov]{}, V. V., [Iacoangeli]{}, A., [Jaffe]{}, A. H., [Lange]{}, A. E., [Martinis]{}, L., [Masi]{}, S., [Mason]{}, P. V., [Mauskopf]{}, P. D., [Melchiorri]{}, A., [Miglio]{}, L., [Montroy]{}, T., [Netterfield]{}, C. B., [Pascale]{}, E., [Piacentini]{}, F., [Pogosyan]{}, D., [Prunet]{}, S., [Rao]{}, S., [Romeo]{}, G., [Ruhl]{}, J. E., [Scaramuzzi]{}, F., [Sforna]{}, D., & [Vittorio]{}, N. 2000, , 404, 955 , K. [et al.]{} 2003, , in press (astro-ph/0212495) , N. W., [Leitch]{}, E. M., [Pryke]{}, C., [Kovac]{}, J., [Carlstrom]{}, J. E., [Holzapfel]{}, W. L., [Dragovan]{}, M., [Cartwright]{}, J. K., [Mason]{}, B. S., [Padin]{}, S., [Pearson]{}, T. J., [Readhead]{}, A. C. S., & [Shepherd]{}, M. C. 2002, , 568, 38 , S., [Ade]{}, P., [Balbi]{}, A., [Bock]{}, J., [Borrill]{}, J., [Boscaleri]{}, A., [de Bernardis]{}, P., [Ferreira]{}, P. G., [Hristov]{}, V. V., [Jaffe]{}, A. H., [Lange]{}, A. E., [Lee]{}, A. T., [Mauskopf]{}, P. D., [Netterfield]{}, C. B., [Oh]{}, S., [Pascale]{}, E., [Rabii]{}, B., [Richards]{}, P. L., [Smoot]{}, G. F., [Stompor]{}, R., [Winant]{}, C. D., & [Wu]{}, J. H. P. 2000, , 545, L5 , P. S. 1971, , 231, 516 , G. F. [et al.]{} 2003, , in press —. 2003, , in press , N. [et al.]{} 2003, , in press , A., [Smoot]{}, G. F., [Bennett]{}, C. L., [Wright]{}, E. L., [Aymon]{}, J., [de Amici]{}, G., [Hinshaw]{}, G., [Jackson]{}, P. D., [Kaita]{}, E., [Keegstra]{}, P., [Lineweaver]{}, C., [Loewenstein]{}, K., [Rokke]{}, L., [Tenorio]{}, L., [Boggess]{}, N. W., [Cheng]{}, E. S., [Gulkis]{}, S., [Hauser]{}, M. G., [Janssen]{}, M. A., [Kelsall]{}, T., [Mather]{}, J. C., [Meyer]{}, S., [Moseley]{}, S. H., [Murdock]{}, T. L., [Shafer]{}, R. A., [Silverberg]{}, R. F., [Weiss]{}, R., & [Wilkinson]{}, D. T. 1992, , 401, 1 Kogut, A. [et al.]{} 2003, , in press Komatsu, E. [et al.]{} 2003, , in press Kuo, C. L. [et al.]{} 2002, , astro-ph/0212289 , J. C., [Cheng]{}, E. S., [Eplee]{}, R. E., [Isaacman]{}, R. B., [Meyer]{}, S. S., [Shafer]{}, R. A., [Weiss]{}, R., [Wright]{}, E. L., [Bennett]{}, C. L., [Boggess]{}, N. W., [Dwek]{}, E., [Gulkis]{}, S., [Hauser]{}, M. G., [Janssen]{}, M., [Kelsall]{}, T., [Lubin]{}, P. M., [Moseley]{}, S. H., [Murdock]{}, T. L., [Silverberg]{}, R. F., [Smoot]{}, G. F., & [Wilkinson]{}, D. T. 1990, , 354, L37 , J. C., [Fixsen]{}, D. J., [Shafer]{}, R. A., [Mosier]{}, C., & [Wilkinson]{}, D. T. 1999, , 512, 511 , S. S. 2000, , 541, L33 , J. V., [Vishwakarma]{}, R. G., [Hajian]{}, A., [Souradeep]{}, T., [Burbidge]{}, G., & [Hoyle]{}, F. 2003, , 585, 1 , S., [Cartwright]{}, J. K., [Mason]{}, B. S., [Pearson]{}, T. J., [Readhead]{}, A. C. S., [Shepherd]{}, M. C., [Sievers]{}, J., [Udomprasert]{}, P. S., [Holzapfel]{}, W. L., [Myers]{}, S. T., [Carlstrom]{}, J. E., [Leitch]{}, E. M., [Joy]{}, M., [Bronfman]{}, L., & [May]{}, J. 2001, , 549, L1 , L. [et al.]{} 2003, , in press —. 2003, , in press Peiris, H. [et al.]{} 2003, , in press , A. A. & [Wilson]{}, R. W. 1965, , 142, 419 , U. & [Zaldarriaga]{}, M. 1996, , 469, 437 , G. F., [Bennett]{}, C. L., [Kogut]{}, A., [Wright]{}, E. L., [Aymon]{}, J., [Boggess]{}, N. W., [Cheng]{}, E. S., [de Amici]{}, G., [Gulkis]{}, S., [Hauser]{}, M. G., [Hinshaw]{}, G., [Jackson]{}, P. D., [Janssen]{}, M., [Kaita]{}, E., [Kelsall]{}, T., [Keegstra]{}, P., [Lineweaver]{}, C., [Loewenstein]{}, K., [Lubin]{}, P., [Mather]{}, J., [Meyer]{}, S. S., [Moseley]{}, S. H., [Murdock]{}, T., [Rokke]{}, L., [Silverberg]{}, R. F., [Tenorio]{}, L., [Weiss]{}, R., & [Wilkinson]{}, D. T. 1992, , 396, L1 , D. N. [et al.]{} 2003, , in press , L. [et al.]{} 2003, , in press , E. L., [Meyer]{}, S. S., [Bennett]{}, C. L., [Boggess]{}, N. W., [Cheng]{}, E. S., [Hauser]{}, M. G., [Kogut]{}, A., [Lineweaver]{}, C., [Mather]{}, J. C., [Smoot]{}, G. F., [Weiss]{}, R., [Gulkis]{}, S., [Hinshaw]{}, G., [Janssen]{}, M., [Kelsall]{}, T., [Lubin]{}, P. M., [Moseley]{}, S. H., [Murdock]{}, T. L., [Shafer]{}, R. A., [Silverberg]{}, R. F., & [Wilkinson]{}, D. T. 1992, , 396, L13
--- abstract: 'Double-lined spectroscopic binaries (SB2s) are one of the main sources of stellar masses, as additional observations are only needed to give the inclinations of the orbital planes in order to obtain the individual masses of the components. For this reason, we are observing a selection of SB2s using the SOPHIE spectrograph at the Haute-Provence observatory in order to precisely determine their orbital elements. Our objective is to finally obtain masses with an accuracy of the order of one percent by combining our radial velocity (RV) measurements and the astrometric measurements that will come from the [Gaia]{} satellite. We present here the RVs and the re-determined orbits of 10 SB2s. In order to verify the masses we will derive from [Gaia]{}, we obtained interferometric measurements of the ESO VLTI for one of these SB2s. Adding the interferometric or speckle measurements already published by us or by others for 4 other stars, we finally obtain the masses of the components of 5 binary stars, with masses ranging from 0.51 to 2.2 solar masses, including main-sequence dwarfs and some more evolved stars whose location in the HR diagram has been estimated.' author: - \| J.-L. Halbwachs$^{1}$[^1], F. Kiefer$^{2,3}$, Y. Lebreton$^{3,4}$, H.M.J. Boffin$^{5}$, F. Arenou$^{6}$, J.-B. Le Bouquin$^{7}$, B. Famaey$^{1}$, D. Pourbaix$^{8}$, P. Guillout$^{1}$, J.-B. Salomon$^{9}$, and T. Mazeh$^{10}$\ $^{1}$Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, 11 rue de l’Université, F-67000 Strasbourg, France\ $^{2}$Institut d’Astrophysique de Paris,CNRS/UMR7095, 98bis boulevard Arago, F-75014 Paris, France\ $^{3}$LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Université, UPMC Univ. Paris 06, Univ. Paris Diderot,\ Sorbonne Paris Cité, F-92195 Meudon, France\ $^{4}$Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35000 Rennes, France\ $^{5}$European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany\ $^{6}$GEPI, Observatoire de Paris, PSL Research University, CNRS, Université Paris Diderot, Sorbonne Paris Cité, Place Jules Janssen,\ F-92195 Meudon, France\ $^{7}$Univ. Grenoble-Alpes, CNRS, IPAG, 38000 Grenoble, France\ $^{8}$FNRS, Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, boulevard du Triomphe, 1050 Bruxelles, Belgium\ $^{9}$Institut UTINAM, CNRS UMR6213, Université Bourgogne Franche-Comté, OSU THETA, Observatoire de Besançon, BP 1615, F-25010\ Besançon Cedex, France\ $^{10}$School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel\ date: 'Accepted . Received 2019 ; in original form 2019' title: 'Masses of the components of SB2 binaries observed with [Gaia]{}. V. Accurate SB2 orbits for 10 binaries and masses of the components of 5 binaries. [^2] [^3] ' --- \[firstpage\] binaries: spectroscopic, stars: fundamental parameters, stars: individual: HIP 104987 Introduction ============ Estimating the mass of stars is a fundamental step in understanding the internal processes that determine how stars work, how bright they shine, and for how long. When the mass is known with sufficient precision, modelling not only enlightens us on the physical processes it depicts [e.g. @Claret19], but also provides constraints on parameters that are not directly accessible, such as age and helium content [e.g. @Lebreton05]. It is only possible to obtain a mass for a few very specific stars, but these are the basis of calibration relationships that allow the masses of other stars to be estimated from more accessible data, such as spectral type, color indices or absolute luminosity [e.g. @Eker15; @Moya18; @Mann19]. These calibrations paved the way for population synthesis models, ultimately allowing to estimate the stellar mass-to-light ratios of galaxies from their observed colours or spectral energy distributions [e.g. @Bell01], which are central to the estimate of their dark matter content [e.g. @Lelli16]. There are only a limited number of direct methods for measuring the mass of a star, and they all apply to components of double stars. Most (but not all) use the double-lined spectroscopic binaries (SB2s), for which the orbital elements allow to calculate a minimum value of the mass of each component, ${\cal M} \sin^3 i$, where $i$ is the inclination of the orbit. This parameter can be obtained from the following complementary techniques: - The eclipsing binaries (EBs) have historically been considered the Royal Road of stellar astrophysics, and this reputation has not been denied. When they are also double-lined spectroscopic binaries (SB2s), photometric and spectroscopic observations make it possible to deduce not only the masses of the components, but also their radii, their effective temperatures [e.g. @Torres19], and even the distance of the system with sufficient precision to test trigonometric parallaxes obtained by astrometric satellites [@Munari04]. Unfortunately, a binary can present eclipses only if its orbital plane is close to the line of sight. This makes the EBs quite rare among binaries, and introduces a bias in favour of short-period systems. As a result, the components of many EBs are affected by the presence of the companion, and only a minority of EBs are representative of single stars. - Visual binaries (VBs) are another case where stellar masses can be obtained, when they are also SB2s. Formerly confined to the domain of long periods, they have moved into the domain of moderate periods (from a few weeks to decades) thanks to the development of interferometry, whether speckle [e.g. @McAlister96; @Balega07] or long-baseline [see e.g. @Pionier11]. The results are even better for EB, VB and SB2 systems combined [e.g. @Lester19; @Gallenne19]. However, the number of short period VB systems is still limited by the considerable resources required to observe an orbit, by the brightness required to observe a star and by the low luminosity contrast between the two components. - Astrometric binaries (ABs). We call here “astrometric binaries” unresolved double stars whose photocentre describes a measurable orbit, such as the 235 orbits observed by the “HIgh Precision PARallax COllecting Satellite” [Hipparcos, @ESA97]. The masses of the AB components may be derived when the system is also a VB [@Martin98], or an SB2 [@Jancart05]. The masses thus obtained have an accuracy of a few percent, at the best. However, much more precise masses should be obtained in the near future, thanks to astrometric measurements from the Gaia satellite. Compilations of the most accurate masses has been given by [@Torres10], and in the references mentioned above about the calibrations. This paper is the fifth in a series dedicated to the determination of precise masses using the [Gaia]{} satellite [@Gaia16]. Although the Gaia collaboration has already published two Data Releases [@GaiaDR1; @GaiaDR2], precise masses can only be calculated when the full data transits are available, with the full release for the nominal mission, according to the Gaia web site[^4]. The first paper of the series [@Halb2014 Paper I hereafter] presented the selection of about 70 SB2s for which the masses of the components could be precisely calculated by combining the astrometric transits of the [Gaia]{} satellite with precise radial velocity (RV) measurements obtained using the “Spectrographe pour l’Observation des PHénomènes des Intérieurs Stellaires et des Exoplanètes” [SOPHIE, @Perruchot] at the Haute-Provence Observatory. The selection included about fifty known SB2s, and about twenty SB1s that our first spectroscopic observations had transformed into SB2s by detecting the secondary component. Simultaneously with the spectroscopic observations, we obtained interferometric observations for five stars of our selection. These observations were carried out with the auxiliary telescopes of the ESO Very Large Telescope with the “Precision Integrated-Optics Near-infrared Imaging ExpeRiment” (PIONIER) instrument. In the second paper [@Halb2015 Paper II hereafter], they were already used to derive preliminary masses for the components of two SB2s, using published RV measurements completed with a few ones that we had obtained from a preliminary reduction of our [sophie]{} observations. We have thus demonstrated the possibility of using these measurements to validate the masses that we will later obtain from [Gaia]{}. In the third and in the fourth paper, [@Kiefer2016; @Kiefer2018 Paper III and Paper IV, respectively], we presented the RV measurements and the revised SB2 orbits of 10 and 14 binaries, respectively. Out of a total of 24 SB2s revised orbits, we found four for which an interferometric orbit had already been published. By combining the interferometric measurements of these stars with our RV measurements, we have calculated the masses of the components of these four binaries (one in Paper III and three in Paper IV). The present paper is particularly in line with the latter two, since we apply the same methods to treat 10 more SB2s. Five of these binary stars were resolved by long-base or speckle interferomery: One was found in the Fourth Catalog of Interferometric Measurements of Binary Stars[^5] [INT4 hereafter; Third catalogue: @Hartkopf01], one was observed by [@Balega07], and three have been observed for us with the PIONIER instrument attached to ESO’s Very Large Telescope Interferometer. The interferometric orbits of two of the latter were derived in Paper II, but the third is calculated here for the first time. Thus, we give here the masses of the components of the five binaries. The article is organized as follows: the observations are presented in Section \[sect:observations\]; this section includes the spectroscopic observations of 10 SB2s, but also the interferometric observations of one of these stars. The derivation of the RVs is in Section \[sect:RV\]. The elements of the spectroscopic orbits are derived in Section \[sect:SB2orbits\]. The masses of the components of 5 binaries separated by interferometry are derived in Section \[sect:interfero\], where we briefly discuss the evolutionary state of these stars and their positions in the HR diagram. Section \[sect:conclusion\] is the conclusion. Observations {#sect:observations} ============ Spectroscopic observations {#sect:observationsSpectro} -------------------------- As before, the observations were performed at the T193 telescope of the Haute-Provence Observatory, with the SOPHIE spectrograph. From 2010 to 2016, they were carried out in visitor mode by assigning priorities to the stars based on ephemerides. Since semester 2014B, we have regularly obtained observations in service mode that we request for selected dates in order to complete phase coverage while avoiding blends that would give unusable RV measurements. Exceptions to this rule are generally stars for which the preliminary orbit was inaccurate, or stars suspected for a time of being multiple systems. The list of stars treated in this article is given in Table \[tab:obs\], where the number of usable spectra and the number of cycles covered by the observations are indicated. A minimum of 11 usable spectra was requested in order to calculate orbital elements with reliable uncertainties. The exposure times have been adapted to the observation conditions in order to have a signal-to-noise ratio (SNR) appropriate for each star. The SNR is a compromise between the need to have sufficiently smooth spectra to distinguish the components, and the need to have exposure times shorter than one hour, which is the limit in service mode. [@l@[ ]{}l@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}l@]{} Name & Alt. name & V & Period$^a$ & $N_\text{spec}$ $^b$ & Span$^c$ & SNR$^d$\ HIP & HD/BD &(mag.) & (day) & &(period) &\ \ &&&&&&\ HIP20601 & HD27935 & 8.93 & 156 & 16 & 14 & 50\ HIP73449 & HD132756 & 7.31 & 2529 & 11 & 0.88 & 97\ HIP76006 & HD138525 & 6.39 & 582 & 12 & 4.6 & 142\ HIP77725 & BD+112874 & 9.36 & 1016 & 13 & 1.9 & 53\ HIP96656 & HD186922 & 8.04 & 4347 & 14 & 1.0 & 102\ HIP104987 & HD202447/8 & 3.93 & 99 & 14 & 12 & 371\ HIP117186 & HD222995 & 7.11 & 86 & 14 & 26 & 96\ \ &&&&&&\ HIP7134 & HD9313 & 7.81 & 53.5 & 16 & 50 & 98\ HIP61732 & BD+172512 & 9.18 & 595 & 11 & 4.3 & 46\ HIP101452 & HD196133 & 6.70 & 88 & 11 & 30 &129\ $^a$ The period values are taken from our solutions.\ $^b$ $ N_\text{spec}$ gives the number of spectra collected with SOPHIE and taken into account in the derivation of the orbital elements.\ $^c$ Span is the total time span of the observation epochs used in the orbit derivation, counted in number of periods.\ $^d$ SNR is the median signal-to-noise ratio of all the SOPHIE spectra of a given star at 5550Å. \[tab:obs\] The spectra are used to derive the RVs of the components, as explained in Section \[sect:RV\] hereafter. Interferometric measurements {#sect:interferObservations} ---------------------------- We obtained additional interferometric observations for one of the 10 SB2s, namely HIP 104987. This star was observed with the four 1.8 m Auxiliary Telescopes of ESO VLTI, using the PIONIER instrument [@Berger10; @Pionier11] in the H-band. Twelve set of observations were made, all resulting in the separation of the components. A first set of observations were done in Visitor Mode under Prog. ID 094.D-0624(A-F) on the nights of 6, 8, 17, 18, and 31 October 2014 for a total of 106 data points. The baseline was A1-G1-K0-J3. In addition, data were obtained in Service Mode, under Prog. ID 097.D-0688(A), on 7 epochs between 29 May 2016 and 25 August 2018. Each time 2 sets of observations were obtained, leading to a total of 70 data points. The baseline was A0-G1-J2-J3. The observations were reduced with the [pndrs]{} package presented by [@Pionier11]. For each epoch, the visibilities and closure phases were fitted to a binary model to determine the relative separation between the components, $\rho$, the position angle of the secondary component with respect to the primary, $\theta$, and the flux ratio in the $H$ band. The binary model is non-linear, and $\chi^2$ minimization can lead to several local minima. Therefore, a classical gridding approach is used to locate the deepest minimum in parameter space. A Levenberg-Marquardt algorithm is then used to derive the best-fitting parameters and the covariance matrix, from which are extracted the following parameters of the astrometric error ellipsoid: the semi-major axis, $\sigma_a$, the semi–minor axis, $\sigma_b$, and the position angle of the semi-major axis, $\theta_a$; by construction, $\theta_a$ is between 0 and 180$^{\rm o}$. These parameters are presented in Section \[sect:HIP104987\], where they are used to calculate the visual orbit of HIP 104987, then the masses of the components. Radial velocity measurements {#sect:RV} ============================ Choice of spectroscopic templates {#sect:templates} --------------------------------- [@lllrrrrr@]{} HIP/Name & $^a$$ T_\text{eff,1}$ & $^b$$ \log g_1$ & $ V_1 \sin i_1$ & $^c$$ [\text{Fe/H}]$ & $\alpha$ & $N_\text{spec}$ & Spectral orders\ & $ T_\text{eff,2}$ & $ \log g_2$ & $ V_2 \sin i_2$& & & & Median wavelength\ & (K) & (dex) & (km s$^{-1}$) & (dex) & (flux ratio) & & (Å)\ HIP7134 & 4754 $\pm$ 48 & 3.75 $\pm$ 0.12 & 4.2 $\pm$ 0.7 & -0.31 $\pm$ 0.04 & 0.045 $\pm$ 0.005 & 4 & 33\ & 5057 $\pm$ 243 & 5.13 $\pm$ 0.18 & 0 (fixed) & & & & 6142\ \ HIP20601 & 5628 $\pm$ 67 & 4.55 $\pm$ 0.07 & 3.4 $\pm$ 1.1 & -0.17 $\pm$ 0.05 & 0.105 $\pm$ 0.004 & 4 & 33\ & 4847 $\pm$ 83 & 5.18 $\pm$ 0.10 & 1.1 $\pm$ 0.7 & & & & 6142\ \ HIP61732 & 6021 $\pm$ 16 & 4.44 $\pm$ 0.00$_{MS}$ & 5.8 $\pm$ 0.1 & 0.16 $\pm$ 0.05 & 0.115 $\pm$ 0.009 & 2 & 24, 33\ & 5070 $\pm$ 326 & 4.59 $\pm$ 0.06$_{MS}$ & 3.0 $\pm$ 0.8 & & & & 5293, 6142\ \ HIP73449 & 5500 $\pm$ 110 & 4.42 $\pm$ 0.08 & 3.8 $\pm$ 0.3 & -0.39 $\pm$ 0.10 & 0.781 $\pm$ 0.115 & 4 & 33\ & 5400 $\pm$ 71 & 4.47 $\pm$ 0.07 & 4.7 $\pm$ 0.9 & & & & 6142\ \ HIP76006 & 6314 $\pm$ 26 & 4.10 $\pm$ 0.04 & 8.8 $\pm$ 1.1 & -0.03 $\pm$ 0.04 & 0.157 $\pm$ 0.034 & 4 & 24, 33\ & 6083 $\pm$ 152 & 4.72 $\pm$ 0.14 & 4.7 $\pm$ 0.4 & & & & 5293, 6142\ \ HIP77725 & 4378 $\pm$ 60 & 5.49 $\pm$ 0.05 & 2.7 $\pm$ 0.6 & -0.11 $\pm$ 0.04 & 0.972 $\pm$ 0.051 & 3 & 33\ & 4323 $\pm$ 29 & 5.45 $\pm$ 0.07 & 3.2 $\pm$ 0.9 & & & & 6142\ \ HIP96656 & 5128 $\pm$ 8 & 4.58$_{MS}$ & 3.2 $\pm$ 0.5 & -0.37 $\pm$ 0.06 & 0.486 $\pm$ 0.033 & 2 & 33\ & 4876 $\pm$ 36 & 4.63$_{MS}$ & 3.8 $\pm$ 0.1 & & & & 6142\ \ HIP101452 & 9767 $\pm$ 288 & 3.08 $\pm$ 0.10 & 21.7 $\pm$ 0.1 & 0.08 $\pm$ 0.06 & 0.254 $\pm$ 0.075 & 4 & 24, 33\ & 7915 $\pm$ 552 & 3.24 $\pm$ 0.13 & 32.2 $\pm$ 2.1 & & & & 5293, 6142\ \ HIP104987 & 5111 $\pm$ 7 & 3.08 $\pm$ 0.11 & 5.1 $\pm$ 0.1 & -0.09 $\pm$ 0.01 & 0.814 $\pm$ 0.032 & 4 & 24, 33\ & 7488 $\pm$ 223 & 3.87 $\pm$ 0.15 & 23.3 $\pm$ 5.9 & & & & 5293, 6142\ \ HIP117186 & 6208 $\pm$ 138 & 3.05 $\pm$ 0.17 & 42.1 $\pm$ 1.4 & -0.70 $\pm$ 0.02 & 0.347 $\pm$ 0.008 & 4 & 24, 33\ & 5785 $\pm$ 110 & 3.27 $\pm$ 0.19 & 13.2 $\pm$ 0.4 & & & & 5293, 6142\ \ Sun & 5836 $\pm$ 40 & 4.58 $\pm$ 0.10 & 4.9 $\pm$ 0.2 & -0.12 $\pm$ 0.04 & & 4 & 33\ & & & & & & 4 & 6142\ $^a$Minimum systematic uncertainties on T$_\text{eff}$ are about 100K.\ $^b$The MS subscript indicates that the $\log g$ did not converge to a realistic value ($>5$) and was fixed to be on the Main Sequence following $\log g = 12 - 2\log T_\text{eff}$ [@Angelov1996].\ $^c$Given the systematic error on \[Fe/H\]$_\text{sun}$, a more reliable value of uncertainty on \[Fe/H\] should be at least 0.1dex. Reliable RV measurements first require the choice of spectroscopic templates. As for the mask used in ordinary 1D-cross correlation function (1D-CCF), the choice of templates with a set of absorption lines as similar as possible to the actual absorption lines in the observed spectrum is crucial to the estimation of velocities, and to the value of the resulting masses. This is even more important in the present case, when the observed spectrum is the combination of two different components with different or similar sets of absorption lines. For that reason, the choice of the stellar parameters that characterise a spectrum was carefully optimised as explained hereafter. Before any further analysis, the SOPHIE multi-order spectra are reduced, flattened and normalised as explained hereafter: The spectra are deblazed, flattened, and the pseudo-continuum are normalized using a $p$-percentile filter [@Hodg1985]. The $\chi^2$ of the residuals of the prepared observed spectrum fitted by the sum of two similarly prepared model atmosphere templates from the PHOENIX database (Huber et al. 2011) is minimised with respect to $T_\text{eff}$, $\log g$, \[Fe/H\], $v\sin i$ and flux ratio $\alpha=F_B/F_A$ at 4916Å. Order 33 around the CaI line at ∼6120Åwas mainly used, but early type stars, such as HIP61732A, HIP76006A&B, HIP101452A&B, HIP104987B and HIP117186A, required the additional use of order 24, bluer and with deep and more numerous lines than on the red wing of their spectrum. When possible and for each binary, the stellar parameters were optimised for up to four observed spectra with the largest RV separation between the two components of the binary. In some case, the recursive algorithm leads to unreasonably low or high $\log g$. In those case, main sequence relation of $\log g$ with $T_\text{eff}$ is assumed. The results of this preliminary step are presented in Table \[tab:stellpar\]. As in paper IV, this table also gives the results of our method applied to the Sun, using Ceres and Vesta spectra. Since we measured the Sun metallicity to be of -0.12 dex, this could be considered as the realistic minimum uncertainty on the metallicities that we derived. It should be noted that, since the minimisation of the parameters is obtained by a recursive algorithm, the resulting parameters are not full-proof against systematics, because metallicity \[Fe/H\] and effective temperature $T_\text{eff}$ can be degenerate on ranges of order $\pm$400K and $\pm$0.5dex, especially when $v \sin i$ strongly departs from 0. The derivation of the Sun’s parameters using spectra observed with SOPHIE on Ceres and Vesta (see Table \[tab:stellpar\]) shows that the effective temperature and surface gravity are correctly derived within 100K and 0.1dex but the metallicity is underestimated by 0.12dex. Nevertheless, the derived model templates are the best-matching with respect to the observed spectra and lead to the best precision possible for RV derivation of the two binary components, as explained below. Derivation of the RVs using [todmor]{} {#Sect:TODMOR} -------------------------------------- After the templates have been fixed, the RVs of the components are derived using [todmor]{}, which is the multi-order version of the Two-Dimensional Cross-Correlation algorithm [todcor]{} [@zucker94; @zucker04]. [todmor]{} consists in cross-correlating the observed spectra with the sum of two templates each shifted with independent values of Doppler shift. All orders of the spectra are taken into account, except the few red orders with strong telluric absorptions. This leads to a direct measurement of each SB2 component radial velocities from the location of the 2D-CCF peak position. The RV uncertainties are given by the Hessian of the CCF peak, as explained in [@zucker04]. These uncertainties are intrinsic to the observed spectrum and do not reflect instrument systematics or Earth atmospheric turbulence effects. Therefore, they are generally underestimated. The resulting RVs and uncertainties are in Table 3. The orbital elements are calculated by correcting these misestimated uncertainties, as described in Section \[sect:SB2orbits\] below. \ \ \ \ \ Derivation of the spectroscopic orbits {#sect:SB2orbits} ====================================== The SB2 orbits are calculated by fitting SB models with a Levenberg–Marquardt algorithm, thanks to the routines in [@NumRec]. However, it is necessary to operate in several steps in order to correct the uncertainties in the RVs. The uncertainties of the RVs derived above are unreliable, which will lead to two types of error: first, the weights are inversely proportional to the squares of the uncertainties, and the RVs of one component could be overweighted relative to those of the other. Second, even if the uncertainties lead to exact relative weights, the uncertainties inferred from the covariance matrix will be false in the same proportions as the measurement uncertainties. For this reason, the uncertainties in Table \[tab:RVs\] are systematically corrected after each orbit computation, by using the $F_2$ estimator of the goodness-of-fit defined in @Kendall: $$\label{def:F2} F_2= \left( \frac{9\nu}{2} \right)^{1/2} \left[ \left( \frac{\chi^2}{\nu} \right)^{1/3}+{\frac{2}{9 \nu}}-1 \right]$$ where $\nu$ is the number of degrees of freedom and $\chi^2$ is the weighted sum of the squares of the differences between the predicted and the observed values, normalized with respect to their uncertainties. When the predicted values are obtained through a linear model, $F_2$ follows the normal distribution ${\cal N}(0,1)$. When non–linear models are used, but when the errors are small in comparison to the measurements, as hereafter, the model is approximately linear around the solution, and $F_2$ follows also ${\cal N}(0,1)$. Therefore, the $\chi^2$ of an orbit calculated with $\nu$ degrees of freedom must be close to the value corresponding to F2=0, which is: $$\label{def:khi20} \chi^2_0 = \nu \left( 1 - \frac{2}{9 \nu} \right)^3$$ The correction of the RV uncertainties is done as explained hereafter: First, the components are treated separately. A noise $\varepsilon$ is added quadratically to the RV uncertainties in order to have, for each component, an SB1 solution with $\chi^2 = \chi^2_0$. This gives the relative weights of the components in the calculation of the SB2 solution. The final uncertainties are derived by applying a multiplying factor, $\varphi=\sqrt{\chi^2/\chi^2_0}$, so that the SB2 solution satisfies the condition $F_2=0$. Thus, the corrected uncertainties are given by the equations \[eq:correction1\] and \[eq:correction2\] hereafter: $$\begin{aligned} \sigma^\text{corr}_{RV, 1} &= \varphi_1 \times \sqrt{\sigma_{RV, 1}^2 + \varepsilon_1^2} \label{eq:correction1}\\ \sigma^\text{corr}_{RV, 2} &= \varphi_2 \times \sqrt{\sigma_{RV, 2}^2 + \varepsilon_2^2} \label{eq:correction2}\end{aligned}$$ With the procedure described above, the coefficients $\varphi_1$ and $\varphi_2$ are necessarily equal. However, it sometimes happens that the [todmor]{} uncertainties of a component produce an SB1 orbit with a ${\chi}^2$ smaller than $\chi^2_0$. In this case, the noise $\varepsilon$ is null and the uncertainties are multiplied by the factor $\sqrt{\chi^2/\chi^2_0}$, which contributes to the $\varphi$–factor of the component. ---------------------- -------- -------------------------- --------------------- ----------------- --------------------- ----------------- --------------------- ----------------- --------------------- ----------------- HIP \# Reference of previous RV $\varepsilon_{1,p}$ $\varphi_{1,p}$ $\varepsilon_{2,p}$ $\varphi_{2,p}$ $\varepsilon_{1,n}$ $\varphi_{1,n}$ $\varepsilon_{2,n}$ $\varphi_{2,n}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ km s$^{-1}$ HIP 7134 [@GrifEm75] 0 0.931 $\ldots$ $\ldots$ 0.0123 1.009 0.1617 1.009 HIP 20601 [@GrifGZG85]$^a$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ 0 0.807 0.1052 1.029 HIP 61732 [@HaMaUd12]$^a$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ 0.0357 1.292 0. 1.292 HIP 73449 [@Goldberg02]$^b$ 0 0.898 0 0.714 0.0898 0.961 0.1183 0.961 HIP 76006 [@Griffin05]$^c$ 0 0.300 0 1.147 0.0693 1.049 0.2621 1.049 HIP 77725 [@Toko00]$^a$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ 0.1277 1.116 0.0742 1.116 HIP 96656 [@Balega07]$^a$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ 0.0060 1.255 0.0363 1.255 HIP 101452 no RV published $\ldots$ $\ldots$ $\ldots$ $\ldots$ 0 0.491 0.8295 0.877 HIP 104987 [@Massarotti08] 0 0.605 0 1.532 0.1167 0.857 2.0016 0.857 HIP 117186 [@Nordstrom97] 0 1.837 0 1.487 1.0851 1.203 0.2761 1.203 \[tab:corsigRVprev\] ---------------------- -------- -------------------------- --------------------- ----------------- --------------------- ----------------- --------------------- ----------------- --------------------- ----------------- $^a$ The previous measurements don’t improve the accuracy of the period, and they were not taken into account.\ $^b$ The components were swapped before calculating the uncertainties.\ $^c$ the radial velocities of the blend have been taken into account by assigning them an uncertainty of 0.340 km s$^{-1}$. The correction terms of the RV uncertainties of the 10 SB2 are listed in Table \[tab:corsigRVprev\]. HIP 20601 is the only exception to the “$\varphi_1=\varphi_2$” rule, for the reason explained above. The previously published RV measurements were also corrected. In most cases, the method was modified in order to maintain the relative weights of the measurements. For this purpose, corrections are based solely on multiplicative coefficients $\varphi_1$ and $\varphi_2$. Since obtained through another process, these RVs are generally biased and they reduce the reliability of the other orbital parameters. Therefore, they are used only to recalculate the period: When the period from our measurements completed by previously published measurements is at least 4 times more accurate than that from our measurements alone, the orbital elements are derived from the SOPHIE RVs, setting the period to this new value. ---------------- ------------ ---------------- --------------- ---------------- --------------- -------------- --------------- ----------------------- -------------- ------------- --------------- ------------------- -- -- -- HIP \# $P$ $T_0$(BJD) $e$ $V_0$ $\omega_1$ $K_1$ ${\cal M}_1 \sin^3 i$ $a_1 \sin i$ $N_1$ $d_{n-p}$ $\sigma(O_1-C_1)$ HD/BD $K_2$ ${\cal M}_2 \sin^3 i$ $a_2 \sin i$ $N_2$ $d_{2-1}$ $\sigma(O_2-C_2)$ (d) 2400000+ (km s$^{-1}$) ($^{\rm o}$) (km s$^{-1}$) (${\cal M}_\odot$) (Gm) (km s$^{-1}$) (km s$^{-1}$) HIP 7134 53.51164 55008.0873 0.39674 -15.1227 279.717 24.6998 0.5242 16.6843 16 0.011 HD 9313 $\pm 0.00012$ $\pm 0.0044$ $\pm 0.00027$ $\pm 0.0038$ $\pm 0.037$ $\pm 0.0058$ $\pm0.0025$ $\pm 0.0037$ 34.674 0.37342 23.422 16 0.4981 0.201 $\pm 0.075$ $\pm0.00096$ $\pm 0.051$ $\pm 0.0528$ HIP 20601 156.380540 56636.67055 0.851280 41.5967 201.984 37.3352 0.9050 42.1207 16 0.0080 HD 27935 $\pm 0.000095$ $\pm 0.00046$ $\pm 0.000031$ $\pm 0.0041$ $\pm 0.015$ $\pm 0.0031$ $\pm0.0016$ $\pm 0.0056$ 50.322 0.67143 56.772 16 0.1544 0.131 $\pm 0.040$ $\pm0.00065$ $\pm 0.045$ $\pm 0.0509$ HIP 61732 595.18 57210.58 0.3393 -15.956 64.85 9.197 0.3326 70.81 11 0.043 BD +17 2512 $\pm 0.20$ $\pm 0.87$ $\pm 0.0019$ $\pm 0.032$ $\pm 0.67$ $\pm 0.021$ $\pm0.0022$ $\pm 0.17$ 13.068 0.2341 100.61 11 0.3466 0.084 $\pm 0.034$ $\pm0.0014$ $\pm 0.27$ $\pm 0.0736$ HIP 73449$^a$ 2528.6 57708.5 0.3752 8.147 99.54 10.175 0.8928 327.9 11 0.3868 0.097 HD 132756 $\pm 1.0$ $\pm 4.0$ $\pm 0.0023$ $\pm 0.038$ $\pm 0.74$ $\pm 0.048$ $\pm0.0080$ $\pm 1.5$ (45) $\pm 0.1434$ (0.862) 10.253 0.8861 330.4 11 0.0041 0.069 $\pm 0.037$ $\pm0.0092$ $\pm 1.2$ (45) $\pm 0.0515$ (0.758) HIP 76006$^a$ 581.816 56717.21 0.6460 -47.331 134.51 13.640 0.4004 83.31 12 -0.9549 0.052 HD 138525 $\pm 0.025$ $\pm 0.23$ $\pm 0.0033$ $\pm 0.031$ $\pm 0.51$ $\pm 0.048$ $\pm0.0052$ $\pm 0.18$ (51) $\pm 0.0521$ (0.310) 16.47 0.3316 100.60 12 0.4104 0.264 $\pm 0.11$ $\pm0.0027$ $\pm 0.61$ (51,15$^b$) $\pm 0.1039$ (1.317$^b$) HIP 77725 1016.2 56911.2 0.3459 -0.335 329.0 6.749 0.1079 88.49 13 0.135 BD +11 2874 $\pm 1.2$ $\pm 2.3$ $\pm 0.0063$ $\pm 0.049$ $\pm 1.3$ $\pm 0.061$ $\pm0.0016$ $\pm 0.75$ 6.780 0.1074 88.90 13 -0.1453 0.070 $\pm 0.044$ $\pm0.0020$ $\pm 0.50$ $\pm 0.0722$ HIP 96656 4350.0 57737.8 0.24294 -3.2920 70.03 7.3500 0.7879 426.49 14 0.008 HD 186922 $\pm 2.2$ $\pm 2.7$ $\pm 0.00066$ $\pm 0.0061$ $\pm 0.26$ $\pm 0.0049$ $\pm0.0035$ $\pm 0.35$ 8.062 0.7183 467.8 14 0.1267 0.053 $\pm 0.017$ $\pm0.0020$ $\pm 1.0$ $\pm 0.0176$ HIP 101452 87.6834 57097.926 0.6806 -7.608 243.39 35.272 1.361 31.158 11 0.050 HD 196133 $\pm 0.0020$ $\pm 0.049$ $\pm 0.0013$ $\pm 0.058$ $\pm 0.14$ $\pm 0.045$ $\pm0.020$ $\pm 0.039$ 51.08 0.9396 45.12 11 0.0295 0.772 $\pm 0.34$ $\pm0.0080$ $\pm 0.30$ $\pm 0.3153$ HIP 104987$^a$ 98.8051 57249.523 0$^c$ -16.445 0$^c$ 16.190 0.239 21.998 14 -0.2766 0.088 HD 202447/8 $\pm 0.0023$ $\pm 0.042$ (fixed) $\pm 0.027$ (fixed) $\pm 0.036$ $\pm0.016$ $\pm 0.049$ (108) $\pm 0.0678$ (0.645) 18.92 0.2043 25.70 14 0.6472 1.470 $\pm 0.61$ $\pm0.0071$ $\pm 0.82$ (108) $\pm 0.4715$ (1.503) HIP 117186$^a$ 85.8244 56402.66 0.3339 -21.03 176.78 32.59 1.636 36.26 14 -0.3657 1.367 HD 202447/8 $\pm 0.0013$ $\pm 0.15$ $\pm 0.0043$ $\pm 0.39$ $\pm 0.67$ $\pm 0.47$ $\pm0.026$ $\pm 0.52$ (19) $\pm 0.5651$ (1.980) 40.79 1.307 45.38 14 0.9052 0.276 $\pm 0.14$ $\pm0.037$ $\pm 0.18$ (19) $\pm 0.4392$ (1.175) \[tab:orbSB2\] ---------------- ------------ ---------------- --------------- ---------------- --------------- -------------- --------------- ----------------------- -------------- ------------- --------------- ------------------- -- -- -- $^a$ The elements were derived fixing $P$ to the value obtained taking also the previous measurements into account.\ $^b$ Fifteen blend measurements were taken into account, with $\sigma(O-C)=0.342$ km s$^{-1}$.\ $^c$ We have assumed a circular orbit since our RVs give the eccentricity $e=0.00000002 \pm 0.0022$. The orbital elements of the 10 stars are presented in Table \[tab:orbSB2\]. They include the following parameters: the period, $P$, the periastron epoch, $T_0$, the eccentricity, $e$, the systemic radial velocity, $V_0$, the periastron longitude, $\omega_1$, the RV semi–amplitudes of both components, $K_1$ and $K_2$, and the offset of the RVs of the secondary component from that of the primary component, $d_{2-1}$. When the previously published RV measurements were taken into account to derive the period, the offset of the SOPHIE RVs, $d_{n-p}$, is indicated. The table also includes the minimum masses, ${\cal M}_1 \sin^3 i$ and ${\cal M}_2 \sin^3 i$, and the minimum semi-major axes, $a_1 \sin i$ and $a_2 \sin i$, which are derived from the solution terms. The spectroscopic orbits are shown in Figure \[fig:orbSB2\] and the residuals are in Figure \[fig:resOrbSB2\]. Among the ten SB2s, HIP 20601 belongs to a multiple system, since [@GrifGZG85] found a faint distant visual component. This star was observed by the Gaia satellite as Gaia DR2 3283823383389256064 [@GaiaDR2]. Its separation relative to HIP 20601 was 7.09 arcsec, well above the diameter of the fibre of the SOPHIE spectrometer, which is 3 arcsec. Although an observation of HIP 20601 by poor seeing observation could still be contaminated at such a distance, the low magnitude of this companion ($G$ = 12.97 mag i.e. 4 magnitudes fainter than HIP 20601) makes such contamination perfectly negligible, as evidenced by the low residuals in the SB2 orbit, Figure \[fig:resOrbSB2\]. Masses and parallaxes of five stars resolved by interferometry {#sect:interfero} ============================================================== Calculation method {#sect:correctionBV} ------------------ Of the ten SB2s, five were sufficiently observed by interferometry to calculate a combined orbit giving the masses of the components and the trigonometric parallax of the system. However, before combining RVs and interferometric measurements, uncertainties must first be corrected. This has been done for the RVs in Section \[sect:SB2orbits\], but remains to be done for interferometric measurements. The method is similar to that applied to RV uncertainties in Section \[sect:SB2orbits\]: The visual orbit is calculated and the $\chi^2$ is considered. From this we deduce the corrective coefficient that must be applied to the uncertainties for $F_2=0$. After this correction, the combined spectroscopic and interferometric orbit is derived from the RVs and from the relative positions. The solutions terms are $P$, $T_0$, $e$, $V_0$, $\omega_1$, and $d_{2-1}$ as for the SB2 orbits, but also the position angle of the ascending node, $\Omega$, the inclination of the orbital plane, $i$, the masses of the components, ${\cal M}_1$ and ${\cal M}_2$, and the trigonometric parallax of the binary star, $\varpi$. The apparent semi–major axis, $a$, is finally derived from $P$, ${\cal M}_1$, ${\cal M}_2$ and $\varpi$. The uncertainties of the solution terms are extracted from the variance-covariance matrix of the Levenberg–Marquardt calculation, but the uncertainty of $a$ is estimated by simulations, as explained hereafter: The solution terms that we have derived are used to calculate the RVs or the relative position for each observation epoch. Simulated measurements are produced by adding, to these model values, errors generated according to the ${\cal N}(0,\sigma_0)$ distribution, where $\sigma_0$ is the measurement uncertainty we finally obtained. A value of $a$ is then calculated from the set of simulated measurements. The uncertainty of $a$ is the standard deviation of the values thus obtained. The simulation program was also used to verify that the uncertainty correction process presented above and in Section \[sect:SB2orbits\] does not introduce an error that would add to the uncertainties estimated from the Levenberg–Marquardt calculation. We have implemented the correction of the uncertainties of the simulated measurements by a multiplicative coefficient in order to have a zero $\chi^2$ after the calculation of the SB1 orbit of each component, then the SB2 orbit, as well as the interferometric orbit. It thus appeared that the standard deviations of the solution terms calculated in the simulations were not affected by this modification, and remained equal to the uncertainties we had found. Although the uncertainty correction implemented in the simulation is slightly simpler than that applied to real measurements, this shows the robustness of the correction process. The simulations also allowed us to verify the absence of anomalies in the correlations between the different orbital parameters. The results obtained for the five binaries are presented in Table \[tab:VB+SBorb\]. In this table, the standard deviation of the astrometric residuals of the combined solution is followed by the standard deviation of the astrometric-only solution, in parentheses. A comparison between these two terms shows that they are quite close, and such a similarity also appears if one compares the standard deviations of the radial velocities, $\sigma_{(o-c)\;RV}$, with the values in Table \[tab:orbSB2\]. This resemblance reflects the compatibility between the astrometric and spectroscopic contributions of the combined solution. HIP20601 HIP77725 HIP96656 HIP104987 HIP117186 ------------------------------------ --------------------------- ------------------------- ----------------------- ------------------------ ----------------------- $P$ (days) 156.380534 $\pm$ 0.000094 1015.53 $\pm$ 0.55 4345.3 $\pm$ 1.4 98.80450 $\pm$ 0.00035 85.8364 $\pm$ 0.0064 $T_0$ (BJD-2400000) 56636.67052 $\pm$ 0.00046 56904.5 $\pm$ 1.6 57739.1 $\pm$ 2.0 57277.7 $\pm$ 1.7 56402.368 $\pm$ 0.094 $e$ 0.851282 $\pm$ 0.000031 0.3415 $\pm$ 0.0017 0.24280 $\pm$ 0.00065 0.00417 $\pm$ 0.00076 0.32778 $\pm$ 0.00073 $V_0$ (km s$^{-1}$) 41.5968 $\pm$ 0.0041 -0.356 $\pm$ 0.043 -3.2884 $\pm$ 0.0053 -16.458 $\pm$ 0.027 -21.11 $\pm$ 0.36 $\omega_1$ ($^{\rm o}$) 201.983 $\pm$ 0.015 325.51 $\pm$ 0.81 70.18 $\pm$ 0.20 102.9 $\pm$ 6.3 175.50 $\pm$ 0.34 $\Omega$($^{\rm o}$; eq. 2000) 340.513 $\pm$ 0.055 120.07 $\pm$ 0.50 292.78 $\pm$ 0.16 216.57 $\pm$ 0.16 16.942 $\pm$ 0.047 $i$ ($^{\rm o}$) 103.133 $\pm$ 0.072 36.49 $\pm$ 0.76 80.377 $\pm$ 0.097 151.52 $\pm$ 0.28 88.047 $\pm$ 0.043 $a$ (mas) 11.338 $\pm$ 0.022 105.59 $\pm$ 0.73 189.38 $\pm$ 0.63 12.105 $\pm$ 0.013 4.677 $\pm$ 0.034 ${\cal M}_1$ (${\cal M}_\odot$) 0.9798 $\pm$ 0.0019 0.510 $\pm$ 0.029 0.8216 $\pm$ 0.0037 2.20 $\pm$ 0.16 1.647 $\pm$ 0.022 ${\cal M}_2$ (${\cal M}_\odot$) 0.72697 $\pm$ 0.00094 0.508 $\pm$ 0.029 0.7491 $\pm$ 0.0022 1.883 $\pm$ 0.083 1.316 $\pm$ 0.034 $\varpi$ (mas) 16.703 $\pm$ 0.034 53.1 $\pm$ 1.3 31.26 $\pm$ 0.11 18.11 $\pm$ 0.24 8.551 $\pm$ 0.080 $H$ (mag) 7.209 $\pm$ 0.047 8.489 $\pm$ 0.010 5.980 $\pm$ 0.023 2.442 $\pm$ 0.196 6.252 $\pm$ 0.031 $\Delta H$ (mag) 0.9990 $\pm$ 0.0158 0.06$^a$ $\pm$ 0.02$^a$ 0.44 $\pm$ 0.24$^b$ 2.1303 $\pm$ 0.0286 0.8914 $\pm$ 0.0074 $d_{2-1}$ (km s$^{-1}$) 0.154 $\pm$ 0.050 -0.104 $\pm$ 0.053 0.119 $\pm$ 0.016 0.676 $\pm$ 0.471 1.067 $\pm$ 0.374 $\sigma_{(o-c)\;VB}$ (mas) 0.031 (0.024) 3.95 (3.75) 2.06 (2.07) 0.652 (0.665) 0.0080 (0.0081) $\sigma_{(o-c)\;RV}$ (km s$^{-1}$) 0.0081, 0.131 0.116, 0.124 0.012, 0.050 0.090, 1.488 1.521, 0.298 $^a$ according to [@Horch17].\ $^b$ from the data of [@Balega07].\ \[tab:VB+SBorb\] HIP 20601 {#sect:HIP20601} --------- The interferometric measurements we obtained for this star have been published in Table 1 of Paper II, where the uncertainties were corrected so that the visual orbit had $F_2=0$. Unlike Paper II, the spectroscopic part now consists only of our [sophie]{} observations, with the reduction by [todmor]{} seen above, which ensures much more reliable results. The parameters of the combined orbit are in Table \[tab:VB+SBorb\]. The masses of the components are determined with a remarkable accuracy of 0.19% and 0.13% (about twice better than in Paper II). The orbit remains visually very close to that shown in Figure 1 of Paper II, and it is useless to reproduce it here. The trigonometric parallax is 4.6 $\sigma$ larger than that found in the second [Gaia]{} DR (DR2), which is (17.32 $\pm$ 0.13) mas. The difference is probably due to the orbital motion, which was ignored in the reduction of the [Gaia]{} DR2. In addition, we note that our uncertainty is 3.8 times smaller than that of [Gaia]{} DR2. HIP 77725 {#sect:HIP77725} --------- --------------- -------- -------------- ------------ ------------ -------------- $T$-2,400,000 $\rho$ $\theta$ $\sigma_a$ $\sigma_b$ $\theta_a$ (BJD) (mas) ($^{\rm o}$) (mas) (mas) ($^{\rm o}$) 49115.344 109.0 56.4 7.66 6.04 146.4 49116.257 103.0 56.2 7.24 6.04 146.2 50178.528 115.3 68.4 4.05 4.03 158.4 50591.726 102.0 135.7 7.17 6.04 45.7 52393.977 132.0 101.9 4.03 2.78 101.9 53898.921 68.0 293.5 4.03 3.11 113.5 54638.756 107.0 140.4 6.04 4.89 140.4 56725.859 90.9 156.2 1.21 0.638 156.2 57085.842 84.8 5.0 0.298 0.201 95.0 57220.616 103.8 50.9 1.21 1.09 50.9 57220.616 103.4 51.7 0.727 0.604 141.7 --------------- -------- -------------- ------------ ------------ -------------- : The interferometric measurements of HIP 77725, taken from the INT4 catalogue and adapted to our purpose. $\rho$ is the apparent separation and $\theta$ is the position angle of the secondary component. $\sigma_a$ and $\sigma_b$ are the semi-major axis and the semi-minor axis of the ellipsoid error, respectively; they are derived as explained in the text. $\theta_a$ is the position angle of the major axis of the ellipsoid error. The position angles are all given for the equinox of the observation epoch. \[tab:HIP77725-interfero\] This star is the visual binary BAG 7 and the Sixth Catalogue of Orbits of Visual Binary Stars[^6] mentions a combined visual and spectroscopic orbit by [@Toko00]. We found several interferometric measurements in the INT4 catalogue, and we selected the 11 of them that had measurement uncertainties. A component inversion was corrected for the observation of 2008.4717, and we put the measurements in the same format as the PIONIER measurements: times in years were converted to Julian days, and the uncertainties on $\theta$ and $\rho$ were converted into uncertainty ellipsoids aligned with the apparent separation, $\rho$. A first calculation of the visual orbit then gives a solution with $F_2=4.82$. The uncertainties $\sigma_a$ and $\sigma_b$ were therefore corrected by multiplying them by 2.016 to obtain a visual orbit of $F_2=0$. The positions thus transformed and the final uncertainties are in Table \[tab:HIP77725-interfero\]. The combined orbit was derived, and the solution terms in Table \[tab:VB+SBorb\] were obtained. The interferometric orbit and the residuals are shown in Figure \[fig:HIP77725\]. The masses of the components are slightly different but much more accurate than those derived by [@Toko00], which were ${\cal M}_1=(0.48 \pm 0.13)$ ${\cal M}_\odot$ and ${\cal M}_2=(0.46 \pm 0.12)$ ${\cal M}_\odot$. The [Gaia]{} DR2 gives the trigonometric parallax $\varpi=47.29 \pm 0.17$ mas, which is 4.4 $\sigma$ smaller than our result. Again, the difference may be due to the orbital motion. HIP 96656 {#sect:HIP96656} --------- This star is the nearby star GJ 765.2, and the double star MLR 224. [@Balega07] observed it with the 6m “large altazimuth telescope” (Russian: Bolshoi Teleskop Alt-azimutalnyi, or BTA6), and obtained high-precision speckle measurements. They also took over visual measurements of lower quality, and combined all these measurements with radial velocities measured with spectrovelocimeters. They thus determined the masses of the components with an accuracy of 2.4 to 2.5 %. We have ignored the visual measurements because of their poor quality, but have taken the BTA6 speckle measurements. As for HIP 77725, the epoch in years were converted in Julian days, and the parameters of the error ellipsoid, $\sigma_a$, $\sigma_b$ and $\theta_a$, were derived from the position angle $\theta$ and from the uncertainties on $\theta$ and $\rho$ given in Section 2 of the paper by Balega et al.; the uncertainties were increased by 2.4 % in order to obtain a visual orbit with $F_2=0$. The measurement in Table \[tab:HIP96656-interfero\] were thus obtained. By combining the speckle measurements and our radial velocities, we found a combined orbit including the visual part presented in Figure \[fig:HIP96656\]. The elements are in Table \[tab:VB+SBorb\]. The mass accuracy is now 0.45 % for the primary component and 0.29 % for the secondary, 5 and 8 times better than in Balega et al., respectively. The parallax of the combined solution is $31.26 \pm 0.11$ mas, in disagreement with the $33.67 \pm 0.53$ mas given by the [Gaia]{} DR2. Again, the difference probably comes from the orbital motion, neglected in the Gaia reduction. --------------- -------- -------------- ------------ ------------ -------------- $T$-2,400,000 $\rho$ $\theta$ $\sigma_a$ $\sigma_b$ $\theta_a$ (BJD) (mas) ($^{\rm o}$) (mas) (mas) ($^{\rm o}$) 49116.111 35.0 151.7 4.08 1.87 151.7 49296.029 52.0 264.7 4.08 2.78 84.7 49613.425 129.0 285.1 1.53 1.15 105.1 50736.983 165.7 303.3 1.53 1.47 123.3 50736.983 165.9 303.0 1.53 1.48 123.0 51097.842 111.0 311.3 1.53 0.988 131.3 51476.160 46.0 348.3 1.53 0.409 168.3 51865.253 68.0 80.8 1.53 0.605 80.8 52184.474 122.0 99.3 1.53 1.09 99.3 52215.483 128.0 100.2 1.53 1.14 100.2 52567.577 165.0 108.3 1.76 1.70 108.3 52567.577 165.0 107.8 1.76 1.70 107.8 52567.577 163.0 107.9 1.76 1.68 107.9 52567.577 163.0 107.8 1.76 1.68 107.8 53303.175 82.0 126.8 1.53 0.730 126.8 53895.963 115.0 282.6 1.53 1.02 102.6 --------------- -------- -------------- ------------ ------------ -------------- : The interferometric measurements of HIP 96656, taken from Balega et al. (2007) and adapted to our purpose. $\rho$ is the apparent separation and $\theta$ is the position angle of the secondary component. $\sigma_a$ and $\sigma_b$ are the semi-major axis and the semi-minor axis of the ellipsoid error, respectively; they are derived as explained in the text. $\theta_a$ is the position angle of the major axis of the ellipsoid error. The position angles are all given for the equinox of the observation epoch. \[tab:HIP96656-interfero\] HIP 104987 {#sect:HIP104987} ---------- --------------- -------- -------------- ------------ ------------ -------------- $T$-2,400,000 $\rho$ $\theta$ $\sigma_a$ $\sigma_b$ $\theta_a$ (BJD) (mas) ($^{\rm o}$) (mas) (mas) ($^{\rm o}$) 47690.973 9.63 306.6 1.31 0.0540 90.9 47695.903 10.30 283.0 0.303 0.0270 104.5 47698.971 10.62 270.7 0.830 0.0607 90.8 47700.944 10.38 262.3 2.15 0.108 94.7 47715.955 12.39 213.8 2.59 0.385 80.6 47720.922 11.79 198.3 2.49 1.51 114.8 47746.781 10.25 96.7 1.69 0.324 106.5 47747.841 10.76 92.2 0.938 0.243 88.9 47751.822 11.33 78.0 0.668 0.121 88.6 47758.762 11.55 53.5 0.303 0.148 78.1 47761.720 11.62 41.5 0.762 0.148 108.2 47765.738 12.36 36.0 1.93 0.0674 99.4 47784.730 10.55 324.1 1.39 0.162 85.5 47813.730 11.21 209.6 2.77 1.11 74.4 47816.652 12.98 213.2 2.05 0.425 75.4 47817.639 11.28 201.8 0.938 0.256 77.9 47832.687 10.85 150.2 0.634 0.0607 69.9 47836.668 10.66 134.5 0.492 0.0540 76.1 48068.889 11.95 10.2 0.911 0.148 103.7 48070.934 13.01 342.0 3.18 0.499 93.2 48101.870 11.48 242.4 0.101 0.0270 86.4 48102.856 11.73 239.7 0.223 0.0540 88.8 48104.829 11.79 232.8 0.175 0.0338 94.9 48130.834 10.68 143.4 0.128 0.0270 82.4 48133.756 10.47 131.1 0.277 0.0472 86.3 48134.778 10.52 127.1 0.196 0.0338 85.8 48136.751 10.54 118.7 0.135 0.0338 96.2 48137.810 11.41 112.4 0.445 0.0540 81.4 48149.753 11.92 69.7 0.229 0.0540 81.4 56937.598 11.20 89.85 0.0600 0.0248 132.0 56939.599 11.37 82.22 0.0583 0.0229 135.0 56948.578 12.06 51.41 0.0724 0.0212 137.0 56949.537 12.09 48.49 0.0795 0.0530 147.0 56962.538 11.69 5.35 0.0707 0.0300 134.0 57537.870 11.73 63.44 0.0795 0.0406 172.0 57569.873 10.63 -50.42 0.1113 0.0883 119.0 57597.710 12.02 -151.36 0.0530 0.0300 4.0 57599.740 11.96 -158.26 0.0848 0.0335 15.0 57600.725 11.88 -161.20 0.1060 0.0459 147.0 57622.707 10.77 115.44 0.0741 0.0371 152.0 57625.659 10.88 103.52 0.0512 0.0318 3.0 --------------- -------- -------------- ------------ ------------ -------------- : The interferometric measurements of HIP 104987. $\rho$ is the apparent separation and $\theta$ is the position angle of the secondary component, defined as the lightest. $\sigma_a$ and $\sigma_b$ are the semi-major axis and the semi-minor axis of the ellipsoid error, respectively; they are corrected as explained in the text. $\theta_a$ is the position angle of the major axis of the ellipsoid error. The position angles are given for the equinox of the observation epoch. Observations dating back to before JD $2\,450\,000$ come from Armstrong et al. (1992), after reversing the components. The others were carried out with the Auxiliary Telescopes of the ESO VLTI, using the PIONIER instrument. \[tab:HIP104987-interfero\] The INT4 catalogue contains many observations of this star, including a series of 29 measurements from the Mark III Optical Interferometer published by [@MkT1992b]. These measurements are individually less accurate than PIONIER measurements, but their number still improves the accuracy of orbital parameters. We have therefore added them to our own measures, as follows: - The uncertainties of each set of measurements are corrected as explained in Section \[sect:correctionBV\], by calculating the interferometric orbit with each of them. A correction coefficient of 0.6086 is thus found for the uncertainties of the Armstrong’s measurements, and a coefficient of 0.1626 for ours. - A comparison between the orbital elements from Armstrong et al. and those from PIONIER shows that the components have been inverted. Since PIONIER’s position angles are compatible with the spectroscopic orbit, we correct the position angles of Armstrong et al. by 180 degrees. - The interferometric orbit is derived again from the two sets together. The $F_2$ estimator of the orbit is 1.03, inferring an acceptable compatibility between the two sets. An additional correction of 1.0856 is still applied in order to have $F_2=0$. The measurements with corrected uncertainties are presented in Table \[tab:HIP104987-interfero\]. They were used to derive the combined spectroscopic and interferometric solution which is presented in Table \[tab:VB+SBorb\], and in Figure \[fig:HIP104987\]. The eccentricity of the orbit is very small for a binary with a period of nearly 100 days, and this is probably due to the evolution of the primary component to the current G6 IV type. Our results provide a relevant insight into the achievement of the circularization of the orbit: The SB2 solution given in Table \[tab:orbSB2\] is circular since the calculation of an eccentric orbit lead to an eccentricity that is clearly not significant, with $e/\sigma_e = 1.8\,10^{-8} \pm 2.24\,10^{-3}$; this is in agreement with [@Eggleton17], who assumed that the orbit is circular. On the other hand, the eccentricity of the interferometric orbit is $(4.76 \pm 0.82)\,10^{-3}$. This value is significant, compatible at the 2 $\sigma$-level with the value of the SB2 solution, and we observe that the interferometric measurements from the INT4 catalogue give nearly the same periastron longitude as the PIONIER measurements: $\omega =(283.8 \pm 8.3)^{\rm o}$ for the former and $\omega =(288 \pm 17)^{\rm o}$ for the latter. Therefore, we conclude that the orbit of HIP 104987 is not perfectly circular, and we adopt the eccentricity of the combined orbit, that is in Table \[tab:VB+SBorb\]; this value is significant at the 5.5 $\sigma$-level. The masses we found are compatible with those of [@Eggleton17], which are respectively 2.000 and 1.847 solar masses. This star is not included in the [Gaia]{} DR2, but the parallax provided by the [Hipparcos 2]{} catalogue [@vanLeeuwen07] is (17.14 $\pm$ 0.21) mas, which is 3.0 $\sigma$ less than our result. The difference may be due to an underestimation of the Hipparcos uncertainty. HIP 117186 {#sect:HIP117186} ---------- As well as HIP 20601 and HIP 104987, this star was observed by interferometry with the PIONIER instrument. Its measurements were presented in Paper II. By combining them with the RVs of this star, we have obtained the elements shown in Table \[tab:VB+SBorb\]. These are not really different from the preliminary elements given in Paper II, but they are more reliable, due to the better quality and homogeneity of the RVs. The parallax is 3.7 $\sigma$ larger than that found in the [Gaia]{} DR2, which is $(8.215 \pm 0.044)$ mas. Hertzsprung-Russell diagram and mass-luminosity relation {#sect:diagHR-ML} -------------------------------------------------------- Figure \[fig:diagHR\] shows the location in the Hertzsprung-Russell (hereafter HR) diagram of the ten stars in the five SB2 systems for which we have inferred the individual masses, while Figure \[fig:ML\] shows their position in the mass-luminosity plane. To draw these figures, we started from the effective temperatures, metallicities, and surface gravities given in Table \[tab:stellpar\]. We corrected the metallicity by adding $+0.12$ to each \[Fe/H\]-value to account for the fact that in our study the solar spectrum is adjusted with $\mathrm{[Fe/H]}_\odot = -0.12$ (as specified in the table caption). Following the conservative guidelines given in the footnotes of Table \[tab:stellpar\], we added quadratically a systematic error of 100 K on the effective temperature and 0.10 on \[Fe/H\]. For two systems, HIP 20601 and HIP 117186, the spectroscopic values in Table \[tab:stellpar\] differ from values in the literature. HIP 20601 is a probable member of the Hyades open cluster. Our corrected value of the metallicity (\[Fe/H\]=$-0.05 \pm 0.13$) differs by $2\sigma$ from the recent determination of the cluster average metallicity by @Dutra-Ferreira16, \[Fe/H\]=0.18$\pm$0.03 which is a robust value. If we fix the metallicity to this latter value and optimize again all other parameters, we get $T_\text{eff,A}$=6100$\pm$180K, $T_\text{eff,B}$=4600$\pm$90K, $\log g_A$=4.9$\pm$0.2, $\log g_B$=5.2$\pm$0.2, and unchanged $V\sin i$’s and flux ratio. As for HIP 117186, we noticed that the effective temperature of the A-component ($T_\mathrm{eff, A}$ = 6208 $\pm$ 138 K) in Table \[tab:stellpar\] is smaller by more than 600 K than the value derived by @Casagrande11 by photometric calibration ($T_\mathrm{eff, A}$ = 6853 $\pm$ 80 K), while @Casagrande11’s metallicity (\[Fe/H\]= $0.11\pm 0.10$) is higher than our corrected value, although still within the error bars. If we take @Casagrande11’s metallicity as a robust value and optimize again, we get $T_\text{eff,A}$=7200$\pm$30K, $T_\text{eff,B}$=7150$\pm$70K, $\log g_A$=3.8$\pm$0.2, $\log g_B$=4.1$\pm$0.2. In the following, we adopt these latter sets of parameters for the two couples. To derive the individual luminosities, we proceeded in two steps. We first derived the individual apparent magnitudes from the magnitude difference between the components of HIP 20601, HIP 96656, HIP 104987, and HIP 117186 measured in the $H$-band by PIONIER. For HIP 77725, we used the magnitude difference in infrared measured by @Horch17 with the WYIN telescope. Then, we calculated the luminosities from the apparent magnitudes, the trigonometric parallaxes of Table \[tab:VB+SBorb\], and the bolometric corrections of @Casagrande18, the latter being functions of the effective temperature, metallicity, and surface gravity. In Figures \[fig:diagHR\] and \[fig:ML\], we show the position of several stellar evolutionary tracks and isochrones taken from the updated BaSTI database [@Hidalgo18]. These models include state-of-the-art input physics and add overshooting of convective cores on the main sequence. As indicated on the figure, the evolutionary tracks correspond to the range of metallicities and masses of the studied SB2 members, while the isochrones fit the position of some of the stars. We now discuss the figure for each SB2 couple: - HIP 20601. Due to their rather low mass, the stars are located in a region of the HR diagram where the evolution proceeds quite slowly. Therefore, they cannot be age-dated in this diagram. Since the system is a member of the Hyades cluster, we can assume that like the Hyades, it is aged $\sim 600-700$ Myr [@Lebreton01]. Indeed, we can see in the mass-luminosity plane (Fig. \[fig:ML\]) that both components can be put on an isochrone of $700$ Myr corresponding to the Hyades metallicity. - HIP 77725. The stars have very similar masses. Due to their low mass, they are located in a region of the HR diagram where the evolution proceeds very slowly. Therefore their age is mostly undetermined. We notice that while their effective temperatures as derived in this study appear to be too hot with respect to those expected from the models, the stars reasonably fit in the theoretical mass-luminosity relation. - HIP 96656. Again, the components have low mass and evolve slowly. In Fig. \[fig:diagHR\] the position of the components are well-fitted on the zero age main sequence at their metallicity, however the system cannot be age–dated accurately. Also, we note that both stars well fit in the theoretical mass-luminosity relation corresponding to their metallicity. - HIP 104987. The two components can be positioned on isochrones of ages in the range $\approx 1000 - 1200$ Myr (the isochrone plotted is of 1100 Myr). This age is older than the age estimated by @Griffin02 which is not surprising since both the observed properties and stellar models have been considerably updated in the meantime. The system is evolved: according to BaSTI stellar models, the A-component lies close to the base of the red giant branch while the B-one is reaching the end of the main sequence. Both components sit on their isochrone in the theoretical mass-luminosity plane. - HIP 117186. First, we point out that if we had taken the spectroscopic values of the effective temperatures and metallicity in Table \[tab:stellpar\], it would not have been possible to place the stars on the same isochrone. On the other hand, with the revised $T_\mathrm{eff}$ based on the metallicity value of @Casagrande11, the components sit on an isochrone of $1500$ Myr in the HR diagram as well as in the mass-luminosity plane. We conclude that the properties of the SB2 couples are overall well retrieved by theoretical models in the mass-luminosity plane. Concerning the HR diagram, the fit is also satisfactory for the 5 systems, once the metallicity and, as a consequence, the effective temperatures of one of them (HIP 117186) have been revised after we adopted robust \[Fe/H\] determination from the literature. The masses we determined in this study are very precise, while the characterization of the stars would benefit from further improvements in the determination of their luminosities, effective temperatures, and metallicities. The error bars on these latter are still too high to constrain the models. Furthermore, although very modern, stellar models are still being affected by uncertainties in their input physics and initial helium abundance. A full characterisation of the stars is well beyond the scope of the paper. Summary and conclusion {#sect:conclusion} ====================== We have obtained 146 spectra for 10 SB2s: 7 SB2s previoulsy known, and 3 binaries that were only SB1s. The RVs of the components were derived with the [todmor]{} cross-correlation algorithm. After discarding the RVs coming from 14 blended spectra, we have derived the orbital elements of the 10 SB2s. We found minimal masses with an accuracy better than 1 % for 11 of the 20 components. Five of the 10 SB2s have received enough long-baseline or speckle interferometric measurements to calculate the masses of their components. The PIONIER measurements of one of them, HIP 104987, were never published before. Thanks to these data, we have derived the masses of the components of these five binary stars with accuracies ranging from 0.13% to a few percents, and we have found that the orbit of HIP 104987 is not circular, although its eccentricity is very small. We were also able to provide an estimate of the state of evolution of these stars by placing them in the HR diagram. Taking into account Paper III and IV, we have now accurate orbits for 34 SB2s, and combined SB2 and interferometric orbits for 9 of them. These latter will be useful to check the masses that will be obtained from [Gaia]{} in the future, and they can also be used to control the forthcoming DR3 parallaxes. Acknowledgments {#acknowledgments .unnumbered} =============== This project was supported by the french INSU-CNRS “Programme National de Physique Stellaire”, and the Centre National des Etudes Spatiales (CNES). We are grateful to the staff of the Haute–Provence Observatory, and especially to Dr H. Le Coroller, Dr M. Véron, and the night assistants, for their kind assistance. [PIONIER]{} is funded by the Université Joseph Fourier (UJF), the Institut de Planétologie et d’Astrophysique de Grenoble (IPAG), and the Agence Nationale pour la Recherche (ANR-06-BLAN-0421, ANR-10-BLAN-0505, ANR-10-LABX56). The integrated optics beam combiner is the result of a collaboration between IPAG and CEA-LETI based on CNES R&T funding. We made use of the SIMBAD database, operated at CDS, Strasbourg, France. Angelov T., 1996, BABel, 154, 13 Armstrong J.T., Mozurkewich D., Vivekanand M., Simon R.S., Denison C.S., et al., 1992, AJ 104, 241 Balega Y.Y., Beuzit J.-L., Delfosse X., Forveille T., Perrier C., et al., 2007, A&A, 464, 635 Bell E.F., de Jong R.S., 2001, ApJ 550, 212 Berger J.-P., Zins G., Lazareff B., Le Bouquin J.-B., Jocou L., et al., 2010, Proc. SPIE, 7734, 35 Casagrande L., Schönrich R., Asplund M., Cassisi S., Ramírez I., Meléndez J., Bensby T., Feltzing S., 2011, A&A, 530, A138 Casagrande L., VandenBerg D. A., 2018, MNRAS, 479, L102 Claret A., Torres G., 2019, ApJ 876, 134 Dutra-Ferreira L., Pasquini L., Smiljanic R., Porto de Mello G. F., Steffen M., 2016, A&A, 585, A756, A&A, 585, A75 Eggleton P.E., Yakut K., 2017, MNRAS 468, 3533 Eker Z., Soydugan F., Soydugan E., Bilir S. et al., 2015, AJ 149, 131 ESA, 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200, vol. 10 Gaia collaboration: Prusti T., de Bruijne J.H.J., Brown A.G.A., Vallenari A., Babusiaux C. et al., 2016, A&A 595, A1 Gaia collaboration: Brown A.G.A., Vallenari A., Prusti T., de Bruijne J.H.J., Mignard F. et al., 2016b, A&A 595, A2 Gaia collaboration: Brown A.G.A., Vallenari A., Prusti T., de Bruijne J.H.J., Babusiaux C. et al., 2018, A&A 616, A1 Gallenne A., Pietrzyński G., Graczyk D., Pilecki B., Storm J. et al., 2019, A&A 632, A31 Goldberg D., Mazeh T., Latham D.W., Stefanik R.P., Carney B.W., Laird J.B., 2002, AJ, 124, 1132 Griffin R.E.M., Griffin R.F., 2002, MNRAS 330, 288 Griffin R.F., 2005, Observatory 125, 367 Griffin R.F., Emerson B., 1975, Observatory 95, 98 Griffin R.F., Gunn J.E., Zimmerman B.A., Griffin R.E.M., 1985, AJ 90, 609 Halbwachs J.-L., Mayor M., Udry S., 2012, MNRAS 422, 14 Halbwachs J.-L., Arenou F., Pourbaix D., Famaey B., Guillout P. et al., 2014, MNRAS 445, 2371 (Paper I) Halbwachs J.-L., Boffin H.M.J., Le Bouquin J.-B., Kiefer F., Famaey B. et al., 2016, MNRAS 455, 3303 (Paper II) Hartkopf W.I., McAlister H.A, Mason B.D., 2001, AJ 122, 3480 Hidalgo S. L., Pietrinferni A., Cassisi S., Salaris M., Mucciarelli A., Savino A., Aparicio A., Silva Aguirre V., Verma K., 2018, ApJ, 856, 125 Hodgson R.M., Bailey D.G., Naylor M.J., Ng A.L.M., McNeil S.J., 1985, Image Vision Comput., 3(1), 4-14 Horch E. P., Casetti-Dinescu D. I., Camarata M. A., Bidarian A., van Altena W.F., et al., 2017, AJ, 153, 212 Jancart S., Jorissen A., Babusiaux C., Pourbaix D., 2005, A&A 442, 365 Kiefer, F., Halbwachs, J.-L., Arenou, F., Pourbaix, D., Famaey, B., et al., 2016, MNRAS, 458, 3272 (Paper III) Kiefer F., Halbwachs J.-L., Lebreton Y., Soubiran C., Arenou F., et al., 2018, MNRAS, 474, 731 (Paper IV) Le Bouquin J.B., Berger J.-P., Lazareff B., Zins G., Haguenauer P., et al., 2011, A&A 535, 67L Lebreton Y., Fernandes J., Lejeune T., 2001, A&A, 374, 540 Lebreton Y., 2005, The Three-Dimensional Universe with Gaia, ESA SP-576, 493 Lelli F., McGaugh S.S., Schombert J.M., 2016, AJ 152, 157 Lester K.V., Gies D.R., Schaefer G.H., Farrington C.D., Monnier J.D. et al., 2019, AJ 157, 140 McAlister H.A., 1996, S&T 92, 28 Mann A.W., Dupuy T., Kraus A.L., Gaidos E., Ansdell M. et al., 2019, ApJ 871, 63 Martin C., Mignard F., 1998, A&A 330, 585 Massarotti A., Latham D.W., Stefanik R.P., Fogel J., 2008, AJ, 135, 209 Moya A., Zuccarino F., Chaplin W.J., Davies G.R., 2018, ApJSS 237, 21 Munari U., Dallaporta S., Siviero A., Soubiran C., Fiorucci M., Girard P., 2004, A&A 418, L31 Nordström B., Stefanik R.P., Latham D.W., Andersen J., 1997, A&ASS, 126, 21 Perruchot S., Kohler D., Bouchy F., Richaud Y., Richaud P., et al. , 2008, Proc. of the SPIE, vol. 7014, 70140J Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1996, Numerical Recipes in Fortran 90 (Cambridge University Press) Stuart A., Ord K., 1994, Kendall’s Advanced Theory of Statistics, vol. 1. Edward Arnold, London Tokovinin A.A., Balega Y.Y., Hofmann K.H., Weigelt G., 2000, Astron. Letters 26, 668 Torres G., Andersen J., Giménez A., 2010, Astron. Astrophys. Rev., 18, 67 Torres G., Sandberg C.H., Fekel F.C., Muterspaugh M.W., 2019, ApJ 876, 41 van Leeuwen F., 2007, A&A 474, 653 Zucker S., Mazeh, T., 1994, ApJ, 420, 806 Zucker S., Mazeh T., Santos N. C., Udry S., Mayor M., 2004, A&A, 426, 695 \[lastpage\] [^1]: E-mail: jean-louis.halbwachs@astro.unistra.fr [^2]: based on observations performed at the Observatoire de Haute–Provence (CNRS), France [^3]: based on data obtained with the ESO Very Large Telescope under programme 094.D-0624 and 097.D-0688. [^4]: https://www.cosmos.esa.int/web/gaia/release [^5]: https://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/wds/int4 [^6]: https://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/wds/orb6
--- abstract: 'A Q-set is an uncountable set of reals all of whose subsets are relative $G_\delta$ sets. We prove that, for an arbitrary uncountable cardinal $\kappa$, there is consistently a Q-set of size $\kappa$ whose square is not Q. This answers a question of A. Miller.' author: - \| Jörg Brendle[^1]\ Graduate School of System Informatics\ Kobe University\ Rokko-dai 1-1, Nada-ku\ Kobe 657-8501, Japan\ email: [brendle@kurt.scitec.kobe-u.ac.jp]{} title: Q --- Introduction ============ A [Q-set]{} is an uncountable set of reals in which every subset is a relative $G_\delta$ set. If $X$ is a Q-set, then clearly $2^{\|X\|} = {{\mathfrak c}}$. In particular, the existence of Q-sets implies $2^{\omega_1} = {{\mathfrak c}}$. On the other hand, under Martin’s Axiom MA every uncountable sets of reals of cardinality $< {{\mathfrak c}}$ is Q [@MS70]. Przymusiński [@Pr80] proved that if there is a Q-set, then there is a Q-set of size $\aleph_1$ all of whose finite powers are Q as well. Furthermore, by the above, it is clear that under MA all finite powers of Q-sets are again Q. This left open the question of whether there can consistently exist a Q-set whose square is not Q. In [@Fl83], Fleissner claimed the consistency of a stronger statement, namely, that there is a Q-set of size $\aleph_2$ while no square of a set of reals of size $\aleph_2$ is Q. However, Miller [@Mi07 Theorem 8] observed that Fleissner’s argument was flawed. More explicitly, he showed that if there is a Q-set of size $\aleph_2$, then there is a set of reals $X = \{ x_\alpha : \alpha < \omega_2 \}$ such that the set of points “above the diagonal", $\{ (x_\alpha, x _\beta ) : \alpha < \beta < \omega_2 \}$, is a relative $G_\delta$ set in the square $X^2$. While this does not contradict the statement of Fleissner’s result (whose correctness remains open as far as we know), it does contradict his method of proof, for he claimed that in his model, for every set of reals $X = \{ x_\alpha : \alpha < \omega_2 \}$, the set $\{ (x_\alpha,x_\beta) : \alpha < \beta < \omega_2 \}$ was not a relative $G_\delta$ in $X^2$. Here we show: Let $\kappa$ be an arbitrary uncountable cardinal. It is consistent that there is a Q-set of size $\kappa$ whose square is not Q. This answers a question of A. Miller [@Mi15 Problem 7.16] (see also [@Mi07 after the proof of Theorem 8 on p. 32]). Our model is, in a sense, very natural: we first add $\kappa$ Cohen reals $c_\alpha $, $\alpha < \kappa $, and then turn the set $C = \{ c_\alpha : \alpha < \kappa \}$ into a Q-set in a finite support iteration of length $\kappa^+$, going through all subsets of $C$ by a book-keeping argument, and turning them into relative $G_\delta$’s by ccc forcing (see, e.g., [@Mi95 Section 5] for this forcing). To have control over the names for conditions arising in the iteration, we present a recursive definition of this forcing with finitary conditions. Incidentally, this model is the same as the one used by Fleissner in [@Fl83], for the case $\kappa =\omega_2$, though our description is somewhat different. Furthermore, it is similar to the one used by Fleissner and Miller in [@FM80], for the case $\kappa =\omega_1$, so that our technique also shows that the square of their Q-set, which is concentrated on the rationals, is not a Q-set. The main point is, of course, to prove that $C^2$ is not a Q-set. To this end, we show that the set of points “above the diagonal", $\{ (c_\alpha, c_\beta) : \alpha < \beta < \kappa \}$, is not a relative $G_\delta$ set in $C^2$. Unlike for Fleissner’s work [@Fl83], there is no contradiction with Miller’s result [@Mi07 Theorem 8], because we prove a much weaker statement. [Acknowledgment.]{} I thank Michael Hrušák for bringing this problem to my attention and for suggesting the model. I am also grateful to the referee for pointing out the reference [@FM80]. Proof of the Theorem ==================== Assume the ground model $V$ satisfies GCH. We perform a finite support iteration $( {{\mathbb P}}_\gamma : \gamma \leq \kappa^+ )$ of ccc forcing. As usual, elements of ${{\mathbb P}}_\gamma$ are functions with domain $\gamma$. Also, a book-keeping argument will hand us down a sequence $(\dot A_\gamma : \gamma < \kappa^+)$ of ${{\mathbb P}}_\gamma$-names of subsets of $\kappa$ such that for every ${{\mathbb P}}_{\kappa^+}$-name $\dot A$ for a subset of $\kappa$ there are $\gamma < \kappa^+$ such that ${{\mathbb P}}_\gamma$ forces that $\dot A = \dot A_\gamma$. Since this is a standard argument we will omit details. The recursive definition of the ${{\mathbb P}}_\gamma$ is as follows: - ${{\mathbb P}}_0$ is the trivial forcing (as usual). - ${{\mathbb P}}_1$ is ${{\mathbb C}}_{\kappa}$, the forcing for adding $\kappa$ Cohen reals: if $p \in {{\mathbb P}}_1$, then $p(0) = ( \sigma_\alpha^p : \alpha \in F^p)$ where $F^p {\subseteq}\kappa$ is finite and $\sigma_\alpha^p \in {2^{<\omega}}$ for $\alpha \in F^p$; the ordering is given by $q \leq p$ if $F^q \supseteq F^p$ and $\sigma_\alpha^q \supseteq \sigma_\alpha^p$ for all $\alpha \in F^p$. - If $\gamma$ is a limit ordinal, ${{\mathbb P}}_\gamma$ consists of all functions $p$ with ${{\mathrm{dom}}}(p) = \gamma$ such that there is $0<\delta < \gamma$ with $p{{\upharpoonright}}\delta \in {{\mathbb P}}_\delta$ and $p (\epsilon) = \emptyset$ for all $\epsilon$ with $\delta \leq\epsilon < \gamma$; the ordering is given by $q \leq p$ if $q {{\upharpoonright}}\delta \leq p {{\upharpoonright}}\delta$ for all $\delta < \gamma$. - If $\gamma \geq 1$, ${{\mathbb P}}_{\gamma + 1}$ consists of all functions $p$ with ${{\mathrm{dom}}}(p) = \gamma + 1$, $p {{\upharpoonright}}\gamma \in {{\mathbb P}}_\gamma$ and $p(\gamma)$ is a finite (possibly empty) subset of $({2^{<\omega}}\cup \kappa) \times \omega$ such that - if $(\alpha,n) \in p(\gamma)$ for some $\alpha \in \kappa$ and $n \in \omega$, then $\alpha \in F^p$ and $p{{\upharpoonright}}\gamma {\Vdash}\alpha \notin \dot A_\gamma$, - if $(\alpha,n) , (\sigma,n) \in p(\gamma)$ for some $\alpha \in\kappa$, $\sigma\in{2^{<\omega}}$ and $n\in\omega$, then $\sigma^p_\alpha$ and $\sigma$ are incompatible in ${2^{<\omega}}$; the ordering is given by $q \leq p$ if $q{{\upharpoonright}}\gamma \leq p{{\upharpoonright}}\gamma$ and $q(\gamma) \supseteq p(\gamma)$. For $\gamma \leq \kappa^+$ and $p \in {{\mathbb P}}_\gamma$, let ${{\mathrm{supp}}}(p) = \{ 0 \} \cup \{ \gamma > 0 : p(\gamma) \neq \emptyset \}$ denote the [support]{} of $p$. First of all, let us note that this is indeed an iteration, that is, ${{\mathbb P}}_\gamma {{<\!\!\circ\;}}{{\mathbb P}}_\delta$ for $\gamma < \delta$. This is straightforward, for if $p \in {{\mathbb P}}_\delta$ and $q' \in {{\mathbb P}}_\gamma$ with $q' \leq p {{\upharpoonright}}\gamma$, then $q \in {{\mathbb P}}_\delta$ defined by $q {{\upharpoonright}}\gamma = q'$ and $q (\epsilon) = p (\epsilon)$ for $\gamma \leq \epsilon <\delta$ is a common extension of $q'$ and $p$. Here is a sufficient criterion for compatibility of conditions. \[compat-crit\] Assume $p,q \in {{\mathbb P}}_\gamma$ are such that - $\sigma_\alpha^p = \sigma_\alpha^q$ for all $\alpha \in F^p \cap F^q$, - for all $\delta \in {{\mathrm{supp}}}(p) \cap {{\mathrm{supp}}}(q)$ with $\delta > 0$ and all $(\sigma, n) \in {2^{<\omega}}\times \omega$, $$(\sigma,n) \in p(\delta) \; {\mbox{$\Longleftrightarrow$}}\; (\sigma,n) \in q (\delta).$$ Then $p$ and $q$ are compatible with a canonical common extension $r$ given by - ${{\mathrm{supp}}}(r) = {{\mathrm{supp}}}(p) \cup {{\mathrm{supp}}}(q)$, - $F^r = F^p \cup F^q$, - $\sigma_\alpha^r = \begin{cases} \sigma_\alpha^p & \mbox{if } \alpha \in F^p \\ \sigma_\alpha^q & \mbox{if } \alpha \in F^q, \\ \end{cases}$ - $r(\delta) = p(\delta) \cup q (\delta)$ for all $\delta \in {{\mathrm{supp}}}(r)$ with $\delta > 0$. By induction on $\delta \leq \gamma$ we simultaneously prove that $r{{\upharpoonright}}\delta$ as defined in the lemma is indeed a condition and that $r{{\upharpoonright}}\delta \leq p {{\upharpoonright}}\delta , q {{\upharpoonright}}\delta$ holds. For $\delta = 1$, this is straightforward by assumption. The limit step is also clear. So suppose this has been proved for $\delta \geq 1$, and we show it for $\delta + 1$. If $(\alpha,n) \in r(\delta)$ for some $\alpha$ and $n$, then, without loss of generality, we may assume $(\alpha,n) \in p(\delta)$. Thus $p {{\upharpoonright}}\delta {\Vdash}\alpha \notin \dot A_\delta$ and, since $r{{\upharpoonright}}\delta \leq p{{\upharpoonright}}\delta$ by induction hypothesis, we also have $r{{\upharpoonright}}\delta {\Vdash}\alpha\notin \dot A_\delta$, as required. Next assume $(\alpha,n)$ and $(\sigma,n)$ belong to $r(\delta)$ for some $\alpha, \sigma$ and $n$. If both belong to either $p(\delta)$ or $q(\delta)$, there is nothing to show. So we may assume without loss of generality $(\alpha,n) \in p(\delta)$ and $(\sigma,n) \in q (\delta)$. In particular, $\delta \in {{\mathrm{supp}}}(p) \cap {{\mathrm{supp}}}(q)$ and, by assumption, $(\sigma,n) \in p(\delta)$ follows. Since $p$ is a condition, $\sigma_\alpha^r = \sigma_\alpha^p$ and $\sigma$ are incompatible, as required. Thus we have proved that $r{{\upharpoonright}}\delta + 1 \in {{\mathbb P}}_{\delta + 1}$, and $r {{\upharpoonright}}\delta + 1 \leq p {{\upharpoonright}}\delta + 1, q {{\upharpoonright}}\delta +1$ now follows easily. This completes the induction and the proof of the lemma. This lemma presents a basic pattern of how to define a common extension of two conditions and then show by induction that the object defined really is a condition. We shall use this pattern several times, see Lemma \[compat-lem\] and Claim \[q-claim\] below. An immediate consequence is: ${{\mathbb P}}_\gamma$ is ccc (and even satisfies Knaster’s condition) for every $\gamma \leq \kappa^+$. Let $\{ p_\zeta: \zeta < \omega_1 \} {\subseteq}{{\mathbb P}}_\gamma$. By a straightforward $\Delta$-system argument we see that we may assume that for any $\zeta < \xi < \omega_1$, $p = p_\zeta$ and $q = p_\xi$ satisfy the assumptions of the previous lemma. Hence the $p_\zeta$ are pairwise compatible. Furthermore, compatible conditions sort of have a “minimal" extension. \[compat-lem\] Assume $p,q \in {{\mathbb P}}_\gamma$ are compatible with common extension $r$. Then there is a condition $s = s^{p,q,r} \in {{\mathbb P}}_\gamma$ with $r \leq s \leq p,q$ such that - ${{\mathrm{supp}}}(s) = {{\mathrm{supp}}}(p) \cup {{\mathrm{supp}}}(q)$, - $F^s = F^p \cup F^q$, - $\sigma_\alpha^s = \sigma_\alpha^r$ for all $\alpha \in F^s$, - $s(\delta) = p(\delta) \cup q(\delta)$ for all $\delta \in {{\mathrm{supp}}}(s)$ with $\delta > 0$. By induction on $\delta\leq\gamma$ we simultaneously prove that $s{{\upharpoonright}}\delta$ as defined in the lemma is indeed a condition and that $r{{\upharpoonright}}\delta \leq s{{\upharpoonright}}\delta \leq p{{\upharpoonright}}\delta,q{{\upharpoonright}}\delta$ holds. For $\delta = 1$ this is obvious. The limit step is also straightforward. So assume this has been proved for $\delta \geq 1$. First suppose that $(\alpha,n) \in s(\delta)$ for some $\alpha$ and $n$. Without loss of generality we may assume that $(\alpha,n) \in p(\delta)$. Thus $p{{\upharpoonright}}\delta {\Vdash}\alpha \notin \dot A_\delta$. Since $s{{\upharpoonright}}\delta \leq p{{\upharpoonright}}\delta$ by induction hypothesis, also $s {{\upharpoonright}}\delta {\Vdash}\alpha \notin \dot A_\delta$, as required. Next suppose $(\alpha,n) , (\sigma,n)$ in $s(\delta)$ for some $\alpha,\sigma$ and $n$. Since $r\leq p,q$, $r(\delta) \supseteq s(\delta)$, and $(\alpha,n) , (\sigma,n) \in r(\delta)$ follows. In particular, $\sigma^s_\alpha = \sigma^r_\alpha$ and $\sigma$ are incompatible in ${2^{<\omega}}$, as required. Thus $s{{\upharpoonright}}\delta + 1 \in {{\mathbb P}}_{\delta + 1}$ and $r {{\upharpoonright}}\delta + 1 \leq s {{\upharpoonright}}\delta + 1 \leq p {{\upharpoonright}}\delta + 1, q {{\upharpoonright}}\delta + 1$ is now obvious. This completes the induction and the proof of the lemma. In particular, two conditions $p,q$ are compatible iff they have a common extension $s$ with ${{\mathrm{supp}}}(s) = {{\mathrm{supp}}}(p) \cup {{\mathrm{supp}}}(q)$, $F^s = F^p\cup F^q$, $\sigma_\alpha^s \supseteq \sigma_\alpha^p, \sigma_\alpha^q$ for all $\alpha \in F^s$, and $s(\delta) = p(\delta) \cup q(\delta)$ for all $\delta \in {{\mathrm{supp}}}(s)$. (Lemma \[compat-lem\] will not be needed but gives some motivation for Claim \[q-claim\] below.) Let $\dot G$ be the name for the ${{\mathbb P}}_{\kappa^+}$-generic filter. For $0 < \gamma < \kappa^+$ and $n \in\omega$ define ${{\mathbb P}}_{\kappa^+}$-names (more explicitly, ${{\mathbb P}}_{\gamma + 1}$-names) $$\dot U_{\gamma,n} = \bigcup \{ [\sigma] : (\sigma,n) \in p(\gamma) \mbox{ for some } p \in \dot G \} \mbox{ and } \dot H_\gamma = \bigcap_{n \in\omega} \dot U_{\gamma , n}.$$ Clearly, the $\dot U_{\gamma,n}$ are names for open sets, and $\dot H_\gamma$ is the name for a $G_\delta$ set. Also let $$\dot c_\alpha = \bigcup \{ \sigma : \sigma = \sigma^p_\alpha \mbox{ for some } p \in \dot G \}$$ be the canonical name for the Cohen real added in coordinate $\alpha$ of stage $1$ of the forcing, for each $\alpha < \kappa$. \[Q-lem\] [(i)]{} Assume $n \in\omega, \alpha < \kappa , \gamma < \kappa^+$ and $p \in{{\mathbb P}}_{\kappa^+}$ are given such that $p{{\upharpoonright}}\gamma {\Vdash}_\gamma \alpha \in \dot A_\gamma$. Then there are $q \leq p$ and $\sigma {\subseteq}\sigma_\alpha^q$ such that $(\sigma,n) \in q(\gamma)$. [(ii)]{} Assume $\alpha < \kappa , \gamma < \kappa^+$ and $p \in{{\mathbb P}}_{\kappa^+}$ are given such that $p{{\upharpoonright}}\gamma {\Vdash}_\gamma \alpha \notin \dot A_\gamma$. Then there are $q \leq p$ and $n \in\omega$ such that $(\alpha,n) \in q(\gamma)$. \(i) Assume $n,\alpha,\gamma, p$ are given as required. Then clearly $(\alpha,n) \notin p(\gamma)$. For $\beta \in F^p$ extend $\sigma^p_\beta$ to $\sigma_\beta^q$ such that they are pairwise incompatible. This defines $q{{\upharpoonright}}1$ (in particular, $F^q = F^p$). For $\delta >0$ let $$q(\delta) = \begin{cases} p(\delta) & \mbox{if } \delta \neq\gamma \\ p(\delta) \cup \{ (\sigma_\alpha^q , n) \} & \mbox{if } \delta = \gamma \\ \end{cases}$$ (in particular, ${{\mathrm{supp}}}(q) = {{\mathrm{supp}}}(p) \cup \{\gamma \}$). It is easy to see that $q$ is a condition and that it strengthens $p$. \(ii) Assume $\alpha,\gamma,p$ are given as required. Let $F^q = F^p \cup \{ \alpha \}$ and let $\sigma_\alpha^q$ be either $\sigma_\alpha^p$ (if $\alpha \in F^p$) or arbitrary (otherwise). Let $n$ be large enough so that no $(\sigma,n)$ belongs to $p(\gamma)$. For $\delta >0$ let $$q(\delta) = \begin{cases} p(\delta) & \mbox{if } \delta \neq\gamma \\ p(\delta) \cup \{ (\alpha , n) \} & \mbox{if } \delta = \gamma \\ \end{cases}$$ (in particular, ${{\mathrm{supp}}}(q) = {{\mathrm{supp}}}(p) \cup \{\gamma \}$). Again, it is easy to see that $q$ is a condition and that it strengthens $p$. [(i)]{} Assume $\alpha < \kappa , \gamma < \kappa^+$ and $p \in{{\mathbb P}}_{\kappa^+}$ are such that $p{{\upharpoonright}}\gamma {\Vdash}_\gamma \alpha \in \dot A_\gamma$. Then $p {{\upharpoonright}}\gamma+1 {\Vdash}_{\gamma+1} \dot c_\alpha \in \dot H_\gamma$. [(ii)]{} Assume $\alpha < \kappa , \gamma < \kappa^+$ and $p \in{{\mathbb P}}_{\kappa^+}$ are such that $p{{\upharpoonright}}\gamma {\Vdash}_\gamma \alpha \notin \dot A_\gamma$. Then $p {{\upharpoonright}}\gamma+1 {\Vdash}_{\gamma+1} \dot c_\alpha \notin \dot H_\gamma$. \(i) This is immediate from Lemma \[Q-lem\] (i). \(ii) Let $q \leq p$ be arbitrary. By Lemma \[Q-lem\] (ii), there are $r \leq q$ and $n \in \omega$ with $(\alpha,n) \in r (\gamma)$. It suffices to show that $r {\Vdash}\dot c_\alpha \notin \dot U_{\gamma,n}$. Assume that $G$ is ${{\mathbb P}}_{\kappa^+}$-generic with $r \in G$ and $c_\alpha \in U_{\gamma,n}$. By definition of $U_{\gamma,n}$, there are $r' \leq r$ and $\sigma \in {2^{<\omega}}$ such that $r' \in G$, $(\sigma,n) \in r'(\gamma)$ and $\sigma {\subseteq}c_\alpha$. Hence there is $r'' \leq r'$ such that $r'' \in G$ and $\sigma {\subseteq}\sigma^{r''}_\alpha$. Since $(\alpha,n), (\sigma,n) \in r'' (\gamma)$, this contradicts the definition of a condition. Hence we must have $c_\alpha \notin U_{\gamma,n}$, as required. As a consequence, we see that $${\Vdash}_{\gamma + 1} \dot H_\gamma \cap \{ \dot c_\alpha : \alpha < \kappa \} = \{ \dot c_\alpha : \alpha \in \dot A_\gamma \}.$$ Thus we obtain: ${\Vdash}_{\kappa^+} `` \{ \dot c_\alpha : \alpha < \kappa \}$ is a Q-set$"$. To see that the square of the set of Cohen reals is not a Q-set, it clearly suffices to establish the following: \[nonQ-lem\] ${\Vdash}_{\kappa^+} ``\{ (\dot c_\alpha,\dot c_\beta ) : \alpha < \beta <\kappa \}$ is not a relative $G_\delta$-set$"$. Assume that $\dot V_n$, $n \in\omega$, are ${{\mathbb P}}_{\kappa^+}$-names of open sets in the plane $({2^\omega})^2$ such that $\{ (\dot c_\alpha, \dot c_\beta) : \alpha < \beta < \kappa \} {\subseteq}\bigcap_{n\in\omega}\dot V_n$ is forced by the trivial condition. We shall find $\beta < \alpha < \kappa$ such that the trivial condition forces $(\dot c_\alpha, \dot c_\beta) \in \bigcap_{n\in\omega}\dot V_n$. This is clearly sufficient. There are $\dot S_n {\subseteq}({2^{<\omega}})^2$ such that for every $n \in \omega$, $\dot V_n = \bigcup \{ [\sigma] \times [\tau] : (\sigma,\tau) \in \dot S_n \}$ is forced. For each $n \in \omega$ and each $(\sigma,\tau) \in ({2^{<\omega}})^2$, let $\{ p^j_{n,\sigma,\tau} : j \in \omega \}$ be a maximal antichain of conditions in ${{\mathbb P}}_{\kappa^+}$ deciding the statement $(\sigma,\tau) \in \dot S_n$. Let $Z = \{ z_i : i \in \omega \}$ be a countable set disjoint from $\kappa$. We say that ${{\mathsf{t}}}= (\Gamma^{{\mathsf{t}}},\Delta^{{\mathsf{t}}}, \bar D^{{\mathsf{t}}}= (D_\gamma^{{\mathsf{t}}}: \gamma \in \Gamma^{{\mathsf{t}}}{\setminus}\{ 0 \} ) , \bar E^{{\mathsf{t}}}= (E^{{\mathsf{t}}}_\gamma ; \gamma \in \Gamma^{{\mathsf{t}}}{\setminus}\{ 0 \}) , \bar\tau^{{\mathsf{t}}}= (\tau_\zeta^{{\mathsf{t}}}: \zeta \in \Delta^{{\mathsf{t}}}))$ is a [pattern]{} if - $\Gamma^{{\mathsf{t}}}{\subseteq}\kappa^+$ is finite, $0 \in \Gamma^{{\mathsf{t}}}$, - $\Delta^{{\mathsf{t}}}{\subseteq}\kappa \cup Z$ is finite and $\Delta^{{\mathsf{t}}}\cap Z$ is an initial segment of $Z$,\ i.e., $\Delta^{{\mathsf{t}}}\cap Z = \{ z_i : i < k \}$ for some $k$, - $D_\gamma^{{\mathsf{t}}}{\subseteq}{2^{<\omega}}\times \omega$ is finite for $\gamma \in \Gamma^{{\mathsf{t}}}$, - $E_\gamma^{{\mathsf{t}}}{\subseteq}\Delta^{{\mathsf{t}}}\times \omega$ is finite for $\gamma \in \Gamma^{{\mathsf{t}}}$, - $\tau_\zeta^{{\mathsf{t}}}\in {2^{<\omega}}$ for $\zeta \in \Delta^{{\mathsf{t}}}$. Usually we will omit the superscript ${{\mathsf{t}}}$. Let ${{\mathsf{PAT}}}$ denote the collection of all patterns. Let $X {\subseteq}\kappa^+$ and $Y {\subseteq}\kappa$. Assume $0\in X$. Also let $p \in {{\mathbb P}}_{\kappa^+}$. We say that $(\Gamma , \Delta , \bar D , \bar E, \bar \tau) \in {{\mathsf{PAT}}}$ is the [$(X,Y)$-pattern of $p$]{} if $\Gamma {\subseteq}X$, $\Delta {\subseteq}Y \cup Z$ and for some (necessarily unique) one-to-one function $\varphi = \varphi^p : \Delta \to \kappa$ with $\varphi {{\upharpoonright}}(\Delta \cap Y) = {{\mathrm{id}}}$ and $\varphi {{\upharpoonright}}(\Delta \cap Z) : Z \to \kappa {\setminus}Y$ order-preserving (i.e., if $i<j$ and $z_i , z_j \in \Delta \cap Z$, then $\varphi (z_i) < \varphi (z_j)$), the following hold: - ${{\mathrm{supp}}}(p) \cap X = \Gamma$, - $F^p = \{ \varphi (\zeta) : \zeta \in \Delta \}$ (in particular, $F^p \cap Y = \Delta \cap Y$ and $F^p {\setminus}Y = \{ \varphi (\zeta) : \zeta \in \Delta \cap Z \}$), - $\sigma^p_{\varphi(\zeta)} = \tau_\zeta$ for $\zeta \in \Delta$, - for $\gamma \in \Gamma$ and $(\sigma,n) \in {2^{<\omega}}\times \omega$: $$(\sigma,n) \in p (\gamma) \;{\mbox{$\Longleftrightarrow$}}\; (\sigma,n) \in D_\gamma,$$ - for $\gamma \in \Gamma$ and $(\zeta,n) \in \Delta \times \omega$: $$(\varphi(\zeta), n) \in p (\gamma) \;{\mbox{$\Longleftrightarrow$}}\; (\zeta,n) \in E_\gamma.$$ Clearly each condition has a unique $(X,Y)$-pattern. Let $\chi \geq (2^{\kappa^+})^+$, and let $M \prec H (\chi)$ be a countable elementary submodel containing $\kappa$, ${{\mathbb P}}_{\kappa^+}$, $Z$, ${{\mathsf{PAT}}}$, all $\dot V_n$, $\dot S_n$, and $\{ p^j_{n,\sigma,\tau} : j \in \omega, (\sigma,\tau) \in ( {2^{<\omega}})^2 \}$. Choose any ordinals $\beta, \alpha$ with $M \cap \omega_1 \leq \beta < \alpha < \omega_1$. We shall see that $\beta$ and $\alpha$ are as required, that is, that ${\Vdash}(\dot c_\alpha, \dot c_\beta) \in \bigcap_{n \in\omega} \dot V_n$. By a standard density argument, it suffices to prove the following: For all $n \in \omega$ and all $p \in {{\mathbb P}}_{\kappa^+}$ there are $q \leq p$ and $\sigma_0, \tau_0 \in {2^{<\omega}}$ such that $\sigma_0 {\subseteq}\sigma_\alpha^q$, $\tau_0 {\subseteq}\sigma_\beta^q$ and $q {\Vdash}(\sigma_0 , \tau_0) \in \dot S_n$. Fix $n \in\omega_2$. Let $p \in {{\mathbb P}}_{\kappa^+}$ be given. By going over to a stronger condition, if necessary, we may assume $\alpha, \beta \in F^p$. Let $X = M \cap \kappa^+$ and $Y = M \cap \kappa$. Let ${{\mathsf{t}}}= (\Gamma, \Delta, \bar D , \bar E, \bar\tau)$ be the $(X,Y)$-pattern of $p$. Put $X_0 := \Gamma$ and $Y_0 := \Delta \cap \kappa$. Then we see that $X_0 = \Gamma = {{\mathrm{supp}}}(p) \cap X = {{\mathrm{supp}}}(p) \cap M$ and $Y_0 = \Delta \cap Y = F^p \cap Y = F^p \cap M$. In particular, ${{\mathsf{t}}}$ is also the $(X_0, Y_0)$-pattern of $p$. Furthermore, unlike $X$ and $Y$, $X_0$ and $Y_0$ belong to $M$, and so does ${{\mathsf{t}}}$. Since $\alpha \in F^p {\setminus}Y$, there is $i \in \omega$ such that $\varphi^p (z_i) = \alpha$. So $$H_\chi \models \exists p' \in {{\mathbb P}}_{\kappa^+} \; \exists \alpha ' < \omega_1 \; ({{\mathsf{t}}}\mbox{ is the } (X_0,Y_0) \mbox{-pattern of } p' \mbox{ and } \varphi^{p'} (z_i) = \alpha ' )$$ because this statement is true for $p ' = p$ and $\alpha ' = \alpha$. By elementarity we obtain $$M \models \exists p' \in {{\mathbb P}}_{\kappa^+} \; \exists \alpha ' < \omega_1 \; ({{\mathsf{t}}}\mbox{ is the } (X_0,Y_0) \mbox{-pattern of } p' \mbox{ and } \varphi^{p'} (z_i) = \alpha ' ).$$ Let $p' \in M \cap {{\mathbb P}}_{\kappa^+}$ and $\alpha ' < M \cap \omega_1 \leq \beta$ be the witnesses. $p$ and $p'$ are compatible, with common extension $p''$ given canonically according to Lemma \[compat-crit\]. We need to verify the two conditions in Lemma \[compat-crit\]. Since $F^{p'} {\subseteq}M \cap \kappa = Y$ and $F^p \cap Y = F^p \cap Y_0 = \Delta \cap Y_0 = F^{p'} \cap Y_0$, we see that $F^p \cap F^{p'} = \Delta \cap Y_0$ and $\sigma_\zeta^p = \tau_\zeta = \sigma_\zeta^{p'}$ for all $\zeta \in \Delta \cap Y_0$. Similarly, since ${{\mathrm{supp}}}(p') {\subseteq}M \cap \kappa^+ = X$ and ${{\mathrm{supp}}}(p) \cap X = {{\mathrm{supp}}}(p) \cap X_0 = \Gamma = {{\mathrm{supp}}}(p') \cap X_0$, we have that ${{\mathrm{supp}}}(p) \cap {{\mathrm{supp}}}(p') = \Gamma$ and, for $\gamma \in \Gamma$, $$(\sigma,n) \in p(\gamma) \; {\mbox{$\Longleftrightarrow$}}\; (\sigma,n) \in D_\gamma \; {\mbox{$\Longleftrightarrow$}}\; (\sigma,n) \in p' (\gamma).$$ Hence the conditions are indeed satisfied. Let $\psi = \varphi^{p'} \circ ( \varphi^p)^{-1}$. Clearly $\psi$ maps $F^p$ one-to-one and onto $F^{p'}$, and we have $\alpha' = \psi (\alpha)$. We also note for later use that for all $\zeta \in F^p$, $\sigma_\zeta^{p''} = \sigma_\zeta^p = \tau_{ ( \varphi^p)^{-1} (\zeta)} = \tau_{ ( \varphi^{p'})^{-1} (\psi (\zeta)) } = \sigma_{\psi(\zeta)}^{p'} = \sigma_{\psi(\zeta)}^{p''}$, and for all $\gamma \in \Gamma$ and $\zeta \in F^p$, $$(\zeta,n) \in p (\gamma) \; {\mbox{$\Longleftrightarrow$}}\; ( (\varphi^p)^{-1} (\zeta) , n) \in E_\gamma \; {\mbox{$\Longleftrightarrow$}}\; ( \psi (\zeta) , n ) \in p' (\gamma).$$ Since ${\Vdash}(\dot c_{\alpha'} , \dot c_\beta) \in \dot V_n$, there are $\tilde p \leq p''$ and $\sigma_0, \tau_0 \in {2^{<\omega}}$ such that $$\tilde p {\Vdash}(\dot c_{\alpha '} , \dot c_\beta) \in [ \sigma_0 ] \times [\tau_0] {\subseteq}\dot V_n,$$ that is, $\sigma_0 {\subseteq}\sigma_{\alpha '}^{\tilde p} , \tau_0 {\subseteq}\sigma_\beta^{\tilde p}$ and $ \tilde p {\Vdash}(\sigma_0 , \tau_0 ) \in \dot S_n$. By construction, for some $j \in \omega$, the condition $r = p^j_{n, \sigma_0, \tau_0} \in M$ is compatible with $\tilde p$. In particular, $r$ must also force $(\sigma_0, \tau_0) \in \dot S_n$. Furthermore we know that ${{\mathrm{supp}}}(r) {\subseteq}X$ and $F^r {\subseteq}Y$. By strengthening $\tilde p$, if necessary, we may assume that $\tilde p \leq r$. \[q-claim\] $p$ and $r$ have a common extension $q$ such that $\sigma_\alpha^q = \sigma_{\alpha '}^{\tilde p}$ and $\sigma_\beta^q = \sigma_\beta^{\tilde p}$. Note that we know already that $p$ and $r$ are compatible because they have common extension $\tilde p$; however, here we construct a different common extension $q$ (in general $\sigma_\alpha^{\tilde p}$ and $\sigma_\alpha^q$ are distinct). As in the proofs of Lemmata \[compat-crit\] and \[compat-lem\], we define $q$ and then show by induction on $\gamma$ that $q {{\upharpoonright}}\gamma \in {{\mathbb P}}_\gamma$ and that $q {{\upharpoonright}}\gamma$ extends both $p{{\upharpoonright}}\gamma$ and $r{{\upharpoonright}}\gamma$. - ${{\mathrm{supp}}}(q) = {{\mathrm{supp}}}(p) \cup {{\mathrm{supp}}}(r)$, - $F^{q} = F^{p} \cup F^{r}$, - for $\zeta \in F^{q}$, $\sigma_\zeta^{q} = \begin{cases} \sigma_\zeta^{\tilde p} & \mbox{if } \zeta\in F^{r} \mbox{ or } \zeta = \beta \\ \sigma_{\psi(\zeta)}^{\tilde p} & \mbox{if } \zeta\in F^{p} {\setminus}F^{r} \mbox{ and } \zeta \neq \beta, \\ \end{cases}$ - for $\gamma \in {{\mathrm{supp}}}(q)$ with $\gamma >0$, $q(\gamma) = p(\gamma) \cup r (\gamma)$. For the induction, first let $\gamma = 1$. If $\zeta \in F^r$, then by $\tilde p \leq r$, $\sigma_\zeta^r {\subseteq}\sigma_\zeta^{\tilde p} = \sigma_\zeta^q$. If, additionally, $\zeta \in F^p$, then by $\tilde p \leq p''$, $\sigma_\zeta^p = \sigma_\zeta^{p''} {\subseteq}\sigma_\zeta^{\tilde p} = \sigma_\zeta^q$. If $\zeta = \beta$, we similarly have $\sigma_\beta^p = \sigma_\beta^{p''} {\subseteq}\sigma_\beta^{\tilde p} = \sigma_\beta^q$. Finally, if $\zeta \in F^p {\setminus}F^r$ with $\zeta \neq \beta$, then, using again $\tilde p \leq p''$, we see that $\sigma_\zeta^p = \sigma_{\psi (\zeta)}^{p'} = \sigma_{\psi (\zeta)}^{p''} {\subseteq}\sigma_{\psi (\zeta)}^{\tilde p} = \sigma_\zeta^q$. Thus $q {{\upharpoonright}}1 \leq p {{\upharpoonright}}1, r {{\upharpoonright}}1$, as required. Note, in particular, that $\sigma_\alpha^q = \sigma_{\psi (\alpha)}^{\tilde p} = \sigma_{\alpha '}^{\tilde p}$. As usual, the limit step of the induction is trivial. Let us assume we have proved the statement for some $\gamma \geq 1$, and let us prove it for $\gamma + 1$. First assume that $(\zeta,n) \in q(\gamma)$. Then we see that $q{{\upharpoonright}}\gamma {\Vdash}\zeta \notin \dot A_\gamma$ as in the proofs of Lemmata \[compat-crit\] and \[compat-lem\]. Hence assume $(\zeta,n), (\sigma,n) \in q (\gamma)$. As in the proof of Lemma \[compat-crit\], we may assume that $\gamma \in {{\mathrm{supp}}}(p) \cap {{\mathrm{supp}}}(r)$. In particular $\gamma \in X_0 = \Gamma$. First suppose $(\zeta,n) \in r (\gamma)$ and $(\sigma,n) \in p(\gamma)$. Then $\zeta \in F^r {\subseteq}F^{\tilde p}$. Since $\tilde p \leq p,r$, we see that $(\zeta,n), (\sigma, n) \in \tilde p (\gamma)$, and since $\tilde p$ is a condition, $\sigma$ and $\sigma_\zeta^{\tilde p} = \sigma_\zeta^q$ are incompatible, as required. Next suppose $(\zeta,n) \in p(\gamma)$ and $(\sigma,n) \in r(\gamma)$. So $\zeta \in F^p$. We split into cases according to whether $\zeta = \beta$ or not. First assume $\zeta \neq \beta$. Then, using $\tilde p \leq p'' \leq p'$, we see $(\psi(\zeta), n) \in p' (\gamma) {\subseteq}p'' (\gamma) {\subseteq}\tilde p (\gamma)$ and $(\sigma,n) \in \tilde p (\gamma)$. Since $\tilde p$ is a condition, $\sigma_{\psi(\zeta)}^{\tilde p}$ and $\sigma$ are incompatible. If $\zeta \in F^r$, then $\zeta\in Y_0$, and $\psi (\zeta) = \zeta$ follows. Hence $\sigma_\zeta^q = \sigma_\zeta^{\tilde p} = \sigma_{\psi (\zeta)}^{\tilde p}$. If $\zeta \notin F^r$, then $\sigma_\zeta^q = \sigma_{\psi (\zeta)}^{\tilde p}$ by definition. In either case, we see that $\sigma_\zeta^q$ and $\sigma$ are incompatible. Now assume $\zeta = \beta$. Since $\tilde p \leq p,r$, we have $(\beta,n) , (\sigma,n) \in \tilde p (\gamma)$, and $\sigma_\beta^q = \sigma_\beta^{\tilde p}$ and $\sigma$ are incompatible because $\tilde p$ is a condition. This shows that $q {{\upharpoonright}}\gamma + 1$ is a condition. Clearly $q {{\upharpoonright}}\gamma + 1 \leq p {{\upharpoonright}}\gamma + 1, r {{\upharpoonright}}\gamma + 1$. This completes the induction. Thus $q\leq p$, $\sigma_0 {\subseteq}\sigma_\alpha^q$, $\tau_0 {\subseteq}\sigma_\beta^q$ and $q$ forces that $(\sigma_0,\tau_0)$ belongs to $\dot S_n$, and the proof of the main claim is complete. This completes the proof of the main lemma and of the theorem. [ABC]{} W.G. Fleissner, [Squares of $Q$ sets]{}, Fund. Math. [118]{} (1983), 223-231. W.G. Fleissner and A.W. Miller, [On $Q$ sets]{}, Proc. Amer. Math. Soc. [78]{} (1980), 280-284. D.A. Martin and R.M. Solovay, [Internal Cohen extensions]{}, Ann. Math. Logic [2]{} (1970), 143-178. A.W. Miller, [Descriptive Set Theory and Forcing]{}, Lecture Notes in Logic [4]{}, Springer 1995. A.W. Miller, [A hodgepodge of sets of reals]{}, Note di Matematica [27]{} (2007), 25-39. A.W. Miller, [Some interesting problems]{}, version of April 2015. T. Przymusiński, [The existence of $Q$-sets is equivalent to the existence of strong $Q$-sets]{}, Proc. Amer. Math. Soc. [79]{} (1980), 626-628. [^1]: Partially supported by Grant-in-Aid for Scientific Research (C) 15K04977, Japan Society for the Promotion of Science. I also acknowledge partial support from Michael Hrušák’s grants, CONACyT grant no. 177758 and PAPIIT grant IN-108014, during my stay at UNAM in spring 2015, when this research was started.
--- author: - Kaća Bradonjić date: 'Submitted on March 31, 2011' title: Quantum Gravity and the Correspondence Principle --- Summary {#summary .unnumbered} ======= Standard approaches to quantum gravity start with a pre-spacetime structure and attempt, in accordance with Bohr’s correspondence principle, to recover the pseudo-Riemannian manifold in the low energy limit. These approaches assume there is a smooth transition from quantum gravity to general relativity common to successful quantum theories. However, as gravitational field, and hence spacetime, cannot be considered in isolation from physical fields, discontinuities in properties of physical fields, such as loss of mass at the electroweak symmetry breaking scale, may result in change of spacetime structure somewhere between the Planck scale and the scale where general relativity holds true. As a result, the correct theory of quantum gravity may not have general relativity as its low energy limit. Standard approaches to the problem of quantum gravity start with a pre-spacetime structure, assumed to exist at the Planck scale, and attempt to recover the full pseudo-Riemannian manifold in the low energy limit [@Thiemann; @Rovelli; @Fotini; @Sorkin; @Loll]. This limit is expected in accordance with Bohr’s correspondence principle which states that the quantum theory must asymptotically approach its classical counterpart in the limit of large quantum number [@Messiah]. Successfully applied in the case of quantum mechanics and quantum electrodynamics, correspondence principle is often used as a guide and a test for a potentially successful quantum theory of gravity. All these approaches assume there is a smooth transition from the Planck scale quantum gravity to the general theory of relativity (GR). However, gravitational field is distinct from the other physical fields in that it, and hence spacetime geometry, cannot be considered in isolation from all the other physical fields. According to Einstein, the special and general theory of relativity, and consequently the attribution of pseudo-Riemannian geometry to spacetime, rest on physically meaningful notions of length, as that which is measured by a rigid rod, and time, as that which which is measured by a physical clock. Einstein goes as far as to say that the entire GR framework rests on the assumption that two line segments defined on rigid bodies equal at some time and place, are equal always and everywhere [@Einstein1961]. Assuming that pseudo-Riemannian (or any other) geometry is independent of the physical fields present is not substantiated. As we consider smaller length scales, we may find that spactime structure changes well before we need a quantum description of gravity. In such case, the limiting case of a correct quantum theory may not be GR as we know it. The first change in the nature of space and time may have been expected to appear at the scales at which notion of a rigid rod breaks down. Einstein himself questioned the applicability of GR to sub-molecular scales and conceded that “physical interpretation of geometry breaks down when applied immediately to spaces of sub-molecular order of magnitude" [@Einstein1922]. At an energy regime where the notion of “rigid body" becomes meaningless, any concept derived from the concept of a “rigid body" must be subject to scrutiny. Today we know that relativity is applicable to sub-molecular scales. Although initially defined as measured by a rigid rod and a physical clock, the notions of length and time can be retained at the particle level if they are redefined and inferred from paths of massless and massive particles. Ehlers, Pirani, and Schild showed that a full pseudo-Riemannian spacetime geometry can be constructed from paths of massless and massive particles by imposing two physically motivated compatibility conditions between the two sets of paths [@Ehlers1972; @Pirani1973]. A second natural scale at which a discontinuity in the nature of spacetime may appear is the electroweak symmetry breaking scale (EWSB) [@Bradonjic]. According to the Standard Model (SM), prior to EWSB, the Higgs field had a vanishing vacuum expectation value and all the particles were massless [@Glashow; @Weinberg; @Salam]. One may object that particles are never really massless due to radiative corrections which induce thermal mass terms and that, due to this induced mass, particles propagate along non-null geodesics in thermal plasma. However, the framework of thermal field theory assumes the existence of the full pseudo-Riemannian structure of spacetime. It also requires that one is working at finite temperature, but accessing a short distance scale, in a high energy collision for example, does not necessarily imply that one is working in a thermal background corresponding to that scale. In addition, one can always consider what happens at length scales which are below the mean free path of the collisions with the thermal background. Recognizing the importance of physical justification for mathematical structures used to describe spacetime, we must admit that the use of full pseudo-Riemannian geometry as the accurate description of spacetime geometry when no massive particles are present is unwarranted unless some physical justification is provided. A hint to a possible description of spacetime at such a scale may be found in the previously mentioned work by Ehlers, Pirani, and Schild who have shown that a full pseudo-Riemannian geometry of spacetime can be constructed from structures defined by paths of massless and massive particles and by imposing two compatibility conditions between them [@Ehlers1972; @Pirani1973]. The construction assumes a differentiable manifold and employs light rays (or free massless particles) and free (not under influence of anything but gravitational effects) massive particles as test bodies. All the considerations of their formalism are local, assume that the manifold and the curves in question are differentiable, and treat particles as classical objects. In broad brushstrokes, the key steps of this axiomatic construction are: - [The propagation of light determines at each point of spactime an inÞnitesimal null cone and hence deÞnes a conformal structure $\mathcal{C}$ on $M$. Light rays are represented by null geodesics which are null curves contained in null hypersurfaces.]{} - [The motions of freely falling massive particles determine a family of preferred unparametrized time-like curves at each point, and such a family at each point of $M$ defines a projective structure $\mathcal{P}$ on $M$. World lines of freely falling particles are said to be $\mathcal{C}$-time-like geodesics of $\mathcal{P}$.]{} - [Requiring two compatibility conditions: 1) that the null geodesics are also geodesics of $\mathcal{P}$, and 2) that the ticking rate of a clock is independent of its history, leads to the full pseudo-Riemannian geometry.]{} While such extrapolation may be possible, one should keep in mind Einstein’s warning that “even when it is a question of describing the electrical elementary particles constituting matter, the attempt may still be made to ascribe physical importance to those concepts of fields that have been physically defined for the purpose of describing the geometrical behavior of bodies that are large as compared with the molecule. Only the outcome can decide the justification of such an attempt, which postulates physical reality for the fundamental principles of Reimann’s geometry outside of the domain of their physical definitions" [@Einstein1922]. If we restrict ourselves to an energy regime where there are no massive particles, the axiomatic construction of the pseudo-Riemannian geometry of Ehlers, Pirani and Schild is not possible because there are no massive particles for the construction of the projective structure. But in Einstein’s own words,“\[a\]ccording to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter" [@Einstein1961]. Since the projective structure cannot manifest itself in such a “massless" regime, maintaining that $\mathcal{P}$, and hence the pseudo-Riemannian geometry, remains a property of spacetime is unjustified. We then have to concede that there may be another description of spactime valid between the Planck scale and the scale where GR holds true. If that is the case, a correct quantum theory of gravity, which may be an accurate description at the Planck scale, would not have GR as its classical limit. Instead, its limit would be the theory which accurately describes spacetime, and hence gravitation, right above the EWBS. The electroweak symmetry breaking scale is interesting, but not the only scale at which a discontinuous shift in spacetime structure may occur. As higher energies are tested experimentally and our existing models are modified to accommodate the empirical observations, we may find that a change in spactime geometry may occur, if at all, at higher energies and due to different physical reasons. Regardless of how and at what scales such change may happen, a possibility that GR may not be the limiting form of a corresponding quantum theory should be taken in consideration in our discussions of what constitutes a good candidate for a correct quantum theory of gravity. I would like to thank John Stachel for introducing me to the work on conformal and projective structures and commenting on an earlier draft of this essay. Thiemann,T.: Lect. Notes Phys. [631]{}, 41 (2003) Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge, (2004) Konopka, T., Markopoulou, F., Severini, S.: Phys. Rev. D [77]{}, 104029 (2008) Bombelli, L., J. Lee, J., Meyer, D., Sorkin, R.: Phys. Rev. Lett. [59]{}(5), 521 (1987) Loll, R., Ambjorn, J., Jurkiewicz, J.: Contemp. Phys. [47]{}, 103 (2006) Messiah, A.: Quantum Mechanics. Dover Publications Inc., Mineola, (1999) Einstein, A.: Relativity: The Special and the General Theory. Three Rivers Press, New York (1961) Einstein, A.: Geometry and Experience. In: P. Pesic (ed.) Beyond geometry: Classic papers from Riemann to Einstein, pp. 147-158. Dover Publications Inc., Mineaola (2007) Ehlers, J. Pirani, A. E. and Schild, A.: The Geometry of Free Fall and Light Propagation. In: O‘Raifertaigh, L. (ed.)General Relativity; Papers in honour of J.L. Synge, pp. 63-82. Clarendon Press, Oxford (1972) Pirani, F. A. E.: Building Space-Time from Light Rays and Free Particles. In: Symposia Mathematica, vol.12, pp. 67-83. Academic Press, London (1973) Bradonjić, K. arXiv:1103.5164 S. L. Glashow, S. L.: Nucl. Phys. [22]{}, 579 (1961) Weinberg, S.: Phys. Rev. Lett. [19]{}, 1264 (1967) Salam, A.: in [Proceedings of the Eighth Nobel Symposium, on Elementary Particle Theory, Relativistic Groups, and Analyticity, Stockholm, Sweden, 1968]{}, edited by N. Svartholm (Almqvist and Wikell, Stockholm, 1968).
--- abstract: 'The success of topological insulators (TI) in creating devices with unique functionalities is directly connected to the ability of coupling their helical spin states to well defined perturbations. However, up to now, TI-based heterostructures always resulted in very disordered interfaces, characterized by strong mesoscopic fluctuations of the chemical potential which make the spin-momentum locking ill-defined over length scales of few nanometers or even completely destroy topological states. These limitations call for the ability to control topological interfaces with atomic precision. Here, we demonstrate that molecular self-assembly processes driven by inherent interactions among the constituents offer the opportunity to create well-defined networks at TIs surfaces. Even more remarkably, we show that the symmetry of the overlayer can be finely controlled by appropriate chemical modifications. By analyzing the influence of the molecules on the TI electronic properties, we rationalize our results in terms of the charge redistribution taking place at the interface. Overall, our approach offers a precise and fast way to produce tailor-made nanoscale surface landscapes. In particular, our findings make organic materials ideal TIs counterparts, since they offer the possibility to chemically tune both electronic and magnetic properties within the same family of molecules, thereby bringing us a significant step closer towards an application of this fascinating class of materials.' author: - 'T.Bathon' - 'P.Sessi' - 'K.A.Kokh' - 'O.E.Tereshchenko' - 'M.Bode' title: \| Systematics of molecular self-assembled networks\ at topological insulators surfaces --- Engineering well-ordered nanostructures by using single atoms or molecules as building blocks offers a convenient opportunity to control matter at the nanoscale. Ultimately, this bottom-up approach represents an alternative or may even replace traditional top-down routes like lithography, which dominated the electronic industry in the past decades but have recently been challenged by the continuos device miniaturization and the related emerging technical difficulties. The extensive study of the mechanisms driving the creation of self-assembled nanostructures, mainly performed in the last decade, resulted in a quite thorough understanding of the relevant interactions at the nanoscale and in the development of guidelines to create tailor-made nanostructures with extremely high precision [@BCK2005; @B2007]. Although well-defined self-assembled superstructures can be obtained by using both, atoms and molecules, the latter are of particular interest [@B2007]. This is mainly caused by the fact that molecules are highly controllable by chemical modifications. Thereby it becomes possible to tune the molecule’s functionality and—at the same time—allowing for a large variety of self-assembled nanostructures [@YYK2001; @SRA2007; @PAS2007; @SDK2007; @OSM2005; @PBP2010]. Since the self-assembly process depends on the detailed balance of intermolecular and molecule–substrate interactions, there are severe limitations regarding the choice of both, molecules and substrates. To date, the most successful ingredients for the emergence of supramolecular order are represented by planar molecules coupled to noble metal surfaces [@B2007; @YYK2001; @SRA2007; @PAS2007; @SDK2007; @OSM2005; @PBP2010], graphite [@ASN2007] and, more recently, to graphene [@MZJ2009; @HSD2012; @JHB2013] and boron nitride [@JHB2013; @SDH2013] Here, we provide evidence that well-ordered molecular superstructures can be obtained on the new class of materials named topological insulators (TI) [@HQW2008; @CAC2009]. Even more remarkably, we demonstrate that both the periodicity and the symmetry of the resulting overlayer can be finely controlled, making self-assembly processes a new and versatile way to engineer periodic landscapes on topological states without introducing strong mesoscopic fluctuations of the chemical potential which make the spin-momentum locking ill defined over length scales of few nanometers or even completely destroy topological states[@BRS2011; @SRB2014]. In particular, by using molecules hosting magnetic moments as building blocks, the self-assembly process results in the creation of a regular network of localized spins. This makes hybrid organic–TI interfaces a promising and reliable platform for the investigation of exotic states of matter and the unconventional magneto-electric effects, which have been predicted to exist when topological states interact with local spins and which may find direct application in devices with new functionalities [@GF2010; @M2010; @NN2011]. More generally, our approach identifies planar molecules as the ideal TI counterpart as they overcome all the problems which have so far limited the study, understanding, and utilization of the response of topological states to external perturbations. As a prototype system, we focus on transition metal (TM)-phthalocyanine (Pc) molecules coupled to the chalcogenide topological insulator Bi$_2$Te$_3$. Pcs are among the most widely studied planar molecules which are already successfully employed in several applications such as sensors and magnets [@H2009]. Furthermore, they have been recently reported to constitute a rich playground hosting unconventional properties associated with their spin-degree of freedom [@FSP2011; @HBP2013]. As shown in Fig.1(a), their structure consists of a central metal atom surrounded by an organic macrocycle. The molecular symmetry is 4-fold and the central atom has a square-planar coordination. In TM-Pcs, this bonding scheme leaves the central atom in a $+2$ state and, within the $D_{4h}$ point symmetry, the $3d$ levels split into $d_{xz,yz}$, $d_{z^2}$, $d_{xy}$, and $d_{x^2-y^2}$ as illustrated in Fig.1(b) [@WSC2009]. Depending on their occupation, energy, and symmetry, these states can mix with the orbitals of the pyrrole-like rings. Since the hybridization between $3d$ levels and molecular orbitals may result in several quasi-degenerate configurations, it is impossible to draw a definitive picture of the energy occupation level based on a single-electron approach. This is particularly true for molecules coupled to substrates, since charge transfer processes and modified ligand fields may further complicate the picture. However, by increasing the number of $3d$ electrons the following trends emerge: (i) the presence, close to the Fermi level, of empty states with $d_{z^2}$ symmetry is gradually reduced; (ii) the TM-substrate distance increases [@WSC2009; @ZDG2011; @MKR2011; @MRK2012]. These observations imply that, contrary to “late” transition metal (Ni and Cu) Pc molecules, where the central atom’s $3d$ orbital does not play any significant role in the adsorption process, the situation is quite different for MnPc, FePc, and CoPc. As described in the following, this has far reaching implications. ![(a) Structure of a Pc molecule, which can host several different elements as central atom, offering the opportunity to tune its properties within the same structure. (b) Crystal field-split energy level for $3d$ orbitals in $D_{4h}$ symmetry. (c-e) Constant-current images obtained for very dilute concentrations of MnPc, CoPc, and CuPc molecules grown on Bi$_2$Te$_3$, respectively (left panel). Atomically resolved images reveal that, despite the very different electronic configuration of the central $3d$ atom, single MnPc, CoPc, and CuPc molecules all sit on top of a Te atom (right panel). This observation indicates the leading role played by the ligand in determining the adsorption geometry. Note that, while the central atom appears as a protrusion in MnPc and CoPc, it corresponds to a depression in CuPc, thereby indicating the absence of $3d$ orbitals projecting outside the surface plane which can effectively couple to tip states. Scanning parameters: $I = 25$pA, $V = 300$mV, 500mV, and $-200$mV for MnPc, CoPc and CuPc, respectively. []{data-label="Fig1"}](Fig1.jpg){width="0.9\columnwidth"} Figure1(c)-(e) display scanning tunneling microscopy (STM) images obtained for a very dilute concentration of three different transition metal-phthalocyanine molecules, i.e. MnPc, CoPc, and CuPc on Bi$_2$Te$_3$, thereby spanning the entire $3d$ occupancy scenario. Three relative orientations with respect to the substrate are visible for all three molecules, resulting from the combined symmetry of the molecules (4-fold) with that of the substrate (6-fold) [@SBK2014]. Inspection of atomically resolved images evidences that, despite their very different $3d$ filling, all molecules have the same adsorption geometry, with the central atom sitting on top of a Te atom of the underlying Bi$_2$Te$_3$ surface. This observation evidences the leading role played by the ligands in determining the adsorption configuration. Nevertheless, a direct participation of the central atom in the molecule–substrate bond can further anchor the molecules to the surface. This is clearly the case for MnPc, whose 2-fold symmetry reduction is indicative of a strong molecule–substrate interaction [@CKB2008; @CCW2010]. In contrast, CoPc and CuPc preserve the 4-fold symmetry typical of the gas phase (see supplementary information). The molecule–substrate interaction observed for MnPc is attributed to $d_{z^2}$ orbitals bonding to the TI surface. However, as the number of $d$ electrons increases, they shift towards negative energies. This is in particular the case for CuPc, where they become double occupied without any significant hybridization with the substrate [@ZDG2011; @MRK2012]. Indeed, while the central atom is imaged as a protrusion on MnPc and CoPc, it appears as a depression in CuPc, indicating the absence—in the probed energy range—of $d_{z^2}$ orbitals, i.e. those electronic states which can effectively couple to the STM tip because of their larger extension into the vacuum (for a discussion on the impact of the central atom in STM images see Ref. and references therein). The different occupation of the strongly directional $3d$ levels thus offers a convenient way to chemically control the delicate balance of forces present at the interface as the molecule coverage increases and molecule–molecule interactions start to play an important role. Indeed, earlier studies on metallic substrates already pointed out the importance of the $d$-filling and the role of the molecule–substrate interaction. While an abrupt transition from a dispersive distribution to a square lattice superstructure was generally observed on metal substrates [@JXL2011; @LHW1996; @CGD2007], we demonstrate below that on topological insulators an appropriate choice of the central atom not only allows to “activate” the self-assembly process, but also to select its symmetry. ![(a) Constant-current image obtained for a MnPc concentration equal to 0.13molecules/nm$^2$. Despite the small intermolecular distance, cluster formation cannot be observed on the surface, indicating a significant molecule–molecule repulsion. (b) STS spectra obtained onto the Bi$_2$Te$_3$ before (green line) and after (blue line) molecule deposition. A rigid energy shift towards negative energies appears after the growth. This indicates that the creation of the molecule–TI interface is accompanied by charge transfer processes which $n$-dope the sample leaving MnPc molecules positively charged. Scanning parameters: $I = 15$pA, $V = 500$mV. []{data-label="Fig2"}](Fig2.jpg){width="0.65\columnwidth"} Figure2(a) shows MnPc molecules on Bi$_2$Te$_3$ for a coverage equal to 0.13molecules/nm$^2$. Note that, despite the reduced molecule–molecule distance, not any signature of molecular clustering is present on the surface. On the contrary, molecules seem to maximize their distance. This behavior can be understood by comparing the electronic properties of the Bi$_2$Te$_3$ surface before and after molecule deposition as inferred by scanning tunneling spectroscopy (STS). Results are reported in Fig.2(b). The positions of the valence band maximum and conduction band minimum have been assigned according to the procedure described in Ref. . The spectra evidence a negative rigid shift of the Bi$_2$Te$_3$ band structure subsequent to deposition which is indicative of an $n$-doping effect caused by MnPc. This charge redistribution leaves the molecules positively charged. Consequently, they are stably anchored to the substrate by the creation of an interfacial dipole which suppresses their mobility on the surface. As a result of the intermolecular repulsive interaction and the relatively strong anchoring to the substrate, the molecules do not show any tendency of ordering, irrespective of the coverage. Furthermore, the second MnPc layer starts to form well before completion of the first one (see Fig.3 in Supplementary Information). These observations show that repulsive-driven self-assembly processes as observed in Ref. are absent for the particular combination of molecule and substrate considered here. ![(a,b) Self-assembled CoPc molecular film. The unit cell is indicated by a black box. (c) Fourier-transformed constant-current image displaying a slightly distorted hexagonal symmetry of the 2D molecular overlayer. (d) In contrast to MnPc \[cf. Fig.2(b)\], STS reveals that no significant charge transfer takes place at the interface, indicating that CoPc molecules are weakly bound to the substrate thus retaining substantial surface mobility. (e) Model illustrating the structure of the self-assembled molecular film with respect to the underlying Bi$_2$Te$_3$ substrate. Scanning parameters: $I = 15$pA, $V = 500$mV. []{data-label="Fig3"}](Fig3.jpg){width="0.99\columnwidth"} A very different scenario appears for CoPc as illustrated in Fig.3(a) and (b). In this case, large regions displaying a very well-ordered molecular superstructure are present on the surface, indicating a relatively weak molecule–substrate interaction as compared to attractive molecule–molecule interactions. The weak molecule–substrate bond is also experimentally signaled by significant tip–sample forces, which, despite the very low set-point currents used when scanning the surface, results in molecules that are occasionally moved by the tip (see Supplementary Information). As for MnPc, the inspection of spectra obtained before and after deposition allows to visualize the charge redistribution processes associated with the creation of the molecule–TI interface. Results are reported in Fig.3(d). The absence of any significant energy shift implies that, contrary to MnPc, CoPc molecules are left in their neutral state. This is the more remarkable as the coverage amounts to 0.32molecules/nm$^2$, i.e. well above the one investigated in Fig.2 for MnPc. Contrary to similar close-packed layers obtained at metal surfaces[@ASK2010], where close-packed layers evolve but charge-transfer nevertheless occur, in the present case molecules have been deposited onto a substrate, i.e. Bi$_2$Te$_3$, with a much lower density of states. As a result, any potential charge transfer taking place at the interface would give rise to a significant band bending, as it has been widely reported for topological insulator surfaces coupled to single adatoms (see Ref. and ) or even simply exposed to residual gases (see Refs. and ). It is worth noticing that the observed difference in charge transfer between MnPc and CoPc is further corroborated by simple electronegativity concepts. Indeed, as compared to Mn (1.55), the electronegativity of Co (1.88) is closer to those of Bi (2.02) and Te (2.10), i.e. the atomic species of the underlying substrate. It thus appears that the combination of CoPc with the Bi$_2$Te$_3$ TI substrate is one of the rare cases where the creation of an interface leaves either of the constituents essentially unaltered. Therefore, we expect that CoPc behaves very differently as compared to other adsorbates which may heavily dope TI surfaces and even create new electronic states that are absent on the pristine samples [@BHM2011; @BLK2011; @VPG2012]. Similarly, many TMPc molecules may—once coupled to a substrate—substantially change their electronic properties through the interplay of substrate screening and hybridization effects [@MKR2011; @MRK2012]. In contrast, the electronic properties detected for CoPc on Bi$_2$Te$_3$ raises the hope that the gas phase functionality may be preserved. The geometrical structure which appears from inspection of Fig.3(b) allows to identify a quasi-hexagonal crystal lattice, which indicates that the symmetry of the substrate has been transferred to the overlayer. Minor periodic distortions are visible along two of the three high symmetry axis, also signaled by the appearance of 4 additional spots in the Fourier transformation \[see Fig.3(c)\]. Although the presence of a well-ordered superstructure indicates that molecule–molecule interactions dominate the delicate balance of forces that drive the self-assembly process, molecule–substrate interactions still appear to play an important role since it determines the axis along which molecules self-assemble. A detailed analysis of the molecular film allows to obtain the structural model illustrated in Fig.3(e). Molecules have a nearest-neighbor distance corresponding to 1.55 nm, with each of them sitting on top (or in very close proximity, see below) of the underlying Te atom, as can also be inferred from Fig.3(b), where all molecules display the very same appearance. Furthermore, the different rotations of the molecules allow to explain the distortions introduced in the lattice as the result of small steric repulsion-induced deviations with respect to the minimum energy adsorption site. Overall, these observations point towards an overlayer unit cell that is commensurate with the substrate. Since MnPc and CoPc have the same structure and ligand, their different behavior in the high coverage regime must be a direct consequence of the different electronic configuration of their $3d$ central atoms. Although a precise analysis of the molecule–substrate interactions would require detailed calculations going beyond the scope of the present work, an heuristic picture that is based on the $3d$ levels occupancy can effectively explain our findings. In particular, the differences are ascribed to the presence of $d_{z^2}$ orbitals close to Fermi level. These electronic states effectively facilitate the bonding to the substrate, which is gradually reduced as the number of electrons residing on the central atom increases, i.e. by changing from a Mn to a Co ion [@WSC2009]. Therefore, changing to a higher atomic number drives the balance of molecule–substrate and molecule–molecule interactions towards the latter. ![(a) Self-assembled CuPc molecular film. (b) Contrary to CoPc \[cf. Fig.3(b)\], CuPc self-assembles with a square symmetry (unit cell indicated by box), as evidenced by the Fourier transformed constant-current image displayed in (c). (d) Similar to CoPc, STS reveals that no significant charge transfer takes place at the interface, indicating that CuPc molecules are weakly bound to the substrate. (e) Model illustrating the structure of the self-assembled molecular film with respect to the underlying Bi$_2$Te$_3$ substrate. Scanning parameters: $I = 15$pA, $V = 500$mV.[]{data-label="Fig4"}](Fig4.jpg){width="0.99\columnwidth"} To test our model, measurements have been performed on CuPc which, because of its double-occupied $d_{z^2}$ orbital that lies well below the Fermi level [@ZDG2011; @MRK2012], is expected to result in an even weaker molecule–substrate interaction with respect to CoPc. By looking at Fig.4(a) it is evident that a well-ordered self-assembled structure is obtained also in this case (coverage 0.46molecules/nm$^2$). As for CoPc, the creation of a molecular film is not associated with any significant charge transfer between molecules and substrate \[see Fig.4(d)\] \[note that, as in the case of CoPc (see above), also for CuPc a simple electronegativity picture supports our findings, being Cu electronegativity equal to 1.90\]. However, a zoomed-in image demonstrates that, contrary to CoPc, CuPc self-assembles in a closed-packed structure with a square lattice \[see Fig.4b\]. This is the typical lattice adopted by Pc molecules deposited on weakly interacting substrates, since it allows to maximize their mutual interaction. Indeed, the nearest-neighbor distance of 1.34 nm is essentially determined by the size of the molecule. An analysis of the molecular film allows to obtain the structural model reported in Fig.4(e). Since the symmetries of the molecular film and the substrate are incompatible, the molecules adopt different adsorption configurations with respect to the underlying Bi$_2$Te$_3$ surface, as also signaled by their different appearance in Fig.4(b). The long wavelength modulation visible in the image suggests an incommensurability between the molecular layer and the substrate, as already discussed in Ref.. Unambiguous determination would require simultaneous atomic resolution of the underlying substrate which was impossible to achieve because of incompatible tunneling parameters. Overall, these observations are in agreement with our model and confirm the leading role played by the central $3d$ orbitals in balancing the different interaction taking place at the molecule–TI interface. In summary, our findings show that appropriate chemical modifications of metal-organic compounds lead to a rich variety of hybrid molecule-TI interfaces. As a result these interfaces represent ideal heterostructures for tailoring potential landscapes at TI surfaces at the atomic scale. Furthermore, molecules with built-in magnetic moments, such as the molecules used in this study, may result in superstructures that constitute well defined spin networks, potentially leading to device concepts with unique functionalities. [Methods]{} Experiments have been performed in a commercial STM operated at $T = 4.8$ K. The Bi$_2$Te$_3$ single-crystal samples were synthesized by mixing stoichiometric amounts of bismuth and tellurium. The crystal structure consists of alternating planes of Bi and Te up to the formation of a quintuple layer with the sequence Te-Bi-Te-Bi-Te. Quintuple layers are weakly coupled by van der Waals forces thus offering a natural cleaving plane. Bi$_2$Te$_3$ single crystals have been cleaved in UHV at a base pressure of $3 \cdot 10^{-11}$ mbar and immediately inserted into the STM. MnPc, CoPc and CuPc molecules (Sigma-Aldrich) were deposited directly onto the cold Bi$_2$Te$_3$ surface by using a home made Knudsen cell and annealed at room temperature. STM measurements were performed using electrochemically etched tungsten tips. Topographic images were acquired in the constant-current mode. Spectroscopic data were obtained by lock-in technique ($f = 793$ Hz, $V_{rms} = 10$ mV). Since the apparent molecular coverage was found to strongly depend on the particular bias voltage applied, an effect that is related to the spatial distribution of certain molecular orbitals, coverages are given in number of molecules per nm$^2$.\ [Associated Content]{}\ [Supporting information]{}\ Description of the adsorption of TM-Pc on Bi$_2$Te$_3$ which lead to a C4 to C2 symmetry reduction; evidence for weak adsorption energy from strong tip-molecule interaction; a coverage-dependent study of MnPc molecules; and the cluster formation at very dilute concentration regime for CoPc and CoPc. This material is available free of charge via the Internet at http://pubs.acs.org. This work has been funded within SPP 1666 “Topologische Isolatoren” (project BO 1468/21-1). K.A.K. and O.E.T. acknowledge financial support by the RFBR (Grant Nos. 13-02-92105 and 14-08-31110). Barth, J.V.; Constantini, G.; Kern, K. [Nature]{} [437]{}, 671-679 (2005). Barth, J.V. [Annu. Rev. Phys. Chem.]{} [58]{}, 375-407 (2007). Yokoyama, T.; Yokoyama, S.; Kamikado, T.; Okuno, Y.; Mashiko, S. [Nature]{} [413]{}, 619-621 (2001). Schiffrin, A.A.; Riemann, A.; Auw[ä]{}rter, W.; Pennec, Y.; Weber-Bargioni, A.; Cvetko, D.; Cossaro, A.; Morgante, A.; Barth, J.V. [Proc. Natl. Acad. Sci.]{} [104]{}, 5279-5284 (2007). Pennec, Y.; Auw[ä]{}rter, W.; Schiffrin, A.; Weber-Bargioni, A.; Riemann, A.; Barth, J.V. [Nature Nanotechnolgy]{} [2]{}, 99-103 (2007). Schlickum, U.; Decker, R.; Klappenberger, F.; Zoppellaro, G.; Klyatskaya, S.; Ruben, M.; Silanes, I.; Arnau, A.; Kern, K.; Brune, H.; Barth, J.V. [Nano Lett.]{} [7]{}, 3813-3817 (2007). Otero, R.; Sch[ö]{}ck, M.; Molina, L.M.; L[æ]{}gsgaard, E.; Stensgaard, I.; Hammer, B.; Besenbacher, F. [Angew. Chem.]{}, [117]{}, 2310-2315 (2005). Pivetta, M.; Bl[ü]{}m, M.-C.; Patthey, F.; Schneider, W.-D. [Chem. Phys. Chem.]{} [11]{}, 1558-1569 (2010). hlund, J.; Schnadt, J.; Nilson, K.; G[ö]{}thelid, E.; Schiessling, J.; Besenbacher, F.; Martensson, N.; Puglia, C. [Surf. Sci.]{} [601]{}, 3661-3667 (2007). Mao, J.; Zhang, H.; Jiang, Y.; Pan, Y.; Gao, M.; Xiao, W.; Gao, H.J. [J. Am. Chem. Soc]{} [131]{}, 14136-14137 (2009). H[ä]{}m[ä]{}l[ä]{}inen, S.K.; Stepanova, M.; Drost, R.; Liljeroth, P.; Lahtinen, J.; Sainio, J. [J. Chem. Phys. C]{} [116]{}, 20433-20437 (2012) J[ä]{}rvinen, P.; H[ä]{}m[ä]{}l[ä]{}inen, S.K.; Banerjee, K.; H[ä]{}kkinen, P.; Ij[ä]{}s, M.; Harju, A.; Liljeroth, P. [Nano Lett.]{} [13]{}, 3199-3204 (2013). Schulz, F.; Drost, R.; H[ä]{}m[ä]{}l[ä]{}inen, S. K.; Liljeroth, P. [ACSNano]{} [7]{}, 11121-11128 (2013). Hsieh, H.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y.S.; Cava, R.J.; Hasan, M.Z. [Nature]{} [452]{}, 970-974 (2008). Chen, Y.L.; Analytis, J.G.; Chu, J.-H.; Liu, Z.K.; Mo, S.-K.; Qi, X.L.; Zhang, H.J.; Lu, D.H.; Dai, X.; Fang, Z.; Zhang, S.C.; Fisher, I.R.; Hussain, Z.; Shen, Z.-X. [Science]{} [325]{}, 178-181 (2009). Beidenkopf, H.; Roushan, P.; Seo, J.; Gorman,L.; Drozdov, I.; Hor, Y.S.; Cava, R.J.; Yazdani, A. [Nature Physics]{} [7]{}, 939-943 (2011). Sessi, P.; Reis, F.; Bathon, T.; Kokh, K.A.; Tereshchenko, O.E.; Bode, M. [Nature Communications]{} [5]{}, 5349 (2014). Garate, I.; Franz, M. [Phys. Rev. Lett.]{} [104]{}, 146802 (2010). Moore, J.E. [Nature Physics]{} [464]{}, 194-198 (2010). Nomura, K.; Nagaosa, N. [Phys. Rev. Lett.]{} [106]{}, 166802 (2011). Heath, J.R. [Annu. Rev. Mater. Res.]{} [39]{}, 1-23 (2009). Franke, K.J.; Schulze, G.; Pascual, J.I. [Science]{} [332]{}, 940 (2011). Heinrich, B.W.; Braun, L.; Pascual, J.I.; Franke, K.J. [Nature Physics]{} [9]{}, 765-768 (2013). Wang, J., Shi, Y., Cao, J.; Wu, R. [Appl. Phys. Lett.]{} [94]{}, 122502 (2009). Zhang, Y.Y., Du, S.X; Gao, H.-J. [Phys. Rev. B]{} [84]{}, 125446 (2011). Mugarza, A.; Krull, C.; Robles, R.; Stepanow, S.; Ceballos, G.; Gambardella, P. [Nature Communications]{} [2]{}, 490 (2011). Mugarza, A.; Robles, R.; Krull, C.; Koryt[á]{}r, R.; Lorente, N.; Gambardella, P. [Phys. Rev. B]{} [85]{}, 155437 (2012). Sessi, P., Bathon, T., Kokh, K.A., Tereshchenko, O.E.; Bode, M. [Nano Lett.]{} [14]{}, 5092-5096 (2014). Auw[ä]{}rter, W.; [É]{}cija, D.; Klappenberger, F.; Barth, J. V. [Nature Chemistry]{} [7]{}, 105-119 (2015). Jiang, Y.H.; Xiao, W.D.; Liu, L.W.; Zhang, L.Z.; Lian, J.C.; Yang, K.; Du, S.X.; Gao, H.-J. [J. Chem. Phys. C]{} [115]{}, 21750-21754 (2011). Lu, X.; Hipps, K.W.; Wang, X.D.; Mazur, U. [J. Am. Chem. Soc.]{} [118]{}, 7197-7202 (1996). Cheng, Z.H.; Gao, L.; Deng, Z.T.; Liu, Q.; Jiang, N.; Lin, X.; He, X.B.; Du, S.X.; Gao, H.-J. [J. Chem. Phys. C]{} [111]{}, 2656-2660 (2007) Chang, S.-H.; Kuck, S.; Brede, J.; Lichtenstein, L.; Hoffmann, G.; Wiesendanger, R. [Phys. Rev. B]{} [78]{}, 233409 (2008). Cuadrado, R.; Cerda, J.I.; Wang, Y.; Xin, G.; Berndt, R.; Tang, H. [J. Chem. Phys.]{} [133]{}, 154701 (2010). Sessi, P.; Otrokov, M.; Bathon, T.; Vergniory, M.G.; Tsirkin, S.S.; Kokh, K.A.; Tereshchenko, O.E.; Chulkov, E.V.; Bode, M. [Phys. Rev. B]{} [88]{}, 161407(R) (2013). Bischoff, F.; Seufert, K.; Auw[ä]{}rter, W; Joshi, S.; Vijayaraghavan, S; [É]{}cija, D.; Diller, K.; Papageorgiou, A.C.; Fischer, S.; Allegretti, F.; Duncan, D. A.; Klappenberger, F.; Blobner, F.; Han, R.; and Barth, J.V. [ACS Nano]{} [7]{}, 3139-3149 (2013). Auw[ä]{}rter, W.; Seufert, K.; Klappenberger, F.; Reichert, J.; Weber-Bargioni, A.; Verdini, A.; Cvetko, D.; Dell’Angela, M.; Floreano, L.; Cossaro, A.; Bavdek, G.; Morgante, A.; Seitsonen, A.P.; Barth, J.V.; [Phys. Rev. B]{} [81]{}, 245403 (2010). Bianchi, M.; Hatch, R.C.; Mi, J.; Iversen, B.B.; Hofmann, P. [Phys. Rev. Lett.]{} [107]{}, 086802 (2011). Benia, H.M.; Lin, C.; Kern, K.; Ast, C.R. [Phys. Rev. Lett.]{} [107]{}, 177602 (2011). Valla, T.; Pan, Z.-H.; Gardner, D.; Lee, Y.S.; Chu, S. [Phys. Rev. Lett.]{} [108]{}, 117601 (2012).
--- abstract: \| This paper deals with stationary Gibbsian point processes on the plane with an interaction that depends on the tiles of the Delaunay triangulation of points via a bounded triangle potential. It is shown that the class of these Gibbs processes includes all minimisers of the associated free energy density and is therefore nonempty. Conversely, each such Gibbs process minimises the free energy density, provided the potential satisfies a weak long-range assumption.\ [Keywords]{}. [Delaunay triangulation, Voronoi tessellation, Gibbs measure, variational principle, free energy, pressure, large deviations.]{}\ [MSC]{}. Primary 60K35; Secondary: 60D05, 60G55, 82B21.\ [Running head]{}. [Delaunay-Gibbs measures]{} author: - 'David Dereudre and Hans-Otto Georgii' title: ' Variational characterisation of Gibbs measures with Delaunay triangle interaction ' --- Introduction ============ It is well-known that stationary renewal processes with a reasonable spacing distribution can be characterised as Gibbs processes for an interaction between nearest-neighbour pairs of points [@HS Section 6]. Here we consider an analogue in two dimensions, viz. Gibbsian point processes on $\mathbb{R}^2$ with an interaction depending on nearest-neighbour triples of points, where the nearest-neighbour triples are defined in terms of the Delaunay triangulation. Recall that the Delaunay triangulation is dual to the Voronoi tessellation, in the sense that two points are connected by a Delaunay edge if and only if their Voronoi cells have a common edge. Since the Voronoi cell of a point consists of the part of space that is closer to this point than to any other point, this means that the Delaunay graph defines a natural nearest-neighbour structure between the points. (Of course, the analogy with renewal processes does not reach too far because the independence of spacings under the Palm distribution, which is characteristic of one-dimensional renewal processes, is lost in two dimensions due to the geometric constraints.) There is a principal difference between the Delaunay interactions considered here and the pair interactions that are common in Statistical Physics. Namely, suppose a point configuration $\o$ is augmented by a new particle at $x$. In the case of pair interactions, $x$ is subject to some additional interaction with the particles in $\o$, but the interaction between the particles of $\o$ is not affected by $x$. In the Delaunay case, however, the particle at $x$ not only gives rise to some new tiles of the Delaunay triangulation, but also destroys some other tiles that were present in the triangulation of $\o$. This so-called non-hereditary nature of the Delaunay triangulation blurs the usual distinction between attractive and repulsive interactions and makes it difficult to use a local characterisation of Gibbs measures in terms of their Campbell measures and Papangelou intensities. Such a local approach to the existence of Gibbs measures for Delaunay interactions was used in the previous work [@BBD99a; @BBD99b; @BBD08; @Dereudre] and made it necessary to impose geometric constraints on the interaction by removing triangles with small angles or large circumcircles. In this paper we address the existence problem from a global point of view, which is based on stationarity and thermodynamic quantities such as pressure and free energy density. Specifically, we show that all minimisers of the free energy density are Gibbsian, which implies the existence of Delaunay-Gibbs measures because the entropy density has compact level sets. The converse part of this variational principle is harder and requires the comparison of different boundary conditions in the thermodynamic limit. It is here that the non-hereditary nature of the interaction shows up again, but it can be controlled with the help of stationarity and an additional condition which is much weaker than the geometric constraints mentioned above. In contrast to [@Dereudre], however, we need to assume throughout that the interaction potential is bounded, and therefore do not cover hard-core interactions that forbid particular shapes of the tiles. We note, however, that some ideas of the present paper can be used to establish the existence of Delaunay-Gibbs measures in a more general setting that includes also the hard-core case, see [@DDG]. The extension of the variational principle to such interactions is left to future work. As a final comment, let us emphasise that we make repeated use of Euler’s polyhedral formula and the resulting linear complexity of the Delaunay triangulations, and are therefore limited to two dimensions, as was already the case in the previous papers mentioned above. Preliminaries ============= Configurations and Delaunay triangulations ------------------------------------------ A subset $\o$ of $\RR^2$ is called locally finite if $\#(\o\cap\D)<\infty$ for all bounded $\D\subset\RR^2$; each such $\o$ is called a configuration. We write $\O$ for the set of all configurations $\o$. The configuration space $\O$ is equipped with the $\s$-algebra $\cF$ that is generated by the counting variables $N_\D:\o\to \#(\o\cap\D)$, with $\D$ an arbitrary bounded Borel subset of $\RR^2$. For each Borel set $\L\subset\RR^2$ we write $\O_\L=\{\o\in\O:\o\subset\L\}$ for the set of configurations in $\L$, $\pr_\L:\o\to\o_\L:=\o\cap\L$ for the projection from $\O$ to $\O_\L$, $\cF_\L'= \cF\|\O_\L$ for the trace $\s$-algebra of $\cF$ on $\O_\L$, and $\cF_\L=\pr_\L\inv\cF_\L'$ for the $\s$-algebra of events in $\O$ that happen in $\L$ only. For each configuration $\o\in\O$ we consider the Delaunay triangulation $\sD(\o)$ associated to $\o$. By definition, (ø)={ ø: \#=3, øB()=}, where $B(\tau)$ is the unique open disc with $\tau\subset\partial B(\tau)$. $\sD(\o)$ is uniquely defined and determines a triangulation of the convex hull of $\o$ whenever $\o$ is in general circular position, in that no four points of $\o$ lie on a circle that contains no further points of $\o$ inside [@Moller]. If this is not the case, one can apply some determistic rule to make the Delaunay triangulation unique. Indeed, let $$\cT:=\big\{\tau\subset\RR^2:\;\#\tau=3\big\}=\big\{(x,y,z)\in(\RR^2)^3:x\prec y\prec z\big\}$$ be the set of all triangles (or tiles) in $\RR^2$ where ‘$\prec$’ stands for the lexicographic order in $\RR^2$. The triangles in $\cT$ can be compared by the lexicographic order of $(\RR^2)^3$, and this in turn induces a lexicographic order on finite collections of triangles. Now, if $n\ge4$ points of $\o$ lie on a circle with no points inside then the associated Delaunay cell is a convex polygon having these $n$ points as vertices. To define a unique triangulation of this polygon one can then simply take the smallest among all possible triangulations. Conflicts with other possible polygons cannot arise because the tessellations inside and outside a fixed convex polygon do not depend on each other. Let us note that the prescription $\o\to\sD(\o)$ is a mapping from $\O$ to the set $\O(\cT)$ of all locally finite subsets of $\cT$. If $\O(\cT)$ is equipped with the $\s$-algebra $\cF(\cT)$ that is defined in analogy to $\cF$, one can easily check that this mapping is measurable. Next we assign to each tile $\tau\in\cT$ a centre and a radius. Specifically, for every $\tau\in\cT$ we write $c(\tau)$ for the centre and $\r(\tau)$ for the radius of the circumscribed disc $B(\tau)$. The centres allow us to consider $\sD(\o)$ as a germ-grain system, i.e., as a marked point configuration of germs in $\RR^2$ and marks in the space $$\cT_0= \{ \tau\in\cT: \;c(\tau)=0\}$$ of centred tiles, by considering the mapping D: ø{(c(),-c()): (ø)} from $\O$ to the point configurations on $\RR^2\times\cT_0$. Here we write $\tau-c(\tau):=\{y-c(\tau):y\in\tau\}$ for the shifted tile. A crucial fact we need in the following is the linear complexity of Delaunay triangulations, which is expressed in the following lemma. This result follows directly from Euler’s polyhedral formula, and is the main reason why we need to confine ourselves to two spatial dimensions; see [@Proofs], Chapter 11, and [@Moller], Remark 2.1.4. \[Euler\] For a simple planar graph on $n$ vertices, the number of edges is at most $3n-6$, and the number of inner faces is at most $2n-5$. In particular, every triangulation with $n$ nodes consists of $2n-2-\partial$ triangles, where $\partial$ is the number of nodes (or: number of edges) along the outer boundary. Stationary point processes and their tile distribution ------------------------------------------------------ Let $\cP_\Th$ be the set of all probability measures $P$ on $(\O,\cF)$ that satisfy the following two properties: $P$ is stationary, that is, $P$ is invariant under the shift group $\Th=(\th_x)_{x\in\RR^2}$ on $\O$, which is defined by $\th_x:\o\to\o-x:=\{y-x:y\in\o\}$. $P$ has a finite intensity $z(P)=\|\D\|\inv\int N_\D\,dP<\infty$. Here, $\D$ is any bounded Borel set in $\RR^2$ (which can be arbitrarily chosen due to stationarity), and $\|\D\|$ its Lebesgue measure. Each $P\in\cP_\Th$ is called a stationary point process on $\RR^2$ with finite intensity. For $\L\subset\RR^2$, we write $P_\L:=P\circ\pr_\L\inv $ for the projection image of $P$ on $(\O_\L,\cF_\L')$, which can of course be identified with the restriction $P\|\cF_\L$ of $P$ to the events in $\L$. Every $P\in \cP_\Th$ defines a germ-grain model $\bar{P}$, namely the distribution of $P$ under the mapping $D$ defined in . That is, $\bar{P}$ is a stationary marked point process on $\RR^2$ with mark space $\cT_0$. Let $\bar{P}^0$ be the associated Palm measure on $\cT_0\times\O$ and $\mu_P=\bar{P}^0(\,\cdot\,\times\O)$ the associated mark distribution, or centred tile distribution, on $\cT_0$. By definition, $$\label{eq:Palm} \int dx \int \mu_P(d\tau) \,f(x,\tau) =\int P(d\o)\sum_{\tau\in\sD(\o)} f(c(\tau),\tau-c(\tau))$$ for all nonnegative measurable functions $f$ on $\RR^2\times\cT_0$. For each $P\in\cP_\Th$, $\mu_P$ has total mass $\\|\mu_P\\|=2z(P)$, as follows from Euler’s polyhedral formula; see, for example, [@Moller Eq. (3.2.11)] or [@SchnWeil Theorem 10.6.1(b)]. Let us say a measure $P\in \cP_\Th$ is tempered if \|B()\| \_P(d) <We write $\cP_\Th^\tmp$ for the set of all tempered $P\in\cP_\Th$. Of course, is equivalent to the condition $\int \r(\tau)^2\, \mu_P(d\tau) <\infty$. Moreover, implies that $$\begin{aligned} \label{eq:tempered2} &&\int \|B(\tau )\| \,\mu_P(d\tau) =\int dx\int \mu_P(d\tau)\, \1_{\{x\in B(\tau)\}}\\ &=&\int \sum_{\tau \in\sD(\o)} \1_{\{c(\tau)\in B(\tau-c(\tau))\}} \,P(d\o) =\int \#\big\{\tau\in\sD(\o): 0\in B(\tau)\big\}\, P(d\o) \,. \nonumber\end{aligned}$$ So, $P$ is tempered if and only if the last expression is finite. A sufficient condition for temperedness will be given in Proposition \[tempered\]. The most prominent members of $\cP_\Th^\tmp$ are the Poisson point processes, which will take the role of reference processes for the models we consider. Recall that the Poisson point process $\Pi^z$ with intensity $z>0$ is characterised by the following two properties: For every bounded Borel set $\D$, the counting variable $N_{\D}$ is Poisson distributed with parameter $z\|\D\|$. Conditional on $N_\D=n$, the $n$ points in $\D$ are independent with uniform distribution on $\D$, for every bounded Borel set $\D$ and each integer $n$. The temperedness of $\Pi^z$ follows from Proposition 4.3.1 of [@Moller], or Proposition \[tempered\] below. Another type of measures in $\cP_\Th^\tmp$ are the [stationary empirical fields]{} that are defined as follows. Let $\L\subset\RR^2$ be an open square of side length $L$, and for $\o\in\O_\L$ let $\o_{\L,\per}=\{x+Li:\,x\in\o,\,i\in\ZZ^2\}$ be its periodic continuation. The associated stationary empirical field is then given by R\_[Ł,ø]{}= 1[\|Ł\|]{}\_[Ł]{} \_[\_x ø\_[Ł,]{}]{}dx. It is clear that $R_{\L,\o}$ is stationary. In addition, it is tempered because $2\r(\tau)\le \text{diam\,}\L$ for each triangle $\tau\in\sD(\o_{\L,\per})$. The topology of local convergence --------------------------------- In contrast to the traditional weak topology on the set $\cP_\Th$ of stationary point processes, we exploit here a finer topology, which is such that the intensity is a continuous function, but nonetheless the entropy density has compact level sets. Let $\cL$ denote the class of all measurable functions $f:\O\to\RR$ which are local and tame, in that there exists some bounded Borel set $\D\subset\RR^2$ such that $f= f\circ\pr_\D$ and $\|f\|\le b(1+N_\D)$ for some constant $b=b(f)<\infty$. The topology $\Tau_\cL$ of local convergence on $\cP_\Th$ is then defined as the weak\* topology induced by $\cL$, i.e., as the smallest topology for which the mappings $P\to\int f\,dP$ with $f\in\cL$ are continuous. By the definition of the intensity, it is then clear that the mapping $P\to z(P)$ is continuous. A further basic continuity property is the fact that the centred tile distribution $\mu_P$ depends continuously on $P$. Let $\cL_0$ be the class of all bounded measurable functions on the space $\cT_0$ of centred tiles, and $\Tau_0$ the associated weak\* topology on the set $\cM(\cT_0)$ of all finite measures on $\cT_0$. (This is sometimes called the $\tau$-topology.) \[muP\_cont\] Relative to the topologies $\Tau_\cL$ and $\Tau_0$ introduced above, the mapping $P\to\mu_P$ from $\cP_\Th$ to $\cM(\cT_0)$ is continuous. This result will be proved in Section \[subsec:energy\]. It takes advantage of the linear complexity of finite Delaunay triangulations, and therefore relies on the planarity of our model. The entropy density ------------------- Given a point process $P\in\cP_\Th$ and a bounded Borel set $\L$ in $\RR^2$, we let $I_\L(P,\Pi^z)$ denote the relative entropy (or Kullback-Leibler information) of $P_\L$ relative to $\Pi^z_\L$. By the independence properties of $\Pi^z$, these quantities are subadditive in $\L$, which implies that the limit I\_z(P) =\_[\|Ł\|]{}I\_Ł(P;\^z)/\|Ł\|exists and is equal to the supremum of this sequence. For our purposes, it is sufficient to take this limit along a fixed sequence of squares; for example, one can take squares with vertex coordinates in $\ZZ+1/2$. The claim then follows from the well-known analogous result for lattice models [@GiiGibbs Chapter 15] by dividing $\RR^2$ into unit squares. $I_z$ is called the (negative) entropy density with reference measure $\Pi^z$. We set $I=I_1$. Each $I_z$ differs from $I$ only by a constant and a multiple of the particle density. In fact, an easy computation shows that I(P) = I\_z(P) + 1-z+z(P) z A crucial fact we need later is the following result obtained in Lemma 5.2 of [@GiiZess]. \[entropy\_cp\] In the topology $\Tau_\cL$, each $I_z$ is lower semicontinuous with compact level sets $\{I_z\le c\}$, $c>0$. Triangle interactions --------------------- This paper is concerned with point processes with a particle interaction which is induced by the associated Delaunay triangulation. We stick here to the simplest kind of interaction, which depends only on the triangles that occur in each configuration. Specifically, let $\ph: \cT_0\to\RR$ be an arbitrary measurable function. It can be extended to a unique shift-invariant measurable function $\ph$ on $\cT$ via $\ph(\tau):= \ph(\tau-c(\tau))$, $\tau\in\cT$. Such a $\ph$ will be called a triangle potential. We will assume throughout that $\ph$ is bounded, in that $$\label{eq:ph_lb} \|\ph\| \le c_\ph$$ for some constant $c_\ph<\infty$. In Theorem \[Gtempered\] we will need the following additional condition to prove the temperedness of Gibbs measures. Let us say that a triangle potential $\ph$ is eventually increasing if there exist a constant $r_\ph<\infty$ and a measurable nondecreasing function $\psi:[r_\ph,\infty\ro\to\RR$ such that $ \ph(\tau) = \psi(\r(\tau))$ when This condition is clearly satisfied when $\ph$ is constant for all triangles $\tau$ with sufficiently large radius $\r(\tau)$. \[examples\]Here are some examples of triangle potentials. For each triangle $\tau\in\cT$ let $b(\tau)=\frac 13\sum_{x\in\tau}x$ be the barycentre and $A(\tau)$ the area of $\tau$. Examples of bounded (and scale invariant) interactions that favour equilateral Delaunay triangles are $$\ph_1(\tau) =\b\, \|c(\tau)-b(\tau)\|/\r(\tau) \quad\text{ or }\quad \ph_2(\tau) = - \b\, A(\tau)/\r(\tau)^2$$ with $\b>0$. Of course, many variants are possible; e.g., one can replace the barycentre by the centre of the inscribed circle. By way of contrast, to penalise regular configurations one can replace the $\ph_i$’s by their negative. The triangle potentials $\ph_i$ above are not eventually increasing. But each triangle potential $\ph$ can be modified to exhibit this property by setting $$\tilde\ph(\tau) =\left\{ \ba{cl}\ph(\tau) & \text{ if } \r(\tau)<r\,,\\ K &\text{ otherwise} \ea\right.$$ with $r>0$ and $K$ a suitable constant. When $K$ is large, one has the additional effect of favouring small circumcircles. \[rem:edges\] The type of interaction introduced above is the simplest possible that is adapted to the Delaunay structure. In particular, we avoid here any explicit interaction $\psi$ along the Delaunay edges. This has two reasons: First, we might add a term of the form $\frac12 \sum_{e\subset\tau:\,\# e=2}\ph_\text{edge}(e)$ to the triangle interaction $\ph$. Such a term would take account for an edge interaction $\ph_\text{edge}$ whenever $\sD(\o)$ is a triangulation of the full plane. Secondly, we often need to control the interaction over large distances; the condition of $\ph$ being eventually increasing is tailored for this purpose. It is then essential to define the range in terms of triangles rather than edges. Namely, if a configuration $\o$ is augmented by a particle at a large distance from $\o$, the circumcircles of all destroyed triangles must be large, but their edges can be arbitrarily short. So, a large-circumcircle assumption on the triangle potential allows to control this effect, but a long-edge asumption on an edge potential would be useless. Results ======= Let $\ph$ be a fixed triangle potential. We assume throughout that $\ph$ is bounded, see , but do not require in general that $\ph$ is eventually increasing. For each bounded Borel set $\L\subset\RR^2$ and each configuration $\o\in\O$, the associated Hamiltonian in $\L$ with boundary condition $\o$ is then defined for arbitrary $\zeta\in\O$ by H\_[Ł,ø]{}()= \_[(\_Łø\_[Ł\^c]{}) :B()Ł]{}(). It is always well-defined since the sum is finite. Note that $H_{\L,\o}(\emptyset)\ne0$ in general. For defining the associated Gibbs distribution we need to impose a condition on the boundary condition $\o$. [[Definition. ]{}]{}\[admissible\] Let us say a configuration $\o\in\O$ is admissible if for every bounded Borel set $\L$ there exists a bounded Borel set $\bar\L(\o)\supset\L$ such that $B(\tau)\subset \bar\L(\o)$ whenever $\zeta\in\O$ and $\tau\in\sD(\zeta_\L\cup\o_{\L^c})$ is such that $B(\tau)\cap\L\ne\emptyset$. We write $\O^$ for the set of all admissible configurations. In Corollary \[temperedness\] we will show that $P(\O^)=1$ for all $P\in\cP_\Th$ with $P(\{\emptyset\})=0$. Suppose now that $\o\in\O^$ and $\L$ is any bounded Borel set. Lemma \[Euler\] then shows that $$H_{\L,\o}\ge -2c_\ph N_\L -2c_\ph N_{\bar\L(\o)\setminus\L}(\o)\,,$$ where $\bar\L(\o)$ is as above. This in turn implies that for each $z>0$ the associated partition function* Z\_[Ł,z,ø]{} = e\^[-H\_[Ł,ø]{}]{} d\^z\_[Ł]{} is finite. We can therefore define the Gibbs distribution with activity $z>0$ by G\_[Ł,z,ø]{}(A) = Z\_[Ł,z,ø]{}\^[-1]{} \_A(ø\_[Ł\^c]{}) e\^[-H\_[Ł,ø]{}()]{} \^z\_[Ł]{}(d),A. The measure $G_{\L,z,\o}$ depends measurably on $\o$ and thus defines a probability kernel from $(\O^, \cF_{\L^c})$ to $(\O,\cF)$. [[Definition.* ]{}]{}A probability measure $P$ on $(\O,\cF)$ is called a Gibbs point process for the Delaunay triangle potential $\ph$ and the activity $z>0$, or a Delaunay-Gibbs measure for short, if $P(\O^)=1$ and $P = \int P(d\o)\,G_{\L,z,\o}$ for all bounded Borel sets $\L\subset\RR^2$. We write $\sG_\Th(z,\ph)$ for the set of all stationary Gibbs measures for $\ph$ and $z$, and $\sG_\Th^\tmp(z,\ph)$ for the set of all tempered* stationary Gibbs measures; recall Eq. . The above definition corresponds to the classical concept of a Gibbs measure, which is based on the location of points. We note that an alternative concept of Gibbs measure that considers the location of Delaunay triangles has been proposed and used by Zessin [@Zess02] and Dereudre [@Dereudre]. Intuitively, the interaction of a configuration in $\L$ with its boundary condition $\o$ reaches not farther than the set $\bar\L(\o)$ above, which guarantees some kind of quasilocality. So one can expect that a limit of suitable Gibbs distributions $G_{\L,z,\o}$ as $\L\uparrow\RR^2$ should be Gibbsian, and the existence problem reduces to the question of whether such limits exist. Our approach here is to take the necessary compactness property from Lemma \[entropy\_cp\], the compactness of level sets of the entropy density. In fact, we even go one step further and show that the stationary Gibbs measures are the minimisers of the free energy density. Since such minimisers exist by the compactness of level sets, this solves in particular the existence problem. The free energy density is defined as follows; recall the definition of the centred tile distribution $\mu_P$ before . [[Definition. ]{}]{}The energy density of a stationary point process $P\in\cP_\Th$ is defined by $$\Phi(P)=\int \ph\,d\mu_P = \|\D\|\inv\int P(d\o)\sum_{\tau\in\sD(\o):\,c(\tau)\in\D} \ph(\tau)\,,$$ where $\D$ is an arbitrary bounded Borel set. The free energy density of $P$ relative to $\Pi^z$ is given by $I_z(P)+\Phi(P)$. The definition of $\Phi$ is justified by Proposition \[prop:Phi\] below. Here are some crucial facts on the free energy density, which will be proved in Subsection \[subsec:energy\]. \[I+Phi\_lsc\] Relative to the topology $\Tau_\cL$ on $\cP_\Th$, $\Phi$ is continuous, and each is lower semicontinuous with compact level sets. In particular, the set $\sM_\Th(z,\ph)$ of all minimisers of $I_z+\Phi $ is a non-empty convex compact set, and in fact a face of the simplex $\cP_\Th$. Next we observe that the elements of $\sM_\Th(z,\ph)$ are nontrivial, in that the empty configuration $\emptyset\in\O$ has zero probability; this result will also be proved in Subsection \[subsec:energy\]. \[prop:non-degenerate\] For all $z>0$ we have $\d_{\emptyset}\notin\sM_\Th(z,\ph)$, and thus $P(\{\emptyset\})=0$ for all $P\in\sM_\Th(z,\ph)$. Our main result is the following variational characterisation of Gibbs measures. \[varprinc\] Let $\ph$ be a bounded triangle potential and let $z>0$. Then every minimiser of the free energy density is a stationary Gibbs measure. That is, the identity $\sM_\Th(z,\ph)\subset\sG_\Th(z,\ph)$ holds. In particular, Gibbs measures exist. Conversely, every tempered stationary Gibbs measure is a minimiser of the free energy density, which means that $\sG^\tmp_\Th(z,\ph)\subset \sM_\Th(z,\ph)$. The proof will be given in Subsections \[subsec:var\_princ1\] and \[subsec:var\_princ2\]. Theorem \[varprinc\] raises the problem of whether all stationary Gibbs measures are tempered. It is natural to expect that $\sG_\Th(z,\ph)=\sG^\tmp_\Th(z,\ph)$, but we did not succeed to prove this in general. In fact, we even do not know whether $\sG^\tmp_\Th(z,\ph)$ is always non-empty. But we can offer the following sufficient condition, which will be proved in Subsection \[Gibbstempered\]. \[Gtempered\] Suppose $\ph$ is eventually increasing and let $z>0$. Then every stationary Gibbs measure is tempered, so that $\sG^\tmp_\Th(z,\ph)=\sG_\Th(z,\ph)$. Combining Theorems \[varprinc\] and \[Gtempered\] we arrive at the following result. \[varprincb\] Suppose $\ph$ is bounded and eventually increasing, and let $z>0$. Then the minimisers of the free energy density are precisely the stationary tempered Gibbs measures. That is, $\sM_\Th(z,\ph)=\sG_\Th(z,\ph)=\sG_\Th^\tmp(z,\ph)$ for all $z>0$. The proof of Theorem \[varprinc\] is based on an analysis of the mean energy and the pressure in the infinite volume limit when $\L\uparrow\RR^2$. For simplicity, we take this limit through a fixed reference sequence, namely the sequence Ł\_n = \]-n-12, n+12\[\^[2]{} of open centred squares. We shall often write $n$ when we refer to $\L_n$. That is, we set $\o_n=\o_{\L_n}$, $P_n=P_{\L_n}$, $R_{n,\o}=R_{\L_n,\o}$, $H_{n,\o}=H_{\L_n,\o}$, and so on. We also write $v_n=\|\L_n\|= (2n+1)^2$ for the Lebesgue measure of $\L_n$. Our first result justifies the above definition of $\Phi(P)$. Besides the Hamiltonian with configurational boundary condition $\o$, we will also consider the Hamiltonian with periodic boundary condition, namely H\_[n,]{}(ø):=v\_n(R\_[n,ø]{})=\_[(ø\_[n,]{}): c()Ł\_n]{}(). By definition, we have $H_{n,\per}(\emptyset)=0$. Applying Lemma \[Euler\] and using , we see that $\|H_{n,\per}\|\le v_nc_\ph\,2\,z(R_n)= 2c_\ph\,N_n$. The following result will be proved in Subsection \[subsec:energy\]. \[prop:Phi\] For every $P\in \cP_\Th$ we have $$\lim_{n\ti} v_n\inv\int H_{n,\per}\,dP= \Phi(P)\,.$$ Moreover, if $P$ is tempered then $$\lim_{n\ti}v_n\inv\int H_{n,\o}(\o)\,P(d\o)=\Phi(P)\,.$$ Finally we turn to the pressure. Let $$Z_{n,z,\per} = \int e^{-H_{n,\per}} \,d\Pi^z_n$$ be the partition function in $\L_n$ with periodic boundary condition. \[prop:pressure\] For each $z>0$, the pressure $$p(z,\ph):=\lim_{n\ti} v_n^{-1}\log Z_{n,z,\per}$$ exists and satisfies p(z,) = - \_[P\_]{}. [[Proof: ]{}]{}This is a direct consequence of Theorem 3.1 of [@GiiZess] because $\Phi$ is continuous by Proposition \[I+Phi\_lsc\].[ $\Diamond$]{} A counterpart for the partition functions with configurational boundary conditions follows later in Proposition \[lowerbound\]. Let us conclude with some remarks on extensions and further results. Large deviations. The following large deviation principle is valid. For every measurable $A\subset\cP_\Th$, $$\limsup_{n\ti} v_n\inv\log G_{n,z,\per}(R_n\in A) \le - \inf I_{z,\ph}(\text{\rm cl\,}A)$$ and $$\liminf_{n\ti} v_n\inv\log G_{n,z,\per}(R_n\in A) \ge - \inf I_{z,\ph}(\text{\rm int\,}A)\,,$$ where $G_{n,z,\per}=Z_{n,z,\per}\inv e^{-H_{n,\per}}\Pi^z_n$ is the Gibbs distribution in $\L_n$ with periodic boundary condition, $I_{z,\ph}=I_z+\Phi +p(z,\ph)$ is the excess free energy density, and the closure cl and the interior int are taken in the topology $\Tau_\cL$. Since $\ph$ is bounded so that $\Phi$ is continuous, this is a direct consequence of Theorem 3.1 of [@GiiZess]. \[rem:marks\]Marked particles. Our results can be extended to the case of point particles with marks, that is, with internal degrees of freedom. Let $E$ be any separable metric space, which is equipped with its Borel $\s$-algebra and a reference measure $\nu$, and $\overline{\O}$ the set of all pairs $\bar\o=(\o,\s_\o)$ with $\o\in\O$ and $\s_\o\in E^\o$. In place of the reference Poisson point process $\Pi^z$, one takes the Poisson point process $\overline\Pi^z$ on $\overline{\O}$ with intensity measure $z\l\otimes\nu$, where $\l$ is Lebesgue measure on $\RR^2$. For $\bar\o\in\overline{\O}$ let $$\sD(\bar\o)=\big\{\bar\tau=(\tau,\s_\tau): \tau\in\sD(\o), \ \s_\tau=\s_\o\,\|_{\/\tau}\big\}\,.$$ Of course, the centre, radius and circumscribed disc of a marked triangle $\bar\tau$ are still defined in terms of the underlying $\tau$. In the germ-grain representation, $\cT_0$ is replaced by the set $\overline{\cT_0}$ of all centred $\bar\tau$. The tile distribution $\mu_{\bar P}$ of a stationary point process $\bar P$ on $\overline{\O}$ is a finite measure on $\overline{\cT_0}$ and is defined by placing bars in . A triangle potential is a bounded function $\ph$ on $\overline{\cT_0}$. Such a $\ph$ is eventually increasing if $ \ph(\bar\tau) = \psi(\r(\tau))$ for some nondecreasing $\psi$ when $\r(\tau)$ is large enough. It is then easily seen that all our arguments carry over to this setting without change. \[rem:hc\]Particles with hard core. There is some interest in the case when the particles are required to have at least some distance $r_0>0$. This is expressed by adding to the Hamiltonian a hard-core pair interaction term $H_{\L,\o}^\hc(\zeta)$ which is equal to $\infty$ if $\|x-y\|\leq r_0$ for a pair $\{x,y\}\subset\zeta_\L\cup\o_{\L^c}$ with $\{x,y\}\cap\L\ne\emptyset$, and zero otherwise. Equivalently, one can replace the configuration space $\O$ by the space $$\O^\hc=\big\{\o\in\O: \|x-y\|> r_0 \text{ for any two distinct }x,y\in\o\big\}$$ of all hard-core configurations. The free energy functional on $\cP_\Th$ then takes the form $F_z^\hc:=I_z+\Phi+\Phi^\hc$, where $$\Phi^\hc(P)= \infty \ P^0\big(\o: 0<\|x\|\le r_0 \text{ for some }x\in\o\big) = \infty \ P(\O\setminus \O^\hc)$$ for $P\in\cP_\Th$ with Palm measure $P^0$; here we use the convention $\infty\, 0 =0$. We claim that our results can also be adapted to this setting. In particular, the minimisers of $F_z^\hc$ are Gibbsian for $z$ and the combined triangle and hard-core pair interaction, and the tempered Gibbs measures for this interaction minimise $F_z^\hc$. We will comment on the necessary modifications in Remarks \[rem:hc1\] and \[rem:hc2\]. Combining the extensions in the last two remarks we can include the following example of phase transition. \[rem:Potts\] The Delaunay-Potts hard-core model for particles with $q\ge2$ colours. In the setup of Remark \[rem:marks\] we have $E=\{1,\ldots,q\}$, and the triangle potential is $$\ph(\bar\tau)=\left\{\ba{cl}\b&\text{if $\r(\tau)\le r_1$ and $\s_\tau$ is not constant,}\\ 0&\text{otherwise,}\ea\right.$$ where $\b>0$ is the inverse temperature and $r_1>0$ is an arbitrary interaction radius. If one adds a hard-core pair interaction with range $r_0<r_1/ \sqrt{2}$ as in Remark \[rem:hc\], this model is similar to the model considered in [@BBD04]. (Instead of a triangle potential, these authors consider an edge potential along the Delaunay edges that do not belong to a tile $\t$ of radius $\r(\tau)>r_1$.) Using a random cluster representation of the triangle interaction as in [@Gr94] and replacing edge percolation by tile percolation one finds that the methods of [@BBD04] can be adapted to the present model. Consequently, if $z$ and $\b$ are sufficiently large, then the simplex $\sM_\Th(z,\ph)=\sG_\Th(z,\ph)$ has at least $q$ distinct extreme points. Proofs ====== Energy and free energy {#subsec:energy} ---------------------- We begin with the proof of Proposition \[muP\_cont\], which states that the centred tile distribution $\mu_P$ depends continuously on $P$. The continuity of the energy density $\Phi$ and the lower semicontinuity of the free energy density $I_z+\Phi$ then follow immediately. [[Proofof Proposition \[muP\_cont\]: ]{}]{} Let $(P_\a)$ be a net in $\cP_\Th$ that converges to some $P\in\cP_\Th$. We need to show that $\int g\,d\mu_{P_\a}\to \int g\,d\mu_{P}$ for all $g\in\cL_0$. We can assume without loss of generality that $0\le g\le 1$. We first consider the case that $g$ has bounded support, in that $g\le\1\{\r\le r\}$ for some $r>0$. Let $\D\subset\RR^2$ be any bounded set of Lebesgue measure $\|\D\|=1$. Also, let $$f(\o)=\sum_{\tau\in\sD(\o)}g(\tau-c(\tau))\,\1_\D(c(\tau))\,.$$ In view of , we then have $\int g\,d\mu_Q=\int f\,dQ$ for all $Q\in\cP_\Th$, and in particular for $Q=P_\a$ and $Q=P$. By the bounded support property of $g$, $f$ depends only on the configuration in the $r$-neigbourhood $\D^r:=\{y\in\RR^2:\|y-x\|\le r\text{ for some }x\in\D\}$ of $\D$. That is, $f$ is measurable with respect to $\cF_{\D^r}$. Moreover, $f \le 2 N_{\D^r}$ by Lemma \[Euler\], so that $f\in\cL$. In the present case, the result thus follows from the definition of the topology $\Tau_\cL$. If $g$ fails to be of bounded support, we can proceed as follows. Let $\e>0$ be given and $r>0$ be so large that $\mu_P(\r > r)<\e$. Since $\\|\mu_{P_\a}\\| = 2z(P_\a)\to 2z(P)= \\|\mu_{P}\\|$ and $\mu_{P_\a}(\r\le r)\to \mu_{P}(\r\le r)$ by the argument above, we have $\mu_{P_\a}(\r> r)\to\break \mu_{P}(\r> r)$. We can therefore assume without loss of generality that $\mu_{P_\a}(\r> r) <\e$ for all $\a$. Using again the first part of this proof, we can thus write $$\begin{split} &\int g\,d\mu_{P}-\e \le \int g\,\1_{\{\r\le r\}}\,d\mu_{P} = \lim_\a \int g\,\1_{\{\r\le r\}}\,d\mu_{P_\a}\\ &\le \liminf_\a \int g \,d\mu_{P_\a} \le \limsup_\a \int g \,d\mu_{P_\a} \\ & \le \lim_\a \int g\,\1_{\{\r\le r\}}\,d\mu_{P_\a} +\e = \int g\,\1_{\{\r\le r\}}\,d\mu_{P} +\e \le \int g\,d\mu_{P} +\e \,. \end{split}$$ Since $\e$ was chosen arbitrarily, the result follows.[ $\Diamond$]{} We now turn to the properties of the free energy density. [[Proofof Proposition \[I+Phi\_lsc\]: ]{}]{}As $\ph$ belongs to $\cL_0$, the continuity of $\Phi$ follows immediately from Proposition \[muP\_cont\]. By Lemma \[entropy\_cp\], we can also conclude that $I_z+\Phi$ is lower semicontiuous. Moreover, hypothesis implies that the level set $\{I_z+\Phi \le c\}$ is contained in $\{I_z\le c+2c_\ph z(\cdot)\}$, which by coincides with the compact set $\{I_{z'}\le c+z'-1\}$ for $z'=z\exp(2c_\ph)$. Let $P=\d_\emptyset\in\cP_\Th$ be the Dirac measure at the empty configuration. Then $\mu_P\equiv 0$ and thus $\Phi(P)=0$. On the other hand, $I_z(P)=z$. This means that $I_z+\Phi $ is not identically equal to $+\infty$ on $\cP_\Th$ and thus, by the compactness of its level sets, attains its infimum. To see that the minimisers form a face of $\cP_\Th$, it is sufficient to note that $I_z+\Phi $ is measure affine; cf. Theorem (15.20) of [@GiiGibbs].[ $\Diamond$]{} Next we show that the minimisers of the free energy are nondegenerate. [[Proofof Proposition \[prop:non-degenerate\]: ]{}]{}The second statement follows from the first because $\cM_\Th(z,\ph)$ is a face of $\cP_\Th$. For, suppose there exists some $P\in\cM_\Th(z,\ph)$ with $P(\{\emptyset\})>0$. Then $\d_{\emptyset}$ appears in the ergodic decomposition of $P$, which would only be possible if $\d_{\emptyset}\in\cM_\Th(z,\ph)$. To prove the first statement we note that $\Phi(\Pi^u)\le c_\ph \\|\mu_{\Pi^u}\\| =2c_\ph u$ for all $u>0$. Therefore, if $z>0$ is given and $u$ is small enough then I\_z(\^u)+(\^[u]{}) z-u+u(u/z) +2c\_u <z = I\_z(\_)+(\_), so that $\d_\emptyset$ is no minimiser of the free energy.[ $\Diamond$]{} Finally we show that the energy density $\Phi$ is the infinite volume limit of the mean energy per volume. [[Proofof Proposition \[prop:Phi\]: ]{}]{}We begin with the case of periodic boundary conditions. For every $P\in\cP_\Th$, we have $ v_n\inv \int H_{n,\per} \,dP=\int \Phi(R_n)\,dP=\Phi(PR_n)$. It is easy to see that $PR_n\to P$, cf. Remark 2.4 of [@GiiZess]. Since $\Phi$ is continuous, it follows that $\Phi(PR_n)\to\Phi(P)$. Next we consider the case of configurational boundary conditions and suppose that $P$ is tempered. Applying we obtain for each $n$ $$\begin{aligned} \label{eq:P(H_n)} \int P(d\o)\, H_{n,\o}(\o) &=& \int P(d\o)\sum_{\tau\in\sD(\o)}\ph(\tau-c(\tau))\,\1_{\{B(\tau-c(\tau))\cap(\L_n-c(\tau))\ne\emptyset\}} \nonumber\\ &=&\int \mu_P(d\tau)\ \ph(\tau)\int dx\ \1_{\{B(\tau)\cap(\L_n-x)\ne\emptyset\}} \\ &=&\int \mu_P(d\tau)\ \ph(\tau) \,\big\|\L_n ^{\r(\tau)}\big\|, \nonumber\end{aligned}$$ where $\L_n ^{\r(\tau)}$ is the $\r(\tau)$-neigbourhood of $\L_n$. Now, for each $\tau$ we have $$\big\|\L_n ^{\r(\tau)}\big\|/v_n= 1+ 4\r(\tau)/\sqrt{v_n} + \pi\r(\tau)^2/v_n\to 1 \text{ as }n\ti.$$ In view of and , we can apply the dominated convergence theorem to conclude that $$\Phi(P) = \lim_{n\ti} v_n\inv\int H_{n,\o}(\o)\,P(d\o),$$ as desired.[ $\Diamond$]{} The variational principle: first part {#subsec:var_princ1} ------------------------------------- In this section we shall prove that each minimiser of the free energy is a Delaunay-Gibbs measure. We start with an auxiliary result on the ‘range of influence’ of the boundary condition on the events within a bounded set. Let $\D\subset\RR^2$ be a bounded Borel set and $\o\in\O$. Writing $B(x,r)$ for the open disc in $\RR^2$ with center $x$ and radius $r$, we define $$\cR_\D(\o) =\big\{r>0: \ \#\,\o_{B(x,r)\setminus\D}\ge1 \text{ for all }x\in\RR^2 \text{ s.t. }B(x,r)\cap\D\ne\emptyset\big\}$$ and $\br_\D(\o) = \inf \cR_\D(\o)$, where $\inf\emptyset :=\infty$. Let $\D^{2r}=\bigcup_{x\in\D}B(x,2r)$ be the open $2r$-neigbourhood of $\D$. We then observe the following. \[rangefct\] (a) For all $r>0$, $\{\br_\D<r\}\in \cF_{\D^{2r}\setminus\D}$. In particular, $\br_\D$ is $\cF_{\D^c}$-measurable. \(b) For all $P\in\cP_\Th$ we have $P(\br_\D=\infty)= P(\{\emptyset\})$. [[Proof: ]{}]{}(a) Let $\tilde \cR_\D(\o)$ be defined as $\cR_\D(\o)$, except that the discs are required to have rational centres $x\in\QQ^2$. Then $\cR_\D(\o)\subset\tilde \cR_\D(\o)$. Moreover, if $r<r'$ then every open $r'$-disc intersecting $\D$ contains an $r$-disc with rational center and intersecting $\D$, so that $r'\in\cR_\D(\o)$ when $r\in \tilde \cR_\D(\o)$. This shows that $$\{\br_\D<r\}=\bigcup_{s\in\QQ: \; 0<s<r}\ \bigcap_{x\in\QQ^2: \; B(x,s)\cap\D\ne\emptyset} \{N_{ B(x,s)\setminus\D}\ge1\}\,,$$ and the last set certainly belongs to $\cF_{\D^{2r}\setminus\D}$. \(b) Since $\{\emptyset\}=\bigcap_{r\in\NN}\{N_{B(0,r)}=0\}\subset \{\br_\D=\infty\}$, it is sufficient to prove that $\{\br_\D=\infty, N_{B(0,r)}\ge1\}$ has measure zero for all $r>0$ and $P\in\cP_\Th$. However, if $\{\br_\D=\infty\}$ occurs then there exists a cone $C$ with apex at some point in the closure of $\D$ and an axis in one of finitely many prescribed directions such that $N_C=0$, while the Poincaré recurrence theorem implies that $N_C=\infty$ for each such $C$ almost surely on $\{ N_{B(0,r)}\ge1\}$. This contradiction gives the desired result.[ $\Diamond$]{} As an immediate consequence we obtain that each nondegenerate stationary point process is concentrated on the set $\O^$ of admissible configurations. \[temperedness\] The set $\O^$ of admissible configurations is measurable (in fact, shift invariant and tail measurable), and $P(\O^)=1$ for all $P\in\cP_{\Th}$ with $P(\{\emptyset\})=0$. [[Proof:* ]{}]{}This is immediate from Lemma \[rangefct\] because $\O^* = \bigcap_{n\ge1}\{\br_{\L_n}<\infty\}$.[ $\Diamond$]{} Next we state a consequence of Proposition \[prop:pressure\]. \[cor:rel-entropy-wrto-Gibbs\] For every $P\in\sM_\Th(z,\ph)$, we have $$\lim_{n\ti} v_n\inv I_n(P;G_{n,z,\per}) = 0\;.$$ [[Proof: ]{}]{}By the definition of relative entropy, $$I_n(P;G_{n,z,\per})= I_n(P;\Pi^z)+\int H_{n,\per} \,dP + \log Z_{n,z,\per}\,.$$ Together with Propositions \[prop:Phi\] and \[prop:pressure\], this gives the result.[ $\Diamond$]{} We are now ready to show that the minimisers of $I_z+\Phi$ are Gibbsian. [[Proofof Theorem \[varprinc\], first part: ]{}]{} We follow the well-known scheme of Preston [@PrestonLNM] (in the variant used in [@GiiJSP], Section 7). Let $P\in\sM_\Th(z,\ph)$, $f$ be a bounded local function, $\D$ a bounded Borel set, and $$f_\D(\o) = \int f(\zeta)\,G_{\D,z,\o}(d\zeta)\,, \quad\o\in \O.$$ We need to show that $\int f\,dP = \int f_\D\,dP$. Let $\br_\D$ be the range function defined above, and for each $r>0$ let $\1_{\D,r}= \1\{\br_\D< r\}$ and $\D^{2r}$ be the $2r$-neigbourhood of $\D$. By Lemma \[rangefct\](a), $\1_{\D,r}$ is measurable with respect to $\cF_{\D^{2r}\setminus\D}$. Moreover, if $\br_\D(\o)< r$ then $H_{\D,\o}(\zeta) = H_{\D,\o_{\D^{2r}\setminus\D}}(\zeta)$ for all $\zeta\in\O_\D$. So, if $r$ is so large that $f$ is $\cF_{\D^{2r}}$-measurable, we can conclude that $\1_{\D,r}\,f_\D$ is $\cF_{\D^{2r}\setminus\D}$-measurable. Now we apply Corollary \[cor:rel-entropy-wrto-Gibbs\], which states that $\lim_{n\ti} v_n\inv I_{\L_n}(P;G_{n,z,\per}) = 0$. By shift invariance, this implies that $P_{\L}\ll G_{\L,z,\per}$ with a density $g_\L$ for each sufficiently large square $\L$. In particular, for any $\D'\subset\L$ we have $P_{\D'}\ll (G_{\L,z,\per})_{\D'}$ wih density $g_{\L,\D'}(\o)=\int G_{\L\setminus\D',z,\o\cap\D'}(d\zeta)\, g_\L(\zeta)$. Corollary \[cor:rel-entropy-wrto-Gibbs\] implies further that for each $\d>0$ there exists a square $\L$ and a Borel set $\D'$ with $\D^{2r}\subset\D'\subset\L$ such that $\int \|g_{\L,\D'}-g_{\L,\D'\setminus\D}\|\,dG_{\L,z,\per} <\d$; cf. Lemma 7.5 of [@GiiJSP]. Now we consider the difference $$\int \1_{\D,r}\,(f-f_\D)\,dP= \int \1_{\D,r}\big(g_{\L,\D'}\,f-g_{\L,\D'\setminus\D}\,f_\D\big)\,dG_{\L,z,\per}\,.$$ Since $G_{\L,z,\per} = \int G_{\L,z,\per}(d\o)\,G_{\D,z,\o}$ and $ \1_{\D,r}\, g_{\L,\D'\setminus\D}$ is $\cF_{\L\setminus\D}$-measurable, we can conclude that $$\int \1_{\D,r}\,g_{\L,\D'\setminus\D}\,f_\D\,dG_{\L,z,\per} =\int \1_{\D,r}\,g_{\L,\D'\setminus\D}\,f\,dG_{\L,z,\per}\,.$$ By the choice of $\L$ and $\D'$, we can replace the density $g_{\L,\D'\setminus\D}$ in the last expression by $g_{\L,\D'}$ making an error of at most $\d$. We thus find that $\int \1_{\D,r}\,(f-f_\D)\,dP<\d$. Letting $\d\to0$ and $r\ti$, we finally obtain by the dominated convergence theorem that $$\int_{\{\br_\D<\infty\}}\,(f-f_\D)\,dP = 0\,.$$ This completes the proof because $P(\br_\D=\infty)= P(\{\emptyset\})=0$ by Lemma \[rangefct\](b) and Proposition \[prop:non-degenerate\].[ $\Diamond$]{} \[rem:hc1\] Here are some comments on the necessary modifications in the hard-core setup of Remark \[rem:hc\]. In analogy to Proposition \[prop:pressure\], one needs that $$\lim_{n\ti} v_n^{-1}\log \int e^{-H_{n,\per}-H_{n,\per}^\hc} \,d\Pi^z_n =- \min_{P\in\cP_\Th}\big [I_z(P)+\Phi(P)+\Phi^\hc(P)\big]\,.$$ This follows directly from Propositions 4.1 and 5.4 of [@GiiPTRF] because $\Phi$ is continuous. Corollary \[cor:rel-entropy-wrto-Gibbs\] therefore still holds for the periodic Gibbs distributions with additional hard-core pair interaction. One also needs to modify the proof of Proposition \[prop:non-degenerate\], in that the Poisson processes $\Pi^u$ should be replaced by the Gibbs measure $P^u$ with activity $u$ and pure hard-core interaction. $P^u$ is defined as the limit of the Gibbs distributions $G_{n,u,\per}^\hc$ for the periodic hard-core Hamiltonians $H_{n,\per}^\hc$. By Proposition 7.4 of [@GiiPTRF], $P^u$ exists and satisfies $$I_u(P^u)=-\lim_{n\ti} v_n^{-1}\log \int e^{-H_{n,\per}^\hc} \,d\Pi^u_n \le-\lim_{n\ti} v_n^{-1}\log \Pi^u_n(\{\emptyset\})= u\,.$$ Together with we find that $$I_z(P^u)+\Phi(P^u)\le z+z(P^u)\, \big[ \log (u/z)+2c_\ph \big]\,,$$ which is strictly less than $z$ when $u$ is small enough. Since $\Phi^\hc(P^u)=\Phi^\hc(\d_\emptyset)=0$, it follows that the minimisers of $I_z+\Phi+\Phi^\hc$ are non-degenerate. No further changes are required for the proof of the first part of the variational principle. Boundary estimates ------------------ We now work towards a proof of the reverse part of the variational principle. In this section, we control the boundary effects that determine the difference of $H_{n,\per}$ and $H_{n,\o}$. The resulting estimates will be crucial for the proof of Proposition \[lowerbound\]. For every $\o\in\O$ and every Borel set $\D$ let $$\sS_\D(\o)=\big\{\tau\in \sD(\o): B(\tau)\cap \D \ne \emptyset \text{ and } B(\tau)\setminus \D\ne \emptyset \big\}$$ be the set of all triangles $\tau\in\sD(\o)$ for which $B(\tau)$ crosses the boundary of $\D$. We start with a lemma that controls the influence on $\sS_\D$ when two configurations are pasted together. \[frontiere\] Let $\Delta$ be a (not necessarily bounded) Borel set in $\RR^2$, $\zeta\in\O^\cup\{\emptyset\}$ a configuraton with $\zeta_{\partial\D}=\emptyset$, and $\o\in\O$. Then for each $\tau \in \sS_\D(\zeta_\D \cup \o_{\D^c})$ and each $x\in \tau_\D$ there exists some $\tau' \in \sS_\D(\zeta)$ with $x \in \tau'$. [[Proof:* ]{}]{}Let $\D$, $\zeta$ and $\o$ be given. If $\zeta$ is empty, there exists no $x\in \tau_\D\subset\zeta_\D$, so that the statement is trivially true. So let $\zeta\in\O^$ and suppose there exists some $\tau\in \sS_\D(\zeta_\Delta \cup \o_{\Delta^c})$ with $\tau_\D\ne\emptyset$. Let $x\in\tau_\D$. Since $\zeta_{\partial\D}=\emptyset$, $x$ does in fact belong to the interior of $\D$. This implies that $B(\tau')\cap \D\ne \emptyset$ for each $\tau'\in\sD(\zeta)$ containing $x$. Therefore we only need to show that $B(\tau')\setminus \D\ne \emptyset$ for at least one such $\tau'$. Suppose the contrary. Then $B(\tau')\subset \D$ whenever $x\in\tau'\in\sD(\zeta)$. This means that the Delaunay triangles containing $x$ are completely determined by $\zeta_\D$. This gives the contradiction $$\emptyset\ne\{\tau\in \sS_\D(\zeta_\D \cup \o_{\D^c}):\tau\ni x\} = \{\tau'\in \sS_\D(\zeta):\tau'\ni x\} =\emptyset\,,$$ and the proof is complete.[ $\Diamond$]{} The following proposition is the fundamental boundary estimate. It bounds the difference of Hamiltonians with periodic and configurational boundary conditions in terms of $S_n:= \#\sS_{\L_n}$. \[boundary\] There exists a universal constant $\g<\infty$ such that $$\|H_{n,\per}(\zeta)-H_{n,\o}(\zeta)\| \le \g c_\ph\,\big(S_n(\o)+S_n(\zeta)\big)$$ for all $n\ge 1$ and all $\zeta,\o \in \O^\cup\{\emptyset\}$ with $\zeta_{\partial\L_n} = \o_{\partial\L_n}=\emptyset$. [[Proof: ]{}]{}Let $n,\zeta,\o$ be fixed and $$\sA= \big\{\tau\in\sD(\zeta_{\L_n}\cup\o_{\L_n^c}): \, B(\tau)\cap\L_n\ne\emptyset\big\} \,,\quad \sB= \big\{\tau\in\sD(\zeta_{n,\per}): \, c(\tau)\in\L_n\big\}\,.$$ In view of (\[eq:cH\_o\]) and (\[eq:H\_per\]) we have $ H_{n,\o}(\zeta)= \sum_{\tau\in\sA}\ph(\tau)$ and $ H_{n,\per}(\zeta)=\sum_{\tau\in\sB}\ph(\tau)$. Since $\ph$ is bounded by $c_\ph$, we only need to estimate the cardinalities of $\sA\setminus\sB$ and $\sB\setminus \sA$. We note that $\sA\setminus \sB \subset \sS_{\L_n}(\zeta_{\L_n}\cup\o_{\L_n^c})$ and $\sB\setminus\sA \subset\sS_{\L_n}(\zeta_{n,\per})$. So we can apply Lemma \[frontiere\] to both $\D=\L_n$ and $\D=\L_n^c$ to obtain that the set of points belonging to a triangle in $\sA\setminus \sB$ is contained in the set of points belonging to a triangle of $\sS_{\L_n}(\zeta)\cup \sS_{\L_n}(\o)$. Hence, $\#\big(\bigcup_{\tau\in\sA\setminus \sB}\tau\big) \le 3(S_n(\zeta) + S_n(\o))$. By Lemma \[Euler\], it follows that $\#(A\setminus B) \le 6 (S_n(\o) + S_n(\zeta))$. To estimate the cardinality of $\sB \setminus\sA$ we may assume that $\zeta_n\ne\emptyset$. The periodic continuation $\zeta_{n,\per}$ then contains a lattice, and this implies that every triangle of $\sD(\zeta_{n,\per})$ has a circumscribed disc of diameter at most $\sqrt{2v_n}$. Hence, each $\tau\in \sB\setminus \sA$ is contained in $\L_{5n+2}$, the union of $5^2$ translates of $\L_n$ (up to their boundaries). Applying Lemma \[frontiere\] to each of these translates we conclude that the number of points that belong to a triangle of $\sB\setminus \sA$ is bounded by $3\cdot5^2\, S_n(\zeta)$. Using Lemma \[Euler\] again we find that $\#(B\setminus A) \le 150 (S_n(\o) + S_n(\zeta))$, and the result follows with $\g=156$.[ $\Diamond$]{} The following immediate corollary will be needed in the proof of Theorem \[varprinc\]. \[vide\] There exists a constant $C<\infty$ such that $ \| H_{n,\o}(\emptyset)\| \le C\, S_n(\o)$ for all $n\ge 1$ and $\o\in\O^$ with $\o_{\partial\L_n}=\emptyset$. The next proposition exhibits the fundamental role of the temperedness condition combined with stationarity for controlling the boundary effects. \[convergence\] For every $P\in\cP_\Th^\tmp$, $v_n\inv S_n\to 0$ in $L^1(P)$ and $P$-almost surely. [[Proof:* ]{}]{} For each $i\in\RR^2$ we consider the shifted unit square $C(i)=\L_0+i$ and define the random variable $$Z_i=\#\big\{\tau\in\sD(\cdot):\, B(\tau)\cap C(i)\ne \emptyset\big\}.$$ Then S\_n \_[iI\_nI\_[n-1]{}]{} Z\_i,where $I_n=\L_n\cap(\ZZ^2+(\frac12,\frac12))$. Note that $\#\, I_n=v_n$. As in we have $$\int Z_0\,dP = \int \mu_P(d\tau)\ \big\|\L_0^{\r(\tau)}\big\| =\int \mu_P(d\tau)\ \big(1+ 4\r(\tau)+ \pi\r(\tau)^2\big)\,.$$ The last term is finite by the temperedness of $P$. So, each $Z_i$ is $P$-integrable. Since $Z_i=Z_0\circ\th_i$, the two-dimensional ergodic theorem implies that $v_n\inv\sum_{i\in I_n} Z_i$ converges to a finite limit $\bar Z$, both $P$-almost surely and in $L^1(P)$. This implies that $v_n\inv \sum_{i\in I_n\setminus I_{n-1}} Z_i$ tends to zero $P$-almost surely and in $L^1(P)$. The result thus follows from .[ $\Diamond$]{} Temperedness and block average approximation {#subsec:block_av} -------------------------------------------- Our first result in this subsection is a sufficient condition for temperedness in terms of vacuum probabilities. For $P\in\cP_\Th$ let V\_k(P) = P(N\_[Ł\_k]{}=0\|\_[Ł\_k\^c]{}) be the essential supremum of the conditional probability that $\L_k$ contains no particle given the configuration outside. \[tempered\] Every $P\in\cP_\Th$ satisfying $$\label{empty} \sum_{k\ge0}v_k \,V_k(P)\,z(P) <\infty$$ is tempered. [[Proof: ]{}]{} We can assume that $P\ne\d_\emptyset$ because otherwise the result is trivial. For each $k\ge 1$ we consider the shifted squares $\L_k(i)=\L_k+(2k+1)i$, $i\in\ZZ^2$, as well as the event $$A_k=\big\{N_{\L_k(i)} \ne0 \text{ for all $i\in\ZZ^2$ with } \\|i\\|_\infty\le1 \big\}.$$ Since $P\ne\d_\emptyset$, it is clear that $P(A_k)\to 1$ as $k\ti$. Thus we can write $$\begin{split} \int \#&\big(\tau\in\sD(\o): 0\in B(\tau)\big) \,P(d\o) \\ &\le \ \sum_{k\ge 1} \int_{A_k \setminus A_{k-1}} \#\big(\tau\in\sD(\o): 0\in B(\tau)\big) P(d\o) \end{split}$$ with the convention $A_0=\emptyset$. Now, if $\o\in A_k$ then each circumscribed disc containing $0$ of a triangle $\tau\in \sD(\o)$ has a diameter not larger than $2\sqrt{2v_k}$, so that each such $\t$ in fact belongs to $\sD(\o_{\L_{7k+3}})$. Lemma \[Euler\] thus shows that the number of such $\t$ is at most $2N_{7k+3}(\o)$. The last sum is therefore not larger than $$\sum_{k\ge 1} \int_{A_{k-1}^c} 2N_{7k+3}\, dP \le 2 \sum_{k\ge 1} \sum_{ i\in\ZZ^2: \\|i\\|_\infty\le1} \int \1_{\{N_{\L_{k-1}(i)}=0\}} N_{7k+3}\, dP \,.$$ In view of the stationarity of $P$, the last integral is bounded by $V_{k-1}(P)\, v_{7k+3}\,z(P)=7^2\,v_k\, V_{k-1}(P)\, z(P)$. So we arrive at the estimate $$\int \#\big(\tau\in\sD(\o): 0\in B(\tau)\big) \,P(d\o)\le 2\cdot 7^2\,3^2\sum_{k\ge 1}v_k\, V_{k-1}(P)\, z(P) \,.$$ Together with and assumption , this implies the temperedness of $P$ because $ v_k\sim v_{k-1}$ as $k\ti$.[ $\Diamond$]{} The second result concerns the approximation of stationary measures in terms of tempered ergodic measures. This approximation uses the block average construction first introduced by Parthasarathy for proving that the ergodic measures are dense in $\cP_\Th$; cf. [@GiiGibbs Theorem (14.12)], for example. \[ergodic\_approx\] Let $z>0$ and $Q\in\cP_\Th$ be such that $I_z(Q)+\Phi(Q)<\infty$. Then for each $\e>0$ there exists some tempered $\Th$-ergodic $\hat Q\in\cP_\Th$ such that $I_z(\hat Q)<I_z(Q)+\e$ and $\Phi(\hat Q)<\Phi(Q)+\e$. [[Proof: ]{}]{}Let $Q\in\cP_\Th$ be given. We can assume that $Q\ne\d_\emptyset$ because otherwise we can choose $\hat Q=Q$. For $n\ge1$ let $Q^{\text{iid}}_n$ denote the probability measure on $\O$ relative to which the particle configurations in the blocks $\L_n+ (2n+1)i$, $i\in \ZZ^2$, are independent with identical distribution $Q_n=Q\circ \pr_{\L_n}\inv$. (In particular, this means that the boundaries of these blocks contain no particles.) Consider the spatial average $$Q^{\text{iid-av}}_n=v_n\inv\int_{\L_n}Q^{\text{iid}}_n\circ\th_x^{-1}dx.$$ It is clear that $Q^{\text{iid-av}}_n\in\cP_\Th$. It is also well-known that $Q^{\text{iid-av}}_n$ is $\Th$-ergodic; cf. [@GiiGibbs Theorem (14.12)], for example. By an analogue of [@GiiGibbs Proposition (16.34)] or [@GiiZess Lemma 5.5], its entropy density satisfies I\_z(Q\_n\^)=v\_nI(Q;\^z\_n). So, $I_z(Q_n^{\text{iid-av}})< I_z(Q)+\e$ when $n$ is large enough. Moreover, the same argument as in [@GiiZess Lemma 5.7] shows that $Q_n^{\text{iid-av}}\to Q$ in the topology $\Tau_\cL$. By Proposition \[muP\_cont\], $\Phi$ is continuous. We thus conclude that $\Phi(Q_n^{\text{iid-av}})\to \Phi(Q)$, whence $\Phi(Q_n^{\text{iid-av}})<\Phi(P)+\e$ for large $n$. It remains to prove that each $Q_n^{\text{iid-av}}$ is tempered. Let $n$ be fixed and $k\ge\ell(2n{+}1)$ for some $\ell\ge1$. We claim that $V_k(Q_n^{\text{iid-av}})\le q^{v_{\ell-1}}$ with $q=Q(N_n=0)$. Indeed, for each $x\in\L_n$ we have $\L_k+x\supset\L_{n+(\ell-1)(2n+1)}$, and the latter set consists of $v_{\ell-1}=(2\ell-1)^2$ distinct blocks as above. Letting $g$ be any nonnegative $\cF_{\L_k^c}$-measurable function and using the independence of block configurations, we thus conclude that $$\begin{split} &\int \1_{\{N_k=0\}}g\,d Q_n^{\text{iid-av}} =v_n\inv\int_{\L_n}dx\int dQ_n^{\text{iid}} \, \1_{\{N_{\L_k+x}=0\}}\,g\circ\th_x\\ &\le v_n\inv\int_{\L_n}dx\int dQ_n^{\text{iid}} \, \1_{\{N_{n+(\ell-1)(2n+1)}=0\}}\,g\circ\th_x =q^{v_{\ell-1}} \int g\,d Q_n^{\text{iid-av}}\,, \end{split}$$ which proves the claim. Now, we have $q<1$ because $Q\ne\d_\emptyset$. It follows that $$\sum_{k> 2n}v_k\,V_k(Q_n^{\text{iid-av}}) \le \sum_{\ell\ge1}q^{v_{\ell-1}}\sum_{\ell(2n+1)\le k<(\ell+1)(2n+1)}v_k \le C_n \sum_{\ell\ge0}v_\ell \,q^{v_\ell}<\infty$$ for some constant $C_n<\infty$. Together with Proposition \[tempered\], this gives the temperedness of $Q_n^{\text{iid-av}}$.[ $\Diamond$]{} The variational principle: second part {#subsec:var_princ2} -------------------------------------- In this section we will complete the proof of the variational principle. The essential ingredient is the following counterpart of Proposition \[prop:pressure\] involving configurational instead of periodic boundary conditions. We only state the lower bound we need. \[lowerbound\] For every $P\in\cP_\Th^\tmp$ and $P$-almost every $\o$ we have $$\liminf_{n\ti}v_n\inv\log Z_{n,z,\o} \ge p(z,\ph)\,.$$ [[Proof: ]{}]{} By and Lemma \[ergodic\_approx\], it is sufficient to show that $$\liminf_{n\ti}v_n\inv\log Z_{n,z,\o} \ge -I_z(Q)-\Phi(Q)$$ for every ergodic $Q\in \cP_\Th^\tmp$. We can assume without loss that the right-hand side is finite. Now, since $I_z(Q)$ is finite, $Q$ is locally absolutely continuous with repect to $\Pi^z$. We fix some $\e>0$, let $f_n=dQ_n/d\Pi^z_n$, and consider for every $\o\in\O^$ the set $$A_{n,\o}=\big\{\|H_{n,\o}-H_{n,\per}\|/v_n\le \e,\, \Phi(R_n)<\Phi(Q)+\e,\, v_n\inv\log f_n< I_z(Q)+\e\big\}\,.$$ Then for sufficiently large $n$ we have $$\begin{aligned} Z_{n,z,\o} & \ge & \int_{A_{n,\o}} e^{-H_{n,\o}} \,f_n\inv \,d Q \\ & \ge & \int_{A_{n,\o}} e^{-H_{n,\per}}e^{-v_n\e} \,f_n\inv \,d Q \\ & \ge & e^{-v_n\,[I_z(Q) +\Phi(Q)+3\e]}\, Q(A_{n,\o})\,. \end{aligned}$$ It is therefore sufficient to show that for $P$-almost $\o$, $Q(A_{n,\o})\to 1$ as $n\ti$. By the ergodic theorem, $\Phi(R_n)$ converges to $\Phi(Q)$ in $Q$-probability; cf. Remark 2.4 of [@GiiZess]. By McMillan’s theorem [@Fritz; @NgZess], $Q(v_n\inv\log f_n < I_z(Q)+\e)\to 1$ when $n$ tends to infinity. Moreover, Propositions \[boundary\] and \[convergence\] imply that, for $P$-almost all $\o$, $\|H_{n,\o}-H_{n,\per}\|/v_n$ converges to $0$ in $L^1(Q)$. This gives the result.[ $\Diamond$]{} We can now show that every tempered stationary Gibbs measure minimises the free energy density. [[Proofof Theorem \[varprinc\], second part:* ]{}]{}We follow the argument of [@GiiJSP], Proposition 7.7. Let $P\in\sG^\tmp_\Th(\ph,z)$. On each $\cF_{\L_n}$, $P$ is absolutely continuous w.r. to $\Pi^z$ with density $$g_n(\zeta) = \int P(d\o)\, \frac{d G_{n,z,\o}}{d\Pi^z_n}(\zeta) = \int P(d\o)\, e^{-H_{n,\o}(\zeta)}/ Z_{n,z,\o} \,.$$ Using Jensen’s inequality and the Gibbs property of $P$ we thus find that $$\begin{split} I_n(P;\Pi^z) &= \int g_n \log g_n \, d\Pi^z\\ &\le \int \Pi^z(d\zeta)\int P(d\o) \frac{d G_{n,z,\o}}{d\Pi^z_n}(\zeta) \big[-H_{n,\o}(\zeta)-\log Z_{n,z,\o}\big]\\ &= - \int P(d\o)\, H_{n,\o}(\o) - \int P(d\o)\,\log Z_{n,z,\o}\,. \end{split}$$ Next we divide by $v_n$ and let $n\ti$. We know from Proposition \[prop:Phi\] that $v_n\inv \int P(d\o)\, H_{n,\o}(\o)\to \Phi(P)$. On the other hand, Corollary \[vide\] implies that $v_n\inv\,\log Z_{n,z,\o}\ge -z- Cv_n\inv S_n(\o)$. Using Propositions \[convergence\] and \[lowerbound\] together with Fatou’s Lemma, we thus find that $$\liminf_{n\ti} v_n\inv \int P(d\o)\,\log Z_{n,z,\o} \ge p(z,\ph).$$ Therefore $I_z(P) \le -\Phi(P) + \min[I_z+\Phi]$, as required.[ $\Diamond$]{} \[rem:hc2\] In the hard-core setting of Remark \[rem:hc\], a slight refinement of Proposition \[ergodic\_approx\] is needed. Namely, under the additional assumption that $\Phi^\hc(Q)=0$ one needs to achieve that also $\Phi^\hc(\hat Q)=0$. To this end we fix an integer $k>r_0/2$ and define $Q^{\text{iid}}_n$ in such a way that the particle configurations in the blocks $\L_n+ (2n+1)i$, $i\in \ZZ^2$, are independent with identical distribution $Q_{n-k}$, rather than $Q_n$. This means that the blocks are separated by corridors of width $2k>r_0$ that contain no particles. It follows that $\Phi^\hc(Q_n^{\text{iid-av}})=0$, and it is still true that $\limsup_{n\ti}I_z(Q_n^{\text{iid-av}})\le I_z(Q)$; cf. [@GiiPTRF Lemma 5.1]. We thus obtain the refined Proposition \[ergodic\_approx\] as before. A similar refinement is required in the proof of Proposition \[lowerbound\]. One can assume that $\Phi^\hc(P)=0$ and $\Phi^\hc(Q)=0$, and in the definition of $A_{n,\o}$ one should introduce an empty corridor at the inner boundary of $\L_n$ to ensure that $H^\hc_{n,\o}=H^\hc_{n,\per}=0$ on $A_{n,\o}$, see [@GiiPTRF Proposition 5.4] for details. In the proof of the second part of Theorem \[varprinc\], one then only needs to note that $\Phi^\hc(P)=0$ when $P$ is a Gibbs measure $P$ for the combined triangle and hard-core pair interaction. The proof of Theorem \[Gtempered\] carries over to the hard-core case without any changes. Temperedness of Gibbs measures {#Gibbstempered} ------------------------------ Here we establish Theorem \[Gtempered\]. By Proposition \[tempered\] it is sufficient to show the following. \[temperedb\] Let $\ph$ be bounded and eventually increasing, $z>0$, and $P$ be any stationary Gibbs measure for $\ph$ and $z$. Then there exists a constant $C>0$ such that $$\label{emptyb} P(N_k=0 \| \cF_{\L_k^c}) \le {C}\,{v_k^{-2}}$$ for all $k\ge 1$. To prove this we need an auxiliary result which states that the radii of all circumcircles in the Delaunay tessellation must decrease when a point is added to the configuration. Specifically, let $\o\in \O^$ and $x\in \RR^2\setminus\o$ be such that $\o\cup\{x\}$ is in general circular position and $x$ is not collinear with two points of $\o$. We consider the sets $$\sC_x(\o):=\sD(\o) \setminus \sD(\o\cup\{x\})=\big\{\tau\in\sD(\o): B(\tau)\ni x\big\}$$ and $$\sC^+_x(\o):=\sD(\o\cup\{x\}) \setminus \sD(\o)= \big\{\tau\in\sD(\o\cup\{x\}):\tau\ni x\big\} .$$ If $\langle\tau\rangle$ denotes the convex hull of a triangle $\tau$, \_x(ø):=\_[\_x(ø)]{}= \_[\_x\^+(ø)]{}is the region on which the triangulations $\sD(\o)$ and $\sD(\o\cup\{x\})$ differ; see Fig. \[DiffGraph\]. Up to the point $x$, the interior $\D_x^o(\o)$ of $\D_x(\o)$ is covered by the discs $B(\tau)$ with $\tau\in\sC_x^+(\o)$, which by definition are free of particles. Consequently, $\D_x^o(\o)$ contains no particle of $\o$, so that the vertices of each $\tau\in\sC_x(\o)$ belong to the boundary $\partial\D_x(\o)$. Next, Lemma \[Euler\] shows that $$\label{card} \# \sC^+_x(\o)= \# \sC_x(\o)+2\,,$$ and for every $\L\ni x$ we have $$\label{energielocale} H_{\L,\o}(\o\cup\{x\})- H_{\L,\o}(\o) = \sum_{\tau \in \sC^+_x(\o)} \ph(\tau) - \sum_{\tau \in \sC_x(\o)} \ph(\tau).$$ Here is the monotonicity result announced above. \[ordre\] Under the conditions above, there exist a subset $\sC_x'(\o)\subset \sC_x(\o)$ with $\#\big(\sC_x(\o)\setminus \sC_x'(\o)\big)\le 4$ and an injection $I$ from $\sC_x'(\o)$ to $\sC^+_x(\o)$ such that $$\label{inegalite} \r(I(\tau)) \le \r(\tau)\quad\text{ for all $\tau\in \sC_x'(\o)$.}$$ We postpone the proof of this lemma until the end, coming first to its use. [[Proofof Proposition \[temperedb\]:* ]{}]{}By assumption, $\ph$ is eventually increasing. So there exists some $r_\ph<\infty$ and a nondecreasing function $\psi$ such that $ \ph(\tau) =\psi(\r(\tau))$ when $\r(\tau) \ge r_\ph$. Combining Lemma \[ordre\] and Equations (\[card\]) and (\[energielocale\]) we thus find that $$\label{energie} H_{k,\o}(\o\cup\{x\}) \le H_{k,\o}(\o) + 10\,c_\ph$$ for all $\o\in \O^$, $k\ge1$, and Lebesgue-almost all $x\in \L_k\setminus\o$ that have at least the distance $2r_\ph$ from all points of $\o$. Next, let $P\in\sG_\Th(z,\ph)$. By definition, $$P(N_k=0 \| \cF_{\L_k^c})(\o)=Z_{k,z,\o}\inv\, e^{-zv_k} \,e^{-H_{k,\o}(\emptyset)}$$ for all $\o\in\O^$. Let $\L_k^{(2)}=\big\{(x,y)\in\L_{k-2r_\ph}^2: \|x-y\|\ge 2r_\ph\big\}$. Applying (\[energie\]) twice (viz. to $\o_{\L_k^c}$ and $x$ as well as $\o_{\L_k^c}\cup\{x\}$ and $y$) and recalling we find that $$\begin{split} Z_{k,z,\o} &\ge e^{-zv_k}\, \frac{z^2}{2} \int_{\L_k^{(2)}} e^{-H_{k,\o}(\{x\}\cup\{y\})}\,dx\, dy\\ &\ge {z^2\,\|\L_k^{(2)}\|}\,e^{-zv_k}\, e^{-H_{k,\o}(\emptyset)-20\,c_\ph}/\,2\,. \end{split}$$ Since $\|\L_k^{(2)}\|\sim v_k^2$ as $k\ti$, the result follows.[ $\Diamond$]{} Finally we turn to the proof of Lemma \[ordre\]. [[Proofof Lemma \[ordre\]: ]{}]{} Let $\tau_x$ be the unique triangle of $\sC_x(\o)$ containing $x$ in its interior, and $\sC_x^{+\wedge}(\o)$ the set of all $\tau\in\sC_x^+(\o)$ that have an acute or right angle at $x$. Note that $\#(\sC_x^+(\o)\setminus\sC_x^{+\wedge}(\o))\le 3$ because the angles at $x$ of all $\tau\in\sC_x^+(\o)$ add up to 360 degrees. We will associate to each triangle $\tau\in\sC_x(\o)$ a triangle $I(\tau)\in\sC^+_x(\o)$, except possibly when $\tau=\tau_x$ or the candidate for $I(\tau)$ does not belong to $\sC_x^{+\wedge}(\o)$. Our definition of $I(\tau)$ depends on the number $k=k(\tau)$ of edges $e\subset\tau$ with $\langle e\rangle\subset\partial\D_x(\o)$. Let $\sC^{(k)}_x(\o)$ be the set of all $\tau\in\sC_x(\o)$ that have $k$ such edges. Since $\sC^{(3)}_x(\o)=\emptyset$ except when $\sC_x(\o)=\{\tau_x\}$, we only need to consider the three cases $k=0,1,2$. The cases $k=1$ and $2$ are easy: For every $\tau\in\sC^{(1)}_x(\o)$ there exists a unique edge $e(\tau)$ such that $e(\tau)\cup\{x\}\in\sC_x^+(\o)$. If in fact $e(\tau)\cup\{x\}\in\sC_x^{+\wedge}(\o)$ we set $I(\tau)=e(\tau)\cup\{x\}$; otherwise we leave $I(\tau)$ undefined. Likewise, every $\tau\in \sC^{(2)}_x(\o)$ has two edges $e_1(\tau)$ and $e_2(\tau)$ in $\partial\D_x(\o)$ (in clockwise order, say) and can be mapped to $I(\tau)=e_1(\tau)\cup\{x\}$, provided this triangle belongs to $\sC^{+\wedge}_x(\o)$. The resulting mapping $I$ is clearly injective. Moreover, $\tau$ and $I(\tau)$ have the edge $e(\tau)$ (resp. $e_1(\tau)$) in common, and $x\in B(\tau)$ because $\tau\in\sC_x(\o)$. Since $I(\tau)\in\sC^{+\wedge}_x(\o)$ whenever it is defined, we can conclude that $\r(I(\tau)) \le \r(\tau)$. The case $k=0$ is more complicated because the tiles $\tau\in\sC^{(0)}_x(\o)$ are not naturally associated to a tile of $\sC^+_x(\o)$. To circumvent this difficulty we define an injection $\tilde I$ from $\sC^{(0)}_x(\o)\setminus \{\tau_x\}$ to $\sC^{(2)}_x(\o)$ such that $\r(\tilde I(\tau)) \le \r(\tau)$. Each triangle $\tau\in\sC^{(0)}_x(\o)$ different from $\tau_x$ can then be mapped to the triangle $I(\tau)=e_2(\tilde I(\tau))\cup\{x\}$, provided the latter belongs to $\sC_x^{+\wedge}(\o)$; otherwise $I(\tau)$ remains undefined. This completes the construction of $I$. (Note that $\tau_x$ does not necessarily belong to $\sC^{(0)}_x(\o)$. However, if it does we have no useful definition of $\tilde I(\tau_x)$.) To construct $\tilde I$ we turn $\sC_x(\o)$ into the vertex set of a graph $G_x(\o)$ by saying that two tiles are adjacent if they share an edge. The set $\sC^{(2)}_x(\o)$ then coincides with the set of all leaves of $G_x(\o)$, and $\sC^{(0)}_x(\o)$ is the set of all triple points of $G_x(\o)$. Consider a fixed $\tau_\in\sC^{(0)}_x(\o)\setminus \{\tau_x\}$. Since $\tau_\subset \partial\D_x(\o)$, the set $\D_x(\o)\setminus\langle\tau_\rangle$ splits into three connected components. Let $W_i=W_i(\tau_,x,\o)$ be the closure of the $i$th component, $i=1,2,3$. Any two of these sets intersect at a point of $\tau_$, and one of them contains $x$ because $\tau_\ne\tau_x$. Suppose $x\in W_3$. For $i=1,2$ let $e_i=\tau_\cap W_i$ be the edge of $\tau_$ that separates $W_i$ from the rest of $\D_x(\o)$; see Fig. \[fig:circles\]. We claim that there exists some $i=i(\tau_)\in\{1,2\}$ such that ()(\_\). The image $\tilde I(\tau_)$ of $\tau_$ can then be defined as the leaf of $G_x(\o)$ in $W_{i(\tau_)}$ with the largest circumcircle. (The largest circumcircle condition takes account of the fact that the path in $G_x(\o)$ from $\tau_$ to $\tilde I(\tau_)$ might contain further triple points.) It remains to prove . Since $\tau_\ne\tau_x$, there exists at least one $i$ such that the triangle $\{x\}\cup e_i$ has an acute angle at $x$. We fix such an $i$ and consider any $\tau\in\sC_x(\o)$ with $\tau\subset W_i$. There exists at least one point $z_0\in\tau$ that is not contained in the closed disc $\overline{B}(\tau_)$. Since $x\in B(\tau)$ and $\langle\tau\rangle$ is covered by the tiles $\langle\tau'\rangle$ for $\tau'\in\sC^+_x(\o)$ with $\langle\tau'\rangle \cap \langle e_i\rangle\ne\emptyset$, we conclude that the line segment $s$ from $z_0$ to $x$ is contained in $\overline{B}(\tau)$ and hits both the circle $\partial B(\tau_)$ and the edge $\langle e_i\rangle$. In particular, $B(\tau)\cap\langle e_i\rangle\ne\emptyset$. Since $B(\tau)$ contains no points of $\o$, we deduce further that the circle $\partial B(\tau)$ hits the edge $\langle e_i\rangle$ in precisely two points $z_1$ and $z_2$. By the choice of $i$, the angle of the triangle $\{z_1,x,z_2\}$ at $x$ is acute. Since $x\in B(\tau)$, it follows that the angle of the triangle $\{z_1,z_0,z_2\}$ at $z_0$ is obtuse. Consequently, if we consider running points $y_k$ such that $y_0$ runs from $z_0$ to the point $s\cap\partial B(\tau_)$ and the edge $\{y_1,y_2\}$ from $\{z_1,z_2\}$ to $e_i$, the associated circumcircles $B(\{y_1,y_0,y_2\})$ run from $B(\tau)$ to $B(\tau_)$, and their radii $\r(\{y_1,y_0,y_2\})$ must grow. This proves that $\r(\tau)\le \r(\tau_)$, and the proof of and the lemma is complete.[ $\Diamond$]{} . We are grateful to Remy Drouilhet who brought us together and drew the interest of H.-O.G. to the subject. [99]{} M. Aigner and G. Ziegler, Proofs from the book, Springer, Berlin etc., 3rd ed., 2004. E. Bertin, J.-M. Billiot and R. Drouilhet, Existence of “nearest-neighbour” spatial Gibbs models, Adv. in Appl. Probab.* [31]{}:895–909 (1999) E. Bertin, J.-M. Billiot and R. Drouilhet, Existence of Delaunay pairwise Gibbs point process with superstable component, J. Statist. Phys. [95]{}:719–744 (1999) E. Bertin, J.-M. Billiot and R. Drouilhet, Phase transition in the nearest-neighbor continuum Potts model, J. Statist. Phys. [114]{}:79–100 (2004) E. Bertin, J.-M. Billiot and R. Drouilhet, $R$-local Delaunay inhibition model, J. Stat. Phys. [132]{}:649–667 (2008) A. Dembo and O. Zeitouni, Large Deviations: Techniques and Applications. Springer, New York, 1998. D. Dereudre, Gibbs Delaunay tessellations with geometric hard core conditions, J. Statist. Phys. [131]{}:127–151 (2008). D. Dereudre, R. Drouilhet and H.-O. Georgii, Existence of Gibbsian graphs for stable interactions, in preparation. J. Fritz, Generalization of McMillan’s theorem to random set functions. Stud. Sci. Math. Hung. [5]{}:369–394 (1970). H.O. Georgii, [Gibbs Measures and Phase Transitions]{}, de Gruyter, Berlin, 1988. H.O. Georgii, Large deviations and maximum entropy principle for interacting random fields on ${\rm Z}^d$, [Ann. Probab.]{} [21]{}:1845–1875 (1993). H.O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction, [Probab. Theory Relat. Fields]{} [99]{}:171–195 (1994). H.O. Georgii, The equivalence of ensembles for classical systems of particles, J. Statist. Phys. [80]{}:1341-1378 (1995). H.O. Georgii and H. Zessin, Large deviations and the maximum entropy principle for marked point random fields, [Probab. Theory Relat. Fields]{} [96]{}:177–204 (1993). G. Grimmett, Potts models and random-cluster processes with many-body interactions, J. Statist. Phys. [75]{}:67–121 (1994). R. Holley and D. W. Stroock, Nearest neighbor birth and death processes on the real line, [Acta Math.]{}Ê[140]{}:103–154 (1978). J. Mller, Lectures on Random Voronoi Tessellations, Lecture Notes in Statistics Vol. 87, Springer, Berlin etc., 1994. X.X. Nguyen and H. Zessin, Ergodic theorems for spatial processes, Z. Wahrscheilichkeitstheor. Verw. Geb. [48]{}:133–158 (1979). C. Preston, [Random fields]{}, Lect. Notes Math. 534, Springer, Berlin, 1976. R. Schneider and W. Weil, Stochastic and integral geometry, Springer, Berlin, 2008. H. Zessin, Specific index and curvature for random simplicial complexes, Inst. Math. Natl. Acad. Sci. Armenia [37]{}:64–81 (2002).
--- abstract: 'The compact, non-thermal emission in DoAr21 has been studied with the VLBA to investigate the possibility that the residuals of the astrometry fitting are due to the reflex motion induced by a possible companion. We find that the fitting of VLBA astrometric observations of DoAr21 improves significantly by adding the orbital motions of three companions. We obtain an improved distance to the source of $134.6\pm1.0$ pc, and estimate that the central star, DoAr21, has a mass of about $2.04\pm0.70~M_{\odot}$. We suggest that DoAr21 represents a unique case where two sub-stellar companions, DoAr21b and DoAr21c ($m_{b}\sim 35.6\pm27.2~M_{jup}$ and $m_{c}\sim44.0\pm13.6~M_{jup}$, respectively), have been found associated to a relatively low mass, pre-main sequence star. In addition, we find that this WTTau star is an astrometric double system, having a low-mass star companion, DoAr21$B$ ($m_{B}\sim0.35\pm0.12~M_{\odot}$), in a relatively eccentric orbit. The orbit of this low-mass stellar companion is compact, while the Brown Dwarfs are located in external orbits. DoAr21$c$ has the strongest astrometric signature in the periodogram, while DoAr21$B$ has a weak but significant signature. On the other hand, the astrometric signature of DoAr21$b$ does not appear in the periodogram, however, this Brown Dwarf was directly detected in some of the VLBA observations. The estimated orbital periods of DoAr21$B$, DoAr21$b$ and DoAr21$c$ are $P_{B}\sim92.92\pm0.02$, $P_{b}\sim450.9\pm3.8$ and $P_{c}\sim1013.5\pm25.3$ days, respectively. Since the estimated age of this young star is about 0.4$-$0.8 Myrs, the detected Brown Dwarf companion is among the youngest companions observed to date.' author: - Salvador Curiel - 'Gisela N. Ortiz-León' - 'Amy J. Mioduszewski' - 'Rosa M. Torres' title: 'Sub-stellar companions of the young weak-line TTauri Star DoAr21' --- Introduction {#sec:intro} ============ The search for extrasolar planets has evolve during the past two decades. Several observational technics have proven to be very useful in the search for extrasolar planets, including radial velocity, transits, gravitational microlensing, direct imaging, and even pulsar timing (e.g., [@wolszczan92], [@mayor95], [@charbonneau00], [@bond04], [@kalas08]). These various technics are sensitive in different ranges of the orbital period and exoplanet masses, as well as the stelar brighness. In particular, radial velocity and transit searches of low-mass, pre-main seques stars (TTauri Stars) are quite challenging due to the faintness of these objects, their broad (molecular) spectral features, and their ubiquitous variability; indeed, very few exoplanets have been discovered around this kind of stars (e.g., [@kraus12], [@sallum15], [@donati16]), [@yu17]. A putative Brown Dwarf was recently found to be orbiting a TTauri star [@ginski18], however its orbit lyes outside the circumbinary disk, with a projected orbit of about 210 $AU$. The estimated mass of this companion is quite uncertain, and it could be a very low-mass star. Astrometry is an additional technique which relies on the positional shift of the star around the center of mass of the orbit due to the gravitational pull of a companion (reflex motion). At present, no firm exoplanet detections have been obtained with this technique. There is only one candidate exoplanet detected with the optical astrometry technique, however its mass is not well established since the source is a binary system and its mass depends to which star it is associated ([@muterspaugh10]). However, GAIA’s astrometric observations have the potential to detect, in the near future, many (probably thousands) exoplanets and Brown Dwarfs associated to solar type and low-mass stars (e.g., [@casertano08], [@sozzetti14], [@perryman14]). Astrometric observations can also be carried out in the optical wavelength range with 10 m class telescopes, but require conversion of relative to absolute parallax (e.g., [@sahlmann16]). Non-thermal radio continuum emission found associated to TTauri stars and low mass stars (e.g., [@phillips91], [@berger01]) opens the possibility to search for sub-stellar companions to this kind of sources. Furthermore, astrometric searches with the required sensitivity to detect sub-stellar companions can be carried out at radio frequencies using very long baseline interferometry (VLBI); see [@bower09], [@forbrich09], [@forbrich13], [@gawronski17]. The principal advantage of VLBI observations would be to obtain absolute accurate stellar positions tied to an extragalactic reference frame, which is crucial for accurately measuring reflex motions, as well as parallaxes and proper motions. In this context, the discovery of radio emission from late M and L dwarfs ([@berger01]; [@berger06]) provides a unique opportunity to uncover exoplanets or Brown Dwarf companions to low mass stars and brown dwarfs with M $\leq$0.2 M$_{\odot}$. At present, low mass Brown dwarfs (several tens of Jupiter masses) have been found orbiting a few ultracool stars (e.g., [@sahlmann13]), however, sub-stellar companions have not been found orbiting TTauri stars. DoAr21 is an unusual X-ray bright ($\sim$$10^{32}$ erg/s) weak-line TTauri star (WTTS; @neuhaeuser95) in the Ophiuchus Molecular Cloud with a spectral type between K0-K2, an age of 0.8 Myrs, a mass of $\sim$$2.2 \,M\odot$ (however, it could also be consistent with a binary system with masses of $\sim$$1.8 \,M\odot$ and an age of 0.4 Myrs) and fairly obscured (A$_{V}$ $\sim$6-7) (@jensen09, and references therein). It also exhibits modest 24-micron excess, suggestive of a circumstellar disk, which is unusual for WTTS, while showing no indication of near-IR excess at shorter wavelengths. An extended asymmetric ring structure is detected at NIR H2 emission [@panic09], which is at $\sim$73-219 AU away from the central star. This is consistent with the estimation of a cavity size of $\sim$70 AU [@vdmarel16]. @jensen09 found that the PAH, H2, and mid-infrared excess are most likely associated with a small-scale photo-dissociation region (PDF) located $\sim$100$'$s of AU from the star that is perhaps excited by the UV emission of DoAr21. @james16 found that the SED of this source is consistent with a $3.5 \,M\odot$ star and a very low mass disk of $\sim$$4.5\times10^{-5} \,M\odot$ with an inclination of about 81 deg. DoAr21 was first detected at radio wavelengths with the VLA by [@feigelson85], who found that the radio flux density of this source had a steep increase in time scales of a few hours. Later on DoAr21 was detected with the VLBA by [@phillips91], showing similar variability in its radio flux density and indicating a non-thermal origin. This source displays an X-ray spectrum typical of pre-MS stars in Orion [@preibisch05] that is characterized by components with temperatures of $\sim$1 and 3 keV. Flares were detected during two independent $\sim$100 ks observations of DoAr21. One epoch displayed small-scale flares superimposed on a slowing declining light curve [@gagne04] whereas the more recent observations displayed an impulsive flare during which the temperature of the X-ray emitting plasma increased by a factor of $\sim$2 [@jensen09]. Indeed, @jensen09 concluded that DoAr 21 exhibits X-ray flares at a rate of nearly one per day. This high rate may reflect the increased likelihood of flaring in small-separation TTS binary systems, wherein the coronal activity of each component may be influenced by the interactions of the stars$'$ magnetospheres [@stelzer00]. It is suggestive that there also seems to be a correlation between excess of non-thermal radio emission in young stars and close binarity, and in fact, DoAr21 is one of two stars which are always significantly brighter than all other YSOs in the Ophiuchus star formation complex in non-thermal radio emission, the other star, S1, is a known binary [@ortizleon17]. The compact, non-thermal emission in DoAr21 has been studied at radio wavelengths with the VLBA (@loinard08; @ortizleon17; @ortizleon18), which provided an accurate estimate of the distance for this radio continuum source of 135.4 +/- 4.3 pc. This error is 2-3 times larger than typical errors obtained for the other sources in @ortizleon17, which suggests that something is going on with this source. One of the VLBA observations reported by @loinard08 shows that DoAr21 could be a binary, with a projected separation of about 5 mas ($\sim$0.6 AU). The orbital parameters are uncertain, but Loinard et al. estimated that the semi-major axis could be of the order of 1-2 AU. Since the binary pair has been in principle detected at only one epoch thus far, there is no information about the mass ratio or the optical luminosity ratio. Ascribing all of the luminosity to the primary star yields a mass of $\sim$$2.2 \,M_\odot$ and an age of $\sim$0.4 Myrs, and if the luminosity is split equally between the two stars, the mass of each star is $\sim$$1.8 \,M_\odot$ and an age of $\sim$0.8 Myrs [@jensen09]. In this paper we investigate the possibility that the reflex motion due to a companion orbiting DoAr21 is the responsible of the large residuals of the astrometric fitting of the multi-epoch non-thermal radio emission. In this paper we present new VLBA observations of this source taken over a period of 9 months. We have also re-calibrated previous VLBA observations of this source to search for evidence of the putative companion of this WTTS. The new and archival observations are presented in Section 2, and the new detections and the reanalyzed data are also described in section 2. In Section 3 we present the algorithms that we use in the search for companions of this source. The results and the discussion are presented in Sections 4 and 5, and we present our main conclusions in Section 6. Observations ============ DoAr21 has been observed with the Very Long Baseline Array (VLBA) during several multi-epoch campaigns. We found in the NRAO archive public data from projects BL128 and BT093. The observations of project BL128 were taken between September 2005 and December 2006, while project BT093 observed between July and September of 2007. Both projects were taken at 8.4 GHz with 32 MHz of bandwidth and observed the quasar J1625$-$2527 as the phase reference calibrator. A subset of these observations were published in [@loinard08]. In total, they covered 17 epochs, from which we use only the 3 epochs that clearly detected the two sources. We also use the positions of DoAr21 published in [@ortizleon17], which correspond to epochs observed from August 2012 to September 2017 under program BL175 (GOBELINS project). Finally, four new observations were obtained under program BC237 on May, August and November 2018, and February 2019. The new data were taken at 8.4 GHz with 256 MHz of bandwidth and, similar to BL175, used J1627–2427 for phase reference calibration, which is located in projection towards the Ophiuchus core [@dzib13]. Project BC237 included two observing blocks of about 30 minutes each spent on calibrators distributed over a wide range of elevations. These scans (the so-called geodetic-like blocks) were used to estimate the multi-band delays, i.e. the phase slope with frequency, which are introduced by tropospheric and clock errors. We conduct an homogeneous calibration for the BC237 data and the archival data (BL128 and BT093) using AIPS [@greisen03]. This means that the same calibration steps are applied to each epoch as follows. First, all scans with elevations below $10^{\rm o}$ are flagged. Ionosphere dispersive delays are removed using Global Positioning System (GPS) models of the electron content in the ionosphere which are downloaded from the Crustal Dynamics Data Information System (CDDIS) archive. Corrections to the Earth Orientation Parameters and amplitude corrections for digital sampling effects of the correlator are also applied. Instrumental single-band delays are then determined and removed using fringes detected on a single scan on the calibrator J1625$-$2527. The correction for the bandpass shape is then done using the same calibrator. To finish the amplitude calibration, we use the provided gain curves and system temperature tables to derive the System Equivalent Flux Density (SEFD) of each antenna. After applying corrections for the rotation of the RCP and LCP feeds, the multi-band delays are derived by fringe-fitting the scans of the geodetic-like blocks and used to estimate the tropospheric and clock errors, which are removed from the data. To finish the phase calibration, fringe fitting is run on the phase reference calibrator to find residual phase rates. In order to take the structure of the reference calibrator into account, this last step is repeated using a self-calibrated image of the calibrator as a source model. Finally, the calibration tables are applied to the data and images of the target are produced using a pixel size of $50-100~\mu$as and pure natural weighting. Table \[tab:jmfit\] lists the positions of DoAr 21 taken from [@ortizleon17]. Source positions at the other epochs were obtained by fitting a Gaussian model to the source brightness distribution in the produced images using the AIPS task JMFIT. The angular resolution of the final images of BC237 was typically $\sim$2 mas and the noise level was $\sim25~\mu$Jy beam$^{-1}$ in the best case. We detect two sources on 2005 November 16, 2006 August 24 and 2007 September 21, while a single source is seen in the remaining epochs of projects BL128 and BT093, and the same for BC237. Counter plot maps of the three epochs are presented in Figure \[fig\_1\]. It is important to mention that the observed epochs in project BL175 were calibrated following the same procedure that we use here (see [@ortizleon17] for more details). Thus, it is expected that the positions of the source detected have a similar precession as those obtained with the new observations of this source and that we report here. In addition, the calibration procedure used for the data observed from projects BL128 and BT093 was different than the one we used for the new observations. However, in this study we only use the relative position of the two sources that we have detected in three epochs. Fitting of the astrometric data =============================== Least-squares Periodograms -------------------------- The most popular method to search for periodicities in data is the so-called Lomb-Scargle periodogram. However, this method performs optimally only under an important implicit assumption: all the other signals (e.g., linear trend, an average offset, etc.) can be subtracted from the data without affecting the significance of the signal. This assumption does not hold for astrometry because the proper motion and the parallax are also a significant part of the signal and they typically correlate with the periodic motion of a companion (see @black82). Following the procedure presented by @anglada10, we use instead a circular least-squares periodogram (CLS periodogram). In this approach, the weighted least-squares solution is obtained by fitting all the free parameters in the model for a given period. The sum of the weighted residuals divided by N is the so-called $\chi^{2}$ statistic, where N is the number of data points. Then, each $\chi_{P}^{2}$ of a given model with $k_{P}$ parameters can be compared to the $\chi_{0}^{2}$ of the null hypothesis with $k_{0}$ free parameters by computing the power, $z$, as $$z(P) = \frac{(\chi^{2}_{0} - \chi^{2}_{P})/(k_{P} - k_{0})}{\chi^{2}_{P}/(N_{obs} - k_{P})},$$ where a large $z$ is interpreted as a very significant solution. The values of $z$ follow a Fisher $F$-distribution with $k_{P} - k_{0}$ and $N_{obs} - k_{P}$ degrees of freedom [@scargle82; @cumming04]. Even if only noise is present, a periodogram will contain several peaks [@scargle82 as an example] whose existence have to be considered in obtaining the probability that a peak in the periodogram has a power higher than $z(P)$ by chance, which is the so-called false alarm probability (FAP): $$FAP = 1 - (1 - Prob[z > z(P)])^{O},$$ where $O$ is the number of independent frequencies. In the case of uneven sampling, $O$ can be quite large and is roughly the number of periodogram peaks one could expect from a data set with only Gaussian noise and the same cadence as the real observations. We adopt the recipe $O \sim 2\Delta T/P_{min}$ given in Cumming (2004, Section 2.2), where $\Delta T$ is the time span of the observations and $P_{min}$ is is the minimum period searched. For instance, assuming that $\Delta$$T$ = 2300 days and $P_{min}$ = 20 days, the astrometric data is expected to have $O \sim 230$ peaks. In our case, the null hypothesis is the basic kinematic model with $k_{0} = 5$: two reference coordinates, two proper motions and the parallax. As a first approach, our simplest non-null hypothesis considers circular orbits. For a given period, the number of free parameters is then $k_{P} = 9$: five kinematic parameters plus the four Thiele Innes elements $X1$, $Y1$, $X2$ and $Y2$ [@green93]. As a second approach, we do a least-squares fit of a full Keplerian orbit to the astrometric data, and if we obtain a reasonable fit for the eccentricity of the orbit, we recalculate the periodogram with the eccentricity fixed to this value. To find the new least-squares periodogram, we find the time passage at the periastron $\tau$ using a Monte Carlo method for each given period (and the eccentricity fixed), and we perform a least-squares periodogram sampling of a grid of fixed eccentricity-period ($eP$; where $e$ is fixed either to 0 or the fitted value) pairs and fitting all other parameters. For each $eP$ pair and estimated $\tau$, $k_{P}$ is 9: the null-hypothesis parameters plus all the other Keplerian elements (using the Thiele Innes elements): $\omega$, $\Omega$, $a_{1}$, and $i$, where $a_{1}$ is the semi-mayor axis of the orbit of the star due to the companion. As mentioned in section 2, in the fitting of the data, we only use the positions of DoAr21 published in [@ortizleon17] and those that we report here, which cover a time span of about 2375 days. Since the model is linear in all 9 parameters, the power can be efficiently computed for many periods between 20 days and 3500 days (i.e., a period larger than the time span of the observations) to obtain the familiar representation of the periodogram (see Figure \[fig\_2\]). Astrometric Fits ---------------- Here, we present a model of $\mu$as astrometric data of the sort that can be provided by Very Long Baseline Interferometry (VLBI), such as the VLBA, as well as GAIA (and in the future by the next generation Very Large Array (ngVLA)). The source barycentric two-dimensional position is described as function of time, accounting for the (secular) effects of proper motions ($\mu_\alpha$ and $\mu_\delta$), the (periodic) effect of the parallax $\Pi$, and the (Keplerian) gravitational perturbation induced on the host star by one or more companions (low mass stars, sub-stellar companions, or planets; mutual interactions between companions are not taken into account). The proper motions and the annual parallax terms can be expressed as follows: $$\begin{aligned} \alpha(t) & = & \alpha_{0} + (\mu_\alpha cos(\delta)) (t - t_{0}) + 0.5 (a_\alpha cos(\delta))(t - t_{0})^2 + \Pi F_\alpha(t) + \sum G_\alpha(t), \\ \delta(t) & = & \delta_{0} + \mu_\delta (t - t_{0}) + 0.5 a_\delta (t - t_{0})^2 + \Pi F_\delta(t) + \sum G_\delta(t),\end{aligned}$$ where ($\alpha_{0}$, $\delta_{0}$) is a reference position, $t_{0}$ is a reference time (usually the mean epoch of the multi-epoch observations), and $a_\alpha$ and $a_\delta$ are acceleration terms needed to take into account the dynamical effect induced by long-period stellar companions to the primary (e.g., when the primary is part of an unseen long-period stellar binary; these acceleration terms can also be necessary for close-by high-velocity stars). However, the acceleration terms are not always necessary. $F_\alpha$ and $F_\delta$ refer to the astrometric displacement due to the parallax in $\alpha$ and $\delta$ directions during observations, which are given by [@seidelman92]: $$\begin{aligned} F_\alpha (t) & = & (X sin(\alpha) - Y cos(\alpha))/(15 cos(\delta)), \\ F_\delta (t) & = & X cos(\alpha) sin(\delta) + Y sin(\alpha) sin(\delta) - Z cos(\delta),\end{aligned}$$ here ($X,Y,Z$) represent the Cartesian components in equatorial coordinates of the position of the observatory at the time of the observations, t, with respect to the solar system barycenter (in units of AU when $\Pi$ is in arcsec) and $\alpha$ and $\delta$ are the coordinates of the barycentric place of the source at each epoch. These values for the Earth are available from the NASA Jet Propulsion Laboratory Solar System ephimerides$\footnote{http://ssd.jpl.nasa.gov}$. The term ($G_\alpha(t)$,$G_\delta(t)$) is the induced Keplerian orbit due to an unseen companion, and the parallax factors are defined using the classic formulation by @green93. The Keplerian orbit of each companion is scaled and projected onto the plane of the sky through (@green93): $$\begin{aligned} G_\alpha(t) & = & r [cos(\nu + \omega) sin(\Omega) + sin(\nu + \omega) cos(\Omega) cos(i)], \\ G_\delta(t) & = & r [cos(\nu + \omega) cos(\Omega) - sin(\nu + \omega) sin(\Omega) cos(i)],\end{aligned}$$ where $i$ is the inclination of the orbital plane (such that $i$ = 0 corresponds to a face-on, anti-clockwise orbit), $\omega$ is the longitude of the periastron, $\Omega$ is the position angle of the line of nodes, $\nu$ is the true anomaly, and $r$ is the radius vector, which can be expressed in terms of the true anomaly $\nu$, or the eccentric anomaly $E$, using the dynamical equations $$r = \frac{a (1 - e^{2})}{(1 + e cos(\nu))} = a (1 - e cos(E)),$$ where $e$ is the eccentricity, and $a$ is the apparent semi-major axis of the star’s orbit around the systemic barycenter, i.e., the astrometric signature. The eccentric anomaly is the solution of Kepler’s equation: $$E - e sin(E) = M,$$ with the mean anomaly $M$, expressed in terms of the orbital period $P$ and the epoch of the periastron passage $\tau$: $$M = \frac{2 \pi (t - \tau)}{P}.$$ Finally, the true anomaly $\nu$ is function of the eccentricity and the eccentric anomaly: $$tan\left(\frac{\nu}{2}\right) = \left(\frac{1 + e}{1 - e}\right)^{1/2} tan\left(\frac{E}{2}\right).$$ Note that since $a$ is the apparent semi-major axis of the star’s orbit in units of arcsec, its relationship to the mass of the secondary depends on the astrometric method used. For astrometric perturbations due to an unseen companion, we have from Kepler’s Third Law: $$a^{3} = \frac{\Pi^{3} m_{2}^{3}}{(m_{1} + m_{2})^{2}} P^{2},$$ where $a$ is measured in arc-seconds when $P$ is measured in years, $\Pi$ is the parallax in arc-seconds and $m_{2}$ is the mass of the unseen companion and $m_{1}$ is the mass of the star, both in solar masses. The unknown parameters are a reference position, proper motions, acceleration terms, parallax, and $7\times$$n_{p}$ orbital elements, where $n_{p}$ is the number of companions that are fitted ($P$, $\tau$, $e$, $\omega$, $\Omega$, $a$ and $i$ for each companion). A total of 14 free parameters in the case of a single companion, 21 free parameters for two companions, etc. In the case of an astrometric binary (when the two stars in the binary are detected), it is necessary to fit the Keplerian parameters of the binary, the proper motions of the system’s barycenter and the parallax simultaneously (see above). For the secondary, $\omega$ is rotated 180 deg, and $a_{2}$ is used instead, which is scaled from $a_{1}$ by the mass ratio $q$=($m_{2}/m_{1}$). In this case, two additional unknown parameters, $a_2$ and $q$, need to be taken into account to fit the secondary component, making, in this case, a total of 16 free parameters. However, in most cases only 14 free parameters are needed. The acceleration terms are only needed when there is something that perturbes the motion of this compact system, which could be the case when the binary is part of a wider multiple system. In the case of having relative astrometry observations of the binary plus absolute observations of only one of the components in the binary (we will call it the main component), the fitting procedure is similar to the case of a binary system, but in this case the two additional parameters ($\omega_{2}$ and $a_{2}$ or $q=(m_{2}/m_{1})$) are used to fit the relative astrometry, the other orbital parameters are the same used for the absolute astrometry: $P$, $\tau$, $e$, $\omega$, $\Omega$, $a$ and $i$. In this case there are also 16 free parameters. In the case that the primary is the most massive component in the system ($q << 1$), 7 additional parameters can be added to fit a second companion, making a total of 23 free parameters. Since the acceleration terms are only needed when the compact system is part of a wider multiple system (e.g., a triple or quadruple system, where the other stellar components are farther away from the observed compact system and have orbital periods much more larger than the orbital period in the compact system), in general only 14 free parameters (in the case a single companion) or 21 free parameters (in the case of two companions) will be fitted. Least-squares Fitting Algorithm ------------------------------- In this case, we follow the procedure described in the previos section, but here the orbital elements ($\omega$, $\Omega$, $a_{1}$ and $i$) are obtained using the Thiele Innes elements $X1$, $Y1$, $X2$, and $Y2$ [@green93] instead of equations 7 and 8. In this case: $$\begin{aligned} G_\alpha(t) & = & X1 x(t) + X2 y(t), \\ G_\delta(t) & = & Y1 x(t) + Y2 y(t),\end{aligned}$$ where the Thiele-Innes constants are expressed as: $$\begin{aligned} X1 & = & a[cos(\omega) sin(\Omega) + sin(\omega) cos(\Omega) cos(i)], \\ Y1 & = & a[cos(\omega) cos(\Omega) - sin(\omega) sin(\Omega) cos(i)], \\ X2 & = & a [-sin(\omega) sin(\Omega) + cos(\omega) cos(\Omega) cos(i)], \\ Y2 & = & a[-sin(\omega) cos(\Omega) - cos(\omega) sin(\Omega) cos(i)].\end{aligned}$$ The elliptical rectangular coordinates $x(t)$ and $y(t)$ are given in terms of the dynamical equations (equation 9) and the true anomaly $\nu$ (equation 12) by $$\begin{aligned} x(t) & = & \left(\frac{r}{a}\right) cos(\nu), \\ y(t) & = & \left(\frac{r}{a}\right) sin(\nu).\end{aligned}$$ The linear parameters (reference position, proper motions, acceleration terms, parallax, and $4\times$$n_{p}$ orbital elements) are fitted by Matrix Inversion, while the non-linear parameters ($P$, $\tau$ and $e$) are found using a Montecarlo Method. We use the correlation matrix to obtain the uncertainty of the fitted parameters. AGA Fitting Algorithm --------------------- The Asexual Genetic Algorithm (AGA) fitting procedure that we use here is similar to that described by @canto09 and @curiel11, where the fitting procedure was used to find Keplerian orbits, using the radial velocities (RV) of the host star. This method can be extended to the problem of fitting the Keplerian elements of an orbit to astrometric data, such as those that can be obtained with the VLBA, GAIA and, in the future, with the ngVLA. As described by @canto09, the curve or model fitting is essentially an optimization problem. Given a discrete set of N data points ($\alpha_{i},\delta_{i}$) with associated measurement errors $\sigma_{i}$, one seeks for the best possible model (in other words, the closest fit) for these data using a specific form of the fitting function, ($\alpha(t),\delta(t)$). This function has, in general, several adjustable parameters, whose values are obtained by minimizing a $''$merit function$''$, which measures the agreement between the data ($\alpha_{i},\delta_{i}$) and the model function ($\alpha(t),\delta(t)$). The maximum likelihood estimate of the model parameters ($c_{1}, ..., c_{k}$) is obtained by minimizing the $\chi^{2}$ function: $$\chi^{2}_{min} = \sum_{i=1}^{N} \left( \frac{\alpha_{i} - \alpha(t_{i}; c_{1}, ..., c_{k})}{\sigma_{i}}\right)^{2} + \sum_{i=1}^{N} \left( \frac{\delta_{i} - \delta(t_{i}; c_{1}, ..., c_{k})}{\sigma_{i}}\right)^{2},$$ where each data point ($\alpha_{i},\delta_{i}$) has a measurement error that is independently random and distributed as a nominal distribution about the $''$true$''$ model with standard deviation $\sigma_{i}$. We then apply the AGA method to search for the best solution using this initial guess. Here we have made two important improvements to the original algorithm. First, we do an initial search in the hipper$-$cube space of possible solutions to find an initial guess for the parameters that are fitted. Second, the error estimates, $\sigma(c_{j})$, for the fitted parameters $c_{j}$ can be estimated as the projection of the confidence region of the m-dimensional space parameter for which $\chi^{2}$ does not exceed the minimum value by an amount $\Delta(m,\alpha)$, where $\alpha$ is the significance level (0 $<$ $\alpha$ $<$ 1). Following @avni76 and @wall03, the probability $$Prob [\chi^{2} - \chi^{2}_{min} \le \Delta(m,\alpha)] = \alpha$$ is that of a chi$-$square distribution with $m$ degrees of freedom. Thus, $\Delta(m, \alpha)$ is the increment of $\chi^{2}$ such that if the observation is repeated a large number of times, a fraction $\alpha$ of times the values of the parameters fitted will be inside the confidence region, i.e., in the interval $c_{j}$ $\pm$ $\sigma(c_{j})$ (see @estalella17, and references therein). In this case, we also follow the procedure described in previos section 3.2. This fitting procedure allow us to fit simultaneously all free parameters. However, in the case of a single star, since we do not know a priori if it has a single or multiple companions, we first fit the proper motions and the parallax of the star using the least-squares method and the AGA method, and if necessary we include the acceleration terms. We then analyze the residuals in order to find if there may be an astrometric companion orbiting around the star. To do this, we compute the least-squares periodogram of the astrometric data comparing the null solution and the Keplerian solution (see Section 3.1). If the periodogram shows possible companions (e.g., significant peaks with FAPs smaller that $1\%$), we then include a possible companion in the fitting procedure of the data, using again both fitting methods: least-squares and AGA algorithms. We apply this procedure for each signal present in the periodogram. In the case that there is more than one signal in the periodogram, and there are enough observations (enough data points) to fit simultaneously all the required free parameters, we fit simultaneously more than one astrometric companion. In the case that more than one component is detected with the VLBA, we fit simultaneously the two components, either by doing full astrometry to the main component, or by combining the relative astrometry of the system and the absolute astrometry of one of main component. Here we follow the standard nomenclature to name the host star and the companions that were detected: we use a capital letter for the host star and low-mass stellar companions, and a lower case letter for sub-stellar companions. Thus, in what follows, DoAr21 or DoAr21$A$ is the host star, DoAr21$B$ is the low-mass stellar companion, DoAr21$b$ is the inner sub-stellar companion, and DoAr21$c$ is the outer sub-stellar companion. Applying the least-squares and the AGA algorithms to other data sets -------------------------------------------------------------------- Before fitting the astrometric data that we present here, we used both algorithms, as well as the least-squares periodogram, to fit the data sets of several sources previously investigated by our team in the Ophiuchus region [@ortizleon17]. We fitted single sources (when only one source was detected with the VLBA) and binary systems (when two sources were detected with the VLBA). We found that our results are in good agreement with those reported by @ortizleon17. As an example, we present in Table \[tab:binary\_fit\] the fit of the binary system LFAM 15. The main difference between our fit and that obtained by @ortizleon17 is the time of the periastro of the orbit, which is offset by two orbital periods of the binary system. We obtained a periastro time near the center of the observing interval, while @ortizleon17 obtained a periastro time near the beginning of the observing interval. Julian date $\alpha$ (J2000.0) $\sigma_\alpha$ $\delta$ (J2000.0) $\sigma_\delta$ Reference --------------- -------------------- ----------------- --------------------- ----------------- ----------- -- Primary 2456158.56345 16 26 3.00879650 0.00000023 $-$24 23 36.532098 0.000008 1 2456271.25813 16 26 3.00899222 0.00000045 $-$24 23 36.541056 0.000014 1 2456537.53271 16 26 3.00730547 0.00000110 $-$24 23 36.559579 0.000037 1 2456718.03765 16 26 3.00766323 0.00000030 $-$24 23 36.575357 0.000010 1 2456938.43345 16 26 3.00579982 0.00000097 $-$24 23 36.589569 0.000036 1 2457081.04343 16 26 3.00621579 0.00000166 $-$24 23 36.602285 0.000058 1 2457300.44295 16 26 3.00441795 0.00000725 $-$24 23 36.615740 0.000215 1 2458257.82118 16 26 3.00114839 0.00000028 $-$24 23 36.688565 0.000009 2 2458335.60840 16 26 3.00026270 0.00000402 $-$24 23 36.692870 0.000088 2 2458433.34082 16 26 3.00019279 0.00000619 $-$24 23 36.700039 0.000131 2 2458534.06442 16 26 3.00050166 0.00000082 $-$24. 23 36.709262 0.000026 2 Binary System Primary 2453691.33229 16 26 3.01909977 0.000005314 $-$24 23 36.343748 0.000153 2 2453971.56511 16 26 3.01741929 0.000004120 $-$24 23 36.368881 0.000124 2 2454365.48657 16 26 3.01575121 0.000002832 $-$24 23 36.402405 0.000107 2 Secondary 2453691.33229 16 26 3.01889886 0.00000304 $-$24 23 36.349153 0.000063 2 2453971.56511 16 26 3.01698794 0.00000267 $-$24 23 36.369931 0.000114 2 2454365.48657 16 26 3.01588946 0.00000202 $-$24 23 36.398070 0.000067 2 \[tab:binary\_fit\] LFAM15 -------------------------- ----------------------- ---------------------- Parameter This work @ortizleon17 Parameters Fitted $\mu_x$ (mas/yr) $-$6.303 $\pm$ 0.020 $-$6.31 $\pm$ 0.02 $\mu_y$ (mas/yr) $-$26.964 $\pm$ 0.050 $-$26.95 $\pm$ 0.05 $\Pi$ (mas) 7.259 $\pm$ 0.079 7.253 $\pm$ 0.054 $P$ (days) 1308.75 $\pm$ 10.39 1311.61 $\pm$ 6.68 $T_{0}$ (days) 2014.1931 $\pm$ 0.076 2007.008 $\pm$ 0.039 $e$ 0.5292 $\pm$ 0.0075 0.528 $\pm$ 0.005 $\omega$ (deg) 56.09 $\pm$ 0.52 55.54 $\pm$ 1.02 $\Omega$ (deg) 338.01 $\pm$ 0.40 337.93 $\pm$ 0.81 $a_1$ (mas) 7.760 $\pm$ 0.070 $...$ $i$ (deg) 110.24 $\pm$ 0.27 110.30 $\pm$ 0.49 $\omega_2$ (deg) 236.09 $\pm$ 0.52 235.54 $\pm$ 1.02 $a_{2}$ (mas) 8.637 $\pm$ 0.091 $...$ $a$ (mas) 16.40 $\pm$ 0.16 16.40 $\pm$ 0.13 Other Parameters $D$ (pc) 137.77 $\pm$ 1.48 137.9 $\pm$ 1.0 $m ~(M_\odot)$ 0.898 $\pm$ 0.042 0.89 $\pm$ 0.01 $m_{1} ~(M_\odot)$ 0.473 $\pm$ 0.022 0.469 $\pm$ 0.015 $m_{2} ~(M_\odot)$ 0.425 $\pm$ 0.019 0.421 $\pm$ 0.010 $a_{1} ~(AU)$ 1.069 $\pm$ 0.021 $...$ $a_{2} ~(AU)$ 1.190 $\pm$ 0.025 $...$ $\chi^2$, $\chi^2_{red}$ 75.53, 3.43 $...$, $..$ \[tab:astrofit\] Parameter Single Star Solution DoAr21$c$ DoAr21$B$ -------------------------- ------------------------- ------------------------- ------------------------- Parameters Fitted $\mu_x$ (mas/yr) $-$19.6953 $\pm$ 0.0020 $-$19.7089 $\pm$ 0.0028 $-$19.5565 $\pm$ 0.0028 $\mu_y$ (mas/yr) $-$26.9477 $\pm$ 0.0050 $-$26.9637 $\pm$ 0.0069 $-$26.8091 $\pm$ 0.0069 $acc_x$ (mas/yr/yr) $... $ $.....$ 0.0335 $\pm$ 0.0020 $acc_y$ (mas/yr/yr) $.....$ $.....$ $-$0.0015 $\pm$ 0.0048 $\Pi$ (mas) 7.5330 $\pm$ 0.0068 7.5001 $\pm$ 0.0095 7.4482 $\pm$ 0.0095 $P$ (days) $.....$ 1018.43 $\pm$ 3.98 92.892 $\pm$ 0.030 $T_0$ (days) $.....$ 2457163.20 $\pm$ 3.89 2457293.95 $\pm$ 0.31 $e$ $.....$ 0.368 $\pm$ 0.036 0.3272 $\pm$ 0.0095 $\omega$ (deg) $.....$ 236.90 $\pm$ 1.91 32.24 $\pm$ 1.05 $\Omega$ (deg) $.....$ 40.41 $\pm$ 2.82 41.43 $\pm$ 1.09 $a_1$ (mas) $.....$ 0.377 $\pm$ 0.024 0.706 $\pm$ 0.014 $i$ (deg) $.....$ 102.25 $\pm$ 1.83 89.41 $\pm$ 0.96 Other Parameters $D$ (pc) 132.74 $\pm$ 0.12 133.33 $\pm$ 0.17 134.26 $\pm$ 0.17 $m ~(M_\odot)$ $.....$ 2.9441 2.8745 $m_1 ~(M_\odot)$ $.....$ 2.8920 $\pm$ 0.0032 2.397 $\pm$ 0.010 $m_2 ~(M_\odot)$ $.....$ 0.0521 $\pm$ 0.0032 0.477 $\pm$ 0.010 $a_1 ~(AU)$ $.....$ 0.0503 $\pm$ 0.0032 0.0947 $\pm$ 0.0021 $a_2 ~(AU)$ $.....$ 2.7890 $\pm$ 0.0041 0.4760 $\pm$ 0.0019 $\chi^2$, $\chi^2_{red}$ 2309.29, 128.29 91.02, 8.27 70.80, 7.87 Parameter DoAr21$c$ and DoAr21$B$ -------------------------- ------------------------- Fitted Parameters $\mu_{x}$ (mas/yr) $-$19.6442 $\pm$ 0.0035 $\mu_{x}$ (mas/yr) $-$26.8756 $\pm$ 0.0084 $\Pi$ (mas) 7.474 $\pm$ 0.012 $P$ (days) 1034.50 $\pm$ 7.74 $T_{0}$ (days) 2457157.64 $\pm$ 6.94 $e$ 0.404 $\pm$ 0.065 $\omega$ (deg) 240.43 $\pm$ 3.52 $\Omega$ (deg) 59.72 $\pm$ 6.98 $a_{1}$ (mas) 0.203 $\pm$ 0.024 $i$ (deg) 109.99 $\pm$ 5.62 $P$ (days) 92.921 $\pm$ 0.091 $T_{0}$ (days) 2457295.80 $\pm$ 0.94 $e$ 0.391 $\pm$ 0.029 $\omega$ (deg) 39.61 $\pm$ 2.78 $\Omega$ (deg) 32.73 $\pm$ 2.50 $a_{1}$ (mas) 0.346 $\pm$ 0.020 $i$ (deg) 93.03 $\pm$ 2.03 Other Parameters $D$ (pc) 133.80 $\pm$ 0.21 $m ~(M_\odot)$ 2.9441 $m_{1} ~(M_\odot)$ 2.681 $\pm$ 0.014 $m_{2} ~(M_\odot)$ 0.0279 $\pm$ 0.0031 $m_{3} ~(M_\odot)$ 0.236 $\pm$ 0.014 $a_{1-2} ~(AU)$ 2.869 $\pm$ 0.014 $a_{1} ~(AU)$ 0.0271 $\pm$ 0.0032 $a_{2} ~(AU)$ 2.842 $\pm$ 0.011 $a_{1-3} ~(AU)$ 0.5736 $\pm$ 0.0004 $a_{1} ~(AU)$ 0.0463 $\pm$ 0.0027 $a_{3} ~(AU)$ 0.5273 $\pm$ 0.0024 $\chi^2$, $\chi^2_{red}$ 31.72, 7.93 Parameter DoAr21$b$ -------------------------- ------------------------- Fitted Parameters $\mu_{x}$ (mas/yr) $-$19.6682 $\pm$ 0.0030 $\mu_{x}$ (mas/yr) $-$26.9746 $\pm$ 0.0074 $acc_x$ (mas/yr/yr) 0.0057 $\pm$ 0.0021 $acc_y$ (mas/yr/yr) $-$0.0091 $\pm$ 0.0052 $\Pi$ (mas) 7.350 $\pm$ 0.010 $P$ (days) 452.97 $\pm$ 0.15 $T_{0}$ (days) 2457507.45 $\pm$ 1.07 $e$ 0.076 $\pm$ 0.012 $\omega$ (deg) 220.90 $\pm$ 0.91 $\Omega$ (deg) 53.20 $\pm$ 1.78 $a_{1}$ (mas) 0.286 $\pm$ 0.015 $i$ (deg) 103.57 $\pm$ 1.17 $q$ (m1/m2) 39.86 $\pm$ 2.67 $\omega_{2}$ (deg) 40.90 $\pm$ 0.91 $a$ (mas) 11.68 $\pm$ 0.44 Other Parameters $D$ (pc) 136.05 $\pm$ 0.19 $m ~(M_\odot)$ 2.6064 $\pm$ 0.0018 $m_{1} ~(M_\odot)$ 2.5426 $\pm$ 0.0029 $m_{2} ~(M_\odot)$ 0.0638 $\pm$ 0.0035 $a_{1-2} ~(AU)$ 1.5886 $\pm$ 0.0007 $a_{1} ~(AU)$ 0.0389 $\pm$ 0.0021 $a_{2} ~(AU)$ 1.5497 $\pm$ 0.0014 $\chi^2$, $\chi^2_{red}$ 903.11, 64.51 Parameter DoAr21$b$ and $c$ DoAr21$b$ and $B$ ---------------------------- ------------------------- ------------------------- Fitted Parameters $\mu_{x}$ (mas/yr) $-$19.6986 $\pm$ 0.0036 $-$19.5478 $\pm$ 0.0036 $\mu_{x}$ (mas/yr) $-$26.9768 $\pm$ 0.0088 $-$26.8373 $\pm$ 0.0088 $acc_x$ (mas/yr/yr) $.....$ 0.0232 $\pm$ 0.0025 $acc_y$ (mas/yr/yr) $.....$ $-$0.0012 $\pm$ 0.0061 $\Pi$ (mas) 7.428 $\pm$ 0.012 7.385 $\pm$ 0.012 $P$ (days) 446.52 $\pm$ 0.19 453.22 $\pm$ 0.22 $T_{0}$ (days) 2457045.94 $\pm$ 1.22 2457063.16 $\pm$ 1.38 $e$ 0.090 $\pm$ 0.036 0.095 $\pm$ 0.018 $\omega$ (deg) 250.02 $\pm$ 1.11 225.86 $\pm$ 1.20 $\Omega$ (deg) 55.80 $\pm$ 2.39 55.41 $\pm$ 2.43 $a_{1}$ (mas) 0.099 $\pm$ 0.015 0.073 $\pm$ 0.018 $i$ (deg) 104.19 $\pm$ 1.52 105.43 $\pm$ 1.67 $q$ (m1/m2) 121.91 $\pm$ 19.16 151.89 $\pm$ 37.59 $\omega_{2}$ (deg) 70.02 $\pm$ 1.11 45.86 $\pm$ 1.20 $a_{1-2}$ (mas) 12.11 $\pm$ 0.54 11.13 $\pm$ 0.50 $P$ (days) 984.93 $\pm$ 6.39 92.935 $\pm$ 0.041 $T_{0}$ (days) 2457158.61 $\pm$ 6.34 2457295.83 $\pm$ 0.43 $e$ 0.234 $\pm$ 0.060 0.355 $\pm$ 0.013 $\omega$ (deg) 243.17 $\pm$ 2.78 41.35 $\pm$ 1.31 $\Omega$ (deg) 44.12 $\pm$ 4.36 46.06 $\pm$ 1.60 $a_{1}$ (mas) 0.321 $\pm$ 0.026 0.650 $\pm$ 0.018 $i$ (deg) 99.53 $\pm$ 2.80 90.07 $\pm$ 1.34 Other Parameters $D$ (pc) 134.62 $\pm$ 0.22 135.40 $\pm$ 0.22 $m ~(M_\odot)$ 2.9477 $\pm$ 0.0025 2.2246 $\pm$ 0.0021 $m_{A}, m_{A} ~(M_\odot)$ 2.8782 $\pm$ 0.0027 1.838 $\pm$ 0.018 $m_{b}, m_{b} ~(M_\odot)$ 0.0236 $\pm$ 0.0036 0.0146 $\pm$ 0.0036 $m_{c}, m_{B} ~(M_\odot)$ 0.0459 $\pm$ 0.0032 0.372 $\pm$ 0.012 $a_{A-b}, a_{A-b} ~(AU)$ 1.6308 $\pm$ 0.0009 1.5074 $\pm$ 0.0010 $a_{A}, a_{A} ~(AU)$ 0.0133 $\pm$ 0.0020 0.0099 $\pm$ 0.0024 $a_{b}, a_{b} ~(AU)$ 1.6175 $\pm$ 0.0011 1.4976 $\pm$ 0.0014 $a_{A-c}, a_{A-B} ~(AU)$ 2.7778 $\pm$ 0.0092 0.3231 $\pm$ 0.0004 $a_{A}, a_{A} ~(AU)$ 0.0433 $\pm$ 0.0036 0.0880 $\pm$ 0.0025 $a_{c}, a_{B} ~(AU)$ 2.7346 $\pm$ 0.0056 0.4350 $\pm$ 0.0021 $\chi^2$, $\chi^2_{red}$ 56.13, 6.24 60.32, 8.62 Parameter DoAr21$B$ DoAr21$b$ DoAr21$c$ ------------------ -------------------- -------------------- ------------------- Orbital Parameters $P$ (days) 92.919 $\pm$ 0.022 450.88 $\pm$ 3.80 1013.5 $\pm$ 25.3 $e$ 0.370 $\pm$ 0.035 0.089 $\pm$ 0.010 0.333 $\pm$ 0.090 $\omega$ (deg) 38.55 $\pm$ 4.94 232.8 $\pm$ 15.6 240.54 $\pm$ 3.18 $\Omega$ (deg) 38.67 $\pm$ 6.98 54.96 $\pm$ 1.41 51.1 $\pm$ 10.9 $a_1$ (mas) 0.55 $\pm$ 0.20 0.15 $\pm$ 0.12 0.301 $\pm$ 0.026 $i$ (deg) 91.31 $\pm$ 2.01 104.50 $\pm$ 0.96 105.75 $\pm$ 5.87 Other Parameters $m_2 ~(M_\odot)$ 0.35 $\pm$ 0.12 0.034 $\pm$ 0.026 0.042 $\pm$ 0.013 $a_2 ~(AU)$ 0.482 $\pm$ 0.046 1.550 $\pm$ 0.060 2.802 $\pm$ 0.056 ------------- -------------------- -------------------- Julian Date Peak Flux Flux Density (mJy) (mJy) Primary 2453621.524 2.302 $\pm$ 0.254 5.939 $\pm$ 0.874 2453744.188 0.425 $\pm$ 0.088 0.480 $\pm$ 0.166 2453755.157 0.926 $\pm$ 0.090 1.003 $\pm$ 0.165 2453822.972 1.215 $\pm$ 0.096 1.451 $\pm$ 0.187 2453890.786 1.980 $\pm$ 0.104 2.930 $\pm$ 0.236 2454092.235 2.541 $\pm$ 0.153 2.756 $\pm$ 0.281 2454321.607 0.881 $\pm$ 0.077 1.874 $\pm$ 0.229 2454331.079 1.724 $\pm$ 0.085 2.056 $\pm$ 0.166 2454353.519 1.283 $\pm$ 0.077 1.756 $\pm$ 0.166 2458257.821 4.402 $\pm$ 0.034 5.231 $\pm$ 0.067 2458335.608 0.517 $\pm$ 0.025 0.681 $\pm$ 0.052 2458534.064 4.388 $\pm$ 0.042 5.793 $\pm$ 0.088 Binary System Primary 2453691.332 6.075 $\pm$ 0.504 17.071 $\pm$ 1.85 2453971.565 0.857 $\pm$ 0.090 1.137 $\pm$ 0.188 2454365.487 2.499 $\pm$ 0.127 3.286 $\pm$ 0.265 2458433.341 2.279 $\pm$ 0.172 5.355 $\pm$ 0.549 Secondary 2453691.332 15.603 $\pm$ 0.510 35.654 $\pm$ 1.60 2453971.565 1.084 $\pm$ 0.089 1.630 $\pm$ 0.203 2454365.487 3.791 $\pm$ 0.129 4.107 $\pm$ 0.236 2458433.341 24.311 $\pm$ 0.183 26.176 $\pm$ 0.333 ------------- -------------------- -------------------- Results ======= Single companion Astrometry --------------------------- The least-squares periodogram with a circular orbit (CLSP) of the astrometric data (see Figure \[fig\_2\]) shows a prominent peak (with 999 days period) with high power and very low False Alarm Probability (FAP). There is another significant peak (with 87 days period) with lower power and higher FAP. These is also a third weaker peak that appears in the CLSP periodogram (see Figure \[fig\_2\], upper panel) with a period of 151 days. However, when we fix the eccentricity to 0.368 (see discussion below), the peak with a period of about 999 days becomes more prominent, while this weak peak remains constant and a somewhat stronger peak appears with a period of 129 days (see Figure \[fig\_2\]). This new peak is quite weak in the upper panel of Figure \[fig\_2\]. Thus these two weaker peaks are dubious, while the peak with a period of about 87 days remains consistent in both periodograms. Thus, we investigate here only the two candidates with the strongest peaks in the periodograms (with periods of about 999 and 87 days) to obtain their $''$significances$''$. We use the two methods described above (least-squares and AGA) to fit the astrometric data. First we use both methods to fit the 11 astrometric observations to obtain the proper motions and the parallax of this source without taking into account any companion (Single Star Solution). The results of a single star solution are shown in Table \[tab:astrofit\] and Figure \[fig\_3\]. We did not find necessary to include an acceleration term for the fitting of the external companion DoAr21$c$. However, in the case of the internal companion DoAr21$B$, we included acceleration terms to take into account the contribution of the external companion. The acceleration terms are small and produce a small effect in the fitting of DoAr21$B$. We find that after the fitting the residuals are large compared with the noise present in the data and the astrometric precision obtained with the VLBA. Figure \[fig\_3\] shows residuals up to about 0.3 mas and the astrometric precision obtained with the VLBA is about 30 $\mu$as. Furthermore, the residuals have a temporal trend that suggests the presence of at least one companion. We then use the least-squares and the AGA algorithms to fit the astrometric observations of this source including a single companion (i.e., we performed an independent fit for each companion candidate). In both cases we obtain consistent solutions. The parallax and the orbital fits for the two candidates are presented in Figure \[fig\_4\]. Table \[tab:astrofit\] summarizes the best fits and their $\chi^{2}_{red}$ per degree of freedom ($\chi^{2}_{red}$ = $\chi^{2}/(N_{data} - N_{par} - 1)$, where $N_{data} = 2 \times N_{points}$ and $N_{par}$ is the number of fitted parameters). The fit of the astrometric data clearly improves when including a companion, as seen by the $\chi^{2}_{red}$. The two components have similar position angle of the line of nodes ($\Omega$), within the errors. The eccentricity of the orbits seem to be reasonably well constrained: component $c$ has an orbital eccentricity $\sim$0.37, while component DoAr21$B$ has an eccentricity of $\sim$0.33. The astrometric signature of the source ($a_{*}$) due to the gravitational pull of the companion is larger in the case of component DoAr21$B$, by almost a factor of two. The inclination of the orbit ($i$) is somewhat different for the two companions. The inclination angle is larger than 90 degrees for component DoAr21$c$ ($\sim$102 deg, which indicates a retrograde orbit), while component $B$ has an orbital inclination of about 90 degrees (indicating an edge-on orbit), which suggests that the orbits of these two companions are not coplanar. This result is somewhat surprising since one would expect a similar inclination for the orbits of all the companions. We do not have an explanation for this result. However, it is important to point out that the combined fit (relative astrometry plus absolute astrometry) of two components (components DoAr21$b$ and DoAr21$c$; see Section 4.3) also give retrograde orbits ($i$ $>$ 90 degrees). With these fits we cannot estimate the dynamical mass of the system, thus to estimate the mass of the companions we use the mass of the system obtained with the combined fit of the astrometric data (a fixed mass of $M$ $\sim$ 2.94 $M_\odot$ for the fit of component DoAr21$c$ and $\sim$ 2.87 $M_\odot$ for the fit of component DoAr21$B$). These masses were obtained with the combined fits of the orbits of components DoAr21$b$ and DoAr21$c$ (see Section 4.3). With this assumption, we obtain that the component with a compact orbit (component DoAr21$B$) has a mass consistent with a low mass star ($m_{B} \sim$ 0.48 $m_\odot$), while component DoAr21$c$ has a mass consistent with a Brown Dwarf ($m_{c} \sim$ 54.57 $m_{jup}$). The orbits of the two companions have semi-mayor axis $a_{c}$ $\sim$ 2.79 AU and $a_{B}$ $\sim$ 0.48 AU, respectively. This suggests that the mass of the companion decreases with the distance to the source. The Brown Dwarf is the component with the wider orbit. The estimated distance to the source is similar in both cases, however, the estimated distance is slightly larger when including a companion than when fitting only the proper motions and the parallax of the host star. The orbital periods of the orbits of components DoAr21$c$ and DoAr21$B$ are consistent with those found in the CLSP (see Figure \[fig\_2\]). The difference in the orbital periods obtained with the astrometric fits and the CLSP is due to the fact that in the case of CLSP we assume circular orbits, while in the astrometric fits we include the eccentricity of the orbits as a free parameter to be fitted. In Figure \[fig\_2\], we also show the least-squares periodogram of the observed astrometric data fixing the eccentricity obtained for companion DoAr21$c$ ($e_{c}$ $\sim$ 0.368). In this case we obtain an orbital period for component DoAr21$c$ that is consistent with that obtained with the astrometry fit. This figure shows that the power of component $c$ increases substantially when fixing the eccentricity of this companion. On the other hand, the power of component DoAr21$B$ does not change substantially when using this eccentricity in the periodogram. Figure \[fig\_4\] shows the orbits of both companions. As it was mentioned before, a companion was detected with the VLBA in 3 different epochs. The relative position of the companion with respect to the host star is included in Figure \[fig\_4\]. This figure shows that the relative position of none of the detected companions coincide with the fitted orbits. Bellow we attempt to fit simultaneously the orbits of the two inferred components (DoAr21$c$ and DoArt21$B$). We also attempt to fit simultaneously the relative astrometry of the companion detected with the VLBA (DoAr21$b$) and the absolute astrometry of the host star (combined fit; Section 4.3). Simultaneous Astrometric fit of DoAr21$c$ and DoAr21$B$ ------------------------------------------------------- The eleven observed epochs are in principle enough to fit simultaneously the orbits of the two companions (DoAr21$c$ and DoAr21$B$), plus the parallax and the proper motions of the host star. We find that the astrometry fit improves when fitting simultaneously both companions. Table \[tab:abs2fit\] summarizes the parameters obtained with the best two-companion astrometry fit of the data. Figure \[fig\_5\] shows the parallax of the host star and the orbital motion of the host star due to the gravitational pull of both companions. This figure also shows the orbits of both companions. For this fit we have also fixed the total mass of the system ($m = 2.9441$ $m_\odot$; see Section 4.3 and Table \[tab:rel2fit\]). The fitted parameters are similar to those found from the single-companion fit. However, some of the fitted parameters change substantially. For instance, the eccentricity, semi-mayor axis and inclination of the orbits are somewhat larger, and the estimated masses of the companions are somewhat smaller in this case than in the case of a single-companion astrometry fit. Furthermore, the position angle of the line of nodes of both companions differs by more than 20 degrees ($\Omega$ $\sim$ 32.7 and 59.7 for DoAr21$B$ and DoArt21$c$, respectively), while in the case of a single-companion fit we obtained a similar $\Omega$ for both companions ($\Omega$ $\simeq$ 41 deg). These new results suggest that the orbits of these two companions are not coplanar. In addition, Figure \[fig\_5\] shows that the companion detected with the VLBA does not coincide with the estimated orbits of these two companions. We obtained a similar result from the single-companion astrometry fit. This suggests that there is probably another companion that does not appear in the periodogram, but that was detected with the VLBA. Bellow we attempt to fit simultaneously the relative astrometry of the companion observed with the VLBA and the absolute astrometry of the host star (combined fit). Relative plus absolute Astrometry: Combined$-$fit ------------------------------------------------- We investigate here the possibility that the relative positions of the detected companion are associated to another companion, different from the two companions that we have already found. We find that the three secondary detections with the VLBA can be fitted by combining relative astrometry and absolute astrometry (combined model) using both the least-squares and the AGA algorithms. These three detections correspond to a new component that we call DoAr21$b$. Table \[tab:relfit\] summarizes the parameters that were obtained with the best fit of the data, and the $\chi^{2}_{red}$ per degree of freedom. The parallax and the astrometric signature of the host star due to the gravitational pull of this companion is presented in Figure \[fig\_6\]. The orbital fit of this companion is also presented in Figure \[fig\_6\]. Since DoAr21$b$ is an internal companion, we also included acceleration terms to take into account the contribution of the external companion DoAr21$c$. The acceleration terms are small and produce a small effect in the fitting of DoAr21$b$. The combined fitted orbit of DoAr21$b$ has a period of $P_{b}$ $\sim$ 453 $days$, a semi-mayor axis of $a_{1}$ $\sim$ 0.29 $mas$ (or $\sim$ 0.04 $AU$), a position angle of the line of nodes $\Omega$ $\sim$ 53 deg, an eccentricity $e$ $\sim$ 0.08, an inclination angle $i$ $\sim$ 104 $deg$, and a longitude of the periastron $\omega$ $\sim$ 221 deg. Component DoAr21$b$ has an orbit with a semi-mayor axis of $a_{b}$ $\sim$ 11.40 $mas$ (or $\sim$ 1.55 $AU$) and a longitude of the periastron $\omega_{b}$ $\sim$ 41 deg. Since we are doing a combined fit (relative plus absolute orbital fit), we obtain the orbital fit around the barycenter position of the system for both, the main source and the companion, thus we obtain the dynamical mass of the system, as well as the masses of each individual component. The dynamical mass of the system is $M_{\star b}$ $\sim$ 2.606 $M_\odot$, the mass of the main source is $M_{\star}$ $\sim$ 2.542 $M_\odot$, and the mass of the companion is $M_{b}$ $\sim$ 0.064 $M_\odot$ (or about 66.8 $M_{jup}$). The orbit of this new companion, DoAr21$b$, lies between the orbits of the other two companions, and its estimated mass is consistent with being a Brown Dwarf. We find that the position angle of the line of nodes ($\Omega$) and the inclination angle ($i$) of the orbit are similar to those found in DoAr21$c$ obtained with the two-companion simultaneous fit (see Table \[tab:abs2fit\]). The eccentricity of the orbit ($e$) is smaller than those found for the other companions, and seems to be well constrained. The astrometric signature of the source ($a_{\star}$) due to the gravitational pull of this companion is somewhat smaller than those found for the other companions (see Table \[tab:astrofit\], \[tab:abs2fit\] and \[tab:relfit\]). Since the astrometric signature of the source is relatively small, the residuals of the fit (see Figure \[fig\_6\]) are larger than those obtained from the fits of the other two companions (see Figure \[fig\_5\]). In addition, since the astrometric signature of DoAr21$b$ is quite small ($<$ 0.1 $mas$; see Tables \[tab:relfit\] and \[tab:rel2fit\]) the least-squares fit of the orbit of this companion basically fails. In other words, the $\chi^{2}$ of the fit is similar to the null solution, and thus the estimated power is small, within the expected $''$noise$''$ in the periodogram (see discussion in section 3.1). These results are consistent with the fact that companion DoAr21$b$ does not appear in the periodogram (see Figure \[fig\_2\]). In other words, we have found DoAr21$b$ only because this companion was detected at several epochs with the VLBA, otherwise the astrometric signature due to this companion would be embedded in the residuals of the fits of the other two companions. The eleven observed epochs and the three direct detections of DoAr21$b$ are in principle enough to fit simultaneously the orbits of two companions (DoAr21$b$ and DoAr21$c$, or DoAr21$b$ and DoAr21$B$), plus the parallax and the proper motions of the host star. Table \[tab:rel2fit\] summarizes the parameters that were obtained with the best fit of the data, and the $\chi^{2}_{red}$ per degree of freedom. The parallax and the astrometric signature of the host star due to the gravitational pull of each companion is presented in Figures \[fig\_7\] and \[fig\_8\]. The orbital fit for each pair of companions is also presented in these Figures. For the simultaneous fit of DoAr21$b$ and DoAr21$B$, we included acceleration terms to take into account the contribution of the external companion DoAr21$c$. The acceleration terms are small and produce a small effect in the fitting of these two companions. We find that the fitted parameters are in general consistent with the previous fits. In particular, the astrometric fit of DoAr21$b$ is quite consistent in all the fits of this component. This is because the relative astrometry of this companion, using the three detected epochs, constrain the orbit of this companion. The astrometric fits of the other two companions are similar to those obtained previously, however, there are some differences. For instance, the position angle of the line of nodes ($\Omega$), the inclination angle ($i$) and the semi-mayor axis ($a_{1}$) of the orbit are somewhat different from those found previously (see Table \[tab:abs2fit\]). It is important to mention that the estimated mass of the system is somewhat different in both fits, even if we take into account the mass of the companion DoAr21$c$ in the simultaneous fit of DoAr21$b$ and DoAr21$B$. The difference in the estimated total mass is $\sim$ 0.72 $M_\odot$ (or $\sim$ 0.68 $M_\odot$, taking into account the mass of DoAr21$c$ in both fits), which is much higher than the estimated mass of DoAr21$c$ ($\sim$ 0.046 $M_\odot$). A better estimate of the masses in this multiple system can be obtained by fitting simultaneously the orbits of all the components in the system. Thus, further observations of this multiple system will be needed in order to obtain a better estimate of the total mass of the system and the individual masses of all the companions. Discusion ========= Mass and spatial distribution in this multiple system ----------------------------------------------------- Figure \[fig\_7\] shows the astrometric signature of the host star due to the companions DoAr21$b$ and DoAr21$c$, and the parallax of the system as function of time, and Figure \[fig\_8\] shows the astrometry of DoAr21$b$ and DoAr21$B$. Figure \[fig\_7\] suggests that the orbits of DoAr21$b$ and DoAr21$c$ cross each other, however, the results presented in Table \[tab:rel2fit\] indicate that this is a projection effect due to the large inclination of the orbits ($i$ $\sim$104 and $\sim$100 $deg$, respectively) and the difference in the position angle of the line of nodes of both orbits ($\Omega$ $\sim$56 and $\sim$44 $deg$, respectively). The semi-mayor axis of the orbit of DoAr21$c$ ($a_{c}$ $\sim$2.73 $AU$) is nearly a factor of two larger than that of DoAr21$b$ ($a_{b}$ $\sim$1.62 $AU$), and their orbits have low eccentricities ($e_{b}$ $\sim$0.09 and $e_{c}$ $\sim$0.23). Thus the orbits of these two companions do not cross each other. In addition, the orbit of DoAr21$B$ ($a_{B}$ $\sim$0.44 $AU$) is about a factor of three smaller than that of DoAr21$b$ ($a_{b}$ $\sim$1.50 $AU$). Therefore the orbits of this multiple system seem to scale with a relationship close to 3:1 between DoAr21$B$ and DoAr21$b$, and close to 2:1 between DoAr21$b$ and DoAr21$c$. The inclination angle of the three companions is not the same. The orbit of DoAr21$B$ appears to be edge-on ($i$$\sim$90.1 deg), while the orbits of DoAr21$b$ and DoAr21$c$ are retrograde, with inclination angles $\sim$104 and 100 deg, respectively. Thus the inclination angle of the system seem to be about 97$\pm$7 deg. However, this large difference in the inclination angle will have to be confirmed with further observations. Since a companion (DoAr21$b$) was detected in three epochs, we were able to fit simultaneously the relative position of the companion and the absolute position of the host star. Furthermore, we were able to fit simultaneously the orbits of the pairs of companions DoAr21$b$$-$DoAr21$c$ and DoAr21$b$$-$DoAr21$B$. With this relative plus absolute simultaneous combined fit, we have obtained the dynamical mass of the system, as well as the mass of the individual companions. The combined fit of the pairs of components give the dynamical masses for the system of $\sim$2.948 $M_\odot$ for the pair DoAr21$b$$-$DoAr21$c$ and $\sim$2.225 $M_\odot$ for the pair DoAr21$b$$-$DoAr21$B$. There is a significative difference in the estimated mass of the system of about 0.68 $M_\odot$ (taking into account the mass of DoAr21$c$ in both fits). This discrepancy is probably due to the astrometric contribution of the other companion (which is not taken into account in the fit), and the fact that DoAr21$b$ was detected at only three epochs. We consider that at a first approximation of the total mass of the system is probably somewhere between these two values. There is also a large difference in the estimates mass of the host star, between 1.84 and 2.51 $M_\odot$ (we have taken into account the mass of DoAr21$B$ in both estimates; see Table \[tab:rel2fit\]). The difference of about 0.67 $M_\odot$ is similar to the difference of about 0.68 $M_\odot$ found for the total mass of the system. Thus, we also consider that the mass of the star is somewhere between these two estimated masses. The estimated mass of DoAr21 is similar to the previously estimated masses of $\sim$$2.2 \,M_\odot$ and $\sim$$1.8 \,M_\odot$ [@jensen09], which were obtained by assuming that all the luminosity of the source is associated to a single star and by splitting the luminosity of this source in two equal stars, respectively. The dynamical mass that we obtained here is, at present, the best estimated mass of the WTTS DoAr21. Tables \[tab:astrofit\] through \[tab:rel2fit\] show that the best fitted parameters of the three companions of DoAr21 do not change considerably when including one or more companions in the astrometric fit. However, taking into account that the different fits give slightly different values for all the orbital parameters, and that the orbits of the companions are not yet fully constrained, we have calculated the weighted average of the orbital parameters for all the companions, as well as the weighted average values for the mass of the host star and the companions. The weighted average parameters are presented in Tables \[waverage\]. The estimated masses of the host star and the three companions are: 2.04 $\pm$ 0.70 $M_\odot$ for the host star, 0.35 $\pm$ 0.12 $M_\odot$ for DoAr21$B$, 0.034 $\pm$ 0.026 $M_\odot$ (35.6 $\pm$ 27.2 $M_{jup}$) for DoAr21$b$ and 0.042$\pm$ 0.013 $M_\odot$ (44.0 $\pm$ 13.6 $M_{jup}$) for DoAr21$c$ (see Table \[waverage\]). The inner companion, DoAr21$B$, has an estimated mass consistent with a low mass star, while the other two companions have masses consistent with being Brown Dwarfs. The masses of DoAr21$b$ and DoAr21$c$ are probably consistent with a spectral types M7-M8 (e.g., @luhman09, and references therein), however, the classification of Brown Dwarfs is based on their spectral type, in particular based on the elements seen in their optical spectrum. These two Brown Dwarfs can not be classified this way since they, in principle, can not be separated from the host star in the optical nor in the infrared. In addition, it is expected that brown Dwarfs change in class type as they burn their deuterium (and their lithium in the case of the most massive Brown Dwarfs). Since DoAr21 is estimated to have an age os about 0.4$-$0.8 Myrs [@jensen09], the two Brown Dwarfs orbiting this WTTS must be extremely young. In addition, these two Brown Dwarfs have broad orbits, with semi-mayor axis of $\sim$1.6 and $\sim$2.8 $AU$ (see Table \[waverage\]). These results suggest that these Brown Dwarfs were formed far away from the host star, probably close to their current orbits. This is consistent with the idea that these Brown Dwarfs were formed by fragmentation of the disk where this young star was formed. On the other hand, it is not clear if DoAr21$B$, being the most massive companion ($\sim$ 0.35 $M_\odot$) in the system and with the most compact orbit ($\sim$ 0.482 $AU$), was also formed by fragmentation of the circumstellar disk, neither if it was formed close to its current orbit. Variability in this multiple system ----------------------------------- DoAr21 presents a large variability at radio wavelengths (see Table \[fluxtab\]). Figure \[fig\_9\] shows that the radio flux of this WTTS changes in more than two orders of magnitude, from a fraction of a mJy to several tens of mJy. This figure shows that there are abrupt changes in the flux in intervals of several weeks. However, this change could occur in much shorter times, the cadence of the observations is not adequate to find how steep are the outbursts observed in this source, neither if there is a frequency in the outbursts. It has been speculated that this variability is probably due to the interaction between the magnetosphere of the star an the magnetosphere of a close stellar companion. We have found three astrometric companions to DoAr21. DoAr21$B$ is the most massive companion ($m_{B}$$\sim$0.35 $M_\odot$) and with the most compact and eccentric orbit ($a_{B}$ $\sim$0.482 $AU$, $e_{B}$$\sim$0.37) in this multiple system. Thus, DoAr21$B$ could be the responsible for the large flux variation observed in DoArt21. Given the large eccentricity of the orbit of this companion, and its proximity to the star, the interaction of their magnetospheres would probably occurs when the companion is closer to the star, which occurs close to the periastron of the orbit. To investigate this possibility, we have first calculated the closest distance between DoAr21$A$ and DoAr21$B$, which occurs close to the periastron of their orbital motions. We obtain that the closest distance between them is $a_{p} = a(1 - e) = 0.34$ $AU$, where $a = a_{\star} + a_{B}$ $\sim$ 0.536 $AU$ is the relative semi-major axis of the orbit of DoAr21$B$ around the host star DoAr21, and $e$ $\sim$ 0.37 is the eccentricity of the orbit (see Table \[waverage\]). The closest separations between the two stars is much larger than the typical magnetosphere sizes of the stars, which are in the range of a few stellar radii (e.g., @feigelson99). Second, we investigated the posibility of a correlation between the period of the orbit of DoAr21$B$ and the variability of the star. We have calculated the expected position of DoAr21$B$ in its orbit for each observed epoch. In Figure \[fig\_9\], we present the observed fluxes of DoAr21 folded with the period of DoAr21$B$ ($P_{B}$$\sim$93.0 days). We do not find a clear correlation between the outbursts observed in DoAr21 and the orbital period of this component. Figure \[fig\_9\] shows that the two strongest peak fluxes nearly coincide in the orbital phase of this companion, about halfway between the periastron and the apoastron in the orbit of DoAr21$B$. However, one would expect that the outbursts would occur when the low-mass stellar companion is closest to the star (near the periastron of its orbit). If DoAr21$B$ were the responsable of the outbursts, then the outburst occurs about 20 days after this low-mass star has passed its periastron. These results indicate that DoAr21$B$ is probably not the responsable for the variability neither the X-ray outburst observed in DoAr21. This suggests that there may be another companion with a more compact orbit than those of the companions we report here, that perturbes the magnetosphere of the host star. Further observations will be needed to search for this putative companion in a very compact orbit around DoAr21. The sub-stellar companion DoAr21$b$ was detected at three epochs at radio wavelengths with the VLBA. This suggests that this Brown Dwarf is highly variable at radio wavelenghts. The other Brown Dwarf in this system, DoAr21$c$, which is in a more extended orbit than DoAr21$b$, was not detected at any of the observed epochs. It is not clear why DoAr21$b$ was directly detected with the VLBA, while DoAr21$c$ was not. This is the first time that a Brown Dwarf orbiting a young star has been directly detected. Several Brown Dwarfs have been detected with the VLA and with the VLBA, but most of them are single stars (they are not part of an stellar system) and are located close by (a few tens of parsecs away), and have ages of more that 100 Myrs (e.g., @berger02, @sahlmann13 and @forbrich16). A few Brown Dwarf candidates have been found at radio wavelengths (e.g., @rodriguez17, @dzib13), and they are also isolated sources. A putative Brown Dwarf was recently found to be orbiting a TTauri star [@ginski18], however its orbit lyes outside the circumbinary disk, with a projected orbit of about 210 $AU$. The estimated mass of this companion is quite uncertain, and it could be a very low-mass star. Distance to DoAr21 ------------------ Tables \[tab:astrofit\] through \[tab:rel2fit\] show that the estimated distance to DoAr21 does not change substantially when including one or more companions in the astrometric fit. Taking into account that the different fits give slightly different values for the distance, and that the orbits of the companions are not yet fully constrained, we have calculated the weighted average of the estimated distances. We obtained that the weighted distance is $d = 134.6 \pm 1.0$ pc, where the estimated error corresponds to the standard deviation of the fitted values, which better reflects the dispersion seen in the different astrometric fits. This estimate is an improvement to the distance to this source of 135.76 $\pm$2.27 pc, previously obtained with the VLBA [@ortizleon18]. The distance to DoAr21 that we obtain is in agreement, within the estimated errors, with that recently obtained by @ortizleon17. The error that we obtain here, however, it is a few times smaller than those obtained previously for this source, and the typical errors for the other sources in the Ophiucus Complex, obtained by @ortizleon17. This is probably due to the larger number of observations used for the present astrometric fit, the accuracy of the observations we present here (see Table \[tab:jmfit\]) and a better coverage of the parallax, as well as the inclusion of the astrometric signal of one or more companions, which was not taken into account by @ortizleon17. Expected Radial Velocities -------------------------- We have obtained the astrometric best fits for the independent Keplerian orbits of the three DoAr21 companions. These solutions can be used to estimate an expected induced radial velocity of the star due to the gravitational pull of each companion. Assuming a simple model of totally independent companions (e.g., @canto09), the expected induced radial velocity would be: $$K_{j} = \left(\frac{2 \pi G}{T_{j}}\right)^{1/3} \frac{m_{j} sin(i_{j})}{(M_{\star} + m_{j})^{2/3}} \frac{1}{\sqrt{1 - e^{2}_{j}}},$$ where $G$ is the Gravitational constant, and $T_{j}$, $M_{\star}$, $m_{j}$ and $e_{j}$ are the orbital period, the star and companion masses and the eccentricity of the orbit of the companion. Using the parameters presented in sections 5.1 and 5.3 and those presented in Table 7, we obtain that the maximum induced velocities on DoAr21 by DoAr21$B$, DoAr21$b$ and DoAr21$c$ are $K$ $\sim$ 9.902, 0.564 and 0.557 km s$^{-1}$, respectively. These radial velocities could in principle be observed with high-spectral resolution spectrographs. However, TTauri stars are magnetically active, may have broad (molecular) spectral features, and present ubiquitous variability, that would make very difficult to observe such radial velocities. Recent optical spectroscopic observations of DoAr21 have shown possible velocity variations of $\sim$4.9 km s$^{-1}$ over the course of about 2 hours, but with a quite low precision ($\sim$1-2 km s$^{-1}$) due to the high rotation velocity of this star [@james16]. This velocity variation, if real, may suggest a much shorter orbital period than that estimated for DoAr21$B$. Future short and long term, high-resolution spectroscopic observations of DoAr21 may show whether this short period signal is real or not, and furthermore, may be able to detect the $\sim$9.9 km s$^{-1}$ radial velocity signature that we find that DoAr21$B$ probably induces on DoAr21. Conclusions =========== The multi-epoch VLBA observations of DoAr21 that we present here allow us to carry out a precise analysis of the spatial wondering of this source due to its parallax, proper motions and the astrometric signature of several companions. The precise astrometric observations obtained with the VLBA are crucial to carry out this kind of study. We find that the determination of the distance to this source improves significantly when the orbital motions of its multiple companions are taken into account. We also find that DoAr21 is a highly variable radio source, with continuos small variations in its flux (within a few mJy), and episodic outbursts (of a few tens of mJy). We present here different ways to analyze the VLBA astrometric observations of the WTTS DoAr21. We have searched for possible companions using a least-squares periodogram and analysing the residuals that result from the astrometric fit of the proper motions and the parallax of the multi-epoch data of this source. We have used two different algorithms (a least-squares algorithm and a Genetic algorithm) to fit the astrometric multi-epoch data obtained with the VLBA, and to obtain the parallax and proper motions of the host star, and the parameters of the orbits of possible companions. The periodogram of the astrometric data shows two astrometric signatures. The strongest signature corresponds to the sub-stellar companion DoAr21$c$ and the weaker signature corresponds to the low-mass stellar companion DoAr21$B$. A third companion, DoAr21$b$, was directly detected with the VLBA. We find that the best astrometric fits of the data are those where we fit simultaneously more than one companion (those that appear in the periodogram), and those were we combine relative and absolute astrometric fits of the data, using simultaneously the relative astrometry of DoAr21$b$ and the absolute astrometry of the host star. However, the multi-companion fit is limited by the reduced number of astrometric data of this source. Further VLBA observations of this source will allow to simultaneously fit the orbits of all the detected companion of the source. Since DoAr21$b$ was directly detected with the VLBA in several epochs, we were able to obtain an accurate determination of the astrometric mass of the system, as well as the masses of the individual components (the star and the companions). We find that the WTTS DoAr21 is a multiple system, formed by a compact binary system (the host star and a low-mass star), and two sub-stellar companions, whose masses are consistent with being Brown Dwarfs and located in external orbits, all of them within 3 $AU$ from the host star. Since this WTTS is very young (less than a million years old), we speculate that the companions of this source must also be extremely young, and that they were probably formed close to their present orbit. We thank the referee for his/her valuable comments that helped to improve this paper. We thank Sergio A. Dzib for a detailed reading of an early version of the manuscript and valuable suggestions. S.C. acknowledges support from DGAPA grant IN103318, UNAM and CONACyT, México. G.N.O.-L. acknowledges support from the von Humboldt Stiftung. The Long Baseline Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Anglada-Escudé, G., et al. 2010, , 711, L24-L29 Avni, Y., 1976, , 210, 642. Berger, E., et al. 2001, , 410, 338. Berger, E., 2002, , 722, 503. Berger, E. 2006, , 648, 629. Black, D. C. & Scargle, J. D. 1982, , 263, 854 Bond, I. A., et al. 2004, , 606, L155. Bower, G. C., et al. 2009, , 701, 1922. Cantó, J., Curiel, S, & Mart[í]{}nez-Gómez, E., 2009, , 501, Issue 3, pp. 1259-1268 Casertano, S., et al. 2008, å, 482, 699. Charbonneau, D., et al. 2000, , 529, L45. Charbonneau, D., et al. 2009, , 462, 891. Cumming, A., 2004, , 354, 1165-1176 Curiel, S., Cantó, J., Georgiev, L., Chávez, C. E., & Poveda, A., 2011, , 525, id.A 78, (5pp) Donati, J. F., et al. 2016, , 534, 662. Dzib, S. A., et al. 2013, , 775, issue 1, article id. 63, 16 pp. Estalella, R. 2017, , 129:025003 (23pp). Feigelson, E. D., & Montmerle, T. 1985, , 289, L19. Feigelson, E. D., & Montmerle, T. 1999, , 37, 363. Forbrich, J., & Berger, E. 2009, , 706, L205. Forbrich, J., et al. 2013, , 777, 70. Forbrich, J., et al. 2016, , 827, 22 (8pp). Gagné, M., et al. 2004, , 613, 393. Gawro[ń]{}ski, M. G., et al. 2017, , 466, 4211. Gillon, M. et al. 2017, , 542, 7642, 456. Ginski C., et al., 2018, , 616, A79 Gomez, M., Stark, D. P., Whitney, B. A., & Churchwell, E. 2003, , 126, 863 Green, R. M. (ed.) 1993, Spherical astronomy (Cambridge University Press) Greisen, 2003, in Information Handling in Astronomy: Historical Vistas, Vol. 285, ed. A. Heck (New York: Springer), 109 James, D.J., et al. 2016, , 452, 1363. Jensen, E. L. N. et al. 2009, , 703, 252. Kalas, P., et al. 2008, Science, 322, 1345. Kraus, A., et al. 2012, , 745:5, 12pp. Loinard, L., et al. 2008, , 675, L29. Luhman, K. L., et al. 2009, , 691, 1265. Mayor, M., & Queloz, D. 1995, , 378, 355. Muirhead, P. S., et al. 2012, , 747, 144. Muterspaugh, M. W., et al. 2010, , 140, 1657. Neuhäeuser, R., et al. 1995, , 297, 391. Ortiz-León, G. N., et al. 2017, , 834, 141. Ortiz-Le[ó]{}n G. N., et al., 2018, , 869, L33 Panic, O. 2009, Ph.D. Thesis. Perryman, M., et al. 2014, , 797:14, 22pp. Phillips, R. B., Lonsdale, C. J., & Feigelson, E. D. 1991, , 382, 261. Preibisch, T., et al. 2005, , 160, 401. Rodr[í]{}guez L. F., Zapata L. A., Palau A., 2017, , 153, 209 Sahlmann, J., et al. 2013, , 556, A133. 10pp. Sahlmann, J., et al. 2016, å, 595, A77, 16pp. Sallum, S., et al. 2015, , 527, 342. Stelzer, B., et al. 2000, , 356, 949. Scargle, J. D., 1982, ApJ, 263, 835 Seidelman, P. K. (ed.) 1992, Explanatory Supplement to the Astronomical Almanac (Nautical Almanac Office, U.S. Naval Observatory/H. M. Nautical Almanac Office, Royal Greenwich Observatory, et al.) (Mill Valley, CA: University Science Books) Sozzetti, A., et al. 2014, , 437, 497. Van der Marel, N., et al. 2016, , 592, A126. Wall, J. V. & Jenkins, C. R. 2003, Practical Statistics for Astronomers (Cambridge: Cambridge Univ. Press). Wolszczan, A., & Frail, D. A. 1992, , 355, 145. Yu, L., et al. 2017, , 467, 1342.
--- abstract: 'This paper is a pedagogical yet critical introduction to the quantum description of unstable systems, mostly at the level of a graduate quantum mechanics course. Quantum decays appear in many different fields of physics, and their description beyond the exponential approximation is the source of technical and conceptual challenges. In this article, we present both general methods that can be adapted to a large class of problems, and specific elementary models to describe phenomena like photo-emission, beta emission and tunneling-induced decays. We pay particular attention to the emergence of exponential decay; we analyze the approximations that justify it, and we present criteria for its breakdown. We also present a detailed model for non-exponential decays due to resonance, and an elementary model describing decays in terms of particle-detection probabilities. We argue that the traditional methods for treating decays face significant problems outside the regime of exponential decay, and that the exploration of novel regimes of current interest requires new tools.' author: - \| Charis Anastopoulos[^1]\ [Department of Physics, University of Patras, 26500 Greece]{} title: Decays of unstable quantum systems --- Introduction ============ Key notions ----------- A decay of a particle $A$ is a process of the type $A \rightarrow B_1 + B_2 + \ldots B_n$, where the particles $B_i$ are the decay products. Decays are ubiquitous in physics. Examples include the emission of photons from excited atoms or nuclei, alpha and beta emission, decays of composite subatomic particles (for example, neutrons or pions) and decays of elementary particles (for example, a muon decaying to one electron and two neutrinos). Most decays follow an exponential law. The probability that a decay takes place within the time-interval $[t, t+ \delta t]$, for $t > 0 $, equals $p(t) \delta t$, where the probability density $p(t)$ is given by $$\begin{aligned} p(t) = \Gamma e^{- \Gamma t}. \label{probdec}\end{aligned}$$ The [decay constant]{} $\Gamma$ is positive and has dimensions of inverse time. We note that for any two instants of time $t_1$ and $t_2$, $$\begin{aligned} p(t_2) = p(t_1) e^{-\Gamma(t_2-t_1)},\end{aligned}$$ Hence, the exponential law remains invariant under a shift of the initial moment of time $t = 0$. The decay of an ensemble of $A$ particles after a moment of time $t$ carries no memory of any properties prion to $t$. In classical probability theory, exponential decays correspond to [Markovian processes]{}. Experiments typically involve a large number of decaying particles with identical preparation and the detection of some of the decay products. Recording the number of detection events within given time intervals $[t, t+\delta t]$, we can reconstruct the probability density $p(t)$ associated to the decay. Hence, $p(t)$ is a [directly observable quantity]{}. The focus of this paper is the derivation of the probability density $p(t)$ from the rules of quantum mechanics. We present different approaches to the problem, we apply them to specific physical problems and we analyze their underlying assumptions and their limitations. The quantum description of decays --------------------------------- The aim of the quantum mechanical description of a decay process is to construct the probability density $p(t)$ from first principles. This probability density is different from the ones usually considered in quantum theory, because the random variable $t$ is temporal. We cannot use Born’s rule, because there is no self-adjoint operator for time in quantum theory [@Pauli]. Indeed, the construction of quantum probabilities in which time is a random variable is an old problem—for reviews, see, Ref. [@ToAbooks]. There are several different approaches that lead to different results, even for elementary problems, for example, constructing probabilities for the time of arrival [@ML] or specifying the time it takes a particle to tunnel through a potential barrier [@tunt]. Most studies of decays avoid a direct construction of $p(t)$. Instead they focus on a slightly different issue, namely, on finding the probability that the quantum system persists in its initial configuration. In quantum theory, the notion of a configuration refers to the set of all quantum states compatible with a specific property. Mathematically, it corresponds to a subspace of the system’s Hilbert space, and it is represented by the associated projection operator. For example, if the defining property of the initial configuration is that a particle is confined by a potential well in a spatial region $U$, the relevant subspace corresponds to the projection operator $\hat{P}_U = \int_U dx \|x\rangle \langle x\|$ that describes the property $"x \in U"$ for the particle position $x$. In some unstable systems, the subspace of the initial configuration is one-dimensional, and it coincides with the quantum state $\|\psi\rangle$ at which the system has been initially prepared. For such systems, it is convenient to employ the [persistence amplitude]{} (or [survival amplitude]{}) $$\begin{aligned} {\cal A}_{\psi}(t) = \langle \psi\|e^{-i\hat{H}t}\|\psi\rangle. \label{persiss}\end{aligned}$$ where $\hat{H}$ is the Hamiltonian of the system—for the mathematical properties of the persistence amplitude, see Refs. [@GhiFo; @Peres80]. The modulus square of ${\cal A}_{\psi}(t)$ is the [persistence probability]{} (or [survival probability]{}), i.e., the probability a system prepared at the state $\|\psi\rangle$ at $t = 0$ will still be found at $\|\psi\rangle$ by a measurement at a later time $t$. The persistence probability is identical with the [fidelity]{} between the initial state and its time-evolution. It is then suggested that the decay probability density be defined as $$\begin{aligned} p(t) = - \frac{d}{dt} \|{\cal A}_{\psi}(t)\|^2, \label{decprob2}\end{aligned}$$ i.e., the decay is assumed to happen when the system leaves $\|\psi\rangle$. Eq. (\[decprob2\]) is a reasonable candidate probability density provided that (i) all states normal to $\|\psi\rangle$ that can be reached via Hamiltonian evolution correspond to decay products, and (ii) the probability of the reverse process to the decay is negligible. If condition (i) is not satisfied, then the propositions A = “the system left $\|\psi\rangle$” and B = “the decay happened” are not identical: B implies A, but A does not imply B. Hence, Eq. (\[decprob2\]) cannot be identified with the decay probability. For example, if $\|\psi\rangle$ evolves to a different state $\|\phi\rangle$ that describes the initial particle $A$, a part of the persistence amplitude will describe Rabi-type oscillations between $\|\psi\rangle $ and $\|\phi \rangle$. The persistence probability may not be a decreasing function of $t$, and, hence, $p(t)$ may take negative values. It will not be a genuine probability density: Eq. (\[decprob2\]) will not predict the number of particles detected at each moment of time. Condition (ii) is satisfied if there are many more states available to the decay products than to the initial particle. Furthermore, the configuration of the experiment must be such as to allow the decay products to ‘explore’ their available states. This means that the decay products leave the locus of their production and they are measured far away. Consider for example an excited atom contained in a cavity. For some cavity geometries, the emitted photon does not exit the cavity immediately, and it may be reabsorbed by the atom at a later time. The persistence amplitude of the excited atomic state may has an oscillating component. In this case, the candidate probability density (\[decprob2\]) does not correlate with photodetection records. For systems that satisfy conditions (i) and (ii), Eq. (\[decprob2\]) provides a reasonable construction of the decay probability density. It works best in model systems that incorporate the conditions (i) and (ii) above into their definition. On such example is the Lee model [@Lee] that is presented in Sec. 3. However, even in such models the candidate probability density (\[persiss\]) may become negative outside the exponential decay regime—see, Sec. 3.1.3. We note that the interpretation of the persistence probability as the fidelity of the initial state makes it a useful tool also for the study of a broader class of physical phenomena than decays [@BPSZ]. Hence, the quantity (\[decprob2\]) may be of physical interest, even if it takes negative values and cannot be identified with the probability density for decay. [Probability currents.]{} An alternative elementary description of decays is available, whenever we can associate a probability-current operator $\hat{\pmb J}({\pmb x}, t)$ to one of the decay products. For example, for non-relativistic particles of mass $m$ satisfying Schrödinger’s equation with Hamiltonian $\hat{H}$, the current operator is $$\begin{aligned} \hat{\pmb J}({\pmb x}, t) = \frac{1}{2m}e^{i\hat{H} t}\left[ \hat{\pmb p} \delta^3(\hat{\pmb x} - {\pmb x}) + \delta^3(\hat{ \pmb x} - {\pmb x}) \hat{\pmb p} \right] e^{-i\hat{H}t},\end{aligned}$$ where $\hat{\pmb x}$ and $\hat{\pmb p}$ are the standard position and momentum operators, respectively. The expectation value of $\hat{\pmb J}({\pmb x}, t)$ on a state $\|\psi\rangle $ reproduces the standard expression for the probability current $\frac{1}{m}[\mbox{Im} \psi^({\pmb x}, t) {\pmb \nabla} \psi({\pmb x}, t)]$ associated to the solution $\psi({\pmb x}, t) := \langle {\pmb x}\|e^{-i \hat{H}t}\|\psi\rangle$ of Schrödinger’s equation. Given a current operator $\hat{\pmb J}({\pmb x}, t)$ for one of the decay products, we can evaluate the flux $\Phi_C(t) = \int_C d^2{\pmb \sigma} \cdot \langle \psi\| \hat{\pmb J}({\pmb x}, t)\|\psi\rangle$ through any surface $C$. In many set-ups, the probability flux coincides with the particle flux through this surface, which is a directly measurable quantity. Then, the flux $\Phi_S(t)$ over a two-sphere surrounding the unstable system is expected to be proportional to the decay probability density $p(t)$. The main limitation of this method is that it cannot be consistently applied to relativistic systems, because of difficulties in defining probability currents that are both Lorentz covariant and causal [@RoHo]. For this reason, it has mainly been used only in non-relativistic settings, in particular, for decays that can be described in terms of tunneling [@LL; @Perel]. Another problem of the probability current method is the possibility of back-flow: the current operator has negative eigenvalues even for states with strictly positive momentum [@Brack]. Hence, in some cases the probability flux may turn out to be negative for some times $t$–especially when the current is evaluated near the locus of the decay [@Winter]. In such cases the flux is not a reliable measure of detected outgoing particles. [Detector models.]{} A more rigorous description of quantum decays requires the incorporation of the measurement process [@EkSi; @GhiFo]. In fact, the idea that quantum decays cannot be explained solely in terms of unitary time evolution was crucial to the early interpretational discussions of quantum theory [@Heismem]. We have to take into account the irreversibility of the quantum measurements on the decay products in order to avoid theoretical discrepancies. This is the case, for example, in systems characterized by competing decay channels. If one ignores the effects of the apparatus, the persistence amplitude exhibits large oscillations, thereby contradicting the standard classical description of sequential decays [@FoGhi]. In quantum measurement theory, the consistent treatment of the measuring apparatus allows of the description of quantum observables in terms of Positive Operator Valued Measures (POVM), i.e., a set of positive operators $\hat{\Pi}(a)$, where $a$ are the values of the physical magnitude recorded by the apparatus. Then, given an initial state $\hat{\rho}$ of the system, the probability that the result $a$ is obtained is given by $Tr\left[\hat{\rho}_0 \hat{\Pi}(a)\right]$. In recent years, a new class of model for particle detection has been developed, allowing for the construction of quantum observables for the time $t$ of a detection event [@AnSav]—see, also [@An08] for an early application to the decay problem. In this paper, we present a study of decays, using one such observable $\hat{\Pi}(t)$ for detection time that can be obtained by elementary arguments similar to the ones of standard photodetection theory [@Glauber]. This paper ---------- The aim of this paper is to provide the reader with tools for addressing a large class of decay problems. We present the most common approximation schemes, as well as criteria for checking when they fail. We prefer to work with models that admit simple solutions and controlled approximations, but we introduce more complex models for the explicit demonstration of important physical issues. A key motivation for this paper is to provide explicit arguments that the traditional methods for describing decays (persistence amplitude and probability current) are inapplicable for many problems of current physical interest. These methods work excellently for exponential decays, but they lead to negative values of the decay probability $p(t)$ outside the exponential regime. The problem is not a failure of some approximation, the very definition of such methods cannot guarantee positivity. We believe that a first-principles construction of decay probabilities is essential for describing decays in novel regimes, for example, in entangled systems [@AnHu2; @CVS17; @C18], attosecond tunneling ionization [@atto] or decay oscillations in nuclear physics [@GSI]. The paper is organized as follows. Sec. 2 presents the simplest method for the study of decays, namely, the evaluation of the persistence amplitude as a line integral on the complex energy space. This method is particularly suitable for perturbative decays, i.e., systems in which the decay originates from a small term in the Hamiltonian of the system. It also applies to relativistic systems described by quantum field theory. We show that exponential decay is common in perturbative decays: it originates from the separation of energy scales in the persistence amplitude. Nonetheless, it is neither exact nor generic. Exponential decay fails at very early and very late times, and also if the energy of the initial state is close to a resonance of the system. In Sec. 3, we present the Lee model that describes the decaying particle as a two-level system. This model can easily be adapted to problems in different branches of physics, including high energy physics, nuclear physics, atom optics, and condensed matter. Here, we present its applications to photo-emission and beta decay. We also employ the model in order to demonstrate the breakdown of the persistence amplitude method outside the exponential decay regime. Sec. 4 describes the decays of an atom in a cavity, providing an explicit example of decays that are non-exponential at all times because of resonance. In Sec. 5, we study non-perturbative decays due to quantum tunneling, using the probability current method. We show that exponential decays originate from a decoherence condition, namely the lack of interferences between different attempts of the particle to tunnel through the barrier. In Sec. 6, we revisit the Lee model using probabilities constructed from a simple temporal observable, analogous to the one used in photodetection theory. We show that the detection probability reproduces the results of the persistence amplitude method, but can also be used in regimes where the latter fails. The material in this article is mostly at the level of a graduate course in quantum mechanics. Familiarity with quantum many-particle systems is presupposed. Secs. 2, 3 and 5 provide a minimal introduction to quantum decays, explaining the emergence of the exponential decay law and its limitations, developing methods that can be applied to different branches of physics, and presenting some important physical examples. The Appendix contains a sketch of additional results that can be used as exercises. Perturbative evaluation of the persistence amplitude ==================================================== In this section, we study decays in which the persistence amplitude can be evaluated perturbatively. All physical properties of the decays are encoded into a function defined on the complex energy plane, the [self-energy function]{}. We identify the conditions under which the exponential law emerges, and we identify regimes for non-exponential decays. Preliminaries -------------- By definition, the initial state $\|\psi\rangle$ of an unstable system is not an eigenstate of the Hamiltonian $\hat{H}$. However, in many problems of physical interest, $\|\psi\rangle$ is close to an eigenstate of $\hat{H}$, in the sense that $\hat{H}$ is of the form $$\begin{aligned} \hat{H} = \hat{H}_0 + \hat{V}, \label{Hamper}\end{aligned}$$ where $\|\psi\rangle$ is an eigenstate of $\hat{H}_0$ and $\hat{V}$ is a small perturbation. For example, $\hat{H}_0$ may be the Hamiltonian of an atom, and $\hat{V}$ the interaction Hamiltonian of the atom with the quantum electromagnetic field. In this case, we evaluate the persistence amplitude perturbatively. Let us denote by $\|b\rangle$ the eigenstates of $\hat{H}_0$ and by $E_b$ the corresponding eigenvalues. Then, $$\begin{aligned} \hat{H}_0 \|b\rangle = E_b\|b\rangle.\end{aligned}$$ We will take the initial state $\|\psi\rangle$ to be one of the eigenstates, say $\|a\rangle$, and we will write $V_{ab} := \langle a\|\hat{V}\|b\rangle$. Without loss of generality, we assume that $\langle a\|\hat{V}\|a\rangle =0$. If $\langle a\|\hat{V}\|a\rangle \neq 0$, we can always redefine $$\begin{aligned} \hat{H}_0' = \hat{H}_0 + \sum_a \langle a\|\hat{V}\|a\rangle \|a\rangle \langle a\| \hspace{1cm} \hat{V}' = \hat{V} - \sum_a \langle a\|\hat{V}\|a\rangle \|a\rangle \langle a\|, \end{aligned}$$ so that $\|a\rangle$ is an eigenstate of $\hat{H}_0'$ and $\langle a\|\hat{V}'\|a\rangle = 0$. [The resolvent.]{} The perturbative calculation of the persistence amplitude is simplified by using the [resolvent]{} associated to the Hamiltonian operator: $(z - \hat{H})^{-1}$, for $z \in {\pmb C}$. We will often write the resolvent as $\frac{1}{z - \hat{H}}$. The resolvent is related to the evolution operator $e^{-i\hat{H}t}$ by $$\begin{aligned} e^{-i\hat{H}t} = \lim_{\epsilon \rightarrow 0} \frac{i}{2\pi } \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E+ i \epsilon - \hat{H}}, \label{propagator2}\end{aligned}$$ where $\epsilon > 0$ and $t > 0$. Eq. (\[propagator2\]) follows from the identity $$\begin{aligned} e^{- i \omega t} = \lim_{\epsilon \rightarrow 0} \frac{i}{2\pi } \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E+ i \epsilon - \omega}. \label{idtycmplx}\end{aligned}$$ To prove Eq. (\[idtycmplx\]) we evaluate the line integral $$\begin{aligned} I(t) = \oint_{C_+} dz \frac{e^{-izt}}{z - \omega}, \label{cont}\end{aligned}$$ along the negative-oriented contour $C_+$ of Fig. 1. At the lower half of the imaginary plane $Im z = - y$, for $y >0$. Therefore, the integrand along the semicircle is proportional to $e^{-yt}$ and vanishes as the radius of the semi-circle tends to infinity. Hence, $$\begin{aligned} I(t) = \lim_{\epsilon \rightarrow 0} \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E+ i \epsilon - \omega}.\end{aligned}$$ The contour $C_+$ includes the single pole of the integrand (\[cont\]), at $z = \omega$. Using Cauchy’s residue theorem, we arrive at Eq. (\[idtycmplx\]). By integrating along the contour $C_-$, we can similarly prove that $$\begin{aligned} \lim_{\epsilon \rightarrow 0} \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E - i \epsilon - \hat{H}} = 0 , \label{propagator2b}\end{aligned}$$ for $\epsilon > 0$ and $t > 0$. The integral vanishes because the contour $C_-$ does not enclose any pole. ![ Integration contours $C_{\pm}$ for calculating the integrals (\[idtycmplx\]) and (\[propagator2b\]). The integrals are evaluated at the limit where the radius $r$ of the semi-circle goes to infinity. Both curves are traversed clock-wise and therefore have negative orientation.](complexcurve){height="6cm"} A crucial property of the resolvent is that it can be expanded in a perturbative series. For a Hamiltonian $\hat{H}$ of the form (\[Hamper\]), $$\begin{aligned} (z - \hat{H})^{-1} = (z - \hat{H}_0 - \hat{V})^{-1} = [(z-\hat{H}_0) (1 - (z-\hat{H}_0)^{-1}\hat{V})]^{-1}\nonumber \\ = (1 - (z-\hat{H}_0)^{-1}\hat{V})^{-1}(z-\hat{H}_0)^{-1}. \nonumber \end{aligned}$$ Using the geometric series formula $(1 - \hat{A})^{-1} = \sum_{n=0}^{\infty} \hat{A}^n$, we obtain $$\begin{aligned} \frac{1}{z - \hat{H}} = \frac{1}{z - \hat{H}_0} + \frac{1}{z - \hat{H}_0} \hat{V} \frac{1}{z - \hat{H}_0} + \frac{1}{z - \hat{H}_0} \hat{V}\frac{1}{z - \hat{H}_0} \hat{V} \frac{1}{z - \hat{H}_0} + \ldots \label{seriesprop}\end{aligned}$$ The random phase approximation ------------------------------ We construct the persistence amplitude ${\cal A}_a(t)$ for an initial state $\|a\rangle$ that is an eigenstate of $\hat{H}_0$. By Eq. (\[propagator2\]), $$\begin{aligned} {\cal A}_a(t) = \lim_{\epsilon \rightarrow 0} \frac{i}{2\pi } \int_{-\infty }^{\infty } dE e^{-iEt} G_a(E+i\epsilon) , \label{ampll} \end{aligned}$$ where $$\begin{aligned} G_a(z) = \langle a\|(z - \hat{H})^{-1}\|a\rangle. \end{aligned}$$ We evaluate $G_a(z)$ using the perturbative series (\[seriesprop\]). We assume that $\hat{V}$ is of first order to some dimensionless parameter $\lambda << 1$. The zeroth-order contribution to $G_a(z)$ is $ \frac{1}{z - E_a}$. The first-order contribution is $ \frac{1}{(z - E_a)^2} \langle a\|\hat{V}\|a\rangle$ = 0. The second-order term is $ \frac{1}{(z - E_a)^2} \langle a\|\hat{V} \frac{1}{z - \hat{H}_0} \hat{V}\|a\rangle$. Writing $ \frac{1}{z - \hat{H}_0} = \sum_b \frac{1}{z - E_b} \|b\rangle \langle b\|$, this term becomes $ \frac{1}{(z - E_a)^2} \Sigma_a(z) $, where $$\begin{aligned} \Sigma_a(z) = \sum_b \frac{\|V_{ab}\|^2}{z - E_b}, \label{sigmaa}\end{aligned}$$ is the [self-energy]{} function of the state $\|a\rangle$. The third-order term is $$\begin{aligned} \sum_{bc} \frac{1}{(z-E_a)^2} \frac{ V_{ab} V_{bc} V_{ca}}{(z-E_b)(z-E_c)}, \nonumber\end{aligned}$$ and the fourth-order term is $$\begin{aligned} \sum_{bcd} \frac{1}{(z-E_a)^2} \frac{ V_{ab} V_{bc} V_{cd}V_{da}}{(z-E_b)(z-E_c)(z-E_d)}. \nonumber\end{aligned}$$ The procedure can be continued ad infinitum. We assume that the Hamiltonian $\hat{H}_0$ has continuous spectrum for $E > \mu$, for some parameter $\mu$. The continuous spectrum corresponds to the kinetic energy of the decay products. With this assumption, we invoke the [Random Phase Approximation]{} (RPA), according to which $$\begin{aligned} \sum_c \frac{V_{ac}V_{cb}}{z-E_c} \simeq \delta_{ab} \Sigma_a(z). \label{rpa}\end{aligned}$$ The reasoning for Eq. (\[rpa\]) is the following. Suppose that the system is contained in a box of volume $V$ with periodic boundary conditions and that it contains a large number $N$ of degrees of freedom. The RPA is the assumption that the phases of the matrix elements $\langle a\| \hat{V}\|b\rangle$, for $b \neq a$ are [randomized]{} in the continuous limit, i.e., in the limit where $N$ and $V$ goes to infinity, with $N/V$ constant. By ‘randomized’, we mean that the phases of $\langle a\| \hat{V}\|b\rangle$ do not exhibit any periodicity, or quasi-periodicity as $b$ varies. Hence, the summation over $b$ is a sum of many random phases, and is expected to be much smaller than the term for $a = b$ that involves no such phases. Eq. (\[rpa\]) then follows. The RPA was initially postulated in systems with a large but finite number of degrees for freedom, in condensed matter [@BohmPi; @GeBr] and in nuclear physics [@Bloch; @Nami]. However, it also applies to systems with an infinite number of degrees of freedom, i.e., quantum fields. Indeed, the name ‘self-energy’ for the function (\[sigmaa\]) originates from quantum field theory. By the RPA, odd-order terms in the expansion of $G_a(z)$ vanish. Furthermore, the terms of order $2n$ for $n$ integer equal $\Sigma_a(z)^n/(z-E_a)^{n+1}$. Hence, $G_a(z)$ is give by a geometric series, $$\begin{aligned} G_a(z) = \frac{1}{z-E_a} \sum_{n=0}^{\infty} \frac{\Sigma_a(z)^n}{(z-E_a)^n} = \frac{1}{z-E_a} \frac{1}{1 - \frac{\Sigma_a(z)}{z-E_a}} = \frac{1}{z - E_a - \Sigma_a(z)}. \label{asas}\end{aligned}$$ Eq. (\[asas\]) is accurate to order $\lambda^2$. Hence, it can be obtained without the RPA, solely by a perturbative analysis. Most treatments of RPA assume a weaker condition that Eq. (\[rpa\]), so that the resulting expression for $G_a(z)$ involves a self-energy function that coincides with Eq. (\[sigmaa\]) only to second order in $\lambda$—for details, see, Ref. [@NNP96]. In the present treatment, the RPA gives the same results with a second-order perturbative expansion. In general, it has a larger domain of validity. Structure of the self-energy function ------------------------------------- Eqs. (\[asas\]) and (\[ampll\]) imply that $$\begin{aligned} {\cal A}_a(t) = \frac{i}{2 \pi } \lim_{\epsilon \rightarrow 0} \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E + i \epsilon- E_a - \Sigma_a(E+i\epsilon)}. \label{ampldec}\end{aligned}$$ If the self-energy function $\Sigma_a(z)$ were analytic, the integral (\[ampldec\]) could be evaluated by integrating along the contour of Fig. 1, and using Cauchy’s theorem. However, this is not the case, the self-energy function is discontinuous and it may contain poles or branch points. To see this, consider the definition (\[sigmaa\]) of $\Sigma_a(z)$. Since the spectrum of $\hat{H}_0$ is continuous for energies larger than $\mu$, $\Sigma_a(z)$ is divergent along the half-line $D = \{ z \in {\pmb C}\|\; \mbox{Re} z > \mu, \mbox{Im}z = 0\}$. Shifting the Hamiltonian by a constant term, we can always choose $\mu = 0$, so that $D = {\pmb R}^+$. To analyze the behavior of $\Sigma_a(z)$ near $D$, we define $$\begin{aligned} \Sigma_a(E^{\pm}) = \lim_{\eta \rightarrow 0} \Sigma_a(E\pm i\eta), \end{aligned}$$ for $\eta > 0$. By definition, $$\begin{aligned} \mbox{Re} \Sigma_a(E\pm i \eta) = \sum_b \frac{\|V_{ab}\|^2(E-E_b)}{(E - E_b)^2 + \eta^2}.\end{aligned}$$ It follows that $\mbox{Re} \Sigma_a(E^+) = \mbox{Re} \Sigma_a(E^-)$, leading to the definition of the [level-shift function]{} $$\begin{aligned} F_a(E) := \mbox{Re} \Sigma_a(E^{\pm}) \label{faE}\end{aligned}$$ On the other hand, the imaginary part of $\Sigma_a(z)$ $$\begin{aligned} \mbox{Im} \Sigma_a(E\pm i \eta) = \mbox{Im} \sum_b \frac{\|V_{ab}\|^2}{E - E_b + i \eta} = \mp \eta \sum_b \frac{\|V_{ab}\|^2}{(E - E_b)^2 + \eta^2}, \label{imsa}\end{aligned}$$ is discontinuous as the half-line $D$ is crossed. Eq. (\[imsa\]) implies that $\mbox{Im} \Sigma(E^-) = - \mbox{Im} \Sigma(E^+)\geq 0$. We define the [decay function]{} $$\begin{aligned} \Gamma_a(E) := 2 \mbox{Im} \Sigma(E^-) > 0, \label{gammaE}\end{aligned}$$ so that $$\begin{aligned} \Sigma_a(E^{\pm}) = F_a(E) \mp \frac{i}{2} \Gamma_a(E). \end{aligned}$$ The discontinuity of $\Sigma_a$ across $D$ is $\Delta \Sigma_a(E) := \Sigma_a(E^+) - \Sigma_a(E^-) = - i \Gamma_a(E)$. Obviously, $\Gamma_a(E) = 0$ for $E < 0$. Eq. (\[ampldec\]) becomes $$\begin{aligned} {\cal A}_a(t) = \frac{i}{2 \pi } \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E - E_a - F_a(E) + \frac{i}{2} \Gamma_a(E)}. \label{ampldecb}\end{aligned}$$ Since $\Gamma(E) = 0$ for $E < 0$, $$\begin{aligned} {\cal A}_a(t) = \frac{i}{2 \pi } \left( \int_{-\infty }^{0} \frac{dE e^{-iEt}}{E - E_a - F_a(E)}+ \int_{0 }^{\infty } \frac{dE e^{-iEt}}{E - E_a - F_a(E) + \frac{i}{2} \Gamma_a(E)} \right). \label{ampldecc}\end{aligned}$$ On the other hand, Eq. (\[propagator2b\]) implies that $$\begin{aligned} \lim_{\epsilon \rightarrow 0} \int_{-\infty }^{\infty } \frac{dE e^{-iEt}}{E - i \epsilon- E_a - \Sigma_a(E^-)} = 0,\end{aligned}$$ hence, $$\begin{aligned} \int_{-\infty }^{0 } \frac{dE e^{-iEt}}{E - E_a - F_a(E)} = - \int_{0 }^{\infty } \frac{dE e^{-iEt}}{E - E_a - F_a(E) - \frac{i}{2} \Gamma_a(E)}.\end{aligned}$$ Substituting into Eq. (\[ampldecb\]), we obtain $$\begin{aligned} {\cal A}_a(t) &=& \frac{i}{2 \pi } \int_{0}^{\infty } dE e^{-iEt} \left[ \frac{1}{E - E_a - F_a(E) + \frac{i}{2} \Gamma_a(E)} - \frac{1}{E - E_a - F_a(E) - \frac{i}{2} \Gamma_a(E)}\right] \nonumber \\ &=& \frac{1}{2 \pi } \int_{0 }^{\infty } \frac{dE \Gamma_a(E) e^{-iEt}}{ [E - E_a - F_a(E)]^2 + \frac{1}{4}[\Gamma_a(E)]^2}. \label{mainampl}\end{aligned}$$ Eq. (\[mainampl\]) is the main result of this section, an explicit formula relating the persistence amplitude to the components of the self-energy function. The Wigner-Weisskopf approximation ---------------------------------- The integral (\[mainampl\]) involves the functions $F_a(E)$ and $\Gamma_a(E)$ in the denominator. These function are second-order with respect the perturbation parameter $\lambda << 1$. If $\|F_a(E)\| << E_a$ and $\Gamma(E) << E_a$ for $E$ in the vicinity of $E_a$, we can evaluate the persistence amplitude using the [Wigner-Weisskopf Approximation]{} (WWA) [@WWA]. The WWA essentially postulates the substitution of the Lorentzian-like function of $E$ in Eq. (\[mainampl\]) with an actual Lorentzian. The justification is the following. The integral (\[mainampl\]) is dominated by values of $E$ within distance of order $\lambda^2$ from $E \simeq E_a$. For these values, the integrand is of order $\lambda^{-2}$, otherwise it is of order $\lambda^0$. Hence, with an error of order $\lambda^2$, we can substitute the energy-shift function $F_a(E)$ with the constant $$\begin{aligned} \delta E := F_a(E_a)\end{aligned}$$ and the decay function $\Gamma_a(E)$ with the constant $$\begin{aligned} \Gamma := \Gamma_a(E_a). \end{aligned}$$ Within an error of the same order of magnitude, we extend the range of integration to $(-\infty, \infty)$. Thus, we obtain an elementary integral $$\begin{aligned} {\cal A}_a(t) = \frac{\Gamma_a}{2 \pi } \int_{-\infty}^{\infty } \frac{dE e^{-iEt}}{ (E - E_a - \delta E )^2 + \frac{1}{4}\Gamma^2} = e^{ -i (E_a +\delta E ) t - \frac{\Gamma }{2}t}, \label{ampl3}\end{aligned}$$ Substituting Eq. (\[ampl3\]) in Eq. (\[decprob2\]), we conclude $$\begin{aligned} p(t) = \Gamma e^{-\Gamma t}.\end{aligned}$$ Hence, the WWA leads to an exponential decay law with a decay constant $\Gamma $ that is determined by the imaginary part of the self-energy function. The real part of the self-energy function leads to a shift $\delta E $ of the energy level $E_a$, usually referred to as the [Lamb shift]{}. The same expression for the decay constant $\Gamma $ is given by [Fermi’s golden rule]{}. To see this, we use Eq. (\[imsa\]), $$\begin{aligned} \Gamma = \lim_{\eta \rightarrow 0} \sum_b \frac{2\eta \|V_{ab}\|^2}{(E_a - E_b)^2 + \eta^2}.\end{aligned}$$ Since $\lim_{\eta \rightarrow 0} \frac{\eta}{x^2 +\eta^2} = \pi \delta(x)$, we obtain Fermi’s decay rate $$\begin{aligned} \Gamma = 2 \pi \sum_{b, E_b = E_a} \|V_{ab}\|^2. \label{fgg}\end{aligned}$$ A more rigorous derivation of Eq. (\[ampl3\]) from Eq. (\[ampldecb\]) employs the notion of the [van Hove limit]{} [@vanHo]. This limit is obtained as follows. First, we change the time variable to $\tilde{t} = \lambda^2 t$, and we define $x = (E-E_a)/\lambda^2$. Then, Eq. (\[ampldecb\]) becomes $$\begin{aligned} e^{iE_at} {\cal A}_a(t) = \frac{i}{2 \pi } \lim_{\bar{\epsilon} \rightarrow 0} \int_{-\infty }^{\infty } \frac{dx e^{-ix \tilde{t}}}{x + i \bar{\epsilon}- \tilde{F}_a(E_a+\lambda^2x ) + \frac{i}{2} \tilde{\Gamma}_a(E_a+\lambda^2x)}, \label{aaavH}\end{aligned}$$ where $\tilde{\Gamma} (E) = \lambda^{-2}\Gamma(E)$ and $\tilde{F} (E) = \lambda^{-2}F (E)$ are of order $\lambda^0$; $\bar{\epsilon} = \epsilon/\lambda^2$ can still be chosen arbitrarily small. The van-Hove limit consists of taking the limit $\lambda \rightarrow 0$ in the r.h.s. of Eq. (\[aaavH\]), while keeping $\tilde{t}$ constant. Then, $$\begin{aligned} e^{iE_at} {\cal A}_a(t) = \frac{i}{2 \pi } \int_{-\infty }^{\infty } \frac{dx e^{-ix \tilde{t}}}{x - \tilde{F}_a(E_a) + \frac{i}{2} \tilde{\Gamma}_a(E_a)}, \label{aaavH2}\end{aligned}$$ Eq. (\[aaavH2\]) can be straightforwardly evaluated using the contour integral of Fig. 1. It leads to Eq. (\[ampl3\]). Hence, the WWA is equivalent to the imposition of the van Hove limit on the decay amplitude. While the van Hove limit of the persistence amplitude is always well defined, we have to keep in mind that in physical systems $\lambda^2$ is always finite and non-zero. In specific systems, the van Hove limit may misrepresent the form of the persistence amplitude. This is the case if $\Gamma_a(E)$ strongly varies with $E$ within a distance of order $\lambda^2$ from $E_a$. Hence, taking the van Hove limit is justified only if the self-energy function is sufficiently ‘flat’ in the vicinity of $E_a$ [@BaRa]. Beyond exponential decay ------------------------- We derived exponential decay as a consequence of two approximations, the RPA and WWA. Since the RPA is redundant for a second-order approximation to the self-energy function, WWA is the only approximation that needs to be considered in the weak coupling regime. [Very early times.]{} First, we note that exponential decay cannot be valid at very early times. This is a general statement that originates from the definition (\[decprob2\]). We Taylor-expand the persistence amplitude around $t = 0$, to obtain ${\cal A}_{\psi} = 1 - i t \langle \hat{H} \rangle - \frac{t^2}{2} \langle \hat{H}^2 \rangle + \ldots$. Keeping terms up to order $t^2$, the probability density becomes $$\begin{aligned} \|{\cal A}_{\psi} \|^2 = 1 - (\Delta H)^2 t^2 + \ldots.\end{aligned}$$ Eq. (\[decprob2\]) implies that $$\begin{aligned} p(t) = (\Delta H)^2 t.\end{aligned}$$ It follows that $p(0) =0$, while in exponential decays, $p(0) = \Gamma$. This violation of exponential decay at early times is a special case of the so called [quantum Zeno effect]{} [@MiSu; @FP08], and it has been verified experimentally [@Zeno]. The early time behavior of a decaying system can also be identified by an uncertainty relation for the persistence probability, first derived by Mandelstam and Tamm [@MaTa45], see, also [@Bhatta; @Dodo]. For a system in a state $\|\psi\rangle$ and Hamiltonian $\hat{H}$, the Kennard-Robertson uncertainty relation gives, $$\begin{aligned} \Delta H \Delta A \geq \frac{1}{2} \|\langle \psi\| [\hat{H}, \hat{A}]\|\psi\rangle\|, \label{kerob} \end{aligned}$$ for any observable $\hat{A}$. For the Heisenberg evolved observable $\hat{A}(t) = e^{i\hat{H}t} \hat{A} e^{-i\hat{H}t}$, Eq. (\[kerob\]) implies the so called Mandelstam-Tamm inequality $$\begin{aligned} \Delta H \Delta A(t) \geq \frac{1}{2} \large\|\langle \psi\| \frac{\partial \hat{A}(t)}{\partial t} \|\psi\rangle \large\| \label{mata1}\end{aligned}$$ Choosing $\hat{A} = \|\psi\rangle \langle \psi\|$, the expectation $ \langle \psi\|\hat{A}(t)\|\psi\rangle$ coincides with the survival probability of $\|\psi\rangle$, which we will denote by $a(t)$. Hence, Eq. (\[mata1\]) becomes $$\begin{aligned} \frac{\|\dot{a}\|}{ \sqrt{a-a^2}} \leq 2\Delta H. \label{mata2} \end{aligned}$$ Eq. (\[mata2\]) holds for all quantum states and Hamiltonians. It is easy to verify that it is not compatible with the exponential decay law at early times. Setting $a(t) = e^{-\Gamma t}$, we find that Eq. (\[mata2\] is satisfied only if $t > \Gamma^{-1} \ln \left[ 1 + \frac{\Gamma^2}{4 (\Delta H)^2}\right]$. Assuming that $\dot{a} \leq 0$, we can integrate inequality (\[mata2\]), to obtain $$\begin{aligned} \cos^{-1} \sqrt{a(t)} \leq \Delta H t \label{mata3}\end{aligned}$$ Eq. (\[mata3\]) is satisfied trivially for $t > \frac{\pi}{2\Delta H}$; for $t < \frac{\pi}{2\Delta H}$, it implies that $a(t) \geq \cos^2(\Delta H t)$. The half-life $\tau_h$ of the state is defined by the requirement $a(\tau_h) = \frac{1}{2}$. Hence, Eq. (\[mata3\]) leads to an uncertainty relation between energy spread and half-life $$\begin{aligned} \Delta H \tau_h \geq \frac{\pi}{4},\end{aligned}$$ that applies even in regimes where exponential decay fails[^2]. [The persistence amplitude in terms of a contour integral.* ]{} In order to establish the range of validity of the WWA, we must evaluate the survival amplitude (\[ampldec\]) without approximations. To this end, we analytically continue $\Gamma_a(E)$ to the lower imaginary plane, and we define $\Sigma^-_a(z) := \Sigma_a(z)$ and $ \Sigma^+_a(z) := \Sigma_a(z) - i \Gamma_a(z)$. The functions $\Sigma^{\pm}_a(z)$ can be viewed as components of a multi-valued complex function: $\Sigma^+$ corresponds to the first Riemann sheet, and $\Sigma^-$ corresponds to the second Riemann sheet [@colth]. We define a line integral over the contour $C_-$ of Fig. 1. The contribution from the circle at infinity vanishes, hence, taking the limit $\epsilon \rightarrow 0$, we obtain $$\begin{aligned} {\cal A}_a(t) = \frac{i}{2 \pi } \oint_{C_-} dz e^{-izt} \left(\frac{1}{z-E_a - \Sigma^+_a(z)} - \frac{1}{z-E_a - \Sigma^-_a(z)} \right)+ I_a(t), \label{amplnew}\end{aligned}$$ where $I_a(t)$ is the contribution of the negative real axis[^3]. $$\begin{aligned} I_a(t) = \frac{1}{2\pi }\int_0^{\infty} \frac{dx e^{ixt} \Gamma_a(-x)}{[x+ E_a +F_a(-x)]^2 + \frac{1}{4}\Gamma_a(-x)^2}. \label{remainder}\end{aligned}$$ If $\Sigma^{\pm}_a(z)$ are meromorphic functions in the region of the complex plane enclosed by the contour $C_-$, the line integral in Eq. (\[amplnew\]) is evaluated by finding the poles of the integrand inside $C_-$. To this end, we must solve the equation $$\begin{aligned} z-E_a - \Sigma_a^{\pm}(z) = 0. \label{poleq}\end{aligned}$$ We will refer to this contribution to the survival amplitude as the [pole term]{}; we will refer to $I_a(t)$ as the [remainder term]{}. [The pole term.]{} Unless $E_a$ is very close to a point of divergence of $\Sigma_a^{\pm} $, we expect that $\|\Sigma_a(E_a^+)\|/ E_a$ is much smaller than unity. Hence, there exist a solution to Eq. (\[poleq\]) within a distance of order $\lambda^2$ from the point $z = E_a$. Setting $z = E + \lambda^2 x$, we find that $ \lambda ^2 x = \Sigma^{\pm}_a(E_x + \lambda^2x) = \Sigma_a^{\pm} (E_a) + O(\lambda^4)$. Hence, Eq. (\[poleq\]) is satisfied for $$\begin{aligned} z = E_a + F_a(E_a) \mp \frac{i}{2} \Gamma_a(E_a) + O(\lambda^4). \label{BBpole}\end{aligned}$$ The pole with the plus sign is outside $C_-$, hence, it does not contribute to the contour integral. The pole with the minus sign reproduces the result of the WWA[^4]. The associated residue is $[1 - F_a'(E_a) + \frac{i}{2}\Gamma'(E_a)]^{-1}$. Let all other solutions to Eq. (\[poleq\]) inside $C_-$ be $z = \alpha_i$ and $R_i$ the associated residues. Then, by Cauchy’s theorem, $$\begin{aligned} {\cal A}_a(t) = \frac{e^{ -i (E_a +\delta E ) t - \frac{\Gamma }{2}t}}{1- F_a'(E_a) + \frac{i}{2}\Gamma_a'(E_a)} + K_a(t) + I_a(t), \label{accuracy} \end{aligned}$$ where $K_a(t) = \sum_i R_i e^{-i\alpha_i t}$. Hence, the WWA is valid if both terms $K_a(t)$ and $I_a(t)$ are negligible. If the self-energy function is analytic in the region enclosed by the contour $C_-$, then, typically, other roots $z = \alpha_i$ to Eq. (\[poleq\]) are further away from the real axis than the perturbative root (\[BBpole\]). This means that $\|Im \alpha_i\| >> \Gamma$. Hence, $K_a(t)$ vanishes much faster than the WWA term, and can be neglected after an initial transient period. If $\Sigma^{\pm}_a(z)$ has divergences near or on the real axis, then there may exist other roots, as close to the real axis as the perturbative root (\[BBpole\]). There is no general rule here, and, in principle, a case-by-case study is required. Of particular importance is the possibility of [resonance]{}. If $E_a$ is sufficiently close to a pole of $\Sigma_a(z)$, then the condition $\|\Sigma_a(E_a^+)\|/ E_a <<1$ will not hold. Then, there is no perturbative pole, the WWA fails, and decays are likely to be non-exponential. [Late times.]{} The contribution of the pole term to the persistence amplitude drops exponentially. The remainder term $I_a(t)$ drops as an inverse power law, hence, it dominates at sufficiently late times [@Helund; @Nami2]. To see this, we change variables to $y = x t$, and write Eq. (\[remainder\]) as $$\begin{aligned} I_a(t) = \frac{1}{2\pi t}\int_0^{\infty} \frac{e^{iy} \Gamma_a(-y/t)}{[y/t + E_a +F_a(-y/t)]^2 + \frac{1}{4}\Gamma_a(-y/t)^2 }, \label{eqyb}\end{aligned}$$ As $t \rightarrow \infty $, the denominator becomes $[E_a+F_a(0)]^2 + \frac{1}{4}\Gamma_a(0)^2 = E_a^2 +O(\lambda^2)$. Hence, $$\begin{aligned} I_a(t) \simeq \frac{1}{2\pi E_a^2 t} \int_0^\infty dy e^{i y} \Gamma_a(-y/t) = \frac{1 }{2\pi E_a^2 } \int_0^\infty dx e^{i xt} \Gamma_a(-x). \label{iat3}\end{aligned}$$ The long -time limit of $I_a(t) $ is determined by the behaviour of $\Gamma_a(z)$ near zero. Let $\Gamma_a(z) \simeq A z^{n}$ as $x \rightarrow 0$, for some positive constant $A$ and integer $n$. We evaluate the integral $\int_0^\infty dx e^{i xt} (-x)^n$ for imaginary time $t = i \tau$, with $\tau >0$, to obtain $(-1)^n n!\tau^{1+n}$. We analytically continue back to $t$, to obtain $$\begin{aligned} {\cal A}_a(t)= - \frac{A n! }{2\pi E_a^2 i^{n+1} } \frac{1}{t^{n+1}}. \label{iat0}\end{aligned}$$ The persistence amplitude drops as an inverse power of $t$. Hence, in the long-time limit, decays are characterized by an inverse-power law and not an exponential one. In most systems, the inverse-power behavior appears is time-scales too large to be measurable. Nonetheless, it has been experimentally confirmed in luminescence decays of dissolved organic material [@longtime], with the exponent depending on the material. We summarize the results of our analysis.\ 1. The exponential law always fails at very short and very long times.\ 2. If the self-energy function has no divergences near or on the real axis, the exponential law is very accurate at all intermediate times.\ 3. If the self-energy function has divergences near or on the real axis, there may be corrections to exponential decay from other poles.\ 4. Exponential decay likely fails on resonance, i.e., for energies near a divergence point of the self-energy function. Lee’s model =========== In this section, we present a general model for decays that can be applied to many different physical situations. This model originates from the work of T.D. Lee [@Lee]. We shall consider two versions of the model, one where the decay is accompanied by the emission of a bosonic particle and one where the decay is accompanied by the emission of two fermions. The former describes spontaneous emission of photons, the latter describes beta decay. Bosonic emission ---------------- ### Definitions and properties We consider decays of the form $A' \rightarrow A + B$, in which the emitted bosonic particle $B$ is much lighter than the particles $A'$ and $A$. Examples of this type of decay are the following. - $A'$ is the excited state of a nucleus, or of an atom, or of a molecule, $A$ is the corresponding ground state and $B$ is a photon. - $A'$ is a heavy nucleus that decays to the nucleus $A$ and an alpha particle. - $A'$ and $A$ are baryons and $B$ is a light meson. The key idea in Lee’s model is to ignore all degrees of freedom pertaining to the motion of the heavy particles. The heavy particles are then treated as a two-level system (2LS).The ground state $\|g\rangle$ corresponds to the particle $A$ and the excited state $\|e\rangle$ to the particle $A'$. The $B$ particle is described by a bosonic Fock space ${\cal F}_B$. We denote the vacuum of ${\cal F}_B$ by $\|0\rangle$ and the creation and annihilation operators by $\hat{a}_r$ and $\hat{a}^{\dagger}_r$. The latter satisfy the canonical commutation relations $$\begin{aligned} [\hat{a}_r, \hat{a}_s] = 0, \hspace{0,3cm} [\hat{a}^{\dagger}_r, \hat{a}^{\dagger}_s] = 0, \hspace{0,3cm} [\hat{a}_r, \hat{a}^{\dagger}_s] = \delta_{rs}. \end{aligned}$$ The indices $r, s, \ldots$ denote the possible states of a single $B$ particle, i.e., they label a (generalized) basis on the associated single-particle Hilbert space. The Hilbert space of the total system is ${\pmb C}^2 \otimes {\cal F}_B$. The Hamiltonian $\hat{H}_L$ of Lee’s model consists of three terms $$\begin{aligned} \hat{H}_L = \hat{H}_A + \hat{H}_B + \hat{V} ,\end{aligned}$$ where $$\begin{aligned} \hat{H}_A = \frac{1}{2} \Omega (\hat{1} + \hat{\sigma}_3) , \label{vlee1}\end{aligned}$$ is the 2LS Hamiltonian, and $\Omega$ stands for the energy difference of the two levels; $$\begin{aligned} \hat{H}_B = \sum_r \omega_r \hat{a}^{\dagger}_r \hat{a}_r, \label{vlee2}\end{aligned}$$ is the Hamiltonian for non-interacting particles $B$ particles; $$\begin{aligned} \hat{V} = \sum_r \left(g_r \hat{\sigma}_+ \hat{a}_r + g^_r \hat{\sigma}_- \hat{a}_r^{\dagger} \right), \label{vlee3}\end{aligned}$$ is the interaction term. It describes the excitation of the 2LS accompanied by the absorption of a B particle, and the decay of the 2LS accompanied by the emission of a $B$ particle. The coefficients $g_r$ depend upon the physical system under consideration. The initial state for Lee’s model is $$\begin{aligned} \|A'\rangle = \|e\rangle \otimes \|0\rangle,\end{aligned}$$ i.e., the 2LS is excited and no $B$-particle is present. We evaluate the survival amplitude ${\cal A}(t) := \langle A' \|e^{-i\hat{H}_L t}\|A'\rangle$ using the series (\[seriesprop\]) for $\hat{H}_0 = \hat{H}_A + \hat{H}_B$. We find that $$\begin{aligned} (z-\hat{H}_0)^{-1}\hat{V}(z-\hat{H}_0)^{-1}\|A'\rangle = \frac{1}{z - \Omega} \|g\rangle \otimes \sum_r \frac{g_r}{z- \omega_r} \hat{a}^{\dagger}_r\|0\rangle,\end{aligned}$$ hence, the matrix elements $\langle \psi\|\hat{V}\|A'\rangle$ are non-zero only for $\|\psi\rangle$ are of the form $\|g\rangle \otimes \hat{a}^{\dagger}_r\|0\rangle$. In particular, $\langle A'\|\hat{V}\|A'\rangle = 0$. The next term in the perturbative series is $$\begin{aligned} (z-\hat{H}_0)^{-1}\hat{V} (z - \hat{H}_0)^{-1} \hat{V}(z - \hat{H}_0)^{-1}\|A'\rangle = \frac{\Sigma(z)}{(z- \Omega)^2} \|A'\rangle, \label{leetwo}\end{aligned}$$ where $$\begin{aligned} \Sigma(z) = \sum_r \frac{\|g_r\|^2}{z-\omega_r}, \label{selee}\end{aligned}$$ is the self-energy function for the initial state $\|A'\rangle$. Since the second-order term is proportional to the zero-th order one, the above expressions are reproduced to all orders of perturbation. We obtain $$\begin{aligned} [(z - \hat{H}_0)^{-1}\hat{V}]^{2n}(z - \hat{H}_0)^{-1} \|A'\rangle = \frac{1}{z- \Omega} \left(\frac{ \Sigma(z)}{z- \Omega}\right)^n\|A'\rangle \nonumber\end{aligned}$$ $$\begin{aligned} [(z - \hat{H}_0)^{-1}\hat{V}]^{2n+1} (z - \hat{H}_0)^{-1} \|A'\rangle = \frac{1}{z- \Omega} \left(\frac{ \Sigma(z)}{z- \Omega}\right)^n\|g\rangle \otimes \sum_r \frac{g_r \hat{a}^{\dagger}_r}{z- \omega_r } \|0\rangle. \nonumber\end{aligned}$$ Hence, $$\begin{aligned} \frac{1}{z -\hat{H}}\|A'\rangle = \frac{1}{z-\Omega} \sum_{n=0}^{\infty} \frac{\Sigma(z)^n}{(z-\Omega)^n} \left(\|A'\rangle + \|g\rangle \otimes \sum_r \frac{g_r \hat{a}^{\dagger}_r}{z- \omega_r} \|0\rangle\right)\nonumber \\ = \frac{1}{z - \Omega - \Sigma(z)}\left(\|A'\rangle + \|g\rangle \otimes \sum_r \frac{g_r \hat{a}^{\dagger}_r}{z- \omega_r} \|0\rangle\right). \label{decay555}\end{aligned}$$ We conclude that $$\begin{aligned} G(z) = \frac{1}{z - \Omega - \Sigma(z)}. \label {asas2}\end{aligned}$$ We obtained Eq. (\[asas\]) by resumming the full perturbative series (\[seriesprop\]), without any approximation. This means that Lee’s model incorporates the RPA in its definition. ### Spontaneous emission from atoms We will evaluate the self-energy function in a simple model where the emitting particles have zero spin and zero rest mass, i.e., scalar photons [@AnHu]. This model ignores the effects of polarization, but otherwise it describes well the emission of photons by excited atoms. The inclusion of polarization changes $\Sigma(z)$ only by a multiplicative factor that can be absorbed in a redefinition of the coupling constant. In this model, the basis $r$ corresponds to photon momenta ${\pmb k}$, $\omega_r$ corresponds to $\omega_{\pmb k } = \|{\pmb k}\|$, and the summation over $r$ corresponds to integration with measure $\frac{d^3k}{(2\pi)^3}$. If the size $a_0$ of the emitting atom is much smaller than the wavelengths of the emitted radiation, we can describe the atom-radiation interaction in the dipole approximation [@QO]. Then, the coupling coefficients are $g_{\pmb k} = \frac{\lambda }{\sqrt{\omega_{\pmb k}}}e^{i {\pmb k}\cdot{\pmb r}}$, where $\lambda $ is a dimensionless constant and ${\pmb r}$ is the position vector of the atom[^5]. We substitute into Eq. (\[selee\]) for the self-energy function, to obtain $$\begin{aligned} \Sigma(z) = \frac{\lambda^2}{2 \pi^2} \int_0^{\infty} \frac{k dk}{z - k}. \label{seel4}\end{aligned}$$ The integral in Eq. (\[seel4\]) diverges as $k \rightarrow \infty$. However, photon energies much larger than $a_0^{-1}$, where $a_0$ is the size of the atom, are not physically relevant. Hence, we regularize the integral (\[seel4\]) by introducing a high-frequency cut-off $\Lambda >> \Omega$, $$\begin{aligned} \Sigma (z) = \frac{\lambda^2}{2 \pi^2} \int_0^{\Lambda} \frac{k dk}{z-k} = - \frac{\lambda^2}{2 \pi^2} \left[ \Lambda + z (\ln(\Lambda - z) - \ln (-z)) \right]. \label{seel5} \end{aligned}$$ There are many other ways to regularize the integral (\[seel4\]), for example, by inserting an exponential cut-off function $e^{-k/\Lambda}$. The choice of regularization does not affect the form of $\Sigma(z)$ for the physical range of values of $z$, i.e., $\|z\|<< \Lambda$. However, it introduces an arbitrariness in the behaviour of $\Sigma(z)$ for $z$ of the order of $\Lambda$. In particular, the apparent branch-point at $z = \Lambda$ in Eq. (\[seel5\]) is a artefact of the regularization. As far as the physically relevant values of $z$ are concerned, there is no error in substituting $\ln(\Lambda - z) \simeq \ln \Lambda$ in Eq. (\[seel5\]). Furthermore, it is convenient to absorb the constant term in $\Sigma(z)$ into a redefinition of the frequency $\tilde{\Omega} = \Omega - \frac{\lambda^2\Lambda}{2 \pi^2}$. With these modifications, Eq. (\[seel5\]) becomes $$\begin{aligned} \Sigma (z) = \frac{\lambda^2 }{2 \pi^2} \left[ - (\ln\Lambda) z + z \ln (-z) \right]. \label{seel6} \end{aligned}$$ The logarithm in Eq. (\[seel6\]) is defined in the principal branch, i.e., its argument lies in $(-\pi, \pi]$. When evaluating $\Sigma(E^-)$, we substitute $z = E - i \eta$, for $\eta > 0$. Hence, $\ln(-z) = \ln E + \ln[-(1-i\eta/E)]$. As $\eta \rightarrow 0$, $1-i\eta/E \simeq e^{-i\eta/E}$. We have two options for the $-1$ term in the logarithm, we can express it either as $e^{i \pi}$ or as $e^{-i \pi}$. The first choice gives $\ln (e^{i (\pi-\eta/E)})$, hence, the argument lies in the principal branch. The second choice gives $\ln (e^{i (-\pi-\eta/E)})$, and the argument lies outside the principal branch. Only the first choice is acceptable. Hence, $\ln[-(1-i\eta/E)] = i(\pi-\eta/E)$, and $\lim_{\eta \rightarrow 0} \ln[-(E-i\eta)] = \ln E + i \pi$. It follows that $$\begin{aligned} \Sigma(E^-) = - \frac{\lambda^2}{2 \pi^2} \left[ E \ln(\Lambda/E) - i \pi E ) \right].\end{aligned}$$ By Eqs. (\[faE\]) and (\[gammaE\]), $$\begin{aligned} F(E) &=& - \frac{\lambda^2}{2 \pi^2} E \ln(\Lambda/E) \\ \Gamma(E) &=& \frac{\lambda^2}{ \pi}E. \label{gammaE2}\end{aligned}$$ Hence, the decay constant in the WWA is $$\begin{aligned} \Gamma = \Gamma(\tilde{\Omega}) = \frac{\lambda^2}{ \pi}\tilde{\Omega}.\end{aligned}$$ The Lamb shift depends on the cut-off parameter $\Lambda$ and it induces a renormalization of the frequency $\tilde{\Omega}$. [Validity of exponential decay.]{} We examine the domain of validity of the WWA. First, we consider the contribution from other poles. To this end, we look for solutions to equation $$\begin{aligned} z - \tilde{\Omega} - \frac{\lambda^2}{2 \pi^2} z \ln(1 - \Lambda/z) = 0. \label{sol3} \end{aligned}$$ Eq. (\[sol3\]) admits one solution for $z$ near $\Lambda$. To see this, we write $z = \Lambda ( 1 +x)$ for $\|x\| << 1$. We obtain $x \simeq e^{-\frac{2\pi^2}{\lambda^2}} << 1$ to leading order in $\lambda^2$. This solution is possibly an artefact of the regularization, but in any case it corresponds to oscillations much more rapid than any physically relevant time scale. It averages to zero in any measurement with temporal resolution of order $\sigma_T >> \Lambda^{-1}$. We conclude that the contribution of other poles to the decay probability is negligible. Next, we evaluate the remainder term, Eq. (\[iat0\]). By Eq. (\[gammaE2\]), $A = \frac{\lambda^2}{\pi}$ and $n = 1$. Hence, $$\begin{aligned} I(t) = \frac{\Gamma}{2 \pi \tilde{\Omega}^3 t^2}. \label{asyem}\end{aligned}$$ For sufficiently large $t$ the remainder term dominates over the exponential term $e^{- \Gamma t/2}$ in the persistence amplitude. The relevant time-scale $\tau$ is found from the solution of the equation $\|I(\tau) \|= e^{- \Gamma \tau/2}$ at large $\tau$. We set $\Gamma \tau/ 2 =x$ and $\alpha = \frac{1}{8\pi} (\Gamma/\tilde{\Omega})^3$, to obtain an equation for $x$, $$\begin{aligned} 2 \ln x - x = \ln \alpha. \label{eqqq}\end{aligned}$$ Solutions to Eq. (\[eqqq\]) for different values of $\Gamma/\tilde{\Omega}$ are given in the following able. $\Gamma/\tilde{\Omega}$ $10^{-2}$ $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ $10^{-7}$ ------------------------- ------------------- -------------------- -------------------- -------------------- -------------------- ------------- x 23.3 30.8 38.1 45.4 52.5 59.8 $ e^{-\Gamma\tau}$ $5\cdot 10^{-21}$ $2 \cdot 10^{-27}$ $8 \cdot 10^{-34}$ $4 \cdot 10^{-40}$ $2 \cdot 10^{-46}$ $ 10^{-52}$ Even for values of $\Gamma/\tilde{\Omega}$ as large as $0.01$, the exponential decay law breaks down at a time where less than $1:10^{20}$ of the initial atoms remains in the excited state. We conclude that any deviations from exponential decay in photo-emission are negligible outside the quantum Zeno regime. ### Emission of a massive boson We adapt the model of Sec. 3.1.2 to describe the emission of a bosonic particle of mass $\mu$. The model is identical to that of Sec. 3.1.2, except for the form of the energy function $\omega_{\pmb k}$. We assume non-relativistic energies for the product particle, hence, $\omega_{\pmb k}= \mu + \frac{{\pmb k}^2}{2 \mu}$. This model is a variation of the one in Refs. [@Ghiold; @AA] that describes the decay of a heavy baryon to a lighter one with the emission of a pion. It allows for an analytic calculation of the persistence amplitude at all times. Hence, we can witness the transition between the exponential and the power-law regimes without worrying about the validity of specific approximation schemes. The self-energy function is $$\begin{aligned} \Sigma(z) = \frac{\lambda^2}{2 \pi^2} \int_0^{\infty} \frac{k^2 dk}{\left(\mu + \frac{k^2}{2\mu}\right) \left(z - \mu - \frac{k^2}{2\mu}\right)} = - \frac{\sqrt{2}\lambda^2\mu}{\pi^2}\int_0^{\infty} \frac{dxx^2}{(1+x^2)(x^2 - \zeta)},\end{aligned}$$ where we set $\zeta = z/\mu - 1$ and $x = k/(\sqrt{2}\mu)$. The integral over $x$ is evaluated to $\frac{\pi}{2}(1 +\sqrt{-\zeta})^{-1}$. Hence, $$\begin{aligned} \Sigma(z) = - \frac{\lambda^2\mu}{\sqrt{2}\pi} \frac{1}{1 + \sqrt{ - \frac{z - \mu}{\mu}}}. \label{sigmazint}\end{aligned}$$ In the non-relativistic regime, the relevant values of $z$ satisfy $\|z - \mu\| << \mu$, so we can approximate $(1 + \sqrt{ - \frac{z - \mu}{\mu}})^{-1}$ with $1 - \sqrt{ -\frac{z - \mu}{\mu}}$. We shift the energy of the ground state by $\mu$, so that the energy of the product particle starts at $z = 0$ rather than at $z = \mu$. We also absorb the constant $\Sigma(0)$ into $\Omega$. Then, the energy of the 2LS is $\tilde{\Omega} = \Omega - \mu - \frac{\lambda^2\mu}{\sqrt{2}\pi}$, and the self-energy function becomes $$\begin{aligned} \Sigma(z) = - \frac{\lambda^2}{\pi} \sqrt{-\frac{\mu z}{2}} . \label{sigmazin}\end{aligned}$$ For $E > 0$, the level-shift function $F(E)$ vanishes because $\Sigma(E^{\pm})$ is purely imaginary. The decay function is $$\begin{aligned} \Gamma(E) = \frac{\lambda^2}{\pi} \sqrt{2m E},\end{aligned}$$ and the decay constant $\Gamma = \frac{\lambda^2}{\pi} \sqrt{2m \tilde{\Omega}}$. The persistence amplitude (\[mainampl\]) becomes $$\begin{aligned} {\cal A}(t) = \frac{\Gamma}{2\pi} \int_0^{\infty} \frac{dE e^{-iEt} \sqrt{E/\tilde{\Omega}}}{(E - \tilde{\Omega})^2 + \frac{\Gamma^2E}{4\tilde{\Omega}}},\end{aligned}$$ where we wrote $\Gamma(E) = \Gamma \sqrt{E/\tilde{\Omega}}$. We change variables to $x = E/\tilde{\Omega}$, to obtain $$\begin{aligned} {\cal A}(t) = \frac{\sqrt{2}\gamma}{\pi} \int_0^{\infty} \frac{dx e^{-ix (\tilde{\Omega t})} \sqrt{x}}{(x-1)^2+ 2 \gamma^2 x},\end{aligned}$$ where $\gamma = \frac{\Gamma}{2\sqrt{2}\tilde{\Omega}} << 1$. The denominator is a binomial with two roots, at $x = x_{\pm}:= 1- \gamma^2 \pm i \gamma\sqrt{2-\gamma^2}$. We use the identity $$\begin{aligned} \frac{x_+-x_-}{(x-x_+)(x-x_-)} = \frac{1}{x-x_+} - \frac{1}{x-x_-},\end{aligned}$$ and change variables to $y = x \tilde{\Omega}t$, to obtain $$\begin{aligned} {\cal A}(t) = \frac{R\left( (1- \gamma^2 + i \gamma\sqrt{2-\gamma^2})\Omega t\right) - R\left( (1- \gamma^2 - i \gamma\sqrt{2-\gamma^2})\Omega t\right)}{2\pi i\sqrt{1 - \frac{1}{2} \gamma^2}\sqrt{\tilde{\Omega t}}}.\end{aligned}$$ We defined the function $$\begin{aligned} R(a) := \int_0^{\infty} \frac{dy e^{-iy}\sqrt{y}} {y - a}. \label{fai}\end{aligned}$$ For $\mbox{Re} \; a > 0$, the integral (\[fai\]) can be expressed analytically in terms of the Fresnel integrals $C(x) = \int_0^x ds \cos(s^2)$ and $S(x) = \int_0^x ds \cos(s^2)$. We find [@ASt], $$\begin{aligned} R(a) = (1-i) \sqrt{\frac{\pi }{2}}-\pi e^{-i a} \sqrt{-a}-(1+i)e^{-i a}\pi \sqrt{a}\left[C\left(\sqrt{\frac{2}{\pi }} \sqrt{a}\right)+i S\left(\sqrt{\frac{2}{\pi }} \sqrt{a}\right)\right]. \label{closedat}\end{aligned}$$ Eq. (\[closedat\]) is a closed expression for the persistence amplitude that is valid for all times. The logarithm of the persistence probability $\|{\cal A}(t)\|^2$ as a function of time is plotted in Fig. 2. The agreement with exponential decay is excellent until $t \simeq 15 \Gamma^{-1}$. The transition to power-law decay is accompanied by increasingly stronger oscillations that originate from the asymptotic behavior of the Fresnel integrals. It is important to emphasize that the oscillations of Fig. 2 signify the breakdown not only of exponential decay, but of the persistence probability method. They imply that the probability density (\[decprob2\]) takes negative values. Hence, the method does not correlate with the experimental records, namely, the number of product particles recorded at each moment of time. The result strongly suggests that the persistence probability method is not reliable outside the regime of exponential decay. ![The persistence probability as a function of $\Gamma t$ for $\gamma = 0.05$. ](per4){height="6cm"} Fermionic emission ------------------ ### The Hamiltonian Next, we consider decays of the form $A' \rightarrow A + B_1 + B_2$, in which $B_1$ and $B_2$ are fermionic particles, much lighter than $A'$ and $A$. The most important example of this type is beta decay, where $A'$ and $A$ are nuclei, $B_1$ is an electron (or positron) and $B_2$ is an anti-neutrino (or a neutrino). Again the nucleus is described as a 2LS, with a ground state $\|g \rangle$ and an excited state $\|e\rangle$. A fermionic Fock space ${\cal F}_1$ is associated to the particle $B_1$ and a fermionic Fock space ${\cal F}_2$ is associated to the particle $B_2$. The corresponding ground states are $\|0\rangle_1$ and $\|0 \rangle_2$, respectively. The creation and annihilation operators on ${\cal F}_1$ will be denoted as $\hat{c}_{r}$ and $\hat{c}_{r}^{\dagger}$, and the creation and annihilation operators on ${\cal F}_2$ as $\hat{d}_{l}$ and $\hat{d}_{l}^{\dagger}$. They satisfy the canonical anti-commutation relations $$\begin{aligned} [\hat{c}_r, \hat{c}_s]_+ = [\hat{c}_r^{\dagger}, \hat{c}^{\dagger}_s]_+ = 0, \hspace{0.4cm} [\hat{c}_r, \hat{c}^{\dagger}_s]_+ = \delta_{rs}\end{aligned}$$ $$\begin{aligned} [\hat{d}_l, \hat{d}_m]_+ = [\hat{d}_l^{\dagger}, \hat{d}^{\dagger}_m]_+ = 0, \hspace{0.4cm} [\hat{d}_l, \hat{d}^{\dagger}_m]_+ = \delta_{lm}.\end{aligned}$$ In the above, $r$ and $s$ are labels of a basis on the Hilbert space of a single $B_1$ particle; $l$ and $m$ are labels of a basis on the Hilbert space of a single $B_2$ particle. The Hilbert space of the total system is ${\pmb C}^2 \otimes {\cal F}_1 \otimes {\cal F}_2$ The Hamiltonian for this system again consists of three parts $\hat{H}_L = \hat{H}_A + \hat{H}_B + \hat{V}$, where $$\begin{aligned} \hat{H}_A &=& \frac{1}{2} \Omega (\hat{1} + \hat{\sigma}_3) , \label{vleeb1}\\ \hat{H}_B &=& \sum_r \omega_r \hat{c}^{\dagger}_r \hat{c}_r + \sum_r \tilde{\omega}_l \hat{d}^{\dagger}_l \hat{d}_l , \label{vleeb2}\\ \hat{V} &=& \sum_{r,l} \left(g_{r,l} \hat{\sigma}_+ \hat{c}_r \hat{d}_l + g_{r,l}^ \hat{\sigma}_- \hat{c}_r^{\dagger} \hat{d}_l^{\dagger} \right). \label{vleeb3}\end{aligned}$$ In Eq. (\[vleeb2\]), $\omega_r$ and $\tilde{\omega}_l$ are energy eigenvalues of a single $B_1$ and $B_2$ particle, respectively. The coefficients $g_{r, l}$ are model-dependent. The self-energy function for an initial state $\|A'\rangle = \|e\rangle \otimes \|0\rangle_1 \otimes \|0\rangle_2$ is $$\begin{aligned} \Sigma(z) = \sum_{r, l} \frac{\|g_{r,l}\|^2}{z - \omega_r - \tilde{\omega}_l}. \label{sig2f}\end{aligned}$$ ### Beta decay We consider a simplified model for beta decay, in which both emitted particles have zero spin and mass. This model is similar to the original Fermi theory of weak interactions [@Fermi; @Wilson]. The zero-mass approximation is reasonable, if the energy $\Omega$ is much larger than the masses of the emitted particles, as is often the case in beta decay. The zero spin approximation is bad; spin is important in the weak interactions. In this model, the basis $r$ corresponds to momenta ${\pmb p}$, the basis $l$ to momenta ${\pmb q}$, $\omega_r$ corresponds to $\omega_{\pmb k } = \|{\pmb p}\|$, and $\tilde{\omega}_l$ corresponds to $\tilde{\omega}_{\pmb q } = \|{\pmb q}\|$. The summation over $r$ corresponds to integration over $\frac{d^3p}{(2\pi)^3}$ and the summation over $s$ corresponds to integration over $\frac{d^3q}{(2\pi)^3}$. The appropriate constants $g_{r,l}$ for beta decay are $$\begin{aligned} g_{{\pmb p}, {\pmb q}} = \frac{1}{\mu^2}e^{i({\pmb p}+{\pmb q})\cdot{\pmb r}} \end{aligned}$$ where $\mu$ has dimensions of mass[^6] and ${\pmb r}$ is the position vector of the atom. We substitute to Eq. (\[sig2f\]), to obtain $$\begin{aligned} \Sigma(z) = \frac{1}{64 \pi^6 \mu^4} \int \frac{d^3p d^3q}{z - \|\pmb p\| - \|{\pmb q}\|} = \frac{1}{16 \pi^4 \mu^4} \int_0^{\infty}p^2dp \int_0^{\infty} q^2 dq \frac{1}{z-p-q}. \end{aligned}$$ We change the integration variables to $y = p+q, \xi = p-q$. Then, $$\begin{aligned} \Sigma(z) = \frac{1}{256 \pi^4 \mu^4} \int_0^{\infty}\frac{dy}{z-y} \int_0^y d\xi (y^2-\xi^2)^2 = \frac{1}{480\pi^4 \mu^4}\int_0^{\infty} \frac{dy y^5}{z-y}, \label{sz2f} \end{aligned}$$ The integral in Eq. (\[sz2f\]) diverges, so we introduce a high-frequency cut-off $\Lambda << \mu$ and restrict the integration over $y$ to the interval $[0, \Lambda]$. Thus, we obtain $$\begin{aligned} \Sigma(z) = - \frac{1}{240 \pi^4 \mu^4} \left[ \Lambda^5\left(\frac{1}{5} + \frac{z}{4 \Lambda} + \frac{z^2}{3\Lambda^2} + \frac{z^3}{2\Lambda^3} + \frac{z^4}{\Lambda^4}\right)+z^5 \ln \Lambda -z^5 \ln(-z) \right].\end{aligned}$$ where we approximated $\ln(\Lambda- z) \simeq \ln \Lambda$. As in Sec. 3.1.2, the branch point $z = 0$ is logarithmic. Following the same procedure, we evaluate the level-shift and decay functions $$\begin{aligned} F(E) &=& - \frac{\Lambda^5}{1200\pi^4 \mu^4} \left(1+ \frac{5E}{4 \Lambda} + \frac{5E^2}{3\Lambda^2} + \frac{5E^3}{2\Lambda^3} + \frac{5E^4}{\Lambda^4}\right) - \frac{E^5}{240 \pi^4 \mu^4} \ln\frac{\Lambda}{E} \\ \Gamma(E) &=& \frac{E^5}{120 \pi^3 \mu^4}. \label{gamma2ef}\end{aligned}$$ The beta decay rate is $$\begin{aligned} \Gamma = \Gamma(\Omega) = \frac{\Omega^5}{120 \pi^3 \mu^4}\end{aligned}$$ Resonant decays =============== In this section, we consider decays in presence of resonance. To this end, we study a variation of the spontaneous emission model of Sec. 3.1.2, in which the 2LS is within a cavity, consisting of two (infinite) parallel metal plates at distance $L$. The cavity is perfect, so the field satisfies Dirichlet boundary conditions on the plates. The model has been studied in Refs. [@AnHu; @Cum], but most of the results presented here are new. The self-energy function ------------------------ In the directions parallel to the cavity the photon momenta take continuous values. The momentum in the perpendicular direction is an integer multiple of the fundamental frequency $$\begin{aligned} \omega_0 = \frac{\pi}{L}.\end{aligned}$$ Thus, the index $r$ of the bosonic Lee model corresponds to the pair $({\pmb k}, n)$, where ${\pmb k}$ is a two-dimensional vector parallel to the plates and $n = 0, 1, 2, \ldots$. The energies are $\omega_{{\pmb k},n} = \sqrt{{\pmb k}^2 +n^2 \omega_0^2}$ and the mode summation corresponds to $\sum_{n=0}^{\infty} \int \frac{d^2k}{(2\pi)^2}$. The coupling constants have the same dependence on energy as in Sec. 3.1.2, but we express them as $$\begin{aligned} g_{{\pmb k},n} = \lambda \sqrt{\frac{\omega_0}{\omega_{{\pmb k},n}} }, \end{aligned}$$ in terms of the dimensionless constant $\lambda$. Then, the self-energy function takes the form $$\begin{aligned} \Sigma(z) = \frac{\lambda^2 \omega_0}{2\pi} \sum_{n=0}^{\infty}\int_0^{\infty} \frac{kdk}{\omega_{{\pmb k},n}(z- \omega_{{\pmb k},n})} = - \frac{\lambda^2 \omega_0}{2\pi} \sum_{n=0}^{\infty} \int_{n \omega_0 -z}^{\infty} \frac{dx}{x}, \label{szcav0}\end{aligned}$$ where in the last step we set $x = \omega_{{\pmb k},n} - z$. The integral for $\Sigma(z)$ in Eq. (\[szcav0\]) diverges at high energies. We regularize by introducing a high-energy cut-off $\Lambda$ in the integral over $x$, and a maximum integer $N = \Lambda/\omega_0$ in the summation over $n$. Then, $$\begin{aligned} \Sigma(z) &=& -\frac{\lambda^2 \omega_0}{2\pi} \left[ N \ln \frac{\Lambda}{\omega_0} - \sum_{n=0}^{N} \ln(n - z/\omega_0)\right] \nonumber \\ &=& -\frac{\lambda^2 \omega_0}{2\pi} \left[\frac{\Lambda}{\omega_0} \ln \frac{\Lambda}{\omega_0} - \ln\Gamma(N - z/\omega_0) + \ln\Gamma( -z/\omega_0) \right], \end{aligned}$$ where $\ln\Gamma(z)$ is the logarithmic gamma function. Since the physical values of $z$ are much smaller than $\Lambda$, we use the asymptotic form of the logarithmic gamma function (Stirling’s formula) $$\begin{aligned} \ln\Gamma(z) \simeq z \ln z -z, \label{stirling} \end{aligned}$$ to obtain $$\begin{aligned} \Sigma(z) = - \frac{\lambda^2\Lambda }{2\pi} - \frac{\lambda^2\log (\Lambda/\omega_0) }{2\pi}z- \frac{\lambda^2 \omega_0}{2\pi} \ln\Gamma( -z/\omega_0). \label{szcav}\end{aligned}$$ As in Sec. 3.1.2, we incorporate the constant part of $\Sigma(z)$ into a frequency redefinition: $\tilde{\Omega} = \Omega - \frac{\lambda^2\Lambda }{2\pi}$, so that $$\begin{aligned} \Sigma(z) = - \frac{\lambda^2\log (\Lambda/\omega_0) }{2\pi}z - \frac{\lambda^2 \omega_0}{2\pi} \ln\Gamma( -z/\omega_0). \label{szcav2}\end{aligned}$$ The logarithmic gamma function has no poles, but it has infinitely many branch points at all negative integers. The identity $\Gamma(z+1) = z \Gamma(z)$ implies that $$\begin{aligned} \ln \Gamma(-x) = - \ln(-x) - \ln(-x+1) - \ldots - \ln(-x + [x]) +\ln\Gamma[-x+[x]+1], \label{lngaid}\end{aligned}$$ for $x > 0$ and where $[x]$ is the integer part of $x$. Eq. (\[lngaid\]) implies that $\Sigma(E^-)$ involves the sum of $[E/\omega_0]+1$ logarithms with negative argument. Each logarithm contributes a term $ \pi$ to the imaginary part of $\Sigma(E^-)$, hence, $$\begin{aligned} \Gamma(E) = \lambda^2 \omega_0 ([E/\omega_0]+1).\end{aligned}$$ The level-shift function $F(E)$ involves a term $ - \ln(E/\omega_0) - \ln(E/\omega_0 - 1) - \ln(E/\omega_0 - [E/\omega_0]) + \ln \Gamma( [E/\omega_0] + 1 - E/\omega_0)$, which can be written as $ \ln \left( \frac{\Gamma(1+ [E/\omega_0] - E/\omega_0) \Gamma(E/\omega_0 - [E/\omega_0]) }{ \Gamma(E/\omega_0) }\right)$. Hence, $$\begin{aligned} F(E) = - \frac{\lambda^2\log (\Lambda/\omega_0) }{2\pi}z + \frac{\lambda^2\omega_0 }{2\pi} \ln \left( \frac{ \Gamma(E/\omega_0) }{ \Gamma(E/\omega_0 - [E/\omega_0])\Gamma(1+ [E/\omega_0] - E/\omega_0) }\right). \end{aligned}$$ Both $F(E)$ and $\Gamma(E)$ have finite discontinuities across the resonances $E = n \omega_0$, for $n = 1, 2, \ldots$. [Large cavity.]{} For a large cavity ($L \Omega >>1$), $\omega_0$ is very small, so we expand the logarithmic gamma function in Eq. (\[szcav2\]) using Stirling’s formula. Then, $$\begin{aligned} \Sigma(z) = \frac{\lambda^2 }{2\pi}\left[ - ( \ln \Lambda - 1)z + z \ln(-z) \right] . \label{szcav2b}\end{aligned}$$ Eq. (\[szcav2b\]) has the same functional dependence on $z$ with the self-energy function of Eq. (\[seel6\]) that is obtained in absence of a cavity. If the coupling constant and the cut-off parameters are properly redefined, the two expressions coincide. We evaluate the persistence amplitude using with Eq. (\[mainampl\]). We define $n_c = [\tilde{\Omega}/\omega_0] $ and $x_0 = \tilde{\Omega}/\omega_0 - n_c $. Then, we express the integral $\int_0^{\infty} dE$ as $\sum_{n=0}^{\infty} \int_{n \omega_0}^{(n+1)\omega_0}dE$, where $n = [E/\omega_0]$. $$\begin{aligned} {\cal A}_a(t) = \frac{\lambda^2 }{2 \pi} \sum_{n=0}^{\infty}(n+1)e^{-i n \omega_0 t} \int_0^1 dx \frac{e^{-i(\omega_0t)x}}{[x + n - n_c - x_0 - \frac{\lambda^2 }{2\pi} f_n(x)]^2 + \frac{\lambda^4 }{4}(n+1)^2 } \label{offres}\end{aligned}$$ where we substituted $ E = \omega_0(n+x)$ and we wrote $$\begin{aligned} f_n(x) = - \ln(\Lambda/\omega_0) (n +x) + \ln \frac{ \Gamma(n+x)}{\Gamma(1-x)\Gamma(x)}.\end{aligned}$$ The function $f_n(x)$ diverges near $x = 0$ and near $x = 1$. Using the expansion, $\Gamma(x) \simeq \frac{1}{x} + \ldots$ near $x = 0$, we obtain $$\begin{aligned} f_n(x) &=& \ln x + \ln (n-1)! - \ln(\Lambda/\omega_0) n \; \; (n > 0), \label{fn0} \\ f_n(1-x) &=& \ln x + \ln n! - \ln(\Lambda/\omega_0) (n+1), \label{fn1}\end{aligned}$$ for $x << 1$. The integrals in Eq. (\[offres\]) are dominated by values of $x$ for which the denominator is of order $\lambda^2$. Their behavior depends crucially on whether the system is near resonance or not. Off resonance -------------- First, we assume that the atomic frequency is far from the cavity’s resonances. Hence, $x_0$ and $1 - x_0$ are much larger than $O(\lambda^2)$. The integrals are dominated by their values near solutions to the equation $$\begin{aligned} x + n - n_c - x_0 - \frac{\lambda^2 }{2\pi} f_n(x) = 0, \label{eqoll}\end{aligned}$$ for $x\in [0, 1]$. For $n = n_c$, Eq. (\[eqoll\]) admits a perturbative solution at $x = x_0 + \frac{\lambda^2 }{2\pi} f_{n_c}(x_0) + O(\lambda^4)$. There are other solutions to Eq. (\[eqoll\]), near the boundaries $x = 0$ and $x = 1$ where $f_n(x)$ diverges. It is easy to show that these solutions lie at distance smaller than $e^{-\frac{2 \pi x_0}{\lambda^2}}$ or $e^{-\frac{2 \pi (1-x_0)}{\lambda^2}}$ from the boundaries, and that their contribution to the integral is negligible. Therefore, the dominant term to the persistence amplitude corresponds to $n = n_c$. Since the integral is dominated by values near the perturbative solution with a width of order $\lambda^2$, we can extend the range of integration to $(-\infty, \infty)$. But then, we recover the integral (\[ampl3\]) of the WWA, with $$\begin{aligned} \delta E &=& - \frac{\lambda^2 \omega_0\log (\Lambda/\omega_0) }{2\pi} (n_c + x_0) + \frac{\lambda^2\omega_0 }{2\pi} \ln \left( \frac{\Gamma(n_c + x_0) }{\Gamma(1- x_0 ) \Gamma(x_0) }\right) \\ \Gamma &=& \lambda^2 \omega_0 (n_c + 1).\end{aligned}$$ As expected, the persistence amplitude off-resonance does not differ significantly from the persistence amplitude of an atom outside a cavity. Resonance --------- The condition for resonance is that either $x_0$ or $1 - x_0$ is of order $\lambda^2$. In the former case, $\Omega_R$ is just above the resonance frequency $n_R \omega_0$, for $n_R = n_c$; we define the detuning parameter $\delta = x_0$. In the latter case, $\Omega_R$ is just below the resonance frequency $n_R \omega_0$, for $n_R = n_c + 1$; we define the detuning parameter as $\delta = x_0 - 1$. Two terms in the series (\[offres\]) dominate. The first corresponds to $n = n_R$. In this term, the denominator of the integral is of order $\lambda^2$ near $x = 0$. Therefore, we can use the approximation (\[fn0\]) for $f_n(x)$. We change variables to $y = \frac{\lambda^2}{2\pi}x$ and we extend the limit of integration for $y$ form $\frac{2\pi}{\lambda^2}>> 1$ to $\infty$. Then, this term equals $e^{-in_R\omega_0 t} G_-(\frac{\Gamma_0t}{\pi}, -d, (n_R+1)\pi)$, where $$\begin{aligned} G_{\pm}(s, a, b) := \frac{b}{\pi} \int_0^{\infty} \frac{dy e^{- i s y}}{(y \pm \ln y - a)^2 + b^2 }, \label{gsab}\end{aligned}$$ $$\begin{aligned} d := \ln \frac{2 \pi}{\lambda^2} - \frac{2\pi\delta}{\lambda^2} - \ln (n_R-1)! + \ln(\Lambda/\omega_0) n_R, \end{aligned}$$ and $\Gamma_0 = \lambda^2 \omega_0$. The second term corresponds to $n = n_R - 1 $. In this term, the denominator is of order $\lambda^2$ near $x = 1$. Again, we can use the approximation (\[fn1\]) for $f_n(x)$. We change variables to $y = \frac{2\pi}{\lambda^2}(1-x)$ and take the limit of integration for $y$ to $\infty$. Then, this term equals $e^{-in_R\omega_0 t} G_+^(\frac{\Gamma_0t}{\pi}, d, \frac{1}{2}n_R\pi)$. Hence, $$\begin{aligned} {\cal A} (t) = e^{-in_R \omega_0 t} \left[ G_-[\frac{\Gamma_0t}{\pi}, -d, (n_R+1)\pi ] + G^_+[\frac{\Gamma_0t}{\pi}, d, n_R \pi ] \right], \label{atres}\end{aligned}$$ For general values of $a$ and $b$ the functions $G_{\pm}(s, a,b)$ can only evaluated numerically. In general, they decay with increasing $s$, with some oscillations for positive $a$. Plots of $G_{\pm}(s, a, b)$ as a function of $s$ for different values of $a$ and $b$ are given in Fig. 3. ![The real and the imaginary part of $G_{\pm}(s, a, b)$ are plotted as functions of $s$ for different choices of $a$ and $b$. ](functionG){height="10cm"} For $a >>b $ , the contribution of the logarithm to the integral (\[gsab\]) is negligible, and the integral is little affected if the range is extended to $(-\infty, \infty)$. Then, $$\begin{aligned} G_{\pm}(s, a, b) = \frac{b}{\pi} \int_{-\infty}^{\infty} \frac{dy e^{-i s y}}{(y- a )^2 + b^2 } = e^{-b s-ias}\end{aligned}$$ In contrast, if $\|a\|>> b$ with $a <0$, $G_{\pm}(s, a, b)$ is of order $(\|a\|/b)^2$. We readily verify that Eq. (\[atres\]) recovers the exponential decay form for $ \|d\| >> \pi n_R$. In all other regimes, the decays are non exponential. This can be seen in Fig. 4, where the logarithm of the persistence probability is plotted as a function of $\Gamma_0t/\pi$ for different values of $d$. The graph becomes a straight line for larger values of $d$, signaling exponential decay. Deviations appear when a tiny fraction of the initial 2LS remains excited. For several values of $d$, the persistence probability is not a decreasing function of $t$. Again, this signifies a failure of the definition (\[decprob2\]) for the decay probability. ![The logarithm of the persistence probability $\ln \|{\cal A}(t)\|^2$ as a function of $\Gamma_0t/\pi$ for different values of $d$ and for $n_R = 1$. ](persistence){height="7cm"} The behavior of the persistence probability derived here is typical for decaying systems with energies close to a branch point of the self-energy function. In quantum field theory, such branch points appear at energy thresholds, i.e., for energies near the activation energy $E_0$ of a new decay channel. For example, if the energy of a photon becomes $2m_e$, where $m_e$ is the electrons’s mass, the photon decay to a electron-positron pair is possible. In such cases, the persistence amplitude receives two distinct contributions, one from energies slightly beneath and one from energies slightly above the threshold. For discussions of non-exponential decays due to threshold effects, see, Refs. [@LZMM; @RZ93; @JMSST; @DJN09]. Decay through barrier tunneling =============================== The methodology developed in Sec. 2 applies to decays that originate from a small perturbation in the Hamiltonian. In this section, we consider non-perturbative decays that can be understood in terms of tunneling. Examples of such decays are the alpha emission of nuclei, tunneling ionization of atoms due to an external field, and leakage of particles from a trap. We will consider a simple model of a particle in one dimension that mimics the classic treatment of alpha decay by Gamow and by Gurney and Condon [@Gamow; @GuCo]. Set-up ------ [Dynamics.]{} We consider a particle in the half-line ${\pmb R}^+ = [0, \infty)$ in presence of a potential $V(x)$. The potential vanishes outside $ [a, b]$, where $a$ and $b$ are microscopic lengths. It is convenient to express the potential in terms of the transmission and reflection coefficients of the Schrödinger operator $\hat{H} = \frac{\hat{p}^2}{2m}+ V(\hat{x})$ over the full real line. $\hat{H}$ has two generalized eigenstates $f_{k\pm}(x)$ for each value of energy $E = \frac{k^2}{2m}$. $$\begin{aligned} f_{k+}(x) &=& \left\{ \begin{array}{cc} \frac{1}{\sqrt{2\pi}} (e^{ikx} + R_k e^{-ikx}), & x < a\\ \frac{1}{\sqrt{2\pi}} T_k e^{ikx}& x > b \end{array}\right. \nonumber \\ f_{k-}(x) &=& \left\{ \begin{array}{cc} \frac{1}{\sqrt{2\pi}} T_k e^{-ikx}, & x < a\\ \frac{1}{\sqrt{2\pi}} (e^{ikx} + \tilde{R}_k e^{ikx})& x > b \end{array}\right. \label{f+-}\end{aligned}$$ where the complex amplitudes $T_k$, $R_k$ and $\tilde{R}_k$ satisfy $$\begin{aligned} \|T_k\|^2+\|R_k\|^2 = 1, \hspace{0.75cm} \|R_k\| = \|\tilde{R}_k\|, \hspace{0.75cm} T^_k R_k +T_k \tilde{R}_k^ = 0. \label{idtyscat}\end{aligned}$$ $T_k$ is the transmission amplitude, $R_k$ is the reflection amplitude for a right-moving particle and $\tilde{R}_k$ is the reflection amplitude for a left-moving particle. When the range of $x$ is restricted into the half-line, the generalized eigenfunction $g_k$ of the Schrödinger operator is the linear combination of $f_{k+}$ and $f_{k-}$ that satisfies $g_E(x) = 0$, i.e., $$\begin{aligned} g_k = \left( -\frac{T_k}{1+R_k} f_{k+} + f_{k-}\right).\end{aligned}$$ This implies that $$\begin{aligned} g_{k}(x) = \left\{ \begin{array}{cc} -\frac{2i}{\sqrt{2\pi}} \frac{T_k}{1+R_k} \sin kx, & x < a\\ \frac{1}{\sqrt{2\pi}} \left[ e^{- ikx} - e^{iS_k} e^{ikx} \right] & x > b \end{array}\right. , \label{gkx}\end{aligned}$$ where $$\begin{aligned} e^{i S_k} = \frac{T_k^2}{1+R_k} - \tilde{R}_k = \frac{1+R^_k}{1+R_k} \left(\frac{T_k^2}{\|T_k\|^2}\right), \label{eithe}\end{aligned}$$ is the reflection amplitude of a left-moving particle. The absolute value of the reflection amplitude is unity, because there is no possibility of transmission to $x < 0$. The eigenfunctions $g_k(x)$ are normalized so that $\int_0^{\infty} g_k(x)^ g_{k'}(x) = \delta(k-k')$. We will represent them by kets $\|k\rangle_D$. [Initial state.]{} We consider an initial state $\psi_0(x)$ with the following four properties. First, it vanishes outside $[0, a]$. Second, it belongs to the Hilbert space of square-integrable harmonic functions on ${\pmb R}^+$ subject to Dirichlet boundary conditions. Hence, it can be expressed as $$\begin{aligned} \psi_0(x) = \sqrt{\frac{2}{\pi}} \int_0^{\infty} \sin(kx) \tilde{\psi}_0(k). \label{Dirich}\end{aligned}$$ The function $\tilde{\psi}_0(k)$ is defined for $k > 0$. Extending to $k <0$ by $\tilde{\psi}_0(-k) = -\tilde{\psi}(k)$, we can write Eq. (\[Dirich\]) as $\psi_0(x) = - \frac{i}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dk e^{ikx} \tilde{\psi}_0(k)$. By Eq. (\[gkx\]), $$\begin{aligned} {}_D\langle k\|\psi_0\rangle = i \frac{T^_k}{1+R^_k} \tilde{\psi}_0(k).\end{aligned}$$ Third, we assume that $\psi_0(x)$ is real-valued. For example, $\psi_0$ may be an eigenstate of a Schrödinger operator $\hat{H}'$ with a different potential $U(x)$. The physical interpretation of this condition is that we prepare the system in an eigenstate of $\hat{H}'$ and at time $t = 0$ we change the potential to $V(x)$, for example, by switching on an external electric field. Fourth, we assume that $\psi_0$ has a sharp energy distribution with respect to the Hamiltonian $\hat{H}$: the energy spread for $\psi_0$ is much smaller than the mean energy. Given an initial state $\psi_0$, we find the state at time $t$ $$\begin{aligned} \psi_t(x) = \int_0^{\infty} dk {}_D\langle k\|\psi_0\rangle g_k(x) e^{ - i\frac{k^2}{2m}t}. \label{evolvt} \end{aligned}$$ The definition (\[decprob2\]) of the decay probability in terms of the persistence probability is not appropriate for this problem, because the initial configuration of the system does not correspond to an one-dimensional subspace, i.e., the particle may remain confined by the potential without remaining in its initial state. A better choice is perhaps to define the decays probability as $p(t) = - \frac{d}{dt} W(t)$, where $W(t) = \int_0^a dx\|\psi_t(x)\|^2$ is the [non-escape probability]{}, i.e., the probability that the particle is found in $[0, a]$ at time $t$—see, Ref. [@CMM95] for the relation between persistence and non-escape probability. We note that the definition of the non-escape probability is arbitrary: we could have equally well taken the integration range to $[0, b]$. The main drawback of both the non-escape and the persistence probability is that they lack a clear operational interpretation. We do not carry out measurements of particles at microscopic scales inside the confining potential, rather we measure the number of particles that have tunneled from the barrier, as recorded by a detector far from the tunneling region. Furthermore, the persistence probability and the non-escape probability are not always increasing functions of $t$ [@Peshk]. Hence, they may lead to negative values for $p(t)$. Here, we will proceed by calculating the probability flux far from the potential region, i.e., we evaluate $$\begin{aligned} J(x, t) = \frac{1}{m}\mbox{Im} \left[ \psi^_t(x) \partial_x \psi_t( x)\right],\end{aligned}$$ for $x >> b$. Eqs. (\[gkx\]) and (\[evolvt\]) imply that $$\begin{aligned} \psi_t(x) = -\frac{i}{\sqrt{2\pi}} \int _0^{\infty} dk \frac{T^_k}{1+R^_k} \left[ e^{iS_k +ikx}- e^{-ikx}\right] e^{-i\frac{k^2}{2m}t} \tilde{\psi}_0(k). \label{psitxx}\end{aligned}$$ For $ x>> b$, there is no stationary phase in the integral involving $e^{-ikx -i\frac{k^2}{2m}t}$. Hence, its contribution to $\psi_t(x)$ is much smaller than that of the integral involving $e^{ikx -i\frac{k^2}{2m}t}$, which has a stationary phase. By Eq. (\[eithe\]), $$\begin{aligned} \psi_t(x) = -\frac{i}{\sqrt{2\pi}} \int _0^{\infty} dk \frac{T_k}{1+R_k} e^{ikx -i\frac{k^2}{2m}t } \tilde{\psi}_0(k). \label{psitx}\end{aligned}$$ The integral (\[psitx\]) contains only out-coming waves, hence, there is no backflow in the probability current. Note that, in general, $J(x, t)$ can take negative values close to the barrier [@Winter], and hence, it is not reliable for the description of near-field experiments, i.e., when a particle detector is placed at a microscopic distance from the tunneling region. We expand $(1+R_k)^{-1} = \sum_{n=0}^{\infty}(-R_k)^n$, to write Eq. (\[psitx\]) as $$\begin{aligned} \psi_t(x) = -\frac{i}{\sqrt{2\pi}}\sum_{n=0}^{\infty}\int _0^{\infty} dk T_k(-R_k)^n e^{ikx -i\frac{k^2}{2m}t } \tilde{\psi}_0(k). \label{psitx1b}\end{aligned}$$ Exponential decay ------------------ Eq. (\[psitx\]) is accurate for the asymptotic behavior of the wave-function at $x >> b$. To proceed further, we exploit the fact that $\psi_0(k)$ is strongly peaked about a specific value $k_0$, and we evaluate Eq. (\[psitx\]) in the saddle-point approximation. To this end, we write $R_k = -\|R_k\|e^{i\phi_k}$ and $T_k = \|T_k\|e^{i\chi_k}$, so that Eq. (\[psitx1b\]) becomes $$\begin{aligned} \psi_t(x) = - \frac{i}{\sqrt{2\pi}} \sum_{n=0}^{\infty} \int _0^{\infty}dk \|T_kR_k^n\| e^{ikx -i\frac{k^2}{2m}t + i \chi_k + i n \phi_k} \tilde{\psi}_0(k). \label{psitx2}\end{aligned}$$ Then, we extend the range of integration of $k$ to $(-\infty, \infty)$ setting $\tilde{\psi}_0(-k) = - \tilde{\psi}_0(k)$ for negative $k$. The integral is not affected, because the additional terms involve a term $e^{-i\|k\|x -i\frac{k^2}{2m}t}$, with no stationary phase. Then, we approximate $\|T_k\| \simeq \|T_{k_0}\|, \|R_k\| \simeq \|R_{k_0}\|$, $k^2 \simeq k_0^2 + 2 k_0 (k-k_0)$, $\phi_k \simeq \phi_{k_0}+ \phi'_{k_0} (k-k_0)$, $\chi_k \simeq \chi_{k_0}+ \chi'_{k_0} (k-k_0)$. The resulting integral is simply the inverse Fourier transform of $\tilde{\psi}_0(k)$. Hence, $$\begin{aligned} \psi_t(x) = T_{k_0} e^{ik_0x -i\frac{k_0^2}{2m}t } \sum_{n=0}^{\infty} (-R_{k_0})^n \psi_0(x - \frac{k_0t}{m} + \chi'_{k_0} + n \phi'_{k_0}) \label{psitx3}\end{aligned}$$ Eq. (\[psitx3\]) has a natural interpretation in terms of classical concepts. The particle makes successive attempts to cross the barrier at $x = a$. On failure, it is reflected back, it is reflected again at $x = 0$, and then it makes a new attempt. The $n$-th term in the sum of Eq. (\[psitx3\]) is the amplitude associated to a particle that succeeded in crossing the barrier at its $(n+1)$-th attempt: it is proportional to $T_{p_0}$ (one success) and to $R^n_{p_0}$ (after $n$ failures). Since $\psi_0(x)$ has support only on $[0, a]$, $\psi_t(x)$ vanishes for $t < t_0 := m (x -a + \chi_{k_0}')/p_0$. The time-scale $t_0$ has an obvious classical interpretation: it is the time it takes a particle inside the barrier region to traverse the distance to point $x$. The term $\chi_{k_0}$ corresponds to the Wigner-Bohm time delay due to the particle crossing the classically forbidden region [@BohmWig]. We rewrite Eq. (\[psitx3\]) as $$\begin{aligned} \psi_t(x) = T_{k_0} e^{ik_0x -i\frac{k_0^2}{2m}t } \theta(t-t_0)\sum_{n=0}^{\infty} (-R_{k_0})^n \psi_0\left(a - \frac{k_0(t - t_0 )}{m} + n\Delta x \right) \label{psitx4}\end{aligned}$$ where we defined $\Delta x = \phi'_{k_0}$ the position-shift between successive terms in the series (\[psitx4\]). If $\|\Delta x\| > a$, the partial amplitudes at different $n$ do not overlap. Hence, there is no quantum interference between different attempts of the particle to cross the barrier. For $\|\Delta x\| < a$, there is quantum interference between $M = [a/\|\Delta x\|]$ successive attempts to cross the barrier. Since we assumed the initial state $\psi_0$ to be almost monochromatic at energy $\frac{k_0^2}{2m}$, the dominant contribution to the current is $ \frac{k_0}{m} \|\psi_t(x)\|^2$. Hence, $$\begin{aligned} J(t,x) &=& \frac{k_0}{m} \|T_{k_0}\|^2 \theta(t-t_0) \sum_{n=0}^{\infty} \sum_{\ell=0}^{\infty}(-R_{k_0})^n(-R_{ k_0}^{})^{\ell}\nonumber \\ &\times& \psi_0(a- \frac{k_0(t - t_0 )}{m} + \ell \Delta x )\psi_0(a- \frac{k_0(t - t_0 )}{m} + n \Delta x ) \label{jxt1}\end{aligned}$$ Terms in the summation with $\|n- \ell\| > M$ vanish because the corresponding wave functions do not overlap. Then, we write $$\begin{aligned} J(t,x) = \frac{k_0}{m} \|T_{k_0}\|^2 \theta(t-t_0) \sum_{N=0}^{\infty}\|R_{k_0}\|^N \rho\left(\frac{k_0(t - t_0 )}{m} - \frac{1}{2} N\Delta x \right), \label{jtx3}\end{aligned}$$ where $$\begin{aligned} \rho(x) = \sum_{j=-M}^M \psi_0(a - x - \frac{j}{2} \Delta x) \psi_0(a - x +\frac{j}{2} \Delta x) e^{i j\phi_{k_0}}.\end{aligned}$$ The function $\rho(x)$ is localized within a width of order $a$. In order to connect Eq. (\[jtx3\]) with experiments, we have to treat both $x$ and $t$ as macroscopic variables. This means that they can be measured with an accuracy of order $\sigma_X$ and $\sigma_T$, respectively, that is much larger than the microscopic scales that characterize the system. Hence, $\sigma_X >> a$ and $\sigma_T >> ma/k_0$. At such scales, the width of $\rho(x)$ is negligible, and we can substitute it with a delta function, $$\begin{aligned} \rho(x) \simeq \alpha \delta(x), \label{Markovtunel}\end{aligned}$$ where $\alpha = \int_{-\infty}^{\infty}dx \rho(x)$ is a number close to unity. In particular, $\alpha =1 $ for $M = 0$. In this regime, we can also approximate the sum over $N$ with an integral, so that $$\begin{aligned} J(t,x) &=& \alpha \frac{k_0}{m} \|T_{k_0}\|^2 \theta(t-t_0) \int_0^{\infty}dN \|R_{k_0}\|^N \delta\left(\frac{k_0(t - t_0 )}{m} - \frac{1}{2} N\Delta x \right)\nonumber \\ &=& \alpha \frac{\|T_{k_0}\|^2}{\Delta t} \theta(t-t_0) e^{\log \|R_{p_0}\|^2 \frac{(t-t_0)}{\Delta t}} \label{jx5}\end{aligned}$$ where $\Delta t = m \Delta x/k_0$ has the classical interpretation as the time between two successive attempts of the particle to cross the barrier. For $\|T_{k_0}\| << 1$, $\log\|R_{p_0}\|^2 \simeq -\|T_{p_0}\|^2$. Then, Eq. (\[jx5\]) describes exponential decay, $$\begin{aligned} J(t,x) = \alpha \Gamma e^{-\Gamma (t-t_0) } \theta (t-t_0), \label{expdec}\end{aligned}$$ with a decay constant $$\begin{aligned} \Gamma = \frac{ \|T_{k_0}\|^2}{\Delta t}. \label{Gammat}\end{aligned}$$ that does not depend on the detailed properties of the initial state. Note that exponential decay fails at early times; the derivation of Eq. (\[expdec\]) requires that $\|t - t_0 \|>> \Delta t$. In deriving the exponential decay law, we employed the saddle point approximation. This it is reasonably accurate for $\cap$-shaped potentials. In general, it does not apply to potentials with multiple transmission and reflection points, like the double well potential [@Matsu]. Such potentials may trap the particle in an intermediate region, and they require an analysis of the escape from this region. The escape satisfies an exponential decay law, except for energies near resonance [@AnSav13]. The exponential decay law also fails at very long times, when wave-function dispersion becomes important. To see this, we change variables to $y = \frac{k^2t}{2m}$ in Eq. (\[psitx\]) for $\psi_t(x)$. The dominant term at $t \rightarrow \infty$ is $$\begin{aligned} \psi_t(x) = - i \sqrt{\frac{m}{4\pi t}} A_0 \int_0^{\infty} \frac{dy}{y} e^{-iy} \tilde{\psi}_0(\sqrt{2my/t}). \label{asymptunel}\end{aligned}$$ where $A_0 $ stands for $\lim_{k\rightarrow \infty} T_k/(1+R_k)$. In general, $A_0 \neq 0$, as can be readily checked in elementary systems. Hence, the asymptotic behavior of $\psi_t(x)$ depends on the infrared behavior of $\tilde{\psi}_0$. For a power law dependence, $\psi_0 \sim k^n$ near $k = 0$, Eq. (\[asymptunel\]) gives $\psi_t(x) \sim t^{-\frac{n+1}{2}}$. It follows that $J(t, x) \sim t^{-(n+1)}$, i.e., the flux decays with an inverse power law. We note here that the asymptotic regime captures some of the information of the initial state [@MDCG]. The key property in deriving the exponential decay law is the lack of interference between different attempts of the particle to cross the barrier. Let us assume that the maximum number of successive attempts that interfere in the probability amplitude is $M$. The decay time scale $\Gamma^{-1}$ corresponds to $\|T_{k_0}\|^{-2}$ attempts to cross the barrier. As long as $$\begin{aligned} \|T_{k_0}\|^{-2} >> M, \label{tkm}\end{aligned}$$ the effects of interference are negligible. This also implies that the particle has very short ‘memory’ about its past attempts to cross the barrier. Hence, the memory time-scale is much shorter than the decay time-scale. This feature is known as the [Markov property]{}. In absence of quantum interferences and memory effects, decays due to tunneling are indistinguishable from classical probabilistic processes that can be described using elementary arguments. Consider a classical particle that attempts to cross a barrier with probability $w << 1$ of success[^7]. After $N$ attempts, the survival probability is $(1-w)^N \simeq e^{-Nw}$. If every attempt takes time $\Delta t$, then for $N >> 1$, the system is described by an exponential decay law with constant $\Gamma = w /\Delta t$, in full agreement with Eq. (\[Gammat\]). Alternative description of tunneling decays ------------------------------------------- We can understand the emergence of exponential decay in tunneling using a different argument that does not rely on the saddle-point approximation. Instead, we use the analyticity properties of the time-evolution operator. For the full development of such methods, see, Ref. [@RNewt], and for applications to tunneling decays, see, Refs. [@CP75; @CMM95; @CCM09]. Assume that we can analytically extend $\tilde{\psi}_0$ to the fourth quadrant of the complex plane. Then, we can write $\psi_t(x) = K(t, x) + I_N(t,x)$, where $$\begin{aligned} K(t, x) = \frac{i}{\sqrt{2\pi}} \oint_C dz \frac{T_z}{1+R_z} e^{izx -i\frac{z^2}{2m}t } \tilde{\psi}_0(z), \label{ktn}\end{aligned}$$ is a line integral along the contour $C$ of Fig. 5, and $I_N(t)$ is the integral across the line segment $N$ of $C$. By Cauchy’s theorem, we can evaluate $K(t,x)$ in terms of the poles of the integrand in the interior of $C$. ![The integration contour of the line-integral (\[ktn\]). The contour is traversed clockwise. ](curven){height="7cm"} Let us denote by $z_n = q_n -i \gamma_n$ the poles of $\frac{T_z}{1+R_z} $ in the interior of $C$. The integer $n$ labels the poles, in the interior of $C$, $q_n$ and $\gamma_n$ are positive. Then, $$\begin{aligned} \psi_t(x) = \sum_n c_n \tilde{\psi}_0(q_n -i \gamma_n) e^{iq_n x - i\frac{q_n^2-\gamma_n^2}{2m}t - \frac{q_n\gamma_n}{m}(t - \frac{mx}{q_n})} + I_N(t),\end{aligned}$$ for some complex constants $c_n$. Each term in the sum, is suppressed by an exponential factor $\exp[\frac{q_n\gamma_n}{m}(t - \frac{x}{q_n})]$, for $t > \frac{x}{q_n}$. For an almost monochromatic state $\tilde{\psi}_0$ with energy $k_0$, only a small number of poles with $q_n$ near $k_0$ contribute. Furthermore, the contribution $I_N$ to $\psi_t(x)$ drops exponentially for $t > mx/k_0$, i.e., after the earliest possible time of detection. For simplicity, let us assume that the contribution of only one pole at $n = n_0$ is significant, and that $q_{n_0} \simeq k_0$. Then, for $t > mx/k_0$, $$\begin{aligned} \psi_t(x) \sim e^{ik_0 x - i\frac{k_0^2-\gamma_{n_0}^2}{2m}t - \frac{k_0\gamma_{n_0}}{m}\|t - \frac{mx}{k_0}\|}. \end{aligned}$$ Therefore, the flux $J(t, x)$ is proportional to $e^{-2 \frac{k_0\gamma_{n_0}}{m} \|t - \frac{mx}{k_0}\|}$. The decay constant is $$\begin{aligned} \Gamma = \frac{2k_0\gamma_{n_0}}{m}, \label{decaytb} \end{aligned}$$ and it is determined [solely]{} by the pole of $\frac{T_z}{1+R_z} $ near $z = k_0$. No information about the initial state other than its energy is required. The above analysis also provides a criterion for the breakdown of exponential decay. If the initial state allows for the contribution of different poles $z_n$, such that there is a significant variation in the values of $\gamma_n$, then corrections to exponential decay, or even its breakdown are possible. For example, consider an initial state $ \psi_0 = a_1 \psi_1 + a_2 \psi_2$ that is a superposition of two almost monochromatic states $\psi_1$ and $\psi_2$ with energies $\frac{k_1^2}{2m}$ and $\frac{k_2^2}{2m}$. Furthermore, assume that $k_1$ is close to one pole of $T_z/(1+R_z)$ at $n = n_1$, and $k_2$ close to another pole of $T_z/(1+R_z)$ at $n = n_2$, and that there is no overlap. Then, for $t > \max \{ mx/k_1, mx/k_2 \}$, $$\begin{aligned} \psi_t(x) \sim c_1 e^{ik_1 x - i\frac{k_1^2-\gamma_{n_1}^2}{2m}t - \frac{k_0\gamma_{n_1}}{m}\|t - \frac{x}{k_1}\|} + c_2 e^{ik_2 x - i\frac{k_2^2-\gamma_{n_2}^2}{2m}t - \frac{k_0\gamma_{n_2}}{m}\|t - \frac{x}{k_2}\|},\end{aligned}$$ for some constants $c_1$ and $c_2$. The dominant contribution to the current is $$\begin{aligned} J(t, x) &=& \|c_1\|^2 k_1 e^{- \Gamma_1 \|t - \frac{mx}{k_1}\|} + \|c_2\|^2 k_2 e^{- \Gamma_2 \|t - \frac{mx}{k_2}\|} \nonumber \\ &+& (k_1 + k_2) e^{- \frac{1}{2} \Gamma_1 \|t - \frac{mx}{k_1}\| - \frac{1}{2} \Gamma_2 \|t - \frac{mx}{k_2}\| } \mbox{Re} \left[ c_1c_2^* e^{i \theta(t, x)}\right],\end{aligned}$$ where $\Gamma_i = \frac{2k_i\gamma_{n_i}}{m}$, and the interference phase is $$\begin{aligned} \theta(t, x) = (k_1 - k_2)x - \frac{k_1^2 -k_2^2}{2m} t + \frac{\gamma_{n_1}^2 - \gamma_{n_2}^2}{2m}t.\end{aligned}$$ The flux is characterized by an exponential decay with a periodic modulation due to the energy difference between the interfering states. This is the well-known phenomenon of [quantum beats]{}. Detection probabilities ======================= In the previous sections, we employed two methods for constructing the decay probability, namely, persistence probabilities and probabilities currents. Both methods work fine for exponential decays, where the decay probability is determined by a single parameter $\Gamma$. Outside exponential decay they have a restricted domain of validity. The key problem is that they are not guaranteed to define positive-definite probabilities. This is due to the fact that they do not express probabilities in terms of measurement outcomes for concrete observables, These probabilities are guaranteed to be positive by the rules of quantum theory. A rigorous description of decays requires a consideration of the explicit measurement scheme through which the decay products are detected, and the construction of appropriate measurement observables. The latter correspond to positive operators $\hat{\Pi}(t)$, in which the detection time $t$ appears as a random variable. Then, given an initial state $\hat{\rho}_0$, the detection probability $p(t)$ is determined by $Tr\left[\hat{\rho}_0 \hat{\Pi}(t)\right]$. A scheme for constructing temporal observables of this type has been developed in [@AnSav]. Here, we will present an elementary example of such observables that generalizes the well-established photodetection model by Glauber [@Glauber]. Suppose that one of the decay products is a particle that is described by quantum field operators $\hat{\phi}({\pmb x})$ and Hamiltonian $\hat{H}$. The field operators are split into a positive frequency part $\hat{\phi}^{(+)}({\pmb x})$ that contains annihilation operators and a negative frequency part $\hat{\phi}^{(-)}({\pmb x})$ that contains creation operators. Consider an elementary apparatus located at a point ${\pmb x}$ that gives a detection signal at time $t$ after having absorbed the incoming particle. The amplitude associated to this process is then proportional to $\hat{\phi}^{(+)}({\pmb x})\|\psi_t\rangle$, where $\|\psi_t \rangle$ is the state of the quantum field at time $t$. The probability of detection is the determined by the modulus square of this amplitude. It is given by Glauber’s formula $$\begin{aligned} P(t, {\pmb x}) = C \langle \psi_t\| \hat{\phi}^{(-)}({\pmb x})\hat{\phi}^{(+)}({\pmb x})\|\psi_t\rangle, \label{Glauber}\end{aligned}$$ where $C$ is a normalization constant. We do not obtain normalized probabilities, because there is a non-zero probability that the particle will not be detected and this probability depends on the field-state. Eq. (\[Glauber\]) was first proposed by Glauber for photodetection. In Glauber’s theory, the role of $\hat{\phi}$ is played by the electric field, and the absorption interaction corresponds to the dipole coupling between the electromagnetic field and a macroscopic detectors. Glauber’s formula is a special case of a larger class of particle detection observables that can be defined in quantum fields [@AnSav]. We will apply Glauber’s formula to the bosonic Lee model. The field operators associated to the bosonic creation and annihilation operators are $$\begin{aligned} \hat{\phi}^{(+)}({\pmb x}) = \sum_r \hat{a}_r \chi_r({\pmb x}) \hspace{1cm} \hat{\phi}^{(-)}({\pmb x}) = \sum_r \hat{a}_r \chi^_r({\pmb x}) \label{scalar}\end{aligned}$$ where $\chi_r(x)$ are eigenfunctions of the single-particle Hamiltonian. For particles in three dimensions, $r$ corresponds to the three-momentum ${\pmb k}$, and $$\begin{aligned} \chi_{\pmb k}({\pmb x}) = \frac{1}{\sqrt{2 \omega_{\pmb k}} }e^{i{\pmb k} \cdot {\pmb x}}. \label{chik}\end{aligned}$$ By Eq. (\[decay555\]), $$\begin{aligned} \hat{\phi}^{(+)}({\pmb x}) \frac{1}{z - \hat{H}} \|A'\rangle = \frac{V(z; {\pmb x})}{z - \Omega - \Sigma(z)} \|g\rangle \otimes \|0\rangle,\end{aligned}$$ where $$\begin{aligned} V_ {\pmb x}(z) = \sum_r \frac{g_r \chi_r({\pmb x}) }{z- \omega_r}. \label{Vz}\end{aligned}$$ Then, Eq. (\[Glauber\]) gives $$\begin{aligned} P(t, {\pmb x}) = C \|{\cal B}(t, {\pmb x})\|^2, \label{Glauber2}\end{aligned}$$ where $$\begin{aligned} {\cal B}(t, {\pmb x}) = \lim_{\epsilon \rightarrow 0} \int_{-\infty+i \epsilon }^{\infty+i \epsilon } \frac{dE V_ {\pmb x}(E) e^{-iEt}}{[E - \Omega - \Sigma_a(E) }.\end{aligned}$$ Following the same steps that lead to Eq. (\[mainampl\]), we find that $$\begin{aligned} {\cal B}(t, {\pmb x}) = \int_{0 }^{\infty } \frac{dE e^{-iEt}}{2\pi} \frac{ \frac{1}{2}\Gamma(E) [V_{\pmb x}^+(E) +V_{\pmb x}^-(E) ] + i [E - \Omega - F(E)][V_{\pmb x}^+(E) - V_{\pmb x}^-(E) ] }{ [E - \Omega - F(E)]^2 + \frac{1}{4}[\Gamma(E)]^2} \label{mainampl44}\end{aligned}$$ where $$\begin{aligned} V_{\pmb x}^{\pm}(E, {\pmb x}) := \lim_{\eta \rightarrow 0 } V_{\pmb x}(E \pm i \eta). \end{aligned}$$ We evaluate the amplitude (\[mainampl44\]) in the WWA, in which the dominant contribution to the integral comes from values of $E$ near $\Omega$. Hence, $$\begin{aligned} {\cal B}(t, {\pmb x}) &\simeq& \frac{1}{2 \pi } \int_{-\infty }^{\infty } dE e^{-iEt} \frac{ \Gamma [V_{\pmb x}^+(\Omega) +V_{\pmb x}^-(\Omega) ] + i [E - \Omega - \delta E][V_{\pmb x}^+(\Omega) - V_{\pmb x}^-(\Omega) ] }{ [E - \Omega - \delta E]^2 + \frac{1}{4}\Gamma^2} \nonumber \\ &=& \frac{1}{2} e^{-i (\Omega t + \delta E)t - \frac{1}{2}\Gamma t} V_+(\Omega; {\pmb x}) , \label{mainampl44b}\end{aligned}$$ where we set $\Gamma = \Gamma(\Omega)$ and $\delta E = F(\Omega)$. Hence, we obtain $$\begin{aligned} P(t, {\pmb x}) = \frac{1}{4} C\|V_{\pmb x}^+(\Omega) \|^2 e^{-\Gamma t}. \label{decayGl}\end{aligned}$$ The constant $C$ can be determined by normalizing over all particle detection events, i.e., by the requirement that $$\begin{aligned} \int_0^{\infty} dt \oint_{S} d^2n P(t, {\pmb x}) =1 , \label{normalk}\end{aligned}$$ where $S$ is a two-sphere at distance $r$ from the location of the 2LS, ${\pmb n}$ is a unit vector such that ${\pmb x} = r {\pmb n}$ on $S$. We conclude that the WWA guarantees exponential decay with constant $\Gamma$, [irrespective of the method used]{}. However, outside the exponential decay regime the probability density (\[Glauber2\]) differs significantly from Eq. (\[decprob2\]). In particular, it is guaranteed to be always positive. As an example, we revisit the photoemission model of Sec. 3.1.2. We employ Eqs. (\[chik\]) and (\[Vz\]), to obtain $$\begin{aligned} V_{\pmb x}(z) = \frac{\lambda}{8\sqrt{2} \pi^3} \int \frac{d^3k}{\|{\pmb k}\|(z - \|{\pmb k}\|)} e^{i {\pmb k} \cdot {\pmb x}}.\end{aligned}$$ We introduce spherical coordinates $(k, \theta, \phi)$ for ${\pmb k}$, so that $$\begin{aligned} V_{\pmb x}(z) = \frac{\lambda}{4\sqrt{2} \pi^2} \int_0^{\infty} \frac{kdk}{z- k} \int_0^{\pi} d\theta \sin \theta e^{ikr \cos \theta} = \frac{\lambda}{2\sqrt{2} \pi^2 r} \int_0^{\infty} \frac{dk \sin(kr)}{z-k}, \label{vzr}\end{aligned}$$ i.e., $V_{\pmb x}(z)$ depends only on the radial coordinate $r = \|{\pmb x}\|$. We evaluate the integral (\[vzr\]) to $$\begin{aligned} V_r(z) = \frac{\lambda}{2\sqrt{2} \pi^2r} \left[ [\gamma + \ln(-rz) + \mbox{Cin}(rz)] \sin rz - \mbox{si}(rz)\cos rz\right], \end{aligned}$$ where the functions $\mbox{Cin}$ and $\mbox{si}$ are defined as $$\begin{aligned} \mbox{Cin}(z) = \int_0^z dt \frac{1-\cos t}{t} \hspace{1cm} \mbox{si}(z) = \int_z^{\infty} dt \frac{\sin t}{t}, \label{CiSi}\end{aligned}$$ and $\gamma$ is the Euler-Mascheroni constant [@ASt]. We straightforwardly evaluate $$\begin{aligned} V_r^{\pm}(E;r) = \frac{\lambda}{2\sqrt{2} \pi^2r} \left[ [\gamma + \ln(rE) + \mbox{Cin}(rE)] \sin rE - \mbox{si}(rE)\cos rE \mp i \pi \sin(rE)\right].\end{aligned}$$ We assume that the detectors are located at macroscopic distance from the decaying atom, so that $\tilde{\Omega}r >> 1$. Then, the terms involving the trigonometric integrals vanish, and the imaginary part of $V_+(\Omega, r)$ dominates. Eq. (\[decayGl\]) gives $$\begin{aligned} P(t, r) = \frac{\lambda^2 \sin^2(\tilde{ \Omega} r)}{32 \pi^2 r^2} C e^{-\Gamma t}.\end{aligned}$$ The sinusoidal dependence on $r$ disappears if we average $P(t, r)$ over a thin shell of width $d >> \tilde{\Omega}^{-1}$ at distance $r$, since $\langle\sin^2(\tilde{\Omega}r)\rangle = \frac{1}{2}$. Then, the normalization condition (\[normalk\]) is satisfied for $C = (16 \tilde{\Omega})^{-1} $. Note that outside the exponential decay regime, the probability density (\[Glauber2\]) leads to different predictions from the persistence probability method. The latter predicts an asymptotic probability density decaying with $t^{-5}$. Eq. (\[mainampl44\]) leads to an asymptotic decay of ${\cal B}$ with $t^{-2}$, hence, the probability density (\[Glauber2\]) decays as $t^{-4}$. Conclusions =========== We presented an overview of the quantum description of decay processes. We showed that the emergence of the exponential decay law is explained in terms of a scale separation. In perturbative decays, the scale separation refers to energy: exponential decays emerge when the released energy associated to the decay is much larger the energy of the interaction, as described by the self-energy function. In non-perturbative decays, the scale separation refers to time: exponential decay emerges when the decay time-scale is much larger than the time-scale of coherence between different attempts of the particle to cross the barrier. Exponential decay may be extremely common, but it is not universal. It is not valid at very early and very late times, and in specific systems, it is not relevant at all. There is good experimental evidence for non-exponential decays, some of which pose persistent theoretical puzzles [@GSI]. Our increasing access and control of multi-partite/multi-particle systems is expected to uncover further unconventional types of decay—for example, memory effects due to interaction with an environment [@nonMark; @BKEC], effects due to the entanglement of the initial state [@AnHu2; @CVS17; @C18], or effects from particle statistics in many-particle systems [@ADCM; @TS11; @CL11; @DC11; @MG11]. Furthermore, studies of decay dynamics in many-particle quantum systems have demonstrated the need of a genuinely many-particle characterization [@PSC12; @HZHB13; @DC16], i.e., going beyond description in terms of single-particle observables. We believe that a significant upgrade of traditional methods for quantum decays will be needed, in order to address such challenges. Acknowledgements {#acknowledgements .unnumbered} ================ Research was supported by Grant No. E611 from the Research Committee of the University of Patras via the ”K. Karatheodoris” program. W. Pauli, [The Principles of Quantum Mechanics]{}, in [Encyclopedia of Physics]{}, edited by S. Flugge, Vol. 5/1 (Springer, Berlin 1958). J. C. Muga, R. S. Mayato, and I. L. Equisquiza, [Time in Quantum Mechanics, vol 1]{} (Springer 2008); J. G. Muga, A Ruschhaupt and A. Del Campo, [Time in Quantum Mechanics, vol 2]{} (Springer 2010). J. C. Muga and J. R. Leavens, [Arrival Time in Quantum Mechanics]{}, Phys. Rep. 338, 353 (2000). E. H. Hauge and J. A. Støvneng, [Tunneling Times: a Critical Review]{}, Rev. Mod. Phys. 61, 917 (1989); V. S. Olkhovsky and E. Recami, [Recent Developments in the Time Analysis of Tunnelling Processes]{}, Phys. Rep. 214, 339 (1992); R. Landauer and T. Martin, [Barrier Interaction Time in Tunneling]{}, Rev. Mod. Phys. 66, 17 (1994). L. Fonda, G. C. Ghirardi and A. Rimini, [Decay Theory of Unstable Quantum Systems]{}, Rep. Prog. Phys. 41, 587 (1978). A. Peres, [Non-exponential Decay Law]{}, Ann. Phys. 129, 33 (1980). T. Gorin, T. Prosen, T. Seligman and N. Znidaric, [Dynamics of Loschmidt echoes and fidelity decay]{}, Phys. Rep. 435, 33 (2006). T. D. Lee, [Some Special Examples in Renormalizable Field Theory]{}, Phys. Rev. 95, 1329 (1954). B. Rosenstein and L. P. Horwitz, [Probability Current versus Charge Current of a Relativistic Particle]{}, J. Phys. A: Math. Gen. 18, 2115 (1985). L. Landau and E. Lifschitz, [Quantum Mechanics: Non-Relativistic Theory (2nd ed.)]{} (Pergamon Press, 1965). A.M. Perelemov, V. S. Popov and M. V. Terent’ev, [Ionization of Atoms in an Alternating Electric Field]{}, Soviet Phys. JETP 23, 924 (1966). A. J. Bracken and G. F. Melloy, [Probability Backflow and a New Dimensionless Quantum Number]{}, J.Phys. A27, 2197 (1994). R. G. Winter, [Evolution of a Quasi-Stationary State]{}, Phys. Rev. 123, 1503 (1961). H. Ekstein and A. J. F. Siegert, [On a Reinterpretation of Decay Experiments]{}, Ann. Phys. (N.Y.) 68, 509 (1971). W. Heisenberg, [Quantum Theory and its Interpretations]{}, reprinted at [Quantum Theory and Measurement]{}, edited by J. A. Wheeler and W. H. Zurek (Princeton University Press, Princeton 1983). L. Fonda, G. C. Ghirardi, C. Omero, A. Rimini and T. Weber, [Quantum Theory of Sequential Decay Processes]{}, Phys. Rev. D18, 4757 (1978). C. Anastopoulos and N. Savvidou, [Time-of-arrival Probabilities for General Particle Detectors]{}, Phys. Rev. A86, 012111 (2012); [Time-of-Arrival Correlations]{}, Phys. Rev. A95, 032105 (2017); [Time of Arrival and Localization of Relativistic Particles]{}, arxiv:1807-06533. C. Anastopoulos, [Time-of-Arrival Probabilities and Quantum Measurements. III. Decay of Unstable States]{}, J. Math. Phys. 49, 022103 (2008). R. J. Glauber, [The Quantum Theory of Optical Coherence]{}, Phys. Rev. 130, 2529 (1963); [Coherent and Incoherent States of the Radiation Field]{}, Phys. Rev. 131, 2766 (1963). C Anastopoulos, S Shresta, and BL Hu, [Non-Markovian Entanglement Dynamics of two Qubits Interacting with a Common Electromagnetic Field]{}, Q. Inf. Proc. 8, 549 (2009). A. del Campo, J. Molina-Vilaplana, J. Sonner, [Scrambling the Spectral Form Factor: Unitarity Constraints and Exact Results]{}, Phys. Rev. D95, 126008 (2017). A. Chenu, I. L. Egusquiza, J. Molina-Vilaplana, and A. del Campo, [Quantum Work Statistics, Loschmidt Echo and Information Scrambling]{}, Sci. Rep. 8, 12634 (2018). A. S. Landsmann and U. Keller, [Attosecond Science and the Tunnelling Time Problem]{}, Phys. Rep. 547, 1 (2015). Yu. A. Litvinov et al.,[Observation of Non-Exponential Orbital Electron Capture Decays of Hydrogen-Like $^{140}Pr$ and $^{142}Pm$ Ions* ]{}, Phys. Lett. B664, 162 (2008); P. Kienle et al, [High-Resolution Measurement of the Time-Modulated Orbital Electron Capture and of the Decay of Hydrogen-like $^{142}Pm^{60+}$ Ions]{}, Phys. Lett. B, 726, 638 (2013). D. Bohm and D. Pines, [A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas]{}, Phys. Rev. 92, 609 (1953). M. Gell-Mann and K. A. Brueckner, [Correlation Energy of an Electron Gas at High Density]{}, Phys. Rev. 106, 364 (1957). C. Bloch, [Une Formulation Unifiée de la Théorie des Réactions Nucléaires]{}, Nucl. Phys. 4, 53 (1957). M. Namiki, [One-Particle Motions in Many-Particle Systems and the Optical Model in Nuclear Reactions]{}, Prog. Theor. Phys. 23, 629 (1960). H. Nakazato, M. Namiki, and S. Pascazio, [Temporal Behavior of Quantum Mechanical Systems]{}, Int. J. Mod. Phys. B10, 247 (1996). W. Weisskopf and E. P. Wigner, [Berechnung der Natürlichen Linienbreite auf Grund der Diracschen Lichttheorie]{}, Zeit. Phys. 63, 54 (1930). S. M. Barnett and P. M. Radmore, [Methods in Theoretical Quantum Optics]{} (Clarendon Press, Oxford 1997), Appendix 6. L. Van Hove, [Quantum-Mechanical Perturbations Giving Rise to a Statistical Transport Equation]{}, Physica 21, 517 (1955). B. Misra and E. C. G. Sudarshan, [The Zeno’s Paradox in Quantum Theory]{}, J. Math. Phys. 18, 756 (1977). P. Facchi, S. Pascazio, [Quantum Zeno Dynamics: Mathematical and Physical Aspects]{}, J. Phys. A: Math. Theor. 41, 493001 (2008). L. Mandelstam and I. Tamm, [The Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics]{}, J. Phys (USSR) 9 , 249 (1945). K. Bhattacharyya, [Quantum decay and the Mandelstam-Tamm energy inequality]{}, J. Phys. A: Math. Gen. 16, 2993 (1983). V. V. Dodonov and A. V. Dodonov, [Energy–Time and Frequency–Time Uncertainty Relations: Exact Inequalities]{}, Phys. Scr. 90, 074049 (2015) S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sundaram and Mark G. Raizen, [Experimental Evidence for non-Exponential Decay in Quantum Tunnelling]{}, Nature 387, 575 (1997); M. C. Fischer, B. Gutiérrez-Medina, and M. G. Raizen, [Observation of the Quantum Zeno and Anti-Zeno Effects in an Unstable System]{}, Phys. Rev. Lett. 87, 040402 (2001). N. Margolus and L. B. Levitin, [The Maximum Speed of Dynamical Evolution]{}, Physica D120, 188 (1998). L. B. Levitin and T. Toffoli, [The Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound is Tight]{}, Phys. Rev. Lett. 103, 160502 (2009). M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, [Quantum Speed Limit for Physical Processes]{}, Phys. Rev. Lett. 110, 050402 (2013); A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, [Quantum Speed Limits in Open System Dynamics]{}, Phys. Rev. Lett. 110, 050403 (2013). B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, [Quantum Speed Limits across the Quantum-to-Classical Transition]{}, Phys. Rev. Lett. 120, 070401 (2018); M. Okuyama and M. Ohzeki, [Quantum Speed Limit is Not Quantum]{} Phys. Rev. Lett. 120, 070402 (2018). S. Deffner and S. Campbell, [Quantum Speed Limits: from Heisenberg’s Uncertainty Principle to Optimal Quantum Control]{}, J. Phys. A: Math. Theor. 50, 453001 (2017). M. L. Goldberger and K. M. Watson, [Collision Theory]{} (John Wiley, 1967). R. G. Newton, [Scattering Theory of Waves and Particles]{} (Springer-Verlag, Berlin 1982). S. Weinberg, [Lectures on Quantum Mechanics]{} (Cambridge University Press, 2015). E. J. Hellund, [The Decay of Resonance Radiation by Spontaneous Emission]{}, Phys. Rev. 89, 919 (1953). M. Namiki and M. Mugibayashi, [On the Radiation Damping and the Decay of an Excited State]{}, Prog. Theor. Phys. 10, 474 (1953). C. Rothe, S. I. Hintschich, and A. P. Monkman, [Violation of the Exponential-Decay Law at Long Times]{}, Phys. Rev. Lett. 96, 163601 (2006). M. O. Scully and M. S. Zubairy, [Quantum Optics]{} (Cambridge University Press, 1997). C. Anastopoulos and B. L. Hu, [Two-Level Atom-Field Interaction: Exact Master Equations for non-Markovian Dynamics, Decoherence, and Relaxation]{}, Phys. Rev. A62, 033821 (2000). L. Fonda, G. C. Ghirardi and A. Rimini, [Interpretation of the Normalizable State in the Lee Model with Form Factor]{} Phys. Rev. 133, B196 (1964). R. Alzetta and E. d’ Ambrogio, [ Evolution of a Resonant State]{}, Nucl. Phys. 82, 683 (1966). M. Abramowitz and I. A. Stegun, [Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]{} (10th ed.). (New York, Dover 1972). E. Fermi, [Tentativo di una Teoria dei Raggi $\beta$]{}, La Ricerca Scientifica 2 (1933). F. L. Wilson, [Fermi’s Theory of Beta Decay]{}, Am. J. Phys. 36, 1150 (1968). W. N. Cottingham and D. A. Greenwood, [An Introduction to Nuclear Physics]{} (Cambridge University Press, 2004). N. I. Cummings and B. L. Hu, [Dynamics of Atom-Field Entanglement: Towards Strong-Coupling and non-Markovian Regimes]{}, Phys. Rev. A77, 053823 (2008). M. Lewenstein, J. Zakrzewski, T. W. Mossberg and J. Mostowski, [Non-Exponential Spontaneous Decay in Cavities and Waveguides]{}, J. Phys. B: At. Mol. Opt. Phys. 21, L9 (1988). A. Raczyfiski and J. Zaremba, [Threshold Effects in Photoionization and Photodetachment]{}, Phys. Rep. 235, 1 (1993). T. Jittoh, S. Matsumoto, J. Sato, Y. Sato, and K. Takeda, [Nonexponential Decay of an Unstable Quantum System: Small-Q-Value s-Wave Decay]{}, Phys. Rev. A 71, 012109 (2005). V. Dinu, A. Jensen, and G. Nenciu, [Nonexponential Decay Laws in Perturbation Theory of Near Threshold Eigenvalues]{}, J. Math. Phys. 50, 013516 (2009). G. Gamow, [Zur Quantentheorie des Atomkernes]{}, Zeit. Phys. 51, 204 (1928). R. W. Gurney and E. U. Condon, [Quantum Mechanics and Radioactive Disintegration]{}, Phys. Rev. 33, 127 (1929). G. García-Calderón, J. L. Mateos, and M. Moshinsky, [Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities]{}, Phys. Rev. Lett. 74 , 337 (1995). M. Peshkin, A. Volya, and V. Zelevinsky, [Non-Exponential and Oscillatory Decays in Quantum Mechanics]{}, Europhys. Lett. 107, 40001 (2014). D. Bohm, [Quantum Theory]{} (Prentice Hall, New York, 1951), pp. 257; E. P. Wigner, [Lower Limit for the Energy Derivative of the Scattering Phase Shift]{}, Phys. Rev. 98, 145 (1955). S. Kudaka and S. Matsumoto, [Questions Concerning the Generalized Hartman Eﬀect]{}, Phys. Lett. A375, 3259 (2011). C. Anastopoulos and N. Savvidou, [Quantum Temporal Probabilities in Tunneling Systems]{}, Ann. Phys. 336, 281 (2013). J. G. Muga, F. Delgado, A. del Campo, and G. García-Calderón, [The Role of Initial State Reconstruction in Short and Long Time Deviations from Exponential Decay]{}, Phys. Rev. A73, 052112 (2006). G. García-Calderón and R. Peierls, [Resonant States and their Uses]{}, Nucl. Phys. A265, 443 (1976). A. del Campo, G. Garcia-Calderon, and J. G. Muga, [Quantum Transients]{}, Phys. Rep. 476, 1 (2009). H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, [Colloquium: Non-Markovian Dynamics in Open Quantum Systems]{}, Rev. Mod. Phys. 88, 021002 (2016). M. Beau, J. Kiukas, I. L. Egusquiza, and A. del Campo, [Nonexponential Quantum Decay under Environmental Decoherence]{}, Phys. Rev. Lett. 119, 130401 (2017). A. del Campo, F. Delgado, G. García-Calderón, and J. G. Muga, [Decay by Tunneling of Bosonic and Fermionic Tonks-Girardeau Gases]{}, Phys. Rev. A74, 013605 (2006). T. Taniguchi and S. I. Sawada, [Escape Behavior of Quantum Two-Particle Systems with Coulomb Interactions]{}, Phys. Rev. E83 , 026208 (2011). G. García-Calderón, L. G. Mendoza-Luna, [Time Evolution of Decay of Two Identical Quantum Particles]{}, Phys. Rev. A84, 032106 (2011). A. del Campo, [Long-time behavior of many-particle quantum decay]{}, Phys. Rev. A 84, 012113 (2011). A. Marchewka and E. Granot, [Role of Quantum Statistics in Multi-Particle Decay Dynamics]{}, Ann. Phys. 355, 348 (2011). M. Pons, D. Sokolovski, A. del Campo, [Fidelity of Fermionic-Atom Number States Subjected to Tunneling Decay]{}, Phys. Rev. A 85, 022107 (2012). S. Hunn, K. Zimmermann, M. Hiller, and A. Buchleitner, [Tunneling Decay of Two Interacting Bosons in an Asymmetric Double-Well Potential: A Spectral Approach]{}, Phys. Rev. A 87, 043626 (2013). A. del Campo, [Exact Quantum Decay of an Interacting Many-Particle System: the Calogero–Sutherland Model]{}, New J. Phys. 18, 015014 (2016). L. A. Khalfin, [Contribution to the Decay Theory of a Quasi-Stationary State]{}, Sov. Phys.–JETP6, 1053 (1958). N. Wiener and R. E. A. C. Paley, [Fourier Transforms in the Complex Domain]{}, Amer. Math. Soc. (1934), Theorem XII, p. 18. A. Leggett, S. Chakravarty, A. Dorsey, M. Fisher, A. Garg, and W. Zwerger, [Dynamics of the Dissipative Two-State System]{}, Rev. Mod. Phys. 59, 1 (1987). Further results =============== The verification of the statements in this Appendix can be used as exercises. 1\. The persistence amplitude cannot give an exponential decay law at all times [@Khalfin]. Let $\|a\rangle$ stand for the eigenvectors of the Hamiltonian $\hat{H}$ with eigenvalues $E_a$. Then, the persistence amplitude (\[persiss\]) can be written as ${\cal A}_{\psi}(t) = \int dE \omega(E)e^{-iEt}$, where $$\begin{aligned} \omega(E) = \sum_a \delta(E-E_a) \|\langle a\|\psi\rangle\|^2.\end{aligned}$$ If the Hamiltonian is bounded from below, with a minimum energy $E_{min}$, then $\omega(E) = 0 $ for $E < E_{min}$. A theorem by Payley and Wiener [@PaWi] asserts that if $\omega(E) = 0 $ for $E < E_{min}$, then its Fourier transform $A(t)$ satisfies $$\begin{aligned} \int_{-\infty}^{\infty} dt \frac{ \log\|A(t)\|}{1+t^2} < \infty.\end{aligned}$$ This implies that $\|{\cal A}_{\psi}(t)\|$ decays at most as $e^{-c t^{1 - p}}$ for $c > 0$ and $p >0$, i.e., more slowly than exponential decay. 2\. A different Hamiltonian defined on the Hilbert space of Lee’s model is the so called spin-boson Hamiltonian, $\hat{H}_{SB}$. The spin-boson model [@spbo] is more fundamental, in the sense that it can be derived from first principles with fewer assumptions than Lee’s Hamiltonian. The spin-boson Hamiltonian is of the form $\hat{H}_{SB} = \hat{H}_A + \hat{H}_B + \hat{V}'$, where $\hat{H}_A$ and $\hat{H}_B$ are given by Eqs. (\[vlee1\]) and (\[vlee2\]) respectively, and $$\begin{aligned} \hat{V}' = \hat{\sigma}_1 \sum_a \left(g_r \hat{a}_r + g_r^* \hat{a}_r^{\dagger} \right). \label{vsb}\end{aligned}$$ Since $\hat{\sigma}_1 = \hat{\sigma}_+ + \hat{\sigma}_-$, the interaction term $\hat{V}'$ includes terms $\hat{\sigma}_+ \hat{a}^{\dagger}_r$ and $\hat{\sigma}_- \hat{a}_r$, in addition to the ones of $\hat{V}$ of Εξ. (\[vlee3\]). Assuming the RPA, we find that the self-energy function $\Sigma(z)$ is given by Eq. (\[selee\]) as is the Lee model. Hence, $$\begin{aligned} \langle A' \|(z -\hat{H}_{SB})^{-1}\|A'\rangle = \langle A \|(z -\hat{H}_L)^{-1}\|A'\rangle = \frac{1}{z - \Omega - \Sigma(z)}.\end{aligned}$$ Within the accuracy of the RPA, the spin-boson and Lee’s Hamiltonian lead to the same predictions. 3\. The correct electromagnetic description of photonic emission has to take into account photon polarization. In this case, the basis $r$ corresponds to momenta ${\pmb k}$ and polarization directions $\sigma = 1, 2$; summation over $r$ corresponds to integration over ${\pmb k}$ with measure $\frac{d^3k}{(2\pi)^3}$ and summation over $\sigma$. The coupling coefficients are the form $$\begin{aligned} g_{{\pmb k}, \sigma} = \frac{\lambda }{\sqrt{\omega_{\pmb k}}} [{\pmb n}\cdot{\pmb \omega}_{\sigma} ({\pmb k})] e^{i {\pmb k}\cdot{\pmb r}}\end{aligned}$$ where $\lambda $ is a dimensionless constant, ${\pmb n}$ is a unit vector, and ${\pmb r}$ is the position vector of the atom. The polarization vectors $\epsilon^i({\pmb k})$ have unit norm, they can be chosen to be real, they are transverse: ${\pmb \epsilon}_{\sigma} ({\pmb k})\cdot {\pmb k} = 0$, and they satisfy $$\begin{aligned} \sum_{\sigma} \epsilon^i_{\sigma}({\pmb k}) \epsilon^j_{\sigma}({\pmb k}) = \delta^{ij} - \frac{k^ik^j}{{\pmb k}^2}. \nonumber \end{aligned}$$ Then, the self-energy function is $$\begin{aligned} \Sigma (z) = \frac{\lambda^2}{3 \pi^2} \int_0^{\Lambda} \frac{k dk}{z-k}, \end{aligned}$$ i.e., it differs from Eq. (\[seel4\]) by a factor $\frac{2}{3}$. 4\. Consider the model of Sec. 3.1.2 in one spatial dimension, describing, for example, the decay of an atom inside an one dimensional cavity of width much smaller than the wave-length of the emitted radiation. Using the same coupling as in Sec. 3.1.2, the self-energy is $$\begin{aligned} \Sigma(z) = \frac{\lambda^2}{\pi} \int_0^{\infty} \frac{dk}{k(z-k)}.\end{aligned}$$ The integral above is finite at large $k$ but diverges at $k = 0$. W change the integration range to $[\kappa, \infty)$, where $\kappa$ is an infrared cut-off. Then, $$\begin{aligned} \Sigma(z) = -\frac{\lambda^2}{\pi z} \ln \left(1 - \frac{z}{\kappa}\right). \label{sigma1s}\end{aligned}$$ Eq. (\[sigma1s\]) implies that $\Gamma(E) = 2\lambda^2/(\kappa + E)$. 5\. We consider a two-level atom of frequency $\Omega$ in an engineered reservoir with a frequency distribution sharply peaked around $\omega_0$. To this end, we employ the model of Sec. 3.1.2 with coupling coefficients $g_{\pmb k} = \frac{\lambda}{\sqrt{\omega_k}} \sqrt{\omega_0 \chi(\omega_{\pmb k}, \omega_0)}$, where $\chi(\omega, \omega_0)$ is a probability distribution on $[0, \infty)$ sharply peaked around $\omega_0$ with a width $\gamma << \omega_0$. The self-energy function equals $$\begin{aligned} \Sigma(z) = -\frac{\lambda^2 \omega_0}{2 \pi^2} + \frac{\lambda^2 \omega_0}{2 \pi^2}z \int_{0}^{\infty} \frac{dk f(k, \omega_0)}{z- k}. \label{2LSres}\end{aligned}$$ Since the integrand in Eq. (\[2LSres\]) is peaked around $\omega_0$, we can extend the $k$ integration to $(-\infty, \infty)$. In this approximation, we can choose a Lorentzian function for $\chi$, $$\begin{aligned} \chi(\omega, \omega_0) = \frac{1}{\pi} \frac{\gamma}{(\omega-\omega_0)^2 + \gamma^2}.\end{aligned}$$ Then, $$\begin{aligned} \Sigma(z) = \Gamma \frac{\omega_0- i \gamma}{z - \omega_0 + i \gamma},\end{aligned}$$ where $\Gamma = \frac{\lambda^2 \omega_0}{2 \pi^2}$. $\Sigma(z)$ has no branch points. Hence, the persistence amplitude is solely determined by the solutions of Eq. $z - \Omega- \Sigma(z) = 0$. They correspond to the two roots of the binomial equation $x^2 - (\delta - i \gamma) x - \Gamma (\omega_0 - i \gamma) = 0$, where $x = z - \Omega$, and $\delta := \omega_0 - \tilde{\Omega}$. Two limits are particularly interesting. For exact resonance, $\delta = 0$, the roots to leading order in $\gamma/\omega_0$ are $z_{\pm} = \Omega \pm \sqrt{\Gamma \omega_0 - \frac{1}{4}\gamma^2} - i \frac{\gamma}{2}$. The persistence amplitude exhibits oscillations at frequency $\sqrt{4 \Gamma \omega_0 - \gamma^2}$, and decays exponentially with a decay constant $\gamma$. For a sharply monochromatic cavity with $\gamma \rightarrow 0$, both roots are real valued. There is no decay, and the persistence amplitude describes vacuum Rabi oscillations of frequency $\sqrt{\delta^2 + 4 \Gamma^2 \omega_0^2}$. 6\. We generalize Lee’s model in order to describe the decay of an entangled pair of 2LSs. The Hamiltonian is of the form $\hat{H}_0 + \hat{V}$ with $$\begin{aligned} \hat{H}_0= \frac{1}{2} \Omega (\hat{1} + \hat{\sigma}^{(1)}_3) + \frac{1}{2} \Omega (\hat{1} + \hat{\sigma}^{(2)}_3) + \sum_r \omega_r \hat{a}^{\dagger}_r \hat{a}_r, \label{2h0} \\ \hat{V} = \sum_r \left(g^{(1)}_{r} \hat{\sigma}_+^{(1} \hat{a}_r + g_{r}^{(1)} \hat{\sigma}_-^{(1)} \hat{a}_r^{\dagger} \right) + \sum_r \left(g_{r}^{(2)} \hat{\sigma}_+^{(2)} \hat{a}_r + g_r^{(2)} \hat{\sigma}_-^{(2)} \hat{a}_r^{\dagger} \right), \label{2v}\end{aligned}$$ where the upper indices $(1)$ and $(2)$ label the 2LS. For identical 2LS the absolute value of the coupling constants $\|g^{(i)}_r\|$ is the same for both atoms: $\|g^{(1)}_r\| = \|g^{(2)}_r\| = g_r$. Hence, we write $g^{(i)}_r = g_r e^{i \Theta^{(i)}_r}$. We consider an initial state $\|B\rangle \otimes\|0\rangle$, where $\|0\rangle$ is the field vacuum and $\|B\rangle$ is a Bell-type state $$\begin{aligned} \|B\rangle = \frac{1}{\sqrt{2}}\left(\|e,g\rangle \pm \|g, e\rangle \right).\end{aligned}$$ The self-energy function $\Sigma_B$ is $$\begin{aligned} \Sigma_B(z) = \Sigma_0(z) + \frac{1}{2} [ \Sigma^{(1)}(z) + \Sigma^{(2)}(z) ], \label{seef2}\end{aligned}$$ where $\Sigma_0(z)$ is the self-energy function for a single 2LS, given by Eq. (\[selee\]), and $$\begin{aligned} \Sigma^{(i)}(z) = \sum_r \frac{g_r^2e^{2i\Theta_r^{(i)}}}{z- \omega_r}.\end{aligned}$$ For the scalar photons of Sec. 3.1.3, we substitute $r$ by the continuous momentum variable ${\pmb k}$ and write $g_{\pmb k} = \lambda/\|\sqrt{{\pmb k\|}}$. We choose a coordinate system so that the first 2LS is located at $+\frac{1}{2} {\pmb r}$ and the second at $+\frac{1}{2} {\pmb r}$. Hence, $\Theta^{(1)}_{\pmb k} = \frac{1}{2} {\pmb k} \cdot {\pmb r} = - \Theta^{(2)}_{\pmb k}$. Then, $\Sigma^{(1)}(z) = \Sigma^{(2)}(z) = \Sigma_r(z)$, where $$\begin{aligned} \Sigma_r(z) = \frac{\lambda^2}{2\pi^2 r} \left[ [\gamma + \ln(-rz) + \mbox{Cin}(rz)] \sin rz - \mbox{si}(rz)\cos rz\right], \end{aligned}$$ where the functions $\mbox{Cin}$ and $\mbox{si}$ are given by Eq. (\[CiSi\]). The decay constant is $$\begin{aligned} \Gamma = \frac{\lambda^2\tilde{\Omega}}{ \pi} (1 \pm \frac{\sin (\tilde{\Omega} r)}{\tilde{\Omega}r}).\end{aligned}$$ 7\. We consider tunneling decays of a particle of mass $m$ through a delta-function potential barrier $V(x) = \eta \delta(x-a)$. The transmission and reflection amplitudes on the real line are, $$\begin{aligned} T_k = \frac{1}{1 - \frac{im\eta}{k}} \hspace{1cm} R_k = - \frac{e^{2ika}}{1+ \frac{ik}{m\eta}}.\end{aligned}$$ For an almost monochromatic initial state with energy $\frac{k_0^2}{2m}$, the opaque barrier condition $\|T_k\|<<1$, implies that $\frac{k_0}{m \eta} << 1$. It follows that $g := m \eta a >> 1$. By Eq. (\[Gammat\]), the dominant contribution to the decay rate is $$\begin{aligned} \Gamma = \frac{k_0^3}{2m^3 \eta^2a}, \label{gammatunnel}\end{aligned}$$ up to corrections of order $g^{-1}$. As shown in Sec. 5.3.1, we can identify the decay rate by finding the poles $q_n - i \gamma_n$ of $\frac{T_k}{1+R_k}$. To leading order in $g^{-1}$, $$\begin{aligned} q_n = \frac{n\pi}{a}( 1 + \frac{1}{2g}) \hspace{1cm} \gamma_n = \frac{n^2\pi^2}{4 a g^2},\end{aligned}$$ where $n$ is a positive integer. For $n$ such that $q_n \simeq k_0$, Eq. (\[decaytb\]) for the decay rate reproduces Eq. (\[gammatunnel\]). [^1]: anastop@physics.upatras.gr [^2]: For $\tau$ such that $a(\tau) = 0$, Eq. (\[mata3\]) leads to the uncertainty relation $\Delta H \tau \geq \frac{\pi}{4}$. The time $\tau$ is interpreted as the minimum time required for the system to arrive to a state orthogonal to $\|\psi\rangle$, and the uncertainty relation is said to define a limit to the ‘speed of quantum evolution’, and, consequently, to the speed of quantum computation. The inequality has been improved [@MaLe; @LeTo] in order to also incorporate the case of $\Delta H \rightarrow \infty$. It has then been broadly generalized, for example, to open-system dynamics [@MLOS] and classical systems [@Shan]—for a review, see, Ref. [@DeCa]. [^3]: We can use a different integration contour, consisting of the positive real axis, an arc of the circle at infinity, and a half line $N$ that starts from the latter and ends at the origin—for an example, see, Fig. 5. As long as the contour encloses the physically relevant poles near the positive real axis, the analysis remains unchanged. Then, $I_a(t)$ is a line integral along $N$. The choice of the negative imaginary axis for $N$ is particularly useful for calculating the long time limit of Eq. (\[amplnew\]). In fact, we use it implicitly in the derivation of Eq. (\[iat0\]). [^4]: Hence, the WWA justifies a common statement of scattering theory, that unstable particles correspond to poles of the $S$-matrix on the second Riemann sheet [@colth; @RNewt]. Indeed, the poles of the S-matrix provide a simple way for identifying the basic properties of an unstable state. However, the $S$-matrix formalism gives a coarse description of a decay process. It enforces the constancy of transition rates [@Weinberg] by averaging over a long time interval, and, for this reason, it cannot discern transient phenomena, including deviation from exponential decay. [^5]: The coupling originates from an interaction Hamiltonian of the form $\lambda[\hat{\sigma}_-\hat{\phi}^{(-)}({\pmb r}) + \hat{\sigma}_+\hat{\phi}^{(+)}({\pmb r})] $, where the scalar field components $\hat{\phi}^{(\pm)} ({\pmb r}) $ are given by Eqs. (\[scalar\], \[chik\]). [^6]: In terms of the standard parameters of the theory of weak interactions, $\mu^{-2} = G_F V_{ud} {\cal M}_n$, where $G_F$ is Fermi’s constant, $V_{ud}$ is an element of the Kobayashi-Maskawa matrix, and ${\cal M}_n$ the amplitude for the nuclear transition [@nuclear]. The dimensionality of $\mu$ comes from the inverse mass squared dimension of Fermi’s constant. [^7]: The non-zero probability to cross the barrier needs not be quantum mechanical in origin. The particle may interact with a stochastic environment, such as a thermal bath. Then, the crossing of the barrier may be due to a random force.
--- abstract: 'We predict a dynamic metallization effect where an ultrafast (single-cycle) optical pulse with a $\lesssim 1$ V/ field causes plasmonic metal-like behavior of a dielectric film with a few-nm thickness. This manifests itself in plasmonic oscillations of polarization and a significant population of the conduction band evolving on a $\sim 1$ fs time scale. These phenomena are due a combination of both adiabatic (reversible) and diabatic (for practical purposes irreversible) pathways.' author: - Maxim Durach - Anastasia Rusina - 'Matthias F. Kling' - 'Mark I. Stockman' title: 'Ultrafast Dynamic Metallization of Dielectric Nanofilms by Strong Single-Cycle Optical Fields' --- Latest advances in the ultrafast optics have recently attracted a great deal of attention. Ultrashort pulses have been successfully employed for monitoring and manipulation of electronic processes in atomic and molecular structures. [@Krausz_Ivanov_RevModPhys.81.163_2009_Attosecond_Review] Significant efforts have been directed toward exploring the potential of ultrashort ($\sim 100$ as to $\sim 1-10$ fs in duration) pulses in application to condensed matter dynamics [@Murnane_et_al_PRL_97_113604_2006_Laser_Assisted_Photoelectric_Effect_from_Surfaces; @Cavalieri_et_al_Nature_2007_Attosecond_Photoemission_from_Solids; @Murnane_Kapteyn_et_al_PRL_2009_Demagnetization_Dynamics; @Murnane_PRA_2009_Dressed_Electronic_Processes_at_Surfaces; @Murnane_et_al_Nature_Materials_2010_Ballistic_Thermal_Transport; @Stockman_et_al_J_Phys_2009_PEEM; @Reis_et_al_Nature_Phys_2011_HHG_from_ZnO_Crystal], in particular, to plasmonic metal and dielectric nanostructures. [@Stockman_Hewageegana_APA_2007_CEP; @Stockman_Kling_Kleineberg_Krausz_Nature_Photonics_2007; @Kling_et_al_Nature_Phot_2011_Dielectric_Sphere_Ultrafast_Photoemission] We have recently predicted that dielectric nanofilms subjected to strong but sufficiently slow (adiabatic) electric fields undergo a reversible change resembling a quantum phase transition to a state that exhibits metallic optical properties. [@Stockman_et_al_PRL_2010_Metallization] We have called this phenomenon metallization. The minimum duration of such an adiabatic field depends exponentially on the thickness of the nanofilm and is in the range from $\gtrsim 10$ fs to $\sim 10$ ns for a film thickness from a few nm to $\sim 10$ nm. [@Stockman_et_al_PRL_2010_Metallization] Both from the fundamental point of view and for applications to ultrafast nanoelectronics, the metallization by much faster optical fields is of great interest. In this Letter, we theoretically predict a new effect that we call [dynamic metallization]{}, where a single-cycle optical pulse incident on a $\sim 2$ nm dielectric nanofilm with a normal polarization and a field $\lesssim 1$ V/ causes a population of the conduction band and metal-like plasmonic polarization oscillations on the optical-period time scale. This effect is caused by both adiabatic (reversible) and diabatic (dissipative) excitation pathways involving band anticrossings and adiabatic evolution between them. A comprehensive solution of the ultrafast electron dynamics in strong optical fields would require many-body quantum kinetics, rendering this problem extremely complicated. To simplify it, we rely on the fact that the characteristic inelastic electron-scattering time is on the order or greater than the surface plasmon decay time $\tau_n$, which is $\tau_n\gtrsim 10$ fs for metals – see, e.g., Fig. 1 (a) in Ref. . Using ultrashort excitation pulses with duration $\tau\ll \tau_n$, we avoid any significant effect of the electron inelastic scattering. This allows us to treat the electron dynamics as Hamiltonian. The evolution of the system in this case is convenient to describe by the density matrix $$\hat\rho(\mathbf r^\prime,\mathbf r; t)=\sum_{i\le i_F} \Psi_i(\mathbf r^\prime, t) \Psi_i^\ast(\mathbf r, t)~, \label{rho}$$ where $i_f$ denotes the Fermi-surface state, i.e., the highest occupied state for the zero-field Hamiltonian $\hat H_0$, and $\Psi_i(\mathbf r^\prime, t)$ are the one-electron wave functions. These satisfy the Schrödinger equation $i\hbar\dot \Psi_i=\hat H(t)\Psi_i$, where $\hat H(t)$ is the Hamiltonian depending on time $t$ due to the optical field, and the dot denotes the derivative over $t$. Consider a thin nanofilm where the energy bands are split into subbands due to the quantum confinement in the direction normal to the film plane. We assume that, due to the material symmetry, the electron wave function can be factorized into normal and parallel to the film. Assuming a normal optical electric field $\mathcal E=\mathcal E(t)$, the one-particle Hamiltonian of the transverse motion is $\hat H(\mathcal E) = \hat H_0 - \mathcal E \hat d$, where $\hat d$ is the dipole operator. Consider the adiabatic basis of states $\psi_i(\mathcal E)$ that diagonalize the instantaneous Hamiltonian, $\hat H(\mathcal E) \psi_i(\mathcal E) = E_i(\mathcal E)\psi_i(\mathcal E) $, where $E_i(\mathcal E)$ are the adiabatic energies. We employ the Kronig-Penney model for an insulator with $E_g=4.8$ eV band gap at the zero field (simulating diamond) whose adiabatic subband energies $E_i(\mathcal E)$ of the valence (red) and conduction (blue) bands are shown in Fig. \[adiabatic\_levels.eps\] (a) as functions of the applied field $\mathcal E$. All calculations are done for a $l=2$ nm thickness film. We expand wave functions $\Psi_i(t)$ in the adiabatic basis, $\Psi_i(t)=\sum_j \mathrm{exp}[-i \varphi_j(t)] a_j^{(i)}(t) \psi_j\left(\mathcal E(t)\right)$, where $a_j^{(i)}(t)$ are the expansion coefficients with the initial condition $a_j^{(i)}(0)=\delta_{ij}$, and $\varphi_j(t)=-(1/\hbar)\int E_j \left(\mathcal E(t)\right) dt$ is the adiabatic phase. Then the Schrödinger equation becomes $$\begin{aligned} &&\dot a_j^{(i)}=-\sum_{k\ne j}\dot\Theta_{jk} \mathrm{exp}\left[-i\varphi_{jk}(t)\right] a_k^{(i)}~,~~~ \label{dota} \\ &&\dot{\Theta}_{jk}\equiv -\dot{\mathcal E} d_{jk}\left(\mathcal E\right)/E_{jk}\left(\mathcal E\right)~, \label{nangle}\end{aligned}$$ where the adiabatic dipole matrix elements, transition energies, and relative phases are $d_{jk}(\mathcal E)=\langle \psi_k(\mathcal E)\|\hat d_0\|\psi_j(\mathcal E)\rangle$ and $E_{jk}(\mathcal E)=E_j(\mathcal E)-E_k(\mathcal E)$, $\varphi_{jk}(t)=\varphi_{j}(t)-\varphi_{k}(t)$, correspondingly. ![(a) Energy spectrum of the nanofilm as a function of the adiabatically applied electric field. The occupied valence subbands are shown in red, the empty conduction subbands are in blue. (b) The diabatic coupling matrix element $\dot \Theta_{ik}$ between band-edge subbands \[see Eq. (\[nangle\])\] for different pulse amplitudes $\mathcal E_0$, as indicated on the panel. []{data-label="adiabatic_levels.eps"}](adiabatic_levels.eps){width=".48\textwidth"} Under the adiabatic conditions [@Stockman_et_al_PRL_2010_Metallization], a strong electric field causes the band gap $E_g$ to decrease. The valence and conduction bands experience anticrossing at the metallization field $\mathcal E_m=0.75$ V/ with the anticrossing gap $\Delta E_m=0.45$ eV – see Fig. \[adiabatic\_levels.eps\] (a). If the field is increased adiabatically above $\mathcal E_m$, the electrons are adiabatically (reversibly) transferred to the conduction band (hole) states and in space across the film. This is the metallization transition where the optical properties of the nanofilm resemble those of a plasmonic metal. [@Stockman_et_al_PRL_2010_Metallization] If the field is adiabatically switched off, the system returns to its ground state. The condition of the adiabaticity is evident from Eqs. (\[dota\])-(\[nangle\]) and is $t_p\gg \hbar/\Delta E_m$, where $t_p$ is the time needed for the field $\mathcal E(t)$ to pass through the anticrossing, in full agreement with Ref. . In the opposite case of a fast diabatic passage of the anticrossing, the Schrödinger equation (\[dota\]) for the valence and conduction band-edge subbands, $v$ and $c$, can be integrated yielding the population of the conduction band $$n_c(t)=\mathrm{sin}^2 \Theta_{vc}\left[\mathcal E(t)\right]n/n_{sb}~,~~ \label{diabatic_excitation}$$ where $n$ is the electron density and $n_{sb}$ is the number of the occupied subbands, $n_{sb}=9$ in the present model. Such rapid fields, in contrast to the adiabatic case, [@Stockman_et_al_PRL_2010_Metallization] do not induce the metal-like polarization: there is no spatial population transfer across the nanofilm. ![Polarization $\mathcal P$ and conduction-band population $n_c$ as functions of time $t$ for various $\mathcal E_0$ and $\tau$. Normalized pulse field $\mathcal E(t)/\mathcal E_0$ is shown by blue line. Normalized population $n_c(t)/n$ (scaled $\times 100$) is displayed by the red curve; the same in DA is shown by the dashed gray curve. Relative polarization $\mathcal P/\mathcal E_0$ is displayed by the green line; the same in AA is given by the dashed yellow line. The pulse length is $\tau=0.85$ fs for (a)-(c) and $\tau=3.4$ fs for (d). []{data-label="amplitude_dependence.eps"}](amplitude_dependence.eps){width=".48\textwidth"} We consider the electron dynamics of a nanofilm subjected to an ultrafast field where both the adiabatic and diabatic processes contribute. The fastest dynamics is driven by single-cycle light pulses with duration $\tau\sim 1$ fs, which have recently been achieved. [@Goulielmakis_et_al_Single_Cycle_Nonlinear_Optics_Science_2008] Here, we model a single-cycle pulse $\mathcal E(t)$ by a waveform [@Rastunkov_Krainov_one_cycle_pulse_shape] $$\mathcal E(t) = \mathcal E_0 e^{-u^2}\left(1-2u^2\right)~,~~~u\equiv t/\tau~, \label{pulse_shape}$$ where the amplitude is $\mathcal E_0$, and the pulse duration is $\tau$. The pulse integral is zero, $\int_{-\infty}^{\infty}\mathcal E(t) dt=0$, as should be. For such a pulse, in Fig. \[adiabatic\_levels.eps\] (b) we display the diabatic coupling matrix element $\dot\Theta_{ik}$ between the valence and conduction band-edge subbands. Note that $\dot\Theta_{ik}\propto 1/\tau$. The peaks of the diabatic coupling element $\dot\Theta_{ik}$ are at the adiabatic metallization points (band-edge anticrossings), and they grow with the excitation-pulse amplitude. Now let us turn to the dynamics of the system excited by an ultrashort pulse with $\tau=0.85$ fs (the mean frequency $\hbar\omega_0=2\hbar/\tau=1.55$ eV). This is illustrated in Figs. \[amplitude\_dependence.eps\] (a)-(c) where we show conduction band population $n_c$ and polarization $\mathcal P=\mathrm{Tr}\{\hat d\rho\}/V$, where $V$ is the nanofilm’s volume, as functions of time $t$ for different pulse-field amplitudes $\mathcal E_0$. For comparison, we also show the excitation waveform and results obtained in the adiabatic (AA) and diabatic (DA) approximations. As Fig. \[amplitude\_dependence.eps\] (a) shows, for field $\mathcal E$ significantly below the adiabatic metallization threshold, $\mathcal E_m=0.75$ V/, the AA polarization [@Stockman_et_al_PRL_2010_Metallization] and DA population (\[diabatic\_excitation\]) follow the pulse. The computed polarization $\mathcal P$ (green curve) is close to the adiabatic case except for a small delay and low-amplitude oscillations on the pulse trailing edge with frequency $\approx E_g/\hbar$. This is due to the short duration of the pulse, which leaves at the end a partial coherence between the valence and conduction bands. The population $n_c$ (red curve) is small and dramatically retarded with respect to both the pulse and the DA curve, which is characteristic of the perturbative excitation. At the threshold of the adiabatic metallization, $E\approx\mathcal E_m$, as illustrated in Fig. \[amplitude\_dependence.eps\] (b), the conduction band population $n_c$ dramatically increases. The calculated dependence $n_c(t)$ (red line) agrees well with the DA, except for the residual population after the pulse. The significant deviation of $n_c(t)$ from the DA (gray dash line) starts at the moment of the second anticrossing on the trailing pulse tail ($t/\tau\approx 0.1$) where the diabatic coupling peaks – cf. Fig. \[adiabatic\_levels.eps\] (d). This adiabaticity violation causes the significant residual population $n_{cr}=n_c(t\gg \tau)$ and is dependent on the adiabatic phase $\varphi$ as will be discussed below in conjunction with Fig. \[dynamics\_summary.eps\]. In Fig. \[amplitude\_dependence.eps\] (b), the polarization $\mathcal P(t)$ is retarded by an almost quarter pulse length ($\approx \pi/2$ in phase) with respect to the driving pulse, which implies a strong absorption. There also coherent oscillations after the end of the excitation pulse. All this is characteristic of plasmonic metal systems. [@Stockman:2002_PRL_control; @Stockman_Hewageegana_APA_2007_CEP] The polarization oscillations exhibit beatings between the frequency of the interband and much slower intraband transitions. The latter are caused by the pulse imprinting its frequency by polarizing the hot carriers in the conduction band. We call this effect [the dynamic metallization]{}. It is an ultrafast and dissipative strong-field transition to a plasmonic metal-like behavior. A similar phase delay between the excitation field and the polarization oscillations has been computed and attributed to the appearance of free electrons in the time-dependent density-functional theory [@Otobe_et_al_PRB_2008_Metalization_in_Diamond] of breakdown in bulk dielectrics subjected to high optical fields. Note that such a breakdown for quasi-stationary fields was introduced by Zener. [@Zener_Proc_Royal_Soc_1934_Breakdown] Importantly, the present dynamic metallization in thin films is fundamentally different. It is based on the adiabatic contribution to polarization, depends critically on the film thickness, and occurs at much lower intensities: our field $\mathcal E\lesssim 1$ V/ corresponds to the radiation intensity $I\lesssim 3\times 10^{13}~\mathrm{W/cm^2}$, in contrast to $I\sim 10^{15}~\mathrm{W/cm^2}$ in Ref. . For the 0.85-fs pulse with amplitude $\mathcal E_0=0.94$ V/, which is significantly greater than the adiabatic metallization-threshold field $\mathcal E_m=0.75$ V/ \[Fig. \[amplitude\_dependence.eps\] (c)\], the dynamic metallization phenomena become even more developed. The magnitudes of population $n_c$ and polarization $\mathcal P$ increase. The field time dependence $n_c(t)$ shows a pronounced saturation between the metallization (anticrossing) points at $t/\tau\approx \pm 0.2$. The residual population forms due to the adiabaticity violation at the anticrossing at $t/\tau\approx 0.2$ and is relatively large because of the large diabatic coupling at this instance – cf. the corresponding (blue) curve in Fig. \[adiabatic\_levels.eps\] (d). The polarization shows a pronounced plasmonic metal-like behavior: an approximately quarter-oscillation delay with respect to the excitation pulse and the oscillations with a lower frequency, which is close to the pulse mean frequency $\omega_0$. The excitation dynamics for a longer pulse with $\tau=3.4$ fs is shown in Fig. \[amplitude\_dependence.eps\] (d). The main difference from panel (c) is that the polarization $\mathcal P(t)$ peaks almost simultaneously with the excitation pulse maximum, as characteristic of the adiabatic metallization – cf. Ref. . Still there are the residual population $n_{cr}$ and oscillations of $\mathcal P(t)$ after the end of the excitation pulse, which imply non-adiabatic processes occurring at the level anticrossings. For the strongly-nonlinear and dispersive problem under consideration, a useful measure of the magnitude of the system’s polarizability can reasonably be defined as the effective permittivity $$\varepsilon_0=1+4\pi \mathcal P_0/\mathcal E_0~, \label{epsilon_0}$$ where $\mathcal P_0$ is the maximum value of the polarization in the process. Note that the maximum of $\mathcal P(t)$ is generally delayed in time with respect to that of $\mathcal E(t)$. For the shorter pulses in Figs. \[amplitude\_dependence.eps\] (b)-(c), this delay is approximately $\approx1/4$ of the oscillation length ($\approx \pi/2$ in phase), which implies a strong dissipation (a significant imaginary part of the complex permittivity). In Fig. \[epsilon0.eps\] (a)-(b), we plot $\varepsilon_0$ as functions of the excitation-pulse amplitude $\mathcal E_0$ and duration $\tau$. For very short pulses with $\tau\le 0.85$ fs, $\varepsilon_0$ slowly increase with $\mathcal E_0$ due to contribution of perturbative nonlinear absorption. The magnitude $\varepsilon_0\sim 20$ is rather large because of the wide, high-frequency spectrum of the short pulses. Close to and above the adiabatic metallization threshold, $\mathcal E_0\gtrsim\mathcal E_m=0.75$ V/, this effective permittivity dramatically increases for longer pulses with $\tau\gtrsim 2$ fs, suggesting that it is dominated by the adiabatic metallization mechanism. In fact, for $\tau=5$ fs, the dependence $\varepsilon_0(\mathcal E_0)$ resembles that for the adiabatic permittivity [@Stockman_et_al_PRL_2010_Metallization] \[cf. the blue and dashed blue lines in Fig. \[epsilon0.eps\] (a)\]. Note that the appreciable oscillations in the dependence of $\varepsilon_0$ on the pulse duration $\tau$ seen for longer times in Fig. \[epsilon0.eps\] (b) are due to the interference of the excitation amplitudes at the two anticrossings (at the leading and trailing edges of the pulse). These are analogous to the Ramsey fringes, as discussed below for the residual conduction-band population $n_{cr}$ in conjunction with Fig. \[dynamics\_summary.eps\]. ![ Effective permittivity $\varepsilon_0$ as a function of pulse amplitude $\mathcal E_0$ (a) and inverse pulse duration $1/\tau$ (b). The label AA denotes a result of the adiabatic approximation. []{data-label="epsilon0.eps"}](epsilon0.eps){width=".48\textwidth"} ![Relative residual population of the conduction band $n_{cr}/n$ as a function of the pulse length $\tau$. The green curve is the result of the full numerical computation, the blue curve is the analytical result with two band-edge subbands, and the red with four. Arrows show the adiabatic phase $\varphi$. (a) $\mathcal E_0=0.94~\mathrm{V/\AA}$ and (b) $\mathcal E_0=1.2~\mathrm{V/\AA}$.[]{data-label="dynamics_summary.eps"}](dynamics_summary.eps){width=".48\textwidth"} In Fig. \[dynamics\_summary.eps\], we display the dependence of the residual (after the pulse end) population $n_{cr}$ on the excitation pulse length $\tau$. A striking feature of this dependence is the presence of high-contrast oscillations. These have a very clear physical origin. In the adiabatic picture, [@Stockman_et_al_PRL_2010_Metallization] when the pulse leading-edge field $\mathcal E(t)$ approaches the metallization threshold (causing the anticrossing of the adjacent valence- and conduction-band subbands), the valence electrons are shifted in space to one surface of the nanofilm in the direction of the field where the electrons occupy the quantum-bouncer states. [@Stockman_et_al_PRL_2010_Metallization] When the field increases above this metallization threshold, the electron population is coherently transferred to the opposite surface. [@Stockman_et_al_PRL_2010_Metallization] This creates polarization oscillating with the transition frequency between the valence- and and conduction-band edges, $\omega_{vc}(t)=\left[E_v(t)-E_c(t)\right]/\hbar$, which then adiabatically evolves with time $t$. The phase accumulated by these oscillations between the time $t_{m1}$ of the anticrossing passage at the leading pulse-edge and that at the trailing edge is $\varphi=\int_{t_{m1}}^{t_{m2}}\omega_{vc}(t)dt$. If $\varphi$ is such that the electrons at the moment $t_{m2}$ are shifted to the initial (in the direction of the maximum pulse field) surface of the nanofilm, then there is a large probability of their return back to the valence band, and the minimum of $n_{cr}$ is observed. Otherwise, the fringe maximum is reached. Thus these oscillations is analogous to the well-known Ramsey fringes. As indicated in Fig. \[dynamics\_summary.eps\], the adjacent minima and maxima of the $n_{cr}(\tau)$ fringes are indeed separated by the phase change $\Delta\varphi=\pi$. As one can see from Fig. \[dynamics\_summary.eps\], these fringes are described analytically reasonably well with two and very well with four band-edge subbbands taken into account. To conclude, in this Letter we have predicted a new effect: ultrafast dynamic metallization of dielectric nanofilms. A single-cycle ultrafast (duration $\sim 1$ fs) optical pulse with the normal electric field of a $\lesssim 1$ V/ amplitude incident on a dielectric nanofilm (here, a diamond-crystal film with thickness $\sim 2$ nm), induces a plasmonic metal-like dynamics that develops during an ultrashort period on the order of the pulse’s duration. For pulses of $1-2$ fs or longer, this dynamics is characterized by a large, metal-like polarization oscillating with optical frequencies. There is also a significant residual population of the conduction band, which strongly depends upon and can be coherently controlled by the adiabatic phase $\varphi$ accumulated between the two metalization instances at the leading and trailing edges of the excitation pulse. The polarization oscillations extend beyond the pulse end and also depend on the accumulated adiabatic phase $\varphi$. Thus the dynamic metallization is due to the combination and mutual influence of both the rapid adiabatic (reversible) and diabatic (dissipative) mechanisms. This dynamic metallization effect can find applications in lightwave electronics, [@Goulielmakis_et_al_Lightwave_Electronics_Science_2007] in particular, to create a field-effect transistor controlled by light’s electric field with a $\sim 100$ THz bandwidth. Useful discussions with F. Krausz are gratefully acknowledged. This work was supported by a grant from the Chemical Sciences, Biosciences and Geosciences Division of the BES Office of the Basic Energy Sciences, Office of Science, U.S. Department of Energy, and by grants from the U.S.–Israel Binational Science Foundation, BaCaTec, and the German Research Foundation (DFG) via the Emmy-Noether program and the Cluster of Excellence: Munich Center for Advanced Photonics. [19]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , **, (). , , , , , , , , (). , , , , , , , , , , , , (). , , , , , , , , , , , , (). , , , , , , , , (). , , , , , , , , (). , , , , , , , , , , , (). , , , , , , , (). , , (). , , , , , (). , , , , , , , , , , , (). , , , , , (). , , (). , , , , , , , , , , , , (). , , (). , , , , (). , , , , , , , (). , , (). , , , , , , , , , **, ().
--- abstract: \| The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n (x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley’s interpretation of $d_n (x)$ as the local $h$-polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition of the Eulerian polynomial of type $B$ whose summands can be expressed in terms of the Eulerian polynomial of type $A$. The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness. address: \| Department of Mathematics (Division of Algebra-Geometry)\ University of Athens\ Panepistimioupolis, Athens 15784\ Hellas (Greece) author: - 'Christos A. Athanasiadis' - Christina Savvidou date: 'March 15, 2013' title: A symmetric unimodal decomposition of the derangement polynomial of type $B$ --- [^1] Introduction and results {#sec:intro} ======================== The derangement polynomial of order $n$ is an interesting $q$-analogue of the number of derangements (elements without fixed points) in the symmetric group $\mathfrak{S}_n$. It is defined by the formula $$\label{eq:dndef} d_n (x) \ = \ \sum_{w \in \dD_n} \ x^{\exc(w)},$$ where $\exc(w)$ is the number of excedances (see Section \[sec:perms\] for missing definitions) of $w \in \mathfrak{S}_n$ and $\dD_n$ is the set of derangements in $\mathfrak{S}_n$. The polynomial $d_n(x)$, first studied by Brenti [@Bre90] in the context of symmetric functions, has a number of pleasant properties. For instance, it has symmetric and unimodal coefficients [@Bre90] (see also [@AS12 Section 4] [@SW10 Section 5] [@Ste92]) and only real roots [@Zh95]. It can also be expressed as $$\label{eq:dnA} d_n(x) \ = \ \sum_{k=0}^n \, (-1)^{n-k} \binom{n}{k} A_k(x),$$ where $A_k(x) = \sum_{w \in \mathfrak{S}_k} x^{\des(w)}$ is the $k$-th Eulerian polynomial. We will be concerned with a natural analogue of $d_n(x)$ for the hyperoctahedral group $B_n$ of signed permutations, introduced and studied independently by Chen, Tang and Zhao [@CTZ09] and by Chow [@Ch09]. It is defined by the formula $$\label{eq:dnBdef} d^B_n (x) \ = \ \sum_{w \in \dD^B_n} \ x^{\exc_B(w)},$$ where $\exc_B(w)$ is the number of type $B$ excedances of $w \in B_n$, introduced by Brenti [@Bre94], and $\dD^B_n$ is the set of derangements in $B_n$. The derangement polynomial $d^B_n(x)$ shares most of the main properties of $d_n (x)$. For instance, it is real-rooted [@CTZ09; @Ch09], hence it has unimodal (but not symmetric) coefficients, and satisfies the analogue $$\label{eq:dnB} d^B_n(x) \ = \ \sum_{k=0}^n \, (-1)^{n-k} \binom{n}{k} B_k(x)$$ of (\[eq:dnA\]), where $B_k(x) = \sum_{w \in B_k} x^{\des_B(w)}$ is the $k$-th Eulerian polynomial of type $B$. Our first main result is the following combinatorial formula for $d^B_n(x)$. \[thm:main\] We have $$\label{eq:main} d_n^B(x) \ = \ \sum \ {n \choose r_0, r_1,\dots,r_k} \, x^{\lfloor \frac{k+1}{2} \rfloor} \, d_{r_0}(x) \, A_{r_1}(x) \cdots A_{r_k}(x)$$ for $n \in \NN$, where $A_0 (x) = 0$, $d_0 (x) = 1$ and the sum ranges over all $k \in \NN$ and over all sequences $(r_0, r_1,\dots,r_k)$ of nonnegative integers which sum to $n$. Chow [@Ch09 Section 4] gave an additional proof of the unimodality of $d^B_n (x)$ by expressing it as a sum of certain nonnegative unimodal polynomials, defined by a symmetric function identity, of a common mode. Theorem \[thm:main\] implies that $d^B_n (x)$ can be written as a sum of two polynomials with nonnegative, symmetric and unimodal coefficients, whose centers of symmetry differ by a half, and thus provides a new proof of its unimodality, as we now explain. Since $d^B_n (x)$ has degree $n$ and zero constant term, it can be written uniquely in the form $$\label{eq:dnBsymsum} d^B_n(x) \ = \ f^+_n (x) \, + \, f^-_n (x),$$ where $f^+_n (x)$ and $f^-_n (x)$ are polynomials of degrees at most $n-1$ and $n$, respectively, satisfying $$\begin{aligned} \label{eq:symf+} f^+_n (x) & = & x^n f^+_n (1/x) \\ \label{eq:symf-} f^-_n (x) & = & x^{n+1} f^-_n (1/x)\end{aligned}$$ (see, for instance, [@BSt10 Lemma 2.4] for this elementary fact). For the first few values of $n$ we have $$f^+_n (x) \ = \ \begin{cases} 1, \ \ & \text{if \ $n=0$} \\ 0, \ \ & \text{if \ $n=1$} \\ 3x, \ \ & \text{if \ $n=2$} \\ 7x + 7x^2, \ \ & \text{if \ $n=3$} \\ 15x + 87x^2 + 15x^3, \ \ & \text{if \ $n=4$} \\ 31x + 551x^2+ 551x^3 + 31x^4, \ \ & \text{if \ $n=5$} \\ 63x + 2803x^2 + 8243x^3 + 2803x^4 + 63x^5, \ \ & \text{if \ $n=6$} \\ 127x + 12867x^2 + 84827x^3 + 84827x^4 + 12867x^5 + 127x^6, \ \ & \text{if \ $n=7$} \end{cases}$$ and $$f^-_n (x) \ = \ \begin{cases} 0, \ \ & \text{if \ $n=0$} \\ x, \ \ & \text{if \ $n=1$} \\ x + x^2, \ \ & \text{if \ $n=2$} \\ x + 13x^2 + x^3, \ \ & \text{if \ $n=3$} \\ x + 57x^2 + 57x^3 + x^4, \ \ & \text{if \ $n=4$} \\ x + 201x^2 + 761x^3 + 201x^4 + x^5, \ \ & \text{if \ $n=5$} \\ x + 653x^2 + 6333x^3 + 6333x^4 + 653x^5 + x^6, \ \ & \text{if \ $n=6$} \\ x + 2045x^2 + 42757x^3 + 106037x^4 + 42757x^5 + 2045x^6 + x^7, \ \ & \text{if \ $n=7$.} \end{cases}$$ The following information for the polynomials $f^+_n (x)$ and $f^-_n (x)$ and for $d^B_n (x)$ can be derived from (\[eq:main\]). \[cor:main\] We have $$\label{eq:main+} f^+_n (x) \ = \ \sum \ {n \choose r_0, r_1,\dots,r_{2k}} \, x^k \, d_{r_0} (x) \, A_{r_1} (x) \cdots A_{r_{2k}}(x)$$ and $$\label{eq:main-} f^-_n (x) \ = \ \sum \ {n \choose r_0, r_1,\dots,r_{2k+1}} \, x^{k+1} \, d_{r_0}(x) \, A_{r_1} (x) \cdots A_{r_{2k+1}}(x)$$ for $n \in \NN$, where the sums range over all $k \in \NN$ and over all sequences $(r_0, r_1,\dots,r_{2k})$ [(]{}respectively, $(r_0, r_1,\dots,r_{2k+1})$[)]{} of nonnegative integers which sum to $n$. Moreover, $f^+_n (x)$ and $f^-_n (x)$ are $\gamma$-nonnegative, meaning there exist nonnegative integers $\xi^+_{n,i}$ and $\xi^-_{n,i}$ such that $$\label{eq:main++} f^+_n (x) \ = \ \sum_{i=0}^{\lfloor n/2 \rfloor} \ \xi^+_{n,i} \, x^i (1+x)^{n-2i}$$ and $$\label{eq:main--} f^-_n (x) \ = \ \sum_{i=0}^{\lfloor (n+1)/2 \rfloor} \ \xi^-_{n,i} \, x^i (1+x)^{n+1-2i}.$$ In particular, $f^+_n (x)$ and $f^-_n (x)$ are symmetric and unimodal, with center of symmetry $n/2$ and $(n+1)/2$, respectively, and $d_n^B(x)$ is unimodal with a peak at $\lfloor (n+1)/2 \rfloor$. Much of the motivation behind this paper comes from the theory of subdivisions and local $h$-vectors, developed by Stanley [@Sta92], and its extension [@Ath12]. We recall that the local $h$-vector is a fundamental enumerative invariant of a simplicial subdivision (triangulation) of the simplex. An example by Stanley (see [@Sta92 Proposition 2.4]) shows that $d_n(x)$ is equal to the local $h$-polynomial of the (first) simplicial barycentric subdivision of the $(n-1)$-dimensional simplex. This fact gives a geometric interpretation to $d_n (x)$ and another proof of its symmetry and unimodality. = 3.0 in Our second main result provides a type $B$ analogue to this interpretation. To state it, we introduce the following notation. We denote by $K_n$ the simplicial barycentric subdivision of the cubical barycentric subdivision of the $(n-1)$-dimensional simplex (Figure \[fig:K3\] shows this subdivision for $n=3$). We also introduce the ‘half Eulerian polynomials’ $$\label{eq:Bn+def} B^+_n (x) \ = \ \sum_{w \in B^+_n} \ x^{\des_B(w)}$$ and $$\label{eq:Bn-def} B^-_n (x) \ = \ \sum_{w \in B^-_n} \ x^{\des_B(w)}$$ for the group $B_n$, where $B^+_n$ and $B^-_n$ are the sets of signed permutations of length $n$ with positive and negative, respectively, last entry, and set $B^+_0 (x) = 1$ and $B^-_0 (x) = 0$ (the set $B^+_n$ has appeared in the context of major indices for classical Weyl groups; see [@BC04 page 613]). \[thm:localint\] The polynomial $f^+_n (x)$ is equal to the local $h$-polynomial of the simplicial subdivision $K_n$ (in particular, $f^+_n (x)$ has nonnegative, symmetric and unimodal coefficients). Moreover, we have $$\label{eq:localint+} f^+_n (x) \ = \ \sum_{k=0}^n \, (-1)^{n-k} {n \choose k} \, B^+_k (x)$$ and $$\label{eq:localint-} f^-_n (x) \ = \ \sum_{k=0}^n \, (-1)^{n-k} {n \choose k} \, B^-_k (x)$$ for $n \in \NN$. We should point out that it is Theorem \[thm:localint\] and the methods of [@Ath12; @Sta92] which led the authors to suspect that formula (\[eq:main\]) holds. Indeed, it follows from the relevant definitions and some more work (see Section \[sec:geom\]) that the local $h$-polynomial of $K_n$ is equal to the right-hand side of (\[eq:localint+\]). By exploiting the symmetry of this polynomial and certain recurrence relations for that and for $d^B_n (x)$ (see Section \[sec:half\]), one can show that the local $h$-polynomial of $K_n$ is equal to $f^+_n (x)$, as defined by the decomposition (\[eq:dnBsymsum\]). A formula for the change in the local $h$-vector of a simplicial subdivision of the simplex after further subdivision [@Ath12 Proposition 3.6] (see also Proposition \[prop:localrelformula\]) can then be used to produce equation (\[eq:main+\]). This suggested that (\[eq:main-\]), and hence (\[eq:main\]), hold as well. The structure and other results of this paper are as follows. Section \[sec:perms\] provides the necessary background on (signed) permutations, simplicial complexes and subdivisions. Section \[sec:proof\] proves Theorem \[thm:main\] and Corollary \[cor:main\]. A bijective proof of Theorem \[thm:main\], as well as one using generating functions, is given and the exponential generating functions of $f^+_n (x)$ and $f^-_n (x)$ are computed. Section \[sec:comb\] gives a combinatorial interpretation to the coefficients of these polynomials. Section \[sec:relative\] proves the main properties of the relative local $h$-vector, a generalization of the concept of local $h$-vector which was introduced in [@Ath12 Section 3] (and, in a variant form, in [@Ni12]) and derives a monotonicity property for local $h$-vectors. These results were stated without proof in [@Ath12]. As an example (used in one of the proofs of Theorem \[thm:localint\]), the relative local $h$-vector of the barycentric subdivision of the simplex is computed. Section \[sec:geom\] gives two proofs of Theorem \[thm:localint\]. A first step towards these proofs is to interpret $B^+_n (x)$ as the $h$-polynomial of the simplicial complex $K_n$ (Proposition \[prop:Knhpoly\]). Given that, one proof uses the theory of (relative) local $h$-vectors, as discussed earlier, while the other uses recurrences and generating functions. Section \[sec:half\] studies the polynomials $B^+_n (x)$ and $B^-_n (x)$. A simple relation between the two is shown to hold (Lemma \[lem:rec\]). Using its interpretation as the $h$-polynomial of $K_n$ and the theory of local $h$-vectors, a simple formula for $B^+_n (x)$ (hence one for $B^-_n (x)$ and one for the Eulerian polynomial $B_n (x)$) in terms of the Eulerian polynomial $A_n (x)$ is proven (Proposition \[prop:Bn+formula\]). Using this formula, it is shown that $B^+_n (x)$ and $B^-_n (x)$ are real-rooted, hence unimodal and log-concave, and a new proof of the unimodality of $B_n (x)$ is deduced. Recurrences and generating functions for $B^+_n (x)$ and $B^-_n (x)$, as well as for $f^+_n (x)$ and $f^-_n (x)$, are also given and a third proof of Theorem \[thm:localint\] is deduced. Permutations and subdivisions {#sec:perms} ============================= This section fixes notation and includes background material on (signed) permutations, simplicial complexes and their subdivisions. For more information on these topics, the reader is referred to [@Bj95; @BB05; @Sta92; @StaCCA; @StaEC1]. Throughout this paper, $\mathbb{N}$ denotes the set of nonnegative integers. For each positive integer $n$ we set $[n]:= \{1, 2,\dots,n\}$ and $\Omega_n = \{1, -1, 2, -2,\dots,n, -n\}$. We denote by $\|S\|$ the cardinality, and by $2^S$ the set of all subsets, of a finite set $S$. Permutations {#subsec:perm} ------------ A permutation of a finite set $S$ is a bijective map $w: S \to S$. We denote by $\mathfrak{S} (S)$ the set of all permutations of $S$ and set $\mathfrak{S}_n := \mathfrak{S} ([n])$. Suppose that $S = \{a_1, a_2,\dots,a_n\}$ has $n$ elements, which are totally ordered by $a_1 \prec a_2 \prec \cdots \prec a_n$. A permutation $w \in \mathfrak{S}(S)$ can be represented as the sequence $(w(a_1), w(a_2),\dots,w(a_n))$, or as the word $w(a_1) w(a_2) \cdots w(a_n)$, or as a disjoint union of cycles [@StaEC1 Section 1.3]. The standard cycle form is defined by requiring that (a) each cycle is written with its largest element (with respect to the total order $\preceq$) first and (b) the cycles are written in increasing order of their largest element [@StaEC1 page 23]. Given $w \in \mathfrak{S} (S)$, an element $a \in S$ is called an excedance of $w$ (with respect to $\preceq$) if $w(a) \succ a$ and an inverse excedance if $w(a) \prec a$. The element $a_i \in S$ is called a descent (respectively, ascent) of $w$ if $i \in [n-1]$ and $w(a_i) \succ w(a_{i+1})$ (respectively, $w(a_i) \prec w(a_{i+1})$). The number of excedances (respectively, inverse excedances, descents or ascents) of $w$ will be denoted by $\exc (w)$ (respectively, by $\iexc(w)$, $\des (w)$ or $\asc (w)$). The $n$th Eulerian polynomial [@StaEC1 Section 1.4] is defined by the formulas $$\label{eq:eulerdef} A_n (x) \ = \ \sum_{w \in \mathfrak{S} (S)} x^{\exc(w)} \ = \ \sum_{w \in \mathfrak{S} (S)} x^{\iexc(w)} \ = \ \sum_{w \in \mathfrak{S} (S)} x^{\des(w)} \ = \ \sum_{w \in \mathfrak{S} (S)} x^{\asc(w)}.$$ Clearly, these sums depend only on $n$ and not on $S$ or the choice of total order $\preceq$. The previous definitions apply in particular to $\mathfrak{S}_n$ (with the standard choice of $\preceq$ obtained by setting $a_i = i$ for $1 \le i \le n$). We will denote by $\dD_n$ the set of all derangements (permutations without fixed points) in $\mathfrak{S}_n$. Signed permutations {#subsec:signed} ------------------- For the purposes of this paper, it will be convenient to define a signed permutation of $[n]$ as a choice of a subset $S = \{a_1, a_2,\dots,a_n\}$ of $\Omega_n$ such that $a_i \in \{i, -i\}$ for $1 \le i \le n$ and permutation $w \in \mathfrak{S} (S)$. We will represent such a permutation $w$ as the sequence $(w(a_1), w(a_2),\dots,w(a_n))$, or as the word $w(a_1) w(a_2) \cdots w(a_n)$, or as a disjoint union of cycles. We will find it convenient to define the standard cycle form of $w$ using the total order on $S$ which is the reverse of the one inherited from the natural total order on $\ZZ$. Thus, cycles of $w$ will be written with their smallest element first and in decreasing order of their smallest element. We will say that $w$ is a derangement if there is no $a \in S \cap [n]$ such that $w(a) = a$. We will denote the set of all signed permutations of $[n]$ by $B_n$ and the set of all derangements in $B_n$ by $\dD^B_n$. Given $w \in B_n$ as before, we say that $i \in \{0, 1,\dots,n-1\}$ is a $B$-descent (respectively, $B$-ascent) of $w$ if $w(a_i) > w(a_{i+1})$ (respectively, $w(a_i) < w(a_{i+1})$), where $w(a_0) = 0$ by convention. The $n$th Eulerian polynomial of type $B$ [@Bre94 Section 3] can be defined by $$\label{eq:eulerBdef} B_n (x) \ = \ \sum_{w \in B_n} x^{\des_B(w)} \ = \ \sum_{w \in B_n} x^{\asc_B(w)},$$ where $\des_B (w)$ stands for the number of $B$-descents and $\asc_B (w)$ for the number of $B$-ascents of $w \in B_n$. Following Brenti [@Bre94 p. 431], we say that $a \in S$ is a $B$-excedance of $w$ if $w(a) > a$, or if $-a \in [n]$ and $w(a) = a$. We say that $a \in S$ is an inverse $B$-excedance of $w$ if $w(a) < a$, or if $-a \in [n]$ and $w(a) = a$. The number of $B$-excedances of $w$ will be denoted by $\exc_B(w)$ and that of inverse $B$-excedances by $\iexc_B(w)$. We then have $\iexc_B(w) = \exc_B(w^{-1})$ and (see Theorem 3.15 and Corollary 3.16 in [@Bre94]) $$\label{eq:eulerBdef2} B_n (x) \ = \ \sum_{w \in B_n} x^{\exc_B(w)}.$$ The $n$th derangement polynomial of type $B$ is defined by (\[eq:dnBdef\]). Since $\exc_B(w) = \iexc_B(w^{-1})$ and the map which sends a permutation $w \in \mathfrak{S} (S)$ to its inverse $w^{-1}$ induces an involution on $B_n$ which preserves fixed points, we have $$\label{eq:dnBidef} d^B_n (x) \ = \ \sum_{w \in \dD^B_n} \ x^{\iexc_B(w)}.$$ For the similar reasons, (\[eq:dndef\]) continues to hold if $\exc$ is replaced by $\iexc$ and (\[eq:eulerBdef2\]) continues to hold if $\exc_B$ is replaced by $\iexc_B$. Polynomials {#subsec:polys} ----------- Let $p(x) = \sum_{k \ge 0} a_k x^k = \sum_{k=0}^d a_k x^k$ be a polynomial with real coefficients. We recall that $p(x)$ is unimodal (and has unimodal coefficients) if there exists an index $0 \le j \le d$ such that $a_i \le a_{i+1}$ for $0 \le i \le j-1$ and $a_i \ge a_{i+1}$ for $j \le i \le d-1$. Such an index is called a peak. The polynomial $p(x)$ is said to be log-concave if $a_i^2 \ge a_{i-1}a_{i+1}$ for $1 \le i \le d-1$ and to have internal zeros if there exist indices $0 \le i < j < k \le d$ such that $a_i, a_k \ne 0$ and $a_j = 0$. We will say that $p(x)$ is symmetric (and that it has symmetric coefficients) if there exists an integer $n \ge d$ such that $a_i = a_{n-i}$ for $0 \le i \le n$. The center of symmetry of $p(x)$ is then defined to be $n/2$ (this is well-defined provided $p(x)$ is nonzero). We will say that $p(x)$ is real-rooted if all its complex roots are real. It is well-known (see, for instance, [@Sta89]) that if $p(x)$ is a real-rooted polynomial with nonnegative coefficients, then $p(x)$ is log-concave and unimodal, with no internal zeros. The following theorem, first proved by Edrei [@Ed53], gives a necessary and sufficient condition for a polynomial with nonnegative real coefficients to be real-rooted. [([@Ed53])]{} \[thm:realroots\] Let $p(x) = \sum_{k \ge 0} a_k x^k \in \RR[x]$ be a polynomial with $a_k \ge 0$ for every $k \in \NN$ and set $a_k = 0$ for all negative integers $k$. Then $p(x)$ is real-rooted if and only if every minor of the lower triangular matrix $(a_{i-j})_{i,j=0}^\infty$ is nonnegative. A (nonzero) symmetric polynomial $p(x) \in \RR[x]$ can be written (uniquely) in the form $$\label{eq:defgamma} p(x) \ = \ (1+x)^n \ \gamma \left( \frac{x}{(1+x)^2} \right) \, = \, \sum_{i=0}^{\lfloor n/2 \rfloor} \, \gamma_i x^i (1+x)^{n-2i}$$ for some polynomial $\gamma(x) = \sum_{i \ge 0} \gamma_i x^i$. We say that $p(x)$ is $\gamma$-nonnegative if $\gamma_i \ge 0$ for every $i$. Clearly, every $\gamma$-nonnegative polynomial is unimodal. For classes of $\gamma$-nonnegative polynomials which appear in combinatorics we refer the reader, for instance, to [@DPK09] and references therein. Simplicial complexes {#subsec:simpcomp} -------------------- An (abstract) simplicial complex $\Delta$ on the ground set $V$ is a collection $\Delta$ of subsets of $V$ such that $F \subseteq G \in \Delta$ implies $F \in \Delta$ (all simplicial complexes considered in this paper will be assumed to be finite). The elements of $\Delta$ are called faces. The dimension of a face is equal to one less than its cardinality. The dimension of $\Delta$ is the maximum dimension of its faces. Faces of dimension 0 and 1 are called vertices and edges, respectively. A facet of $\Delta$ is a face which is maximal with respect to inclusion. The complex $\Delta$ is said to be pure if all its facets have the same dimension. The face poset $\mathcal{F}(\Delta)$ of a simplicial complex $\Delta$ is the set of nonempty faces of $\Delta$, partially ordered by inclusion. The open star ${\rm st}_{\Delta}(F)$ of a face $F \in \Delta$ is the collection of all faces of $\Delta$ containing $F$. The link of a face $F$ in $\Delta$ is the subcomplex of $\Delta$ defined as $\link_{\Delta}(F) = \{ G \sm F: G \in \Delta, F \subseteq G\}$. Suppose that $\Delta_1$ and $\Delta_2$ are simplicial complexes on disjoint ground sets. The simplicial join of $\Delta_1$ and $\Delta_2$ is the simplicial complex $\Delta_1 * \Delta_2$ whose faces are the sets of the form $F_1 \cup F_2$, where $F_1 \in \Delta_1$ and $F_2 \in \Delta_2$. The order complex [@Bj95 Section 9.3] [@StaEC1 Section 3.8] of a (finite) partially ordered set $Q$ is defined as the simplicial complex of chains (totally ordered subsets) of $Q$. All topological properties or invariants of $\Delta$ mentioned in the sequel will refer to those of its geometric realization $\\|\Delta\\|$ [@Bj95 Section 9.1]. For example, $\Delta$ is a simplicial ball if $\\|\Delta\\|$ is homeomorphic to a ball. For a simplicial $d$-dimensional ball $\Delta$, we denote by $\partial \Delta$ the subcomplex consisting of all subsets of the $(d-1)$-dimensional faces which are contained in a unique facet of $\Delta$. We call $\partial \Delta$ the boundary and $\inte(\Delta) := \Delta \sm \partial \Delta$ the interior of $\Delta$. Subdivisions {#subsec:sub} ------------ Let $\Delta$ be a simplicial complex. A (topological) simplicial subdivision of $\Delta$ [@Sta92 Section 2] is a simplicial complex $\Delta'$ together with a map $\sigma: \Delta' \to \Delta$ such that the following hold for every $F \in \Delta$: (a) the set $\Delta'_F := \sigma^{-1} (2^F)$ is a subcomplex of $\Delta'$ which is a simplicial ball of dimension $\dim(F)$; and (b) the interior of $\Delta'_F$ is equal to $\sigma^{-1}(F)$. The subcomplex $\Delta'_F$ is called the restriction of $\Delta'$ to $F$. The face $\sigma(G)\in \Delta$ is called the carrier of $G \in \Delta'$. The subdivision $\Delta'$ is called quasi-geometric [@Sta92 Definition 4.1 (a)] if no face of $\Delta'$ has the carriers of its vertices contained in a face of $\Delta$ of smaller dimension. Moreover, $\Delta'$ is called geometric [@Sta92 Definition 4.1 (b)] if there exists a geometric realization of $\Delta'$ which geometrically subdivides a geometric realization of $\Delta$, in the way prescribed by $\sigma$. Clearly, all geometric subdivisions (such as the barycentric subdivisions considered in this paper) are quasi-geometric. We now describe two common ways to subdivide a simplicial complex $\Delta$. The order complex of the face poset $\fF(\Delta)$, denoted by $\sd (\Delta)$, consists of the chains of nonempty faces of $\Delta$. This complex is naturally a (geometric) simplicial subdivision of $\Delta$, called the barycentric subdivision, where the carrier of a chain $\cC$ of nonempty faces of $\Delta$ is defined as the maximum element of $\cC$. Given a face $F \in \Delta$ we set $\Delta' = (\Delta \sm \textrm{st}_{\Delta}(F)) \cup (\{v\} * \partial(2^F) * \textrm{lk}_{\Delta}(F))$, where $v$ is a new vertex added and $\partial(2^F) = 2^F \sm F$. Then $\Delta'$ is a simplicial complex which is a simplicial subdivision of $\Delta$, called the stellar subdivision of $\Delta$ on $F$. Face enumeration {#subsec:enumface} ---------------- Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. We denote by $f_i (\Delta)$ the number of $i$-dimensional faces of $\Delta$. A fundamental enumerative invariant of $\Delta$ is the $f$-polynomial, defined by $$f_{\Delta}(x) \ = \ \sum_{i=0}^{d-1} f_i (\Delta) x^i.$$ The $h$-polynomial of $\Delta$ is defined by $$h_{\Delta}(x) \ = \ \sum_{i=0}^d h_i(\Delta) \, x^i \ = \ (1-x)^d f_{\Delta} \left( \frac{x}{1-x} \right).$$ For the importance of $h$-polynomials, the reader is referred to [@StaCCA Chapter II]. For the simplicial join $\Delta_1 * \Delta_2$ of two simplicial complexes we have $h(\Delta_1 * \Delta_2, x) = h(\Delta_1, x) h(\Delta_2, x)$. Let $\Gamma$ be a simplicial subdivision of a $(d-1)$-dimensional simplex $2^V$. The polynomial $\ell_V (\Gamma, x) = \ell_0 + \ell_1 x + \cdots + \ell_d x^d$ defined by $$\label{eq:deflocalh} \ell_V (\Gamma, x) \ = \sum_{F \subseteq V} \ (-1)^{d - \|F\|} \, h (\Gamma_F, x)$$ is the local $h$-polynomial of $\Gamma$ (with respect to $V$) [@Sta92 Definition 2.1]. The sequence $\ell_V (\Gamma) = (\ell_0, \ell_1,\dots,\ell_d)$ is the local $h$-vector of $\Gamma$ (with respect to $V$). The following theorem summarizes some of the main properties of local $h$-vectors (see Theorems 3.2 and 3.3 and Corollary 4.7 in [@Sta92]). For the definition of regular subdivision we refer the reader to [@Sta92 Definition 5.1]. [(Stanley [@Sta92])]{} \[thm:stalocal\] - For every simplicial subdivision $\Delta'$ of a pure simplicial complex $\Delta$ we have $$\label{eq:hformula} h (\Delta', x) \ = \ \sum_{F \in \Delta} \, \ell_F (\Delta'_F, x) \, h (\link_\Delta (F), x).$$ - The local $h$-polynomial $\ell_V (\Gamma, x)$ is symmetric for every simplicial subdivision $\Gamma$ of the simplex $2^V$, i.e. we have $\ell_i = \ell_{d-i}$ for $0 \le i \le d$. - The local $h$-polynomial $\ell_V (\Gamma, x)$ has nonnegative coefficients for every quasi-geometric simplicial subdivision $\Gamma$ of the simplex $2^V$. - The local $h$-polynomial $\ell_V (\Gamma, x)$ has unimodal coefficients for every regular simplicial subdivision $\Gamma$ of the simplex $2^V$. Proof of the main formula {#sec:proof} ========================= This section gives two proofs of Theorem \[thm:main\], one bijective and one using generating functions, and deduces Corollary \[cor:main\]. As a byproduct of the second proof, the exponential generating functions of $f^+_n (x)$ and $f^-_n (x)$ are computed. Let us denote by $\cC_n$ the collection of sequences $(\sigma_0, \sigma_1,\dots,\sigma_k)$ of permutations, where $k \in \NN$ and $\sigma_i \in \mathfrak{S}(S_i)$ for $0 \le i \le k$, such that $(S_0, S_1,\dots,S_k)$ is a weak ordered partition of $[n]$ with $S_i$ nonempty for $1 \le i \le k$ and $\sigma_0$ is a derangement of $S_0$. We will describe a one-to-one correspondence $\varphi: \dD^B_n \to \cC_n$ such that $$\label{eq:bicondition} \iexc_B (w) \ = \ \iexc(\sigma_0) \, + \, \sum_{i=1}^k \, f(\sigma_i) \, + \, \lfloor \frac{k+1}{2} \rfloor$$ for every $w \in \dD^B_n$, where $(\sigma_0, \sigma_1,\dots,\sigma_k) = \varphi (w)$ and $f(\sigma_i)$ stands for $\des(\sigma_i)$ or $\asc(\sigma_i)$, if $i$ is even or odd, respectively. Given this, using (\[eq:dnBidef\]) and recalling that there are ${n \choose r_0, r_1,\dots,r_k}$ weak ordered partitions $(S_0, S_1,\dots,S_k)$ of $[n]$ satisfying $\|S_i\| = r_i$ for $0 \le i \le k$, we get $$\begin{aligned} d^B_n (x) &=& \sum \ {n \choose r_0, r_1,\dots,r_k} \, x^{\lfloor \frac{k+1}{2} \rfloor} \sum_{\sigma_0 \in \dD_{r_0}} x^{\iexc(\sigma_0)} \, \left( \, \prod_{i=1}^k \, \sum_{\sigma_i \in \mathfrak{S}_{r_i}} x^{\des(\sigma_i)} \right) \\ & & \\ &=& \sum \ {n \choose r_0, r_1,\dots,r_k} \, x^{\lfloor \frac{k+1}{2} \rfloor} \, d_{r_0}(x) \, A_{r_1}(x) \cdots A_{r_k}(x) \end{aligned}$$ and the proof follows. To define $\varphi$, consider a derangement $w \in \dD^B_n$ and let $C_1 C_2 \cdots C_m$ be the standard cycle form of $w$. Then there is an index $j \in \{0, 1,\dots,m\}$ such that all elements of $C_1, C_2,\dots,C_j$ are positive and the first (smallest) element of $C_{j+1}$ is negative. We define $\sigma_0$ as the product of $C_1, C_2,\dots,C_j$ and $S_0$ as the set of all elements which appear in these cycles, so that $\sigma_0 \in \mathfrak{S}(S_0)$ is a derangement. The remaining cycles $C_{j+1},\dots,C_m$ form a word $u$ whose first element is negative. This word decomposes uniquely as a product $u = u_1 u_2 \cdots u_k$ of subwords $u_i$ so that for $1 \le i \le k$, all elements of $u_i$ are negative if $i$ is odd and positive if $i$ is even. We define $S_i$ as the set of absolute values of the elements of $u_i$ and $\sigma_i \in \mathfrak{S}(S_i)$ as the permutation which corresponds to the word $u_i$. For instance, if $n = 9$ and $w = (3 \ 7) (1 \ 4) (-5 \ \, 9 \ -2) (-8 \ -6)$ in standard cycle form, then $\sigma_0 = (1 \ 4) (3 \ 7)$ in cycle form, $k = 3$ and $\sigma_1 = (5)$, $\sigma_2 = (9)$, $\sigma_3 = (2, 8, 6)$, as sequences. We set $\varphi(w) = (\sigma_0, \sigma_1,\dots,\sigma_k)$ and leave it to the reader to verify that the map $\varphi: \dD^B_n \to \cC_n$ is a well defined bijection. To verify (\[eq:bicondition\]) we let $w \in \dD^B_n$ with $\varphi (w) = (\sigma_0, \sigma_1,\dots,\sigma_k)$ and $u = a_1 a_2 \cdots a_p$ be the word defined in the previous paragraph. Then, by the definitions of standard cycle form and (inverse) $B$-excedance, $a \in \Omega_n$ is an inverse $B$-excedance of $w$ if and only if $a$ is an inverse excedance of $\sigma_0$, or $a = a_i$ for some index $1 \le i < p$ with $a_i > a_{i+1}$, or $a = a_p$. Thus, equation (\[eq:bicondition\]) follows. For the second proof of Theorem \[thm:main\] we set $$\label{eq:Anexp} \aA(t) \ := \ \sum_{n \ge 1} \ A_n (x) \, \frac{t^n}{n!} \ = \ \frac{e^t - e^{xt}} {e^{xt} - xe^t}$$ and (see [@Bre90 Proposition 5]) $$\label{eq:dnexp} \dD(t) \ := \ \sum_{n \ge 0} \ d_n (x) \, \frac{t^n}{n!} \ = \ \frac{1-x} {e^{xt} - xe^t},$$ where $d_0 (x) = 1$. We also recall (see [@CTZ09 Theorem 3.3] [@Ch09 Theorem 3.2]) that $$\label{eq:dnBexp} \sum_{n \ge 0} \ d^B_n (x) \, \frac{t^n}{n!} \ = \ \frac{(1-x) e^{xt}} {e^{2xt} - xe^{2t}},$$ where $d^B_0 (x) = 1$. We denote by $S_n (x)$ (respectively, by $S^+_n (x)$ and $S^-_n (x)$) the right-hand side of (\[eq:main\]) (respectively, of (\[eq:main+\]) and (\[eq:main-\])), so that $S_n (x) = S^+_n (x) + S^-_n (x)$ for $n \in \NN$. We compute that $$\begin{aligned} \sum_{n \ge 0} \ S^+_n (x) \, \frac{t^n}{n!} &=& \sum_{k, \, r_i \ge 0} \ x^k \, d_{r_0} (x) \, \frac{t^{r_0}}{r_0!} \ A_{r_1} (x) \, \frac{t^{r_1}}{r_1!} \cdots A_{r_{2k}} (x) \, \frac{t^{r_{2k}}}{r_{2k}!} \\ &=& \sum_{n \ge 0} \ d_n (x) \, \frac{t^n}{n!} \ \, \sum_{k \ge 0} \ x^k \left( \sum_{r \ge 1} \ A_r (x) \, \frac{t^r}{r!} \right)^{2k} \\ & & \\ &=& \frac{\dD(t)}{1 - x (\aA(t))^2} \end{aligned}$$ and similarly that $$\begin{aligned} \sum_{n \ge 0} \ S^-_n (x) \, \frac{t^n}{n!} &=& \sum_{k, \, r_i \ge 0} \ x^{k+1} \, d_{r_0} (x) \, \frac{t^{r_0}}{r_0!} \ A_{r_1} (x) \, \frac{t^{r_1}}{r_1!} \cdots A_{r_{2k+1}} (x) \, \frac{t^{r_{2k+1}}}{r_{2k+1}!} \\ &=& \sum_{n \ge 0} \ d_n (x) \, \frac{t^n}{n!} \ \, \sum_{k \ge 0} \ x^{k+1} \left( \sum_{r \ge 1} \ A_r (x) \, \frac{t^r}{r!} \right)^{2k+1} \\ & & \\ &=& \dD(t) \cdot \frac{x \aA(t)}{1 - x (\aA(t))^2} \end{aligned}$$ and conclude that $$\sum_{n \ge 0} \ S_n (x) \, \frac{t^n}{n!} \ = \ \dD(t) \cdot \frac{1 + x \aA(t)}{1 - x (\aA(t))^2}.$$ Combining the previous equation with (\[eq:Anexp\]) and (\[eq:dnexp\]) we get, after some straightforward algebraic manipulations, that $$\sum_{n \ge 0} \ S_n (x) \, \frac{t^n}{n!} \ = \ \frac{(1-x) e^{xt}} {e^{2xt} - xe^{2t}} \ = \ \sum_{n \ge 0} \ d^B_n (x) \, \frac{t^n}{n!}$$ and the proof follows. As in the second proof of Theorem \[thm:main\], we denote by $S^+_n (x)$ and $S^-_n (x)$ the right-hand side of (\[eq:main+\]) and (\[eq:main-\]), respectively. Theorem \[thm:main\] shows that $d^B_n (x) = S^+_n (x) + S^-_n (x)$ for every $n \in \NN$. From the symmetry properties $A_n (x) = x^{n-1} A_n (1/x)$ and $d_n (x) = x^n \, d_n (1/x)$ of the Eulerian and derangement polynomials for $\mathfrak{S}_n$ it follows that $S^+_n (x)$ and $S^-_n (x)$ satisfy (\[eq:symf+\]) and (\[eq:symf-\]), respectively. The uniqueness of the defining properties of $f^+_n (x)$ and $f^-_n (x)$ imply that $f^+_n (x) = S^+_n (x)$ and $f^-_n (x) = S^-_n (x)$ for every $n \in \NN$. This proves equations (\[eq:main+\]) and (\[eq:main-\]). The $\gamma$-nonnegativity of $f^+_n (x)$ and $f^-_n (x)$ follows from equations (\[eq:main+\]) and (\[eq:main-\]) and the $\gamma$-nonnegativity of $A_n (x)$ and $d_n (x)$ (see Proposition \[prop:xi+-formula\] in the sequel). The last statement in the corollary follows from (\[eq:main++\]) and (\[eq:main–\]). Since the polynomials $A_n (x)$ and $d_n (x)$ have nonnegative and symmetric coefficients and only real roots, we can write $$\label{eq:Angamma} A_n (x) \ = \ (1+x)^{n-1} \ \gamma_n \left( \frac{x}{(1+x)^2} \right)$$ and $$\label{eq:dngamma} d_n (x) \ = \ (1+x)^n \ \xi_n \left( \frac{x}{(1+x)^2} \right)$$ for some polynomials $\gamma_n (x)$ and $\xi_n (x)$ with nonnegative coefficients. Explicit combinatorial interpretations to these coefficients are known (see, for instance, [@FSc70 Theorem 5.6] and [@AS12 Section 4]). Equations (\[eq:main+\]), (\[eq:main-\]), (\[eq:Angamma\]) and (\[eq:dngamma\]) imply explicit combinatorial formulas for the polynomials $\xi^+_n (x) = \sum \xi^+_{n,i} x^i$ and $\xi^-_n (x) = \sum \xi^-_{n,i} x^i$, appearing in Corollary \[cor:main\], which we record in the following proposition. \[prop:xi+-formula\] We have $$\label{eq:fn+gamma} f^+_n (x) \ = \ (1+x)^n \ \xi^+_n \left( \frac{x}{(1+x)^2} \right)$$ and $$\label{eq:fn-gamma} f^-_n (x) \ = \ (1+x)^{n+1} \ \xi^-_n \left( \frac{x}{(1+x)^2} \right),$$ where $$\label{eq:xin+} \xi^+_n (x) \ = \ \sum \ {n \choose r_0, r_1,\dots,r_{2k}} \, x^k \, \xi_{r_0} (x) \, \gamma_{r_1} (x) \cdots \gamma_{r_{2k}}(x),$$ $$\label{eq:xin-} \xi^-_n (x) \ = \ \sum \ {n \choose r_0, r_1,\dots,r_{2k+1}} \, x^{k+1} \, \xi_{r_0}(x) \, \gamma_{r_1} (x) \cdots \gamma_{r_{2k+1}}(x),$$ the sums in the previous equations range as in [(\[eq:main+\])]{} and [(\[eq:main-\])]{}, respectively, and $\xi_0 (x) = 1$, $\gamma_0 (x) = 0$. For the first few values of $n$ we have $$\xi^+_n (x) \ = \ \begin{cases} 1, \ \ & \text{if \ $n=0$} \\ 0, \ \ & \text{if \ $n=1$} \\ 3x, \ \ & \text{if \ $n=2$} \\ 7x, \ \ & \text{if \ $n=3$} \\ 15x + 57x^2, \ \ & \text{if \ $n=4$} \\ 31x + 458x^2, \ \ & \text{if \ $n=5$} \\ 63x + 2551x^2 + 2763x^3, \ \ & \text{if \ $n=6$} \\ 127x + 12232x^2 + 46861x^3, \ \ & \text{if \ $n=7$} \end{cases}$$ and $$\xi^-_n (x) \ = \ \begin{cases} 0, \ \ & \text{if \ $n=0$} \\ x, \ \ & \text{if \ $n=1$} \\ x, \ \ & \text{if \ $n=2$} \\ x + 11x^2, \ \ & \text{if \ $n=3$} \\ x + 54x^2, \ \ & \text{if \ $n=4$} \\ x + 197x^2 + 361x^3, \ \ & \text{if \ $n=5$} \\ x + 648x^2 + 4379x^3, \ \ & \text{if \ $n=6$} \\ x + 2039x^2 + 34586x^3 + 24611x^4, \ \ & \text{if \ $n=7$.} \end{cases}$$ We are not aware of any combinatorial interpretations for the coefficients of $\xi^+_n(x)$ or $\xi^-_n (x)$. The second proof of Theorem \[thm:main\] and the proof of Corollary \[cor:main\] yield the following explicit formulas for the exponential generating functions of $f^+_n (x)$ and $f^-_n (x)$. \[prop:f+-exp\] We have $$\label{eq:fn+exp} \sum_{n \ge 0} \ f^+_n (x) \, \frac{t^n}{n!} \ = \ \frac{e^{xt} - xe^t} {e^{2xt} - xe^{2t}}$$ and $$\label{eq:fn-exp} \sum_{n \ge 0} \ f^-_n (x) \, \frac{t^n}{n!} \ = \ \frac{x(e^t - e^{xt})} {e^{2xt} - xe^{2t}}.$$ We noticed in the proof of Corollary \[cor:main\] that $f^+_n (x) = S^+_n (x)$ and $f^-_n (x) = S^-_n (x)$. Thus, the result follows from the formulas in the second proof of Theorem \[thm:main\] and straightforward algebraic manipulations. A combinatorial interpretation {#sec:comb} ============================== This section gives a combinatorial interpretation to the coefficients of $f^+_n (x)$ and $f^-_n (x)$ by exploiting the first proof of Theorem \[thm:main\], given in Section \[sec:proof\]. Consider a signed permutation $w \in \mathfrak{S} (S)$, where $S = \{a_1, a_2,\dots,a_n\}$ is as in Section \[subsec:signed\]. We denote by $m_w$ the minimum element of $S$ with respect to the natural total order inherited from $\ZZ$ and set $B^_n = \{ w \in B_n: w(m_w) > 0\}$. \[prop:f+-interpret\] We have $$\label{eq:fn+interpret} f^+_n (x) \ = \ \sum_{w \in \dD^B_n \cap B^_n} \ x^{\exc_B (w)}$$ and $$\label{eq:fn-interpret} f^-_n (x) \ = \ \sum_{w \in \dD^B_n \sm B^_n} \ x^{\exc_B (w)}$$ for every $n \ge 1$. We will follow the setup of the first proof of Theorem \[thm:main\]. Given $w \in \dD^B_n$ with $\varphi(w) = (\sigma_0, \sigma_1,\dots,\sigma_k)$, we observe that $k$ is even if and only if the last element in the standard cycle form of $w$ is positive. Therefore, equation (\[eq:main+\]) and the argument in the proof of Theorem \[thm:main\] show that $$f^+_n (x) \ = \ \sum \ x^{\iexc_B (w)},$$ where the sum ranges over all $w \in \dD^B_n$ for which the last element in the standard cycle form is positive. Since this element equals $w^{-1} (m_w)$, we get $$f^+_n (x) \ = \ \sum_{w \in \dD^B_n: \ w^{-1} (m_w) > 0} \ x^{\iexc_B (w)} \ = \ \sum_{w \in \dD^B_n: \ w (m_w) > 0} \ x^{\iexc_B (w^{-1})} \ = \ \sum_{w \in \dD^B_n \cap B^_n} \ x^{\exc_B (w)}.$$ Equation (\[eq:fn-interpret\]) follows from (\[eq:fn+interpret\]) and (\[eq:dnBdef\]), or by a similar argument. The relative local $h$-vector {#sec:relative} ============================= This section reviews the definition of the relative local $h$-polynomial of a simplicial subdivision of a simplex, introduced in [@Ath12 Section 3] and, independently (in a different level of generality), in [@Ni12], and establishes some of its main properties (most of them stated without proof in [@Ath12 Section 3]). The relative local $h$-polynomial of the barycentric subdivision of the simplex is also computed (Example \[ex:barycentrel\]). This computation will be used in Section \[sec:geom\]. We will fix a field $\kk$ in this section and work with the notion of a homology (rather than topological) simplicial subdivision over $\kk$, as in [@Ath12]. Thus, in the definition of a subdivision $\sigma: \Delta' \to \Delta$ we require that the subcomplex $\Delta'_F := \sigma^{-1} (2^F)$ of $\Delta'$ is a homology (rather than topological) ball over $\kk$ of dimension $\dim(F)$, for every $F \in \Delta$; see [@Ath12 Section 2] for details. The following concept was introduced in [@Ath12 Remark 3.7] and (for regular triangulations of polytopes) in [@Ni12]. [([@Ath12 Section 3])]{} \[def:relative\] Let $\Gamma$ be a homology subdivision of a $(d-1)$-dimensional simplex $2^V$, with subdivision map $\sigma: \Gamma \to 2^V$, and let $E \in \Gamma$. The polynomial $$\label{eq:deflocalhrel} \ell_V (\Gamma, E, x) \ = \sum_{\sigma(E) \subseteq F \subseteq V} \ (-1)^{d - \|F\|} \, h (\link_{\Gamma_F} (E), x)$$ is the relative local $h$-polynomial of $\Gamma$ (with respect to $V$) at $E$. Thus, $\ell_V (\Gamma, E, x)$ reduces to the local $h$-polynomial $\ell_V (\Gamma, x)$ for $E = \varnothing$. \[ex:barycentrel\] Let $\Gamma = \sd(2^V)$ be the barycentric subdivision of an $(n-1)$-dimensional simplex $2^V$ and $E = \{S_1, S_2,\dots,S_k\}$ be a face of $\Gamma$, where $S_1 \subset S_2 \subset \cdots \subset S_k \subseteq V$ are nonempty sets. We will show that $$\label{eq:barycentrel} \ell_V (\Gamma, E, x) \ = \ d_{r_0}(x) \, A_{r_1}(x) A_{r_2}(x) \cdots A_{r_k}(x),$$ where $r_0 = \|V \sm S_k\|$ and $r_i = \|S_i \sm S_{i-1}\|$ for $1 \le i \le k$ (with the convention $S_0 = \varnothing$). We recall from Section \[subsec:sub\] that the carrier of $E$ in $\Gamma$ is given by $\sigma(E) = S_k$. Thus the right-hand side of (\[eq:deflocalhrel\]) is a sum over all $S_k \subseteq F \subseteq V$. The restriction $\Gamma_F$ is the barycentric subdivision of $2^F$ and the link of $E$ in this restriction satisfies $\link_{\Gamma_F} (E) = \Delta_0 \ast \Delta_1 \ast \cdots \ast \Delta_k$, where $\Delta_i$ is the simplicial complex of all chains of subsets of $V$ which strictly contain $S_{i-1}$ and are strictly contained in $S_i$, for $1 \le i \le k$, and $\Delta_0$ is the simplicial complex of all chains of subsets of $V$ which strictly contain $S_k$ and are strictly contained in $F$. As a result, we have $$\begin{aligned} h(\link_{\Gamma_F} (E), x) &=& h(\Delta_0, x) \, h(\Delta_1, x) \cdots h(\Delta_k, x) \\ &=& A_{\|F \sm S_k\|}(x) \, A_{r_1}(x) A_{r_2}(x) \cdots A_{r_k}(x). \end{aligned}$$ Multiplying this equation with $(-1)^{d - \|F\|}$, summing over all $S_k \subseteq F \subseteq V$ and using (\[eq:dnA\]) we get (\[eq:barycentrel\]). Our motivation for introducing the relative local $h$-polynomial comes from the following statement (for another motivation, see [@Ni12 Section 3]). [([@Ath12 Proposition 3.6])]{} \[prop:localrelformula\] For every homology subdivision $\Gamma$ of the simplex $2^V$ and every homology subdivision $\Gamma'$ of $\Gamma$ we have $$\label{eq:localrelformula} \ell_V (\Gamma', x) \ = \, \sum_{E \in \Gamma} \ \ell_E (\Gamma'_E, x) \, \ell_V (\Gamma, E, x).$$ We now confirm that the polynomial $\ell_V (\Gamma, E, x)$ shares two of the main properties of $\ell_V (\Gamma, x)$ and deduce a monotonicity property of local $h$-vectors. These results were stated without proof in [@Ath12 Remark 3.7]. Here we will sketch the proof, which follows closely ideas of [@Sta92] and their refinements in [@Ath10]. For that reason, we will assume familiarity with the corresponding proofs in [@Ath10; @Sta92]. \[thm:relative\] Let $V$ be a set with $d$ elements. - The relative local $h$-polynomial $\ell_V (\Gamma, E, x)$ has symmetric coefficients, in the sense that $$\label{eq:relsymm} x^{d-\|E\|} \, \ell_V (\Gamma, E, 1/x) \ = \ \ell_V (\Gamma, E, x),$$ for every homology subdivision $\Gamma$ of the simplex $2^V$ and every $E \in \Gamma$. - The relative local $h$-polynomial $\ell_V (\Gamma, E, x)$ has nonnegative coefficients for every quasi-geometric homology subdivision $\Gamma$ of the simplex $2^V$ and every $E \in \Gamma$. \(a) The proof of [@Ath10 Theorem 4.2] can be adapted as follows. Using the defining equation (\[eq:deflocalhrel\]) and [@Ath10 Proposition 2.1], we find that $$\begin{aligned} x^{d-\|E\|} \, \ell_V (\Gamma, E, 1/x) &=& \sum_{\sigma(E) \subseteq F \subseteq V} (-1)^{d - \|F\|} \, x^{d-\|E\|} \, h (\link_{\Gamma_F} (E), 1/x) \\ & & \\ &=& \sum_{\sigma(E) \subseteq F \subseteq V} (-x)^{d - \|F\|} \, h (\inte(\link_{\Gamma_F} (E)), x). \end{aligned}$$ An inclusion-exclusion argument, similar to the one in the proof of [@Ath10 (4.3)], shows that $$h (\inte(\link_{\Gamma_F} (E)), x) \ = \ \sum_{\sigma(E) \subseteq G \subseteq F} (x-1)^{\|F\|-\|G\|} \, h (\link_{\Gamma_G} (E), x).$$ Replacing $h (\inte(\link_{\Gamma_F} (E)), x)$ in the first formula by the right-hand side of the previous equation and changing the order of summation, as in the proof of [@Ath10 Theorem 4.2], results in (\[eq:relsymm\]). \(b) The special case $E = \varnothing$ is equivalent to part (iii) of [@Ath12 Theorem 3.3] (essentially, part (c) of Theorem \[thm:stalocal\]). The general case follows by the argument in the proof of [@Ath10 Theorem 5.1] (generalizing that in the proof of [@Sta92 Theorem 4.6]), where the role of $\Delta$ in that proof is played by $\link_\Gamma (E)$, the role of $d$ is played by $d - \|E\| = \dim \link_\Gamma (E) + 1$ and the role of $e$ is played by the rank $d - \|\sigma(E)\|$ of the interval $[\sigma(E), V]$ in the lattice of subsets of $V$. For polynomials $p(x), q(x) \in \RR[x]$ we write $p(x) \ge q(x)$ if the difference $p(x) - q(x)$ has nonnegative coefficients. \[cor:localmonotone\] For every quasi-geometric homology subdivision $\Gamma$ of the simplex $2^V$ and every quasi-geometric homology subdivision $\Gamma'$ of $\Gamma$, we have $\ell_V (\Gamma', x) \ge \ell_V (\Gamma, x)$. The right-hand side of (\[eq:localrelformula\]) reduces to $\ell_V(\Gamma, x)$ for $E = \varnothing$. The other terms in the sum are nonnegative by Theorems \[thm:stalocal\] (c) and \[thm:relative\] (b) and the proof follows. A geometric interpretation {#sec:geom} ========================== This section formally defines the simplicial subdivision $K_n$ and gives two proofs of Theorem \[thm:localint\], one using the theory of (relative) local $h$-vectors (specifically, Proposition \[prop:localrelformula\]) and another using generating functions. Let $\Delta$ be a simplicial complex. The cubical barycentric subdivision (see, for instance, [@BBC97 Section 2.3]) of $\Delta$, denoted $\sd_c (\Delta)$, is defined as the set of all nonempty closed intervals $[F, G]$ in the face poset $\fF(\Delta)$, partially ordered by inclusion. It follows from [@Wa88 Theorem 6.1 (a)] and [@StaEC1 Equation (3.24)] that the order complex, say $\Delta'$, of $\sd_c (\Delta)$ is homeomorphic to $\Delta$. Moreover, $\Delta'$ is naturally a simplicial subdivision of $\Delta$: the carrier of a face of $\Delta'$ is the maximum element of the largest of the intervals in the corresponding chain of intervals of $\fF(\Delta)$. We will denote by $K_n$ the order complex of $\sd_c (2^{[n]})$, so that $K_n$ is a simplicial subdivision of the simplex $2^{[n]}$ (see Figure \[fig:K3\] for the case $n=3$). We note that $K_n$ is the special case $N=1$ of a subdivision of the simplex considered in [@CMS84 p. 414]. The following statement is an essential step for both proofs of Theorem \[thm:localint\] which will be given in this section. \[prop:Knhpoly\] We have $h (K_n, x) = B^+_n (x)$ for $n \in \NN$. The poset $\sd_c (2^{[n]})$ consists of all intervals of the form $[A, B]$, where $\varnothing \neq A \subseteq B \subseteq [n]$, partially ordered by inclusion. To describe this poset differently, we consider the following poset $(P_n, \preceq)$. The elements of $P_n$ are the subsets of $\Omega_n$ which contain at least one positive number and at most one number from each set $\{i , -i\}$ for $i \in \{1, 2,\dots,n \}$; the partial order is reverse inclusion. We observe that the map $\varphi: \sd_c (2^{[n]}) \to P_n$ defined by $\varphi([A, B]) = \ A \, \cup \, (-([n] \sm B) )$ is a poset isomorphism. Thus, we may identify $K_n$ with the order complex of $P_n$. For $S = \{s_1, s_2,\dots,s_k\} \subseteq [n]$ with $s_1 < s_2 < \cdots < s_k$, we define $\alpha_{P_n}(S)$ as the number of chains $F_1 \prec F_2 \prec \cdots \prec F_k$ in $P_n$ such that $\|F_i\| = s_{k-i+1}$ for $i \in \{1, 2,\dots,k\}$. The map $\alpha_{P_n}: 2^{[n]} \rightarrow \NN$ is the flag $f$-vector of $P_n$; see [@StaEC1 Section 3.13]. The chains of $P_n$ enumerated by $\alpha_{P_n}(S)$ are in one-to-one correspondence with the elements $w \in B^+_n$ for which $\Des_B(w) \subseteq n-S := \{ n-s: \, s \in S\}$. Indeed, given such a chain, the corresponding element of $B^+_n$ consists of the elements of $[n] \sm \{\|s\|: s \in F_1\}$ in increasing order, followed by those of $F_1 \sm F_2$ in increasing order and so on, followed at the end by the elements of $F_k$ in increasing order. Recall that the flag $h$-vector $\beta_{P_n}:2^{[n]} \rightarrow \ZZ$ of $P_n$ is defined by $$\beta_{P_n}(S) \ = \ \sum_{T \subseteq S} (-1)^{\|S \sm T\|} \ \alpha_{P_n}(T),$$ for $S \subseteq [n]$, or equivalently, by $$\alpha_{P_n}(S) \ = \ \sum_{T\subseteq S} \beta_{P_n}(T)$$ for $S\subseteq [n]$. Since $\alpha_{P_n}(S)$ enumerates signed permutations $w \in B^+_n$ for which $\Des_B(w) \subseteq n -S$, by the Principle of Inclusion-Exclusion we get that $\beta_{P_n}(S)$ enumerates signed permutation $w \in B_n^+$ for which $\Des_B(w) = n-S$. The result follows from this interpretation by recalling [@StaEC1 Section 3.13] that $$h_k(K_n) \ = \ \sum_{S\subseteq [n], \|S\|=k} \beta_{P_n}(S)$$ and switching $S$ to $n-S$ in the previous equation. Our first proof of Theorem \[thm:localint\] will be based on the fact that $K_n$ can be viewed as a subdivision of the barycentric subdivision $\sd (2^{[n]})$. To explain how, we consider the following setup. Let $V = \{v_1, v_2,\dots,v_d\}$ be a set totally ordered by $v_1 < v_2 < \cdots < v_d$. We recall that $\sd_c (V)$ denotes the poset of intervals in $V$ of the form $[v_i, v_j] = \{v_i, v_{i+1},\dots,v_j\}$ for $1 \le i \le j \le n$, partially ordered by inclusion. We denote by $\Gamma$ the order complex of $\sd_c (V)$, consisting of all chains of such intervals. For such a chain $G \in \Gamma$, we define $\sigma(G)$ as the set of all endpoints of the intervals in $G$. Thus we have a well defined map $\sigma: \Gamma \to 2^V$. = 2.0 in \[lem:Knbary\] Under the previous assumptions and notation, the map $\sigma: \Gamma \to 2^V$ turns $\Gamma$ into a geometric subdivision of $2^V$. The number of facets of $\Gamma$ is equal to $2^{d-1}$, where $d$ is the number of elements of $V$. Let $\Sigma_V$ be a geometric $(d-1)$-dimensional simplex whose vertices are labeled by the singleton subsets of $V = \{v_1, v_2,\dots,v_d\}$. We will construct a geometric simplicial subdivision (triangulation) $\Gamma_V$ of $\Sigma_V$ whose vertices are labeled (in a one-to-one fashion) with the closed intervals in the total order $V$, so that: (a) the singleton intervals label the vertices of $\Sigma_V$; (b) the point labeled by a non-singleton interval $I = [v_i, v_j] \in \sd_c (V)$ lies in the relative interior of the edge of $\Sigma_V$ whose endpoints are labeled by $\{v_i\}$ and $\{v_j\}$; and (c) the faces of $\Gamma_V$ correspond to the chains of intervals (see Figure \[fig:Int3\] for the case $d=3$). We proceed by induction on $d$. The triangulation $\Gamma_V$ is a single point for $d=1$ and the triangulation of a line segment with one interior point (labeled by $\{v_1, v_2\}$) for $d=2$. We assume $d \ge 3$ and set $U = V \sm \{v_d\}$ and $W = V \sm \{v_1\}$. We choose the simplices $\Sigma_U$ and $\Sigma_W$ as the codimension one faces of $\Sigma_V$ which correspond to $U$ and $V$ and, using the inductive hypothesis, triangulations $\Gamma_U$ and $\Gamma_W$ of these two simplices having properties (a), (b) and (c) with respect to the totally ordered subsets $U$ and $W$ of $V$, respectively. Clearly, we may choose these triangulations to have the same restriction on the face $\Sigma_U \cap \Sigma_W$ of $\Sigma_V$. We then label by $V$ an arbitrary point $p$ in the relative interior of the edge of $\Sigma_V$ whose endpoints are labeled with $\{v_1\}$ and $\{v_d\}$ and define $\Gamma_V$ as the collection consisting of all simplices in $\Gamma_U \cup \Gamma_W$ and the cones of these on the vertex $p$. We leave it to the reader to verify that $\Gamma_V$ has properties (a), (b) and (c) and that it realizes an abstract simplicial subdivision of $2^V$ with the required properties. We now recall that $K_n$ consists of all chains of intervals of the form $[A, B]$, where $\varnothing \ne A \subseteq B \subseteq [n]$. We define the carrier of such a chain $\cC$ as the set of all endpoints of the intervals in $\cC$ and note that this set is a chain in the poset $\fF(2^{[n]})$ of nonempty subsets of $[n]$ and hence belongs to the barycentric subdivision $\sd (2^{[n]})$. Applying Lemma \[lem:Knbary\] to an arbitrary chain $V \in \sd (2^{[n]})$ we conclude that $K_n$ is a subdivision of $\sd (2^{[n]})$ and that the restriction of this subdivision to a nonempty face $V \in \sd (2^{[n]})$ of dimension $d-1$ has exactly $2^{d-1}$ facets. \[lem:2\^n-1facets\] Let $\Gamma$ be a quasi-geometric simplicial subdivision of a $(d-1)$-dimensional simplex $2^V$. If the restriction $\Gamma_F$ has exactly $2^{\dim(F)}$ facets for every nonempty face $F$ of $2^V$, then $$\ell_V (\Gamma, x) \ = \ \begin{cases} x^{d/2}, & \text{if $d$ is even} \\ 0, & \text{if $d$ is odd.} \end{cases}$$ We recall that the number of facets of a simplicial complex $\Delta$ is equal to the value of the $h$-polynomial $h(\Delta, x)$ at $x=1$. Thus, setting $x=1$ in the defining equation (\[eq:deflocalh\]) and using the assumption on $\Gamma$, we find that $$\ell_V (\Gamma, 1) \ = \ (-1)^d \, + \, \sum_{k = 1}^d \ (-1)^{d - k} {d \choose k} \, 2^{k-1} \ = \ \begin{cases} 1, & \text{if $d$ is even} \\ 0, & \text{if $d$ is odd} \end{cases}$$ and the result follows from parts (b) and (c) of Theorem \[thm:stalocal\]. Let us denote by $\ell^+_n (x)$ the local $h$-polynomial of $K_n$. To compute this polynomial, we will apply Proposition \[prop:localrelformula\] to $\Gamma' = K_n$ and $\Gamma = \sd (2^{[n]})$. Let $E = \{S_1, S_2,\dots,S_k\}$ be a face of $\Gamma$ with $k$ elements, where $S_1 \subset S_2 \subset \cdots \subset S_k \subseteq [n]$ are nonempty sets. We have already noted that the restriction $\Gamma'_E$ satisfies the assumptions of Lemma \[lem:2\^n-1facets\]. Thus, by Lemma \[lem:2\^n-1facets\] we have $$\label{eq:2^k-1facets} \ell_E (\Gamma'_E, x) \ = \ \begin{cases} x^{k/2}, & \text{if $k$ is even} \\ 0, & \text{if $k$ is odd.} \end{cases}$$ The relative local $h$-vector of $\Gamma$ was computed in Example \[ex:barycentrel\]. Thus, in view of (\[eq:2\^k-1facets\]) and (\[eq:barycentrel\]), Proposition \[prop:localrelformula\] yields that $$\ell^+_n (x) \ = \ \sum \ {n \choose r_0, r_1,\dots,r_k} \, x^{k/2} \, d_{r_0} (x) \, A_{r_1} (x) \cdots A_{r_k}(x),$$ where the sum ranges over all even numbers $k \in \NN$ and over all sequences $(r_0, r_1,\dots,r_k)$ of nonnegative integers which sum to $n$. This equation and (\[eq:main+\]) imply that $\ell^+_n (x) = f^+_n (x)$ and the first statement of Theorem \[thm:localint\] follows. We leave to the reader to verify that $K_n$ can be obtained from the trivial subdivision of the simplex by successive stellar subdivisions. This implies that $K_n$ is a regular subdivision. The claim that $f^+_n (x)$ has nonnegative, symmetric and unimodal coefficients follows from the main properties of local $h$-polynomials [@Sta92] (see Theorem \[thm:stalocal\]). Equation (\[eq:localint+\]) follows from the fact that $f^+_n (x) = \ell^+_n (x)$, the defining equation (\[eq:deflocalh\]) of local $h$-polynomials and Proposition \[prop:Knhpoly\]. Given that $d^B_n (x) = f^+_n (x) + f^-_n (x)$ and $B_n (x) = B^+_n (x) + B^-_n (x)$ for every $n$, equation (\[eq:localint-\]) is a consequence of (\[eq:dnB\]) and (\[eq:localint+\]). For the second proof of Theorem \[thm:localint\] we will need the exponential generating functions of $B^+_n (x)$ and $B^-_n (x)$. These will be computed in Section \[sec:half\]. Let us denote by $\ell^+_n (x)$ and $\ell^-_n (x)$ the right-hand side of (\[eq:localint+\]) and (\[eq:localint-\]), respectively. Proposition \[prop:Knhpoly\] and (\[eq:deflocalh\]) imply that $\ell^+_n (x)$ is equal to the local $h$-polynomial of $K_n$. Thus, we need to show that $\ell^+_n (x) = f^+_n (x)$ and $\ell^-_n (x) = f^-_n (x)$ for every $n$. From the definition of $\ell^+_n (x)$ and $\ell^-_n (x)$ and Proposition \[prop:B+-nexp\] we get $$\sum_{n \ge 0} \ \ell^+_n (x) \, \frac{t^n}{n!} \ = \ e^{-t} \ \sum_{n \ge 0} \ B^+_n (x) \, \frac{t^n}{n!} \ = \ \frac{e^{xt} - xe^t} {e^{2xt} - xe^{2t}}$$ and $$\sum_{n \ge 0} \ \ell^-_n (x) \, \frac{t^n}{n!} \ = \ e^{-t} \ \sum_{n \ge 0} \ B^-_n (x) \, \frac{t^n}{n!} \ = \ \frac{x(e^t - e^{xt})} {e^{2xt} - xe^{2t}}.$$ The result follows from these equations and Proposition \[prop:f+-exp\]. A decomposition of the Eulerian polynomial of type $B$ {#sec:half} ====================================================== This section studies the decomposition of the Eulerian polynomial $B_n (x)$ as a sum of $B^+_n (x)$ and $B^-_n (x)$. First, it is observed that a simple relation between the two summands holds. Then, using the theory of local $h$-vectors and results of Section \[sec:geom\], a simple formula for $B^+_n (x)$ in terms of the Eulerian polynomial $A_n (x)$ is proven (Proposition \[prop:Bn+formula\]). From this formula, it is deduced that $B^+_n (x)$ and $B^-_n (x)$ are real-rooted (Corollary \[cor:B+-nrealroots\]), hence unimodal and log-concave, and a new proof of the unimodality of $B_n (x)$ is derived. Finally, recurrences and generating functions for $B^+_n (x)$ and $B^-_n (x)$ are given. These lead to recurrences and generating functions for $f^+_n (x)$ and $f^-_n (x)$ and to yet another proof of Theorem \[thm:localint\]. We recall that $B^+_n (x)$ and $B^-_n (x)$ are defined by (\[eq:Bn+def\]) and (\[eq:Bn-def\]). For the first few values of $n$ we have $$B^+_n(x) \ = \ \begin{cases} 1, & \ \text{if \ $n=0$} \\ 1, & \ \text{if \ $n=1$} \\ 1 + 3x, & \ \text{if \ $n=2$} \\ 1 + 16x + 7x^2, & \ \text{if \ $n=3$} \\ 1 + 61x + 115x^2 + 15x^3, & \ \text{if \ $n=4$} \\ 1 + 206x + 1056x^2 + 626x^3 + 31x^4, & \ \text{if \ $n=5$} \\ 1 + 659x + 7554x^2 + 11774x^3 + 2989x^4 + 63x^5, & \ \text{if \ $n=6$} \end{cases}$$ and $$B^-_n(x) \ = \ \begin{cases} 0, & \ \text{if \ $n=0$} \\ x, & \ \text{if \ $n=1$} \\ 3x + x^2, & \ \text{if \ $n=2$} \\ 7x + 16x^2 + x^3, & \ \text{if \ $n=3$} \\ 15x + 115x^2 + 61x^3 + x^4, & \ \text{if \ $n=4$} \\ 31x + 626x^2 + 1056x^3 + 206x^4 + x^5, & \ \text{if \ $n=5$} \\ 63x + 2989x^2 + 11774x^3 + 7554x^4 + 659x^5 + x^6, & \ \text{if \ $n=6$.} \end{cases}$$ The previous data suggest the following statement. \[lem:rec\] We have $B^-_n (x) = x^n B^+_n (1/x)$ for $n \ge 1$. Given a signed permutation $w = (w(a_1), w(a_2),\dots,w(a_n)) \in B_n$, where the notation is as in Section \[subsec:signed\], we set $-w := (-w(-a_1), -w(-a_2),\dots,-w(-a_n)) \in B_n$. Then the induced map $\varphi: B^+_n \to B^-_n$ defined by $\varphi(w) = -w$ is a bijection. Moreover, for every $w \in B^+_n$, an index $i \in \{0, 1,\dots,n-1\}$ is a $B$-ascent of $w$ if and only if $i$ is a $B$-descent of $\varphi(w)$ and the proof follows. To prove the formula for $B^+_n (x)$ promised, we will use the construction of the $r$th edgewise subdivision $\Delta^{\langle r \rangle}$ of a simplicial complex $\Delta$. We refer the reader to [@BR05; @BW09] for the definition and history of this subdivision and recall the following known facts. First, the restriction $\Delta^{\langle r \rangle}_F$ of $\Delta^{\langle r \rangle}$ has exactly $r^{\dim(F)}$ facets for every nonempty face $F \in \Delta$. Second, combining [@BR05 Corollary 6.8] with [@BW09 Corollary 1.2], one gets the explicit formula $$\label{eq:hedgewise} h(\Delta^{\langle r \rangle}, x) \ = \ {\rm E}_r \left( (1 + x + \cdots + x^{r-1})^d \, h (\Delta, x) \right)$$ for the $h$-polynomial of $\Delta^{\langle r \rangle}$, where $d-1$ is the dimension of $\Delta$ and ${\rm E}_r$ is the operator on polynomials (more generally, on formal power series) defined by $${\rm E}_r \left( \, \sum_{k \ge 0} \, c_k x^k \right) \ = \ \sum_{k \ge 0} \, c_{rk} x^k \ = \ c_0 + c_r x + c_{2r} x^2 + \cdots.$$ Figure \[fig:KK3\] shows the second edgewise subdivision of the barycentric subdivision of the 2-dimensional simplex. = 1.5 in \[prop:Bn+formula\] We have $B^+_n (x) = {\rm E}_2 \left( (1 + x)^n A_n (x) \right)$ for every $n \ge 1$. We consider the subdivision $K_n$ and the second edgewise subdivision $K'_n$ of the barycentric subdivision $\sd (2^{[n]})$ (see Figures \[fig:K3\] and \[fig:KK3\] for the special case $n=3$). Applying (\[eq:hformula\]) for $\Delta' = K_n$ or $K'_n$, respectively, and $\Delta = \sd (2^{[n]})$ we get $$\begin{aligned} h(K_n, x) &=& \sum_{F \in \Delta} \ \ell_F ((K_n)_F, x) \, h (\link_\Delta (F), x), \\ h(K'_n, x) &=& \sum_{F \in \Delta} \ \ell_F ((K'_n)_F, x) \, h (\link_\Delta (F), x). \end{aligned}$$ Since both restrictions $(K_n)_F$ and $(K'_n)_F$ have exactly $2^{\dim(F)}$ facets for every nonempty face $F \in \Delta$, it follows from the previous formulas and Lemma \[lem:2\^n-1facets\] that $h(K_n, x) = h(K'_n, x)$. Combining this equality with Proposition \[prop:Knhpoly\] we get $B^+_n (x) = h(K'_n, x)$ for every $n \ge 1$. Formula (\[eq:hedgewise\]) implies that $$B^+_n (x) \ = \ {\rm E}_2 \left( (1 + x)^n h (\Delta, x) \right) \ = \ {\rm E}_2 \left( (1 + x)^n A_n (x) \right)$$ for $n \ge 1$ and the proof follows. \[rem:cry\] We thank Mirkó Visontai for informing us that a formula similar to the one in Proposition \[prop:Bn+formula\] can be derived from [@ABR01 Theorem 4.4], for which a bijective proof was given in [@LP11]. We will use the following lemma to deduce the real-rootedness of $B^+_n (x)$ and $B^-_n (x)$. \[lem:Er\] Let $p(x)$ be a polynomial with real coefficients and let $r$ be a positive integer. - If $p(x)$ has unimodal coefficients, then so does ${\rm E}_r (p(x))$. - If $p(x)$ has nonnegative and log-concave coefficients, with no internal zeros, then so does ${\rm E}_r (p(x))$. - If $p(x)$ is real-rooted, then so is ${\rm E}_r (p(x))$. Part (a) is trivial and part (b) can be left as an excercise. For part (c) we set $p(x) = \sum_{k \ge 0} a_k x^k$ and note that the matrix $(a_{ri - rj})_{i,j=0}^\infty$ is a submatrix of $(a_{i - j})_{i,j=0}^\infty$. Therefore, every minor of the former is also a minor of the latter and the result follows from Theorem \[thm:realroots\]. \[cor:B+-nrealroots\] The polynomials $B^+_n (x)$ and $B^-_n (x)$ are real-rooted for every $n \ge 1$. They are unimodal with peaks at $\lfloor n/2 \rfloor$ and $\lfloor (n+1)/2 \rfloor$, respectively, for every $n \ge 2$. The first statement follows from Lemma \[lem:rec\], Proposition \[prop:Bn+formula\] and the fact that the Eulerian polynomial $A_n(x)$ is real-rooted for $n \ge 1$, via part (c) of Lemma \[lem:Er\]. The second statement follows from Lemma \[lem:rec\], Proposition \[prop:Bn+formula\] and the fact that $(1 + x)^n A_n (x)$ is a polynomial of degree $2n-1$ with symmetric and unimodal coefficients, via part (a) of Lemma \[lem:Er\]. \[rem:Bnformula\] Since $B_n (x) = B^+_n (x) + B^-_n(x)$, Lemma \[lem:rec\] and Proposition \[prop:Bn+formula\] express the Eulerian polynomial $B_n (x)$ as a sum of two unimodal polynomials with peaks which differ by at most one (see Corollary \[cor:B+-nrealroots\]). This decomposition shows that the unimodality of $B_n (x)$ is a consequence of the unimodality of $A_n (x)$. For a $\gamma$-nonnegativity proof of the unimodality of $B_n (x)$, see [@Pe07 Proposition 4.16]. For an equation relating the Eulerian polynomials of types $A$, $B$ and $D$, see [@Ste94 Lemma 9.1]. We will now give recurrences and generating functions for $B^+_n (x)$ and $B^-_n (x)$. \[prop:B+-nexp\] We have $$\label{eq:B+nrec} B^+_n(x) \ = \ 2(n-1)x\, B^+_{n-1}(x) \, + \, 2x(1-x)\, \frac{\partial B^+_{n-1}}{\partial x} (x) \, + \, B_{n-1}(x)$$ for every $n \ge 1$, $$\label{eq:B+nexp} \sum_{n \ge 0} B^+_n(x) \frac{t^n}{n!} \ = \ \frac{e^t (e^{xt} - xe^t)} {e^{2xt} - xe^{2t}}$$ and $$\label{eq:B-nexp} \sum_{n \ge 0} B^-_n(x) \frac{t^n}{n!} \ = \ \frac{xe^t (e^t - e^{xt})} {e^{2xt} - xe^{2t}}.$$ Let $w = u_1 u_2 \cdots u_{n-1} \in B_{n-1}$ be a signed permutation, represented as a word. For $i \in \{1, 2,\dots,n\}$, we will denote by $w_i$ (respectively, $w_{-i}$) the signed permutation in $B_n$ obtained from $w$ by inserting $n$ (respectively, $-n$) between $u_{i-1}$ and $u_i$. For $1 \le i \le n-1$ we have $w_i \in B^+_n$ (respectively, $w_{-i} \in B^+_n$) if and only if $w \in B^+_{n-1}$. On the other hand, $w_n \in B^+_n$ and $w_{-n} \in B^-_n$ for every $w \in B_{n-1}$. Moreover, for $1 \le i \le n-1$ we have $$\des_B(w_{\pm i}) \ = \ \begin{cases} \des_B(w), & \textrm{if $i-1 \in \Des_B(w)$}\\ \des_B(w) + 1, & \textrm{if $i-1 \notin \Des_B(w)$} \end{cases} \nonumber$$ and $\des_B(w_n) \ = \ \des_B(w)$. Thus, we compute that $$\begin{aligned} B^+_n(x) & = & \sum_{\sigma \in B^+_n} x^{\des_B(\sigma)} \ = \ \sum_{i=1}^{n-1} \left(\sum_{w \in B^+_{n-1}} x^{\des_B(w_i)} \, + \, x^{\des_B (w_{-i}) } \right) + \sum_{w\in B_{n-1}} x^{\des_B(w_n)} \\ & & \\ & = & 2 \sum_{w \in B^+_{n-1}} \left( \des_B(w) \, x^{\des_B(w)} + (n-1-\des_B(w)) \, x^{\des_B(w) + 1} \right) \ + \ B_{n-1}(x) \\ & & \\ & = & 2(n-1) \sum_{w \in B^+_{n-1}} x^{\des_B(w) + 1} \ + \ 2 (1-x) \sum_{w \in B^+_{n-1}} \des_B(w) \, x^{\des_B(w)} \ + \ B_{n-1}(x) \\ & & \\ & = & 2(n-1) x \, B^+_{n-1}(x) \ + \ 2x(1-x)\, \frac{\partial B^+_{n-1}}{\partial x}(x) \ + \ B_{n-1}(x),\end{aligned}$$ which proves (\[eq:B+nrec\]). We now claim that $$\label{eq:ratB+n} \frac{B^+_n(x)}{(1-x)^n} \ = \ \sum_{i \ge 0} \ \left( (2i+1)^n - (2i)^n \right) \, x^i.$$ Given that $B^+_0 (x) = 1$, equation (\[eq:B+nexp\]) then follows by straightforward computations. To prove (\[eq:ratB+n\]), denote by $a_n(i)$ the coefficient of $x^i$ in the expansion of $B^+_n(x)/(1-x)^n$ as a formal power series. Dividing (\[eq:B+nrec\]) by $(1-x)^n$ and using the equality $$\frac{\partial}{\partial x} \left( \frac{B^+_{n-1}(x)}{(1-x)^{n-1}} \right) \ = \ \frac{{\displaystyle \frac{\partial B^+_{n-1}} {\partial x}(x)}} {(1-x)^{n-1}} \ + \ (n-1) \, \frac{B^+_{n-1}(x)}{(1-x)^{n}}$$ we find that $$\frac{B^+_n(x)}{(1-x)^n} \ = \ 2x \ \frac{\partial}{\partial x} \left( \frac{B^+_{n-1}(x)}{(1-x)^{n-1}}\right) \ + \ \frac{B_{n-1}(x)}{(1-x)^n}.$$ Comparing the coefficients of $x^i$ in the two sides of the previous equation and using [@Bre94 Theorem 3.4 (ii)], we get $a_n(i) = 2i a_{n-1}(i) + (2i+1)^{n-1}$. The claim then follows by induction on $n$. Equation (\[eq:B-nexp\]) follows from (\[eq:B+nexp\]) and Lemma \[lem:rec\]. Alternatively, it follows from (\[eq:B-nexp\]) and the formula for the exponential generating function of $B_n (x)$ [@Bre94 Theorem 3.4 (iv)]. We now deduce recurrence relations for $f^+_n (x)$ and $f^-_n (x)$. \[prop:recf+n\] For $n \ge 1$ we have $$f^+_n (x) \ = \ (2(n-1)x-1)\, f^+_{n-1}(x) \, + \, 2x(1-x) \, \frac{\partial f^+_{n-1}} {\partial x}(x) \, + \, 2(n-1) x\, f^+_{n-2}(x) \, + \, d^B_{n-1} (x).$$ Using equation (\[eq:localint+\]), we compute that $$\begin{aligned} f^+_n (x) & = & \sum_{k=0}^n \, (-1)^{n-k} \binom{n}{k} \ B^+_k (x) \\ & & \\ &=& \sum_{k=1}^n \, (-1)^{n-k} \binom{n-1}{k-1} \ B^+_k (x) \ + \ \sum_{k=0}^{n-1} \, (-1)^{n-k} \binom{n-1}{k} \ B^+_k (x) \\ & & \\ &=& \sum_{k=1}^n \, (-1)^{n-k} \binom{n-1}{k-1} \ B^+_k(x) \ - \ f^+_{n-1} (x). \end{aligned}$$ Substituting for $B^+_k(x)$ the right-hand side of (\[eq:B+nrec\]), setting $$S_n(x) \ = \ 2x \, \sum_{k=1}^n \, (-1)^{n-k} \, (k-1) \binom{n-1}{k-1} \ B^+_{k-1}(x)$$ and using (\[eq:dnB\]), we get $$\begin{aligned} f^+_n(x) & = & S_n(x) \, + \, 2x(1-x) \ \sum_{k=1}^n \, (-1)^{n-k} \binom{n-1}{k-1} \, \frac{\partial B^+_{k-1}}{\partial x}(x) \, \\ & & \\ & & + \, \sum_{k=1}^n \, (-1)^{n-k} \binom{n-1}{k-1} \, B_{k-1}(x) \, - \, f^+_{n-1} (x) \\ & & \\ &=& S_n(x) \, + \, 2x(1-x) \, \frac{\partial f^+_{n-1}}{\partial x}(x) \, + \, d^B_{n-1}(x) \, - \, f^+_{n-1}(x). \end{aligned}$$ Finally, using equation (\[eq:localint+\]), we compute that $$\begin{aligned} S_n(x) & = & 2x \sum_{k=1}^n \, (-1)^{n-k} \, k \, \binom{n-1}{k-1} \, B^+_{k-1}(x) \, - \, 2x \sum_{k=1}^n \, (-1)^{n-k} \binom{n-1}{k-1} \, B^+_{k-1}(x) \\ & & \\ &=& 2x \sum_{k=1}^n \, (-1)^{n-k} \, k \,\binom{n}{k} \, B^+_{k-1}(x) \, - \, 2x \sum_{k=1}^{n-1} \, (-1)^{n-k} \, k \, \binom{n-1}{k} \, B^+_{k-1}(x) \\ & & \\ & & - \, 2x \, f^+_{n-1}(x) \\ & & \\ &=& 2nx \sum_{k=1}^n \, (-1)^{n-k} \binom{n-1}{k-1} \, B^+_{k-1}(x) \, - \, 2(n-1)x \sum_{k=1}^n \, (-1)^{n-k} \binom{n-2}{k-1} \, B^+_{k-1}(x) \\ & & \\ & & - \, 2x \, f^+_{n-1}(x) \\ & & \\ &=& 2(n-1)x \, f^+_{n-1}(x) \, + \, 2(n-1)x \, f^+_{n-2}(x)\end{aligned}$$ and the proof follows. We will denote by $a^+_{n,k}$, $a^-_{n,k}$ and $d^B_{n,k}$ the coefficient of $x^k$ in $f^+_n (x)$, $f^-_n (x)$ and $d^B_n (x)$, respectively. The following recurrence relations can be derived from Proposition \[prop:recf+n\] and [@CTZ09 Corollary 4.3]. \[cor:ank\] For $n \ge 2$ and $k \ge 1$ we have $$\label{eq:a+nk} a^+_{n,k} \ = \ (2k-1) a^+_{n-1,k} \, + \, 2(n-k) a^+_{n-1,k-1} \, + \, 2(n-1) a^+_{n-2,k-1} \, + \, d^B_{n-1, k}$$ and $$\label{eq:a-nk} a^-_{n,k} \ = \ (2k-1) a^-_{n-1,k} \, + \, 2(n-k) a^-_{n-1,k-1} \, + \, 2(n-1) a^-_{n-2,k-1} \, + \, d^B_{n-1, k-1}.$$ Equation (\[eq:a+nk\]) follows from the formula of Proposition \[prop:recf+n\] by comparing the coefficients of $x^k$. Since $a^-_{n, k} = d^B_{n, k} - a^+_{n, k}$, equation (\[eq:a-nk\]) follows from (\[eq:a+nk\]) and the recurrence relation for $d^B_{n, k}$ given in [@CTZ09 Corollary 4.3]. As in the second proof, we denote by $\ell^+_n (x)$ and $\ell^-_n (x)$ the right-hand sides of (\[eq:localint+\]) and (\[eq:localint-\]), respectively, and note that $\ell^+_n (x) + \ell^-_n (x) = d^B_n (x)$ and that $\ell^+_n (x)$ is equal to the local $h$-polynomial of $K_n$. In particular, we have $\ell^+_n (x) = x^n \ell^+_n (1/x)$ by Theorem \[thm:stalocal\] (b). The proofs of Proposition \[prop:recf+n\] and Corollary \[cor:ank\] show that the coefficients of $\ell^+_n (x)$ and $\ell^-_n (x)$ satisfy (\[eq:a+nk\]) and (\[eq:a-nk\]), respectively. Since $a^-_{n, k} = d^B_{n, k} - a^+_{n, k}$, we may rewrite (\[eq:a+nk\]) as $$a^+_{n,k} \ = \ 2k a^+_{n-1,k} \, + \, 2(n-k) a^+_{n-1,k-1} \, + \, 2 (n-1)a^+_{n-2,k-1} \, + \, a^-_{n-1, k}.$$ Switching $k$ to $n-k$ in this equality and using the symmetry $a^+_{n,k} = a^+_{n,n-k}$ shows that $$a^-_{n-1,k} \ = \ a^-_{n-1,n-k}.$$ Equivalently, we have $a^-_{n,k} = a^-_{n,n-k+1}$ for all $n$ and $k$ and hence $\ell^-_n (x) = x^{n+1} \ell^-_n (1/x)$ for every $n \in \NN$. The uniqueness of the defining properties of $f^+_n (x)$ and $f^-_n (x)$ shows that $\ell^+_n (x) = f^+_n (x)$ and $\ell^+_n (x) = f^-_n (x)$ for every $n \in \NN$. We have verified that $f^+_n (x)$ and $f^-_n (x)$ are real-rooted for $2 \le n \le 10$. Thus, it is natural to conjecture the following statement. \[conj:f+-n\] The polynomials $f^+_n (x)$ and $f^-_n (x)$ are real-rooted for every $n \ge 2$. Acknowledgments {#acknowledgments .unnumbered} =============== The authors wish to thank Ron Adin, Benjamin Nill, Yuval Roichman, John Stembridge, Mirkó Visontai and Volkmar Welker for useful pointers to the literature. The second author also thanks Francesco Brenti and Mirkó Visontai for useful discussions. The second author was co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. [99]{} R.M. Adin, F. Brenti and Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. [ 27]{} (2001), 210–224. C.A. Athanasiadis, Cubical subdivisions and local $h$-vectors, Ann. Comb. [ 16]{} (2012), 421–448. C.A. Athanasiadis, Flag subdivisions and $\gamma$-vectors, Pacific J. Math. [ 259]{} (2012), 257–278. C.A. Athanasiadis and C. Savvidou, The local $h$-vector of the cluster subdivision of a simplex, Sém. Lothar. Combin. [ 66]{} (2012), Article B66c, 21pp (electronic). E.K. Babson, L.J. Billera and C.S. Chan, Neighborly cubical spheres and a cubical lower bound conjecture, Israel J. Math. [ 102]{} (1997), 297–315. M. Beck and A. Stapledon, On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series, Math. Z. [ 264]{} (2010), 195–207. R. Biagoli and F. Caselli, Invariant algebras and major indices for classical Weyl groups, Proc. Lond. Math. Soc. [ 88]{} (2004), 603–631. A. Björner, Topological methods, in Handbook of Combinatorics (R.L. Graham, M. Grötschel and L. Lovász, eds.), North Holland, Amsterdam, 1995, pp. 1819–1872. A. Björner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics [ 231]{}, Springer-Verlag, New York, 2005. F. Brenti, Unimodal polynomials arising from symmetric functions, Proc. Amer. Math. Soc. [ 108]{} (1990), 1133–1141. F. Brenti, $q$-Eulerian polynomials arising from Coxeter groups, European J. Combin. [ 15]{} (1994), 417–441. F. Brenti and V. Welker, The Veronese construction for formal power series and graded algebras, Adv. in Appl. Math. [ 42]{} (2009), 545–556. M. Brun and T. Römer, Subdivisions of toric complexes, J. Algebraic Combin. [ 21]{} (2005), 423–448. J. Cheeger, W. Müller and R. Schrader, On the curvature of piecewise flat spaces, Commun. Math. Phys. [ 92]{} (1984), 405–454. W.Y.C. Chen, R.L. Tang and A.F.Y. Zhao, Derangement polynomials and excedances of type $B$, Electron. J. Combin. [ 16]{} (2) (2009), Research Paper 15, 16pp (electronic). C.-O. Chow, On derangement polynomials of type $B$. II, J. Combin. Theory Series A [ 116]{} (2009), 816–830. K. Dilks, T.K. Petersen and J.R. Stembridge, Affine descents and the Steinberg torus, Adv. in Appl. Math. [ 42]{} (2009), 423–444. A. Edrei, Proof of a conjecture of Schoenberg on the generating function of a totally positive sequence, Canad. J. Math. [ 5]{} (1953), 86–94. D. Foata and M.-P. Schützenberger, Théorie Géometrique des Polynômes Eulériens, Lecture Notes in Mathematics [ 138]{}, Springer-Verlag, 1970. L.M. Lai and T.K. Petersen, Euler-Mahonian distributions of type $B_n$, Discrete Math. [ 311]{} (2011), 645–650. B. Nill, Combinatorial questions related to stringy $E$-polynomials of Gorenstein polytopes, in Toric Geometry (K. Altmann et. al., eds.), Oberwolfach Reports (2012), no. 21, pp. 62–64. T.K. Petersen, Enriched $P$-partitions and peak algebras, Adv. Math. [ 209]{} (2007), 561–610. J. Shareshian and M.L. Wachs, Eulerian quasisymmetric functions, Adv. Math. [ 225]{} (2010), 2921–2966. R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph Theory and its Applications: East and West, Annals of the New York Academy of Sciences [ 576]{}, New York Acad. Sci., New York, 1989, pp. 500–535. R.P. Stanley, Subdivisions and local $h$-vectors, J. Amer. Math. Soc. [ 5]{} (1992), 805–851. R.P. Stanley, Combinatorics and Commutative Algebra, second edition, Birkhäuser, Basel, 1996. R.P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics [ 49]{}, Cambridge University Press, second edition, Cambridge, 2011. R.P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics [ 62]{}, Cambridge University Press, Cambridge, 1999. J.R. Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Math. [ 99]{} (1992), 307–320. J.R. Stembridge, Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math. [ 106]{} (1994), 244–301. J. Walker, Canonical homeomorphisms of posets, European J. Combin. [ 9]{} (1988), 97–107. X. Zhang, On $q$-derangement polynomials, in Combinatorics and Graph Theory ’95, Vol. 1 (Hefei), pp. 462–465, World Sci. Publishing, River Edge, NJ, 1995. [^1]: 2000 Mathematics Subject Classification. Primary 05A05; Secondary 05A15, 05E45.
--- abstract: 'The $A$- dependence of $\rho^0$ meson production in neutrino-induced reactions is investigated for the first time, using the data obtained with SKAT bubble chamber. The nuclear medium influence on the $\rho^0$ total yield and inclusive distributions (on $z = E_{\rho}/\nu$ and Feynman $x_F$ variables) is found to be approximately the same as for pions. It is shown, that these distributions, with increasing $A$, tend to shift toward smaller values of $z$ and $x_F$, thus indicating on an increasing role of secondary intranuclear interactions. The predictions of a simplified model incorporating the latter are found to be in qualitative agreement with experimental data.' --- 1.0cm [SKAT Collaboration]{} N.M. Agababyan[$^{\mathrm{1}}$]{}, V.V. Ammosov[$^{\mathrm{2}}$]{}, M. Atayan[$^{\mathrm{3}}$]{},\ N. Grigoryan[$^{\mathrm{3}}$]{}, H. Gulkanyan[$^{\mathrm{3}}$]{}, A.A. Ivanilov[$^{\mathrm{2}}$]{},\ Zh. Karamyan[$^{\mathrm{3}}$]{}, V.A. Korotkov[$^{\mathrm{2}}$]{} $^{\mathrm{1}}$ Joint Institute for Nuclear Research, Dubna, Russia $^{\mathrm{2}}$ Institute for High Energy Physics, Protvino, Russia $^{\mathrm{3}}$ Yerevan Physics Institute, Armenia YEREVAN 2005 Introduction ============ The total yields and inclusive spectra of different species of hadrons in leptonuclear interactions reflect the complicated space-time structure of the quark string fragmentation, the hadron formation and the secondary intranuclear interactions. The experimental investigations on this topic concern mainly stable hadrons (pions, kaons, (anti)protons). Meanwhile, the data concerning hadronic resonances (being, predominantly, direct products of the quark string fragmentation) can provide a valuable additional information about the nuclear medium influence on the leptoproduction processes. Hitherto, no data are available for the A- dependence of the total yields and inclusive spectra of hadronic resonances in lepton-induced reactions.\ This work is devoted to the study, for the first time, of the A- dependence of $\rho^0$ meson neutrinoproduction on nuclei. To this end, the data from the SKAT bubble chamber are used. In Section 2, the experimental procedure is described. The experimental data on the A- dependence of the total yield and inclusive spectra of $\rho^0$ are presented in Section 3 and discussed in Section 4. The results are summarized in Section 5. Experimental procedure ====================== The experiment was performed with SKAT bubble chamber [@ref1], exposed to a wideband neutrino beam obtained with a 70 GeV primary protons from the Serpukhov accelerator. The chamber was filled with a propane-freon mixture containing 87 vol% propane ($C_3H_8$) and 13 vol% freon ($CF_3Br$) with the percentage of nuclei H:C:F:Br = 67.9:26.8:4.0:1.3 %. A 20 kG uniform magnetic field was provided within the operating chamber volume.\ Charged current interactions containing a negative muon with momentum $p_{\mu} >$0.5 GeV/c were selected. Other negatively charged particles were considered to be $\pi^-$ mesons. Protons with momentum below 0.6 GeV$/c$ and a fraction of protons with momentum 0.6-0.85 GeV$/c$ were identified by their stopping in the chamber. Non-identified positively charged particles were considered to be ${\pi}^+$ mesons. Events in which errors in measuring the momenta of all charged secondaries and photons were less than 60% and 100%, respectively, were selected. Each event is given a weight which corrects for the fraction of events excluded due to improperly reconstruction. More details concerning the experimental procedure, in particular, the reconstruction of the neutrino energy $E_{\nu}$ can be found in our previous publications [@ref2; @ref3].\ The events with $3 < E_{\nu} <$ 30 GeV were accepted, provided that the reconstructed mass $W$ of the hadronic system exceeds 2 GeV. No restriction was imposed on the transfer momentum squared $Q^2$. The number of accepted events was 4353 (5508 weighted events). The mean values of the kinematical variables were $<E_{\nu}>$ = 10.7 GeV, $<W>$ = 3.0 GeV, $<W^2>$ = 9.6 GeV$^2$, $<Q^2>$ = 2.8 (GeV/c)$^2$.\ Further, the whole event sample was subdivided, using several topological and kinematical criteria [@ref3; @ref4], into three subsamples: the ’cascade’ subsample $B_S$ with a sign of intranuclear secondary interaction, the ’quasiproton’ ($B_p$) and ’quasineutron’ ($B_n$) subsamples. About 40% of subsample $B_p$ is contributed by interactions with free hydrogen. Weighting the ’quasiproton’ events with a factor of 0.6, one can compose a ’pure’ nuclear subsample $B_A = B_S + B_n + 0.6 B_p$ and a ’quasinucleon’ subsample $B_N = B_n + 0.6 B_p$. It has been verified [@ref4; @ref5], that the multiplicity and spectral characteristics of secondary particles in the $B_p (B_N)$ subsample are in satisfactory agreement with those measured with a pure proton (deuteron) target. The effective atomic weight corresponding to the subsample $B_A$ is estimated [@ref6] to be approximately equal to $A_{eff} = 21 \pm 2$, when taking into account the probability of secondary intranuclear interactions in the composite target.\ In order to extract the A- dependence of the $\rho^0$ neutrinoproduction, we use in the next section the data obtained for subsamples $B_N$ and $B_A$, as well as the published data [@ref7] on neutrino-freon interactions (with $A_{eff} = 45 \pm 2$). The A- dependence of the $\rho^0$ mean multiplicity and inclusive spectra ========================================================================= The $(\pi^+ \pi^-)$ effective mass distribution for subsamples $B_N$ and $B_A$ is plotted in Fig. 1. Clear signals near the $\rho^0$ mass, as well as faint signals near the $f_0(980)$ mass are seen. The distributions are fitted by expression $$dN/dM_{\pi^+\pi^-} = BG \cdot (1 + \alpha_{\rho} \cdot BW_{\rho} + \alpha_{f} \cdot BW_{f}) \, ,$$ where the mass dependence of the background distribution is parametrized as $$BG = \alpha_1 \cdot exp (\alpha_2 \cdot M + \alpha_3 \cdot M^2) \, ,$$ while for $BW_{\rho}$ and $BW_f$ the corresponding relativistic Breit-Wigner functions [@ref8] for $\rho^0$ and $f_0(980)$ were used, taking into account the experimental mass resolution $\sigma(M)$ = 35 MeV. The mass and width of resonances are fixed as: $M_{\rho}$ = 775 MeV and $\Gamma_{\rho}$ = 150 MeV from the PDG data [@ref9] and $M_f$ = 963 MeV and $\Gamma_f$ = 35 MeV from the recent NOMAD measurements [@ref10].\ The resulting total yields of $\rho^0$ and $f_0(980)$ are presented in Table 1, where the SKAT data for $\rho^0$ in neutrino-freon interactions [@ref7] are also given. The data on $f_0(980)$ are corrected for the undetectable mode $(\pi^0\pi^0)$. For comparison, the data on the $\pi^-$ yields are shown too. $A$ $<n_{\rho^0}>$ $<n_{f_0}(980)>$ $<n_{\pi^-}>$ $<n_{\rho^0}>/<n_{\pi^-}>$ ----- ----------------- ------------------ --------------- ---------------------------- 1 0.070$\pm$0.031 0.030$\pm$0.015 0.73$\pm$0.02 0.096$\pm$0.043 21 0.075$\pm$0.023 0.019$\pm$0.011 0.80$\pm$0.01 0.094$\pm$0.029 45 0.09$\pm$0.02 $-$ 0.90$\pm$0.01 0.10$\pm$0.02 : The mean multiplicities of $\rho^0$, $f_0(980)$ and $\pi^-$ and the ratio $<n_{\rho^0}>/<n_{\pi^-}>$. The A-dependences of the $\rho^0$ and $\pi^-$ yields, plotted in Fig. 2, are rather similar. An exponential parametrization ($\sim A^{\beta}$) of the yields leads to compatible values of ${\beta}_{\rho^0}$ = 0.07 $\pm$ 0.13 and ${\beta}_{\pi^-}$ = 0.052 $\pm$ 0.007. As a result, no A- dependence of the $<\rho^0/\pi^->$ ratio is observed (the last column of Table 1).\ Note, that a comparison with higher-energy neutrino-induced data shows that this ratio tends to increase with increasing $W$, reaching up to 0.128 $\pm$ 0.030 at $<W>$ = 4.8 GeV [@ref11] and 0.156 $\pm$ 0.028 at $<W>$ = 6.1 GeV [@ref12].\ Our estimations of the $f_0(980)$ total yield are rather rough. They do not contradict, within the experimental uncertainties, the only published data [@ref10], $<n_{f_0}(980)>$ = 0.018 $\pm$0.004, obtained at higher neutrino energies ($<E_{\nu}>$ = 45 GeV). $A_{eff}$ $<n_{\rho^0}>$ $<n_{\pi^-}>$ $<n_{\rho^0}>/<n_{\pi^-}>$ ----------- ----------------- ----------------- ---------------------------- $x_F >0$ 1 0.066$\pm$0.026 0.435$\pm$0.014 0.15$\pm$0.06 21 0.050$\pm$0.016 0.399$\pm$0.009 0.12$\pm$0.04 $x_F <0$ 1 0.005$\pm$0.017 0.239$\pm$0.012 0.02$\pm$0.06 21 0.025$\pm$0.016 0.402$\pm$0.009 0.06$\pm$0.04 : The yields of $\rho^0$ and $\pi^-$ and their ratio in the forward ($x_F > 0$) and backward ($x_F < 0$) hemispheres. Table 2 presents the yields of $\rho^0$ and $\pi^-$ and their ratio in the forward ($x_F > 0$, $x_F$ being the Feynman variable) and backward ($x_F < 0$) hemispheres in the hadronic c.m.s. The data on $\pi^-$ indicate, in accordance with our previous studies (see details in [@ref3; @ref5; @ref6]) a clear depletion in the forward hemisphere (due to the nuclear attenuation) and a significant enhancement in the backward hemisphere (due to the secondary inelastic intranuclear interactions). A faint indication on similar effects are also seen for $\rho^0$.\ As it is seen from Table 2 and Fig. 3 where the distributions on $x_F$ for $\rho^0$ and $\pi^-$ are plotted, the overwhelming part of $\rho^0$ in $\nu N$ interactions and the most part of that in nuclear interactions are produced in the forward hemisphere, as a result of the current quark ($u$ or $\bar{d}$) fragmentation into a favorable hadron $\rho^0$ (which can contain the current quark), while the production of the (unfavorable) $\pi^-$ meson occurs mainly in the central region. As a result, the ratio $<\rho^0/\pi^->$ for the fastest hadrons (with $x_F > 0.5$) significantly exceeds that in other ranges of the variable $x_F$, as it can be seen from Fig. 4. Note, that a comparison of our data with those obtained at higher energies does not reveal any $W$- dependence of the $<\rho^0/\pi^->$ ratio in the forward hemisphere. Indeed, the values of ${<\rho^0/\pi^->}_N$ = 2.42$\pm$1.08 at $x_F >$ 0.5 for $\nu N$ - interactions and ${<\rho^0/\pi^->}_A$ = 0.12$\pm$0.04 at $x_F >$ 0 for $\nu A$ - interactions obtained in this work are compatible, respectively, with ${<\rho^0/\pi^->}_p$ = 1.99$\pm$0.44 measured in $\nu p$ - interactions at $<W>$ = 6.1 GeV [@ref12] and ${<\rho^0/\pi^->}_{Ne}$ = 0.13$\pm$0.03 measured in $\nu {Ne}$ - interactions at $<W>$ = 4.8 GeV [@ref11].\ It is interesting to compare the relative yield of strange and non-strange favorable mesons, $K^+(890)$ and $\rho^0$, with that of unfavorable ones, $K^0$ and $\pi^-$. The former can be extracted using our recent data on $K^+(890)$ neutrinoproduction [@ref13]. The ratio of the $K^+(890)$ and $\rho^0$ total yields is estimated to be 0.19$\pm$0.14 (for $A$ = 1) and 0.20$\pm$0.11 (for $A \approx$ 21) which seems to exceed that for $K^0$ and $\pi^-$ measured in [@ref6]: 0.055$\pm$0.013 (for $A$ = 1) and 0.070$\pm$0.011 (for $A \approx$ 21). These data, therefore, indicate, that the strangeness content in favorable mesons in neutrino-induced reactions is higher than that for unfavorable ones. In Figs. 5 - 7, the inclusive spectra of $\rho^0$ for ’quasinucleon’ and nuclear interactions are compared with those measured in neutrino-freon interactions [@ref7].\ Fig. 5 shows the invariant distribution on $x_F$. It is seen, that the data do not exhibit a significant $A$- dependence in the forward hemisphere, while at $x_F <$ 0 the $\rho^0$ yield in $\nu A$ - interactions is enhanced as compared to $\nu N$ - interactions.\ The distributions on the variable $z = E_{\rho^0}/\nu$ are presented in Fig. 6. It is seen, that they tend to shift towards lower values of $z$ with increasing A, as expected due to the effects of secondary intranuclear interactions. Fig. 7 shows the $z$- dependence of the ratio $\rho^0/({\pi^+} +\pi^-)/2$. The data exhibit no significant dependence on $A$, thus indicating that the nuclear effects are approximately the same for $\rho^0$ and pions (averaged over $\pi^+$ and $\pi^-$). Discussion ========== The data presented in the previous section indicate on a small but not negligible role of nuclear effects in the $\rho^0$ neutrinoproduction. It is interesting to clarify whether these effects are compatible with expectations based on the accounting for intranuclear interactions of secondary pions, $\pi N \rightarrow \rho^0 X$, resulting in a $\rho^0$ multiplicity gain $\delta_{\rho^0}^{sec}$ (mainly at $x_F < 0$), as well as the absorption processes, $\rho^0 N \rightarrow$ (no $\rho^0$), resulting in a yield reduction, $\delta_{\rho^0}^{abs}$ (mainly at $x_F > 0$). The details of a simple model incorporating these processes can be found in [@ref6] and references therein.\ The gain $\delta_{\rho^0}^{sec}(p_{\pi})$ induced by pions with momenta $p_{\pi} \pm {\Delta}p_{\pi}$ is determined by the differential multiplicity of pions, ${\Delta}n(p_{\pi})$, the mean probability of their secondary inelastic interactions $<w_A(p_{\pi})>$ averaged over the nuclei of the composite target, and by the mean multiplicity $\bar{n}_{\rho^0}(p_{\pi})$ of $\rho^0$ in inelastic $\pi N$ interactions. The values of $\bar{n}_{\rho^0}(p_{\pi})$ extracted (with an uncertainty of about $\pm$ 15%) from the available experimental data [@ref14] vary from 0.002 at $p_{\pi}$ = 0.9 - 1 GeV$/c$ up to 0.22 at $p_{\pi}$ = 10 - 15 GeV$/c$ (the end of the pion spectrum in this experiment).\ The product $\delta_{\rho^0}^{sec}(p_{\pi}) = {\Delta}n(p_{\pi}) \cdot <w_A(p_{\pi})> \cdot \, \bar{n}_{\rho^0}(p_{\pi})$ was integrated over the momentum spectra of charged pions measured in ’quasinucleon’ interactions. The contribution from secondary interactions of $\pi^0$ mesons is assumed to be the average of those for $\pi^+$ and $\pi^-$ mesons. The resulting values of ${\delta}^{sec}_{\rho^0}$ are found to be 0.031$\pm$0.005 for $A_{eff} \approx$ 45 and 0.023$\pm0.003$ for $A_{eff} \approx$ 21. The latter value can be compared with the experimental value of ${\delta}^{exp}_{\rho^0}(x_F < 0)$ = 0.020$\pm$0.012, inferred from the last two lines of Table 2 (note, that in the evolution of the error in ${\delta}^{exp}_{\rho^0} = {<n_{\rho^0}>}_A - {<n_{\rho^0}>}_N$ the correlation between the values of ${<n_{\rho^0}>}_A$ and ${<n_{\rho^0}>}_N$ was taken into account here and below).\ In order to estimate the $\rho^0$ yield decreasing, ${\delta}^{abs}_{\rho^0}$, one needs to know the $\rho^0$ absorption cross section ${\sigma}^{abs}_{\rho^0}$ via the channel $\rho^0 N \rightarrow$ (no $\rho^0$). As ${\sigma}^{abs}_{\rho^0}$ is not known, we use tentative values in between $5 < {\sigma}^{abs}_{\rho^0} < 10$ mb. At these values, the probability $w^{abs}_{\rho^0}$ of the $\rho^0$ absorption is estimated to be 23$\pm$6% for $A_{eff} \approx$ 45 and 18$\pm$5% for $A_{eff} \approx$ 21 (see [@ref3] for details of the absorption probability calculations). With these probabilities, the value of ${\delta}^{abs}_{\rho^0} = -w^{abs}_{\rho^0} \cdot \, {<n_{\rho^0}(x_F > 0)>}_N$ is equal to -0.016$\pm$0.008 and -0.012$\pm$0.006, respectively. The latter value can be compared with ${\delta}^{exp}_{\rho^0}(x_F > 0)$ = -0.016$\pm$0.020 inferred from the first two lines of Table 2.\ The total multiplicity gain, ${\delta}_{\rho^0} = {\delta}^{sec}_{\rho^0}(x_F < 0) + {\delta}^{abs}_{\rho^0}(x_F > 0 )$, is predicted to be ${\delta}_{\rho^0}$ = 0.011$\pm$0.007 for $A_{eff} \approx$ 21 and 0.015$\pm$0.009 for $A_{eff} \approx$ 45. These values are compatible with a rather weak variation of $<n_{\rho^0}>$ with $A$ (cf. Table 1), resulting in a total multiplicity gain ${\delta}^{exp}_{\rho^0}$ = 0.005$\pm$0.020 for $A_{eff} \approx$ 21 and 0.020$\pm$0.037 for $A_{eff} \approx$ 45. The large errors in ${\delta}^{exp}_{\rho^0}$ do not allow to check quantitatively the predictions of the model incorporating secondary intranuclear interactions. Summary ======= New experimental data on $\rho^0$ production in $\nu N$ and $\nu A$ ($A \approx 21$) interactions are obtained at intermediate energies ($<E_{\nu}>$ = 10.7 GeV, $<W>$ = 3.0 GeV). For the first time, nuclear effects in $\rho^0$ neutrinoproductions are observed. These effects for the total yield $<n_{\rho^0}>$ of $\rho^0$ are found to be rather small and compatible with those for pions. Using also the SKAT data for $A_{eff} \approx$ 45, the slope parameter $\beta$ in the exponential parametrization of the total yields ($<n> \sim A^{\beta}$) is found $\beta_{\rho^0} = 0.07 \pm 0.13$ which agrees with that for $\pi^-$ mesons, $\beta_{\pi^-} = 0.052 \pm 0.007$. The ratio of $<n_{\rho^0}> / <n_{\pi^-}>$ is found to be independent of $A$ being equal about 0.1. A comparison with the data at higher energies reveals a slight increase of this ratio with increasing energy, while no energy dependence is observed for that in the forward hemisphere ($x_F > 0$).\ A comparison of the $\rho^0$ inclusive spectra in $\nu N$ and $\nu A$- interactions indicates, that the major influence of the nuclear medium consists in their shifting towards smaller values of $x_F$ and $z$, thus indicating on a non-negligible role of the secondary intranuclear interactions. The predictions of a simplified model incorporating the latter are found to be in qualitative agreement with experimental data.\ A comparison of the relative yield of the favorable strange and non-strange mesons ($K^+(890)$ and $\rho^0$) with that for unfavorable ones ($K^0$ and $\pi^-$) indicates, that the strangeness content in the former is higher than in the latter.\ [Acknowledgement.]{} The activity of one of the autors (H.G.) is supported by Cooperation Agreement between DESY and YerPhI signed on December 6, 2002. The autors from YerPhI acknowledge the supporting grants of Calouste Gulbenkian Foundation and Swiss Fonds “Kidagan”. [00]{} =0.pt =0.pt =0.pt V.V.Ammosov et al., Fiz. Elem. Chastits At. Yadra [23]{}, 648, 1992 \[Sov. J. Part. Nucl. [23]{}, 283, (1992)\]. N.M.Agababyan et al., (SKAT Coll), YerPhI Preprint N 1535(9) (Yerevan, 1999). N.M.Agababyan et al., (SKAT Coll), Yad. Fiz. [66]{}, 1350 (2003). \[Phys. of At. Nucl. [66]{}, 1310 (2003)\]. N.M.Agababyan et al., (SKAT Coll),YerPhI Preprint N 1578(3) (Yerevan, 2002). N.M.Agababyan et al., (SKAT Coll), Yad. Fiz. [68]{}, 1241 (2005). N.M.Agababyan et al., (SKAT Coll),YerPhI Preprint N 1597(1) (Yerevan, 2005); hep-ex/0504024. V.V.Ammosov et al., (SKAT Coll.), Yad. Fiz. [46]{}, 131 (1987). \[Sov. J. Nucl. Phys. [46]{}, 80 (1987)\]. J.D.Jackson, Nuovo Cim. [34]{}, 1644 (1964). Review of Particle Physics, Phys. Lett. B[592]{}, 39 (2004). P.Astier et al., (NOMAD Coll.), Nucl. Phys. B[601]{}, 3 (2001). W.Wittek et al., (BEBC WA59 Coll.), Z. Phys. C[44]{}, 175 (1989). G.T.Jones et al., (WA21 Coll.), Z. Phys. C[51]{}, 11 (1991). N.M.Agababyan et al., (SKAT Coll), YerPhI Preprint N 1548(2) (Yerevan, 2005); hep-ex/0504040. V.Flaminio et al., Compilation CERN-HERA 83-01, 1983.
--- abstract: 'This paper presents a survey of energy efficiency of Medium Access Control (MAC) protocols for Wireless Body Area Sensor Networks (WBASNs). We highlight the features of MAC protocols along with their advantages and limitations in context of WBASNs. Comparison of Low Power Listening (LPL), Scheduled Contention and Time Division Multiple Access (TDMA) is also elaborated. MAC protocols with respect to different approaches and techniques which are used for energy minimization, traffic control mechanisms for collision avoidance are discussed.We also present a survey of path loss models for In-body, On-body and Off-body communications in WBASNs and analytically discuss that path loss is maximum in In-body communication because of low energy levels to take care of tissues and organs located inside the body. Survey of Power model for WBANs of CSMA/CA and beacon mode is also presented.' author: - \| S. Hayat, N. Javaid, Z. A. Khan$^{\S}$, A. Shareef, A. Mahmood, S. H. Bouk\ Department of Electrical Engineering, COMSATS\ Institute of Information Technology, Islamabad, Pakistan\ $^{\S}$Faculty of Engineering, Dalhousie University, Halifax, Canada title: Energy Efficient MAC Protocols --- Medium Access Control protocol; Wireless Body Area Networks; Energy-Efficiency. Introduction ============ of wireless, medical and computer networking technology has merged into an emerging horizon of science and technology called Wireless Body Area Networks (WBANs). However, applications of WBANs are not limited to medical field only. Miniaturization and connectivity are notable parameters of this field. WBANs consist of three levels; first level is low power sensors or nodes which are battery powered and need to be operated for a long time without repairing and maintenance. These nodes may be placed on the body, around the body or implanted in the body. Second level is called master node, gateway or coordinator which controls its child nodes; its power requirements may be less strengthened than nodes due to its applications and flexibility. Third level is the local or metropolitan or internet network that serves for monitoring purposes. Energy efficiency or effective power consumption of a system is one of the basic requirements for WBANs because of limited power of batteries. The most suitable layer for discussing energy and power issues is MAC Layer. The basic way of saving power or enhancing energy efficiency is to minimize the energy wastage. There are several sources of energy wastage including packet collisions, over hearing, idle listening, control packet overhead, etc. Major source of energy inefficiency among the above listed sources is packet collision for WBANs. Fig. 1. best explains that how a node’s battery is consumed, in the process of communication. Collision avoidance for energy efficiency, minimum latency, high throughput, and communication reliability, are basic requirements in the design of MAC protocol. The fundamental way of saving power or enhancing energy efficiency is to minimize the energy wastage. Simulations are performed in MATLAB for different scenarios to compute path loss. Results show that path loss is maximum in In-body communication, as compared to On-body and Off-body communication because human body is composed of tissues and organs in which communication is difficult and thus results in high path loss. On-body and Off-body also results some variations when the source and destination sensors or nodes are placed Line of Sight (LoS) and Non Line of Sight (NLoS). In this paper, we therefore, provide a survey of energy efficient MAC protocols for WBANs. First, we elaborate the protocol features and then their advantages and limitations are discussed. Sources that contribute to the energy inefficiency in a particular protocol is also identified. Moreover, comparisons of MAC protocols in the context of WBANs are tabulated in detail. Related Work ============ Gopalan et al. \[1\] survey MAC protocols for WBANs along with the comparison of four protocols i.e., Energy Efficient MAC, MedMac, Low Duty Cycle MAC, and Body MAC. Some key requirements and sources of energy wastage are also discussed. They also discussed some open research issues in this survey. Still a lot of work has to be done in data link layer, network layer and cross layer design. In \[2\], Shahjahan kutty et al. discuss the design challenges for MAC protocols for WBANs. They classify data traffic for WBANs into three categories: energy minimization techniques, frame structures and network architecture. However, the comparison of protocols is not provided by them. Sana Ullah et al. in \[3\] provide relatively a comprehensive study of MAC protocols for WBANs. Comparison of the low power listening, scheduled contention and Time Division Multiple Access (TDMA) is provided. MAC requirements, frame structures and comparison of different protocols and their trade-offs are discussed in detail. Energy Minimization Techniques in MAC Protocols for WBANs ========================================================= Low power mechanisms play an important role in performance enhancement of MAC protocol for WBANs. In this section, different approaches and techniques that provide energy efficiency in MAC protocols for WBANs are discussed and compared. Energy efficiency is an important issue because the power of nodes in WBANs is limited and long duration of operation is expected. The key concept for low power consumption is to minimize the energy consumption in the following sources: sensing, data processing and communication. Most of the energy wastage is caused during communication process because of the collision of packets, idle listening, over hearing, over-emitting, control packet overhead and traffic fluctuations. Idle listening can be reduced through duty cycling. To reduce energy waste in order to increase network’s life time and to enhance the performance of MAC protocol, different wake-up mechanisms are used. There are three main approaches adopted for the energy saving mechanisms in MAC protocols for WBANs, which are: Low Power Listening (LPL), Scheduled Contention, and TDMA. ![Sources of Power Consumption](flow_chart.eps){height="9cm" width="9cm"} Low Power Listening ------------------- LPL procedure is that “node awakes for a very short period to check activity of channel". If the channel is not idle then the node remains in active state to receive data and other nodes go back to sleeping mode. This is also termed as channel polling \[3\]. This procedure is performed regularly without any synchronization among the nodes. A long preamble is used by the sender to check polling of the receiver. LPL is sensitive to traffic rates which results in degradation of performance in the scenario of highly varying traffic rates. However, it can be optimized effectively for already known periodic traffic rates. Wise-MAC \[3\] is one of the MAC protocols which is based on LPL. This protocol reduce Idle listening using non-persistent CSMA and preamble sampling technique. Scheduled Contention -------------------- Scheduled Contention is the combination of the scheduling and contention based mechanisms to effectively cope with the scalability and collision problems. In contention based protocols, contending nodes try to access the channel for data transmission therefore, ability of collision of packet is greatly increased. Example of contention based MAC protocol is Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA) in which Clear Channel Assessment (CCA) is performed by the nodes before transmitting data. Scheduling or contention free means that each node has the schedule of transmission in the form of bandwidth or time slot assignment. TDMA, CDMA and FDMA schemes are some examples of scheduling mechanisms. However, CDMA and FDMA are not suitable for WBANs because of high computational overhead and frequency limitations, respectively. TDMA is the most suitable scheduling scheme, even though it requires extra power consumption due to its sensitivity for synchronization. The scheduled contention is the combination of scheduling and contention based mechanisms. In scheduled contention, a common schedule is adopted by all the nodes to transmit data. This schedule is exchanged periodically among the nodes to make communication adaptive, flexible and scalable. Sensor MAC (S-MAC) is one of a MAC protocol based on the scheduled contention. In this protocol, low duty mode is set as default mode for all the nodes which assures the coordinated sleeping among neighboring nodes. The energy wastage due to collision, overhearing, idle listening etc. is minimized because the node is turned on only for transmission of data and remains in sleep mode, otherwise. Time Division Multiple Access ------------------------------ In TDMA mechanism, a super frame consists of a fixed number of time slots is used. Time slots are allocated to the sensor nodes by a central node and is known as Master Node (MN), Cluster Head (CH), coordinator or Base Station Transceiver (BST). Traffic rate is one of the key parameter used by the coordinator to allocate time for each contending node. The scheme is power efficient because a node gets time slot for transmission of data and remains in sleep mode for rest of the time. However, the synchronization requirements may degrade performance in terms of power consumption. Therefore, it is highly sensitive to clock drift, which may result in limited throughput. Preamble-Based TDMA (PB-TDMA) protocol is one of the TDMA based protocol. Other examples include Body-MAC (B-MAC) \[5\], MedMAC \[3\] etc. These techniques are briefly compared in Table.I. [ \| m [2cm]{}\| m [5cm]{}\| m [4cm]{}\| m[5cm]{}\| ]{}\ Energy Saving Mechanisms & LPL & Scheduled Contention & TDMA\ Adaptability to traffic and delay & Scalable and adaptive to traffic load and low delay & Better delay performance due to sleep schedules & Better end-to-end reliability, smaller delays, high reliability\ Transmission latency and throughput & Flexible, high throughput, tolerable latency, and low power consumption & High transmission latency, loosely synchronized, low throughput & Good for energy efficiency, prolonged network’s lifetime, load balancing\ Synchronous/ Asynchronous & Asynchronous & Synchronous & Synchronous-Fine grained time synchronization\ Traffic heterogeneity requirements & Low duty cycle nodes do not accommodate aperiodic traffic. Very hard to satisfy the WBANs traffic heterogeneity requirements & Low duty cycle nodes do not require frequent synchronization of schedules. Hard to satisfy the WBANs traffic heterogeneity requirements & Low duty cycle nodes do not require frequent synchronization at the beginning of each superframe. Easy to satisfy the WBANs traffic heterogeneity requirements\ Sensitivity & Sensitive to tuning for neighborhood size and traffic rate & Sensitive to clock drift & Very sensitive to clock drift\ Performance with respect to traffic rates& Poor performance when traffic rates changes & With the increase in traffic, performance is improved & Throughput and number of active nodes are limited\ Cost incurred by sender and receiver & Receiver and polling efficiency is gained at much greater cost of senders & Similar cost incurred by sender and receiver & Require clustering\ Extravagant & It does not listen for full contention period as a result it is less expensive & Listening for full contention period & Low duty cycle\ Scalability and adaptability & Challenging to adapt LPL directly to new radios like IEEE 802.15.4 & Scalable, adaptive, and flexible & Limited scalability and adaptability to changes on number of nodes\ Energy Efficient MAC Protocols ============================== In this section, we briefly discuss the energy efficient MAC protocols for WBAN. Okundu MAC Protocol ------------------- An energy efficient MAC protocol for single hop WBANs is proposed by Okundu et al. in \[4\]. This protocol consists of three main processes: link establishment, wakeup service, and alarm process. Basic energy saving mechanism of this protocol consists of central control of wakeup/sleep time and Wakeup Fall-back Time (WFT) processes. WFT mechanism is used to avoid collision due to continuous time slot. This mechanism states that, if a slave node wants to communicate with a MN and it fails in its task due to MN’s other activities, then it goes back to sleep mode for a specific time computed by WFT. However, data is continuously being buffered during the sleep time. To minimize time slot collision, the concept of WFT has been introduced. This concept helps every slave node to maintain a guaranteed time slot even if it fails to communicate with the MN. In this protocol, problems like idle listening and over-hearing can be reduced because of central management of traffic. In one cluster, only $8$ slave nodes can be connected to MN which restricts inclusion of other slave nodes. In link establishment, wakeup service, and alarm processes, communication is initiated by the MN. Another main problem is that, only one slave node can join network at a time. MedMac Protocol --------------- N. F. Timmons et al. in \[5\] propose a TDMA-Based MAC protocol for WBANs called MedMAC. This protocol consists of two schemes for the power saving: Adaptive Guard Band Algorithm (AGBA) and Drift Adjustment Factor (DAF). AGBA along with time stamp is used for synchronization among coordinator and other nodes. This synchronization is introduced using Guard Band (GB) between time slots to allow the node to sleep for many beacon periods. DAF is used to minimize bandwidth. GB is calculated by AGBA and shows the worst cases. However, practically gaps may be different between time slots depending upon application scenarios. DAF adjusts GB according to practical situation and avoids overlapping between consecutive slots. MedMac outperforms IEEE 802.15.4 for Class 0 (lower data rate applications such as health monitoring and fitness) and Class 1 (medium data rate medical applications such as EEG). Energy waste due to collision is reduced by introducing Guaranteed Time Slot (GTS). Each device has exclusive use of a channel for a fixed time slot, therefore, synchronization overhead is also reduced. This protocol works efficiently for low data rate applications, and work inefficiently for high data rate applications. However, In-body and On-body applications of WBAN are usually of higher data rate. Low Duty Cycle MAC Protocol --------------------------- Low Duty Cycle MAC protocol for WBANs is designed in \[6\]. In this protocol, analog to digital conversion is performed by slave nodes while the other complex tasks such as digital signal processing is carried out at MN. MNs are supposed to be less power than slave nodes. This protocol introduces the concept of Guard Time ($Tg$) to avoid overlapping between consecutive time slots. After T frames a Network Control (NC) packet is used for general network information. Power saving is achieved by using effective TDMA strategy. This protocol is energy efficient because it sends data in short bursts. By using TDMA strategy, this protocol effectively overcomes the collision problem. It allows monitoring patient’s condition and can reduce the work load on medical staff, while keeping minimum power usage. TDMA strategy is used in WBANs, and it is found that TDMA is more suitable for static type of networks with a limited number of sensors generating data at a fixed rate therefore, this protocol may not respond well in a dynamic topology. B-MAC Protocol -------------- B-MAC protocol achieves energy efficiency by using three bandwidth management schemes: Burst, Periodic and Adjust Bandwidth. Burst bandwidth consists of temporary period of the bandwidth, which includes several MAC frames and recycled by the gateway (coordinator). Bandwidth is reduced to half if it does not fully utilized by the nodes, which is also informed about reduction of bandwidth. Periodic bandwidth is a provision for a node to have access to the channel exclusively within a portion of each MAC frame or few MAC frames. It is also allocated by the gateway based on node’s QoS requirements and current availability of the bandwidth \[7\]. Adjust bandwidth defines the amount of bandwidth to be added to or reduced from previous Periodic Bandwidth \[7\]. Nodes can enter into sleep mode and wake up only when they have to receive and transmit any data to the gateway, because the nodes and the gateway are synchronized in time. The time slot allocation in Contention Free Period (CFP) is collision free, which improves packet transmission and thus, saves energy. The protocol uses CSMA/CA in the uplink frame of Contention Access Period (CAP) period, which is not reliable scheme due to its unreliable CCA and collision issues. Ta-MAC Protocol --------------- Traffic aware MAC (Ta-MAC) protocol utilizes traffic information to enable low-power communication. It introduces two wakeup mechanisms: a traffic-based wakeup mechanism, and a wakeup radio mechanism. Former mechanism accommodates normal traffic by exploiting traffic patterns of nodes whereas, later mechanism accommodates emergency and on-demand traffic by using a wakeup radio signal. In the traffic-based wakeup mechanism, the operation of each node is based on traffic patterns. The initial traffic pattern is defined by the coordinator and can be changed later. The traffic patterns of all nodes are organized into a table called traffic-based wakeup table. In wakeup radio mechanism, a separate control channel is used to send a wakeup radio signal. The coordinator and the member node send wakeup radio signal in on-demand and emergency case. In Ta-MAC, a node wakes up, whenever it has a packet to send/receive. Since the traffic patterns are pre-defined and known to the coordinator, it does not have to wait for resource allocation information/beacon. As a result, delay is minimized comparitive to other MAC protocols. This protocol accommodates normal, emergency and on-demand traffic in a reliable manner. To achieve energy efficiency in MAC protocol, the central coordination and resource allocation is based upon the traffic patterns of the nodes. As, in this protocol, the traffic pattern are defined by the coordinator, in a static topology. Therefore, it does not work efficient in dynamic topology (in dynamic topology, traffic patterns are changed frequently). [ \| m[2.5cm]{}\| m[2.5cm]{}\| m[2.5cm]{}\| ]{}\ Protocols & Advantages & Disadvantages\ Okundu MAC & Minimize time slot collision, reduce idle listening and overhearing & Only 8 slave nodes can be communicated to MN\ MedMAC & Energy waste due to collision is reduced & Do not support high data rate applications\ Low Duty Cycle & Collision problem is reduced, allows patients’ monitoring & Not suitable for dynamic type of networks\ B-MAC & Improves packet transmission hence saves energy & Uses CSMA/CA in the uplink frame of CAP period, which is not a reliable scheme\ Ta-MAC & Accommodates normal, emergency and on-demand traffic, energy efficient, reasonable delay & Not suitable for dynamic topologies\ S-MAC & High latency and time synchronization overhead may be prevented due to sleep schedules & Low throughput, overhearing and collision may cause if packet is not destined to listening node\ T-MAC & Packets are sent in burst and with low latency which collectively gives better result under variable load & Suffers from sleeping problems\ H-MAC & Improves BSN’s energy efficiency and reduces extra energy cost & Does not support sporadic events and posseseslow spectral/bandwidth efficiency\ DTDMA & Reduce packet dropping rate, less energy consumption & Does not support emergency and on-demand traffic\ S-MAC Protocol --------------- S-MAC \[8\] protocol is proposed for WBASNs. This protocol uses fixed duty cycles to solve idle listening problem. Nodes wakeup after a specific time, as assigned by coordinator, sends data and goes back to sleep mode again. As, all the nodes are synchronized, therefore, collision can also be easily avoided. S-MAC gives considerably low latency. In this protocol, time synchronization overhead may be prevented due to sleep schedules. Fluctuating traffics are not supported and no priority is given to the emergency traffic scenarios by S-MAC. Therefore, it is not a reliable for WBANs. Overhearing and collision may occur if the packet is not destined to the listening node. T-Mac Protocol --------------- Mihai et al. \[9\] suggested Time-out MAC (T-MAC) for WBASNs. It uses flexible duty cycles for increasing energy efficiency. In T-MAC, the node wakes up after time slot assignment, sends pending messages. If there is no activation event for Time Interval (TA), the node goes back to sleep mode again. If a node sends Route To Send (RTS) and does not receive Clear To Send (CTS), then sends RTS two more times before going to sleep. To solve early sleep problem, it uses future RTS for taking priority on full buffer. In T-MAC, packets are sent in burst, as a result delay is minimized. It also outperforms other MAC protocols under variable load. The main disadvantage in this protocol is that it suffers from sleeping problems. H-MAC Protocol --------------- Heartbeat Driven MAC (H-MAC) uses heart beat rhythm information for synchronization of nodes. This avoids the use of external clock and thus reducing the power consumption. Also guaranteed time slot (GTS) provision to each node helps to avoid collision. H-MAC aims to improve BSNs energy efficiency by exploiting heartbeat rhythm information, instead of using periodic synchronization beacons to perform time synchronization \[3\]. Although, H-MAC protocol reduces extra energy cost of synchronization, however, it does not support sporadic events. Since TDMA slots are dedicated and are not traffic adaptive, H-MAC protocol encounters low spectral/bandwidth efficiency in case of low traffic. The heartbeat rhythm information varies depending on patient’s condition. It may not reveal valid information for synchronization all the time \[3\]. DTDMA Protocol -------------- Reservation based dynamic TDMA (DTDMA) protocol uses slotted ALOHA in CAP field of super frame to reduce collisions and to enhance power efficiency. Through the adaptive allocation of the slots in a DTDMA frame, WBAN’s coordinator adjusts the duty cycle adaptively with traffic load. Comparing with IEEE 802.15.4 MAC protocol, DTDMA provides more dependability in terms of lower packet dropping rate and low energy consumption especially for an end device of WBAN \[3\]. It does not support emergency and on-demand traffic. Furthermore, DTDMA protocol has several limitations when considered for the Medical Implant Communication Service (MICS) band. The MICS band has ten sub-channels and each sub-channel has $300$ Kbps bandwidth. DTDMA protocol can operate on one sub-channel, however, cannot operate on ten sub-channels simultaneously \[3\]. The main purposes of a MAC protocol are to provide energy efficiency, network stability, bandwidth utilization and reduce packet collision. The energy minimization techniques and mechanism in MAC protocols are summarized in Table. 3. [\| m [3cm]{}\| m[4cm]{}\| ]{}\ Protocol & Energy Efficiency Mechanism\ Okundu MAC & Wake up Fall back Time (WFT)\ MedMAC & TDMA, Adaptive Guard Band Algorithm (AGBA) and Drift Adjustment Factor (DAF)\ Low Duty Cycle & TDMA, concept of Guard Time ($Tg$)\ B-MAC & TDMA, Bandwidth mechanism\ Ta-MAC & Central coordination according to traffic patterns of the nodes\ S-MAC & Scheduled based, organized in slots and operation based on schedules\ T-MAC & Have slots and operation is based on schedules\ H-MAC & Heartbeat Rhythm information is used for synchronization\ DTDMA & TDMA based, use of slotted aloha in CAP field\ Performance Trade-offs made by MAC Protocols ============================================ In this section, we discuss the performance of the MAC protocols they achieve and price they pay. In other words, trade-offs, the MAC protocols have to make. Okundu MAC Protocol ------------------- Network’s scalability is mainly application dependent, e.g., ECG can support upto maximum of 8 slave nodes because of 8 percent duty cycle. However, in practice this is 6 to allow for possible retransmissions. Therefore, we have a trade-off, for retransmission, slave nodes attached to the MN are reduced to attain scalability of network. MedMac Protocol --------------- The low data rate applications of Class $0$ medical devices include monitoring of respiration system, temperature of human body, pulse monitoring etc. Power consumed by respiration transceiver is slightly high in MedMAC protocol with respect to other protocols, while temperature and pulse node show much low power consumption, as compared to other protocols. MedMAC trade-offs power consumption of respiration for less power of other two applications. Low Duty Cycle MAC Protocol --------------------------- The number of extra slots needed for protocol robustness is dependent on Packet Error Rate (PER) and Packet Loss Ratio (PLR). When PER is high, it will increase PLR. However, PLR, may be reduced by using extra slots in the time frame. Therefore, this protocol can trade-offs extra slots for less PLR. B-MAC Protocol -------------- B-MAC trade-offs idle listening for a reduced time to transmit and reception of data. As, we know that reducing duty cycle increases sleep time which in turn reduces idle listening. Another trade-off is between idle listening and packet length, because this overhead dominates the energy consumption. Ta-MAC Protocol --------------- Ta-MAC uses two wakeup mechanisms one for handling data traffic and other for emergency traffic. By using these two mechanisms this protocol outperforms all other protocols in terms of power consumption because problems like idle listening, collision and overhearing are reduced. However, by sending frequent control messages to the nodes increases node’s overhead, which is a trade-off. The initial traffic patterns of all the nodes are defined by the coordinator, as a result delay is also slightly increased. S-MAC Protocol --------------- For transmission and reception of data in S-MAC, an extremely low duty cycle is used. When throughput increases SAC’s duty cycle also increases , which further increases the overhead of SYNChronization (SYNC) period, as a result, power consumption is increase linearly. S-MAC can trade-offs throughput for energy, also it can trade-offs energy for latency. T-Mac Protocol --------------- T-MAC uses adaptive duty cycle, implemented as a time out after the last event. At lower transmission rates, throughput increases because probability of packet loss is much less than received packet, however, the latency is increased between source and destination node. H-MAC Protocol -------------- In H-MAC a Guard Band is introduced in time slots to avoid collision by overlapping of data, however, when time slots are completely aligned then there will be no data transmission in Guard Band, therefore, it reduces bandwidth utilization. The coordinator of BSN then uses this GB for synchronization, by sending re-synchronization control packets, hence achieving energy efficiency. Thus making a trade-offs between energy efficiency and bandwidth utilization efficiency. DTDMA Protocol -------------- DTDMA is a TDMA based protocol which uses time slots for data transmission and as a result low power is consumed. However, TDMA requires synchronization between nodes and the coordinator, as a result, overhead is increased. This overhead is a trade-off for energy. The trade-offs of each selected protocol are summarized and given in Table. IV. [\| m [3cm]{}\| m[4cm]{}\| ]{}\ Protocol & Trade-Offs\ Okundu MAC &Trade-offs number of slave nodes attached to the MN are reduced for scalability of network\ MedMAC & Trade-offs idle listening for a reduced time to transmit and reception of data\ Low Duty Cycle & Can trade-off extra slots for less PLR\ B-MAC & Trade-off is between idle listening and packet length\ Ta-MAC & Trade-off delay for low power consumption\ S-MAC & Can trade-off energy for latency.\ T-MAC & Can trade-off latency for high throughput\ H-MAC & Trade-offs between energy efficiency and bandwidth utilization efficiency\ DTDMA & Trade-offs overhead for a low power consumption\ MAC Frame Structure =================== MAC frame structure consists of control portion or control packet and data portion. Control portion is responsible for the management and control messages (beacon period, request period, topology management period) to control and manage dynamic topology and varying data rate traffic. Data portion consist of two sub parts: CAP and Contention Free Period (CFP). CAP consists of CSMA/CA while the nodes contend in CAP transmit MAC control packets. Similarly, small size data packets can also be transmitted in CAP. In \[7\], the allocation of time slots is controlled by the coordinator. The coordinator arrange the duration of control and data packet on the basis of current traffic of topology that is why the slots are allocated to CFP and are collision free. In each frame, bandwidth allocation in CFP can be changed. In \[5\], the GB is used to maintain synchronization among devices even if a node is sleeping for many beacon periods. In \[4\]\[9\]\[10\]\[11\], MAC protocols use slotted ALOHA in its frame structure to divide a slot into 4 equal mini slots. In \[6\], $Tg$ is introduced in its frame structure to reduce overlapping between the two following nodes. ![image](frame2.eps){height="11cm" width="14cm"} Technique for Collision Avoidance for Traffic Control ===================================================== The main schemes of MAC protocol for WBANs are divided into two groups: contention based i.e., CSMA and contention free i.e., TDMA. Most of the traffic is interrelated in WBANs, therefore, contention based solutions are not suitable for them. For example, if a patient is suffering from fever, the body temperature increases which increases blood pressure, hence, the sensor sensing temperature variation and the sensor that senses blood pressure variation, both become active. Along with them other respiration sensors also become active at the same time and try to access the channel/coordinator. However, in this situation, collision occurs in CSMA. In TDMA, each node communicate to MN according to the assigned pattern by the coordinator. As a result, collision in data traffic is low, as compared to CSMA. Okundu MAC ----------- This protocol controls traffic using centrally controlled wakeup/sleep time. Slots are assigned to sensors change every time when coordinator detects any change in traffic pattern. Assignment of different time slots, decreases collision between the nodes . It makes the system to handle fluctuating traffic. The sensor nodes establishe link with the coordinator after listening to the Radio Frequency (RF)-channel for a fixed time period. MN sends request to the sensor node for information by setting and communicating the next wakeup time after establishing the link. [ \| c\| c \| ]{}\ IEEE 802.15.4 MAC & Original IEEE 802.15.4\ Low power consumption & High power consumption\ Higher data rate & Low data rate\ Higher flexibility & Low flexibility\ TDMA based & Contention based\ Collision Free & Greater collisions\ Sleep mode & Idle listening\ MedMAC ------ MedMac reduces the collision by using AGBA. AGBA allows the sensor nodes to sleep for a GB time period between each time slot. Each node has specific time slot to communicate with master node/coordinator, which means there is no collision. Thus minimizes the synchronization overhead. Ta-MAC ------ Ta-MAC protocol uses two channel access mechanisms for traffic control i.e., traffic based wakeup mechanism for normal traffic, and wakeup radio mechanism for on-demand and emergency traffic. In traffic-based wakeup mechanism, all nodes have traffic patterns that are assigned by the coordinator. The initial patterns are defined and updated by the coordinator. The traffic patterns of all nodes are synchronized and arranged in a specific table, known as $ Traffic $ $ Based $ $ Wakeup $ $ Table $. Node’s ID and its respective traffic patterns are stored in this table. Normally, all the nodes become active/wakeup according to their traffic patterns. If two or more nodes have same wakeup pattern then the node with high priority is treated first by the coordinator, as shown in Fig. 3. By assigning these patterns, load at the coordinator is minimized, and chances of collision is also reduced. B-MAC ----- B-MAC uses downlink and uplink schemes along with sleeping mode for data traffic control. Downlink is only used by MN, therefore, traffic and data load on downlink are reduced. Uplink is divided into CAP and CFP. MN allocates time slots to CFP according to data traffic which makes CFP collision free. In case, when nodes have no data to transmit or receive then they go to sleeping mode. ![image](data-traffic-ctrl.eps){width="16cm" height="12cm"} Low Duty Cycle -------------- Low duty cycle MAC protocol is based on TDMA. In TDMA, time slots are assigned to the sensor nodes by the coordinator. To avoid collision between the data traffic, the concept of $Tg$ is introduced. Use of $Tg$ between every consecutive slots prevents the transmission overlaps and controls data traffic. IEEE 802.15.4 MAC ----------------- The basic requirement of QoS is to minimize delay and maximize the probability of successful transmission. CFP scheme is used to control data traffic to guarantee the QoS. If a node wants to send data, first it listens for the network beacon. After node finds the beacon that is sent by the coordinator, the node synchronizes to the super frame structure. IEEE 802.15.4 supports up to $250$ Kbps data rate with possible coverage of $10$ meters. This data rate is not enough to support the required rates of WBANs that is up to $10$ Mbps. According to IEEE 802.15.4, packets are transmitted in the contention period, which may result longer delays in real time critical applications. When traffic is increased, the nodes compete for the contention based slots, resulting in long delays and the actual size of the network is almost doubled \[7\]. In \[10\], to satisfy the requirements of WBANs including QoS, network scalability, support for multiple PHY’s and multiple application traffics, IEEE 802.15.4 MAC is proposed. It is the modified version of original IEEE 802.15.4. QoS means to decrease the packet latency and increase the probability of successful transmission of data packets without collision and loss of data. In original IEEE 802.15.4, GTS mechanism is provided to support the emergency data. GTS is very effective for data transfer, however, inherently the limit of GTS in a super frames is seven. As a result, it cannot support more than seven devices simultaneously in CFP. Whereas, in IEEE 802.15.4 MAC, the coordinator may allocate more than seven GTS simultaneously to the sensor devices. IEEE 802.15.4 MAC and 802.15.4 original is compared briefly in Table. 5. Conclusion ========== We present a survey of different MAC protocols with respect to energy efficiency and their advantages and disadvantages in WBANs. Low power listening, scheduled contention and TDMA are also compared. It is observed that TDMA is more power efficient, however, suffers with synchronization sensitivity. Techniques for collision avoidance of different MAC protocols are also comparatively analyzed. Path loss model for In-body, On-body and Off-body communication in WBANs is also described. Because human body is composed of tissues and organs in which communication is difficult and thus results in high path loss. On-body and Off-body also show some variations in results when the source and destination sensors or nodes are LoS and NLoS. [00]{} Anand Gopalan, S. and Park, J.T., “Energy-efficient MAC protocols for wireless body area networks: Survey”, ICUMT, 2010. Kutty, S. and Laxminarayan, JA., “ Towards energy efficient protocols for wireless body area networks”, ICIIS, 2010. Ullah, S. and Shen, B. and Riazul Islam, SM and Khan, P. and Saleem, S. and Sup Kwak, K., “ A study of MAC protocols for WBANs”, SENSOR, 2009. Omeni, O. and Wong, A. and Burdett, A.J. and Toumazou, C, “Energy efficient medium access protocol for wireless medical body area sensornetworks", IEEE, 2008. Timmons, NF and Scanlon, WG., “An adaptive energy efficient MAC protocol for the medical body area network”, VITAE, 2009. Marinkovic, S.J. and Popovici, E.M. and Spagnol, C. and Faul, S. and Marnane, W.P., “ Energy-efficient low duty cycle MAC protocol for wireless body area networks”, IEEE, 2009. Fang, G. and Dutkiewicz, E., “BodyMAC: Energy efficient TDMA-based MAC protocol for wireless body area networks”, ISCIT, 2009. W. Ye, J. Heidemann, and D. Estrin, “An energy-efficient MAC protocol for wireless sensor networks”, In Proceedings of the IEEE Infocom, New York, USA, pp. 1567-1576, Jun. 2002. T. Van Dam and K. Langendoen, “An adaptive energy-efficient MAC protocol for wireless sensor networks”, In ACM Conference on Embedded Networked Sensor Systems (Sensys), Los Angeles, USA, pp. 171-180, Nov. 2003. Li, C. et al., “Scalable and robust medium access control protocol in wireless body area networks”, IEEE, 2009. Ullah, S. and Kwak, K.S., “An ultra low-power and traffic-adaptive medium access control protocol for wireless body area network", Journal of Medical Systems, 2010.
--- author: - 'C. Maier' - 'M. Hayashi' - 'B. L. Ziegler' - 'T. Kodama' date: 'Received ; accepted' title: 'Cluster induced quenching of galaxies in the massive cluster XMMXCSJ2215.9-1738 at $z \sim 1.5$ traced by enhanced metallicities inside half $R_{200}$ ' --- [Cluster environments at $z<0.5$ were found to increase the gas metallicities of galaxies which enter inner regions of the clusters where the density of the intracluster medium is high enough to remove their hot halo gas by ram-pressure stripping effects and to stop the inflow of pristine gas. To extend these studies to $z>1$, the most massive clusters known at these redshifts are the sites where these environmental effects should be more pronounced and more easily observed with present day telescopes.]{} [We explore the massive cluster XMMXCSJ2215.9-1738 at $z \sim 1.5$ with KMOS spectroscopy of and covering a region that corresponds to about one virial radius. Using published spectroscopic redshifts of 108 galaxies in and around the cluster we computed the location of galaxies in the projected velocity-versus-position phase-space to separate our cluster sample into a virialized region of objects accreted longer ago (roughly inside half $R_{200}$) and a region of infalling galaxies. We measured oxygen abundances for ten cluster galaxies with detected lines in the individual galaxy spectra and compared the mass–metallicity relation of the galaxies inside half $R_{200}$ with the infalling galaxies and a field sample at similar redshifts.]{} [We find that the oxygen abundances of individual $z \sim 1.5$ star-forming cluster galaxies inside half $R_{200}$ are comparable, at the respective stellar mass, to the higher local SDSS metallicity values. We compare our measurements with a field galaxy sample from the KMOS3D survey at similar redshifts. We find that the / line ratios inside half $R_{200}$ are higher by 0.2dex and that the resultant metallicities of the galaxies in the inner part of the cluster are higher by about 0.1dex, at a given mass, than the metallicities of infalling galaxies and of field galaxies at $z \sim 1.5$. The enhanced metallicities of cluster galaxies at $z \sim 1.5$ inside $0.5 R_{200}$ indicate that the density of the intracluster medium in this massive cluster becomes high enough toward the cluster center such that the ram pressure exceeds the restoring pressure of the hot gas reservoir of cluster galaxies. This can remove the gas reservoir and initiate quenching; although the galaxies continue to form stars, albeit at slightly lower rates, using the available cold gas in the disk which is not stripped. ]{} Introduction {#sec:intro} ============ While local and low-redshift clusters host large fractions of passive galaxies, the star-forming (SF) population becomes larger with increasing redshift. Since clusters of galaxies grow by accreting mass from their surroundings, the SF galaxies observed in high-z clusters must be the progenitors of the local passive galaxies. The $1 < z < 2$ redshift range hosts the emergence of the Hubble sequence of disks and elliptical galaxies and the buildup of a significant fraction of the stellar mass in the universe [e.g., @dickins03; @drory05], which is a transition epoch for clusters. At higher redshifts, forming clusters (protoclusters) consisting of mostly SF galaxies are forming a vast majority of their stellar content [e.g., @chiang17]. At later epochs ($z<1$), many massive clusters virialize and most galaxies in their central parts are passive. Therefore, this $1 < z < 2$ transition period when cluster galaxies are at the onset of environmental influences is a crucial time to investigate their properties and the mechanisms driving their evolution. While studies of the mass-metallicity relation (MZR), chemical evolution, and physical conditions of the interstellar medium (ISM) of SF field galaxies at these redshifts are now based on larger samples [e.g., @zahid14; @kashino17], the samples of cluster SF galaxies at $z>1$ with studied metallicities are still very small. There is ongoing debate about environmental signatures in the chemical enrichment of cluster galaxies, especially at $z>1$. The few available samples are small (compared to field samples) and are affected by selection biases and the metallicity calibrator used, and produced some contradictory results. @kulas13 found a 0.15dex metallicity enhancement for $z\sim 2.3$ proto-cluster galaxies with respect to field counterparts at lower masses (by stacking galaxies with nondetected \[NII\] emission lines), but no difference at higher masses. @shimakawa15 found higher metallicities in proto-cluster members at $z>2$ than in the field for $10<\rm{log}(\rm{M/M}_{\odot})<11$, by measuring metallicities based on \[NII\]/ from stacked spectra. On the other hand, @valent15 found lower \[NII\]/H$\alpha$ ratios in $z \sim 2$ protocluster galaxies than in the field. @kacprzak15 and @tran15 used stacked \[NII\]/H$\alpha$ ratios (including upper limits for \[NII\]) and found a similar MZR relation for field and cluster galaxies at $z \sim 2.1$ and $z \sim 1.62$, respectively. At lower redshifts there is still debate over the amplitude of environmental effects on metallicities, with studies affected by sample selections and metallicity estimators, but also by the definition of environment. Defining environment by local densities, @mouhcine07 reported variation in metallicity at a given mass ranging from 0.02dex for massive to 0.08dex for lower-mass galaxies. Studying central and satellite SDSS galaxies, @pasq12 and @pengmaio14 both reported an average metallicity of satellites higher than that of centrals, especially for low-stellar-mass galaxies. Another investigation of 1318 galaxies in local clusters with a range of halo masses of $\sim 10^{13}-10^{15}M_{\odot}$ found an increase of 0.04dex in oxygen abundances for galaxies in clusters compared to the field [@elli09]. @cooper08 used the SDSS sample and claimed that there is a stronger relationship between metallicity and environment (largely driven by galaxies in high-density regions such as groups and clusters), such that more metal-rich galaxies favor regions of higher overdensity. On the other hand, @hughes13 found a similar MZR in field galaxies compared to Virgo cluster galaxies with a slight trend of increasing metallicities in cluster galaxies compared to field galaxies at lower masses (their Fig.7), but not significant due to their small number statistics. @gupta16 studied two clusters at $z \sim 0.35$ finding a higher MZR in cluster galaxies compared to field galaxies in one cluster, and no difference in the MZR of field and cluster galaxies in the other cluster. As discussed below and driven by our metallicity studies in $z<0.5$ $M_{200} \sim 10^{15}M_{\odot}$ clusters [@maier16; @maier19], it seems that environmental effects are stronger in more dense environments at $z<0.5$ and that a higher cluster mass produces stronger effects on metallicities. Most of the studies of the MZR of cluster galaxies at $z>1$ have used the / ratio to determine metallicities with the N2-calibration [@petpag04]. The \[NII\]/ ratio involves two emission lines (ELs) that are close to each other in wavelength and has some benefits: it can be observed at the same time with higher-resolution spectroscopy, the ratio is relatively insensitive to extinction correction, and, from the ground, and can be followed all the way to redshift $z \sim 2.5$ (the limit of the near-infrared (NIR) K-band). On the other hand, one main issue with the \[NII\]/ ratio is that the line is often too weak to be detected in individual spectra at higher redshifts, therefore the stacking of spectra of several cluster galaxies was often used in the literature to study their chemical enrichment. Another problem is that the contamination of \[NII\] by night-sky lines can produce spurious metallicities relying only on \[NII\]/. This issue was investigated by @magdis16, who used K-band Multi Object Spectrograph (KMOS) observations of KROSS (KMOS Redshift One Spectroscopic Survey) galaxies at $z \sim 1$. These latter authors found that imperfect subtraction of OH sky lines affected 113 galaxies with no \[NII\] detection in their sample with distances $<7$Å between the expected central $\lambda$ of and sky lines, producing incorrect \[NII\] upper limits on flux measurements and spurious metallicities from stacked spectra; they concluded that very weak lines partly overlapping with strong sky lines may not be used to determine reliable metallicities based on using \[NII\]/ only. In the samples of cluster galaxies at $z>1.6$ with MZR studies mentioned above, it is possible that there are galaxies with no detection of \[NII\] and a likely problematic measurement of the EL flux (from stacking these spectra) due to the OH sky-line residuals at the position of \[NII\], which could bias the resulting metallicity measurement from stacked spectra. ![ \[fig:HaNII\] The six galaxies of the main SFvirialized sample. The panels show for each galaxy the extracted spectrum (black solid line) over an area of 25 spaxels with the fitted Gaussians by KUBEVIZ to and shown in green and the noise spectrum as a solid red line. It can be seen that the measurements of and are not affected by strong sky lines (peaks of the solid red lines). ](fig1.eps){width="9cm"} At lower redshifts, using a sample of SF cluster galaxies at $z \sim 0.2$ and $z \sim 0.4$, @maier16 [@maier19] found higher metallicities at a given mass for cluster galaxies in the inner parts of seven massive LoCuSS clusters and one CLASH cluster. @maier16 [@maier19] compared their observational findings with metallicity-star-formation rate (SFR)-mass bathtub model predictions with inflowing gas [@lilly13] and deduced a slow-quenching (strangulation) scenario in which the gas metallicities can increase after removal of the hot halo gas reservoir because the interstellar medium (ISM) is no longer diluted by the inflow of pristine gas. During strangulation, the inflow of new gas is cut out, but the cold ISM disk is not directly perturbed. In this case the star formation can continue, using the gas available in the disk (resulting in higher gas-phase metallicities) until the cold gas is completely used up. These observational studies were compared with high-resolution cosmological hydrodynamic simulations of @bahe13, who computed the density of the intracluster medium (ICM) and derived the ram pressure in clusters. The conclusion was that the ICM becomes dense enough at $R \sim R_{200}$, in $z<0.5$ massive clusters with $M_{200}\sim 10^{15}M_{\odot}$, to remove the [hot]{} halo gas surrounding massive galaxies ($10<\rm{log}(\rm{M/M}_{\odot})<11$) by ram-pressure stripping (RPS) effects. $R_{200}$ is the radius that encloses a mean density 200 times the critical density at a given redshift. For lower-mass galaxies ($9<\rm{log}(\rm{M/M}_{\odot})<10$), the ICM becomes dense enough to remove their [hot]{} halo gas already further out at $R \sim 2R_{200}$. @maier19 further deduced that, while slowly quenching for 1-2Gyrs, galaxies travel to denser inner regions of the cluster where the RPS also exceeds the restoring pressure of the cold gas, eventually completely quenching star formation by stripping the [cold]{} gas in a [rapid]{} phase. This [slow-then-rapid]{} quenching scenario is also in agreement with the theoretical study of @stein16, who used ram-pressure stripping simulations employing the moving-mesh code AREPO [@springel10] to follow at high resolution the interaction of a galaxy cluster with infalling galaxies. @stein16 found that typically their model galaxies continue to form stars with only slightly modified rates as a result of the stripping of the hot gaseous halos (strangulation) of the galaxies. On the other hand, the cold gas of their simulated galaxies is stripped only during pericenter passages with small pericenter distances leading to a full quenching of star formation on a short timescale. Id spect. z log\[NII\]/ log(M/M$_{\odot}$) O/H (PP04) $R/R_{200}$ ------- ---------- ------------------ -------------------------- ----------------- ------------- -- 29284 1.461 -0.57 $\pm$ 0.08 10.36 $^{-0.03}_{+0.03}$ 8.57 $\pm$ 0.05 0.74 29609 1.459 \[NII\] on OH 10.49 $^{-0.08}_{+0.11}$ \* 0.44 30183 1.458 \[NII\] on OH 10.18 $^{-0.04}_{+0.03}$ \* 0.27 As discussed also in @maier19, there is a significant difference between this [slow-then-rapid]{} scenario and the “delayed-then-rapid” scenario of @wetzel13. In the “slow” quenching phase the galaxies are already affected by the environment because strangulation was inititated, while in the “delayed” phase of the @wetzel13 scenario the galaxies are completely unaffected by environment. A more recent study of @roberts19 used X-ray data for $z<0.1$ clusters to study the influence of the ICM on the quenching of satellite galaxies. They found that the quenched fraction of galaxies increases modestly at ICM densities below a threshold before increasing sharply beyond this threshold toward the cluster center. These latter authors found their results to be consistent with a picture where cluster galaxies experience an initial, slow-quenching mode driven by steady gas depletion, followed by rapid quenching associated with ram pressure of cold-gas stripping near the cluster center, in agreement with the slow-then-rapid quenching scenario of @maier19. These investigations of lower-redshift clusters demonstrated that the density of the ICM is an important parameter influencing the quenching of galaxies and that enhanced metallicities in massive clusters are a valuable quenching tracer. To extend these studies to $z>1$ we aim to study clusters of galaxies found as overdensities of galaxies which are unambiguosly identified through their hot diffuse medium, manifested as X-ray emission. One of the most massive galaxy clusters at $z>1.5$ is XMMXCSJ2215.9-1738 (hereafter XMM2215) at $z \sim 1.46$ with extended X-ray emission from the hot gas, found from XMM Newton X-ray data as part of the XMM Cluster Survey [@stanford06]. Given its high mass and high ICM density possibly favoring stronger RPS, this is one of the best cluster targets to study environmental effects on metallicities and quenching at $z>1$, because more pronounced environmental effects than in other less-massive clusters at $z>1$ are expected. A pioneering study of the MZR in XMM2215 was done by @hayashi11. They claimed to find a similar MZR relation for cluster and field galaxies at $z \sim 1.5$. However, a caveat of their study was the use of relatively low-resolution ($R \sim 700$) Subaru-MOIRCS spectroscopy which made night-sky line-correction difficult, and therefore night-sky-line contamination of \[NII\] influencing metallicity measurements (as mentioned above) could not be always avoided. For most of their observed galaxies the signal-to-noise ratio (S/N) of the \[NII\] fluxes was relatively low (S/N $<2-3$) or only upper limits for \[NII\] line fluxes could be measured. The new KMOS observations which we present in this study also contain some of the galaxies studied by @hayashi11. The much better resolution of KMOS ($R \sim 4000$ in H-band vs. $R \sim 700$ with MOIRCS) and better S/N from the higher sensitivity Integral Field Unit (IFU) data enables us to improve the S/N of and \[NII\] in all galaxies in common with @hayashi11. Additionally, we revise the study of the chemical enrichment in XMM2215 by @hayashi11 now additionally using the information from the phase-space diagram about the position of galaxies in the cluster, by separating galaxies from the infalling and virialized regions. The paper is structured as follows: In Sect. 2 we present the selection of the XMM2215 cluster EL galaxies at $z \sim 1.5$, their KMOS spectroscopy, and data reduction. We describe the EL flux measurements and present the derivation of SFRs, metallicities, and stellar masses of the observed cluster galaxies. In Sect.3 we present the phase-space diagram, the mass$-$specific SFR (SSFR) relation, and the MZR at $z\sim 1.5$. We investigate how the cluster environment affects the chemical enrichment. In Sect.4 we discuss the slow quenching scenario in a massive cluster at $z>1$ implied by our findings and the comparison with literature metallicity studies in clusters at $z>1$. Finally, we summarize our conclusions. A concordance cosmology with $\rm{H}_{0}=70$ , $\Omega_{0}=0.25$, $\Omega_{\Lambda}=0.75$ is used throughout this paper. We assume a Salpeter [@salp55] initial mass function (IMF) for all derived stellar masses and SFRs and correct existing measurements used in this paper to a Salpeter IMF. We note that metallicity and abundance are taken to denote oxygen abundance, O/H, throughout this paper, unless otherwise specified. In addition, we use dex throughout to denote the antilogarithm, that is, 0.3dex is a factor of two. Data and measurements {#sec:data} ===================== ![image](fig2.eps){width="7cm"} KMOS observations and data reduction of cluster SF galaxies in XMM2215 ---------------------------------------------------------------------- The cluster XMM2215 is a massive $z \sim 1.46$ cluster discovered in the XMM Cluster Survey [@stanford06]. A previous estimation of the mass gave $M_{200} = 2.1^{-0.8}_{+1.9} \cdot 10^{14}M_{\odot}$ [@hilton10; @stott10], and a more recent estimation in this study gives $M_{200} = (6.3 \pm 1.2) \cdot 10^{14}M_{\odot}$ (see Sect.\[sec:phasespace\]). One advantage of observational environmental studies in this cluster compared to other clusters at $z>1$ is its wealth of spectroscopic redshifts published not only for SF but also for passive galaxies [@hilton09; @hilton10; @hayashi11; @beifiori17; @chan18]. This enables the accretion state of cluster member galaxies to be characterized by identifying virialized and infalling regions (see Sect.\[sec:phasespace\]). The targets for the KMOS H-band observations were selected from the list of $1.44<z<1.48$ \[OII\] emitters, with a flux larger than $2 \times 10^{-17} \rm{ergs/s/cm}^{2}$, identified from narrow-bands NB912 and/or NB921 by @hayashi14. Most of the targets had already been spectroscopically confirmed before the KMOS observations. VLT/KMOS H-band observations were carried out in 2016 on the night between October 19 and 20 (ESO program ID 098.A-0204(B)) with a nod-to sky strategy and each observing block (OB) consisting of several ABA ABA sequences, where “A” signifies that the IFUs were placed on the science targets, and “B” indicates that the IFUs were observing blank sky. The integration time of each exposure was 300s with a total exposure time of 9900s. Seeing conditions measured in the optical by the DIMM seeing monitor ranged between 0.6 and 1.0 arcsec. The KMOS data reduction was carried out with the official ESO-KMOS pipeline version 1.4.3 with the default settings of the pipeline, with the exception of the default of the pipeline to first combine exposures from a single OB before then combining these OBs into final data cubes. Since the best data quality in the co-addition of individual exposures is achieved using sigma clipping, we modified the default approach and ran the sigma clipping and cube combination on all 33 individual exposures (from four OBs) at once. KMOS emission-line measurements ------------------------------- To measure the main ELs, and , from the KMOS H-band observations, we use the Interactive Data Language (IDL) publicly available software KUBEVIZ [@fossati16] applied to each extracted 1D spectrum. To measure the ratio of the and the (typically) fainter EL flux to compute O/H metallicities, we extracted 1D spectra summed over 25 spaxels centered on the EL of the respective KMOS data cube, corresponding to an aperture of about 1 square arcsecond. This typically corresponds to the area where KUBEVIZ measures $S/N>3$ per spaxel in . For the total fluxes used to compute SFRs we extracted 1D spectra over slightly larger areas, including all spaxels where KUBEVIZ measures $S/N>3$ in . Due to differences in resolution in each IFU [@davies13], the instrumental resolution at the wavelength of was computed from the skylines for every IFU in order to estimate the line widths of and . Starting from an initial redshift, KUBEVIZ simultaneuosly fits the “lineset” and . This “lineset” is described by a combination of 1D Gaussian functions keeping the velocity separation of the lines fixed according to the line wavelengths (see Fig.\[fig:HaNII\]). The continuum level is evaluated inside two symmetric windows between 80 and 200Å redward and blueward of each line, omitting regions with other emission lines and using only values between the 40th and 60th percentiles. During the fit, KUBEVIZ takes into account the noise from the noise data cube, thus optimally suppressing sky line residuals. We additionally fitted the and lines including also the second (weaker) line. We checked that the fit of three ELs produces and flux values in agreement (given the error bars) with the and values using fits of two lines. Because the fainter line is more often heavily affected by night-sky lines than the brighter , we decided to use the measurements of and fluxes from the fit of two lines. From the 20 observed XMM2215 targets we could detect the EL for 19 galaxies (see Table \[MeasXMM2215\]). The EL could be detected in twelve galaxies, while in the remaining seven galaxies \[NII\] was either affected by a strong OH line or was too faint to be detected, yielding only an upper limit in two cases. For one galaxy among the seven, the line was detected but was heavily affected by a strong night-sky line, and therefore no reliable measurement of the EL flux was possible. Two of the twelve galaxies with detected \[NII\] turn out to be AGNs because they have an flux comparable with the flux (see Table \[MeasXMM2215\]). The remaining ten galaxies with detected \[NII\] include seven galaxies in the inner region of the cluster ($R<0.5R_{200}$) and three galaxies outside it. We note that two additional galaxies with upper limits on \[NII\] (see Table\[MeasXMM2215\]) are also shown in Fig.\[fig:PhaseSpace\] with blue symbols. Stellar masses, SFRs, and oxygen abundances {#SFRsox} ------------------------------------------- Stellar masses of the XMM2215 cluster galaxies were calculated using the code Lephare of @arnilb11, which fits stellar population synthesis models [@bruzcharl03] to the available SUBARU B, $R_{c}$, i’, z’, and WFCAM/UKIRT K-band photometry described in @hayashi14. This code is a simple $\chi^{2}$ minimization algorithm that finds the best match of templates for the given data. We do not fit templates with stellar ages of less than 0.5Gyr and larger than 5Gyr. This is a valid assumption since at $z \sim 1.5$ the Universe is only $\sim$4.5 Gyr old. In addition, we kept the redshift fixed, limited the number of extinction $E_{(B-V)}$ values (0 to 0.5, in 0.1 steps), and used a @calzetti00 extinction curve. We are confident in the robustness of the calculated masses since the $z-K$ color encompasses the redshifted 4000Å break and thus is sensitive to galaxy mass-to-light ratios [@kaufm03]. It turns out that one galaxy (39683, red filled circle in Fig.\[fig:PhaseSpace\]) of our sample of ten SF galaxies with detected \[NII\] has a high log(\[NII\]/)$=-0.20 \pm 0.06$ ratio indicating possible AGN activity. For our study of environmental effects on metallicities we therefore restrict our cluster sample to nine galaxies with \[NII\]/ measured and $\rm{log}(M/M_{\odot})<11$, including six objects at $R<0.5R_{200}$ (our SFvirialized sample), but we also show the O/H measurement for 39683 in the MZR diagram, although with a different symbol (red filled circle). Since we do not have measurements of we cannot use an observed Balmer decrement to derive extinction. Therefore, the line luminosities $\rm{L}(\rm{H}\alpha)$ were corrected for three values of assumed extinction $A_{V}=0,0.5,1$ and then transformed into SFRs by applying the @ken98 conversion: $\rm{SFR} (M_{\odot}\rm{yr}^{-1}) = 7.9 \times 10^{-42} \rm{L}(\rm{H}\alpha)\rm{ergs/s}$. To estimate the chemical abundances, a number of diagnostics have been developed based on strong ELs, ${\rm [O\,II]\,}{\lambda\,3727}$, H$\beta$, ${\rm [O\,III]\,}{\lambda\,5007}$, H$\alpha,$ and \[NII\]${\lambda\,6584}$. At higher redshifts, these ELs move to the NIR, and studies of metallicities up to $z \sim 2.5$ use mostly the N2 calibration (or sometimes the O3N2 calibration) of @petpag04 to derive oxygen abundances. For easier comparison with existing publications of the MZR at $z>1$ we therefore use the N2 metallicity calibration for this study. We derive oxygen abundances from \[NII\]/using the N2 method of @petpag04. We note that one galaxy from our SFvirialized sample has an \[OIII\]/ measurement (with large error bars) and another galaxy has an upper limit for \[OIII\]/, both from the work of @hayashi11, and both indicating a ratio of \[OIII\]/ smaller than one. Results ======= Spatial distribution and phase-space diagram {#sec:phasespace} -------------------------------------------- ![ \[fig:SSFRXMM2215\] Mass$-$SSFR relation for the \[OII\] emitters in and around the XMM2215 cluster (green open circles), the six XMM2215 cluster galaxies of the SFvirialized sample (red triangles) and four infalling cluster galaxies (blue symbols). The oblique solid line shows the MS at $z \sim 1.46$ and its dispersion (indicated by the dotted lines) using Eq.1 in @peng10. ](fig3.eps){width="6cm"} ![image](fig4.eps){width="7cm"} A phase-space diagram is a useful tool to characterize the accretion state of cluster member galaxies relatively free from effects due to the 2D projected positions with respect to the cluster center (see Fig.\[fig:PhaseSpace\], comparing the three panels). To investigate cluster membership for XMM2215 we collected redshift information from the literature [@hilton09; @hilton10; @hayashi11; @beifiori17; @chan18] yielding a sample of 108 galaxies with published spectroscopic redshifts. We then used the $3 \sigma$-clipping technique of @yahvid77, which assumes that clusters are relaxed isothermal spheres and that the velocity distribution of cluster galaxies follows an underlying Gaussian distribution. Out of the 108 galaxies, 74 lie inside $3 \sigma$. A mass model of the cluster was computed assuming that the cluster is a singular isothermal sphere. $R_{200}$ is roughly equivalent to the virial radius of @carlberg97 and is calculated as $\sqrt(3)/10\cdot \sigma_{z}/H(z)$, where $\sigma_{z}$ is the velocity dispersion of the cluster and H(z) the redshift dependent Hubble parameter. From this iterative method based on the technique of @carlberg97 we obtained $R_{200} = (1.23 \pm 0.18)Mpc$ and $M_{200} = (6.3 \pm 1.2) \cdot 10^{14}M_{\odot}$. The uncertainties of $M_{200}$ and $R_{200}$ were determined by randomly taking out ten galaxies of the spectroscopic sample of 108 galaxies, and then repeating the phase-space analysis a few hundred times. Our derived $M_{200}$ and $R_{200}$ are slightly higher than those derived by @hilton10 using a similar analysis ($M_{200} = 2.1^{-0.8}_{+1.9} \cdot 10^{14}M_{\odot}$, $R_{200} = (0.8 \pm 0.1)Mpc$), but it should be noted that @hilton10 used only 44 spectroscopic redshifts (compared to 74 used here), and we also used spectroscopic redshifts of passive galaxies more recently published by @beifiori17 and [@chan18]. From the 74 objects with spectroscopic redshifts inside $3 \sigma$ we found 58 galaxies to lie inside $R_{200}$, as depicted in Fig.\[fig:PhaseSpace\]. The red triangle symbols represent our main SFvirialized sample of cluster galaxies for which \[NII\]/ has been measured and which lie in the virialized region of the cluster, as indicated by the large triangle in the lower left-hand corner derived by @rhee17 using cosmological hydrodynamic simulations of groups and clusters. Star-forming XMM2215 cluster galaxies with enhanced metallicities ----------------------------------------------------------------- Figure\[fig:SSFRXMM2215\] shows the mass–SSFR relation of SF galaxies in and around the XMM2215 cluster. @peng10 derived a formula of the evolution of the SSFR as a function of mass and time that we use to calculate the mean SSFR as a function of stellar mass at $z \sim 1.5$ (main sequence, MS). For this, we assume a dependence of the SSFR on mass as observed for local SDSS galaxies, an increase in the mean SSFR from $z \sim 0 $ to $z \sim 1.5$ as derived by @peng10, and a dispersion (indicated by the dotted oblique lines) of a factor of 0.3dex about the mean relation. Star-formation rates for \[OII\] emitters, identified from narrow-bands NB912 and/or NB921 by @hayashi14, are derived from their \[OII\] fluxes using the relation recommended by @maier15. For the six galaxies in the $SFvirialized$ sample and the four infalling cluster galaxies we assumed three values of extinction, $A_{V}=0,0.5$, and 1, and then transformed their fluxes into SFRs by applying the @ken98 conversion. The error bars of the six red triangles and of the blue symbols in Fig.\[fig:SSFRXMM2215\] indicate the range of SFRs for $A_{V}$ values between 0 and 1. We use the @calzetti00 $E_{B-V} = A_{V}/3.1$ (or 4.05) formula and their $E_{B-Vstellar} / E_{B-Vemission} = 0.44$ conversion between the color excess of the stellar continuum $E_{B-Vstellar}$ and the color excess derived from the gas emission lines $E_{B-Vemission}$. Using the minimum derived $E_{B-Vstellar}=0.2$ value for our SFvirialized galaxies we obtain $A_{V}=1.4$ (or $A_{V}=1.8$). We note that our assumed $A_V=0$, 0.5, and 1 values are not higher than or in contradiction with the $E_{B-Vstellar}$ values. Five out of the six $SFvirialized$ galaxies lie in the lower SSFR part of the MS region, while the most massive one lies just below the MS region assuming a maximum of $A_{V}=1$; this latter would however lie in the lower SSFR part of the MS region for a plausible higher extinction value at higher masses of $A_{V}>1$. Therefore, we conclude that the six $SFvirialized$ galaxies are still forming stars, although with slightly lower SSFRs, lying in the lower SSFR part of the MS region. We note that their slightly lower SFRs and higher metallicities (as discussed below) are in relatively good agreement with expectations of the link between SFRs and O/Hs described by the fundamental metallicity relation [e.g., @mannu10]. For the four galaxies outside the virialized region (blue symbols) no clear trend of high/low SSFR is seen, due to the small number statistics. The left panel of Fig.\[fig:MZR2215\] shows the MZR, using the N2-calibration of @petpag04, of the six $SFvirialized$ cluster galaxies (triangles) compared to the infalling galaxies (blue asterisks), and the local SDSS MZR (red solid line). To compare our sample with a field sample at similar redshifts with similar data quality, we used field galaxies at $1.42<z<1.52$ with KMOS H-band spectroscopy from the KMOS3D survey, with \[NII\]/ measurements published by @wuyts16 shown as filled blue circles. We note, from looking at Fig.2 of @wuyts16, that their $z \sim 1.5$ stacked \[NII\]/ measurements seem consistent with $z \sim 1.5$ literature results from the FMOS-COSMOS survey [@zahid14]. The dotted red lines show the dispersion of the SDSS relation, $\pm 0.09$dex, as given by @keweli08, while the dotted blue line is displaced by further 0.11dex to lower metallicities compared to the lower dotted red line, and corresponds to the location of the stacked measurements of a larger sample of $z \sim 1.5$ KMOS3D field galaxies (large blue filled squares). Because the MZR of cluster galaxies inside $0.5R_{200}$ is roughly consistent with the SDSS position of the lower dotted red line, the offset between the blue dotted line and the lower red dotted line indicates an increase in metallicities by a factor of 0.11dex for SFvirialized cluster galaxies compared to field galaxies. Due to concerns surrounding the N2-metallicity indicator (see Sect.\[sec:intro\]), we also show our results in the right panel of Fig.\[fig:MZR2215\] in terms of the observed / ratios. We performed a linear fit of the log(/)-versus-stellar mass relation for the SFvirialized sample (red dotted line), and show the linear relation implied by the stacked values of the KMOS3D sample as a blue dotted line. The log(\[NII\]/) offset between the blue and red dotted lines is 0.2dex. Given the mean uncertainties of the stacked values from @wuyts16 of about 0.07dex, we conclude that this enhancement in log(\[NII\]/) ratios has a 2.9$\sigma$ significance. Thus, Fig.\[fig:MZR2215\] clearly reveals an enhancement of the log(/) ratios by 0.2dex and of the derived oxygen abundances by 0.11dex for XMM2215 cluster galaxies inside $0.5R_{200}$ compared to field. The different values for the enhancement (0.2dex vs. 0.11dex) reflect the factor 0.57 in the N2 conversion, $12+log(O/H)=8.90+0.57 \cdot log([NII]/\Ha)$. Discussion and Summary {#sec:disc} ====================== Slow quenching implied by enhanced metallicities of cluster galaxies at $z \sim 1.5$ ------------------------------------------------------------------------------------ The enhanced metallicities of $z \sim 1.5$ cluster galaxies inside $0.5R_{200}$ compared to $z \sim 1.5$ field galaxies can be interpreted as a quenching indicator based on the comparison of observed quantities with metallicity-SFR-mass bathtub model predictions with inflowing gas [@lilly13] performed for lower-redshift clusters by @maier16 [@maier19]. The interpretation is that the gas metallicities increase because their ISM is no longer diluted by the inflow of pristine gas after the hot halo gas reservoir has been stripped in the cluster environment, likely around $0.5R_{200}$, initiating strangulation. The enhancement of O/Hs of cluster galaxies compared to field galaxies is in agreement with the enhancement in metallicities found by @shimakawa15 in a protocluster at $z>2$ compared to field galaxies based on stacked measurements of \[NII\]/ for protocluster galaxies. @shimakawa15 also discussed one strangulation scenario based on their result: for infalling cluster galaxies, once an infalling galaxy is incorporated into a common cluster halo, the gas reservoir of the galaxy may be stripped or truncated due to the interactions with the cluster environment. This can not only expel low-metallicity gas trapped in the outer region of the galaxy, but also terminate the fresh pristine gas accretion on to the galaxy, increasing the retained gas metallicity. On the other hand, the studies of @kacprzak15 and @tran15 for cluster galaxies at $z>1.6$ used the N2 calibration from stacking and found a similar MZR for cluster and field galaxies. Since the clusters studied by @kacprzak15 and @tran15 have a much lower total mass than our studied XMM2215 cluster, it is possible that environmental effects are not strong enough in these two clusters to produce an observable effect on metallicities, due to a possible overly low ICM density, as discussed below. The current KMOS study of cluster galaxies in XMM2215 shows, using [individual]{} metallicity measurements, that environmental effects on metallicities implying quenching are already starting to act at $z>1$. @maier19 discussed how the hot gas reservoir and the cold gas in the disk may be influenced for infalling galaxies into clusters, depending on the strength of the cluster potential and the ICM density. Gas can be removed from infalling galaxies if the ram pressure exceeds the restoring force per unit area (gravitational restoring pressure) exerted by the galaxy, as first derived by @gunngott72. XMM2215 contains only a few known passive galaxies for which spectroscopic redshifts have been determined [@beifiori17; @chan18]. Therefore, it is plausible that the ICM density in this cluster implies a ram pressure higher than the restoring pressure of the [hot]{} halo gas reservoir of cluster galaxies, but not higher than the restoring pressure of the [cold]{} gas in the disk out of which stars are currently born. Compared to the slow-then-rapid quenching scenario discussed by @maier19 and @roberts19, we think that we see only the “slow” quenching in XMM2215 SF cluster galaxies at $z\sim 1.5$ traced by enhanced metallicities. As discussed in @maier19 based on comparisons with high-resolution cosmological hydrodynamic simulations of @bahe13, the ram pressure is probably only slightly larger than $3 \times 10^{-14} N/m^{2}$ in XMM2215, larger than the restoring pressure of hot gas and enough to strip the more extended, less dense, and less tightly bound hot gas. This slow quenching is also in relatively good agreement with results of @hayashi17 [@hayashi18] who used ALMA to measure CO(2-1) ELs and dust continuum and derived molecular gas masses for several XMM2215 cluster galaxies. These latter authors discussed that the main component of the stripped gas is the neutral gas reservoir, while the molecular gas (observed with ALMA) is relatively much less affected by the ram pressure. This means that the galaxies continue to form stars from the molecular gas, in agreement with our slow quenching scenario. The ICM density in XMM2215 has probably not yet reached the threshold found by @roberts19 to initiate the “rapid” quenching, a ram pressure threshold required to strip the cold gas of cluster galaxies; cold gas which is denser and sits much closer to the galactic center. Summary ------- To study environmental effects on the chemical enrichment and quenching of cluster galaxies, we used KMOS spectroscopy of and in the massive cluster XMM2215 at $z \sim 1.5$ to measure oxygen abundances and estimate SFRs. Using the phase-space diagram results of Fig.\[fig:PhaseSpace\] we identified XMM2215 SF cluster galaxies in a virialized region with $R<0.5R_{200}$ and smaller line-of-sight velocities. Studying the metallicities of a sub-sample of the cluster galaxies with KMOS H-band observations and located in the virialized region (SFvirialized sample), we found evidence for enhanced metallicities by about $0.1$dex for these galaxies compared to infalling and field galaxies at similar redshifts (Fig.\[fig:MZR2215\]). These cluster galaxies are still forming stars, although at slightly lower SFRs (see Fig.\[fig:SSFRXMM2215\]). These findings indicate a strangulation scenario in which the ICM density toward the cluster center becomes high enough such that RPS can remove the hot halo gas reservoir of cluster galaxies, thus enhancing metallicities and initiating slow quenching, while the galaxies continue to form stars using cold gas in the disk and enhance their metallicities because no fresh pristine gas accretion dilutes their ISM. Additional spectroscopic observations, not only of \[NII\] and but also of \[OIII\] and , for a larger sample of cluster galaxies at $z>1$ are needed to reinforce these interesting results, which for now are based on a small sample. We would like to thank the anonymous referee for providing constructive comments and help in improving the manuscript. Arnouts, S., & Ilbert, O., 2011, Astrophysics Source Code Library, ascl:1108.009 Bah[é]{}, Y. M., McCarthy, I. G., Balogh, M. L., & Font, A. S., 2013, MNRAS, 430, 3017 Beifiori, A., Mendel, J. T., Chan, J. C. C., et al. 2017, ApJ, 846, 120 Bruzual & Charlot, 2003, MNRAS, 344, 1000 Calzetti, D., Armus, L., Bohlin, R. C., et al., 2000, ApJ, 533, 682 Carlberg, R. G., Yee, H. K. C., & Ellingson, E., 1997, ApJ, 478, 462 Chan, J. C. C., Beifiori, A., Saglia, R. P. et al. 2018, ApJ, 856, 8 Chiang, Y.-K., Overzier, R. A., Gebhardt, K., & Henriques, B., ApJ, 844, 23 Cooper, M. C., Tremonti, C. A., Newman, J. A., & Zabludoff, A. I., 2008, MNRAS, 390, 245 Davies, R. I., Agudo Berbel, A., Wiezorrek, E., et al., 2013, A&A, 558, A56 Dickinson, M., Papovich, C., Ferguson, H. C., et al., 2003, ApJ, 587, 25 Drory, N., Salvato, M., Gabasch, A., et al., 2005, ApJ, 619, 131 Ellison, S. L., Simard, L., Cowan, N. B., et al. 2009, MNRAS, 396, 1257 Fossati, M., Fumagalli, M., Boselli, et al., 2016, MNRAS, 455, 2028 Gunn J. E., Gott J. R. I., 1972, ApJ, 176, 1 Gupta, A., Yuan, T., Tran, K.-V. H., et al., 2016, ApJ, 831, 104 Hayashi, M., Kodama, T., Koyama, Y., et al., 2011, MNRAS, 415, 2670 Hayashi, M., Kodama, T., Koyama, Y., et al., 2014, MNRAS, 439, 2571 Hayashi, M., Kodama, T., Kohno, K., et al., 2017, ApJ, 841, L21 Hayashi, M., Tadaki, K.-i., Kodama, T., et al., 2018, ApJ, 856, 118 Hilton, M., Stanford, S. A., Stott, J. P., et al. 2009, ApJ, 697, 436 Hilton, M., Lloyd-Davies, E., Stanford, S. A., et al. 2010, ApJ, 718, 133 Hughes, T. M., Cortese, L., Boselli, A.,et al., 2013, A&A, 550, A115 Kauffmann, G., Heckman, T. M., White, S. D. M., et al., 2003, MNRAS, 341, 33 Kauffmann, G., Heckman, T. M., Tremonti. C. et al., 2003b, MNRAS, 346, 1055 Kacprzak, G. G., Yuan, T., Nanayakkara, T., et al., 2015, ApJ, 802, 26 Kashino, D., Silverman, J. D., Sanders, D., et al., 2017, ApJ, 835, 88 Kennicutt, R. C., Jr. 1998, ARA&A, 36, 189 Kewley, L.J. & Dopita, M.A. 2002 ApJSS, 142, 35 Kewley, L. J., & Ellison, S. L., 2008, ApJ, 681, 1183 Kulas, K. R., McLean, I. S., Shapley, A. E., et al., 2013, ApJ, 774, 130 Lilly, S. J., Carollo, C. M., Pipino, A. et al., 2013, ApJ, 772, 119 Maier, C., Ziegler, B. L., Lilly, S. J., et al., 2015, A&A, 577, 14 Maier, C., Kuchner, U., Ziegler, B. L., et al., 2016, A&A, 590, A108 Maier, C., Ziegler, B. L., Haines, C. P., & Smith, G. P., 2019, A&A, 621, A131 Magdis, G. E., Bureau, M., Stott, J. P., et al., 2016, MNRAS, 456, 4533 Mannucci, F., Cresci, G., Maiolino, R. et al. 2010, MNRAS, 408, 2115 Mouhcine, M., Baldry, I. K., & Bamford, S. P. 2007, MNRAS, 382, 801 Pasquali, A., Gallazzi, A., & van den Bosch, F. C. 2012, MNRAS, 425, 273 Pettini & Pagel 2004, MNRAS, 348, 59 Peng, Y.-j., Lilly, S. J., Kova[č]{}, K., et al., 2010, ApJ, 721, 193 Peng, Y.-j., & Maiolino, R., 2014, MNRAS, 438, 262 Rhee, J., Smith, R., Choi, H., et al., 2017, ApJ, 843, 128 Roberts, I. D, Parker, L. C, Brown, T., et al., 2019, arXiv:1902.02820 Salpeter, E. E., 1955, ApJ, 121, 161 Shimakawa, R., Kodama, T., Tadaki, K.-i., et al., 2015, MNRAS 448, 666 Springel, V. 2010, MNRAS, 401, 791 Stanford, S. A., Romer, A. K., Sabirli, K., et al. 2006, ApJL, 646, L13 Steinhauser, D., Schindler, S., & Springel, V., 2016, A & A, 591, A51 Stott, J. P., Collins, C. A., Sahl[é]{}n, M., et al. 2010, ApJ, 718, 23 Tran, K.-V. H., Nanayakkara, T., Yuan, T., Kacprzak, G. G., et al., 2015, ApJ, 811, 28 Valentino, F., Daddi, E., Strazzullo, V., et al., 2015, ApJ, 801, 132 Wetzel, A. R., Tinker, J. L., Conroy, C., & van den Bosch, F. C., 2013, MNRAS, 432, 336 Wuyts, E., Wisnioski, E., Fossati, M., et al., 2016, ApJ, 827, 74 Yahil, A., & Vidal, N. V., 1977, ApJ, 214, 347 Zahid, H. J., Kashino, D., Silverman, J. D., et al., 2014, ApJ, 792, 75 Kewley, L. J., Daddi, E.,
--- author: - \| Hong Qian\ Department of Applied Mathematics, University of Washington\ Seattle, WA 98195, USA.\ Hao Ge\ Beijing International Center for Mathematical Research\ and Biodynamic Optical Imaging Center\ Peking University, Beijing, 100871, PRC. title: 'Mesoscopic Biochemical Basis of Isogenetic Inheritance and Canalization: Stochasticity, Nonlinearity, and Emergent Landscape' --- = 6.5 in = 8.5 in = -0.75 in = -0.5 in = 0.29 in Introduction ============ Epigenetic inheritance at the cellular level preserves certain phenotypes through cell divisions \[[@leibler_05]\]. Since the definition of “epigenetic” means the inheritability is not due to genes, i.e., DNA sequences, the epigenetic phenomenon must be a biochemical process \[[@ptashne_2007; @spudich_76]\]. DNA and histone modifications, gene regulations by transcriptional factors, signaling networks, and metabolic pathways are all parts of cellular biochemistry. Current research chiefly focuses on the “code” of epigenetic inheritance in terms of DNA methylation and/or histone acetylation \[[@turner_2000; @jones_2001; @zhang_cocb_07]\]. While the gene expressions and their regulations are a central component of epigenetic processes, it is less certain what the roles of cellular signaling networks, or metabolic networks, are. Even more importantly, are different gene expressions the [cause]{} of the epigenetic phenomenon, or consequences of the dynamics of a larger intracellular biochemical network as a whole? In the present paper, we advance a theory for biochemical reaction systems in a mesoscopic volume \[[@qian_jsp]\]. Taking a broader perspective that is rooted in stochastic, nonlinear dynamical systems \[[@qian_iop]\], we illustrate that it is likely that the code for epigenetic inheritance is [distributive]{} \[[@hopfield_84]\]. Using two classes of biochemical networks (see Fig. \[fig1\]) as illustrations, we investigate mesoscopic, nonlinear biochemical reaction networks with multiple [stochastic attractors]{} \[[@wolynes_pnas_05]\]. More importantly, we shall show that two very different types of mesoscopic bistabilities exist: the [stochastic bistability]{} which has no macroscopic counterpart \[[@elston_bj_01; @wolynes_pnas_02; @wolynes_pre_05; @arkin_pnas_05; @wolynes_nobel_09; @qian_bj_10]\], and the [nonlinear bistability]{} which exhibits deterministic bistability when the system’s volume tends to macroscopic scale \[[@qian_jrsi_09; @qian_prl_09; @liang_qian; @shi_qian_11]\]. There is a growing awareness of mesoscopic biochemical bistability \[[@wang_xj_05; @dodd_cell_07; @Turcotte_pnas; @Arup_cell; @cooper_pnas_09]\]. Multistability in a mesoscopic chemical reaction system can be mathematically represented by the Delbrück-Gillespie processes (DGP) whose probability distribution function follows chemical master equation \[[@delbruck]\] and whose exact stochastic trajectories can be obtained by the widely known Gillespie algorithm \[[@arkin_98; @gillespie]\]. This approach has found numerous applications in recent studies. For an introduction to this chemical reaction system theory, see \[[@qian_iop; @qian_jsp; @liang_qian; @arkin; @wilkinson; @qian_book]\]. A review by [@qian_bishop] is particularly accessible. In principle, a biochemical reaction system in a small volume of the size of a cell, on a very large time scale (years, hundreds years), will have a steady state probability distribution which reflects the continuous jumping among the multiple stochastic attractors: regions with high local probability \[[@ge_qian_jrsi; @qian_prl_09; @ao_2004]\]. In terms of this [stationary distribution]{}, the dynamics of a mesoscopic biochemical system can be cogently visualized and even quantified by a landscape representation \[[@wolynes; @ao_2004; @wang_pnas_08; @hugang; @ge_qian]\]. $$\psfig{figure=fig1.eps,width=2.75in}$$ It will be illustrated in this paper that multistability is not only stable against noise, i.e., robust, but also could be readily inherited during the process of cell volume change and division. Within a cell cycle, it is not necessary to directly control the concentration of a biochemical species: As soon as cellular concentrations are deviated from their locally most stable values, the dynamics will cause them to spontaneously relax back as long as the deviations are within the limit of the basin of the attraction (stable state). Indeed, this restoring phenomenon should be regarded as a form of “self-organization”. Such state naturally has stability and robustness. On the larger landscape scale, the system is “digital”. This kind of “inheritable code” is not only distributive but also dynamic \[[@hopfield_84]\], in contrast to Watson-Crick basepairng. Some mathematical analysis of the DGP is at the foundation of our current landscape theory \[[@qian_jsp; @qian_iop]\]. In Sec. \[on\_lands\] we offer some discussions. Intuitive and appealing as it is, the justification of using the stationary probability distribution as the landscape involves subtle mathematical ideas which deserves further investigations \[[@ge_qian_jrsi; @ge_qian]\]. Biosynthesis of Self-regulating Repressor with Slow On-and-off Gene Fluctuations ================================================================================ We first consider a simple model for the biosynthesis and degradation of a repressor with stochastic gene expression, which is regulated by the repressor. The canonical kinetic scheme for this model, neglecting the intermediate stage of mRNA, is in Fig. \[fig1\]b. For simplicity, we assume that the repressor-gene binding rate $h(n)=h_on$ which involves a monomer, and $g_0>g_1$. See [@elston_bj_01], [@wolynes_pnas_02] [@wolynes_pre_05], and [@wolynes_pnas_05] for extensive studies of this model, and related systems with self-activation dimer ($h(n)\propto n(n-1)$, $g_1>g_0$). We choose this model to demonstrate bistability due to slow, nonadiabatic fluctuations in the gene state. This is a stochastic effect due to single-molecule behavior \[[@bai_pnas_99; @wolynes_nobel_09]\] which disappears in the macroscopic Law of Mass Action \[[@wolynes_pre_05; @qian_pccp_09]\] (see Methods). Stochastic bimodality and bistability ------------------------------------- It is generally accepted that, in biochemistry, [noise-induced bistability]{} means a small kinetic system has bimodal distribution while its macroscopic counterpart has only uni-stability \[[@wolynes_pre_05; @arkin_pnas_05; @qian_bj_10]\]. Usually, the meaning of “macroscopic counterpart” is defined as the same chemical or biochemical reaction systems at same concentrations. For a large volume, the concentrations are deterministic variables; but in a mesoscopic volume, the copy numbers fluctuate. This “correspondence principle” is consistent with the experimental practices: In the past most biochemical experiments on gene expression were carried out with DNA measured in concentrations \[[@PvH_07]\]. $$\includegraphics[width=2.2in,angle=270]{stoch_bi.eps}$$ With increasing volume of the a reaction system ($V$) and the copy numbers DNA ($M$), stability of one of the two states decreases while the other increases. For a sufficiently large $V$, the bistability disappears all together \[[@wolynes_pre_05; @qian_pccp_09]\]. We call such bistability [stochastic bistability]{}. Fig. \[v\_dep\_4\_sb\] shows how the stationary probability distributions change with the increasing $V$ and DNA copy number $M$ while keeping its concentration $x_t=M/V=1$. In agreement with the visual landscapes in Figs. \[v\_dep\_4\_sb\]A-D, \[v\_dep\_4\_sb\]E also shows how the mean sojourn time for the all-off, low transcription state ($m=0, n\le 80$ in Eq. \[the\_cme\_4\_srg\]) decreases as a function of $V$.[^1] This is in sharp contrast to the nonlinear bistability (Fig. \[fig:switchtime\]) we shall discuss next. Phosphorylation Dephosphorylation Cycle with Nonlinear Feedbacks ================================================================ A gene regulatory network with dimeric activator also exhibits bistability \[[@elston_bj_01; @wolynes_pnas_05; @shi_qian_11]\], but by a different mechanism. To illustrate this, and also to broaden the scope of our discussions, we shall consider a cellular phosphorylation dephosphorylation cycle (PdPC) signaling network in Fig. \[fig1\]a. In particular, we shall consider the case of positive feedback with a dimer ($\chi=2$). The same analysis can be applied to the gene expression system with $\chi=2$ and $g_1>g_0$ in Fig. \[fig1\]b. It is easy to show that for $\chi=0,1$, the cellular signaling network in Fig. \[fig1\]a, with slow fluctuating kinase activity, exhibits stochastic bistability \[[@qian_bj_10; @qian_pccp_09]\]. The theory we present here is general to both types of biochemical networks in terms of nonlinear chemical kinetics. The class of networks in Fig. \[fig1\]a has been widely implicated, such as in Src family kinase membrane signaling \[[@cooper_qian]\], Rab 5 GTPase in endocytic pathway \[[@liguangpu]\], [Xenopus oocytes]{} regulation for cell fate \[[@ferrell_sci]\], and long-term neural memory \[[@wang_xj_05; @cooper_pnas_09]\]. The network has been studied in [@ferrell_chaos; @qian_prl_05; @qian_prl_09; @qian_bj_10]. The detailed kinetic scheme is given in Eq. \[fig3a\] for which a DGP is uniquely specified (see Methods): $$\begin{aligned} && E+K^{\dag}+ATP \overset{k_1}{\underset{k_{-1}}{\rightleftharpoons}} E^+K^{\dag}+ADP, \label{fig3a}\\ && K+2E^ \overset{k_2}{\underset{k_{-2}}{\rightleftharpoons}} K^{\dag}, \ \ \ E^* + P \overset{k_3}{\underset{k_{-3}}{\rightleftharpoons}} E+P+Pi. \nonumber\end{aligned}$$ Note that the stochastic Delbrück-Gillespie approach is not an alternative to the traditional enzyme kinetic modeling (Eqs. \[the\_ode\], \[fixed\_pt\]). When copy numbers in a chemical reaction system are large, a Delbrück-Gillespie process (DGP) automatically yields the deterministic dynamics predicted by the traditional Law of Mass Action \[[@qian_book; @qian_bishop]\]. One of the most important predictions of the DGP theory is a dynamic landscape for the system, as shown in Fig. \[fig2\] as well as Figs. \[v\_dep\_4\_sb\]A-D. Such a landscacpe can only be rigorously defined in a stochastic model; it can be computed using a chemical master equation. $$\psfig{figure=fig2.eps,width=2.5in,angle=270}$$ Time Scales, Emergent Landscape and Implications to Epigenetics {#on_lands} =============================================================== Three time scales of cellular dynamics -------------------------------------- Double-well landscapes shown in Figs. \[v\_dep\_4\_sb\] and \[fig2\] suggest multiple time scales in the biochemical dynamics. In fact, there are three distinct time scales in such systems. Note that while fluctuations modify the “down-hill” deterministic kinetics, they lead to “up-hill” dynamics which is impossible in a macroscopic system. The time for “barrier crossing”, however, is extremely slow in comparison to the down-hill kinetics. Therefore, the fast time scale in the system is the individual biomolecular reactions in Fig. \[fig1\]. For the present work, they are given in terms of the rate parameters $k$’s in Fig. \[fig1\]a and $f,g,h,k$ in Fig. \[fig1\]b (or equivalently, the $\alpha,\beta,\epsilon,\delta$ in Eq. \[pdpcwfb\_2\].) Millisecond are not unreasonable, even though individual biochemical reactions inside a cell could be much faster or slower. The middle time scale is the relaxation kinetics of a network to its steady states, as illustrated in the Fig. \[fig2\] by the downhill dynamics. Note that the very existence of a steady state (an attractor), or steady states, is a consequence of “self-organization” of a complex reaction network. The slow time scale is the transition rates between the two basins of attraction, or “wells”. Both the middle (deterministic) and slow (stochastic) time scales are [emergent properties]{} of the biochemical network. Following [@gqq] we shall denote them molecular signaling time scale (MSts), biochemical network time scale (BNts), and cellular evolution time scale (CEts), respectively. $$\psfig{figure=r2.eps,width=2.25in,angle=270}$$ Again taking the signaling system in Fig. \[fig1\]a (also Eq. \[fig3a\]) as an example. For biochemically realistic situations, $\epsilon\ll\alpha$. We thus have the landscape given in Eq. \[phi\_x\] simplified into $$\phi(x) = x_t\ln (x_t-x)-x\ln\left[\frac{(\alpha x^2+\delta)(x_t-x)} {\beta x}\right]$$ $$-2\sqrt{\frac{\delta}{\alpha}}\arctan \left(\sqrt{\frac{\alpha}{\delta}}x\right) +2x. \label{phix}$$ The parameters $\alpha,\beta,\delta$ in Eq. \[phix\] define the MSts. We can now use the model to investigate the role of $\alpha,\beta$ and $\delta$ on the BNts and CEts. The BNts is given by the $r_1$ and $r_2$ in Eq. \[rs\], and the CEts is given by the $T_{1\rightarrow 2}$ and $T_{2\rightarrow 1}$ in Eqs. \[T12\_exact\]-\[T21\]. The BNts changes with total concentrations of the regulators $E$ inside the system, as well as the concentrations of other factors. Fig. \[fig:r2\] shows how in the simple model the time-scale decrease, i.e., rate increases, with the total concentration of $E$, $x_t$. Within less than one order of magnitude change of $x_t$, from 1 to 6$\mu M$, the relaxation rate in the state 2, i.e., the well on the right in Fig. \[fig2\], increase by a factor of 100. Eq. \[rs\] confirms that there is a square dependence of the relaxation rate to the concentration. The CEts is extremely sensitive to the number of molecules in the biochemical system. Fig. \[fig:switchtime\] shows that with the MSts on the order of milli- and microsecond, and with concentration of $E$ on the order of micromolar , the transition times between the two states in Fig. \[fig2\] can be as long as thirty thousand years! Hence, the stability of the emergent attractors could be extremely robust against spontaneous concentration fluctuations (i.e., intrinsic noise) in the system. We also notice that both transition rates decrease with the $V$. This will be explained in Sec. \[sec:on\_land\] below. More interesting biologically, we note that the range of 700-1000 copy number of $E$ corresponds to a time range of 10 hours to 30 years. In yeast, [@oshea] have reported that the copy numbers for most of the transcription factors are centered arround $2^{10}=1024$ per cell. $$\psfig{figure=switchtime.eps,width=2.in,angle=270}$$ Epigenetic inheritance and canalization on the CEts --------------------------------------------------- Let us consider two replica of a mesoscopic biochemical reaction system in a laboratory, for example one of those in Fig. \[fig1\]a which do not involve gene expression. If the two systems have same total $E$ but different initial values for $E^$, one near zero and one near the total $E$, then these two systems settle to the two different attractors. In the time scale much shorter than the evolutionary transitions, the numbers of $E^$ fluctuate around the $n_1^$ and $n_2^$, respectively, or equivalently around the $c_1^$ and $c_2^$ if the volume of systems do not change. (If the volumes are changing, then the fluctuations are around the $c_1^$ and $c_2^$, but not $n_1^$ and $n_2^$!) However, at the time scale greater than the cellular evolution, there will be transitions between the two attractors. This is shown in Fig. \[fig3\]A. The probability distribution for the concentration of $E^$ is shown in Fig. \[fig3\]C. $$\psfig{figure=fig3.eps,width=2.1in,angle=270}$$ Fig. \[fig3\]B shows the identical reaction system, except its volume and total $E$ are twice as large (keeping the concentration invariant). What we observe from the Fig. \[fig3\]C is that the “concentration” distribution for $E^$ in the two cases have essentially the same locations for the peaks and trough. This implies that if the size of the biochemical reaction system increases in the time scale sufficiently short, then the identities of the attractors can be preserved. The stable state of the system is not only stable against intrinsic noise, but also could be readily transferred to the two daughter cells. During the cell cycle, the concentrations of biochemical substances might become approximately one half of the original value, in the extreme cases due to cell volume increase, and the system deviates from its corresponding stable state. The kinetic law of these two daughter systems is just the same as their mother cell except they have not relaxed to any of the stable states. As we have shown previously that the relaxation scale is not very large, so they will spontaneously relax back to the same corresponding stable state as long as not leaving an basin. Many previous work of epigenetics always searched for a stable chemical substance like DNA, which could be self-propagated and inherited to the daughter cells, while here we give another alternative possibility that the code of epigenetic is at the dynamic level of the whole biochemical network. Further, there are clear upper and lower bounds for the rate of volume increase: it can not be too large such that the instantaneous changing concentration is outside the basin of an attractor; it can not be too slow such that it is on the cellular evolution time scale. Surely, the stability of epigenetic code is weaker than that for a stable chemical substance and it is more flexible facing the influence of the fluctuating environment, but it is suffient for a normal cell, a chemical system, to survive and inherit even in a fluctuating environment. Fig. \[fig3\]D shows precisely two of such simulations. Consider the volume, and the total $E$, double within the time of 5 units. This is a duration much shorter than the cellular evolution time. The top and bottom traces are the number of $E^$ from two simulations. At the end of the doubling, if each system is divided into two, both “daughter systems” will also inherit the biochemical state of the “mother system”. The biochemistry of a mesoscopic reaction system is inheritable! The epigenetic stability, i.e., canalization \[[@zhang_cocb_07]\] could be directly related to the CEts. Mesoscopic stochastic dynamics on a landscape {#sec:on_land} --------------------------------------------- The foregoing discussion clearly illustrates the power of the “landscape” perspective in visualizing and representing the global dynamics in bistable systems. We see that for stochastic bistability, at least one of the “barrier heights” decrease with volume $V$ (Fig. \[v\_dep\_4\_sb\]), while for nonlinear bistability, the both barrier heights increase with the volume (Figs. \[fig:switchtime\] and \[fig3\]). What determines the entire landscape? Since it is defined on the space for all possible concentrations and/or copy numbers of all the molecular species in the reaction system, itself can not be determined by the concentrations and copy numbers. Rather, it is determined by the all possible molecules involved and their interaction/reaction rate constants. In other words, biochemical reaction networks. Since the molecular interaction/reaction rate constants are properties of molecular structures, which in turn is determined by the primary sequences in the case of proteins, we conclude that the landscape, conceptually, is encoded in the DNA sequence, together with the extracellular environment including the volume $V$, but is independent of the expression patterns of transcription factors. They are the consequences of a biochemical reaction system’s dynamics \[[@wolynes_pnas_02]\]. There are many similarities between the energy landscape for a protein in equilibrium \[[@wolynes]\] and the landscape for an open mesoscopic chemical system in a nonequilibrium steady state \[[@ao_2004; @qian_arpc_07; @gqq]\]. We, however, want to emphasize a key difference: Recall that an energy landscape exists [a priori]{} for a dynamical protein [@wolynes]. The open-chemical systems are fundamentally different in this respect \[[@qian_jpc_06; @qian_arpc_07; @gqq]\]. Specifically, for any finite volume $V$, a mesoscopic reaction system has a stationary probability distribution for the number of copies of all its biochemical species, $p_V(\vn)$, where $\vn=(n_X,n_Y,n_Z,\cdots)$ are the copy numbers of the molecules $X,Y,Z$ etc. It can be shown that such a distribution can be expressed as $$p_V(\vn) = \exp\left[ -V\phi(\vx)+\phi_1(\vx) + \frac{\phi_2(\vx)}{V} + \cdots\right], \label{land_scape}$$ where $\vx=\vn/V$. Furthermore, it can be shown that the function $\phi(\vx)$ is a meaningful landscape for the complex dynamics of the nonlinear biochemical system. That is, the dynamics always goes “down the hill” of $\phi(\vx)$ \[[@hugang; @ge_qian]\], though usually not by the steepest descent path. As we have seen, the landscape provides an very useful organizational device for thinking about cellular biochemical dynamics at widely different time scales, ranging from individual signaling reactions to cellular phenotype switching. Eq. \[land\_scape\] shows that the stationary distribution $p_V(\vx)$ changes with $V$. When the $V$ becomes macroscopic volume, the probabilities are concentrated only at the global minima of the function $\phi(\vx)$. However, with increasing $V$, $-(1/V) \ln p_V(\vn)$ approach to the function $\phi(\vx)$ which is defined on the entire space of $\vx$. It is an important insight that such a landscape exist and it is independent of the system’s volume, as long as the volume is reasonably large \[[@hugang; @ge_qian]\]. The nonlinear bistability, therefore, is the mesoscopic manifestation of a double well in the $\phi(\vx)$. Its macroscopic counterpart has two stable steady states. However, stochastic bistability is something very different: The bimodal distribution only exists when the $V$ is sufficiently small; when $\phi_1$ and $\phi_2$ in Eq. \[land\_scape\] contribute to the $p_V(\vn)$. $\phi(\vx)$ has only a single well. With increasing volume: the barrier in the nonlinear bistability increases, while that of stochastic bistability decreases. The emergent landscape of cellular interaction network dynamics and the landscape for protein dynamics are fundamentally different. The lack of detailed balance due to the open chemical nature of the former gives rise to the cycle flux underneath the landscape \[[@wang_pnas_08; @gqq]\]. The cycle flux makes the landscape non-local. When such a flux is sufficiently strong (i.e., mathematically characterized by the emerging of complex eigenvalue and eigenvectors), a synchronized dynamics arises \[[@qian_qian_prl; @ge_mbs_08]\]. The emergence of synchronized dynamics requires an entirely new kind of phenomenology. In the macroscopic classical world, this is the birth of oscillatory behavior and wave phenomena that have ruled classical engineering for a century. Discussion ========== Adaptive landscape ------------------ While the concept of energy landscape becoming a very useful term in protein dynamics, the concept of adaptive landscape in evolutionary dynamics is still highly controversial \[[@hartl_pnas_10]\]. We believe one of the main reasons for this situation is that the landscape in the latter, as the landscape in the present work, is an emergent entity, which is not given [a priori]{}. There are two issues related to this important distinction: (1) The mathematical existence of such a landscape in a general, nonlinear stochastic dynamics which does not have detailed balance; and (2) How is such a landscape, even exists, related to the dynamics, both deterministic and stochastic. The most nontrivial issue here is the logical relationship between the landscape and the dynamics: It is rather clear that in systems with detailed balance, the landscape exists [a priori]{} and the dynamics is a [consequence]{} of the landscape. However, for system without detailed balance, the dynamics, as defined by the reaction networks and all the individual rate constants, define the overall dynamics [as well as]{} define the landscape. Dynamics and the landscape have [correlations]{} but no [causality]{}. Hence, logically, it is not correct to view the dynamics as a consequence of the landscape. Nevertheless, the landscape is still a very useful device to understand and characterize the overall dynamics. The concept of emergent landscape, thus, should only be understood in this “historically retrospective” sense. Genocentric epigenetic inheritable memory and a possible alternative -------------------------------------------------------------------- Currently, methylation of DNA is considered to be the leading candidate for epigenetic inheritance. Even though this mechanism is not based on Watson-Crick basepairing, its function is still intimately dependent on the discovery of 1950s. Namely, the “code” is still a part of an extended, modified DNA structure. While certainly methylation is a part of the “whole picture” of epigenetic regulation, such a view might be too genocentric and too limited. It is not unlikely that the code of epigenetic inheritance is dynamics rather than static, distributive rather than localized: It is an emergent property of the whole system rather than depend on a very few number of substances or regulatory mechanisms. The emergent landscape of a mesoscopic chemical reaction system share certain features with the content addressable memory proposed by J.J. Hopfield years ago, but eliminated the technical need for detailed balance \[[@hopfield_84]\]. We would like to point out the fundamental difference of our proposal is a non-genome, pure biochemical based epigenetic mechanism. While DNA methylation is a part of this picture, our proposed mechanism moves the focus away from DNA basepair recognition and memory, and shift it to biochemical networks. The memory in our model, in principle, can be independent of DNA. We understand that immunological diversity and memory has now been shown to be DNA based \[[@tonegawa_83]\]. Still, the digital “quantal” nature of immunity at individual cell level has been noted \[[@smith_06]\]. Also, in the field of neuroplasticity, cellular mechanism for memory has been proposed to be very similar to our PdPC model \[[@wang_xj_05; @cooper_pnas_09]\]. A biochemical based memory is too natural to be completely neglected by evolution. At single cell level, bimodality has been regularly observed in diverse cell types. Recently, for example, [@xie_08] demonstrated in their Fig. 1C bimodal expression level of lactose permease in [E. coli.]{} corresponding to the two states of a bacteria cell, uninduced or induced by lactose analog TMG (methyl-b-D-thiogalactoside). [@zhang] in their Figure 4 reported bimodal Raman spectra as an indicator for DNA fragmentation in apoptosis of DAOY cell line (human brain tumor medulloblatoma), and in Figures 6 and 7, [@xu] showed bimodal FITC-Annexin V protein in apoptosis of U2OS cell line (human osteosarcoma). What is a pathway, and what are cross-talks? -------------------------------------------- The concept of biochemical regulatory pathway is widely accepted in the molecular cellular biology. In general applications, a pathway provides a sequential events of activation/deactivation in terms of the regulatory/signaling proteins. However, from our biochemical reaction system perspective, the state of a cell, as a stochastic attractor, is defined by the states of [all]{} the regulatory/signaling proteins \[[@ge_mbs_08]\]. Hence, from a more rigorous theoretical standpoint, one needs to know, when a protein $Y$ is activated, what are the states of its upper-stream and down-stream proteins, $X$ and $Z$. In this sense, the “pathway view” of cellular signaling is similar to the “structural pathway” view of protein conformational change. While it is useful, it can be mis-leading \[[@cui_06]\]. Almost universally in the discussion of cellular signaling, the concept of “certain pathway leading to certain response” is an established language. But knowing a great deal of “cross-talks” in signaling pathways \[[@zhang_ncb_08; @rama_06]\], this linear thinking is incorrect. In fact, a response is really a changing state of a cell, be it proliferation versus growth arrest, apoptosis versus senescence. So they should not be associated with [only]{} one pathway or another. Rather, they are different states of [entire integrated network]{}. It is true, by “activating certain pathway” while keeping other the same, one might promote a particular state, but that does not imply the response is only associated with that pathway. Now if we take the view these responses are different “states” of a cell, then many pathways are involved. And the most important things about biological complexity is how many these stable states are “available” in the system \[[@wolynes_pnas_02]\]. More states also means more possible transitions between them (whether a transition actually occurs is an issue of time scales), and thus the complexity is associated with multi-stability. This view is consistent with the definition of cellular complexity by possible number of responses to combinatorial stimuli \[[@rama_06]\]. If there is only one state, then the system always goes back to somewhere it starts, then no complexity. The key idea here is not to associate a particular cellular response to a particular pathway; but rather activating particular pathway promotes certain response which is defined as state change. Living matter: Mesoscopic open chemical system as a biochemical machine ----------------------------------------------------------------------- [@Laughlin] have discussed the possible new phenomena at mesoscopic scale which they called the middle way. Others has asked “what is and how is living matter different from the three (or five) known states of matters from physics \[[@cele_07]\]. The study of mesosopic chemical systems offers some insights: 1\) If a chemical system is too large (for example, grinding all the cells into a single tube without the small volume of a cell), then the chemistry is different. It completely loses the possibility of multi-stability at that CEts \[[@qian_prl_09]\], which is a defining feature of complex dynamics \[[@qian_pccp_09]\]. The size of biological cells might indeed be a consequnce of living matters possessing mesoscopic complexity. 2\) This idea is generally understood but its implication has not been widely appreciated: A living matter has to be in continuous exchange of materials, with chemical gradient, with its environment; The driven nature of a living matter is completely different from the classical way of thinking a “matter” which is being in isolation. To define a living matter, one can not completely separate the system from its environment: This is the origin of systems view of holistic biology. 3\) The classical thesis of “irreversibility” of Boltzmann is to understand the spontaneous processes leading to equilibrium. This is a different problem as that suggested by Schrödinger in his “what is life”. The former is to understand the macroscopic irreversibility in a system with Newtonian mechanics, while the latter is a completely different problem; Life is an open system: To a first-order approximation, it is an isothermal system sustained by an active input and output of chemicals with Gibbs free energy difference. The irreversibility of such a system is self-evident. Hence, essential thesis in nonequilibrium physics of living matter is not that of Boltzmann, but more quantitative understanding of how such chemically driven systems give rise to complex “living” behavior such as self-organization and inheritability. 4\) The significance of the bistability arising from the Delbrück-Gillespie is not that it has two different “phenotypical” states for relatively low and relatively high levels of “inducer”, borrowing the language from Lac operon \[[@xie_08]\], but that both states co-exist for an range of intermediate level of inducer! This is reflected by the distribution in this intermediate range being bimodal. Such a realization was emphasized in protein folding by [@lattman_pnas_93]. This realization can have important implications: A pre-cancerous state might already exist “on the other side of the mountain”, which is encoded in our genome \[[@aoping_med_hy]\]. Methods ======= Self-regulating gene and stochastic bistability ----------------------------------------------- We consider the coupled birth-death process for the gene regulatory network in Fig. \[fig1\]b with $\chi=1$. We note that in a macroscopic biochemical experiment the rate of protein synthesis per DNA is $g_i$ $(i=0,1)$. Furthermore, we denote the concentration of the DNA with repressor $x=m/V$ and the concentration of the repressor $y=n/V$. Then $$\begin{aligned} && \frac{dp(m,n)}{dt} \nonumber\\ &=& g(m)p(m,n-1)\ +\ (n+1)kp(m,n+1) \nonumber\\ &+& \frac{h_on}{V}(M-m-1)p(m-1,n) \label{the_cme_4_srg}\\ &-& \left(g(m)+nk+\frac{h_on}{V}(M-m)+mf\right) p(m,n) \nonumber\\ &+& (m+1)fp(m+1,n), \nonumber\end{aligned}$$ where $g(m)=g_0(M-m)+g_1m$, $M=x_tV$ is the copy number of DNA. This model, in the limit of $V\rightarrow\infty$, yields the Mass Action kinetic equation $$\frac{dx}{dt} = h_oy(x_t-x)-fx, \ \ \ \frac{dy}{dt} = g_0(x_t-x) + g_1x - ky. \label{ode_4_srg}$$ Eq. \[ode\_4\_srg\] has two roots but one in the interval $(0,x_t)$. For parameters $g_0=80,g_1=1,k=1,h_o=0.007,f=0.1,k=1.0,x_t=1$, we have the steady state $x^=0.66$ and $y^=27.8$. Using nonadiabatic approximation, we have $p(n\|m)$ being a Poisson distribution with mean $g(m)/k$. Then the 2-d model is reduced to 1-d with birth rates $h_o(M-m)g(m)/(kV)$ and death rate $mf$. The stationary distribution for the 1-d model can be analytically studied following [@qian_bj_10]. Mathematically, one can understand the problem as an eigenvalue perturbation: The Markov operator involved has the block structure \[[@qian_pccp_09]\]: $$\left(\begin{array}{cc} \mathbf{L_0}-\mathbf{h} & \mathbf{f} \\ \mathbf{h} & \mathbf{L_1}-\mathbf{f} \end{array}\right),$$ in which $\mathbf{h}$ and $\mathbf{f}$ are small and treated as a perturbation. The unperturbed operator has a degenerated eigenvalue zero, with eigenvectors on the left $(\mathbf{1},\mathbf{1})$ and $(\mathbf{1},-\mathbf{1})$, and on the right $(\mathbf{p}_0,\mathbf{p}_1)^T$ and $(\mathbf{p}_0,-\mathbf{p}_1)^T$. $\mathbf{p}_0$ and $\mathbf{p}_1$ are Poisson distributions with mean $\overline{n}_0 =g_0/k$ and $\overline{n}_1=g_1/k$, which are the stationary distributions for the nonperturbed problem. Now for the perturbed system, it is easy to verify that zero is still an eigenvalue $\lambda_0=0$; however, there is also a nonzero, smallest eigenvalue $$\lambda_1 = \frac{\left(\begin{array}{cc} \mathbf{1},-\mathbf{1} \end{array}\right)}{2} \left(\begin{array}{cc} -\mathbf{h} & \mathbf{f} \\ \mathbf{h} & -\mathbf{f} \end{array}\right) \left(\begin{array}{c} \mathbf{p}_0 \\ -\mathbf{p}_1 \end{array}\right) = -(h_o\overline{n}_0+f).$$ Furthermore, the approximated right eigenvectors associated with $\lambda_0$ and $\lambda_1$ are precisely $$\left(\frac{f}{h_o\overline{n}_0+f} \mathbf{p}_0,\frac{h_o\overline{n}_0}{h_o\overline{n}_0+f} \mathbf{p}_1\right) \ \ \textrm{ and } \ \ (\mathbf{p}_0,-\mathbf{p}_1).$$ A more accurate approximations for the two eigenvectors can be obtained, if needed, by carrying through the first-order perturbation calculations \[[@shi_qian_11]\]. The eigenvectors for the $\lambda_1$ has a [nodal decomposition]{} which yields two connected domains with positive and negative values.[^2] This provides a rigorous mathematical definition for the two states of the system. Since all the other eigenvalues $\lambda_i\gg\lambda_1$ $(i\ge 2)$, the dynamics within each of the two domains are on a different time scale, and fast equilibrated. The remaining slow dynamics corresponds precisely to a two state system \[[@qian_jrsi_09]\] with transition rates $h$ and $f$. The nonadiabaticity plays a decisive role in this problem.[^3] Chemical master equation and landscape $\phi(x)$ ------------------------------------------------ The PdPC with feedback in Eq. \[fig3a\] is the same as that in Fig. \[fig1\]a with $k_0=0$. With rapid binding $K+2E^\rightleftharpoons K^{\dagger}$ and assuming $\frac{k_{-2}}{k_2} \gg [E^]^2$, we have the following autocatalytic, nonlinear chemical reaction system: $$E+2E^ \overset{\alpha}{\underset{\epsilon}{\rightleftharpoons}} 3E^, \ \ E^ \overset{\beta}{\underset{\delta}{\rightleftharpoons}} E, \label{pdpcwfb_2}$$ in which $\alpha = \frac{k_1k_2}{k_{-2}}[K][ATP]$, $\beta =k_3[P]$, $\epsilon=\frac{k_{-1}k_2}{k_{-2}}[K][ADP]$, and $\delta=k_{-3}[P][Pi]$, where $[K]$ and $[P]$ are the concentrations of the kinase and the phosphatase. One can see that this kinetic model is intimately related to the Schlögl model which has been the prototype of nonlinear chemical bistability \[[@qian_book; @qian_jrsi_09]\]. Let $x$ be the concentration of $E^$, then the macroscopic kinetic equation for (\[pdpcwfb\_2\]) in terms of the Law of Mass Action is $$\frac{dx}{dt} =\alpha x^2(x_t-x)-\beta x -\epsilon x^3 +\delta (x_t-x), \label{the_ode}$$ where $x_t$ is the total concentration of the $E$ and $E^$, assumed to be constant in the system. The dynamic has three positive fixed points. For most biochemical applications $\epsilon\ll\alpha$ and $\delta\ll\beta$. Hence, one can approximately have the three steady states $$x_1^* = \frac{\delta x_t}{\beta},\ \ x_2^* = \frac{x_t+\sqrt{x_t^2-4\beta/\alpha}}{2},$$ and $$x_3^* = \frac{x_t-\sqrt{x_t^2-4\beta/\alpha}}{2}. \label{fixed_pt}$$ Hence, when $\alpha x_t^2 >4\beta$, there is bistability. $x_1^$ and $x_2^$ are stable steady states, and $x_3^$ is unstable. The basins of attraction for $x_1^$ and $x_2^$ are $[0,x_3^)$ and $(x_3,\infty)$, respectively. The corresponding linear relaxation rate for the steady state $x_i^$, $(i=1,2)$, is $r_i = 3\alpha\left(x^_i\right)^2-2\alpha x_t x_i^+\beta$. That is, $$r_1 =\beta \ \textrm{ and } \ r_2 =\alpha x_2^(2x_2^-x_t). \label{rs}$$ The chemical master equation for the system in Eq. \[pdpcwfb\_2\] is $$\begin{aligned} && \frac{dp(n)}{dt} = \nonumber\\ && \left(\frac{\alpha}{V^2}(n-1)(n-2)+\delta\right)(n_t-n+1)p(n-1) \nonumber\\ &+& \left(\beta+\frac{\epsilon}{V^2}n(n-1) \right)(n+1)p(n+1) \nonumber\\ &-& \left[\left(\frac{\alpha}{V^2}n(n-1)+\delta \right)(n_t-n) \right. \nonumber\\ && \left. +\left(\beta+\frac{\epsilon}{V^2}(n-1)(n-2) \right)n\right]p(n),\end{aligned}$$ where $n_t$ is the total number of $E$ and $E^$ molecules, and $V$ is the volume of the mesoscopic system. The steady state probability distribution for the number of $E^$ is $$p^{ss}(n) = C\prod_{k=2}^n \frac{[\alpha k(k-1)+\delta V^2](n_t-k)} {\left[\beta V^2+\epsilon k(k-1)\right](k+1)},$$ where $C$ is a normalization factor. If $V$ and $n_t$ are large, but their ratio is hold constant, then one can develop an approximated formula for the probability density function for the concentration $x=n/V$: $$f^{ss}(x) \propto e^{-V\phi(x)},$$ where the $\phi(x)$ $$\phi(x) = x_t\ln (x_t-x)-x\ln\left[\frac{(\alpha x^2+\delta)(x_t-x)} {(\beta+\epsilon x^2) x}\right] \label{phi_x}$$ $$- 2\sqrt{\frac{\delta}{\alpha}}\arctan \left(\sqrt{\frac{\alpha}{\delta}}x\right) +2\sqrt{\frac{\beta}{\epsilon}} \arctan \left(\sqrt{\frac{\epsilon}{\beta}}x\right).$$ One can check the extrema of $\phi(x)$ by setting its derivative being zero: $$\frac{d\phi(x)}{dx} = -\ln\frac{(\alpha x^2+\delta)(x_t-x)} {(\beta+\epsilon x^2)x} = 0.$$ We see that the extrema of $\phi(x)$ are precisely the roots of Eq. \[the\_ode\], $x_1^,x_2^$ and $x_3^$: The ratio in the logarithm being 1 corresponds to the right-hand-side of Eq. \[the\_ode\] being 0. The switching time ------------------ The mean time for switching from $x_1^$ attractor to $x_2^$ attractor can be analytically computed for 1-d model. For discrete case, this formulae has been widely used in the various lattice hopping models with birth-death processes [@vanKampen]: $$T_{1\rightarrow 2} = \sum_{n=0}^{n_1^}p^{ss}(n) \sum_{m=n_1^+1}^{n_2}\frac{1}{w_mp^{ss}(m)}$$ $$+\sum_{n=n_1^+1}^{n_2^-1}p^{ss}(n) \sum_{m=n+1}^{n_2^}\frac{1}{w_mp^{ss}(m)}. \label{T12_exact}$$ where $n_i^=[x_i^V]$, and $$u_n = \left(\frac{\alpha}{V^2}(n-1)(n-2)+\delta\right)(n_t-n+1),$$ $$w_n = \left(\beta+\frac{\epsilon}{V^2}(n-1)(n-2)\right)n.$$ When $V\rightarrow\infty$, $w_m\sim w\left(\frac{m}{V}\right)V$ where $w(x)=\beta x+\epsilon x^3$, and $p^{ss}(n)\sim (C/V)e^{-V\phi\left(\frac{n}{V}\right)}$, we have (see Appendix) $$T_{1\rightarrow 2} \approx \frac{2\pi e^{V(\phi(x_3^)-\phi(x_1^))}} {w(x_3^)\sqrt{-\phi^{''}(x_1^)\phi^{''}(x_3^)}}. \label{T12}$$ and similarly, $$T_{2\rightarrow 1}\approx \frac{2\pi e^{V(\phi(x_3^)-\phi(x_2^))}} {w(x_3^)\sqrt{-\phi^{''}(x_2^)\phi^{''}(x_3^)}}. \label{T21}$$ This result, as expected, is very similar to Kramers formula for overdamped barrier crossing in the energy landscape $\phi(x)$ with “frictional coefficient” being $1/w(x_3^)$. Note that at $x_3^$, $w(x_3^)=u(x_3^)$; hence a symmetric expression for the “fractional coefficient” is $2/(w(x_3^)+u(x_3^))$. Acknowledgements ================ We thank Peter Wolynes and X. Sunney Xie for many helpful discussions. HQ was partially supported by NSF grant EF0827592 (PI: Dr. H. Sauro). [00]{} Abad-Zapatero, C.* (2007) Notes of a protein crystallographer: [quo vadis]{} structural biology. [Acta Cryst. D]{}, [63]{}, 660–664. Ao, P.; Galas, D.; Hood, L.; Zhu, X.-M. (2007) Cancer as robust intrinsic state of endogenous molecular-cellular network shaped by evolution. [Med. Hypotheses]{}, [70]{}, 678–684. Arkin, A.P.; Ross, J.; McAdams, H.H. (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected [[E]{}scherichia coli]{} cells. [Genetics]{}, [149]{}, 1633–1648. Bai, C.-L.; Wang, C.; Xie, X.S.; Wolynes, P.G. (1999) Single molecule physics and chemistry. [Proc. Natl. Acad. Sci. USA]{}, [96]{}, 11075–11076. Beard, D.A.; Qian, H. (2008) [Chemical Biophysics: Quantitative Analysis of Cellular Systems.]{} Cambridge Texts in Biomedical Engineering, Cambridge Univ. Press. Bishop, L.M.; Qian, H. (2010) Stochastic bistability and bifurcation in a mesoscopic signaling system with autocatalytic kinase. [Biophys. J.]{}, [98]{}, 1–11. Buckmaster, R.; Asphahani, F.; Thein, M.; Xu, J.; Zhang, M.-Q. (2009) Detection of drug-induced cellular changes using confocal Raman spectroscopy on patterned single-cell biosensors. [Analyst]{}, [134]{}, 1440–1446. Carneiro, M.; Hartl, D.L. (2010) Adaptive landscapes and protein evolution. [Proc. Natl. Acad. Sci. USA]{}, [107]{}, 1747–1751. Castellani, G.C.; Bazzani, A.; Cooper, L.N. (2009) Toward a microscopic model of bidirectional synaptic plasticity. [Proc. Natl. Acad. Sci. USA]{}, [106]{}, 14091–14095. Choi, P.J.; Cai, L.; Frieda, K.; Xie, X.S. (2008) A stochastic single-molecule event triggers phenotype switching of a bacterial cell. [Science]{}, [322]{}, 442–446. Cooper, J.A.; Qian, H. (2008) A mechanism for SRC kinase-dependent signaling by noncatalytic receptors. [Biochemistry]{}, [47]{}, 5681–5688. Das, J.; Ho, M.; Zikherman, J.; Govern, C.; Yang, M.; Weiss, A.; Chakraborty, A.K.; Roose, J.P. (2009) Digital signaling and hysteresis characterize Ras activation in lymphoid cells. [Cell]{}, 136:337–351. Delbrück, M. (1940) Statistical fluctuations in autocatalytic reactions. [J. Chem. Phys.]{}, [8]{}, 120–124. Dodd, I.B.; Micheelsen, M.A.; Sneppen, K.; Thon, G. (2007) Theoretical analysis of epigenetic cell memory by nucleosome modification. [Cell]{}, [129]{}, 813–822. Ferrell, J.E.; Machleder, E.M. (1998) The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes. [Science]{}, [280]{}, 895–898. Ferrell, J.E.; Xiong, W. (2001) Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible. [Chaos]{}, [11]{}, 227–236. Formaneck, M.S.; Ma, L.; Cui, Q. (2006) Reconciling the “old” and “new” views of protein allostery: A molecular simulation study of Che Y. [Prot. Struct. Funct. Bioinf.]{}, [63]{}, 846–867. Frauenfelder, H.; Sliger, S.; Wolynes, P.G. (1991) The energy landscapes and motions of proteins. [Science]{}, [254]{}, 1598–1603. Ge, H.; Qian, H. (2009) Thermodynamic limit of a nonequilibrium steady-state: Maxwell-type construction for a bistable biochemical system. [Phys. Rev. Lett.]{}, [103]{}, 148103. Ge, H.; Qian, H. (2011) Nonequilibrium phase transition in mesoscoipic biochemical systems: From stochastic to nonlinear dynamics and beyond. [J. R. Soc. Interf.]{}, [8]{}, 107–116. Ge, H.; Qian, H. (2012) Asymptotic limit of a singularly perturbed stationary diffusion equation: The case of a limit cycle. http://arxiv.org/abs/1011.4049. Ge, H.; Qian, H.; Qian, M. (2008) Synchronized dynamics and nonequilibrium steady states in a stochastic yeast cell-cycle network. [Math. Biosci.]{}, [211]{}, 132–152. Ge, H.; Qian, M.; Qian, H. (2012) Stochastic theory of nonequilibrium steady states (Part II): Applications in chemical biophysics. [Phys. Rep.]{}, [510]{}, 87–118. Ghaemmaghami, S.; Huh, W.-K.; Bower, K.; Howson, R.W.; Belle, A.; Dephoure, N.; O’Shea, E.K.; Weissman, J.S. (2003) Global analysis of protein expression in yeast. [Nature]{}, [425]{}, 737–741. Gillespie, D.T. (2007) Stochastic simulation of chemical kinetics. [Ann. Rev. Phys. Chem.]{} [58]{}, 35–55. Hopfield, J.J. (1984) Neurons with graded response have collective computational properties like those of two state neurons. [Proc. Natl. Acad. Sci. USA]{}, [81]{}, 3088–3092. Hornos, J.E.M.; Schultz, D.; Innocentini, G.C.P.; Wang, J.; Walczak, A.M.; Onuchic, J.N.; Wolynes, P.G. (2005) Self-regulating gene: An exact solution. [Phys. Rev. E.]{}, [72]{}, 051907. Hu, G. (1986) Lyapounov function and stationary probability distributions. [Zeit. Phys. B]{}, [65]{}, 103–106. Jones, P.A.; Takai, D. (2001) The Role of DNA methylation in mammalian epigenetics. [Science]{}, [293]{}, 1068–1070. Kepler, T.B.; Elston, T.C. (2001) Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations. [Biophys. J.]{}, [81]{}, 3116–3136. Kussell, E.; Kishony, R.; Balaban, N.Q; Leibler, S. (2005): Bacterial persistence: a model of survival in changing environments. [Genetics]{}, [169]{}, 1804–1807. Lattman, E.E.; Rose, G.D. (1993) Protein folding — what’s the question? [Proc. Natl. Acad. Sci. USA]{}, [90]{}, 439–441. Laughlin, R.B.; Pines, D.; Schmalian, J.; Stojković, B.P.; Wolynes, P.G. (2000) [Proc. Natl. Acad. Sci. USA]{}, [97]{}, 32–37. Lee, N.; Zhang, Y. (2008) Chemical answers to epigenetic crosstalk. [Nature Cell Biol.]{}, [4]{}, 335–337. Liang, J.; Qian, H. (2010) Computational cellular dynamics based on the chemical master equation: A challenge for understanding complexity. [J. Comput. Sci. Tech.]{}, [25]{}, 154–168. Martin, C.; Zhang, Y. (2007) Mechanisms of epigenetic inheritance. [Curr. Opin. Cell Bio.]{}, [19]{}, 266–272. Miller, P.; Zhabotinsky, A.; Lisman, J.; Wang, X.-J. (2005). The stability of a stochastic CaMKII switch: Dependence on the number of molecules and protein turnover. [PLoS Biol.]{}, [3]{}, 705–717. Natarajan, M.; Lin, K.-M.; Hsueh, R.C.; Sternweis, P.C.; Ranganathan, R. (2006) A global analysis of cross-talk in a mammalian cellular signalling network. [Nature Cell Biol.]{}, [8]{}, 571–580. Ptashne, M. (2007) On the use of the word ‘epigenetic’. [Curr. Biol.]{}, [17]{}, R233–R236. Qian, H. (2006) Open-system nonequilibrium steady-state: Statistical thermodynamics, fluctuations and chemical oscillations. [J. Phys. Chem. B]{}, [110]{}, 15063–15074. Qian, H. (2007) Phosphorylation energy hypothesis: Open chemical systems and their biological functions. [Ann. Rev. Phys. Chem.]{}, [58]{}, 113–142. Qian, H. (2010) Cellular biology in terms of stochastic nonlinear biochemical dynamics: Emergent properties, isogenetic variations and chemical system inheritability. [J. Stat. Phys.]{}, [141]{}, 990–1013. Qian, H. (2011) Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reactions systems - An analytical theory. [Nonlinearity]{}, [24]{}, R19–R49. Qian, H.; Qian, M. (2000) Pumped biochemical reactions, nonequilibrium circulation, and stochastic resonance. [Phys. Rev. Lett.]{}, [84]{}, 2271–2274. Qian, H.; Reluga, T.C. (2005) Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch. [Phys. Rev. Lett.]{}, [94]{}, 028101. Qian, H.; Bishop, L.M. (2010) The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: Linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks (Review). [Int. J. Mol. Sci.]{}, [11]{}, 3472–3500. Qian, H.; Shi, P.-Z.; Xing, J. (2009) Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity. [Phys. Chem. Chem. Phys.]{} [11]{}, 4861–4870. Samoilov, M.S.; Arkin, A.P. (2006) Deviant effects in molecular reaction pathways. [Nat. Biotechnol.]{}, [24]{}, 1235–1240. Samoilov, M.; Plyasunov, S.; Arkin, A.P. (2005) Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. [Proc. Natl. Acad. Sci. USA]{}, [102]{}, 2310–2315. Sasai, M.; Wolynes, P.G. (2002) Stochastic gene expression as a many-body problem. [Proc. Natl. Acad. Sci. USA]{}, [100]{}, 2374–2379. Shi, P.-Z.; Qian, H. (2011) A perturbation analysis of rate theory of self-regulating genes and signaling networks. [J. Chem. Phys.]{}, [134]{}, 065104. Smith, K.A. (2006) The quantal theory of immunity. [Cell Res.]{}, [16]{}, 11–19. Spudich, J.L.; Koshland, D.E. (1976) Non-genetic individuality: chance in the single cell. [Nature]{}, [262]{}, 467–471. Tonegawa, S. (1983) Somatic generation of antibody diversity. [Nature]{}, [302]{}, 575–581. Turcotte, M.; Garcia-Ojalvo, J.; Süel, G.M. (2008) A genetic timer through noise-induced stabilization of an unstable state. [Proc. Natl. Acad. Sci. USA]{}, [105]{}, 15732–15737. Turner, B.M. (2000) Histone acetylation and an epigenetic code. [Bioessays]{}, [22]{}, 836–45. van Kampen, N.G. (1992) [Stochastic Processes in Physics and Chemistry]{}, Elsevier Science, North-Holland. Vellela, M.; Qian, H. (2009) Stochastic dynamics and nonequilibrium thermodynamics of a bistable chemical system: The Schlögl model revisited. [J. R. Soc. Interf.]{}, [6]{}, 925–940. von Hippel, P.H. (2007) From ‘simple’ protein-DNA interactions to the macromolecular machines of gene expression. [Ann. Rev. Biophys. Biomol. Struct.]{}, [36]{}, 79–105. Walczak, A.M.; Onuchic, J.N.; Wolynes, P.G. (2005) Absolute rate theories of epigenetic stability. [Proc. Natl. Acad. Sci. USA]{}, [102]{}, 18926–18931. Wang, J.; Xu, L.; Wang, E. (2008) Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations. [Proc. Natl. Acad. Sci. USA]{}, [105]{}, 12271–12276. Wilkinson, D.J. (2006) [Stochastic Modeling for Systems Biology.]{} New York: Chapman & Hall/CRC. Wolynes, P.G. (2009) Gene regulation: Single molecule chemical physics in a natural context. In [Single Molecule Spectroscopy in Chemistry, Physics and Biology (Nobel Symposium)]{}, Springer Series in Chemical Physics, vol. 96, A. Gräslund, R. Rigler and J. Widengren, eds. Springer, pp. 553–560. Xu, L.; Chen, Y.; Song, Q.; Wang, Y.; Ma, D. (2009) PDCD4 interacts with Tip60 and functions as a cooperator in acetyltransferase activity and DNA damage-induced apoptosis. [Neoplasia]{}, [11]{}, 345–354. Zhu, H.; Qian, H.; Li, G.-P. (2010) Delayed onset of positive feedback activation of Rab5 by Rabex-5 and Rabaptin-5 in endocytosis. [PLoS ONE]{}, [5]{}, e9226. Zhu, X.-M.; Yin, L.; Hood, L.; Ao, P. (2004) Robustness, stability and efficiency of phage lambda genetic switch: Dynamical structure analysis. [J. Bioinf. Compt. Biol.]{}, [2]{}, 785–817. Mean Switching Time for 1-D CME =============================== The mean time for switching from $x_1^$ attractor to $x_2^$ attractor can be analytically computed. For discrete case, according to [@vanKampen], let $$u_n = \left(\frac{\alpha}{V^2}(n-1)(n-2)+\delta\right)(n_t-n+1),$$ $$w_n = \left(\beta+\frac{\epsilon}{V^2}(n-1)(n-2)\right)n,$$ we have $$T_{1\rightarrow 2} = \sum_{n=0}^{x_1^V}p^{ss}(n) \sum_{m=x_1^V+1}^{x_2^V}\frac{1}{w_mp^{ss}(m)}$$ $$+\sum_{n=x_1^V+1}^{x_2^V-1}p^{ss}(n) \sum_{m=n+1}^{x_2^V}\frac{1}{w_mp^{ss}(m)}.$$ $w_m\approx w\left(\frac{m}{V}\right)V$, where $w(x)=\beta x+\epsilon x^3$, and $p^{ss}(n)\approx (C/V)e^{-V\phi\left(\frac{n}{V}\right)}$, we have $$\begin{aligned} T_{1\rightarrow 2} &\approx& \sum_{n=0}^{x_1^V}\frac{e^{-V\phi(\frac{n}{V})}}{V} \sum_{m=x_1^V+1}^{x_2^V}\frac{1}{w(\frac{m}{V}) e^{-V\phi(\frac{m}{V})}} \nonumber\\ && +\sum_{n=x_1^V+1}^{x_2^V}\frac{e^{-V\phi(\frac{n}{V})}}{V}\sum_{m=n+1}^{x_2^V}\frac{1}{w(\frac{m}{V})e^{-V\phi(\frac{m}{V})}} \nonumber\\ &\approx&\sum_{n=0}^{x_1^V}e^{-V\phi(\frac{n}{V})}\int_{x_1^}^{x_2^}\frac{1}{w(x)e^{-V\phi(x)}}dx \nonumber\\ && +\sum_{n=x_1^V+1}^{x_2^V-1}e^{-V\phi(\frac{n}{V})}\int_{\frac{n}{V}}^{x_2^}\frac{1}{w(x)e^{-V\phi(x)}}dx \nonumber\\ &\approx&V\int_{0}^{x_1^}e^{-V\phi(x)}dx\int_{x_1^}^{x_2^}\frac{1}{w(x)e^{-V\phi(x)}}dx \nonumber\\ &&+V\int_{x_1^}^{x_2^}e^{-V\phi(y)}\int_{y}^{x_2^}\frac{1}{w(x)e^{-V\phi(x)}}dxdy. \nonumber\end{aligned}$$ Applying Laplace’s method, one has $$\int_{0}^{x_1^}e^{-V\phi(x)}dx\approx\frac{\sqrt{2\pi} e^{-V\phi(x_1^)}}{2\sqrt{V\phi^{''}(x_1^)}},$$ $$\int_{x_1^}^{x_2^}\frac{1}{w(x)e^{-V\phi(x)}}dx \approx\frac{{\frac{\sqrt{2\pi}}{w(x_3^)e^{-V\phi(x_3^)}}}} {\sqrt{-V\phi^{''}(x_3^)}},$$ $$\int_{y}^{x_2^}\frac{1}{w(x)e^{-V\phi(x)}}dx \approx \left\{\begin{array}{ll} \frac{{\frac{\sqrt{2\pi}}{w(x_3^)e^{-V\phi(x_3^)}}}}{\sqrt{-V\phi^{''}(x_3^)}} & x_1^\leq y\leq x_3^, \\[8pt] -\frac{\frac{1}{w(y)e^{-V\phi(y)}}}{V\phi^{'}(y)} & x_3^\leq y\leq x_2^. \end{array}\right.$$ Then $$T_{1\rightarrow 2} \approx \frac{\pi e^{V(\phi(x_3^)-\phi(x_1^))}}{w(x_3^) \sqrt{-\phi^{''}(x_1^)\phi^{''}(x_3^)}}$$ $$+V\int_{x_1^}^{x_3^}e^{-V\phi(y)}dy\frac{{\frac{\sqrt{2\pi}}{w(x_3^)e^{-V\phi(x_3^)}}}}{\sqrt{-V\phi^{''}(x_3^)}}$$ $$+V\int_{x_3^}^{x_2^}e^{-V\phi(y)}-\frac{\frac{1}{w(y)e^{-V\phi(y)}}}{V\phi^{'}(y)}dy.$$ Furthermore, $$\int_{x_1^}^{x_3^}e^{-V\phi(y)}dy\approx \frac{\sqrt{2\pi}e^{-V\phi(x_1^)}}{2\sqrt{V\phi^{''}(x_1^)}},$$ hence $$T_{1\rightarrow 2}\approx \frac{2\pi e^{V(\phi(x_3^)-\phi(x_1^))}}{w(x_3^)\sqrt{-\phi^{''}(x_1^)\phi^{''}(x_3^)}} -\int_{x_3^}^{x_2^}\frac{1}{w(y)\phi^{'}(y)}dy,\nonumber$$ in which the second term is comparable to $\int_{x_3^}^{x_2^}\frac{1}{u(y)-w(y)}dy$ which is the time relaxing from the unstable fixed point $x_3^$ to $x_2^$. It should be neglectable when the noise is present. Finally, when $V\rightarrow\infty$, we have the switching time in terms of $\phi(x)$: $$T_{1\rightarrow 2}\approx \frac{2\pi e^{V(\phi(x_3^)-\phi(x_1^))}}{w(x_3^)\sqrt{-\phi^{''}(x_1^)\phi^{''}(x_3^)}}, \label{time_1_to_2}$$ and $$T_{2\rightarrow 1}\approx \frac{2\pi e^{V(\phi(x_3^)-\phi(x_2^))}}{w(x_3^)\sqrt{-\phi^{''}(x_2^)\phi^{''}(x_3^*)}}.$$ Comparison with Kramers’ theory {#comparison-with-kramers-theory .unnumbered} ------------------------------- The Kramers theory considers a diffusing particle that obeys the Fokker-Planck equation $$\frac{\partial p(x,t)}{\partial t} = \frac{1}{\eta} \left( \frac{\partial}{\partial x} [U^{'}(x)p(x,t)] +k_BT\frac{\partial^2}{\partial x^2} p(x,t) \right).$$ For energy function $U(x)$ with two energy wells, say well $a$ and well $b$, he derived [@vanKampen] the rate constant for transition from $a$ to $c$ by crossing barrier $c$, known as the celebrated Kramers’ formula: $$k_{a\rightarrow b} = \frac{\sqrt{\mu_a\mu_c}}{2\pi\eta} e^{(U_a-U_c)/k_BT},$$ where $\mu_a$ and $\mu_c$ are the curvatures of the energy function $U(x)$ at $a$, the well, and $c$, the transition state. Similarly one has $k_{b\rightarrow a}$. Then their ratio, which relates the populations in the two wells at equilibrium: $$\frac{k_{a\rightarrow b}}{k_{b\rightarrow a}} = \sqrt{\frac{\mu_a}{\mu_b}} e^{-\frac{U_b-U_a}{k_BT}} = e^{-\frac{F_b-F_a}{k_BT}},$$ where $F_a = U_a+k_BT\ln\sqrt{\mu_a}$ is the free energy of energy well $a$. Comparison with diffusion approximation to CME {#comparison-with-diffusion-approximation-to-cme .unnumbered} ---------------------------------------------- The result in Eq. \[time\_1\_to\_2\], as far as we know, is new. In the past, analysis of barrier crossing in the CME, in the limit of large $V$, have been based on diffusion approximation to the CME. In that approach, one derives a one-dimensional Fokker-Planck equation for large $V$: $$\frac{\partial p(x,t)}{\partial t} =-\frac{\partial}{\partial x} (u(x)-w(x))p(x,t)$$ $$+\frac{\partial^2}{\partial x^2} \left(\frac{u(x)+w(x)}{2V}\right)p(x,t).$$ Here, it suggests a potential $$\psi(x)=-2\int^x \frac{u(x)-w(x)}{u(x)+w(x)}\ dx,$$ which is different from our $$\phi(x)=-\int^x \log\frac{u(x)}{w(x)}\ dx.$$ $\phi(x)$ is the correct one while $\psi(x)$ could give incorrect result for the $T_{1\rightarrow 2}$ and $T_{2\rightarrow 1}$. See \[[@qian_jrsi_09]\] for an extensive discussion. Transition Time in Nonadiabatic Gene Switching ============================================== We consider the coupled birth-death process with probability distribution $\left[p_0(n),p_1(n)\right]$, where $0$ and $1$ represent the gene states, with and without bound transcription factor, and $n$ represent the copy number of the transcription factor which is the gene product. The distribution satisfies the chemical master equation $$\begin{aligned} \frac{dp_0(n)}{dt} &=& g_0p_0(n-1)-\left(g_0+nk\right)p_0(n) \\ &+& (n+1)kp_0(n+1)- h(n) p_0(n) + fp_1(n) \\ \frac{dp_1(n)}{dt} &=& g_1p_0(n-1)-\left(g_1+nk\right)p_0(n) \\ &+& (n+1)kp_0(n+1)+ h(n) p_0(n) - fp_1(n).\end{aligned}$$ We shall consider $h(n)=hn(n-1)/2$. If the gene switching between $0$ and $1$ is slow, i.e., it is nonadiabatic, then we have the rapid pre-steady states (conditional probability) in the state $i$ ($=0,1$) following their respective Poisson distribution $$p(n\|i) = \frac{1}{n!}\left(\frac{g_i}{k}\right)^n e^{-g_i/k}.$$ Then the mean transition rate from state $0$ to $1$, and $1$ to $0$, are $$k_{0\rightarrow 1} = \sum_{n=0}^{\infty}h(n) p(n\|0) = \frac{hg_0^2}{2k^2}, \ \ \ k_{1\rightarrow 0} = \sum_{n=0}^{\infty} f p(n\|1) = f. \label{abs_nonadia}$$ Perturbation method for eigenvalue problem {#perturbation-method-for-eigenvalue-problem .unnumbered} ------------------------------------------ We can also solve the eigenvalue problem using the method of perturbation theory. Note that the unperturbed system, i.e., when $h=f=0$, has degeneracy. For the perturbed system, we still have an eigenvalue 0, corresponding to the stationary distribution $[p(n\|0),p(n\|1)]^T$. The other eigenvector on the left is $[1,-1]$ and on the right is $[p(n\|0),-p(n\|1)]^T$, with the corresponding nonzero eigenvalue: $$\lambda_1 = -f -\frac{hg_0^2}{2k^2}, \label{lambda_1}$$ which is exactly the $k_{0\rightarrow 1}+k_{1\rightarrow 0}$ in Eq. \[abs\_nonadia\]. Expressing the $\lambda_1$ in terms of the nondimensionalized parameters in \[[@wolynes_pnas_05; @shi_qian_11]\], we have $$-\frac{\lambda_1}{k} =\frac{h}{2k}\left(n_N^{\dag}\right)^2+ \frac{hg_1^2}{2k^3}\left(\frac{g_0}{g_1}\right)^2$$ $$= \kappa \left[\left(\frac{k}{g_1}n_N^{\dag}\right)^2 +\left(\frac{g_0}{g_1}\right)^2\right] = 0.29\kappa.$$ The stochastic separatrix is well defined in this case by the domains of positive and negative values of the eigenvector associated with eigenvalue $\lambda_1$ \[[@qian_jrsi_09]\]. Nonadiabatic reduction to 1-d with $M$ copies of DNA {#nonadiabatic-reduction-to-1-d-with-m-copies-of-dna .unnumbered} ---------------------------------------------------- The 1-d model has the transition rates: $$m \rightarrow m+1: \ \ \frac{h_o(g_0(M-m)+g_1m)}{Vk}(M-m),$$ $$m+1 \rightarrow m: \ \ f(m+1).$$ If we let $x=m/V$ then we have $$(g_0-g_1)x^2+\left(g_1x_t-2g_0x_t-\frac{kf}{h_o}\right)x +g_0x_t^2 = 0.$$ This should be compared with the quadratic equation for the model in [@qian_bj_10]: $$(k_1+k_{-1})x^2-(k_1x_t-k_2-k_{-2})x-k_{-2}x_t = 0.$$ [^1]: The landscape representation in the nonadiabatic analysis is valid quantitatively: The transition rates between any two states $a$ and $b$, $k_{ab}$ and $k_{ba}$, satisfy $k_{ab}/k_{ba}=p_b^{ss}/p_a^{ss}$ where $p^{ss}_x$ is the probability of state $x$. Let’s assume the shape of each peak is approximately Gaussian. Then the logarithms of the peak values are $\ln(p_a^{ss}/\sqrt{2\pi}\sigma_a)$ and $\ln(p_b^{ss}/\sqrt{2\pi}\sigma_b)$, where $\sigma$’s are the variances of the Gaussian distributions. Thus the “energy difference” between the two wells $E_b-E_a=-\ln(k_{ab}/\sigma_b)+\ln(k_{ba}/\sigma_a)$, or $\ln(k_{ab}/k_{ba})=-(E_b-\ln\sigma_b)+(E_a-\ln\sigma_a)$. The right-hand-side are the “free energy” difference of the two wells. [^2]: The mathematical theory for the bistability proceeds with the following argument: If one treats the $\lambda_1$ as a small parameter, then for system with $\lambda_1=0$, the stochastic dynamics can be reduced to two independent subsystems, each with a unique stationary distribution. Therefore, with the error on the order of $\lambda_1$, both eigenvectors for the $\lambda_0=0$ and $\lambda_1\neq 0$ are linear combinations of the two stationary distributions defined on the subspaces. This explains the origin of the two-state behavior on the slow time scale (CEts) [@qian_jrsi_09]. [^3]: The problem appears very similar to the quantum mechanical eigenvalue perturbation with degeneracy. However, it is worth pointing out that in quantum mechanics one computes the eigenvalues which serves as the “energy landscape” in Heitler-London theory; here we compute the eigenvectors as the “energy landscape”. This distinction has been noted by [@qian_pccp_09].
--- abstract: 'We use Green’s transference principle to show that any subset of the $d$th powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in $d^2+1$ or more variables, with explicit quantitative bounds.' address: 'School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, United Kingdom' author: - Sam Chow title: 'Roth–Waring–Goldbach' --- [^1] Introduction {#intro} ============ Classical methods address problems of type Waring, Goldbach, Roth and Waring–Goldbach. In 2005, Green [@Gre2005] famously solved a problem of Roth–Goldbach type for three primes. Using Bohr sets, he was able to transfer Roth-type results from the integers to the primes. Recently Browning and Prendiville [@BP2016] have shown Green’s transference method to be versatile, establishing a theorem of Roth–Waring type for five squares. They were able to transfer results from the integers to the squares. We present some results of type Roth–Waring–Goldbach. Let $c_1, \ldots, c_s$ be nonzero integers such that $$\label{zerosum} c_1 + \ldots + c_s = 0.$$ Let $K$ be a union of $k$ proper subspaces of the rational hyperplane $$\label{LinearEquation} c_1 x_1 + \ldots + c_s x_s = 0,$$ each of which contains the diagonal $$\label{diagonal} \{(x,\ldots,x): x \in \bQ\}.$$ Let $d \ge 2$ be an integer, and let $A$ be a set of primes in $[X] := \{1,2,\ldots,X\}$ such that the only solutions $\bx \in A^s$ to $$\label{ActualEquation} c_1 x_1^d + \ldots + c_s x_s^d = 0$$ have $(x_1^d, \ldots, x_s^d) \in K$. \[thm1\] Assume $s \ge C(d)$, where $C(2) = 5$, $C(3) = 9$, $C(4) = 15$ and $$C(d)= d^2 + 1 \qquad (d \ge 5).$$ Then $$\label{DensityBound} \|A\| \ll_{\bc, k, \eps} \frac X{\log X} (\log \log \log \log X)^{(2-s)/d+\eps}.$$ Loosely, this says that any subset of relative density $(\log \log \log \log)^{(2-s)/d+\eps}$ within the primes contains nontrivial solutions to . One could choose $$K = \bigcup_{i \ne j} \{ \bx \in \bQ^s: x_i = x_j, \: \bc \cdot \bx = 0 \},$$ for instance, which was the choice of Keil [@Kei2014] and Henriot [@Hen2015] in their work on diagonal quadrics in dense variables. Thus, any positive density subset of the primes contains a solution to with pairwise distinct coordinates. The reason for having a notion of trivial solutions is that, by , the diagonal lies within the solution set of . We shall use Green’s transference technology [@Gre2005] to transfer Roth-type results from the integers to the set of $d$th powers of primes. The protagonist shall be a measure $\nu$ on some interval $[N]$, where $N$ can be thought of as $X^d$. Morally $\nu(n)$ should be $dp^{d-1} \log p$, if $n = p^d$ for some prime $p \le X$, and zero otherwise. The measure $\nu$ has become known as a majorant; a majorant on $[N]$ is a function $\nu: \bZ \to [0,\infty)$ with support in $[N]$. Our majorant shall have the additional normalisation property that $$\label{normalisation} \\| \nu\\|_1 \sim N.$$ We refer the curious reader to the expository article [@Pre2015] for more on the history and terminology of the transference principle. The point is that our set $A$ can be lifted to $[N]$ and weighted by $\nu$ to behave like a dense subset — not of $\cP_X := \{ \text{prime } p \le X\}$, but of $[N]$. Bloom’s theorem [@Blo2012] then ensures that the $\nu$-weighted solution count is large in terms of the density — see [@BP2016 §2] and [@Pre2015 §1.2]. Since $A$ has only $K$-trivial solutions to , in the sense that the only solutions $\bx \in A^s$ to have $(x_1^d, \ldots,x_s^d) \in K$, we also obtain an upper bound for this count. Combining the two inequalities reaps a density bound of the shape . Browning and Prendiville [@BP2016] have distilled the method into the following ingredients. 1. Density transfer. We shall lift our set $A \subset \cP_X$ to a set $\cA \subset [N]$ in the support of our majorant $\nu$. With $$\label{delta} \del := \|A\| \frac{\log X}X,$$ we will show that $\cA$ has a $\nu$-weighted density of at least $\del^d$ in $[N]$. In other words, we shall establish the bound $$\label{DensityTransfer} \sum_{n \in \cA} \nu(n) \gg \del^d N.$$ 2. Fourier decay. The majorant $\nu$ has Fourier decay of level $\tet$ if $$\\| \hat{\nu} - \widehat{1_{[N]}} \\|_\infty \le \tet N.$$ We shall demonstrate a quantitatively $o(1)$ level of Fourier decay. 3. Restriction estimate. The majorant $\nu$ satisfies a restriction estimate at exponent $u$ if $$\sup_{\|\phi\| \le \nu} \int_\bT \|\hat{\phi}(\alp)\|^u \d \alp \ll_u \\| \nu \\|_1^u N^{-1}.$$ 4. $K$-trivial saving. For $\eta > 0$, the majorant $\nu$ saves $\eta$ on $K$-trivial solutions if $$\sum_{\bx \in K} \prod_{i=1}^s \nu(x_i) \ll_{k,s,d,\eta} \\| \nu \\|_1^s N^{-1-\eta}.$$ Of these, the most technically demanding are Fourier decay and the restriction estimate. For both, it is necessary to have good pointwise estimates for certain exponential sums over primes. These exponential sums are given by the Fourier transform of our majorant. To ensure the necessary Fourier decay, we shall use the $W$-trick [@Gre2005], which circumvents technical difficulties arising from the fact that the prime $d$th powers are not equidistributed in congruence classes to small moduli. The number of variables required in Theorem \[thm1\] is determined by the restriction estimate. Since we have good control on the growth of the weights involved, we shall see that the restriction estimate can be derived from a moment estimate for a simpler exponential sum. This leads to the following strengthening of Theorem \[thm1\]. \[MainThm\] Let $t \ge d$ be an integer such that the number of solutions $\bz \in [X]^{2t}$ to $$\label{hyp} z_1^d + \ldots + z_t^d = z_{t+1}^d + \ldots + z_{2t}^d$$ is $O_{t,d,\eps}(X^{2t-d+\eps})$, and assume $s > 2t$. Then we have . Theorem \[thm1\] follows from Theorem \[MainThm\] with $$t = \begin{cases} \lfloor d^2/2 \rfloor, &\text{if } d \ne 4 \\ 7, &\text{if } d=4. \end{cases}$$ For $d \ge 5$, one can show that has $O(X^{2t-d})$ solutions $\bz \in [X]^{2t}$. This is a consequence of the main conjecture in Vinogradov’s mean value theorem, which was recently established by Bourgain, Demeter and Guth [@BDG2016]; one mimics the proof of [@Woo2012 Theorem 4.1]. For $(d,t) = (4,7)$, we again follow that proof, interpolating on minor arcs between an eighth and a twentieth moment, to show that the number $\cN$ of solutions $\bz \in [X]^{14}$ to satisfies $$\cN \ll (X^{5+\eps})^{1/2} (X^{15+\eps})^{1/2} = X^{10+\eps}.$$ For $(d,t) = (2,2)$, it is known that has $O(X^2L)$ solutions $\bz \in [X]^4$. For $(d,t) = (3,4)$ it is known that there are $O(X^5)$ solutions — this follows, for instance, from the methods of [@Vau1986]. Theorem \[MainThm\] also enables a septenary result for cubes, assuming the so-called Hooley Riemann hypothesis (HRH); see [@BW2014 §6]. The statement below follows easily from Theorem \[MainThm\] and [@BW2014 Lemma 6.2]. \[Hooley\] Assume HRH, $d = 3$ and $s \ge 7$. Then we have . The existing literature on problems of Waring, Goldbach or Roth flavour is truly vast, so we shall only mention the most relevant highlights. Waring’s problem [@War1770] dates back to 1770, and asks how large $s$ has to be in terms of $d$ to ensure that if $n$ is a large positive integer then $$\label{WaringEq} x_1^d + \ldots + x_s^d = n$$ has a solution $\bx \in \bN^s$. The Hardy–Littlewood circle method has been a particularly effective approach to such problems, with the best results due to Wooley — see [@VW2002; @Woo2015], as well as [@Bou2016 §2] and [@Woo2016]. The circle method has also been used to solve the ternary Goldbach problem, and other problems concerning the addition of primes [@Hel2014; @Vin1937]. Since Hua [@Hua1939], many authors have enjoyed working on the Waring–Goldbach problem, which considers prime solutions to — see [@KW2001; @KW2015; @LZ2015; @Tha1987; @Tha1989], for instance. The circle method has again been the weapon of choice, with the main technical issue being the study of exponential sums over primes [@Hua1965]. Roth’s theorem [@Rot1953] states that if $A \subset [N]$ contains no nontrivial three-term arithmetic progressions then $\|A\| \ll \frac N {\log \log N}$. This bound has since been improved, most recently by Bloom [@Blo2015]. Such results are interesting because they identify patterns in the set $A$ without assuming anything about its structure. Three-term arithmetic progressions pertain to the diophantine equation $$x - 2y + z = 0,$$ and much of the arithmetic combinatorics literature surrounds linear equations. Smith [@Smi2009], Keil [@Kei2014] and Henriot [@Hen2015; @Hen2016] have considered higher degree systems with the property that the solution set is invariant under translations and dilations. This property allows the use of a density increment strategy, which is the standard approach to Roth’s theorem. This brings us to the aforementioned breakthrough of Browning and Prendiville [@BP2016], who used the transference principle to obtain a Roth-type bound for a quadratic equation without the property of translation-dilation invariance. Their result is like a hybrid of Roth’s theorem and Waring’s problem. The present article combines aspects of Roth’s theorem, Waring’s problem and Goldbach problems. We comment briefly on the relevance of restriction theory [@Bou1989; @Bou1993; @Hen2016; @Tao2004]. The number of variables required to implement the circle method is often governed by the exponent at which we know a sharp moment estimate for an exponential sum. When the variables are restricted to lie in a set $A$, the relevant exponential sums necessarily come with weights supported on $A$. The key ingredient for such problems, therefore, is a moment estimate for an exponential sum with fairly arbitrary weights. Restriction theory concerns inequalities between norms of Fourier transforms, which is the same as bounding moments of weighted exponential sums. We organise thus. In §\[W\], we shall construct our majorant $\nu$, confirm , define our lifted set $\cA$, and establish the density transfer inequality . In §\[exponential\], we use the circle method to study the Fourier transform $\hat{\nu}$. The analysis therein will allow us to establish Fourier decay in §\[decay\]. In §\[Restriction\], we use Bourgain’s methods [@Bou1989] to prove that $\nu$ satisfies the relevant restriction estimate. We check in §\[trivial\] that $\nu$ saves $1/s$ on $K$-trivial solutions, before putting it all together to prove Theorem \[MainThm\] in §\[density\]. We adopt the convention that $\eps$ denotes an arbitrarily small positive real number, so its value may differ between instances. The symbol $p$ shall be reserved for primes. For $x \in \bR$ and $q \in \bN$, put and $e_q(x) = e^{2 \pi i x / q}$. Boldface will be used for vectors, for instance we abbreviate $(x_1,\ldots,x_n)$ to $\bx$, and define $\|\bx\| = \max(\|x_1\|, \ldots, \|x_n\|)$. For $x \in \bR$, let $\\| x \\|$ be the distance from $x$ to the nearest integer. Let $\cP$ denote the set of primes. For $Y \in \bN$, let $[Y] = \{ 1,2, \ldots, Y \}$ and $\cP_Y = \cP \cap [Y]$. We shall make use of the offset logarithmic integral $\Li(x) = \int_2^x \frac {\d t}{\log t}$. We write $\bT$ for the torus $\bR / \bZ$. We shall use Landau and Vinogradov notation: for functions $f$ and positive-valued functions $g$, write $f \ll g$ or $f = O(g)$ if there exists a constant $C$ such that $\|f(x)\| \le C g(x)$ for all $x$. If $S$ is a set, we denote the cardinality of $S$ by $\|S\|$ or $\# S$. The pronumeral $X$ denotes a large positive integer, and we shall put $L = \log X$ throughout. We write $C_1, C_2, \ldots$ for positive constants that appear in the course of our proofs. For $r \ge 1$ and $f: \bZ \to \bC$, we define the $L^r$-norm by $$\\| f \\|_r = \Bigl( \sum_n \|f(n)\|^r \Bigr)^{1/r}.$$ When $\\| f\\|_1 < \infty$, we also define the Fourier transform of $f$ by $$\begin{aligned} \hat{f}: \bT &\to \bC, \\ \hat{f}(\alp) &= \sum_n f(n) e(\alp n).\end{aligned}$$ The author would like to thank his advisor Trevor Wooley very much for his guidance. Thanks also to Tim Browning and Sean Prendiville for fruitful conversations. The $W$-trick {#W} ============= We begin by defining our majorant $\nu$. As discussed, we shall apply the $W$-trick [@Gre2005] from the outset, so that we will later obtain sufficient Fourier decay. Let $$\label{Wdef} w = \frac12 \log \log X, \qquad W= 4d^3 \prod_{p \le w} p.$$ Since $X$ is large, it follows from the prime number theorem that $$\label{Wbound} W \le e^{2w} = \log X = L.$$ For $b \in [W]$ with $$\label{bcond} -b \in (\bZ/W\bZ)^{\times d} := \{ z^d: z \in (\bZ / W \bZ)^\times \},$$ let $$\label{sigdef} \sig(b) = \# \{z \in [W]: z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}\}.$$ We begin with the observation that $\sig(b)$ does not, in fact, depend on $b$. To verify this it suffices, by the Chinese remainder theorem, to show that if $r \in \bN$ and $p^r \\| W$ then $$\# \{z {{\,\,\mathrm{mod}\,\,p}}^r: z^d \equiv -b {{\,\,\mathrm{mod}\,\,p}}^r \}$$ is the same for each $b$ satisfying . If $p \ne 2$ then this follows easily using a primitive root (see [@Apo1976 Ch. 10]). For $p =2$, we can instead use the fact that any odd residue class is representable uniquely as $(-1)^u 5^v$, with $u {{\,\,\mathrm{mod}\,\,2}}$ and $v {{\,\,\mathrm{mod}\,\,2}}^{r-2}$ (see [@Dav2000 Ch. 4]). We conclude that $\sig(b)$ is the same for each $b \in -(\bZ/W\bZ)^{\times d}$. To ensure density transfer, we shall choose $b$ to maximise the $\nu_b$-measure of $\cA_b$, where $$\label{nudef} \nu_b(n) = \begin{cases} \frac {\varphi(W)}{W \sig(b)} dp^{d-1} \log p, & \text{if } Wn-b = p^d \text{ with } p \in \cP_X \\ 0, &\text{ otherwise} \end{cases}$$ and $$\label{cAdef} \cA_b = \{ n \in \bZ: Wn-b = p^d \text{ for some } p \in A \}.$$ Let $$\label{Ndef} N = \lfloor X^d/W \rfloor + 1.$$ \[Density transfer\] \[DT\] Assume $\del > (\log X)^{-1}$. Then there exists $b \in [W]$ such that $-b \in (\bZ/W\bZ)^{\times d}$ and $$\sum_{n \in \cA_b} \nu_b(n) \gg \del^d N.$$ We shall implicitly embed $-(\bZ/W\bZ)^{\times d}$ into $[W]$, in the obvious way. We use a standard averaging argument, noting first that $$\sum_{b \in -(\bZ/W\bZ)^{\times d} } \sum_{n \in \cA_b} \nu_b(n) = \sum_{p \in A} \frac {\varphi(W)}{W \sig(b)} dp^{d-1} \log p - \sum_{p \in A, \: p \le w} \frac {\varphi(W)}{W \sig(b)} dp^{d-1} \log p.$$ By over-counting, and by recalling that $\sig(b)$ is the same for each $b \in -(\bZ/W\bZ)^{\times d}$, we deduce that $$\|-(\bZ/W\bZ)^{\times d}\|= \frac{\varphi(W)}{\sig(b)}.$$ Thus, if $b$ is chosen to maximise $\sum\limits_{n \in \cA_b} \nu_b(n)$, then $$\sum_{n \in \cA_b} \nu_b(n) \ge -1 + \sum_{p \in A} W^{-1} dp^{d-1} \log p.$$ A crude lower bound for $\sum\limits_{p \in A} p^{d-1} \log p$ is given by the sum of $p^{d-1} \log p$ over the first $\|A\|$ primes $p$. By and the prime number theorem, we now have $$\sum_{p \in A} p^{d-1} \log p \ge \sum_{p \le (1-\eps) \del X} p^{d-1} \log p \gg \del^d X^d.$$ Hence $$\sum_{n \in \cA_b} \nu_b(n) \gg \del^d X^d / W \gg \del^d N.$$ The assumption that $\del > (\log X)^{-1}$ is harmless in the context of Theorem \[MainThm\], for if $\del \le (\log X)^{-1}$ then we certainly have . We henceforth fix $b$ as in Lemma \[DT\], and write $\cA = \cA_b$, $\nu = \nu_b$, so that we have . Note that our majorant $\nu$ is supported on $[N]$, and that $$\label{supremum} \\| \nu \\|_\infty \ll X^{d-1} L.$$ Next, we verify . The proof is standard, but we nonetheless present it, as it will prepare us well for the next section. We compute: $$\begin{aligned} \notag \\| \nu \\|_1 &= \sum_{\substack{p \le X: \\ p^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} \frac {\varphi(W)}{W \sig(b)} dp^{d-1} \log p \\ \label{L1calc} &= \sig(b)^{-1} \sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} \frac{\varphi(W)}W \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} dp^{d-1} \log p.\end{aligned}$$ The inner sum is treated using Abel summation. For $n \in [X]$, put $$A_n = \sum_{\substack{p \le n: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} 1.$$ Note that since $(b,W) = 1$ we must also have $(z,W) = 1$. The bound allows us to apply Siegel–Walfisz [@Hua1965 Lemma 7.14], so $$A_n = \frac{\Li(n)}{\varphi(W)} + O(X e^{-C_1 \sqrt{L}}).$$ With $g(n) = dn^{d-1} \log n$, we have $$\begin{aligned} \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} dp^{d-1} \log p &= \sum_{n =2}^X (A_n - A_{n-1}) g(n) \\ &= A_X g(X+1) + \sum_{n=2}^X A_n (g(n) - g(n+1)).\end{aligned}$$ The mean value theorem tells us that $g(n) - g(n+1) \ll X^{d-2}L$. In light of and , we now have $$\begin{aligned} \varphi(W) \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} dp^{d-1} \log p &= \Li(X)g(X+1) + \sum_{n=2}^X \Li(n) \cdot (g(n) - g(n+1)) \\ &\qquad + O(Ne^{-C_2 \sqrt{L}}).\end{aligned}$$ As $\Li(2) = 0$, we therefore have $$\begin{aligned} \varphi(W) \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} dp^{d-1} \log p &= \sum_{n=3}^X g(n) \int_{n-1}^n \frac{\d x}{\log x}+ O(Ne^{-C_2 \sqrt{L}}) \\ &= \sum_{n=3}^X dn^{d-1} \int_{n-1}^n \frac{ \log n}{\log x} \d x+ O(Ne^{-C_2 \sqrt{L}}).\end{aligned}$$ When $2 \le n-1 < x < n$, the mean value theorem tells us that $$\log n = \log x + O(1/n).$$ Hence $$\varphi(W) \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} dp^{d-1} \log p = \sum_{n \le X} dn^{d-1} + O(Ne^{-C_2 \sqrt{L}}) = X^d + O(Ne^{-C_2 \sqrt{L}}).$$ Substituting this into , and recalling and , yields $$\\| \nu \\|_1 = X^d/W + O(Ne^{-C_2 \sqrt{L}}) = N + O(Ne^{-C_2 \sqrt{L}}),$$ confirming . Exponential sums {#exponential} ================ We wish to investigate $$\begin{aligned} \notag \hat{\nu}(\alp) &= \frac{\varphi(W)}{W\sig(b)} \sum_{\substack{p \le X:\\ p^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp (p^d+b)/W) \\ \label{hatcalc} &= \frac{\varphi(W) e(\alp b/W)}{W\sig(b)}\sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} \:\: \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W),\end{aligned}$$ so we focus on the inner sum. We begin with a Hardy–Littlewood dissection, dissecting $\bT$ into major arcs $\fM$ and minor arcs $\fm$. Let $\sig_0$ be a large positive constant, and let $\sig$ be a much larger positive constant. For $q \in \bN$ and $a \in \bZ$, let $\fM(q,a)$ be the set of $\alp \in \bT$ such that $\|\alp - a/q\| \le L^\sig X^{-d}$. Let $\fM(q)$ be the union of the sets $\fM(q,a)$ over integers $a$ such that $(a,q) = 1$, and let $\fM$ be the union of the sets $\fM(q)$ over $q \le L^\sig$. Put $\fm = \bT \setminus \fM$. By identifying $\bT$ with a unit interval, we may write$$\fM(q) = \bigcup_{\substack{{a=0}\\(a,q)=1}}^{q-1} \fM(q,a).$$ \[minorbound\] If $\alp \in \fm$ then $\hat{\nu}(\alp) \ll NL^{-\sig_0}$. Let $\alp \in \fm$. By Dirichlet’s approximation theorem [@Vau1997 Lemma 2.1], we obtain relatively prime integers $q$ and $a$ such that $1 \le q \le X^d L^{-\sig}$ and $\|q \alp - a\| \le L^\sig X^{-d}$. Now $\|\alp-a/q\| \le L^\sig X^{-d}$ so, as $\alp \notin \fM$, we must have $q > L^\sig$. Thus, with $\bet = \alp - a/q$, we have $$\|\bet\| \le \frac{L^\sig}{qX^d} \le X^{-d}.$$ Let $z \in [W]$ with $z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}$. By partial summation, we have $$\begin{aligned} \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W) &= \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} e(\bet p^d/W) (dp^{d-1} \log p) e_{Wq}(a p^d) \\ &= A_\diam(X) f(X) - \int_1^X A_\diam(t) f'(t) \d t,\end{aligned}$$ where $f(t) = e(\bet t^d/W) dt^{d-1} \log t$ and $$A_\diam(t) = \sum_{\substack{p \le t: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} e_{Wq}(ap^d).$$ Note that $f'(t) \ll X^{d-2}L$, and that $\|A_\diam(t)\| \le t$. Thus, $$\begin{aligned} \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W) &= A_\diam(X) f(X) - \int_{XL^{-2\sig_0}}^X A_\diam(t) f'(t) \d t \\ &\quad+ O(NL^{-\sig_0}).\end{aligned}$$ In view of and , we now have $$\hat{\nu}(\alp) \ll NL^{-\sig_0} + X^{d-1}L \sup_{XL^{-2\sig_0} < t \le X} \|A_\diam(t)\|.$$ It remains to estimate $A_\diam(t)$ when $XL^{-2\sig_0} < t \le X$. We shall use [@Hua1965 Theorem 10] for this. In order to apply this result, we need to control size of the denominator $$q^* := \frac{Wq}{(a,Wq)} = \frac{Wq}{(a,W)},$$ in terms of $t$. Recalling , we have $$L^\sig < q \le q^* \le Wq \le X^d L^{1-\sig} \le t^d L^{-\sig/2}.$$ As $L \ll \log t \ll L$, we may thus invoke [@Hua1965 Theorem 10], which tells us that $$A_\diam(t) \ll XL^{-\sig_0-1}W^{-1}.$$ Hence $\hat{\nu}(\alp) \ll NL^{-\sig_0}$. On major arcs we can decompose our Fourier transform into archimedean and non-archimedean components. When $(z,W) = 1$, let $$\label{Sdef} S^_q(a,z) = \sum_{\substack{r {{\,\,\mathrm{mod}\,\,\it}}{q}: \\ (z+Wr,Wq)=1}} e_q\Bigl(a \frac{(z+Wr)^d+b}W \Bigr)$$ and $$\label{Idef} I(\bet) = \int_0^N e(\bet t) \d t.$$ Let $\alp \in \fM(q,a)$ with $(a,q) = 1$ and $q \le L^\sig$, and put $$\label{beta} \bet = \alp - a/q \in [-L^\sig X^{-d}, L^\sig X^{-d}].$$ Then $$\label{majorarcs} \hat{\nu}(\alp) = I(\bet) \sig(b)^{-1} \sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} \frac{\varphi(W)}{\varphi(Wq)} S^_q(a,z) + O(Ne^{-C_4 \sqrt{L}}).$$ With in mind, we initially fix $z \in [W]$ with $z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}$, and study $$\sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W).$$ For $n \in [X]$, let $$S_n = \sum_{\substack{p \le n: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} e_{Wq}(ap^d).$$ Then $$S_n = O(Wq) + \sum_{\substack{r {{\,\,\mathrm{mod}\,\,q}}: \\ (z+Wr, Wq) = 1}} e_{Wq}(a(z+Wr)^d) \sum_{\substack{p \le n: \\ p \equiv z+Wr {{\,\,\mathrm{mod}\,\,\it}}{Wq}}} 1.$$ As $n \le X$ and $Wq \le L^{\sig+1}$, the inner sum is amenable to Siegel–Walfisz [@Hua1965 Lemma 7.14], and so $$S_n = \frac{\Li(n)}{\varphi(Wq)} V_q(a,z) + O(Xe^{-C_3 \sqrt{L}}),$$ where $$\label{Vdef} V_q(a,z) = \sum_{\substack{r {{\,\,\mathrm{mod}\,\,q}}: \\ (z+Wr, Wq) = 1}} e_{Wq}(a(z+Wr)^d) .$$ With $f(t) = e(\bet t^d/W) dt^{d-1} \log t$, we have $$\begin{aligned} \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W) &= \sum_{n=2}^X (S_n - S_{n-1}) f(n) \\ &= S_X f(X+1) + \sum_{n=2}^X S_n (f(n) - f(n+1)).\end{aligned}$$ As $\|\bet\| \le L^\sig X^{-d}$, the mean value theorem implies that $$f(n) - f(n+1) \ll X^{d-2} L^{\sig+1}.$$ Hence $$\begin{aligned} & \sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W) \\ & = \frac{V_q(a,z)}{\varphi(Wq)} \Bigl[ \Li(X)f(X+1) + \sum_{n=2}^X \Li(n) \cdot (f(n) - f(n+1)) \Bigr] +O(Ne^{-C_4 \sqrt{L}}).\end{aligned}$$ As $\Li(2) = 0$, we thus have $$\begin{aligned} \notag &\sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W)\\ \label{innercalc} &= \frac{V_q(a,z)}{\varphi(Wq)} \sum_{n=3}^X \int_{n-1}^n \frac{f(n)} {\log x} \d x +O(Ne^{-C_4 \sqrt{L}}).\end{aligned}$$ When $n-1 < x < n$, the mean value theorem reveals that $$f(n) = f(x) + O(X^{d-2}L^{\sig+1}),$$ and so $$\begin{aligned} \sum_{n=3}^X \int_{n-1}^n \frac{f(n)} {\log x} \d x &= \int_2^X dx^{d-1} e(\bet x^d/W) \d x + O(X^{d-1}L^{\sig+1}) \\ &= WI(\bet) + O(X^{d-1} L^{\sig+1}).\end{aligned}$$ Substituting this into , and noting that $\|V_q(a,z)\| \le q \le L^\sig$, gives $$\sum_{\substack{p \le X: \\ p \equiv z {{\,\,\mathrm{mod}\,\,\it}}{W}}} (dp^{d-1} \log p) e(\alp p^d/W) \\ = \frac{W}{\varphi(Wq)} V_q(a,z) I(\bet)+O(Ne^{-C_4 \sqrt{L}}).$$ Substituting this into , and recalling , gives $$\hat{\nu}(\alp) = \frac{\varphi(W) e(\alp b/W)}{\varphi(Wq) \sig(b)} \sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} V_q(a,z) I(\bet) +O(Ne^{-C_4 \sqrt{L}}).$$ From and , we see that $$e_{Wq}(ab) V_q(a,z) = S_q^(a,z).$$ Since $$e(\bet b/W) I(\bet) = \int_0^N e(\bet (t+b/W)) \d t = I(\bet) + O(1),$$ we finally have . Fourier decay {#decay} ============= In this section, we will establish that $\nu$ has Fourier decay of level $w^{\eps - 1/2}$. In other words, we shall prove that if $\alp \in \bT$ then $$\label{FourierDecay} \hat{\nu}(\alp) - \widehat{1_{[N]}} (\alp) \ll_\eps w^{\eps-1/2}N.$$ By a geometric series, we have $$\label{geom} \widehat{1_{[N]}} (\alp) = \sum_{n \le N} e(\alp n) \ll \\| \alp \\|^{-1}.$$ First suppose $\alp \in \fm$. By Dirichlet’s approximation theorem, we obtain relatively prime integers $q$ and $a$ such that $1 \le q \le L^\sig$ and As $\alp \notin \fM$, we must have $\|\alp - a/q\| > L^\sig X^{-d}$, so $$\widehat{1_{[N]}} (\alp) \ll \\| \alp \\|^{-1} \ll \frac q { \\| q \alp \\|} \ll X^d L^{-\sig}.$$ Recalling and , we now have $\widehat{1_{[N]}} (\alp) \ll N L^{1-\sig}$. Coupling this with Lemma \[minorbound\], using the triangle inequality, yields $$\hat\nu (\alp) - \widehat{1_{[N]}} (\alp) \ll N L^{1-\sig} + NL^{-\sig_0}.$$ Upon recalling the definition of $w$, we conclude that holds for $\alp \in \fm$. Next we consider the case in which $q=1$ and $\alp \in \fM(q)$. In other words, $$\\|\alp\\| \le L^\sig X^{-d}.$$ From and , we see that $$\hat \nu (\alp) = I(\alp) + O(Ne^{-C_4 \sqrt L}).$$ By Euler–Maclaurin summation [@Vau1997 Eq. (4.8)], we have $$\widehat{1_{[N]}} (\alp) - I(\alp) \ll 1 + N \\|\alp\\| \ll 1 + NL^\sig X^{-d} \ll L^\sig.$$ The triangle inequality now gives $$\hat \nu (\alp) - \widehat{1_{[N]}} (\alp) \ll Ne^{-C_4 \sqrt L}.$$ Recalling , we conclude that holds whenever $\alp \in \fM(1)$. Finally, let $\alp \in \fM(q,a)$ with $2 \le q \le L^\sig$ and $(a,q) = 1$, and put . Since $q \ge 2$, we must have $\|a\| \ge 1$. Substituting $$\\| \alp \\| \ge q^{-1} - \|\bet\| \ge q^{-1} - L^\sig X^{-d} \gg q^{-1}$$ into gives $$\label{basic} \widehat{1_{[N]}} (\alp) \ll q \ll L^\sig.$$ By , and the trivial estimate $\|I(\bet)\| \le N$, we have $$\label{majorcalc} \hat \nu (\alp) \ll Ne^{-C_4 \sqrt L} + \frac{N}{\varphi(q)} \sup_z \|S_q^(a,z)\|,$$ where the supremum is over $z \in [W]$ such that $(z,W) = 1$. We now study the sums $S_q^(a,z)$. Let $z \in [W]$ with $(z,W) = 1$. By , we have $$\label{twist} S_q^(a,z) = e_{Wq}(a(z^d+b)) S^\diam_q(a,z),$$ where $$S^\diam_q(a,z) = \sum_{\substack{r {{\,\,\mathrm{mod}\,\,\it}}{q}: \\ (z+Wr,Wq)=1}} e_q \Bigl(a \sum_{\ell = 1}^d {d \choose \ell} W^{\ell - 1} z^{d- \ell} r^\ell \Bigr).$$ As $(z+Wr,W) = (z,W) = 1$, we have the slightly simpler expression $$S^\diam_q(a,z) = \sum_{\substack{r {{\,\,\mathrm{mod}\,\,\it}}{q}: \\ (z+Wr,q)=1}} e_q \Bigl(a \sum_{\ell = 1}^d {d \choose \ell} W^{\ell - 1} z^{d- \ell} r^\ell \Bigr).$$ Let $q = uv$, where $u$ is $w$-smooth and $(v,W) = 1$. Since $(u,v) = 1$, a standard calculation reveals that $$\label{decomp} S^\diam_q(a,z) = S^\diam_u(a_1,z) S^\diam_v(a_2,z),$$ where $a_1 = av^{-1} \in (\bZ / u\bZ)^\times$ and $a_2 = au^{-1} \in (\bZ / v\bZ)^\times$ (see [@Vau1997 Lemma 2.10]). First consider $$S^\diam_u(a_1,z) = \sum_{\substack{r {{\,\,\mathrm{mod}\,\,\it}}{u}: \\ (z+Wr,u)=1}} e_u \Bigl(a_1 \sum_{\ell = 1}^d {d \choose \ell} W^{\ell - 1} z^{d- \ell} r^\ell \Bigr).$$ As $u$ is $w$-smooth and $(z,W) = 1$, the condition $(z+Wr,u)=1$ is always met, and so $$S^\diam_u(a_1,z) = \sum_{r {{\,\,\mathrm{mod}\,\,\it}}{u}} e_u \Bigl(a_1 \sum_{\ell = 1}^d {d \choose \ell} W^{\ell - 1} z^{d- \ell} r^\ell \Bigr).$$ We now borrow a strategy employed in [@BP2016 §5]. Let $h = (u,W)$, and put $u = hu'$ and $W= hW'$, noting that $(u',W') = 1$. Writing $r = r_1 + u' r_2$, with $r_1 {{\,\,\mathrm{mod}\,\,u}}'$ and $r_2 {{\,\,\mathrm{mod}\,\,h}}$, yields $$\begin{aligned} S^\diam_u(a_1,z) &= \sum_{\substack{r_1 {{\,\,\mathrm{mod}\,\,u}}' \\ r_2 {{\,\,\mathrm{mod}\,\,h}}}} e_{hu'} \Bigl(a_1 \sum_{\ell = 1}^d {d \choose \ell} (hW')^{\ell - 1} z^{d- \ell} (r_1+u'r_2)^\ell \Bigr) \\ &= \sum_{r_1 =0}^{u'-1} e_{hu'} \Bigl(a_1 \sum_{\ell = 1}^d {d \choose \ell} (hW')^{\ell - 1} z^{d- \ell} r_1^\ell \Bigr) \\ & \qquad \sum_{r_2 = 0}^{h-1} e_h \Bigl(a_1 \sum_{\ell = 1}^d {d \choose \ell} (hW')^{\ell - 1} z^{d- \ell} (u')^{\ell-1} r_2^\ell \Bigr).\end{aligned}$$ The inner sum is $$\sum_{r_2 {{\,\,\mathrm{mod}\,\,h}}} e_h(da_1 z^{d-1} r_2),$$ which vanishes unless $h \mid d a_1 z^{d-1}$. As $(h,a_1) = (h,z) = 1$, we conclude that $$\label{bigu} S^\diam_u(a_1,z) =0, \quad \text{if } (u,W) \nmid d,$$ while if $h \mid d$ then $$\label{critical} S^\diam_u(a_1,z) = h \sum_{r_1 = 0}^{u'-1} e_u \Bigl(a_1 \sum_{\ell = 1}^d {d \choose \ell} W^{\ell - 1} z^{d- \ell} r_1^\ell \Bigr).$$ Next consider $$e_{Wv}(a_2z^d) S^\diam_v(a_2,z) = \sum_{\substack{r {{\,\,\mathrm{mod}\,\,\it}}{v}: \\ (z+Wr,v)=1}} e_v\Bigl(a_2 \frac{(z+Wr)^d}W\Bigr).$$ As $(v,W) = 1$, we can change variables by $t = zW^{-1} + r \in \bZ / v \bZ$, which gives $$e_{Wv}(a_2z^d) S^\diam_v(a_2,z) = \sum_{\substack{t {{\,\,\mathrm{mod}\,\,\it}}{v}: \\ (t,v)=1}} e_v(a_2 W^{d-1} t^d).$$ Since $(a_2 W^{d-1}, v) = 1$, we may apply [@Hua1965 Lemma 8.5], which tells us that $$\label{vbound} S^\diam_v(a_2,z) \ll v^{1/2+\eps} \ll q^{1/2+\eps}.$$ As $q \ge 2$, we must have (i) $u \nmid d$, (ii) $1 \ne q \mid d$, or (iii) $u \mid d$ and $q > w$. Case: $u \nmid d$. Suppose for a contradiction that $(u,W) \mid d$. Then for all primes $p$ we have $$\min(\ord_p(u), \ord_p(W)) \le \ord_p(d).$$ Since $\ord_p(W) > \ord_p(d)$ whenever $p \le w$, and since $u$ is $w$-smooth, this tells us that $u \mid d$, contradicting this case. Hence $(u,W) \nmid d$, so by we have $S^\diam_u(a_1,z) = 0$. Therefore $S^_q(a,z)$ vanishes, by and . Now gives $$\label{simple} \hat \nu (\alp) \ll Ne^{-C_4 \sqrt L}.$$ Case: $1 \ne q \mid d$.* In this case $v=1$ and $q = u$. Further, $$h = (u, W) = (q, W) = q,$$ since $q \mid d \mid W$. So $u' = 1$, and from we see that $S^\diam_q(a,z) = q$. By , we therefore have $$\sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} S^_q(a,z) = q \sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} e_{Wq}(a(z^d+b)).$$ We will find that this sum vanishes, which is the point of the $W$-trick. Write $$z \equiv x + \frac W d y {{\,\,\mathrm{mod}\,\,W}}$$ with $x {{\,\,\mathrm{mod}\,\,W}}/d$ and $y {{\,\,\mathrm{mod}\,\,d}}$. Note that $$z^d \equiv \Bigl(x+\frac W d y \Bigr)^d \equiv x^d {{\,\,\mathrm{mod}\,\,W}},$$ in view of the definition of $W$. Hence $$q^{-1}\sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} S^_q(a,z) = \sum_{\substack{x \in [W/d]:\\ x^d \equiv -b {{\,\,\mathrm{mod}\,\,W}}}} e_{Wq}(a(x^d+b)) \sum_{y {{\,\,\mathrm{mod}\,\,d}}} e_q(ax^{d-1}y).$$ As $(b,W) = 1$ and $q \mid W$, we have $(x,q) = 1$. Since $q \ne 1$ and $(a,q) = 1$, and since $q \mid d$, we deduce that the inner sum vanishes. Therefore $$\sum_{\substack{z \in [W]: \\ z^d \equiv -b {{\,\,\mathrm{mod}\,\,\it}}{W}}} S^_q(a,z) = 0.$$ Substituting this into yields . Case: $u \mid d$ and $q > w$.* Since $u \le d$, we have $S^\diam_u(a_1,z) \ll 1$. Now , and give $$\label{sqrt} S^_q(a,z) \ll q^{1/2+\eps}.$$ Substituting this into yields $$\hat \nu (\alp) \ll Ne^{-C_4 \sqrt L} + q^{2\eps - 1/2}N \ll w^{2 \eps - 1/2}N.$$ Recalling , we see that we have $$\hat \nu (\alp) \ll w^{\eps - 1/2}N$$ in all three cases. Coupling this with yields . We conclude that the majorant $\nu$ has Fourier decay of level $w^{\eps - 1/2}$. Note that the inequality is valid in all three cases. We record the following estimate for later use. Let $\alp \in \fM(q,a)$ with $1 \le q \le L^\sig$ and $(a,q) = 1$. Then $$\label{record} \hat \nu(\alp) \ll q^{\eps- 1/2} \min\{ N, \\| \alp - a/q \\| ^{-1} \} + Ne^{-C_4 \sqrt L}.$$ Put . By and , we have $$\hat \nu (\alp) \ll \varphi(q)^{-1} \sup_z \|I(\bet) S^_q(a,z)\| + Ne^{-C_4 \sqrt L}.$$ The integral admits the standard estimate $$I(\bet) \ll \min \{ N, \\|\bet\\|^{-1} \} = \min \{ N, \\| \alp-a/q \\|^{-1} \},$$ so by we have $$\begin{aligned} \hat \nu (\alp) &\ll \varphi(q)^{-1} q^{1/2+\eps} \min \{ N, \\| \alp-a/q \\|^{-1} \} + Ne^{-C_4 \sqrt L} \\ &\ll q^{2\eps -1/2} \min \{ N, \\| \alp-a/q \\|^{-1} \} + Ne^{-C_4 \sqrt L}.\end{aligned}$$ The restriction estimate {#Restriction} ======================== Let $t \ge d$ be an integer such that the number of solutions $\bz \in [X]^{2t}$ to is $O_{t,d,\eps}(X^{2t-d+\eps})$. In this section, we show that $\nu$ satisfies a restriction estimate at any exponent $u > 2t$. The following lemma suffices, by . \[RestrictionEstimate\] Let $\phi: \bZ \to \bC$ with $\|\phi\| \le \nu$, and let $u > 2t$ be a real number. Then $$\int_\bT \|\hat \phi (\alp)\|^u \d \alp \ll_u N^{u-1}.$$ We proceed in stages. The factor of $X^\eps$ in the assumed bound on the number of solutions to is a formidable hurdle, as Lemma \[minorbound\] fails to provide a power saving on minor arcs. We shall reduce this to a logarithmic factor, at some intermediate exponent $v$; here $v$ will be a real number with $$\label{vrange} 2t < v < u.$$ In order to obtain a power saving on minor arcs, we will introduce thicker major arcs, and use them to study the auxiliary majorant $$\mu(n) = \sig(b)^{-1} \sum_{\substack{x \in [X]:\\ Wn - b = x^d}} dx^{d-1},$$ wherein we recall . Observe that $$\nu(n) \le L \cdot \mu(n) \qquad (n \in \bZ).$$ We shall prove the following restriction estimate for $\mu$, which will serve as a platform from which to attack Lemma \[RestrictionEstimate\]. \[platform\] Let $\psi: \bZ \to \bC$ with $\|\psi\| \le \mu$, and let $v > 2t$ be a real number. Then $$\int_\bT \|\hat \psi (\alp)\|^v \d \alp \ll_v N^{v-1} L^v.$$ Let us explain how this implies Lemma \[RestrictionEstimate\], following Bourgain’s strategy [@Bou1989 §4]. We apply Lemma \[platform\] with $\psi = L^{-1} \phi$, and with $v$ in the range , obtaining $$\label{inter} \int_\bT \|\hat \phi (\alp)\|^v \d \alp \ll N^{v-1} L^{2v}.$$ In this section only, we denote by $\del$ an arbitrary parameter in the range $$0 < \del < 1,$$ for consistency with previous literature. This is not to be confused with the density $\del$ defined in . Consider the large spectra $$\cR_\del = \{ \alp \in \bT: \|\hat \phi(\alp)\| > \del N \},$$ which may be regarded as level sets. Note from that $$\\| \hat \phi \\|_\infty \le \\| \phi \\|_1 \le \\| \nu \\|_1 < (1+\eps)N.$$ By a standard argument involving dyadic intervals, it suffices to show that if $\eps_0 > 0$ then $$\label{toshow} \meas (\cR_\del) \ll_{\eps_0} \frac1 {\del^{v+\eps_0}N}$$ (see the discussion surrounding [@BP2016 Lemma 6.3]). In our quest to establish , we begin by noting that if $\del \le L^{-2v/\eps_0}$ then by we have $$(\del N)^v \meas(\cR_\del) \le \int_\bT \|\hat \phi (\alp)\|^v \d \alp \ll N^{v-1} L^{2v},$$ whereupon $$\meas(\cR_\del) \ll \frac{L^{2v}}{\del^v N} \ll \frac1 {\del^{v+\eps_0}N}.$$ Thus, we may assume that $$\label{wma} L^{-2v/\eps_0} < \del < 1.$$ Let $\tet_1, \ldots, \tet_R$ be $N^{-1}$-spaced points in $\cR_\del$. As $v > 2t \ge 2d$, it suffices to show that $$\label{goal} R \ll_{\eps_0} \del^{-2d-\eps_0}.$$ Put $$\label{gamdef} \gam = d + \eps_0/3.$$ Routinely, as in [@Bou1989 §4] and [@BP2016 §6], we have $$\label{Bourg} \del^{2 \gam} N^\gam R^2 \ll \sum_{1 \le r,r' \le R} \|\hat \nu (\tet_r - \tet_{r'})\|^\gam.$$ Recall our Hardy–Littlewood dissection from §\[exponential\]. In this dissection, we now specify that $\sig_0$ is large in terms of $\eps_0$ and $u$. Consider $$\tet = \tet_r - \tet_{r'}$$ in the summand on the right hand side of . By Lemma \[minorbound\], the contribution from $\tet \in \fm$ to the right hand side of is $O(R^2 N^\gam L^{-\sig_0 \gam})$, with $\sig_0$ large. By , this is $o(\del^{2 \gam} N^\gam R^2)$. Hence $$\label{Bourg2} \del^{2 \gam} N^\gam R^2 \ll \sum_{\substack{1 \le r,r' \le R:\\ \tet \in \fM}} \|\hat \nu (\tet_r - \tet_{r'})\|^\gam.$$ Let $Q = C_5 + \del^{-5}$, with $C_5$ a large positive constant. By , the contribution to the right hand side of from denominators $q > Q$ is bounded, up to a constant, by $$R^2 N^\gam (Q^{\eps-\gam/2} + e^{-C_4 \gam \sqrt L}).$$ This is negligible compared to the left hand side of , by and the fact that $C_5$ is large. We thus conclude from , and that $$\del^{2 \gam} R^2 \ll \sum_{q \le Q} \: \sum_{\substack{a {{\,\,\mathrm{mod}\,\,q}} \\ (a,q)=1}} \: \sum_{1 \le r,r' \le R} \frac{q^{\eps-\gam/2}}{(1+N\\| \tet_r - \tet_{r'}-a/q\\|)^\gam}.$$ Hence $$\label{BourgainExpression} \del^{2 \gam} R^2 \ll \sum_{1 \le r,r' \le R} G(\tet_r - \tet_{r'}),$$ where $$G(\alp) = \sum_{q \le Q} \: \sum_{a=0}^{q-1} \frac{q^{\eps - \gam/2}}{(1+N\|\sin(\alp-a/q)\|)^\gam}.$$ The inequality is very similar to [@Bou1989 Eq. (4.16)], but with $N^2$ replaced by $N$. We have an additional factor of $q^\eps$ in the definition of $G(\alp)$, and we save a $\gamma$th power in the denominator, whereas Bourgain saves only a ($\gamma/2$)nd power. Bourgain’s argument carries through, and we obtain . We have shown that Lemma \[platform\] implies Lemma \[RestrictionEstimate\]. To prove Lemma \[platform\], we begin by establishing the following ‘$\eps$-sharp’ $(2t)$th moment bound. \[intermediate\] Let $\psi: \bZ \to \bC$ with $\|\psi\| \le \mu$. Then $$\int_\bT \|\hat \psi (\alp)\|^{2t} \d \alp \ll N^{2t-1+\eps}.$$ By orthogonality, we have $$\begin{aligned} \int_\bT \|\hat \psi (\alp)\|^{2t} \d \alp &= \int_\bT \sum_{\bn \in \bZ^{2t}} \psi(n_1) \cdots \psi(n_t) \overline{\psi(n_{t+1})} \cdots \overline{\psi(n_{2t})} \\ & \qquad \qquad e(\alp(n_1+ \ldots + n_t - n_{t+1} - \ldots - n_{2t})) \d \alp \\ &= \sum_{\substack{\bn: \\ n_1 + \ldots + n_t = n_{t+1} + \ldots + n_{2t}}} \psi(n_1) \cdots \psi(n_t) \overline{\psi(n_{t+1})} \cdots \overline{\psi(n_{2t})} .\end{aligned}$$ The triangle inequality now gives $$\begin{aligned} \int_\bT \|\hat \psi (\alp)\|^{2t} \d \alp &\le \sum_{\substack{\bn: \\ n_1 + \ldots + n_t = n_{t+1} + \ldots + n_{2t}}} \mu(n_1) \cdots \mu(n_{2t}) \\ &\ll X^{2t(d-1)} \sum_{\substack{\bz \in [X]^{2t}: \\ z_1^d + \ldots + z_t^d = z_{t+1}^d + \ldots + z_{2t}^d}} 1.\end{aligned}$$ Our hypothesis on $t$ now yields $$\int_\bT \|\hat \psi (\alp)\|^{2t} \d \alp \ll X^{2t(d-1)} X^{2t-d+\eps} = X^{2td-d+\eps}$$ so, by and , we finally have $$\int_\bT \|\hat \psi (\alp)\|^{2t} \d \alp \ll (NL)^{2t-1} X^\eps \ll N^{2t-1+\eps}.$$ We now prove Lemma \[platform\], again using Bourgain’s strategy. We begin by obtaining pointwise estimates for $\hat \mu$. The triangle inequality gives $$\begin{aligned} \sig(b) \hat \mu(\tet) &= e(\tet b/W) \sum_{\substack{x \le X:\\ x^d \equiv -b {{\,\,\mathrm{mod}\,\,W}}}} dx^{d-1} e(\tet x^d / W) \\ &\ll \Biggl \|\sum_{\substack{x \le X:\\ x^d \equiv -b {{\,\,\mathrm{mod}\,\,W}}}} x^{d-1} e(\tet x^d / W) \Biggr \|.\end{aligned}$$ Hence, by partial summation, we obtain $$\label{mupartial} \hat \mu(\tet) \ll X^{d/2} + X^{d-1} \sup_{X^{1/2} \le P \le X} \|g(\tet;P)\|,$$ where $$g(\tet; P) = \sig(b)^{-1} \sum_{\substack{x \le P:\\ x^d \equiv -b {{\,\,\mathrm{mod}\,\,W}}}} e(\tet x^d / W).$$ Moreover, by , we have $$\label{gsup} g(\tet; P) \ll \sup_{z \in [W]} \Biggl\| \sum_{\substack{x \le P:\\ x \equiv z {{\,\,\mathrm{mod}\,\,W}}}} e(\tet x^d / W) \Biggr\|.$$ Writing $x = Wy + z$, we find that $$\label{descend} \sum_{\substack{x \le P:\\ x \equiv z {{\,\,\mathrm{mod}\,\,W}}}} e(\tet x^d / W) = \sum_{y \le P/W} e(W^{d-1} \tet h(y)) + O(1),$$ for some monic polynomial $h$ of degree $d$. Let $P \in [X^{1/2}, X]$ and $z \in [W]$. The Weyl sums $$g_1(\alp) := \sum_{y \le P/W} e(\alp h(y))$$ are very classical, and are discussed in many texts. For reasons of economy, we employ Baker’s estimates [@Bak1986], as packaged in [@Cho2016 §2]. The bounds apply to monic polynomials $h$ of degree $d$, and are uniform in the other coefficients of $h$; in particular, they are uniform in $z$. It is plain from the proof of [@Cho2016 Lemma 2.3] that the quantity $\sig(d)$ therein may be replaced by $2^{1-d}$. We conclude thus. \[wps\] If $$\|g_1(\alp)\| > (P/W)^{1-2^{1-d}+\eps}$$ then there exist relatively prime integers $r > 0$ and $b$ such that $$g_1(\alp) \ll r^{\eps-1/d}PW^{-1} (1+(P/W)^d \|\alp - b/r\|)^{-1/d}.$$ From Lemma \[wps\], we deduce that if $$\|g_1(W^{d-1}\tet)\| > X^{1-2^{1-d}+\eps}$$ then there exist relatively prime integers $r > 0$ and $b$ such that $$g_1(W^{d-1}\tet) \ll r^{\eps - 1/d} XW^{-1} (1+(X/W)^d\|W^{d-1}\tet - b/r\|)^{-1/d}.$$ In this case we can put $$a = \frac b{(b,W^{d-1})}, \qquad q = \frac{rW^{d-1}}{(b,W^{d-1})},$$ thus obtaining relatively prime integers $q > 0$ and $a$ such that $$\label{g1bound} g_1(W^{d-1}\tet) \ll Xq^{\eps-1/d} (1+X^d W^{-1} \|\tet - a/q\|)^{-1/d}.$$ Write $$\label{minor2} \fn = \{ \tet \in \bT: \|\hat \mu (\tet)\| \le X^{d-2^{-d}} \}.$$ In light of and , we can collect , , and to obtain the following ‘major arc estimate’: if $\tet \notin \fn$ then there exist relatively prime integers $q$ and $a$ such that $0 \le a \le q-1$ and $$\label{major2} \hat \mu (\tet) \ll NLq^{\eps-1/d}(1+N \\| \tet - a/q \\|)^{-1/d}.$$ Now that we have made the necessary preparations, we complete the proof of Lemma \[platform\]. This will parallel our proof that Lemma \[RestrictionEstimate\] follows from Lemma \[platform\]. Consider the large spectra $$\cR_\del = \{ \alp \in \bT: \|\hat \psi(\alp)\| > \del NL \},$$ noting from and the crude bound $$\label{mu1} \\| \hat \psi \\|_\infty \le \\| \psi \\|_1 \le \\| \mu \\|_1 \le \sum_{x \le X} dx^{d-1} \sim X^d \le NL.$$ Similarly to before, it suffices to show that if $\eps_0 > 0$ then we have $$\meas (\cR_\del) \ll_{\eps_0} \frac1 {\del^{2t+\eps_0}N}.$$ This time, we can use Lemma \[intermediate\] to reduce consideration to $\del$ in the range $$\label{wma2} N^{-\eps} < \del < 1,$$ wherein we recall our notational convention for $\eps$. With $\tet_1, \ldots, \tet_R$ be as $N^{-1}$-spaced points in $\cR_\del$, it remains to show . Again with , we will find that $$\label{Bourg3} \del^{2\gam} N^\gam L^{\gam} R^2 \ll \sum_{1 \le r,r' \le R} \|\hat \mu (\tet_r - \tet_{r'})\|^\gam.$$ We now verify this inequality by following the corresponding argument in the proof of [@BP2016 Lemma 6.3]. Let $a_n \in \bC$ be such that $\|a_n\| \le 1$ and $\psi(n) = a_n \mu(n)$, for $n \in [N]$. Furthermore, let $c_1, \ldots, c_R \in \bC$ be such that $\|c_r\| = 1$ and $$c_r \hat \psi(\tet_r) = \|\hat \psi(\tet_r)\| \qquad(1 \le r \le R).$$ It follows from the Cauchy-Schwarz inequality and that $$\begin{aligned} \del^2 N^2 L^2 R^2 &\le \Biggl( \sum_{r \in [R]} \|\hat \psi(\tet_r)\| \Biggr)^2 = \Biggl( \sum_{r \in [R]} c_r \sum_n a_n \mu(n) e(n \tet_r) \Biggr)^2 \\ &\le \\| \mu\\|_1 \sum_n \mu(n) \Biggl \| \sum_{r \in [R]} c_r e(n\tet_r) \Biggl \|^2 \ll NL \sum_n \mu(n) \Biggl \| \sum_{r \in [R]} c_r e(n\tet_r) \Biggl \|^2,\end{aligned}$$ and so $$\del^2 NL R^2 \ll \sum_{1 \le r, r' \le R} \|\hat \mu(\tet_r - \tet_{r'})\|.$$ An application of Hölder’s inequality now harvests . Consider $\tet = \tet_r - \tet_{r'}$ in the summand on the right hand side of . By , and , the contribution from $\tet \in \fn$ to the right hand side of is $O(R^2 N^{\gam(1 + \eps - 2^{-d}/d)})$. By , this is $o(\del^{2 \gam} N^\gam L^{\gam} R^2)$. Hence $$\label{Bourg4} \del^{2 \gam} N^\gam L^\gam R^2 \ll \sum_{\substack{1 \le r,r' \le R:\\ \tet \notin \fn}} \|\hat \mu (\tet_r - \tet_{r'})\|^\gam.$$ Let $Q = C_6 + \del^{-3d}$, with $C_6$ a large positive constant. By , the contribution to the right hand side of from denominators $q > Q$ is bounded, up to a constant, by $$R^2 N^\gam L^{\gam} Q^{\eps-\gam/d}.$$ This is negligible compared to the left hand side of , as $C_6$ is large. We thus conclude from and that $$\del^{2 \gam} R^2 \ll \sum_{q \le Q} \: \sum_{\substack{a {{\,\,\mathrm{mod}\,\,q}} \\ (a,q)=1}} \: \sum_{1 \le r,r' \le R} \frac{q^{\eps-\gam/d}}{(1+N\\| \tet_r - \tet_{r'}-a/q\\|)^{\gam/d}}.$$ Hence $$\label{BourgainExpression2} \del^{2 \gam} R^2 \ll \sum_{1 \le r,r' \le R} G_2(\tet_r - \tet_{r'}),$$ where $$G_2(\alp) = \sum_{q \le Q} \: \sum_{a=0}^{q-1} \frac{q^{\eps- \gam/d}}{(1+N\|\sin(\alp-a/q)\|)^{\gam/d}}.$$ The inequality is very similar to [@Bou1989 Eq. (4.16)], but with $N^2$ replaced by $N$, and with $Q \sim \del^{-3d}$ rather than $Q \sim \del^{-5}$. The exponents differ but, since $\gam > d$, Bourgain’s argument carries through, and provides the desired bound for $R$. This completes the proof of Lemma \[platform\]. We have established all of the results in this section. In particular, we know from Lemma \[RestrictionEstimate\] that $\nu$ satisfies a restriction estimate at any exponent $u > 2t$. The $K$-trivial count {#trivial} ===================== In this section we show that $\nu$ saves $1/s$ on $K$-trivial solutions. Let $t \in \bN$ be such that the number of solutions $\bz \in [X]^{2t}$ to is $O_{t,d,\eps}(X^{2t-d+\eps})$, and assume $s > 2t$. \[trivialstep\] The number of $\bx \in [X]^s$ with $(x_1^d, \ldots, x_s^d) \in K$ is $$O_{k,s,d,\eps}(X^{s-d-d/(s-1)+\eps}).$$ The set $K$ lies in the union of $k$ subspaces of the form $$\{ \by \in \bQ^s: \bc \cdot \by = \bd \cdot \by = 0 \},$$ where $\bd \in \bQ^s$ is a fixed vector that is not proportional to $\bc$. Our task, therefore, is to count solutions $\bx \in [X]^s$ to the system $$\label{system} c_1 x_1^d + \ldots + c_s x_s^d = d_1 x_1^d + \ldots + d_s x_s^d = 0.$$ From we obtain $$e_1 x_1^d + \ldots + e_{s-1} x_{s-1}^d = 0,$$ where $(e_1, \ldots, e_{s-1}) \ne \bzero,$ and by rescaling we may assume that $$(e_1, \ldots, e_{s-1}) \in \bZ^{s-1}.$$ Let $u$ be the number of nonzero $e_i$, and note that $1 \le u \le s-1$. Without loss of generality $e_1, \ldots, e_u \in \bZ \setminus \{ 0 \}$ and $e_{u+1} = \ldots = e_{s-1} = 0$, so that $$\label{reduced} e_1 x_1^d + \ldots + e_u x_u^d = 0.$$ By orthogonality, the number $\cM$ of solutions $(x_1, \ldots, x_u) \in [X]^u$ to is $$\int_\bT \sum_{\by \in [X]^u} e\Bigl(\alp \sum_{i \le u} e_i y_i^d \Bigr) \d \alp = \int_\bT \Bigl(\prod_{i \le u}\sum_{y_i \le X} e(\alp e_i y_i^d)\Bigr) \d \alp.$$ By Hölder’s inequality we now have, for some $i \in [u]$, $$\cM \ll \int_\bT \|f(e_i \alp)\|^u \d \alp,$$ where $$f(\tet) = \sum_{x \le X} e(\tet x^d).$$ Note that $e_i \ne 0$, since $i \in [u]$. By periodicity, a change of variables reveals that $$\int_\bT \|f(e_i \alp)\|^u\d \alp = \int_\bT \|f(\alp)\|^u \d \alp.$$ Another application of Hölder’s inequality now gives $$\cM \ll \int_\bT \|f(\alp)\|^u \d \alp \ll \Bigl(\int_\bT \|f(\alp)\|^{s-1} \d \alp \Bigr)^{u/(s-1)}.$$ As $s-1 \ge 2t$ and $\|f(\alp)\| \le X$, we now have $$\label{Mbound} \cM \ll \Bigl(X^{s-1-2t} \int_\bT \|f(\alp)\|^{2t} \d \alp \Bigr)^{u/(s-1)}.$$ From we have $$\label{determine} c_{u+1} x_{u+1}^d + \ldots + c_s x_s^d = -(c_1 x_1^d + \ldots + c_u x_u^d).$$ Given integers $x_1, \ldots, x_u$, the number of solutions $(x_{u+1}, \ldots, x_s) \in \bZ^{s-u}$ to is, by orthogonality, at most $$\int_\bT \Biggl\| \sum_{\bz \in [X]^{s-u}} e\Bigl(\alp \sum_{i \le s-u} c_{u+i} z_i^d \Bigr) \Biggr\| \d \alp.$$ By following our calculation bounding $\cM$, we deduce that this quantity is bounded by $$\Bigl(X^{s-1-2t} \int_\bT \|f(\alp)\|^{2t} \d \alp \Bigr)^{(s-u)/(s-1)}.$$ Coupling this information with , we find that the number $\cN$ of solutions $\bx \in [X]^s$ to satisfies $$\cN \ll \Bigl(X^{s-1-2t} \int_\bT \|f(\alp)\|^{2t} \d \alp \Bigr)^{s/(s-1)}.$$ By orthogonality, the integral $\int_\bT \|f(\alp)\|^{2t} \d \alp$ equals the number of solutions $\bz \in [X]^{2t}$ to which, by hypothesis, is $O(X^{2t-d+\eps})$. Hence $$\cN \ll (X^{s-1-2t} X^{2t-d+\eps})^{s/(s-1)} \ll X^{(s-1-d)s/(s-1)+2\eps} = X^{s-d - d/(s-1) + 2 \eps},$$ which proves the lemma. The majorant $\nu$ saves $1/s$ on $K$-trivial solutions. By , our task is to establish the inequality $$\sum_{\by \in K} \prod_{i=1}^s \nu(y_i) \ll_{k,s,d} N^{s-1-1/s}.$$ By and , we have $$\sum_{\by \in K} \prod_{i=1}^s \nu(y_i) \ll (X^{d-1}L)^s \sum_{\substack{\bx \in [X]^s:\\ (x_1^d + b, \ldots, x_s^d + b) / W \in K}} 1.$$ By the definition of $K$, the condition $(x_1^d + b, \ldots, x_s^d + b) / W \in K$ is equivalent to the condition $$(x_1^d, \ldots, x_s^d) \in K.$$ Now Lemma \[trivialstep\] yields $$\sum_{\by \in K} \prod_{i=1}^s \nu(y_i) \ll (X^{d-1}L)^s X^{s-d-d/(s-1)+\eps} \ll X^{d(s-1)-d/(s-1)+2\eps},$$ so by and we have $$\sum_{\by \in K} \prod_{i=1}^s \nu(y_i) \ll N^{s-1-1/(s-1)+\eps} \ll N^{s-1-1/s}.$$ The density bound {#density} ================= Finally, we have all of the ingredients needed to prove Theorem \[MainThm\]. Note that $\cA$ has only $K$-trivial solutions to , in the sense that if $\bn \in \cA^s$ and $\bc \cdot \bn = 0$ then $\bn \in K$. Indeed, suppose $n_1, \ldots, n_s \in \cA$ and $\bc \cdot \bn = 0$. Then, by and , we have with $$x_i = (Wn_i - b)^{1/d} \in A \qquad (1 \le i \le s).$$ Our hypothesis on $A$ then tells us that $$(Wn_1 -b, \ldots, Wn_s - b) = (x_1^d, \ldots, x_s^d) \in K.$$ From its construction, we see that $K$ is invariant under translations and dilations, so we now have $\bn \in K$, which confirms that $\cA$ has only $K$-trivial solutions to . We apply [@BP2016 Proposition 2.8] to the majorant $\nu$, noting that $\cA \subseteq \supp(\nu)$. We showed in §\[decay\] that $\nu$ has Fourier decay of level $w^{\eps-1/2}$. We showed in §\[Restriction\] that $\nu$ satisfies a restriction estimate at the exponent $s-1/2$. We showed in §\[trivial\] that $\nu$ saves $1/s$ on trivial solutions. Hence $$\sum_{n \in \cA} \nu(n) \ll \frac N{ \min \{ \log \log (w^{1/2-\eps}), \log N \}^{s-2-\eps}},$$ so by we have $$\sum_{n \in \cA} \nu(n) \ll N(\log \log \log \log X)^{2+\eps-s}.$$ Coupling this with yields $$\del^d N \ll N(\log \log \log \log X)^{2+\eps-s},$$ and so $\del \ll (\log \log \log \log X)^{(2-s)/d + \eps}$. By , this gives , completing the proof of Theorem \[MainThm\]. [50]{} T. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. R. C. Baker, Diophantine Inequalities, London Math. Soc. Monographs (N.S.) 1, Clarendon Press, Oxford, 1986. T. F. Bloom, Translation invariant equations and the method of Sanders, Bull. Lond. Math. Soc. 44 (2012), 1050–1067. T. F. Bloom, [A quantitative improvement for Roth’s theorem on arithmetic progressions]{}, arXiv:1405.5800. J. Bourgain, On $\Lam(p)$-subsets of squares, Israel J. Math. 67 (1989), 291–311. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107–156. J. Bourgain, On the Vinogradov mean value, arXiv:1601.08173. J. Bourgain, C. Demeter and L. Guth, Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, arXiv:1512.01565. T. D. Browning and S. M. Prendiville, A transference approach to a Roth-type theorem in the squares, Int. Math. Res. Not., to appear, arXiv:1510.00136. J. Brüdern and T. D. Wooley, Subconvexity for additive equations: pairs of undenary cubic forms, J. Reine Angew. Math. 696 (2014), 31–67. S. Chow, Waring’s problem with shifts, Mathematika 62 (2016), 13–46. H. Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. B. J. Green, Roth’s theorem in the primes, Ann. of Math. (2) 161 (2005), 1609–1636. H. A. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748. K. Henriot, Logarithmic bounds for translation-invariant equations in squares, Int. Math. Res. Not. 2015, 12540–12562. K. Henriot, Additive equations in dense variables via truncated restriction estimates, arXiv:1508.05923. L.-K. Hua, On the representation of numbers as the sums of the powers of primes, Math. Z. 44 (1939), 335–346. L.-K. Hua, Additive Theory of Prime Numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965. K. Kawada and T. D. Wooley, On the Waring–Goldbach problem for fourth and fifth powers, Proc. London Math. Soc. (3) 83 (2001), 1–50. E. Keil, On a diagonal quadric in dense variables, Glasg. Math. J. 56 (2014), 601–628. A. V. Kumchev and T. D. Wooley, On the Waring–Goldbach problem for eighth and higher powers, J. Lond. Math. Soc. (2), to appear, arXiv:1510.00982. J. Y. Liu and T. Zhan, New Developments in the Additive Theory of Prime Numbers, World Scientific, Singapore, to appear. S. M. Prendiville, Four variants of the Fourier-analytic transference principle, arXiv:1509.09200. K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109. M. L. Smith, On solution-free sets for simultaneous quadratic and linear equations, J. Lond. Math. Soc. (2) 79 (2009), 273–293. T. Tao, Some recent progress on the restriction conjecture, Fourier analysis and convexity, Birkhäuser Boston, Boston, MA, 2004, pp. 217–243. K. Thanigasalam, Improvement on Davenport’s iterative method and new results in additive number theory, III, Acta Arith. 48 (1987), 97–116. K. Thanigasalam, On sums of positive integral powers and simple proof of $G(6) \le 31$, Bull. Calcutta Math. Soc. 81 (1989), 279–294. R. C. Vaughan, On Waring’s problem for cubes, J. Reine Angew. Math. 365 (1986), 122–170. R. C. Vaughan, The Hardy–Littlewood method, 2nd edition, Cambridge University Press, Cambridge, 1997. R. C. Vaughan and T. D. Wooley, Waring’s problem: a survey, Number theory for the millennium III, 301–340, A. K. Peters, Natick, MA, 2002. I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 291–294 (in Russian). E. Waring, Meditationes Arithmeticæ, 2nd edition, Archdeacon, Cambridge, 1770. T. D. Wooley, The asymptotic formula in Waring’s problem, Int. Math. Res. Not. 2012, 1485–1504. T. D. Wooley, Multigrade efficient congruencing and Vinogradov’s mean value theorem, Proc. Lond. Math. Soc. (3) 111 (2015), 519–560. T. D. Wooley, On Waring’s problem for intermediate powers, arXiv:1602.03221. [^1]:
--- abstract: 'We investigate how dark energy properties impact the cosmological limits on the total mass of active neutrinos. We consider two typical, simple dark energy models (that have only one more additional parameter than $\Lambda$CDM), i.e., the $w$CDM model and the holographic dark energy (HDE) model, as examples, to make an analysis. In the cosmological fits, we use the Planck 2015 temperature and polarization data, in combination with other low-redshift observations, including the baryon acoustic oscillations, type Ia supernovae, and Hubble constant measurement, as well as the Planck lensing measurements. We find that, once dynamical dark energy is considered, the degeneracy between $\sum m_\nu$ and $H_0$ will be changed, i.e., in the $\Lambda$CDM model, $\sum m_\nu$ is anti-correlated with $H_0$, but in the $w$CDM and HDE models, $\sum m_\nu$ becomes positively correlated with $H_0$. Compared to $\Lambda$CDM, in the $w$CDM model the limit on $\sum m_\nu$ becomes much looser, but in the HDE model the limit becomes much tighter. In the HDE model, we obtain $\sum m_\nu<0.113$ eV (95% CL) with the combined data sets, which is perhaps the most stringent upper limit by far on neutrino mass. Thus, our result in the HDE model is nearly ready to diagnose the neutrino mass hierarchy with the current cosmological observations.' author: - Xin Zhang title: Impacts of dark energy on weighing neutrinos after Planck 2015 --- The solar and atmospheric neutrino experiments have shown that neutrinos are massive and that there is significant mixing between different neutrino species (see [@Lesgourgues:2006nd] for a review). However, the measurement of the absolute neutrino mass scale is a challenge for experimental particle physics. The neutrino oscillation experiments can only measure the squared mass differences between the neutrino mass eigenstates. Cosmological observations are, nevertheless, more prone to be capable of detecting the effects of the absolute neutrino mass. The cosmic microwave background (CMB) observations, combined with large-scale structure and distance measurements, have been providing tight limits on the total mass of neutrinos (see, e.g., [@Ade:2013zuv; @Ade:2015xua], and references therein). Massive neutrinos could affect the CMB anisotropies and matter clustering, thus providing a potential way to weigh them through the CMB and large-scale structure observations. The fact that neutrinos have masses changes the redshift of matter-radiation equality epoch $z_{\rm eq}$, and thus affects the position and amplitude of the acoustic peaks in the CMB power spectrum. The neutrino mass will change the angular diameter distance to the last scattering surface, $D_A(z_\ast)$. To ensure the same observed acoustic peak scale $\theta_\ast$, the effect on the background cosmology can be compensated by changing other background parameters, such as the Hubble constant $H_0$ and the equation-of-state parameter of dark energy $w$ (or other parameters characterizing dark energy properties). This is why the neutrino mass $\sum m_\nu$ (summed over the three neutrino families) is degenerated with $H_0$ and $w$. The neutrino mass also affects the slope of the CMB power spectrum at low multipoles due to the integrated Sachs-Wolfe (ISW) effect that describes the energy change of CMB photons caused by the decay of the gravitational potentials during radiation domination (early ISW effect) or dark energy domination (late ISW effect). In addition, due to the large thermal velocity of neutrinos which leads to a large free-streaming scale, the massive neutrinos could free-stream out of matter perturbations, and thus introduce a scale-dependent suppression of the clustering amplitude. This also affects the late-time effect of lensing on the CMB power spectrum, i.e., massive neutrinos also suppress the lensing potential. The Planck 2015 CMB data have provided the tight limits on the total mass of active neutrinos, $\sum m_\nu$ [@Ade:2015xua]. The base $\Lambda$CDM model assumes a normal mass hierarchy of neutrinos with $\sum m_\nu\approx 0.06$ eV (dominated by the heaviest neutrino mass eigenstate). When the model is extended to allow for larger neutrino masses, a reasonable assumption is that the three species of neutrinos have degenerate masses, neglecting the small differences between mass eigenstates. For the $\Lambda$CDM model, the Planck data (Planck TT+lowP) give the constraint $\sum m_\nu<0.72$ eV.[^1] Here, “lowP” denotes the Planck low-$\ell$ temperature-polarization data. It is known that the CMB data alone have a limitation to constrain the neutrino mass due to the acoustic scale degeneracy with $H_0$. So, it is necessary to combine the CMB data with other late-time cosmological probes in order to break the degeneracy. Adding the baryon acoustic oscillation (BAO) data could help to break the acoustic scale degeneracy and tighten the constraint on $\sum m_\nu$ substantially. The Planck TT+lowP+BAO data combination changes the limit to: $\sum m_\nu<0.21$ eV. Since the full Planck mission released the first analysis of the Planck polarization data, one could add the polarization data to the constraint, which will further tighten the neutrino mass limit. The combination of Planck TT, TE, EE+lowP+BAO leads to the limit $\sum m_\nu<0.17$ eV. Note that all the upper limit values for neutrino mass quoted in this paper refer to the 95% confidence level (CL). It should also be noticed that dark energy properties would impact the constraints on neutrino mass with cosmological observations. The cosmological limits on neutrino mass in some dynamical dark energy models have been discussed in the literature [@Li:2012vn; @Wang:2012uf; @Zhang:2014ifa; @Zhang:2015rha]. In this paper, we will investigate how a dynamical dark energy impacts the measurements of neutrino mass $\sum m_\nu$ with the Planck 2015 data, compared to the case of $\Lambda$CDM. Though the cosmological constant (or vacuum energy) $\Lambda$ is the simplest candidate of dark energy and the $\Lambda$CDM model can fit various cosmological observations quite well, it always suffers from the severe theoretical challenges, and actually some other possibilities for dark energy could not be excluded currently. But there exists too many seemingly viable dark energy models originating from various physical considerations. Of course, in practice, it is not possible, and not necessary, to discuss them one by one. A feasible scheme is to choose some typical dark energy models, as the simple extensions to $\Lambda$CDM, to detect the effects of dark energy properties on weighing neutrinos. In this paper we only consider the simplest dynamical dark energy models, which means that the models we choose are those having only one more parameter compared to $\Lambda$CDM. The first one is the $w$CDM model in which dark energy has a constant equation of state (EoS) $w$. Though it is simple, it seems that we have no reason to let $w$ remain constant in the actual physical consideration. Usually, one can test a time-varying EoS by adopting the parametrization of $w(a)=w_0+w_a(1-a)$. But this will introduce one more additional parameter, and so we do not consider such a case in this paper. Actually, a more reasonable way in this regard to test a time-varying EoS is to consider the holographic dark energy (HDE) model [@Li:2004rb; @Huang:2004ai; @Zhang:2014ija] which has the only additional parameter $c$ in the definition of its energy density, $\rho_{\rm de}=3c^2 M_{\rm Pl}^2 R_{\rm EH}^{-2}$, where $M_{\rm Pl}$ is the reduced Planck mass and $R_{\rm EH}$ is the event horizon size of the universe. Note here that $c$ is not the speed of light (actually we adopt in this paper the natural units in which the speed of light equals one), but the parameter of HDE. In the HDE model, $c$ is a dimensionless parameter that solely determines the evolution of dark energy [@Zhang:2005hs; @Zhang:2007sh; @Zhang:2006av; @Zhang:2006qu; @Li:2009bn]; see the equations (2.4)–(2.7) in [@Zhang:2015rha] for the evolution of dark energy in the HDE model with massive neutrinos and dark radiation. We use the Planck 2015 CMB power spectra data, in combination with other astrophysical data, to place constraints on the neutrino mass in the two considered dynamical dark energy models, and then make a comparison with the case of $\Lambda$CDM. We use the Planck 2015 CMB temperature and polarization data [@Aghanim:2015xee] in our calculations. We consider the combination of the likelihood at $30\leq \ell\leq 2500$ in the TT, TE, and EE power spectra and the Planck low-$\ell$ likelihood in the range of $2\leq\ell\leq 29$, which is denoted as “Planck TT,TE,EE+lowP,” following the nomenclature of the Planck collaboration [@Ade:2015xua]. In order to break the geometric degeneracy, it is necessary to consider the BAO data. Following [@Ade:2015xua], we use the BAO data of the six-degree-field galaxy survey (6dFGS) at $z_{\rm eff}=0.106$ [@Beutler:2011hx], the SDSS main galaxy sample (MGS) at $z_{\rm eff}=0.15$ [@Ross:2014qpa], the baryon oscillation spectroscopic survey (BOSS) “LOWZ” at $z_{\rm eff}=0.32$ [@Anderson:2013zyy], and the BOSS CMASS (i.e., “constant mass” sample) at $z_{\rm eff}=0.57$ [@Anderson:2013zyy]. Our basic data combination adopted in this paper is the Planck TT,TE,EE+lowP+BAO combination. To simultaneously constrain dark energy parameters, one needs to include more low-redshift measurements. We thus consider the type Ia supernova (SN) data and the Hubble constant measurement. For the SN data, we use the “joint light-curve analysis” (JLA) sample [@Betoule:2014frx]. For the Hubble constant direct measurement, we use the value given by Efstathiou [@Efstathiou:2013via], $H_0=70.6\pm3.3$ km s$^{-1}$ Mpc$^{-1}$ (derived from a re-analysis of the Cepheid data of Riess et al. [@Riess:2011yx] using the revised geometric maser distance to NGC 4258). In addition, since the CMB lensing can provide additional information at low redshifts, it is also useful to employ the Planck lensing likelihood [@Ade:2015zua] in our calculations. In addition, to probe the neutrino mass and dark energy properties, the measurements of growth of structure are rather important. For example, the observations of redshift space distortions (RSD), weak gravitational lensing (WL), and galaxy cluster counts have been used to search for massive neutrinos [@Zhang:2014dxk; @Dvorkin:2014lea; @Zhang:2014nta; @Li:2014dja; @Archidiacono:2014apa; @Bergstrom:2014fqa; @Leistedt:2014sia; @Beutler:2014yhv; @Mantz:2014paa; @DiValentino:2015wba; @Rossi:2014nea; @DiValentino:2015sam] and to distinguish between effects of dark energy and modified gravity [@Ade:2015rim; @Samushia:2012iq; @Beutler:2013yhm; @Zhang:2014lfa; @Li:2015poa], and the combination of Planck, RSD and WL data does prefer extensions to the base $\Lambda$CDM cosmology. However, it is believed that currently significant, uncontrolled systematics still remains, more or less, in these measurements. Therefore, we do not use RSD, WL, or cluster counts measurements for combined constraints in this paper. We use the [CosmoMC]{} package [@Lewis:2002ah] to infer the posterior probability distributions of parameters. We set flat priors for the base parameters; the prior ranges for the parameters are chosen to be much wider than the posterior in order not to affect the results of parameter estimation. The perturbations in dark energy are also considered in our calculations, which is physically necessary, though for smooth dark energy the clustering of dark energy inside the horizon is strongly suppressed. To deal with the perturbation divergence problem at the $w=-1$ crossing [@Zhao:2005vj] in some dynamical dark energy models such as the HDE model, we employ the “parametrized post-Friedmann” (PPF) framework [@Hu:2007pj; @Hu:2008zd] as implemented in [CAMB]{} [@Fang:2008sn] (see also [@Li:2014eha; @Li:2014cee; @Li:2015vla]). In the following, we report the results of the parameter estimation. [ccccccccc]{}Data &&&&\ Model & $\Lambda$CDM &$ w$CDM&HDE &&$\Lambda$CDM &$w$CDM &HDE\ $\Omega_{\rm b}h^2$&$0.02228\pm0.00015$&$0.02223^{+0.00016}_{-0.00015}$&$0.02228\pm0.00015$&&$0.02229\pm0.00014$&$0.02226\pm0.00015$&$0.02237\pm0.00015$\ $\Omega_{\rm c}h^2$&$0.1192\pm0.0011$&$0.1197\pm0.0014$&$0.1191\pm0.0013$&&$0.1187\pm0.0011$&$0.119\pm0.0012$&$0.1177^{+0.0011}_{-0.0012}$\ $100\theta_{\rm MC}$&$1.04083^{+0.00030}_{-0.00031}$&$1.04075^{+0.00032}_{-0.00031}$&$1.04086\pm0.00032$&&$1.0409\pm0.00029$&$1.04086\pm0.0003$&$1.04105\pm0.0003$\ $\tau$&$0.082\pm0.017$&$0.081\pm0.018$&$0.086\pm0.017$&&$0.068^{+0.014}_{-0.016}$&$0.068^{+0.015}_{-0.016}$&$0.083\pm0.014$\ $w/c$&...&$-1.068^{+0.077}_{-0.070}$&$0.533^{+0.048}_{-0.056}$&&...&$-1.043^{+0.056}_{-0.047}$&$0 .633^{+0.032}_{-0.039}$\ $\sum m_\nu\,[\rm eV]$&$< 0.177$&$< 0.328$&$< 0.168$&&$<0.197$&$<0.304$&$<0.113$\ $n_{\rm s}$&$0.9659\pm0.0041$&$0.9645\pm0.0046$&$0.9663\pm0.0045$&&$0.9669^{+0.0041}_{-0.0040}$&$0.9656\pm0.0043$&$0.9697\pm0.0044$\ ${\rm{ln}}(10^{10}A_{\rm s})$&$3.097\pm0.033$&$3.096^{+0.035}_{-0.034}$&$3.105^{+0.032}_{-0.033}$&&$3.067^{+0.027}_{-0.029}$&$3.067^{+0.027}_{-0.031}$&$3.096^{+0.027}_{-0.026}$\ $\Omega_{\rm m}$&$0.3128^{+0.0073}_{-0.0075}$&$0.304\pm0.014$&$0.276^{+0.016}_{-0.015}$&&$0.3109^{+0.0070}_{-0.0079}$&$0.3068^{+0.0092}_{-0.0093}$&$0.3008^{+0.0090}_{-0.0098}$\ $H_0$&$67.6\pm0.6$&$68.7^{+1.6}_{-1.9}$&$71.9^{+2.0}_{-2.4}$&&$67.55^{+0.64}_{-0.56}$&$68.2\pm1.0$&$68.4\pm1.0$\ $\sigma_8$&$0.829^{+0.019}_{-0.016}$&$0.836\pm0.023$&$0.862\pm0.025$&&$0.811^{+0.015}_{-0.011}$&$0.813^{+0.017}_{-0.014}$&$0.818\pm0.013$\ $\chi^{2}_{\rm min}$ &12940.94 &12939.28 &12945.50 && 13659.04&13655.89 &13671.50\ \[tab1\] ![image](data1.pdf){width="17cm"} ![image](data2.pdf){width="17cm"} ![image](wcdm.pdf){width="6cm"} ![image](hde.pdf){width="6cm"} We discuss the fitting results of the three models, $\Lambda$CDM, $w$CDM, and HDE, under the constraints from the two data combinations, i.e., Planck TT, TE, EE + lowP + BAO and Planck TT, TE, EE + lowP + BAO + lensing + SN + $H_0$. In these models, we consider the total mass of three species of active neutrinos, $\sum m_\nu$. The degenerate-mass assumption is made for the three species of neutrinos, as mentioned above. Note also that the three cosmological models have different numbers of parameters: the $\Lambda$CDM model has seven base parameters, and the $w$CDM and HDE models have eight base parameters. The fitting results are shown in Table \[tab1\] and Figs. \[fig1\]–\[fig3\]. In Table \[tab1\], we present the fit values with $\pm1\sigma$ errors for the parameters, but for the total mass of neutrinos $\sum m_\nu$, we give the 95% CL upper limits. In Figs. \[fig1\]–\[fig3\], we show the 68% and 95% CL posterior distribution contours for the models. In the case of Planck TT,TE,EE+lowP+BAO constraints, we have $\sum m_\nu<0.177$ eV for $\Lambda$CDM, $\sum m_\nu<0.328$ eV for $w$CDM, and $\sum m_\nu<0.168$ eV for HDE. We find that in the $w$CDM model, the upper limit of $\sum m_\nu$ is much bigger than that in the $\Lambda$CDM model. But in this case, the HDE model yields a slightly smaller limit value of neutrino mass than the $\Lambda$CDM model. Comparing the $\chi^2$ values of the three models in the fit, we find that the $w$CDM model performs slightly better than the $\Lambda$CDM model by $\Delta\chi^2=-1.66$, at the expense of adding one more parameter. But the HDE model performs worse than $\Lambda$CDM by $\Delta\chi^2=4.56$ though it has one more parameter. A careful check reveals that the reason for this is that the HDE model cannot fit the BAO point at $z_{\rm eff}=0.57$ well in the global fit, of which $\Delta\chi^2$ contributes solely about 3. In Fig. \[fig1\], we compare the constraint results of the three models in the fit to the Planck TT,TE,EE+lowP+BAO data combination. We find that the fit values of $\Omega_{\rm b}h^2$ and $\Omega_{\rm c}h^2$ are similar for the three models, but the results of $\Omega_{\rm m}$, $H_0$, and $\sigma_8$ are quite different. This indicates that the dark energy properties play an important role in changing the fit results. The Planck data have accurately measured the acoustic peaks and thus the observed angular size of acoustic scale $\theta_\ast=r_s/D_A$ is determined to a high precision (much better than 0.1% precision at 1$\sigma$). This places tight constraints on some combinations of the cosmological parameters that determine $r_s$ and $D_A$. In the cosmological fit using the Planck data, the parameter combinations must be constrained to be close to a surface of constant $\theta_\ast$. This surface depends on the models that are assumed. In the $\Lambda$CDM model, the precise determination of $\theta_\ast$ yields a nearly constant $\Omega_{\rm m}h^3$. But for the $w$CDM and HDE models, this is not true; the distributions of $\Omega_{\rm m}h^3$ in these two models are much broader than in the $\Lambda$CDM model [@Li:2013dha]. However, the three models have the similar distributions of $\Omega_{\rm m}h^2$ [@Li:2013dha]. The constraints on $\Omega_{\rm m}h^3$ and $\Omega_{\rm m}h^2$ lead to the fact that $H_0$ can be tightly constrained in the $\Lambda$CDM model, but cannot be so well constrained in dynamical dark energy models. Therefore, we find that dark energy properties significantly affect the determination of $\Omega_{\rm m}h^3$ and thus the value of $H_0$. From Fig. \[fig1\], we also find that $\sum m_\nu$ is anti-correlated with $H_0$ in the $\Lambda$CDM model, but is positively correlated with $H_0$ in the $w$CDM and HDE models. The degeneracy in the parameter space gives rise to consistent changes in parameters including $H_0$, $\Omega_{\rm m}$, $\sum m_\nu$, and $w$ (or $c$), so that the ratio of the sound horizon and angular diameter distance remains nearly constant. Changes in the dark energy density due to its dynamical properties (characterized by $w$ or $c$ in our cases) could have some effects on the parameters such as $H_0$ and $\sum m_\nu$ because they would change to compensate. Hence, the impacts of dark energy lead to the changes of correlation between $\sum m_\nu$ and $H_0$. Neutrino masses could suppress the powers of CMB, which can be compensated by increasing the amplitude of primordial spectrum $A_s$. The measurements of the CMB temperature power spectrum provide a highly accurate measurement of the combination $A_s e^{-2\tau}$, where $\tau$ is the reionization optical depth parameter, which leads to a strong degeneracy (positive correlation) between $A_s$ and $\tau$. Thus we can infer that $\sum m_\nu$ is positively correlated with $\tau$, which is confirmed in Fig. \[fig1\] for all the three models. We find that the current CMB+BAO data prefer a high value of $\tau$ ($\tau\sim 0.081-0.086$) for all the models. As shown in [@Ade:2015xua], the CMB lensing data could break the degeneracy between $A_s$ and $\tau$, and in the case of $\Lambda$CDM, the value of $\tau$ could be lowered by adding lensing data. In addition, due to the free-streaming of massive neutrinos, larger masses tend to prefer lower $\sigma_8$, thus $\sum m_\nu$ is anti-correlated with $\sigma_8$, as confirmed by the constraint results for all the models in Fig. \[fig1\]. Adding the BAO data helps partly break the geometric degeneracies, but not enough. In order to further break the degeneracies, in particular, to accurately probe the dark energy properties and measure neutrino mass, one needs to add more low-redshift observations, such as SN and $H_0$, as well as CMB lensing. Figure \[fig2\] shows the comparison of the three models under the constraints from Planck TT,TE,EE+lowP+BAO+lensing+SN+$H_0$. In this case, we have $\sum m_\nu<0.197$ eV for $\Lambda$CDM, $\sum m_\nu<0.304$ eV for $w$CDM, and $\sum m_\nu<0.113$ eV for HDE. Thus, we see that adding the low-redshift data changes the values of upper limit of $\sum m_\nu$. For the $\Lambda$CDM model, the constraint becomes slightly weaker; for the $w$CDM model, the constraint becomes slightly tighter; and for the HDE model, the constraint is significantly tightened. We also find that adding the CMB lensing data indeed helps improve the measurement of $\tau$. For the $\Lambda$CDM and $w$CDM models, the values of $\tau$ are significantly lowered by adding lensing data, but for the HDE data, the central value of $\tau$ is still at 0.083 in this case. To directly show how dark energy properties affect the constraints on neutrino mass, we plot the two-dimensional posterior distribution contours (68% and 95% CL) in the $\sum m_\nu$–$w$ plane for $w$CDM and in the $\sum m_\nu$–$c$ plane for HDE, under the constraints from the both two data combinations, in Fig. \[fig3\]. We find that, in the $w$CDM model, $\sum m_\nu$ is anti-correlated with $w$, and in the HDE model, $\sum m_\nu$ is also anti-correlated with $c$. Evidently, adding low-redshift data tightens the constraints on dark energy parameters and thus changes the limits on neutrino mass. For the $w$CDM model, we have $w=-1.068^{+0.077}_{-0.070}$ from CMB+BAO, and it changes to $w=-1.043^{+0.056}_{-0.047}$ when the low-redshift data are added. For the HDE model, we have $c=0.533^{+0.048}_{-0.056}$ from CMB+BAO, and it changes to $c=0.633^{+0.032}_{-0.039}$ when the low-redshift observations are included. We find that, for the HDE model, the Planck CMB data prefer a low value of $c$, and the low-redshift observations tend to drive $c$ upward to a higher value, which is in accordance with the conclusions in previous studies [@Zhang:2015rha; @Li:2013dha]. The changes of $c$ strikingly impact the measurements of neutrino mass, and thus we obtain the extremely stringent limit, $\sum m_\nu<0.113$ eV, in this case. Since $\sum m_\nu$ must be greater than approximately 0.1 eV in the inverted mass hierarchy (degenerate hierarchy in cosmology) [@GonzalezGarcia:2012sz], our result in the HDE model is nearly ready to diagnose the neutrino mass hierarchy with the current cosmological probes. In conclusion, the dark energy properties could significantly impact the constraint limits on neutrino mass $\sum m_\nu$. To test how dynamical dark energy would affect the constraints on $\sum m_\nu$, we employ two typical, simple dark energy models as examples, i.e., the $w$CDM model and the HDE model, to make an analysis. They both have only one more parameter than the $\Lambda$CDM model. We use the Planck 2015 temperature and polarization data, in combination with other low-redshift observations, to constrain the models. The acoustic scale degeneracy leads to consistent changes in parameters such as $H_0$, $\sum m_\nu$, and $w$ (or $c$), which ensures that the ratio of the sound horizon and angular diameter distance remains nearly constant. Thus the dark energy parameters could have effects on the parameters such as $H_0$ and $\sum m_\nu$ since they would change to compensate. This leads to the fact that, in the $\Lambda$CDM model, $\sum m_\nu$ is anti-correlated with $H_0$, but once dynamical dark energy is introduced, $\sum m_\nu$ becomes positively correlated with $H_0$, as shown in the $w$CDM and HDE models (Figs. \[fig1\] and \[fig2\]). Our analysis also shows that $\sum m_\nu$ is anti-correlated with $w$ in the $w$CDM model and is anti-correlated with $c$ in the HDE model (Fig. \[fig3\]). We find that in the $w$CDM model the limits on $\sum m_\nu$ become much looser, but in the HDE model the limits become much more stringent. Using the Planck TT,TE,EE+lowP+BAO data, we obtain $\sum m_\nu<0.177$ eV for $\Lambda$CDM, $\sum m_\nu<0.328$ eV for $w$CDM, and $\sum m_\nu<0.168$ eV for HDE. Using the Planck TT,TE,EE+lowP+BAO+lensing+SN+$H_0$ data, we obtain $\sum m_\nu<0.197$ eV for $\Lambda$CDM, $\sum m_\nu<0.304$ eV for $w$CDM, and $\sum m_\nu<0.113$ eV for HDE. Therefore, we conclude that in the HDE model we can get perhaps the most stringent upper limit by far on the total mass of active neutrinos, $\sum m_\nu<0.113$ eV, with the combined cosmological data sets. Our study shows that, if dark energy is not a cosmological constant, then the allowed neutrino mass window could become much tighter and would be further closed by forthcoming observations. We will leave further relevant discussions on this subject to a forthcoming paper [@zhaomm]. The author would like to thank Yun-He Li, Jing-Fei Zhang, Ming-Ming Zhao, and Shun Zhou for helpful discussions. This work is supported by the Top-Notch Young Talents Program of China, the National Natural Science Foundation of China (Grants No. 11522540 and No. 11175042), and the Fundamental Research Funds for the Central Universities (Grants No. N140505002, No. N140506002, and No. L1505007). J. Lesgourgues and S. Pastor, Phys. Rept. [429]{}, 307 (2006) \[astro-ph/0603494\]. P. A. R. Ade [et al.]{} \[Planck Collaboration\], Astron. Astrophys. [571]{}, A16 (2014) \[arXiv:1303.5076 \[astro-ph.CO\]\]. P. A. R. Ade [et al.]{} \[Planck Collaboration\], arXiv:1502.01589 \[astro-ph.CO\]. H. Li and X. Zhang, Phys. Lett. B [713]{}, 160 (2012) \[arXiv:1202.4071 \[astro-ph.CO\]\]. Y. H. Li, S. Wang, X. D. Li and X. Zhang, JCAP [1302]{}, 033 (2013) \[arXiv:1207.6679 \[astro-ph.CO\]\]. J. F. Zhang, J. J. Geng and X. Zhang, JCAP [1410]{}, no. 10, 044 (2014) \[arXiv:1408.0481 \[astro-ph.CO\]\]. J. F. Zhang, M. M. Zhao, Y. H. Li and X. Zhang, JCAP [1504]{}, 038 (2015) \[arXiv:1502.04028 \[astro-ph.CO\]\]. M. Li, Phys. Lett. B [603]{}, 1 (2004) \[hep-th/0403127\]. Q. G. Huang and M. Li, JCAP [0408]{}, 013 (2004) \[astro-ph/0404229\]. J. F. Zhang, M. M. Zhao, J. L. Cui and X. Zhang, Eur. Phys. J. C [74]{}, no. 11, 3178 (2014) \[arXiv:1409.6078 \[astro-ph.CO\]\]. X. Zhang and F. Q. Wu, Phys. Rev. D [72]{}, 043524 (2005) \[astro-ph/0506310\]. X. Zhang and F. Q. Wu, Phys. Rev. D [76]{}, 023502 (2007) \[astro-ph/0701405\]. X. Zhang, Phys. Lett. B [648]{}, 1 (2007) \[astro-ph/0604484\]. X. Zhang, Phys. Rev. D [74]{}, 103505 (2006) \[astro-ph/0609699\]. M. Li, X. D. Li, S. Wang and X. Zhang, JCAP [0906]{}, 036 (2009) \[arXiv:0904.0928 \[astro-ph.CO\]\]. N. Aghanim [et al.]{} \[Planck Collaboration\], \[arXiv:1507.02704 \[astro-ph.CO\]\]. F. Beutler [et al.]{}, Mon. Not. Roy. Astron. Soc. [416]{}, 3017 (2011) \[arXiv:1106.3366 \[astro-ph.CO\]\]. A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, Mon. Not. Roy. Astron. Soc. [449]{}, no. 1, 835 (2015) \[arXiv:1409.3242 \[astro-ph.CO\]\]. L. Anderson [et al.]{} \[BOSS Collaboration\], Mon. Not. Roy. Astron. Soc. [441]{}, no. 1, 24 (2014) \[arXiv:1312.4877 \[astro-ph.CO\]\]. M. Betoule [et al.]{} \[SDSS Collaboration\], Astron. Astrophys. [568]{}, A22 (2014) \[arXiv:1401.4064 \[astro-ph.CO\]\]. G. Efstathiou, Mon. Not. Roy. Astron. Soc. [440]{}, no. 2, 1138 (2014) \[arXiv:1311.3461 \[astro-ph.CO\]\]. A. G. Riess [et al.]{}, Astrophys. J. [730]{}, 119 (2011) \[Astrophys. J. [732]{}, 129 (2011)\] \[arXiv:1103.2976 \[astro-ph.CO\]\]. P. A. R. Ade [et al.]{} \[Planck Collaboration\], arXiv:1502.01591 \[astro-ph.CO\]. J. F. Zhang, Y. H. Li and X. Zhang, Phys. Lett. B [740]{}, 359 (2015) \[arXiv:1403.7028 \[astro-ph.CO\]\]. C. Dvorkin, M. Wyman, D. H. Rudd and W. Hu, Phys. Rev. D [90]{}, no. 8, 083503 (2014) \[arXiv:1403.8049 \[astro-ph.CO\]\]. J. F. Zhang, Y. H. Li and X. Zhang, Eur. Phys. J. C [74]{}, 2954 (2014) \[arXiv:1404.3598 \[astro-ph.CO\]\]. Y. H. Li, J. F. Zhang and X. Zhang, Sci. China Phys. Mech. Astron. [57]{}, 1455 (2014) \[arXiv:1405.0570 \[astro-ph.CO\]\]. M. Archidiacono, N. Fornengo, S. Gariazzo, C. Giunti, S. Hannestad and M. Laveder, JCAP [1406]{}, 031 (2014) \[arXiv:1404.1794 \[astro-ph.CO\]\]. J. Bergström, M. C. Gonzalez-Garcia, V. Niro and J. Salvado, JHEP [1410]{}, 104 (2014) \[arXiv:1407.3806 \[hep-ph\]\]. B. Leistedt, H. V. Peiris and L. Verde, Phys. Rev. Lett. [113]{}, 041301 (2014) \[arXiv:1404.5950 \[astro-ph.CO\]\]. F. Beutler [et al.]{} \[BOSS Collaboration\], Mon. Not. Roy. Astron. Soc. [444]{}, 3501 (2014) \[arXiv:1403.4599 \[astro-ph.CO\]\]. A. B. Mantz [et al.]{}, Mon. Not. Roy. Astron. Soc. [446]{}, 2205 (2015) \[arXiv:1407.4516 \[astro-ph.CO\]\]. E. Di Valentino, E. Giusarma, M. Lattanzi, O. Mena, A. Melchiorri and J. Silk, Phys. Lett. B [752]{}, 182 (2016) \[arXiv:1507.08665 \[astro-ph.CO\]\]. G. Rossi, C. Yèche, N. Palanque-Delabrouille and J. Lesgourgues, Phys. Rev. D [92]{}, no. 6, 063505 (2015) \[arXiv:1412.6763 \[astro-ph.CO\]\]. E. Di Valentino, E. Giusarma, O. Mena, A. Melchiorri and J. Silk, arXiv:1511.00975 \[astro-ph.CO\]. P. A. R. Ade [et al.]{} \[Planck Collaboration\], arXiv:1502.01590 \[astro-ph.CO\]. L. Samushia [et al.]{}, Mon. Not. Roy. Astron. Soc. [429]{}, 1514 (2013) \[arXiv:1206.5309 \[astro-ph.CO\]\]. F. Beutler [et al.]{} \[BOSS Collaboration\], Mon. Not. Roy. Astron. Soc. [443]{}, no. 2, 1065 (2014) \[arXiv:1312.4611 \[astro-ph.CO\]\]. J. F. Zhang, Y. H. Li and X. Zhang, Phys. Lett. B [739]{}, 102 (2014) \[arXiv:1408.4603 \[astro-ph.CO\]\]. Y. H. Li, J. F. Zhang and X. Zhang, Phys. Lett. B [744]{}, 213 (2015) \[arXiv:1502.01136 \[astro-ph.CO\]\]. A. Lewis and S. Bridle, Phys. Rev. D [66]{}, 103511 (2002) \[astro-ph/0205436\]. G. B. Zhao, J. Q. Xia, M. Li, B. Feng and X. Zhang, Phys. Rev. D [72]{}, 123515 (2005) \[astro-ph/0507482\]. W. Hu and I. Sawicki, Phys. Rev. D [76]{}, 104043 (2007) \[arXiv:0708.1190 \[astro-ph\]\]. W. Hu, Phys. Rev. D [77]{}, 103524 (2008) \[arXiv:0801.2433 \[astro-ph\]\]. W. Fang, W. Hu and A. Lewis, Phys. Rev. D [78]{}, 087303 (2008) \[arXiv:0808.3125 \[astro-ph\]\]. Y. H. Li, J. F. Zhang and X. Zhang, Phys. Rev. D [90]{}, no. 6, 063005 (2014) \[arXiv:1404.5220 \[astro-ph.CO\]\]. Y. H. Li, J. F. Zhang and X. Zhang, Phys. Rev. D [90]{}, no. 12, 123007 (2014) \[arXiv:1409.7205 \[astro-ph.CO\]\]. Y. H. Li, J. F. Zhang and X. Zhang, Phys. Rev. D [93]{}, no. 2, 023002 (2016) \[arXiv:1506.06349 \[astro-ph.CO\]\]. M. Li, X. D. Li, Y. Z. Ma, X. Zhang and Z. Zhang, JCAP [1309]{}, 021 (2013) \[arXiv:1305.5302 \[astro-ph.CO\]\]. M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP [1212]{}, 123 (2012) \[arXiv:1209.3023 \[hep-ph\]\]. M. M. Zhao and X. Zhang, in preparation. [^1]: The CMB data alone can only loosely constrain the total neutrino mass. For dynamical dark energy models, the situation is almost the same. For example, for the $w$CDM model and the holographic dark energy model, our numerical calculations show that the Planck TT+lowP data give $\sum m_\nu<0.76$ eV and $\sum m_\nu<0.61$ eV, respectively.
--- abstract: 'The acceleration of the universe can be explained either through dark energy or through the modification of gravity on large scales. In this paper we investigate modified gravity models and compare their observable predictions with dark energy models. Modifications of general relativity are expected to be scale-independent on super-horizon scales and scale-dependent on sub-horizon scales. For scale-independent modifications, utilizing the conservation of the curvature scalar and a parameterized post-Newtonian formulation of cosmological perturbations, we derive results for large scale structure growth, weak gravitational lensing, and cosmic microwave background anisotropy. For scale-dependent modifications, inspired by recent $f(R)$ theories we introduce a parameterization for the gravitational coupling $G$ and the post-Newtonian parameter $\gamma$. These parameterizations provide a convenient formalism for testing general relativity. However, we find that if dark energy is generalized to include both entropy and shear stress perturbations, and the dynamics of dark energy is unknown a priori, then modified gravity cannot in general be distinguished from dark energy using cosmological linear perturbations.' author: - Edmund Bertschinger and Phillip Zukin title: Distinguishing Modified Gravity from Dark Energy --- Introduction {#sec:intro} ============ Cosmic acceleration has been revealed by measurements of the redshift-distance relation $\chi(z)$ where $\chi$ is the comoving radial distance. The Hubble expansion rate follows from $H(z)=(d\chi/dz)^{-1}$ (in units where $c=1$). This determination assumes only that the observable universe is adequately described by the Robertson-Walker metric, an assertion that is testable empirically [@Hogg04; @LuHellaby07] and does not imply the validity of general relativity (GR). The inference of dark energy follows once the Einstein field equations of general relativity are imposed on the Robertson-Walker metric yielding the Friedmann equations. These equations imply that a stress-energy-momentum component with negative pressure is needed to explain cosmic acceleration. This substance may be vacuum energy (i.e., a cosmological constant, giving rise to the $\Lambda$CDM model) or a scalar field [@Copeland06]. The dark energy equation of state for uniform expansion, $p(\rho)$, can be determined from measurement of $\rho(z)$, which itself follows from $H(z)$ combined with the first Friedmann equation. Measuring $w(z)=p/\rho$ is the primary goal of dark energy experiments [@DETF]. Another possibility is that general relativity requires modification on large distance scales and at late times in the universe. In this case cosmic acceleration would arise not from dark energy as a substance but rather from the dynamics of modified gravity. Modified gravity is not particularly attractive theoretically, but the observed cosmic acceleration is so surprising that all plausible explanations should be considered. Observations of the cosmic expansion history cannot distinguish dark energy from modified gravity [@fRexphist; @Song06]. Testing gravity requires exploring the evolution of spatial inhomogeneity (e.g. [@ZLBD07; @inhom] and references therein). Modified gravity theories must pass tests within the solar system and in relativistic binaries [@Will06]. They are expected to show significant departures from general relativity only on cosmological distance scales. The combination of cosmic microwave background anisotropy, weak gravitational lensing, and the growth of clustering of dark matter and galaxies provides an opportunity to discriminate between dark energy and modified gravity. Performing cosmological tests of modified gravity requires a set of predictions. There are two approaches to generating these predictions. In the first, a theory is specified by its Lagrangian (or other fundamental description) which provides the equations of motion for both homogeneous expansion and cosmological perturbations. A class of theories can be specified by giving a Lagrangian with free parameters, e.g. $f(R)$ theories where $R$ is the Ricci scalar [@Song06]. A second approach is inspired by the parameterized post-Newtonian framework for solar system tests [@ThorneWill71]. Here one begins with the solution of the gravitational field equations (i.e., the metric) instead of the Lagrangian. Several authors have recently adopted this framework or a similar one [@HuSawicki07b; @Caldwell07; @Amendola07; @Amin07; @Linder07]. The difficulty here is to find a good “Newtonian” description in cosmology to which one adds “Post-Newtonian” parameters. Metric perturbations in the scalar sector governing the growth of cosmic structure are characterized by two spatial scalar fields, $\Phi$ (the Newtonian potential) and $\Psi$ (the spatial curvature potential). Even if we introduce the relationship $\Psi=\gamma\Phi$ with Eddington parameter $\gamma$ [@Eddington22], there remains one unknown function of space and time. In the solar system case, by contrast, $\Phi=-GM/r$ is known to provide an excellent approximation to planetary dynamics. On solar system scales, and even within galaxies, one can use test-particle orbits to determine $\Phi$, and light rays to determine $\Phi+\Psi$. This comparison yields impressive limits on $\|\gamma-1\|$ in the solar system [@Will06]. However, on a scale of several kpc, gravitational lensing combined with stellar dynamics in elliptical galaxies yields a current best result $\|\gamma-1\|=0.02\pm0.07$ [@Bolton06]. At the scale of Gpc where dark energy appears to drive accelerated expansion, there are no longer any bound test-particle orbits to measure $\Phi$, so a different approach, based on cosmological perturbation theory, is needed. Previous work in the cosmological parameterized post-Newtonian framework has either assumed that some of the Einstein field equations remain valid with modified gravity [@Caldwell07] or has examined the dynamics of individual theories, e.g.[@Dore07]. Neither approach is ideal. One would prefer to sample all possible theories in a broad class, and for each theory to constrain the potentials by a consistency condition that does not assume general relativity or any particular modification thereof. Such a consistency condition was found recently in Ref.[@Bertschinger06] for the long-wavelength perturbations of a Robertson-Walker spacetime. This result was derived assuming that gravity is described by a classical four-dimensional metric theory having a well-defined infrared limit (i.e., the theory is well behaved for very long wavelength perturbations). For practical application, assumptions must also be made about the background spatial curvature and entropy perturbations. Assuming an inflationary or equivalent origin of perturbations, long-wavelength isentropic perturbations are imprinted in the spatial curvature on a flat background. In general relativity, these spatial curvature fluctuations, represented by the gauge-invariant $\zeta$ field of Bardeen et al. [@BST83] or the ${\cal R}$ field of Lyth [@Lyth85], are time-independent in the long-wavelength limit. Ref. [@Bertschinger06] presented a derivation of the conserved curvature perturbation (calling it $\kappa$) making no assumption about the field equations except that they have a well-defined infrared limit. Physically this means that curvature perturbations are small and that all waves propagate causally. In what follows, the curvature perturbation introduced in Ref. [@Bertschinger06] will be called $\zeta$ although its definition differs from that of Ref. [@BST83]. For long wavelength perturbations on a flat background, ${\cal R}=\zeta$. In the long-wavelength limit all cosmological perturbations factorize into functions of time multiplying the curvature perturbation or its spatial derivatives. This is true in GR and in modified gravity theories that are well-behaved in the infrared limit. One might naively expect this factorization to hold only on scales larger than the Hubble length. In general relativity, however, signals in the scalar sector propagate at the speed of sound, not the speed of light [@Bertschinger96], leading to conservation of $\zeta$ on scales larger than the Jeans length. To satisfy solar system tests, modified gravity theories for cosmic acceleration must introduce a length scale $L_G$ below which general relativity is recovered. This length scale might be associated, for example, with the dynamics of new scalar degrees of freedom. The value of this length scale, compared with the size of the systems investigated, plays a crucial role in characterizing the behavior of modified gravity theories. If $L_G$ is much smaller than the length scales over which linear cosmological structure formation is measured (e.g., $L_G=1$ Mpc), then the factorization of cosmological perturbations on scales larger than the Jeans length remains valid. We denote this case scale-independent modified gravity. These theories are like GR in that the curvature perturbation is conserved for the relevant length scales. This condition yields a great simplification of the dynamics, reducing cosmological perturbations to quadratures. In GR, waves propagating in the scalar sector travel only at the speed of sound, so that scale-dependence of transfer functions arises only below the Jeans length. Modified gravity theories, however, typically have additional fields supporting waves that travel at the speed of light. In this case, $L_G$ is the Hubble length and the factorization of cosmological perturbations no longer holds. Theories of this type are called scale-dependent modified gravity theories. Now two quantities, $\gamma$ and the gravitational coupling $G_\Phi$ (the generalization of Newton’s constant in the Poisson equation for $\Phi$), are needed to characterize gravity, and both will vary with length scale as well as with time. Even such complicated models can still be approximated by parameterizations, as we will discuss below. Ref. [@HuSawicki07b] found a way of bridging super- and sub-horizon modifications to GR. Our parametrization, while not as general, is simpler because it only involves a few free parameters and no free functions. Because we do not start with a Lagrangian, we cannot explain cosmic acceleration. We take the cosmic expansion history as given from observations. Rather than providing a complete theory of modified gravity, we provide a framework for observational tests of gravity in cosmology. This paper is organized as follows. Section \[sec:long\] describes the curvature perturbation and its use to build scale-independent modified gravity theories. Section \[sec:observe\] works out the growth of structure on sub-horizon scales, shows that the Poisson equation is modified, and derives results for cosmic microwave background anisotropy and weak gravitational lensing for scale-independent modified gravity theories. Section \[sec:f(R)\] then examines the sub-horizon behavior of a currently popular class of theories known as $f(R)$ models and shows that they are scale-dependent. Section \[sec:shear\] considers the alternative hypothesis that dark energy has a peculiar stress tensor while gravity is governed by GR. Finally, results are summarized and conclusions are presented in Section \[sec:discuss\]. Gravity at long wavelengths {#sec:long} =========================== Our starting point is the perturbed Robertson-Walker metric in conformal Newtonian gauge [@MFB92]: $$\label{pertFRW} ds^2=a^2(t)[-(1+2\Phi)dt^2+(1-2\Psi)\gamma_{ij}dx^idx^i]\ ,$$ where $t$ is conformal time, $a(t)=1/(1+z)$ is the expansion scale factor, and $\gamma_{ij}({\bf x},K)$ is the three-metric for a space of constant spatial curvature $K$, e.g. $\gamma_{ij}dx^idx^j=d\chi^2+r^2(\chi,K)d\Omega^2$ where $r(\chi,K)\sqrt{K}=\sin(\chi\sqrt{K})$ for $K>0$ and is analytically continued for $K\le0$. Note that different conventions appear in the literature for the metric perturbations: $\Phi=\Psi_{\rm Hu}=\psi$ and $\Psi=-\Phi_{\rm Hu}=\phi$ where $(\Psi_{\rm Hu},\Phi_{\rm Hu})$ are the potentials of Ref. [@HuSawicki07b] and $(\psi,\phi)$ are the potentials of Ref. [@MaBert95]. Linear perturbation theory is assumed to be valid throughout this paper. The evolution of the scale factor can depend, in principle, on any quantities characterizing the geometry and composition of the Robertson-Walker background, for example the spatial curvature $K$ and the entropy density (or equivalently, parameters characterizing the equation of state). We neglect entropy perturbations and consider only curvature perturbations on a flat ($K=0$) background. Assuming that the unknown gravity theory has a well-defined infrared limit obeying causality, long wavelength curvature perturbations should evolve like separate Robertson-Walker universes. In this case it is possible to transform to a new set of coordinates, $t\to t-\alpha(t)$ and $\chi\to\chi(1+\zeta)$ where $\zeta$ is constant and $\dot\alpha=\Phi+\Psi-\zeta$, such that the new line element is eq. (\[pertFRW\]) with $\Phi=\Psi=0$ and having spatial curvature $K(1+2\zeta)$. Thus, $\zeta$ is one-half the spatial curvature perturbation. Enforcing the coordinate transformation leads to the consistency condition [@Bertschinger06] $$\label{consistent} \frac{1}{a^2}\frac{\partial}{\partial t}\left(\frac{a^2\Psi} {\cal H}\right)+\Phi-\Psi=\left[\frac{1}{a}\frac{\partial} {\partial t}\left(\frac{a}{\cal H}\right)+\frac{K}{{\cal H}^2} +O(k^2)\right]\zeta\ ,$$ where ${\cal H}=\dot a/a=aH$ and $k$ is the comoving wavenumber. Although Ref. [@Bertschinger06] states that large-scale shear stress is neglected in this result, in fact eq. (\[consistent\]) is valid for $k\to0$ in general relativity (and presumably in modified gravity theories) even if shear stress is present. The curvature term $K/{\cal H}^2$ has been computed assuming the Friedmann equation is valid; in modified gravity theories this term might be different but it must vanish when $K=0$. Hereafter we assume $K=0$ and drop the curvature term. Eq. (\[consistent\]) may be regarded as a definition of the curvature perturbation $\zeta$ for arbitrary theories of gravity. For long wavelengths, $\zeta$ is independent of time. Sound waves in the matter sector or wave propagation in the modified gravity sector cause $\zeta$ to change with time on small scales. These changes are implied by the neglected terms proportional to $k^2\zeta$ in eq. (\[consistent\]). For now we ignore such terms, in effect assuming that both the Jeans length and $L_G$ are smaller than the cosmological scales of interest. During the radiation-dominated era the Jeans length is comparable to the Hubble length, and $\zeta$ (and $\Phi$ and $\Psi$) is damped for scales smaller than the Jeans length. We assume that during this early period of evolution general relativity is an excellent approximation so that the damping is well described by the transfer functions computed using standard codes [@MaBert95; @cmbfast]. When modified gravity becomes important at low redshift, the Jeans length has dropped to a few Mpc or less. In practice, we modify CMBFAST only for $z<30$ and then do so in such a way as to enforce eq. (\[consistent\]) with $\zeta$ corrected from its primeval value using the GR transfer function at $z=30$. Assuming a well-defined infrared limit, the time and space dependence of perturbations must factorize for wavelengths longer than the Jeans length or $L_G$, e.g. $$\label{factorize} \Phi({\bf k},t)=F(a)\zeta({\bf k})+O(k^2\zeta)$$ where ${\bf k}$ is the wavevector. Factorization implies that the ratio of the two gravitational potentials depends only on time as $k\to0$. Therefore we may write, for any causal theory of gravity having a well-defined infrared limit, $$\label{gammadef} \Psi({\bf k},t)=\gamma(a)\Phi({\bf k},t)+O(k^2\zeta)\ .$$ In modified gravity theories, $\gamma(a)$ is the only degree of freedom important for long-wavelength scalar perturbations. The conditions of causality and a well-defined infrared limit greatly restrict the dynamics of modified gravity theories. We now make the key assumption that the terms proportional to $k^2\zeta$ in eqs. (\[consistent\])–(\[gammadef\]) can be neglected not only on super-horizon scales $k<{\cal H}^{-1}$ but also, as they can be in general relativity, on sub-horizon scales down to the Jeans length. This assumption defines a class of theories we call scale-independent modified gravity models. Under these assumptions, modified gravity is completely specified on large scales by the scale-independent function $\gamma(a)$. At high redshift when dark energy is unimportant we require $\gamma\to1$ in order to retain the success of general relativity in explaining the cosmic microwave background anisotropy [@Spergel07]. Thus we adopt the following parameterization for scale-independent modified gravity: $$\label{gamma} \gamma(a) = 1+\beta a^s\ ,$$ where $\beta$ and $s>0$ are constants. Eqs.(\[consistent\])–(\[gamma\]) now give $$\label{Fquad} \gamma F(a)=a^{-2}{\cal H}\gamma^{(1/s)} \int^a_0 a\gamma^{-(1/s)}\frac{d}{da} \left(\frac{a}{\cal H}\right)\,da\ .$$ Changing the lower limit of integration introduces a rapidly decaying solution which we ignore. In general relativity with negligible shear stress, $\gamma=1$. When a component with constant equation of state parameter $w>-\frac{1}{3}$ is dominant, $a\propto t^n$ with $n=2/(1+3w)$ yielding $F=(3+3w)/(5+3w)$. Thus, for long wavelengths the potential drops from $\Phi=\frac{2}{3}\zeta$ during the radiation-dominated era to $\Phi=\frac{3}{5}\zeta$ during the matter-dominated era. Of greater interest here is the evolution of the potentials during the matter-dominated era with modified gravity parameterized by (\[gamma\]). Figure \[fig:F\] shows the results for $s=1$ and $s=3$ as well as the GR case $\beta=0$. The background expansion history is chosen to match GR with $\Omega_m=0.284$ and a cosmological constant with $\Omega_\Lambda=0.716$. The choice $s=3$ matches Caldwell et al. in the limit $\beta\ll 1$ [@Caldwell07]. However, as we will see below, our results differ from theirs because they assumed the validity of some components of the Einstein field equations, while we instead required consistent causal evolution on large scales. This difference will be discussed further below in Section \[sec:shear\]. Figure \[fig:F\] shows that the Newtonian potential $\Phi$ is enhanced and the spatial curvature $\Psi=\gamma\Phi$ is diminished for $\gamma<1$, compared with general relativity. For $s=3$ the modifications occur later because $\|\gamma-1\|$ is smaller at earlier times. The quantitative results depend on the validity of eq.(\[consistent\]) but this qualitative behavior (the Newtonian potential being enhanced for $\gamma<1$) should persist in general. Observables for scale-independent modified gravity {#sec:observe} ================================================== With the time evolution of the metric in hand for long wavelengths we are now able to calculate observable quantities for scale-independent modified gravity theories parameterized by the constants $(\beta,s)$. The effects considered here are the growth of structure in the dark matter, microwave background anisotropy, and weak gravitational lensing. Growth of structure ------------------- Until now, no assumptions have been made about dynamics in the matter sector except for causality and consistency with a spatially homogeneous, uniformly expanding Robertson-Walker solution. To follow the growth of structure we must specify how the matter fields are coupled to gravity. Here we assume that the dark matter obeys the weak equivalence principle, i.e. collisionless dark matter particles follow geodesics. This choice explicitly forces scalar-tensor theories to the Jordan frame in which matter fields are minimally coupled to gravity. In the conformal Newtonian gauge, on sub-horizon scales where $\|\delta\|\gg\|\Psi\|$ with $\delta\equiv\delta\rho/\rho_m$, and $\rho_m$ is the average mass density, cold dark matter fluctuations obey the evolution equation $$\label{cdmevol} \ddot\delta+{\cal H}\dot\delta=-k^2\Phi\ .$$ This equation follows from particle number conservation and geodesic motion or, equivalently, from energy-momentum conservation. The density perturbation field can be written as $$\label{Dfactor} \delta({\bf k},t)=-k^2D(a,k)\zeta_i({\bf k})$$ where $\zeta_i$ is the curvature perturbation at $a=a_i$ which we take to be $z=30$ so that the Jeans length is smaller than the scales of interest and modified gravity has not yet become important. For $a>a_i$, $\Phi$ factorizes and eq. (\[cdmevol\]) can be reduced to quadratures for $D(a,k)$. With initial conditions $D(a,k)=D_i(k)$ and $\partial_t D(a,k)=\dot D_i(k)$ at $a=a_i$, the solution is $$\label{Density} D(a,k)=D(a)+D_i(k)+a_iy(a)\dot D_i(k)\ ,$$ where \[Dysol\] $$\begin{aligned} D(a)&\equiv&y\int^a_{a_i}\frac{F}{\cal H}\,da - \int^a_{a_i}\frac{yF}{\cal H}\,da\ ,\label{Dsol}\\ y(a)&\equiv&\int^a_{a_i}{\frac{da}{a^2{\cal H}}}\ . \label{ydef}\end{aligned}$$ The function $y(a)$ asymptotes to a constant but $D(a)\propto a$ for $a\gg a_i$ in the matter-dominated era. Thus the late-time solution for density perturbation growth factorizes, $D(a,k)=D(a)$ in eq.(\[Dfactor\]). This perturbation growth is often represented as a function of redshift by defining $g(z)\equiv D(a)/D(1)$ with $a=1/(1+z)$. Figure \[fig:I\] shows the logarithmic derivative of the density perturbation growth versus time for our parameterized modified gravity models. In the $\Lambda$CDM model, $d\ln D/d\ln a\approx[\Omega_m(a)]^{6/11}$ [@Nesseris07]. The transition to a cosmological constant-dominated universe leads to a suppression of growth. If instead gravity is modified, the growth rate can be increased or decreased relative to the GR case. The qualitative effects are easy to understand. We already saw that models with $\gamma<1$ ($\beta<0$) get an enhanced Newtonian potential $\Phi$. A stronger potential increases the gravitational force on density perturbations leading to an enhanced growth rate. The simplest test of growth of perturbations is the total perturbation growth by redshift zero, which is characterized by the variance of density fluctuations in spheres of radius $R_8=8\,h^{-1}$ Mpc, $$\label{sigma8} \sigma^2_8=\int^{\infty}_0\frac{d^3k}{(2\pi)^3}P_m(k)W^2(kR_8)\ ,$$ where $W(x)=3j_1(x)/x$. The power spectrum of matter density fluctuations is $$\label{powerspec} P_m(k)=\frac{2\pi^2}{k^3}\Delta_{\cal R}^2\left(\frac{k}{k_0}\right)^{n_s-1} T^2_m(k,z=30)\frac{D^2(z=0)}{D^2(z=30)}\ ,$$ where $\Delta_{\cal R}$ is the amplitude of the initial scalar curvature fluctuations on scale $k_0$ and $T_m$ is the transfer function for matter fluctuations in the synchronous gauge relative to $\zeta = {\cal{R}}$ computed by CMBFAST (which accounts for the suppression of growth during the radiation-dominated era). We adopt $\Delta_{\cal R}^2=2.4\times 10^{-9}$, $k_0=0.002$ Mpc$^{-1}$, and $n_s=0.958$ [@Spergel07]. Our modification of gravity is scale-invariant in that the $k$-dependence of the dark matter power spectrum is unchanged relative to GR. Thus the amplitude of density perturbations $\sigma_8$ depends on modified gravity only through the enhancement or diminution of growth shown in Fig. \[fig:I\]. The specific results obtained here assume that the gravitational potentials factorize on scales larger than a few Mpc. If this is true, then CMB and galaxy clustering results on all scales can be fit by a single modified gravity theory with one function $\gamma(a)$ or, equivalently, $D(a)$. If, however, modified gravity introduces a length scale $L_G$ intermediate between $R_8$ and the Hubble length, then the growth of structure will depend on wavenumber in a way not described by a scale-independent $D(a)$. In Section \[sec:scaledep\] below we consider an alternative scale-dependent parameterization of modified gravity and examine the observable consequences. For now, we assume $L_G<R_8$ or equivalently that super-horizon relations apply, as they do in GR, to sub-horizon scales all the way down to the larger of the Jeans and nonlinear lengths. Figure \[fig:Sigma\] shows a contour plot of $\sigma_8$ (normalized to the GR case) for different choices of the modified gravity parameters. As expected, smaller values of $\gamma$ (i.e., $\beta<0$) lead to larger amplitude. Thus, modified gravity changes the amplitude of galaxy clustering relative to the CMB, and could explain any apparent discrepancy between the values of $\sigma_8$ inferred from CMB analysis and galaxy clustering or lensing measurements. Modified Poisson Equation ------------------------- Rearranging eqs. (\[factorize\]) and (\[Dfactor\]), we arrive at a modified Poisson equation relating the Newtonian potential to the overdensity: $$\label{Poisson} \nabla^2\Phi = \frac{F(a)}{D(a)}\delta \equiv 4\pi G_{\Phi}(a) a^2 \rho_m \delta\ .$$ The space curvature potential obeys a similar Poisson equation, $$\label{Poisson-Psi} \nabla^2\Psi = 4\pi G_{\Psi}(a) a^2 \rho_m \delta\ ,$$ where $G_{\Psi}(a) = \gamma G_{\Phi}(a)$. Plots of $G_{\Phi}$ and $G_{\Psi}$ are shown in Figure \[fig:G\]. Their time dependence is dominated by the potentials $F(a)$ and $\gamma(a)F(a)$ since $\delta$ is less sensitive to our modified gravity parameters ($\beta,s$). As a result, we see the same qualitative behavior as in Fig. \[fig:F\]. Models with $\gamma < 1$ produces a greater value of $G_{\Phi}$ relative to the $\Lambda$CDM model, and larger values of $s$ produce larger late time behavior. In GR, the gravitational coupling $G$ is constant. In many alternative theories of gravity, the strength of gravity varies with time (and also with place, for length scales smaller than $L_G$). Time-varying $G$ is well known for scalar-tensor theories [@Acquaviva05; @MotaEmail], but we find that it is a generic feature of all modified gravity theories with $\gamma\ne1$ on cosmological scales. The time-variation of $G$, represented by $\dot G/G$, has been severely constrained by measurements in the solar system, in stars, and in the early universe [@TestG]. Limits on larger scales are provided by the microwave background [@TestGCMB]. The CMB acoustic peak structure will be unaffected if variations occur only long after recombination. Modified gravity explanations of cosmic acceleration suggest the need to constrain $\dot G/G$ on large length scales in the late universe. It would be very interesting to know, for example, to what extent the structure of clusters of galaxies and their X-ray emission can be used to constrain $\dot G/G$. Note that the method used to derive our modified Poisson equations is roundabout. Had we started with a Lagrangian, the gravitational field equations would directly yield the gravitational coupling strength. Because we started with a phenomenological description of modified gravity, we instead deduce the dynamics of this coupling from the requirements of causal evolution and the weak equivalence principle. On small scales our treatment breaks down as modifications of general relativity must become scale-dependent. Nonetheless, the motivation to investigate limits on $\dot G/G$ on scales much larger than the solar system remains valid. CMB temperature anisotropy -------------------------- Modified gravity (or dark energy) affect the microwave background only at late times through the integrated Sachs-Wolfe (ISW) effect: $$\frac{\Delta T}{T}(\hat{n})= \int (\dot{\Psi} + \dot{\Phi})\,d\chi$$ where $\chi$ is the comoving radius and $\hat n$ is the photon direction. The ISW effect arises when the gravitational potentials change with time, as occurs during transitional periods in cosmic evolution. One such contribution occurs during the transition from radiation to matter domination. The other contribution is occurring today during the current transition to an accelerating expansion. The physics governing the matter-radiation transition is well explained by GR, while the physics governing the transition today is (for the models considered here) dependent on modified gravity parameters. Because the recent ISW effect arises relatively nearby ($z<2$), it shows up only at large angular scales. The ISW contribution from some modified gravity models has been studied in several recent papers [@Song06; @Caldwell07]. We computed the CMB temperature anisotropy spectrum by modifying CMBFAST [@cmbfast] to replace the ISW contribution of the $\Lambda$CDM model with that for our models assuming the factorization of the potential. The results are shown in Figure \[fig:T\]. As expected, only the low-order multipoles are affected. Models with higher $s$ produce larger changes because they lead to larger time derivatives. Models with $0.2<\gamma<1$ ($-0.8<\beta<0$) produce less anisotropy because of destructive interference between the ISW and primary anisotropy contributions. Although decreasing the quadrupole moment improves agreement with observations, little statistical weight can be given to this conclusion because the modifications, at least for $0.2\le\gamma \le1.5$, are smaller than cosmic variance. However, cross correlating the CMB with galaxy surveys could potentially be a more discriminating probe of modified gravity [@HoEtAl08; @Cooray02] It was recently found [@Daniel08] that our results are consistent with recipe R1 of Caldwell et al. [@Caldwell07]. However, we expect differences from our Figure \[fig:T\] for recipe R3 of [@Caldwell07] since $\zeta$ is not conserved on large scales in this scheme. This difference will be discussed below in Section \[sec:shear\]. Weak lensing ------------ Metric perturbations $\Phi+\Psi$ affect both the energy of photons (ISW effect) and their direction of travel (gravitational lensing). Gravitational lensing causes both magnification (or de-magnification) and differential stretching (shear) of background images. The correlation function or power spectrum of weak gravitational lens shear is an observable measure of large-scale structure. The weak lensing power spectrum is given by [@WeakLens] $$\label{shearPS} P^{\kappa}_{l}=\int^{\chi_\infty}_0d\chi\, {W^2(\chi)}\frac{l^4}{\chi^4}P_{\Psi+\Phi}(k=\frac{l} {\chi}, \chi)\ ,$$ where $$\frac{P_{\Psi+\Phi}}{(1+\gamma)^2} =\frac{2\pi^2\triangle^2_{\cal{R}}}{k^3} \left(\frac{k}{k_0}\right)^{n_s-1}T_{\Phi}^2 (k,z=30)\frac{F^2(z)}{F^2(z=30)}\ ,\ \$$ and $$W(\chi) = \int^{\chi_\infty}_\chi d\chi' \frac{\chi' - \chi}{\chi'} \eta(\chi')\ .$$ Here $\chi_\infty$ is the comoving distance to $z=10$ (the results change by a negligible amount if the maximum redshift lies anywhere between $6\le z\le15$), $T_{\Phi}$ is the transfer function of the Newtonian potential relative to $\zeta$ computed at $z=30$ using CMBFAST and $\eta(\chi)$ is the radial distribution of sources, normalized with $\int\eta(\chi)\,d\chi=1$. Note that these formulae assume a flat space. Our lensing analysis uses the source distribution $$\eta(z)\propto z^2\exp[{-(1.41z/z_{\rm med})}^{1.5}]$$ with $z_{\rm med}=1.26$ [@Massey]. This distribution approximates the galaxy redshift distribution of the COSMOS survey, if there were no clumping. Measurements of weak gravitational lens shear, for galaxies separated by angle $\theta$ on the sky, provide an estimate of shear correlation functions including $$C_+(\theta)=\frac{1}{2\pi}\int^{\infty}_0 P^{\kappa}_{l}J_0(l\theta) l\,dl\ .$$ Modifying gravity changes $F(z)$ thereby changing $C_+(\theta)$. Figure \[fig:C+\] plots $C_+(\theta)$ for modified gravity, normalized at each $\theta$ by its GR value. At $z=1$, 1 Mpc corresponds to 2.15 arcmin. Hence, for some of the scales shown in Figure \[fig:C+\] structures are nonlinear and thus beyond the regime of validity of the current framework. However, a scheme that takes a linear power spectrum to a nonlinear power spectrum would correct this flaw [@PeacockDodds]. On such small scales, our assumption of scale-independent modified gravity may also be invalid, so Figure \[fig:C+\] should be regarded as suggestive, but not definitive, of modified gravity effects on weak lensing. As expected, models with $\gamma<1$ have larger shear correlations because they have a larger $F(z)$ and therefore more growth of structure (despite having a smaller $1+\gamma$). For angular scales less than about 10 arcmin, the effect is almost equivalent to a constant change in the normalization of the power spectrum, i.e., in the value of $\sigma_8$. At larger angular scales, the redshift-dependence of $F(z)$ at small redshift translates to a dependence on distance and hence on angular scale, however this is in a regime where the shear correlations are small and difficult to measure. Thus, scale-independent modified gravity theories predict an amplitude of weak lensing different from GR with the same CMB primary anisotropy. The acoustic peak amplitudes tightly constrain $\Delta_{\cal R}$. In principle, measurements of $\sigma_8$ based on the CMB acoustic peaks (which are unaffected by modified gravity) could differ both from measurements based on galaxy clustering \[which depend on $D(z)$\] and those based on weak lensing \[which depend on $F(z)$\]. Current error bars are inconclusive [@Spergel07] but this comparison of different $\sigma_8$ values could eventually provide a powerful test of GR. Comparison with $f(R)$ theories {#sec:f(R)} =============================== Substantial work has already been done investigating modified gravity effects for $f(R)$ theories. Here we consider such theories where the Ricci scalar $R$ is replaced by $R+f(R)$ in the Einstein-Hilbert action, and where the action is extremized with respect to the metric. In these models, the field equations are generically fourth-order. In effect, modified gravity introduces a new propagating scalar degree of freedom coupled to gravity, the scalaron $f_R\equiv df/dR$ [@Starobinsky]. The Compton wavelength of the scalaron imprints a physical length scale, which is made dimensionless by combining with the wavenumber $k$: $$\label{Qdef} Q\equiv\frac{3k^2}{a^2}\frac{f_{RR}}{1+f_R}\ ,$$ where $f_{RR}\equiv d^2f/dR^2$. Several papers have recently discussed cosmological perturbation evolution for metric $f(R)$ theories [@f(R); @HuSawicki07a; @PS07]; our notation most closely follows that of ref. [@PS07] except our potentials $\Phi$ and $\Psi$ are exchanged from theirs. In $f(R)$ theories, $\zeta$ is conserved on super-horizon scales [@HuSawicki07b]. However, the scalaron obeys a nonlinear Klein-Gordon equation with two length scales: the Hubble length and the scalaron Compton wavelength. The interesting case for large-scale structure is the quasi-static regime of linear, sub-horizon perturbations ($k^2\gg{\cal H}^2$) where [@PS07] $$\label{fPoisson} \nabla^2\Phi\approx\frac{4\pi Ga^2\rho_m}{1+f_R} \left(\frac{3+4Q}{3+3Q}\right)\delta\ ,\ \ \gamma\approx\frac{3+2Q}{3+4Q}\ .$$ Differentiating eq. (\[consistent\]) and substituting eqs.(\[cdmevol\]) and (\[fPoisson\]) along with the background evolution equations (5) and (6) of ref. [@PS07] for a universe containing only nonrelativistic matter and (optionally) a cosmological constant yields $$\label{zetadot} \dot\zeta=U\dot\Psi+V\Psi\ ,$$ where $$\label{Udef} U\equiv\frac{a{\cal H}}{\Gamma^2}\frac{\partial}{\partial t} \left(\frac{\Gamma^2B}{a{\cal H}^2}\right)+\frac{2QB}{3+2Q}$$ and $$\begin{aligned} \label{Vdef} V\equiv\frac{4\pi Ga^2\rho_m\Gamma B}{\gamma {\cal H}}+\frac{\partial}{\partial t}\left(\frac{B}{\gamma} \right) -\frac{\Gamma B}{{\cal H}a^2}\frac{\partial}{\partial t} \left[a\frac{\partial}{\partial t}\left(\frac{a}{\Gamma} \right)\right]\ ,\nonumber\\\end{aligned}$$ where we have defined the auxiliary variables \[BGammadef\] $$\begin{aligned} \Gamma&\equiv&\frac{G_\Psi}{G}=\frac{1}{1+f_R}\left( \frac{3+2Q}{3+3Q}\right)\ ,\label{Gammadef}\\ B&\equiv&a\left[\frac{\partial}{\partial t}\left(\frac{a} {\cal H}\right)\right]^{-1}=\frac{2(1+f_R){\cal H}^2}{8\pi Ga^2\rho_m+a^2\partial_t(\dot f_R/a^2)} \label{Bdef}\ .\qquad\end{aligned}$$ General relativity with a cosmological constant corresponds to the case $f=2\Lambda$, $f_R=0$, and $\gamma=\Gamma=1$, yielding $U=V=0$. Thus, $\zeta$ is conserved even on sub-horizon scales in a $\Lambda$CDM universe. However, this is no longer true if $f_R\ne0$. Two distinct effects modify the curvature perturbation. First, the $1+f_R$ factor in (\[fPoisson\]) modifies the evolution on sub-horizon scales. In practice, this effect is small if $\|f_R\|\ll1$, as is favored by galactic structure considerations [@HuSawicki07a]. In this case, the background expansion history is nearly identical to GR with a cosmological constant and gravity is significantly modified only at wavelengths approaching the scalaron Compton wavelength, where $Q\sim1$. For long wavelengths such that $Q\ll1$, $\gamma\approx1-\frac{2}{3}Q$ and the corrections introduced to eq. (\[consistent\])–(\[gammadef\]) by scalaron dynamics are $O(k^2)$. The treatment given in the preceding sections remains valid for $\|f_R\|\ll1$ and $Q\ll1$. However, this limit corresponds to general relativity. Unfortunately, the treatment presented in Section \[sec:long\], which was based on a scale-invariant modification of gravity, breaks down just where $f(R)$ theories begin to deviate significantly from GR. The $f(R)$ models generically have $\gamma-1\propto k^2$ for sub-horizon wavelengths longer than the scalaron Compton wavelength. These models have a scale-dependent modification of gravity. Although we cannot use the results of Section \[sec:long\] to describe them, it is still possible to parameterize scale-dependent modified gravity models so as to obtain useful results for the sub-horizon growth of large-scale structure. A simple parameterization inspired by $f(R)$ theories is presented in the next section. Scale-dependent modified gravity {#sec:scaledep} ================================ For a wide class of theories, modified gravity leads generically to a Poisson equation with variable gravitational coupling. In the scale-invariant modifications of Section \[sec:long\], the Newton constant is replaced by the time-varying $G_\Phi(t)$ which follows from the scale-invariant potential ratio $\gamma(t)$. In scale-dependent modified gravity theories, on the other hand, $G_\Phi(k,t)$ and $\gamma(k,t)=G_\Psi/G_\Phi$ are functions of length scale as well as time, and there is no simple relation between them. Thus, more parameters are needed to characterize such theories [@Amin07]. Despite their generality, $f(R)$ theories with $f_R\ll1$ have, for a wide range of sub-horizon length scales, a very simple form for $G_\Phi$ and $\gamma$ given by eq. (\[fPoisson\]). To arrive at a simple phenomenological model we simplify the time dependence as follows: $$\label{scaledepMG} \frac{G_\Phi}{G}=\frac{1+\alpha_1k^2a^s}{1+\alpha_2k^2a^s}\ ,\ \ \gamma=\frac{1+\beta_1k^2a^s}{1+\beta_2k^2a^s}\ .$$ We assume that these relationships hold only in the linear regime of cosmological density perturbations, and that $G_\Phi/G\to1$ and $\gamma\to1$ on solar system scales. We also require GR to hold at early times, implying $s>0$. Eq. (\[scaledepMG\]) describes $f(R)$ theories with $\|f_R\|\ll1$ if $\alpha_1=\frac{4}{3}\alpha_2=2\beta_1=\beta_2=4f_{RR}/a^{2+s}$. As a simple post-Newtonian model we will now assume that $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ are arbitrary constants with units of length squared. In order to ensure that $G_\Phi/G$ and $\gamma$ are finite for all $k$, we require $\alpha_2$ and $\beta_2$ to be non-negative. Moreover, we need $G_\Phi > 0$ in order to ensure that gravity is attractive. Hence, $\alpha_1$ must be non-negative as well. This scale-dependent parameterization has a different dependence on length scale than that of Amin et al.[@Amin07]. It is chosen to reproduce the scale-dependence of $f(R)$ theories. For some modified gravity theories $\gamma=1$, e.g., Einstein plus Yukawa gravity. For this model $G_\Phi/G$ in eq. (\[scaledepMG\]) is multiplied by an overall factor $\alpha_2/\alpha_1$ [@Dore07] so that the deviation from Einstein gravity shows up only at large distances. The class of theories considered here has at least three physical length scales: the horizon scale $a/{\cal H}$, the transition scale $a^{1+s/2}\sqrt{\alpha_1}$ where gravity changes strength (for simplicity, we consider models where the $\alpha_i$ and $\beta_i$ are all of comparable magnitude), and the nonlinear length scale for structure formation (e.g., approximately 10 Mpc today). If $a^{1+s/2}\sqrt{\alpha_i}$ and $a^{1+s/2}\sqrt{\beta_i}$ are smaller than the nonlinear scale, then for purposes of large-scale structure formation, gravity is adequately described by GR. The parameterization of eq. (\[scaledepMG\]) applies only to intermediate length scales between the horizon scale and the smaller transition scale $a^{1+s/2}\sqrt{\alpha_1}$. However, it implies that for long wavelengths and at early times, gravity reduces to GR (with constant gravitational coupling). This assumption can be relaxed at the cost of introducing additional parameters, which seems premature given the difficulty of measuring any post-Newtonian parameters. Also, for wavelengths short compared with $a^{1+s/2}\sqrt{\alpha_i}$ and $a^{1+s/2}\sqrt{\beta_i}$ but large compared with the nonlinear scale, the gravitational couplings are constant but differ from GR, e.g., $\gamma=\frac{1}{2}$ for $f(R)$. From the perspective of model testing, scale-dependent modified gravity is much more complicated than the scale-independent case considered in Sect. \[sec:long\]. The models have four dimensional parameters plus an exponent giving the time dependence. However, the situation is not so bleak, because structure formation depends only on $G_\Phi(k,t)$ and not on $\gamma(k,t)$. In particular, matter density perturbations on scales larger than the Jeans length and smaller than the Hubble length follow from integration of $$\label{pertevol} \ddot\delta+{\cal H}\dot\delta=4\pi G_\Phi(k,t)a^2\rho_m\delta\ .$$ At early times, $G_\Phi\to G$ and $\delta$ evolves as in the GR solution until the scale-dependent terms in eq. (\[scaledepMG\]) become important. The density transfer function $D(k,t)$ given by eq. (\[Dfactor\]) is now scale-dependent at late times, implying a change in the shape of the matter power spectrum. It is easy to see that the transfer function can depend only on the dimensionless variables $(k\sqrt{\alpha_1},\alpha_1/\alpha_2,a)$. The most interesting new feature of scale-dependent modified gravity is the change of shape of the matter density transfer function. Figure \[fig:fundep\] shows $D(k,a=1)$ normalized to the GR result, obtained by numerically integrating eq. (\[pertevol\]) with initial conditions given by the GR result ${\cal H}^2D(a)\to\frac{2}{3}F= \frac{2}{5}$ for a matter-dominated universe at $a=0.03$. As expected, at large length scales ($k\sqrt{\alpha_1}\ll1$) the results converge to the GR limit. At short length scales, gravity is weaker than GR if $\alpha_1/\alpha_2<1$, leading to reduced growth; the growth is enhanced for $\alpha_1/\alpha_2>1$. Thus, scale-dependent gravity changes the shape of the matter power spectrum [@Starobinsky07]. Ultimately, measuring this scale-dependence (and doing so at several redshifts) can constrain scale-dependent modified gravity theories. However, the interpretation of the galaxy power spectrum shape is complicated by scale-dependent biased galaxy formation and by the dark-matter-dependent transfer function (e.g., the neutrino fraction). Thus, while the linear growth of structure offers a potentially powerful test of GR versus modified gravity, it must be combined with other tests. The second function characterizing scale-dependent modified gravity, $\gamma(k,t)$, is (in our analysis, which assumes no particular Lagrangian) unrelated to $G_\Phi(k,t)$. This function is best constrained by combining weak gravitational lensing and galaxy clustering measurements made at the same redshift. Care is required because the lensing amplitude is proportional to $(1+\gamma)\Phi$ while the galaxy density is proportional to $D$ and also depends on biasing. Galaxy peculiar velocity measurements could be used, in principle, to reduce or ideally eliminate the dependence on biasing [@ZLBD07]. However, one must be careful not to assume the velocity-density relation obtained in GR. The continuity equation gives $$\label{velden} {\bf v}=-\frac{i{\bf k}}{k^2}\frac{\partial\ln D}{\partial\ln a} {\cal H}\delta\ ,$$ where the logarithmic growth rate $\partial\ln D/\partial\ln a$ is now scale-dependent, as shown in Figure \[fig:S\]. As in the case of galaxy clustering, measurement of this effect is contingent upon knowing the composition of dark matter (hot dark matter has a free-streaming scale, and its abundance determines the suppression of growth at small scales) and correcting for any velocity bias. The greater freedom allowed by scale-dependent modified gravity models, and the fact that astrophysics (biased galaxy formation and dark matter dynamics) may also introduce scale-dependence into transfer functions, makes it challenging to test GR using growth of structure and weak gravitational lensing. It is likely that a combination of galaxy clustering, peculiar velocities, and weak lensing will be needed to obtain strong constraints on scale-dependent modified gravity theories. Modified gravity versus shear stress {#sec:shear} ==================================== A difference between the two longitudinal potentials $\Phi$ and $\Psi$ need not signal modified gravity; it might arise from shear stress [@Amendola07; @Mota07]. For scalar mode fluctuations, the shear stress is fully characterized by a scalar potential $\pi$, such that the spatial stress tensor components are $$\label{shearstress} T^i_{\ \,j}=p\delta^i_{\ \,j}+\frac{3}{2}(\bar\rho+\bar p)\left( \nabla^i\nabla_j-\frac{1}{3}\delta^i_{\ \,j}\Delta\right)\pi$$ where $(\bar\rho+\bar p)$ is the background enthalpy and $\Delta=\nabla^i\nabla_i$. In linearized GR, one of the Einstein field equations yields $$\label{GRshear} \Psi-\Phi=12\pi Ga^2(\bar\rho+\bar p)\pi\ .$$ All of the results obtained in Sections \[sec:long\] and \[sec:observe\] for modified gravity apply equally to GR with shear stress if $\gamma$ is replaced by $12\pi Ga^2(\bar\rho+\bar p)\pi/\Phi$. In standard cosmology, the only significant source of shear stress is relativistic neutrinos after neutrino decoupling during the radiation-dominated era. For long wavelengths during the radiation-dominated era, neutrino shear stress gives [@MaBert95] $$\label{nushear} \gamma-1=\frac{2}{5}\left(\frac{\rho_\nu+p_\nu}{\bar\rho+\bar p} \right)\ .$$ During the matter-dominated era, $\gamma-1\propto a^{-1}$ and shear stress is unimportant at late times in the $\Lambda$CDM model. It is also unimportant in simple quintessence models because shear stress vanishes for linear perturbations of a minimally-coupled scalar field. Shear stress might nonetheless be important if cosmic acceleration is driven by an imperfect fluid. Without specifying the dynamics of this fluid, few constraints can be placed on $\pi$. One possible bound comes from the dominant energy condition, which states that each of the eigenvalues of $T^i_{\ \,j}$ must be smaller in absolute value than the energy density. If this condition holds, then eqs.(\[shearstress\]) and (\[GRshear\]) can be combined to give rather weak bounds on $\Psi/\Phi-1$. Additional constraints follow from the initial-value constraints of GR and energy-momentum conservation, which for a spatially flat background become [@Bertschinger06] \[ivcon\] $$\begin{aligned} -k^2\Psi&=&4\pi Ga^2(\bar\rho+\bar p)(\delta+3{\cal H}u)\ , \qquad\label{Energycon}\\ \dot\Psi+{\cal H}\Phi&=&4\pi Ga^2(\bar\rho+\bar p)u\ , \label{Momcon}\\ \dot\delta+3{\cal H}\sigma&=&3\dot\Psi-k^2u\ , \label{Econs}\\ \dot u+(1-3c_s^2){\cal H}u&=&c_s^2\delta+\sigma+\Phi-k^2\pi\ . \label{Momcons}\end{aligned}$$ The first of these equations is the usual Poisson equation in conformal Newtonian gauge. The density and velocity potential perturbations are defined from the energy-momentum tensor components by $T^0_{\ \,0}=-\bar\rho-(\bar\rho+\bar p)\delta$, $T^0_{\ \,i}= -(\bar\rho+\bar p)\nabla_iu$, while the entropy perturbation is defined by $\sigma\equiv(\delta p-c_s^2\delta\rho)/(\bar\rho+\bar p)$ with $c_s^2=d\bar p/d\bar\rho$. For a single perfect fluid like cold dark matter before its trajectories intersect, $\sigma=0$. However, in general $\sigma\ne0$ for a multi-component fluid, e.g. dark matter and a non-constant dark energy. The freedom introduced by entropy and shear stress perturbations is, unfortunately, sufficient in principle to reproduce any observations of large-scale structure and gravitational lensing. Consider, for example, perfect measurements of galaxy peculiar velocities everywhere and at all times assuming that galaxies exactly trace cold dark matter. Then, eq. (\[Momcons\]) with $c_s^2=\sigma=\pi=0$ for CDM yields $\Phi({\bf k},t)$. Assume furthermore that complete and ideal gravitational lensing measurements are available to yield $\Phi({\bf k},t)+\Psi({\bf k},t)$. Now, the GR initial-value constraints (\[Energycon\])–(\[Momcon\]) suffice to yield $\delta({\bf k},t)$ and $u({\bf k},t)$ for the multi-component fluid of dark matter and dark energy. Requiring this fluid to obey energy-momentum conservation (\[Econs\])–(\[Momcons\]) yields $\sigma$ and $\pi$. In short, perfect measurements of $\Phi$ and $\Psi$ can be used to determine $\sigma$ and $\pi$ of the combined fluid of dark matter and dark energy. When combined with measurements of the dark matter density and velocity fields (assuming galaxies trace dark matter), one can, in principle, determine the energy-momentum tensor components for dark energy. One can think of this energy-momentum tensor for dark energy as the difference between the Einstein tensor for the observed metric and the energy-momentum tensor of the ordinary matter [@HuSawicki07b]. This approach can be used, in principle, to determine the dark energy entropy and shear stress needed to explain any observations of large-scale structure (including peculiar velocities) and weak lensing — even if there is no dark energy, but instead gravity is modified. In effect, the two observable metric fields $\Phi$ and $\Psi$ can be exchanged for either $\sigma$ and $\pi$ (GR with exotic dark energy) or $G_\Phi$ and $\gamma$ (modified gravity). Although the initial-value constraints of GR can be used to determine the properties of dark energy, they cannot be used to model modified gravity. As we have seen in previous sections, modified gravity leads generically to a modified Poisson equation with a variable gravitational coupling $G_\Phi(k,t)$. In order not to break local Lorentz invariance by the selection of a preferred frame, the other components of the Einstein equation should also be modified. For example, in $f(R)$ theories the left-hand side of (\[Momcon\]) is multiplied by $(1+f_R)$ while the right-hand side acquires an extra term $\frac{1}{2}\dot f_R$. Neglecting these modifications leads to violations of eq. (\[consistent\]) on large scales. For example, Caldwell et al. [@Caldwell07] modeled the dynamics of modified gravity with three recipes that assume the validity of some combination of eqs. (\[Energycon\]) and (\[Momcon\]). While recipes R1 and R2 satisfy the conservation of $\zeta$, recipe R3 does not. As a result, it leads to a different prediction for the gravitational potentials and therefore the ISW effect. Discussion {#sec:discuss} ========== Parameterizing modified gravity theories in cosmology is much more difficult than parameterizing post-Newtonian gravity in the solar system because the gravitational potentials $\Phi$ and $\Psi$ in the GR limit are not static Coulomb potentials in cosmology. In GR, on scales larger than the Jeans or nonlinear length scales, $\Psi=\Phi$ factorizes into a product of a time-dependent growth function and a spatially-varying curvature perturbation. If this factorization persists in modified gravity theories, then a simple post-Newtonian parameterization can be obtained. This is the approach we introduced in Section \[sec:long\]. It has the practical virtue of yielding easily calculated predictions for all observables in the linear regime, including the growth of matter clustering, peculiar velocities, microwave background anisotropy, and weak gravitational lensing. Introducing the Eddington parameterization $\Psi=\gamma\Phi$, where $\gamma$ depends on time but not on space, we showed that structure grows faster (gravity is stronger) in models with $\gamma<1$. However, the shape of the transfer functions (i.e., their dependence on spatial wavenumber) for these scale-independent models is unchanged compared with GR. Unfortunately, realistic modified gravity theories such as the $f(R)$ models have scale-dependent effects and can no longer be described by only one post-Newtonian quantity $\gamma$. Instead, the strength of gravity, described by $G_\Phi$ in the Poisson equation, can vary independently of $\gamma$. Even so, because galaxy clustering and dynamics depends on $G_\Phi$ but not on $\gamma$, while weak lensing depends on $\gamma$, observations could, in principle, measure modified gravity parameters assuming there is no dark energy. If dark energy is complex enough to require two additional fields to characterize its stress tensor (e.g., shear stress potential and entropy), then it appears that there is enough information in the dark energy model to account for any $\Phi$ and $\Psi$ without modifying gravity. One cannot prove gravity is modified unless one can account for all significant contributions to the stress-energy tensor. Thus, our hope to describe all modified gravity models with two parameters, yielding predictions measurably different from all dark energy models, has not been realized. Distinguishing modified gravity from dark energy will require making additional assumptions. Nevertheless, the $(\beta,s)$ parameterization of scale-independent modified gravity presented in Section \[sec:long\], and the $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ parameterization of scale-dependent modified gravity models presented in Section \[sec:scaledep\], are still useful for characterizing observational data. If measurements of galaxy clustering, peculiar velocities, and weak lensing are all consistent with $\beta=0$ and $\alpha_1=\alpha_2=\beta_1=\beta_2=0$, for example, then modified gravity and exotic dark energy models can both be excluded. If measurements require nonzero parameters, however, dark energy and modified gravity remain viable explanations until additional assumptions are made to distinguish them,e.g. restriction of the Lagrangian to a particular form. A generic prediction of modified gravity theories in cosmology is that the gravitational coupling $G_\Phi$ in the Poisson equation should vary with time and with length scale. Departures from GR could be important not only in the linear regime of cosmological perturbations but perhaps also in the nonlinear regime (albeit on scales much larger than the solar system). Nonlinear effects may allow modified gravity to be distinguished from exotic dark energy, assuming that the dark energy fluctuations are small. For this reason it would be valuable to perform N-body simulations of structure formation using variable $G_\Phi$, extending previous work [@Nbody] to the scale-independent and scale-dependent modified gravity models discussed in the current paper. We thank Richard Gott for helpful comments and Scott Tremaine for the hospitality of the Institute for Advanced Study. This work was supported by the Kavli Foundation, by a Guggenheim Fellowship to EB, and by NSF grant AST-0708501. [99]{} D. W. Hogg, D. J. Eisenstein, M. R. Blanton, N. A. Bahcall, J. Brinkmann, J. E. Gunn, and D. P. Schneider, Astrophys. J. [624]{}, 54 (2005), arXiv:astro-ph/0411197. T. Lu and C. Hellaby, Class. Quantum Grav. [24]{}, 4107 (2007), arXiv:0705.1060 \[gr-qc\]. E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. [D15]{} 1753 (2006), arXiv:hep-th/0603057. A. Albrecht et al., arXiv:astro-ph/0609591 (2006). S. Capozziello, V. F. Cardone and A. Troisi, Phys. Rev. D [71]{}, 043503 (2005), arXiv:astro-ph/0501426; S. Nojiri and S. D. Odintsov, Phys. Rev. D [74]{}, 086005 (2006), arXiv:hep-th/0608008. Y. S. Song, W. Hu and I. Sawicki, Phys. Rev. D [75]{}, 044004 (2007), arXiv:astro-ph/0610532. P. Zhang, M. Liguori, R. Bean, and S. Dodelson, Phys. Rev. Lett. [99]{}, 141302, arXiv:0704.1932 \[astro-ph\] (2007). S. Wang, L. Hui, M. May, and Z. Haiman, Phys. Rev. D [76]{}, 063503 (2007), arXiv:0705.0165 \[astro-ph\]; B. Jain and P. Zhang (2007), arXiv:0709.2375 \[astro-ph\]; J. Uzan (2006), arXiv:astro-ph/0605313. C. M. Will, Living Rev. Rel. [9]{} (2006), arXiv:gr-qc/0510072. K. S. Thorne and C. M. Will, Astrophys. J. [163]{}, 595 (1971). W. Hu and I. Sawicki, Phys. Rev. D [76]{}, 104043, arXiv:0708.1190 \[astro-ph\] (2007). R. Caldwell, A. Cooray and A. Melchiorri, Phys. Rev. D [76]{}, 023507 (2007), arXiv:astro-ph/0703375. L. Amendola, M. Kunz and D. Sapone, JCAP [4]{}, 13, arXiv:0704.2421 \[astro-ph\]. M. A. Amin, R. V. Wagoner and R. D. Blandford, arXiv:0708.1793 \[astro-ph\] (2007). E. Linder, R. Cahn, APh [28]{}, 481, arXiv:astro-ph/0701317 A. S. Eddington, [The Mathematical Theory of Relativity]{} (Cambridge University Press, 1922). A.S. Bolton, S. Rappaport, and S. Burles, Phys. Rev. D [74]{}, 061501(R) (2006); arXiv:astro-ph/0607657. O. Doré et al., arXiv:0712.1599 \[astro-ph\] (2007). E. Bertschinger, Astrophys. J. [638]{}, 797 (2006), arXiv:astro-ph/0604485. J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D [28]{}, 679 (1983). D. H. Lyth, Phys. Rev. D [31]{}, 1792 (1985). E. Bertschinger, in [Cosmology and Large Scale Structure]{}, ed. R. Schaeffer, J. Silk, M. Spiro, and J. Zinn-Justin (Amsterdam: Elsevier Science), 273 (1996), arXiv:astro-ph/9503125. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rep. [251]{}, 1 (1992). C.-P. Ma and E. Bertschinger, Astrophys. J. [455]{}, 7 (1995), arXiv:astro-ph/9401007. U. Seljak and M. Zaldarriaga, Astrophys. J. [469]{}, 437 (1996), arXiv:astro-ph/9603033. D. N. Spergel et al., Astrophys. J. Suppl. [170]{}, 377 (2007), arXiv:astro-ph/0603449. S. Nesseris and L. Perivolaropoulos, Phys. Rev. D [77]{}, 023504, arXiv:0710.1092 \[astro-ph\] (2008). V. Acquaviva, C. Baccigalupi, S. M. Leach, A. R. Liddle, and F. Perrotta, Phys. Rev. D [71]{}, 104025, arXiv:astro-ph/0412052. T. Clifton, D. Mota, J. Barrow, MNRAS [358]{}, 601, arXiv:gr-qc/0406001 J.-P. Uzan, Rev. Mod. Phys. [75]{}, 403 (2003), arXiv:hep-ph/0205340; P. Jofre, A. Reisenegger, and R. Fernandez, Phys. Rev. Lett. [97]{}, 131102 (2006), arXiv:astro-ph/0606708; P. G. Krastev and B. A. Li, Phys. Rev. C [76]{}, 055804 (2007), arXiv:nucl-th/0702080. O. Zahn and M. Zaldarriaga, Phys. Rev. D [67]{}, 063002 (2003), arXiv:astro-ph/0212360; K. I. Umezu, K. Ichiki, and M. Yahiro, Phys. Rev. D [72]{}, 044010 (2005), arXiv:astro-ph/0503578; K. C. Chan and M.-C. Chu, Phys. Rev. D [75]{}, 083521 (2007), arXiv:astro-ph/0611851. S. Ho, C. Hirata, N. Padmanabhan, U. Seljak, N. Bahcall, arXiv:astro-ph/0801.0642 A. Cooray, Phys. Rev. D [65]{}, 083510 (2002), arXiv:astro-ph/0112408 S. Daniel, R. Caldwell, A. Cooray, A. Melchiorri, arXiv:astro-ph/0802.1068 N. Kaiser, Astrophys. J. [498]{}, 26 (1998), arXiv:astro-ph/9610120; W. Hu, Astrophys. J. [522]{}, L21 (1999), arXiv:astro-ph/9904153. R. Massey et al., Astrophys. J. Suppl. [172]{}, 239 (2007), arXiv:astro-ph/0701480. J.A. Peacock, S.J. Dodds, MNRAS [280]{}, L19, arXiv:astro-ph/9603031 A. A. Starobinsky, JETP Lett. [30]{}, 682 (1979); Phys. Lett. B [91]{}, 99 (1980). R. Bean, D. Bernat, L. Pogosian, A. Silvestri, and M. Trodden, Phys. Rev. D [75]{}, 064020 (2007), arXiv:astro-ph/0611321; B. Li and J. D. Barrow, Phys. Rev. D [75]{}, 084010, arXiv:astro-ph/0701111 (2007); S. Tsujikawa, K. Uddin and R. Tavakol, Phys. Rev. D [77]{}, 043007, arXiv:0712.0082 \[astro-ph\] (2007). W. Hu and I. Sawicki, Phys. Rev. D [76]{}, 064004, arXiv:0705.1158 \[astro-ph\] (2007). L. Pogosian and A. Silvestri, Phys. Rev. D [77]{}, 023503, arXiv:0709.0296 \[astro-ph\] (2007). A. A. Starobinksy, JETP Lett. [86]{}, 157 (2007), arXiv:0706.2041 \[astro-ph\]. D. F. Mota, J. R. Kristiansen, R. Koivisto, and N. E. Groeneboom, MNRAS [382]{}, 793, arXiv:0708.0830 \[astro-ph\] (2007). H. F. Stabenau and B. Jain, Phys. Rev. D [74]{}, 084007 (22006), arXiv:astro-ph/0604038; I. Laszlo and R. Bean, Phys. Rev. D [77]{}, 02404, arXiv:0709.0307 \[astro-ph\] (2007).
--- abstract: 'We introduce a new approach to LZ77 factorization that uses $\O(n/d)$ words of working space and $\O(dn)$ time for any $d\ge 1$ (for polylogarithmic alphabet sizes). We also describe carefully engineered implementations of alternative approaches to lightweight LZ77 factorization. Extensive experiments show that the new algorithm is superior in most cases, particularly at the lowest memory levels and for highly repetitive data. As a part of the algorithm, we describe new methods for computing matching statistics which may be of independent interest.' author: - 'Juha K[ä]{}rkk[ä]{}inen' - Dominik Kempa - 'Simon J. Puglisi' bibliography: - 'lz.bib' title: 'Lightweight Lempel-Ziv Parsing[^1]' --- Introduction {#sec-intro} ============ The Lempel-Ziv factorization [@ZL77], also known as the LZ77 factorization, or LZ77 parsing, is a fundamental tool for compressing data and string processing, and has recently become the basis for several compressed full-text pattern matching indexes [@kn2011; @ggknp2012]. These indexes are designed to efficiently store and search massive, highly-repetitive data sets — such as web crawls, genome collections, and versioned code repositories — which are increasingly common [@n2012]. In traditional compression settings (for example the popular [ gzip]{} tool) LZ77 factorization is kept timely by factorizing relative to only a small, recent window of data, or by breaking the data up into blocks and factorizing each block separately. This approach fails to capture widely spaced repetitions in the input, and anyway, in many applications, including construction of the above mentioned LZ77-based text indexes, whole-string LZ77 factorizations are required. The fastest LZ77 algorithms (see [@kp2013; @kkp2013]) use a lot of space, at least $6n$ bytes for an input of $n$ symbols and often more. This prevents them from scaling to really large inputs. Space-efficient algorithms are desirable even on smaller inputs, as they place less burden on the underlying system. One approach to more space efficient LZ factorization is to use compressed suffix arrays and succinct data structures [@nm2007]. Two proposals in this direction are due to Kreft and Navarro [@kn2010] and Ohlebusch and Gog [@og2011]. In this paper, we describe carefully engineered implementations of these algorithms. We also propose a new, space-efficient variant of the recent $\ISA$ family of algorithms [@kp2013]. Most compressed index implementations are build from the uncompressed suffix array (SA) which requires $4n$ bytes. Our implementations are instead based on the Burrows-Wheeler transform (BWT), constructed directly in about $2$–$2.5n$ bytes using the algorithm of Okanohara and Sadakane [@os2009]. There also exists two online algorithms based on compressed indexes [@os2008; @s2012] but they are not competitive in practice in the offline context. The main contribution of this paper is a new algorithm to compute the LZ77 factorization without ever constructing SA or BWT for the whole input. At a high-level, the algorithm divides the input up into blocks, and processes each block in turn, by first computing a pattern matching index for the block, then scanning the prefix of the input prior to the block through the index to compute longest-matches, which are then massaged into LZ77 factors. For a string of length $n$ and $\sigma$ distinct symbols, the algorithm uses $n\log\sigma + \O(n\log n/d)$ bits of space, and $\O(d n t_{\rank})$ time, where $d$ is the number of blocks, and $t_{\rank}$ is the time complexity of the rank operation over sequences with alphabet size $\sigma$ (see e.g. [@bgnn2010]). The $n\log\sigma$ bits in the space bound is for the input string itself which is treated as read-only. Our implementation of the new algorithm does not, for the most part, use compressed or succinct data structures. The goal is to optimize speed rather than space in the data structures, because we can use the parameter $d$ to control the tradeoff. Our experiments demonstrate that this approach is in most cases superior to algorithms using compressed indexes. As a part of the new algorithm, we describe new techniques for computing matching statistics [@cl1994] that may be of independent interest. In particular, we show how to invert matching statistics, i.e., to compute the matching statistics of a string $\B$ w.r.t. a string $\A$ from the matching statistics of $\A$ w.r.t. $\B$, which saves a lot of space when $\A$ is much longer than $\B$. All our implementations operate in main memory only and thus need at least $n$ bytes just to hold the input. Reducing the memory consumption further requires some use of external memory, a direction largely unexplored in the literature so far. We speculate that the scanning, block oriented nature of the new algorithm will allow efficient secondary memory implementations, but that study is left for the future. Basic Notation and Algorithmic Machinery {#sec-preliminaries} ======================================== #### Strings. Throughout we consider a string $\X = \X[1,n] = \X[1]\X[2]\ldots \X[n]$ of $\|\X\| = n$ symbols drawn from the alphabet $[0,\sigma-1]$. We assume $\X[n]$ is a special “end of string” symbol, \$, smaller than all other symbols in the alphabet. The reverse of $\X$ is denoted $\reverse{\X}$. For $i=1,\ldots,n$ we write $\X[i,n]$ to denote the [suffix]{} of $\X$ of length $n-i+1$, that is $\X[i,n] = \X[i]\X[i+1]\ldots \X[n]$. We will often refer to suffix $\X[i,n]$ simply as “suffix $i$”. Similarly, we write $\X[1,i]$ to denote the [prefix]{} of $\X$ of length $i$. $\X[i,j]$ is the [substring]{} $\X[i]\X[i+1]\ldots \X[j]$ of $\X$ that starts at position $i$ and ends at position $j$. By $\X[i,j)$ we denote $\X[i,j-1]$. If $j < i$ we define $\X[i,j]$ to be the empty string, also denoted by $\varepsilon$. #### Suffix Arrays. The suffix array [@mm1993] $\SA_{\X}$ (we drop subscripts when they are clear from the context) of a string $\X$ is an array $\SA[1,n]$ which contains a permutation of the integers $[1,n]$ such that $\X[\SA[1],n] < \X[\SA[2],n] < \cdots < \X[\SA[n],n]$. In other words, $\SA[j] = i$ iff $\X[i,n]$ is the $j^{\mbox{{\scriptsize th}}}$ suffix of $\X$ in ascending lexicographical order. The inverse suffix array $\ISA$ is the inverse permutation of $\SA$, that is $\ISA[i] = j$ iff $\SA[j] = i$. Let $\lcp(i,j)$ denote the length of the longest-common-prefix of suffix $i$ and suffix $j$. For example, in the string $\X = zzzzzapzap$, $\lcp(1,4) = 2 = \|zz\|$, and $\lcp(5,8) = 3 = \|zap\|$. The longest-common-prefix (LCP) array [@klaap2001; @KarkkainenMP09], $\LCP_{\X}=\LCP[1,n]$, is defined such that $\LCP[1] = 0$, and $\LCP[i] = \lcp(\SA[i],\SA[i-1])$ for $i \in [2,n]$. For a string $\Y$, the $\Y$-interval in the suffix array $\SA_{\X}$ is the interval $\SA[s,e]$ that contains all suffixes having $\Y$ as a prefix. The $\Y$-interval is a representation of the occurrences of $\Y$ in $\X$. For a character $c$ and a string $\Y$, the computation of $c\Y$-interval from $\Y$-interval is called a left extension and the computation of $\Y$-interval from ${\Y}c$-interval is called a right contraction. Left contraction and right extension are defined symmetrically. #### BWT and backward search. The Burrows-Wheeler Transform [@bw1994] $\BWT[1,n]$ is a permutation of $\X$ such that $\BWT[i] = \X[\SA[i]-1]$ if $\SA[i]>1$ and $\$$ otherwise. We also define $\LF[i] = j$ iff $\SA[j] = \SA[i]-1$, except when $\SA[i] = 1$, in which case $\LF[i] = \ISA[n]$. Let $\C[c]$, for symbol $c$ be the number of symbols in $\X$ lexicographically smaller than $c$. The function $\rank(\X,c,i)$, for string $\X$, symbol $c$, and integer $i$, returns the number of occurrences of $c$ in $\X[1,i]$. It is well known that $\LF[i] = \C[\BWT[i]] + \rank(\BWT,\BWT[i],i)$. Furthermore, we can compute the left extension using $\C$ and $\rank$. If $\SA[s,e]$ is the $\Y$-interval, then $\SA[\C[c]+\rank(\BWT,c,s),\C[c]+\rank(\BWT,c,e)]$ is the $c\Y$-interval. This is called backward search. #### NSV/PSV and RMQ. For an array $\A$, the next and previous smaller value (NSV/PSV) operations are defined as $\NSV[i] = \min \{ j\in [i+1,n] \mid \A[j] < \A[i]\}$ and $\PSV[i] = \max \{ j\in [1,i-1] \mid \A[j] < \A[i]\}$. A related operation on $\A$ is range minimum query: $\RMQ(\A,i,j)$ is $k\in[i,j]$ such that $\A[k]$ is the minimum value in $\A[i,j]$. Both NSV/PSV operations and RMQ operations over the LCP array can be used for implementing right contraction (see Section \[sec-ms\]). #### LZ77. Before defining the LZ77 factorization, we introduce the concept of a [longest previous factor]{} (LPF). The LPF at position $i$ in string $\X$ is a pair $\LPF_{\X}[i]=(p_i,\ell_i)$ such that, $p_i < i$, $\X[p_i,p_i+\ell_i) = \X[i,i+\ell_i)$, and $\ell_i$ is maximized. In other words, $\X[i,i+\ell_i)$ is the longest prefix of $\X[i,n]$ which also occurs at some position $p_i < i$ in $\X$. Note that if $\X[i]$ is the leftmost occurrence of a symbol in $\X$ then $p_i$ does not exist. In this case we adopt the convention that $p_i = \X[i]$ and $\ell_i = 0$. When $p_i$ does exist we call $\X[p_i,p_i+\ell_i)$ the [source]{} for position $i$. Note also that there may be more than one potential source (that is, $p_i$ value), and we do not care which one is used. The LZ77 factorization (or LZ77 parsing) of a string $\X$ is then just a greedy, left-to-right parsing of $\X$ into longest previous factors. More precisely, if the $j$th LZ factor (or [phrase]{}) in the parsing is to start at position $i$, then we output $(p_i,\ell_i)$ (to represent the $j$th phrase), and then the $(j+1)$th phrase starts at position $i+\ell_i$, unless $\ell_i = 0$, in which case the next phrase starts at position $i+1$. When $\ell_i > 0$, the substring $\X[p_i,p_i+\ell_i)$ is called the [source]{} of phrase $\X[i,i+\ell_i)$. We denote the number of phrases in the LZ77 parsing of $\X$ by $z$. #### Matching Statistics. Given two strings $\Y$ and $\Z$, the matching statistics of $\Y$ w.r.t. $\Z$, denoted $\MS_{\YZ}$ is an array of $\|\Y\|$ pairs, $(p_1,\ell_1)$, $(p_2,\ell_2)$, ..., $(p_{\|\ssY\|},\ell_{\|\ssY\|})$, such that for all $i \in [1,\|\Y\|]$, $\Y[i,i+\ell_i) = \Z[p_i,p_i+\ell_i)$ is the longest substring starting at position $i$ in $\Y$ that is also a substring of $\Z$. The observant reader will note the resemblance to the LPF array. Indeed, if we replace $\LPF_{\Y}$ with $\MS_{\YZ}$ in the computation of the LZ factorization of $\Y$, the result is the relative LZ factorization of $\Y$ w.r.t. $\Z$ [@RLZspire2010]. Lightweight, Scan-based LZ77 Parsing {#sec-algorithm} ==================================== In this section we present a new algorithm for LZ77 factorization called . #### Basic Algorithm. Conceptually divides $\X$ up into $d=\lceil n/b \rceil$ fixed size blocks of length $b$: $\X[1,b]$, $\X[b+1,2b]$, ... . The last block could be smaller than $b$, but this does not change the operation of the algorithm. In the description that follows we will refer to the block currently under consideration as $\B$, and to the prefix of $\X$ that ends just before $\B$ as $\A$. Thus, if $\B = \X[kb+1,(k+1)b]$, then $\A = \X[1,kb]$. To begin with we will assume no LZ factor or its source crosses a boundary of the block $\B$. Later we will show how to remove these assumptions. The outline of the algorithm for processing a block $\B$ is shown below. 1. Compute $\MS_{\A\|\B}$ 2. Compute $\MS_{\B\|\A}$ from $\MS_{\A\|\B}$, $\SA_{\B}$ and $\LCP_{\B}$ 3. Compute $\LPF_{\A\B}[kb+1,(k+1)b]$ from $\MS_{\B\|\A}$ and $\LPF_{\B}$ 4. Factorize $\B$ using $\LPF_{\A\B}[kb+1,(k+1)b]$ Step 1 is the computational bottleneck of the algorithm in theory and practice. Theoretically, the time complexity of Step 1 is $\O((\|A\|+\|B\|)t_{\rank})$, where $t_{\rank}$ is the time complexity of the rank operation on $\BWT_{\B}$ (see, e.g., [@bgnn2010]). Thus the total time complexity of is $\O(dnt_{\rank})$ using $\O(b)$ words of space in addition to input and output. The practical implementation of Step 1 is described in Section \[sec-ms\]. In the rest of this section, we describe the details of the other steps. #### Step 2: Inverting Matching Statistics. We want to compute $\MS_{\B\|\A}$ but we cannot afford the space of the large data structures on $\A$ required by standard methods. Instead, we compute first $\MS_{\A\|\B}$ involving large data structures on $\B$, which we can afford, and only a scan of $\A$ (see Section \[sec-ms\] for details). We then invert $\MS_{\A\|\B}$ to obtain $\MS_{\B\|\A}$. The inversion algorithm is given in Fig. \[fig-msinvert\]. 00: ======\ 1:$i \la 1$ $\|\B\|$ $\MS_{\B\|\A}[i] \la (0,0)$\ 2:$i \la 1$ $\|A\|$\ 3:$(p_{\A},\ell_{\A}) \la \MS_{\A\|\B}[i]$\ 4:$(p_{\B},\ell_{\B}) \la \MS_{\B\|\A}[p_{\A}]$\ 5:$\ell_{\A} > \ell_{\B}$ $\MS_{\B\|\A}[p_{\A}] \la (i,\ell_{\A})$\ 6:$(p,\ell) \la \MS_{\B\|\A}[\SA_{\B}[1]]$\ 7:$i \la 2$ $\|\B\|$\ 8:$\ell \la \min(\ell,\LCP_{\B}[i])$\ 9:$(p_{\B},\ell_{\B}) \la \MS_{\B\|\A}[\SA_{\B}[i]]$\ 10:$\ell > \ell_{\B}$ $\MS_{\B\|\A}[\SA_{\B}[i]] \la (p,\ell)$\ 11:$(p,\ell) \la (p_{\B},\ell_{\B})$\ 12:$(p,\ell) \la \MS_{\B\|\A}[\SA_{\B}[\|\B\|]]$\ 13:$i \la \|B\|-1$ $1$\ 14:$\ell \la \min(\ell,\LCP_{\B}[i+1])$\ 15:$(p_{\B},\ell_{\B}) \la \MS_{\B\|\A}[\SA_{\B}[i]]$\ 16:$\ell > \ell_{\B}$ $\MS_{\B\|\A}[\SA_{\B}[i]] \la (p,\ell)$\ 17:$(p,\ell) \la (p_{\B},\ell_{\B})$ Note that the algorithm accesses each entry of $\MS_{\A\|\B}$ only once and the order of these accesses does not matter. Thus we can execute the code on lines 3–5 immediately after computing $\MS_{\A\|\B}[i]$ in Step 1 and then discard that value. This way we can avoid storing $\MS_{\A\|\B}$. #### Step3: Computing LPF. Consider the pair $(p,\ell)=\LPF_{\A\B}[i]$ for $i\in[kb+1,(k+1)b]$ that we want to compute and assume $\ell>0$ (otherwise $i$ is the position of the leftmost occurrence of $\X[i]$ in $\X$, which we can easily detect). Clearly, either $p\le kb$ and $\LPF_{\A\B}[i]=\MS_{\B\|\A}[i]$, or $kb < p < i$ and $\LPF_{\A\B}[i]=(kb+p_{\B},l_{\B})$, where $(p_{\B},l_{\B})=\LPF_{\B}[i-kb]$. Thus computing $\LPF_{\A\B}$ from $\MS_{\B\|\A}[i]$ and $\LPF_{\B}$ is easy. The above is true if the sources do not cross the block boundary, but the case where $p\le kb$ but $p+\ell > kb+1$ is not handled correctly. An easy correction is to replace $\MS_{\A\|\B}$ with $\MS_{\A\B\|\B}[1,kb]$ in all of the steps. #### Step 4: Parsing. We use the standard LZ77 parsing to factorize $\B$ except $\LPF_{\B}$ is replaced with $\LPF_{\A\B}[kb+1,(k+1)b]$. So far we have assumed that every block starts with a new phrase, or, put another way, that a phrase ends at the end of every block. Let $\X[i,(k+1)b]$ the last factor in $\B$, after we have factorized $\B$ as described above. This may not be a true LZ factor when considering the whole $\X$ but may continue beyond the end of $\B$. To find the true end point, we treat $\X[i,n]$ as a pattern, and apply the constant extra space pattern matching algorithm of Crochemore [@c1992], looking for the longest prefix of $\X[i,n]$ starting in $\X[1,i-1]$. We must modify the algorithm from [@c1992] so that it matches prefixes rather than whole occurrences of the pattern, but this is possible without increasing its time or space complexity. Computation of matching statistics {#sec-ms} ================================== In this section, we describe how to compute the matching statistics $\MS_{\A\|\B}$. As mentioned in Section \[sec-algorithm\], what we really want is $\MS_{\A\B\|\B}[1,kb]$. However, the only difference is that the starting point of the computation is the $\B$-interval in $\SA_{\B}$ instead of the $\varepsilon$-interval. Similarly to most algorithms for computing the matching statistics, we first construct some data structures on $\B$ and then scan $\A$. During the whole LZ factorization, most of the time is spend on the scanning and the time for constructing the data structures is insignificant in practice. Thus we omit the construction details here. The space requirement of the data structures is more important but not critical as we can compensate for increased space by reducing the block size $b$. Using more space (per character of $\B$) is worth doing if it increases scanning speed more than it increases space. Consequently, we mostly use plain, uncompressed arrays. #### Standard approach. The standard approach of computing the matching statistics using the suffix array is to compute for each position $i$ the longest prefix $\Pr_i=\A[i,i+\ell_i)$ of the suffix $\A[i,\|A\|]$ such that the $\Pr_i$-interval in $\SA_{\B}$ is non-empty. Then $\MS_{\A\|\B}[i]=(p_i,\ell_i)$, where $p_i$ is any suffix in the $\Pr_i$-interval. This can be done either with a forward scan of $\A$, computing each $\Pr_i$-interval from $\Pr_{i-1}$-interval using the extend right and contract left operations [@ako2004], or with a backward scan computing each $\Pr_i$-interval from $\Pr_{i+1}$-interval using the extend left and contract right operations [@og2011]. We use the latter alternative but with bigger and faster data structures. The extend left operation is implemented by backward search. We need the array $\C$ of size $\sigma$ and an implementation of the rank function on $\BWT$. For the latter, we use the fast rank data structure of [@fgm2012], which uses $4b$ bytes. The contract right operation is implemented using the NSV and PSV operations on $\LCP_{\B}$ as in [@og2011], but instead of a compressed representation, we store the NSV and PSV values as plain arrays. As a nod towards reducing space, we store the NSV/PSV values as offsets using 2 bytes each. If the offset is too large (which is very rare), we obtain the value using the NSV/PSV data structure of C[á]{}novas and Navarro [@cn2010], which needs less than $0.1b$ bytes. Here the space saving was worth it as it had essentially no effect on speed. The peak memory use of the resulting algorithm is $n+(24.1)b+\O(\sigma)$ bytes. #### New approach. Our second approach is similar to the first, but instead of maintaining both end points of the $\Pr_i$-interval, we keep just one, arbitrary position $s_i$ within the interval. In principle, we perform left extension by backward search, i.e., $s_i=\C[\X[i]]+\rank(\BWT,\X[i],s_{i+1})$. However, checking whether the resulting interval is empty and performing right contractions if it is, is more involved. To compute $s_i$ and $\ell_i$ from $s_{i+1}$ and $\ell_{i+1}$, we execute the following steps: 1. Let $c=\X[i]$. If $\BWT[s_{i+1}]=c$, set $s_i=\C[c]+\rank(\BWT,c,s_{i+1})$ and $\ell_i=\ell_{i+1}+1$. 2. Otherwise, let $\BWT[u]$ be the nearest occurrence of $c$ in $\BWT$ before the position $s_{i+1}$. Compute the rank of that occurrence $r=\rank(\BWT,c,u)$ and $\ell_u=\LCP[\RMQ(\LCP,u+1,s_{i+1})]$. If $\ell_u\ge \ell_{i+1}$, set $s_i=\C[c]+r$ and $\ell_i=\ell_{i+1}+1$. 3. Otherwise, let $\BWT[v]$ be the nearest occurrence of $c$ in $\BWT$ after the position $s_{i+1}$ and compute $\ell_v=\LCP[\RMQ(\LCP,s_{i+1}+1,v)]$. If $\ell_v \le \ell_u$, set $s_i=\C[c]+r$ and $\ell_i=\ell_u+1$. 4. Otherwise, set $s_i=\C[c]+r+1$ and $\ell_i=\min(\ell_{i+1},\ell_v)+1$. The implementation of the above algorithm is based on the arrays $\BWT$, $\LCP$ and $\R[1,b]$, where $\R[i]=\rank(\BWT,\BWT[i],i)$. All the above operations can be performed by scanning $\BWT$ and $\LCP$ starting from the position $s_{i+1}$ and accessing one value in $\R$. To avoid long scans, we divide $\BWT$ and $\LCP$ into blocks of size $2\sigma$, and store for each block and each symbol $c$, the values $r$, $\ell_u$ and $\ell_v$ that would get computed if scans starting inside the block continued beyond the block boundaries. The peak memory use is $n+27b+\O(\sigma)$ bytes. This is more than in the first approach, but this is more than compensated by increased scanning speed. #### Skipping repetitions. During the preceding stages of the LZ factorization, we have built up knowledge of repetition present in $\A$, which can be exploited to skip (sometimes large) parts of $\A$ during the matching-statistics scan. Consider an LZ factor $\A[i,i+\ell)$. Because, by definition, $\A[i,i+\ell)$ occurs earlier in $\A$ too, any source of an LZ factor of $\B$ that is completely inside $\A[i,i+\ell)$ could be replaced with an equivalent source in that earlier occurrence. Thus such factors can be skipped during the computation of $\MS_{\A\|\B}$ without an effect on the factorization. More precisely, if during the scan we compute $\MS_{\A\|\B}[j]=(p,k)$ and find that $i \leq j < j + k \leq i+\ell$ for an LZ factor $\A[i,i+\ell)$, we will compute $\MS_{\A\|\B}[i-1]$ and continue the scanning from $i-1$. However, we will do this only for long phrases with $\ell\ge 40$. To compute $\MS_{\A\|\B}[i-1]$ from scratch, we use right extension operations implemented by a binary search on $\SA$. To implement this “skipping trick” we use a bitvector of $n$ bits to mark LZ77 phrase boundaries adding $0.125n$ bytes to the peak memory. Algorithms Based on Compressed Indexes {#sec-oldalgs} ====================================== We went to some effort to ensure the baseline system used to evaluate $\LZSCAN$ in our experiments was not a “straw man”. This required careful study and improvement of some existing approaches, which we now describe. #### FM-Index. The main data structure in all the algorithms below is an implementation of the FM-index (FMI) [@fm2005]. It consists of two main components: - $\BWT_{\X}$ with support for the rank operation. This enables backward search and the LF operation as described in Section \[sec-preliminaries\]. We have tried several rank data structures and found the one by Navarro [@nav2004 Sect. 7.1] to be the best in practice. - A sampling of $\SA_{\X}$. This together with the LF operation enables arbitrary $\SA$ access since $\SA[i]=\SA[\LF^k[i]]+k$ for any $k<\SA[i]$. The sampling rate is a major space–time tradeoff parameter. In many implementations of FMI, the construction starts with computing the uncompressed suffix array but we cannot afford the space. Instead, we construct $\BWT$ directly using the algorithm of Okanohara and Sadakane [@os2009]. The method uses roughly $2$–$2.5n$ bytes of space but destroys the text, which is required later during LZ parsing. Thus, once we have $\BWT$, we build a rank structure over it and use it to invert the $\BWT$. During the inversion process we recover and store the text and gather the $\SA$ sample values. #### CPS2 simulation. The CPS2 algorithm [@cps2008] is an LZ parsing algorithm based on $\SA_{\X}$. To compute the LZ factor starting at $i$, it computes the $\X[i,i+\ell)$-interval for $\ell=1,2,3,\ldots$ as long as the $\X[i,i+\ell)$-interval contains a value $p<i$, indicating an occurrence of $\X[i,i+\ell)$ starting at $p$. The key operations in CPS2 are right extension and checking whether an $\SA$ interval contains a value smaller than $i$. Kreft and Navarro [@kn2010] as well as Ohlebusch and Gog [@og2011] are using $\FMI$ for $\reverse{\X}$, the reverse of $\X$, which allows simulating right extension on $\SA_{\X}$ by left extension on $\SA_{\reverse{\X}}$. The two algorithms differ in the way they implement the interval checks: - Kreft and Navarro use the RMQ operation. They use the RMQ data structure by Fischer and Heun [@fh2007] but we use the one by C[á]{}novas and Navarro [@cn2010]. The latter is easy and fast to construct during BWT inversion but queries are slow without an explicit $\SA$. We speed up queries by replacing a general RMQ with the check whether the interval contains a value smaller than $i$. This implementation is called $\LZFMICN$. - Ohlebusch and Gog use NSV/PSV queries. The position $s$ of $i$ in $\SA$ must be in the $\X[i,i+\ell)$-interval. Thus we just need to check whether either $\NSV[s]$ or $\PSV[s]$ is in the interval too. They as well as we implement NSV/PSV using a balanced parentheses representation (BPR). This representation is initialized by accessing the values of $\SA$ left-to-right, which makes the construction slow using $\FMI$. However, NSV/PSV queries with this data structure are fast, as they do not require accessing $\SA$. This implementation is called $\LZFMIBPR$. #### ISA variant. Among the most space efficient prior LZ factorization algorithms are those of the ISA family [@kp2013] that use a sampled $\ISA$, a full $\SA$ and a rank/LF implementation that relies on the presence of the full $\SA$. We reduce the space further by replacing $\SA$ and the rank/LF data structure with the FM-index described above to obtain an algorithm called $\LZFMIISA$. Experiments {#sec-experiments} =========== We performed experiments with the files listed in Table \[tab:files\]. All tests were conducted on a 2.53GHz Intel Xeon Duo CPU with 32GB main memory and 8192K L2 Cache. The machine had no other significant CPU tasks running. The operating system was Linux (Ubuntu 10.04) running kernel 3.0.0-26. The compiler was g++ (gcc version 4.4.3) executed with the [-O3 -static -DNDEBUG]{} options. Times were recorded with the C [ clock]{} function. All algorithms operate strictly in-memory. #### LZscan vs. other algorithms. We compared the $\LZSCAN$ implementation using our new approach for matching statistics boosted with the “skipping trick” (Section \[sec-ms\]) to algorithms based on compressed indexes (Section \[sec-oldalgs\]). The experiments measured the time to compute LZ factorization with varying amount of available working space. The results are shown in Figure \[fig-all\]. In almost all cases $\LZSCAN$ outperforms other algorithm across the whole space spectrum. Moreover, it can operate with very small available memory (close to $n$ bytes) unlike other algorithms, which all require at least $2n$ space to compute $\BWT$. It achieves a superior performance for highly repetitive data even at very low memory levels. #### Variants of LZscan. The second experiment measured the improvement of our new matching statistics computation over standard approach (see Section \[sec-ms\]). Additionally, each variant was tested with and without the “skipping trick”, giving 4 combinations in total. The results are plotted in Figure \[fig-variants\]. In nearly all cases applying any of our new techniques improves the runtime over the standard approach, but the best effect in all cases is achieved when the techniques are combined together. The total speedup then varies from a factor of 2 (dna) up to 12 (einstein), clearly depending on the repetitiveness of input. [^1]: Supported by Academy of Finland grant 118653 (ALGODAN)
--- abstract: 'While quark-hadron duality is well-established experimentally, the current theoretical understanding of this important phenomenon is quite limited. To expose the essential features of the dynamics behind duality, we use a simple model in which the hadronic spectrum is dominated by narrow resonances made of valence quarks. We qualitatively reproduce the features of duality as seen in electron scattering data within our model. We show that in order to observe duality, it is essential to use the appropriate scaling variable and scaling function. In addition to its great intrinsic interest in connecting the quark-gluon and hadronic pictures, an understanding of quark-hadron duality could lead to important benefits in extending the applicability of scaling into previously inaccessible regions.' address: - '$^{(1)}$ Jefferson Lab, 12000 Jefferson Ave, Newport News, VA 23606' - \| $^{(2)}$ Special Research Centre for the Subatomic Structure of Matter,\ Adelaide University, Adelaide 5005, Australia - '$^{(3)}$ Department of Physics, Old Dominion University, Norfolk, VA 23529' author: - 'Nathan Isgur$^{(1)}$, Sabine Jeschonnek$^{(1)}$, W. Melnitchouk$^{(1,2)}$, and J. W. Van Orden$^{(1,3)}$' title: 'Quark-Hadron Duality in Structure Functions' --- Introduction ============ Background ---------- Duality is a much used and much abused concept. In some cases it is used to describe an equivalence between quark- and hadron-based pictures which is trivial; in others an equivalence which is impossible. In almost all cases, the conceptual framework in which duality is discussed and used is either hopelessly muddled or hopelessly abstract. Nevertheless, the data indicate that some extremely interesting and potentially very important “duality” phenomena are occurring at low energy. We begin by making the trivial observation that any hadronic process can be correctly described in terms of quarks and gluons, assuming that Quantum Chromodynamics (QCD) is the correct theory for strong interactions. While this statement is obvious, it rarely has practical value, since in most cases we can neither perform nor interpret a full QCD calculation. We will refer to the above statement that any hadronic process can be described by a full QCD calculation as “degrees of freedom duality”: if one could perform and interpret the calculations, it would not matter at all which set of states — hadronic states or quark and gluon states — was used. On the other hand, there are rare cases where the average of hadronic observables is described by a perturbative QCD (pQCD) calculation. We reserve the use of the term “duality” to describe these rare correspondences, in contrast to the trivial “degrees of freedom duality” described above. In these rare cases, a quark-gluon calculation leads to a very simple description of some phenomenon even though this phenomenon “materializes” in the form of hadrons. Deep inelastic scattering is the prototypical example, and the one on which we focus here. These rare examples are all characterized by a special choice of kinematic conditions which serve to expose the “bare” quarks and gluons of the QCD Lagrangian. In the case of deep inelastic scattering, the kinematics are such that the struck quark receives so much energy over such a small space-time region that it behaves like a free particle during the essential part of its interaction. This leads to the compellingly simple picture that the electron-nucleon cross section is determined in this kinematic region by free electron-quark scattering, i.e. duality is exact for this process in this kinematic regime. For inclusive inelastic electron scattering from a proton in the scaling region, the cross section is determined by the convolution of a non-perturbative and currently difficult to calculate parton distribution function with an electron-quark scattering cross section determined by perturbative QCD (pQCD). For semileptonic decays of heavy quarks, [e.g.]{} $\bar{B} \rightarrow X_c l \bar \nu_l$, one can prove using pQCD that the decay rate is determined by that of the underlying heavy quark, in this case obtained from the process $b \rightarrow c l \bar \nu_l$ [@isgurwise]. In $e^+e^- \rightarrow hadrons$, it is the underlying $e^+e^- \rightarrow q \bar q$ process that applies because of pQCD. However, while duality applies to all of these phenomena, we will see that even in these special processes we must invoke an averaging procedure to identify the hadronic results with the quark-gluon predictions. In addition to its need of an averaging procedure, it is easy to see that the pQCD picture of inelastic electron scattering must fail for $Q^2 \rightarrow 0$. For duality to hold for the nucleon structure functions in this case, the elastic electric proton and neutron form factors, which take the value of the nucleon charges for $Q^2 \rightarrow 0$, would have to be reproduced by electron scattering off the corresponding $u$ and $d$ quarks. This is possible for the proton since the squares of the charges of two $u$ quarks and one $d$ quark add up to 1 [@gottfried]. However, for the neutron, the squared quark charges cannot add up to 0, so it is clear that local duality in inclusive inelastic electron scattering from a neutron must fail for $Q^2 \rightarrow 0$. Also, we know that duality must fail for polarized structure functions at low $Q^2$, as the Ellis-Jaffe sum rule and the Gerassimov-Drell-Hearn sum rule, which can be written as integrals over $g_1(\nu,Q^2)$ at different $Q^2$, are negative (GDH sum rule for $Q^2 = 0$) and positive (Ellis-Jaffe sum rule at $Q^2$ of several GeV), respectively [@g1dual]. Thus duality in inelastic electron scattering has to hold in the scaling regime and must in general break down at low energy. Obviously, a very interesting question is what happens in between these regimes, [i.e.]{} how does duality break down? This paper answers this question, which is not only interesting in itself, but also crucial for practical, quantitative applications of duality. Introducing Local Averaging and Our Model ----------------------------------------- We begin by discussing the issue of averaging. If duality is relevant at all at low energy, then it is quite obvious that we need to perform some sort of average: the smooth, analytic pQCD prediction cannot in general correspond exactly to the generally highly structured hadronic data. For low energies this requirement is universally accepted; however, even in the “scaling” region one must average in principle. To see this, consider QCD in the large-$N_c$ limit [@largenc]. We can do this because no element of the pQCD results for deep inelastic scattering depends on the number of colors. However, in this limit the hadronic spectrum consists entirely of infinitely narrow noninteracting resonances [@baryoncaveat], [i.e.]{}, there are only infinitely narrow spikes in the $N_c \rightarrow \infty$ hadronic world. Since the quark level calculation still yields a smooth scaling curve, and the kinematic conditions for being in the scaling region are unchanged as $N_c \rightarrow \infty$, we see that we must average even in the scaling region. While in Nature, the resonances have fairly broad decay widths so that the averaging takes place automatically in the data, the large $N_c$ limit shows us that averaging is always required in principle. It is thus clearly important to be able to define this averaging procedure, [[e.g.]{}]{}, how large the intervals must be and which resonances have to be included. It is easy to see that this procedure will not be universal, and will certainly not simply be that the resonances one-by-one locally average the pQCD-derived scaling curve: the averaging method will depend on the process and on the target. Consider, as an illustration of these points, the case of a spinless quark and antiquark with charges $e_1$ and $e_2$ and equal masses bound into a nonrelativistic $q_1 \bar q_2$ system. The inelastic electron scattering rate calculated at the quark level in leading twist will then be proportional to $e^2_1+e^2_2$. Since the elastic state will be produced with a rate proportional to $(e_1+e_2)^2$, it clearly cannot in general be locally dual to the scaling curve [@schmidt]. How then is duality realized in this system? Consider the charge operator $\sum_i e_i e^{i\vec q \cdot \vec r_i}$: from the ground state it excites even partial wave states with an amplitude proportional to $e_1+e_2$ and odd ones with an amplitude proportional to $e_1-e_2$. Thus the resonances build up a cross section of the form $\alpha_1 (e_1+e_2)^2 + \alpha_2 (e_1-e_2)^2 + \alpha_3 (e_1+e_2)^2 + \cdots$ and one can see by explicit calculations in models that (up to phase space factors) the cross terms in this sum will cancel to give a cross section proportional to $e^2_1+e^2_2$ once averaged over nearby even and odd parity resonances. It is clear that such target- and process-dependence is worthy of study. However, in this paper we will restrict ourselves to a model with $e_2=0$ so that local duality might apply [@CloseIsgur]. The question of the validity of low energy duality, [i.e.]{}, duality in electron scattering at finite beam energies in inelastic electron scattering after suitable averaging, is as old as the first inclusive electron scattering experiments themselves. It begins with the seminal paper of Bloom and Gilman [@bgduality], which made the observation that the inclusive $F_2$ structure function in the resonance region at low $Q^2$ generally oscillates about and averages to a global scaling curve which describes high $Q^2$ data. More recently, interest in Bloom-Gilman duality has been revived with the collection of high precision data on the $F_2$ structure function from Jefferson Lab [@JLAB]. These data not only confirmed the existence of Bloom-Gilman duality to rather low values of $Q^2$, but also seem to demonstrate that for the proton the equivalence of the averaged resonance and scaling structure functions holds also for each resonance so that duality also exists locally. Here we present a model for the study of quark-hadron duality in electron scattering that uses only a few basic ingredients. Namely, in addition to requiring that our model be relativistic, we assume confinement and that it is sufficient to consider only valence quarks (this latter simplification being underwritten, as mentioned previously, by the large $N_c$ limit). In addition, since our model is designed to explore conceptual issues and not to be compared to data, and since we postpone addressing spin-dependent issues to later work, for simplicity we also take the quarks, electrons and photons to be scalars. A model with these features will not give a realistic description of any data, but it should allow us to study the critical questions of when and why duality holds. While this model is extremely simple, we see no impediment to extending it to describe a more realistic situation since we find that duality arises from the most basic properties of our model. We make several more convenient simplifications. Although it is our aim to study duality in electron scattering from the nucleon, [i.e.]{} from a three-quark-system, as a first step we study these issues in what is effectively a one quark system by considering such a quark to be confined to an infinitely massive antiquark. In the case of scalar quarks considered here, we can therefore describe the system by the Klein-Gordon equation. We also select for our confining potential one which is linear in $r$, namely $V^2 (\vec r \,) = \alpha \, r^2$, where $\alpha$ is a generalized, relativistic string constant. This choice allows us to obtain analytic solutions, without which the required numerical work for this study would be daunting. Indeed, the energy eigenvalues, $E_N = \sqrt{ 2 \sqrt{\alpha}(N + 3/2) + m^2 }$, where $m$ is the mass of interacting quark, can be readily obtained by noting the similarity to the Schrödinger equation for a non-relativistic harmonic oscillator potential: the solutions for the wave functions are the same as for the non-relativistic case. In the next Section we construct the structure function out of resonances described by form factors, each of which individually gives vanishing contributions at large momenta, and show that it both scales and, when suitably averaged, is equal to the “free quark” result. An analysis in terms of structure function moments is presented in Section III. In Section IV we examine the onset of scaling, and the appearance of Bloom-Gilman duality, while in Section V we discuss the connection of Bloom-Gilman duality with duality in heavy quark systems. Finally, in Section VI we summarize our results and mention some possible directions for future research. Quark-Hadron Duality in the Scaling Limit ========================================= The differential cross section for inclusive inelastic scattering of a “scalar electron” [via]{} the exchange of a “scalar photon” is $$\frac{d \sigma} {dE_f d\Omega_f} = \frac{g^4}{16\pi^2} \frac{E_f}{ E_i} \frac{1}{Q^4}\ {\cal W} \,, \label{defscalw}$$ where the scalar coupling constant $g$ carries the dimension of a mass, and the factor multiplying the scalar structure function ${\cal W}$ corresponds to the Mott cross section. In a model where the only excited states are infinitely narrow resonances, ${\cal W}$ is given entirely by a sum of squares of transition form factors weighted by appropriate kinematic factors: $$% {\cal W} = \sum_{N=0}^{\infty} \, \frac{1}{4 E_0 E_N} \, \, {\cal W}(\nu,\vec q \, ^2) = \sum_{N=0}^{N_{\rm max}} \, \frac{1}{4 E_0 E_N}\, \, \left\| F_{0N}(\vec q) \right\|^2 \, \, \delta(E_N - E_0 - \nu)\, , \label{wscalho}$$ where $\vec q \equiv \vec p_i - \vec p_f$, the form factor $F_{0N}$ represents a transition from the ground state to a state characterized by the principal quantum number $N$, and the sum over states $N$ goes up to the maximum $N_{\rm max}$ allowed kinematically. Note that for fixed, positive $Q^2 \equiv \vec q \, ^2 - \nu^2$, $N_{max} = \infty$. The excitation form factors can be derived using the recurrence relations of the Hermite polynomials. One finds: $$\label{eqff} F_{0N} (\vec q \, ^2) = \frac{1}{\sqrt{N!}} \, i^N \, \left ( \frac{\|\vec q\|}{\sqrt{2} \, \beta} \right ) ^N \exp (- \vec q \, ^2 / 4 \, \beta^2) \, ,$$ where $\beta = \alpha^{1/4}$. This form factor is in fact the sum of all form factors for excitations from the ground state to degenerate states with the same principal quantum number $N$. As a precursor to our discussion of duality, we note that it will be a necessary condition for duality that these form factors (or more generally those corresponding to some other model potential) can represent the pointlike free quark. It is in fact the case that $\sum_{N=0}^{N_{\rm max}} \|F_{0N} (\vec q \,)\|^2 \rightarrow 1$ as $N_{\rm max} \rightarrow \infty$, a relation which follows from the completeness of the confined wave functions. Incidentally, an examination of the convergence of this sum as a function of $\vert \vec q \vert^2$ is sufficient to make the point that reproducing the behavior of a free quark requires more and more resonances as $\vert \vec q \vert^2$ increases (details of this will be discussed in a forthcoming publication). Scaling in the presence of confining final state interactions has previously been investigated in Refs. [@ioffe; @gurvitzrinat; @greenberg; @psl], where similar conclusions are reached. This suggests that scaling may indeed be a trivial feature of a large class of simple quantum mechanical models. Some sense of how this can occur can be obtained by considering some of the properties of the relativistic oscillator model used in this paper. In particular, consider the properties of the square of the form factors. For a fixed principal quantum number, $N$, the form factor has a maximum in $\|\vec q\|$ at $\vec q \, ^2_N=2\beta^2\,N$. Using $\nu_N=E_N-E_0$ and $E_N=\sqrt{2\beta^2\,N+E_0^2}$, it can be shown that $$\nu_N=\frac{Q^2_N}{2E_0}$$ where $Q^2_N=\vec q \, ^2_N - \nu_N^2$. So the position of the peak in the averaged structure function occurs at $u_{Bj}={m/E_0}$ where $u_{Bj}={Q^2/2m\nu}$ is a scaled Bjorken scaling variable $u_{Bj}\equiv\frac{M}{m}x_{Bj}$ which takes into account that as the mass of the antiquark $M_{\bar Q}\rightarrow\infty$, the constituent quark will carry only a fraction of order $m/E_0$ of the hadron’s infinite-momentum-frame momentum. Furthermore, for fixed $\vec q$ the structure function falls off smoothly for energy transfers away from the peak value. The width of this peak as a function of energy transfer also becomes constant for large $\|\vec q\|$. Now consider the integral of the structure function $$\Sigma(\vec q \, ^2)=\int_0^\infty d\nu\ {\cal W}(\nu,\vec q \, ^2) =\sum_{nlm}\frac{1}{4E_0E_N}<\psi_{000}\|\rho(-\vec q)\|\psi_{nlm}> <\psi_{nlm}\|\rho(\vec q)\|\psi_{000}>$$ where $N=2(n-1)+l$ with $n=1,2,3,\cdots$, and where $\rho(\vec q)=e^{i\vec q\cdot\vec x}$. Since the form factor sum for a fixed $\vec q$ peaks about $E_{N_{max}}=\sqrt{\vec q \, ^2+E_0^2}$, we can substitute $E_N\rightarrow E_{N_{max}}$ and then sum over the complete set of final states to give $$\Sigma(\vec q \, ^2)\cong\frac{1}{4E_0E_{N_{max}}}\cong\frac{1}{4E_0q}$$ for large momentum transfer. Therefore, if we define the scaling function as ${\cal S}\equiv \|\vec q\|\ {\cal W}$, as will be done below, the area of the scaling function becomes constant at large momentum transfer. Since the scaling function peaks at fixed $u_{Bj}$, smoothly falls about the peak, has fixed width and constant area at large momentum transfer, the model scales. It is a common misconception that the presence of scaling implies that the final states must become plane waves. In fact, the argument above makes it clear that scaling occurs when the structure function becomes independent of the final states as in the closure approximation used here. To see duality clearly both experimentally and theoretically, one needs to go beyond the Bjorken scaling variable $x_{Bj}$ and the scaling function ${\cal{S}}_{Bj} = \nu \cal{W}$ that goes with it. This is because in deriving Bjorken’s variable and scaling function, one not only assumes $Q^2$ to be larger than any mass scale in the problem, but also that high $Q^2$ (pQCD) dynamics controls the interactions. However, duality has its onset in the region of low to moderate $Q^2$, and there masses and violations of asymptotic freedom do play a role. Bloom and Gilman used a new, [ad hoc]{} scaling variable $\omega'$ [@bgduality] in an attempt to deal with this fact. In most contemporary data analyses, the Nachtmann variable [@greenbergb; @nachtmann] is used together with ${\cal{S}}_{Bj}$. Nachtmann’s variable contains the target mass as a scale, but neglects quark masses. For our model, the constituent quark mass (assumed to arise as a result of spontaneous chiral symmetry breaking) is vital at low energy, and a scaling variable that treats both target and quark masses is desirable. Such a variable was derived more than twenty years ago by Barbieri [et al.]{} [@barbieri] to take into account the masses of heavy quarks; we use it here given that after spontaneous chiral symmetry breaking the nearly massless light quarks have become massive constituent quarks, calling it $x_{cq}$: $$% x_{cq} = \frac{1}{2 M} \left ( \sqrt{\nu^2 + Q^2} - \nu \right ) x_{cq} = \frac{1}{2 M} \left ( \sqrt{\nu^2 + Q^2} - \nu \right ) \left ( 1 + \sqrt{1 + \frac{4 m^2}{Q^2}} \right ) \, . \label{defxdis}$$ The scaling function associated with this variable is given by: $$\label{S} {\cal{S}}_{cq} \equiv \|\vec q\|\ {\cal W} = \sqrt{\nu^2 + Q^2}\ {\cal W}\,.$$ This scaling function and variable were derived for scalar quarks which are free, but have a momentum distribution. The derivation of a new scaling variable and function for bound quarks will be published elsewhere. Numerically, this scaling variable does not differ very much from the one in Eq. (\[defxdis\]). Of course all versions of the scaling variable must converge to $x_{Bj}$ and all versions of the scaling function must converge towards ${\cal{S}}_{Bj}$ for large enough $Q^2$. One can also easily verify that in the limit $m \to 0$ one obtains from (\[defxdis\]) the Nachtmann scaling variable. In the following, we use the variable $x_{cq}$ and the scaling function ${\cal{S}}_{cq}$. We are now ready to look at scaling and duality in our model. Since the target has mass $M \rightarrow \infty$, it is convenient to rescale the scaling variable $x_{cq}$ by a factor $M/m$: $$\begin{aligned} u & \equiv & {M \over m}\ x_{cq}\ ~~.\end{aligned}$$ The variable $u$ takes values from 0 to a maximal, $Q^2$ dependent value, which can go to infinity. The high energy scaling behavior of the appropriately rescaled structure function ${\cal {S}}_{cq}$ is illustrated in Fig. 1. The structure function has been evaluated using the phenomenologically reasonable parameters $m = 0.33$ GeV and $\alpha = (0.4 ~{\rm GeV})^{1/4}$, though we remind the reader not to compare our results, which might resemble electron scattering from a $B$ meson, to nucleon data! To display it in a visually meaningful manner, the energy-dependent $\delta$-function has been smoothed out by introducing an unphysical Breit-Wigner shape with an arbitrary but small width, $\Gamma$ chosen for purposes of illustration: $$\delta(E_N - E_0 - \nu) \rightarrow \frac{\Gamma}{2 \pi} \, \, \frac{f}{(E_N - E_0 - \nu)^2 + (\Gamma/2)^2}\, ,$$ where the factor $ f = {\pi}/[{\frac{\pi}{2} + \arctan {2 (E_N - E_0) \over \Gamma}] } $ ensures that the integral over the $\delta$-function is identical to that over the Breit-Wigner shape. The curves in Fig. 1 show that scaling sets in rather rapidly. The resonances show up as bumpy structures in the low $Q^2$ region (which will be discussed in Section IV below), a trace of which is visible for the $Q^2 = 5$ GeV$^2$ curve. By taking the continuum limit for the energy and applying Stirling’s formula, one can obtain an analytic expression for the scaling curve, valid in the scaling region, for the transition of the quark from the ground state to the sum of all excited states: $$% {\cal{S}}(u) = m^2 \, u^2 \,\frac{1}{\pi^\frac{1}{2}\beta E_0} {\cal{S}}_{cq}(u) = \frac{E_0}{ \sqrt{\pi} \beta } \exp{\frac{(E_0-m u)^2}{\beta^2}} \, . \label{sfanalyt}$$ Of course we still need to verify that this scaling curve as seen in Fig. 1 found by summing over hadrons is the same as the one which we would obtain from deep inelastic scattering off the quark, [[i.e.]{}]{}, if we were to switch off the potential in the final state. In this case, the tower of hadronic states is replaced by the free quark continuum. Duality predicts that the results should be the same in the scaling limit, and by direct calculation we confirm this. Moments of Structure Functions {#secglo} ============================== Bloom-Gilman duality relates structure functions at low and high $Q^2$ averaged over appropriate intervals of the hadronic mass $W$. As a quantitative measure of this feature of the data, one conventionally examines the $Q^2$-dependence of moments of structure functions. The moments offer the cleanest connection with the operator product expansion of QCD, and provide a natural connection between duality in the high- and low-$Q^2$ regions. By considering the moments, we also remove artifacts introduced through the smoothing procedure described above for the structure function itself. The moments of the structure function ${\cal{S}}_{cq}(u,Q^2)$ are defined as: $$\label{moment} M_n(Q^2) = \int_0^{u_{\rm max}} du \, \, u^{n-2} \, {\cal{S}}_{cq}(u,Q^2)\, ,$$ where $u_{\rm max}$ corresponds to the maximum value of $u$ which is kinematically accessible at a given $Q^2$. Evaluating the moments of the structure function (\[S\]) explicitly one has (provided the kinematics allow us to access all excited states): $$\begin{aligned} M_n(Q^2) &=& \left ( \frac{r}{2 m} \right )^{n-1} \, \sum_{N=0}^{\infty} \, \left( \sqrt{\nu_N^2 + Q^2} - \nu_N \right)^{n-1} \, \, \frac{E_0}{ E_N} \, \left\|F_{0N} \left(\sqrt{\nu_N^2 + Q^2} \right) \right\|^2 ~~ ,\end{aligned}$$ where $\nu_N = E_N - E_0$ and $r = 1 + \sqrt{1 + 4 m^2/Q^2}$. The elastic contribution to the moments is $$M_n^{\rm elastic} (Q^2) = \left ( \frac{r}{2 m} \right )^{n-1}\, Q^{n-1} \, \, \left\| F_{00} (Q^2) \right\|^2\ = u^{n-1}_0 \left\| F_{00} (Q^2) \right\|^2\, ~~,$$ where $u_0(Q^2)$ is the position in $u$ of the ground state. Note that $M_n^{\rm elastic} (Q^2)$ becomes independent of $n$ in the limit $Q^2 \rightarrow 0$, approaching unity and that the inelastic contributions to the moments vanish for vanishing $Q^2$. In Fig. 2 we show the $n=2$, 4, 6 and 8 moments $M_n$ as a function of $Q^2$. All the moments appear qualitatively similar, rising to within about 10% of their asymptotic values by $Q^2=1$ GeV$^2$. Also evident is the fact that the lower moments reach their asymptotic values earlier than the higher moments. This is qualitatively consistent with the expectation from the operator product expansion discussed in [@DGP], where it was argued that the effective expansion parameter in the twist expansion $\sim n/Q^2$, so that for higher moments, $n$, the higher twist terms survive to larger values of $Q^2$. Unfortunately, these moments do not have such useful interpretations here as they do in real deep inelastic scattering. For example, the analog of the Gross-Llewellyn Smith sum rule is not applicable here because the scalar current which couples to our quark is not conserved. Nonetheless, the moments in Fig. \[figmom1dist\] do serve to demonstrate that scaling is a natural consequence of our model, and illustrate the relative onset of scaling for different moments. Onset of Scaling and Bloom-Gilman Duality ========================================= After studying the scaling behavior of the structure functions in our model at high $Q^2$ and the moments over a range of four-momentum transfers, we now study the structure functions at low $Q^2$ where not only in the large $N_C$ limit but also in nature resonances are visibly dominant over a wide range in the scaling variable. Here, we consider a target where only one quark carries all the charge of the system, so there is no forced breakdown of duality at $Q^2 = 0$ of the type noted earlier for the neutron. Still, one cannot expect that the perturbative QCD result will describe even averaged hadronic observables well at very low $Q^2$: these are after all strong interactions! If local duality holds, one might expect the resonance “spikes" to oscillate around the scaling curve and to average to it, once $Q^2$ is large enough. (We remind the reader that while scaling in deep-inelastic electron scattering from the nucleon is known from experiment to set in by $Q^2 \sim 2$ GeV$^2$, the target considered here corresponds to an infinitely heavy “meson” composed of scalar quarks interacting with a scalar current, so one should not expect numerically realistic results, only qualitative ones.) Figure 3 shows the onset of scaling for the structure function ${\cal{S}}_{cq}$ as a function of $u$, as $Q^2$ varies from 0.5 GeV$^2$ to 2 GeV$^2$. As in Fig. 1, for each of the resonances (excluding the elastic peak) the energy $\delta$-function has been smoothed out using the Breit-Wigner method with a width $\Gamma=100$ MeV. With increasing $Q^2$, each of the resonances moves out towards higher $u$, as dictated by kinematics. At $Q^2 = 0$, the elastic peak is the only allowed state and contributes about 44% of the asymptotic value of $M_2$. It remains rather prominent for $Q^2$ = 0.5 GeV$^2$, though most of $M_2$ is by this point built up of excited states, and it becomes negligible for $Q^2 \geq$ 2.0 GeV$^2$. Remarkably, the curves at lower $Q^2$ do tend to oscillate (at least qualitatively) around the scaling curve, as is observed in proton data. Note that these curves are at fixed $Q^2$, but sweep over all $\nu$. In a typical low energy experiment, $\nu $ will also be limited; in such circumstances these curves still apply, but they get cut off at the minimum value of $u$ that is kinematically allowed. For another perspective on these curves, note that $\vert \vec q \vert^2 = Q^2+\nu^2$ so for fixed $Q^2$, as $\nu$ is increased so that more and more highly excited states are created, the struck quark is being hit harder and harder. In contrast, the structure function ${\cal{S}}_{Bj}$ when plotted as a function of the scaled Bjorken variable $u_{Bj}$ shows very poor duality between its low- and high-$Q^2$ behaviors, as seen in Fig. 4. One of the reasons for this failure is that $x_{Bj}$ and ${\cal{S}}_{Bj}$ know nothing about the constituent quark mass, while low energy free quark scattering certainly does, so the corresponding pQCD cross section calculated neglecting the quark mass is simply wrong at low energy. Duality in Semileptonic Decays of Heavy Quarks =============================================== We have seen that low-energy (Bloom-Gilman) duality is displayed by our model in terms of the appropriate low-energy variable $u$ and described some of the physics behind this duality (completeness of the bound state wave functions to expand a plane wave and an approximate closure based on the required expansion states being in a narrow band of $\nu$ relative to those that are kinematically allowed). To obtain a deeper understanding of the physics behind low energy duality, it is instructive to compare and contrast duality in electron scattering with that in heavy quark decays. We will begin by carefully examining duality in heavy-light systems, where it is exact in the heavy quark limit even down to zero recoil, and where the mechanisms behind this exact duality are very clear. Duality in heavy quark systems is easily understood intuitively. Consider a $Q^* \bar q$ system where $m_Q^* >> \Lambda_{QCD}$, and imagine that $Q^$ can decay to $Q$ by emitting a scalar particle $\phi$ of mass $\mu$: $Q^ \rightarrow Q+\phi$. (Note that in this case it is the heavy quark that interacts with the current and not the light quark as in our model!) At the free quark level, the decay of $Q^$ at rest will produce the $\phi$ with a single sharp kinetic energy $T_{free}$ and corresponding $Q$ recoil velocity $\vec v$. (We use the standard variables $T_{free}$ and $\vec v$, but others, like the $\phi$ recoil momentum, could be chosen.) In reality, since the heavy quarks are bound into mesons, $\phi$ will (in the narrow resonance approximation) emerge from the decay at rest of the initial meson’s ground state $(Q^ \bar q)_0$ with any of the sharp kinetic energies allowed by the processes $(Q^* \bar q)_0 \rightarrow (Q \bar q)_n + \phi$ as determined by the strong interaction spectra of these two mesonic systems. Since in the heavy quark limit $m_{(Q^* \bar q)_n}-m_{(Q \bar q)_n} \simeq m_{Q^}-m_Q$, $m_{(Q^ \bar q)_n} \simeq m_{Q^}$, and $m_{(Q \bar q)_n} \simeq m_{Q}$, the hadronic spectral lines are guaranteed to cluster around $T_{free}$, and to coincide with it exactly as $m_Q \rightarrow \infty$. Moreover, since $m_Q^,m_Q >> \Lambda_{QCD}$, one can show using an analog of the operator product expansion [@inclorig] that the strong interactions can be neglected in calculating the total decay rate ([[i.e.]{}]{}, the heavy quarks $Q^$ and $Q$ are so heavy that the decay proceeds as though it were free.) Thus the sum of the strengths of the spectral lines clustering around $T_{free}$ is the free quark strength: there is perfect low energy duality as $m_Q^, m_Q \rightarrow \infty$. What is now especially interesting is to unravel this duality to understand how the required “conspiracy” of spectral line strengths arises physically. Because the heavy quark is so massive, if it would as a free particle recoil with a velocity $\vec v$, then this velocity would be changed only negligibly by the strong interaction since in the heavy quark limit it carries off a negligible kinetic energy, but a momentum much larger than $\Lambda_{QCD}$. In the rest frame of the recoiling meson, this configuration requires that the two constituents have a [relative]{} momentum $\vec q$ which grows with $\vec v$. [Thus the strong interaction dynamics is identical to that of our model in which the relative momentum $\vec q$ is supplied by the scattered electron.]{} Moreover, in this case, with duality exact at all energies, we can reconstruct exactly how it arises. What one sees is remarkably simple [@BjSumRule; @IWonBj]. At low $\vec v$ corresponding to low $\vec q$, only the ground state process $(Q^* \bar q)_0 \rightarrow (Q \bar q)_0 + \phi$ occurs. Since the masses and matrix elements for the transitions $(Q^* \bar q)_0 \rightarrow (Q \bar q)_0 + \phi$ and $Q^* \rightarrow Q + \phi$ are identical (the elastic form factor goes identically to unity as $\vec q \rightarrow 0$), the hadronic and quark spectral lines and strengths are also identical and duality is valid at $\vert \vec q \vert^2=0$! Next consider duality at a different kinematic point (which one might reach by choosing a smaller $\phi$ mass) where $\vec v$ and therefore $\vec q$ have increased. The elastic form factor will fall, so its spectral line (which is still found at exactly the new value of $T_{free}$ in the heavy quark limit) will carry less strength. However, once $\vec q $ differs from zero, excited states $(Q \bar q)_n$ can be created, and indeed are created with a strength that exactly compensates for the loss of elastic rate. These excited state spectral lines also coincide with $T_{free}$ and duality is once again exact. Indeed, no matter how large $\|\vec q\|^2$ becomes, all of the excited states produce spectral lines at $T_{free}$ with strengths that sum to that of the free quark spectral line. Heavy quark theory also allows one to go beyond the heavy quark limit to the case of quarks of finite mass. In this case one of course finds that duality-violation occurs, but that it is formally suppressed by two powers of $\Lambda_{QCD}/m_Q$ [@inclorig; @DualityControversy], with the spectral lines now clustered about $T_{free}$ but not coinciding with it. A remarkable feature of this duality violation is that the spectral line strengths differ from those of the heavy quark limit in ways that tend to compensate for the duality-violating phase space effects from the spread of spectral lines around $T_{free}$. An additional source of duality-violation is that some of the high mass resonances that are required for exact duality are kinematically forbidden since for finite heavy quark masses $m_{Q^}-m_Q$ is finite. From this discussion it is clear that the strong interaction dynamics of heavy-light decays is the same as that of scattering a probe off of the $Q$ of a $Q \bar q$ system [@XpQCD]: what is relevant is that the system must in each case respond to a relative momentum kick $\vec q$. Needless to say, one must still carefully organize the kinematics to expose duality: in a decay to a fixed mass $\phi$ only a single magnitude $\vert \vec q \vert^2 $ is produced at the quark level, while in electron scattering a large range of $\vert \vec q \vert^2 $ and $\nu$ is produced by a given electron beam. Given these connections, it is relevant to note that in addition to the obvious conceptual relevance of heavy-light systems, model studies indicate that in these systems heavy quark behavior continues to hold qualitatively even for $m_Q \sim m$. These models are, as one might expect, similar to ours which displays the same clustering of spectral lines, the same tendency for excited state spectral lines to compensate for the fall with $\vert \vec q \vert^2 $ of lighter states, and the same sources of duality violation such as kinematically forbidden states and mismatches between the mass of the recoiling hadrons and the struck quark. We believe that these elements of the dynamics are clearly in operation and that we have understood through our model that the qualitative applicability of duality for real systems should indeed extend all of the way down to zero recoil as seen in Nature. Summary and Outlook =================== We have presented a simple, quantum-mechanical model in which we were able to qualitatively reproduce the features of Bloom-Gilman duality. The model assumptions we made are the most basic ones possible: we assumed relativistic, confined, valence scalar quarks and treated the hadrons in the infinitely narrow resonance approximation. To further simplify the situation, we did not consider a three quark “nucleon” target, but a target made up by an infinitely heavy antiquark and a light quark. The present work does not attempt to quantitatively describe any data, but to give qualitative insight into the physics of duality. Our work complements previous work on duality, where the experimental data were analyzed in terms of the operator product expansion (OPE) [@JI; @DGP]. There, it was observed that at moderate $Q^2$, the higher twist corrections to the lower moments of the structure function are small. The higher twist corrections arise due to initial and final state interactions of the quarks and gluons. Hence, the average value of the structure function at moderate $Q^2$ is not very different from its value in the scaling region. While true, this statement is merely a rephrasing in the language of the operator product expansion of the experimentally observed fact that the resonance curve averages to the scaling curve. However, the operator product expansion does not explain why a certain correction is small or why there are cancellations: the expansion coefficients which determine this behavior are not predicted. The numerical confirmation of these coefficients will eventually come from a numerical solution of QCD on the lattice, but an [understanding]{} of the physical mechanism that leads to the small values of the expansion coefficient will almost certainly only be found in the framework of a model like ours. For example, one clear lesson from our study of duality is that the commonly made sharp distinction between the “resonance region”, corresponding to an invariant mass $W < 2 $ GeV for scattering from a proton, and the deep inelastic region, where $W > 2 $ GeV, is completely artificial. Finally, we remind the reader that our model, with all the charge on a single quark, with scalar currents, and with no spin degrees of freedom, leaves much to be done in model-building. The next step is to use more realistic currents. While making the calculations more complicated, coupling to the conserved quark current will allow one to study the $Q^2$-evolution of the Gross-Llewellyn Smith and momentum sum rules. To use a spin-${1\over 2}$ target will also be a useful step forward, but it may require foregoing the great advantages of the analytic solutions of the Klein-Gordon equation. As we have emphasized, the local duality seen here [cannot]{} be expected for more complicated targets and processes, and pursuing this issue is also clearly very important [@CloseIsgur]. Here we have taken a first small step which nevertheless has been enough to strongly suggest that for these more realistic models and more general processes there will be a generalization of local averaging — a theoretically well-defined procedure for integrating over regions of $x_{cq}$ — which will also display low energy duality. If so, we will not only have understood quark-hadron duality. We will also have opened the door to extending studies of a variety of structure functions into previously unreachable kinematic regimes. We thank C. Carlson, F. Close, R. Ent, J. Goity, C. Keppel, R. Lebed, I. Niculescu and S. Liuti for stimulating discussions, and N.N. Nikolaev and S. Simula for pointing out several references. This work was supported in part by DOE contract DE-AC05-84ER40150 under which the Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility (Jefferson Lab), and by the Australian Research Council. [99]{} N. Isgur and M.B. Wise, Phys. Lett. B[232]{}, 113 (1989); Phys. Lett. B[237]{}, 527 (1990). For an overview of Heavy Quark Symmetry see N. Isgur and M.B. Wise, “Heavy Quark Symmetry" in [B Decays]{}, ed. S. Stone (World Scientific, Singapore, 1991), p. 158, and in [“Heavy Flavors"]{}, ed. A.J. Buras and M. Lindner (World Scientific, Singapore, 1992), p. 234. For the original paper on heavy-light spectroscopy, see N. Isgur and M.B. Wise, Phys. Rev. Lett. [66]{}, 1130 (1991). K. Gottfried, Phys. Rev. Lett. [18]{}, 1174 (1967). F. Close, in [Excited baryons 1988]{}, Proc. of the Topical Workshop on Excited Baryons, G. Adams, N.C. Mukhopadhyay and P. Stoler, editors (World Scientific, Singapore, 1989); V.D. Burkert and B.L. Ioffe, J. Exp. Theor. Phys. [78]{}, 619 (1994); C.E. Carlson and N.C. Mukhopadhyay, hep-ph/9801205. G. ’t Hooft, Nucl. Phys. [B72]{}, 461 (1974); E. Witten, Nucl. Phys. [B160]{}, 57 (1979); S. Coleman. in [Aspects of Symmetry]{}, Cambridge University Press, New York (1985); A.V. Manohar, in [Probing the Standard Model of Particle Interactions]{} (ed. by F. David and R. Gupta), hep-ph/9802419; E. Jenkins, Ann. Rev. Nucl. Part. Sci. [48]{}, 81 (1998); R. Lebed, Czech. J. Phys. [49]{}, 1273 (1999). That the meson spectrum consists of narrow, noninteracting resonances in the large-$N_c$ limit is clear, but the situation for baryons is somewhat complicated. See, [e.g.]{} the discussions in R. Dashen and A. Manohar, Phys. Lett. B [315]{}, 425 (1993); B [315]{}, 438 (1993); R. Dashen, E. Jenkins and A. Manohar, Phys. Rev. D [49]{}, 4713 (1994); D [51]{}, 2489 (1995); E. Jenkins and R.F. Lebed, Phys. Rev. D [52]{}, 282 (1995); R. Lebed, Czech. J. Phys. [49]{}, 1273 (1999). I.A. Schmidt and R. Blankenbecler, Phys. Rev. D [16]{}, 1318 (1977). For a quark-model-based discussion of such issues, see F.E. Close and N. Isgur, “The Origins of Quark-Hadron Duality: How Does the Square of the Sum Become the Sum of the Squares?", JLab-THY-01-01, hep-h/0102067. E.D. Bloom and F.J. Gilman, Phys. Rev. Lett. [25]{}, 1140 (1970); Phys. Rev. D [4]{}, 2901 (1971). I. Niculescu et al., Phys. Rev. Lett. [85]{}, 1182 (2000); [85]{}, 1186 (2000); R. Ent, C.E. Keppel and I. Niculescu, Phys. Rev. D [62]{}, 073008 (2000). B.L. Ioffe, V.A. Khoze and L.N. Lipatov, [Hard Processes]{} (North-Holland, Amsterdam, 1984). B.L. Ioffe, JETP Lett. [58]{}, 876 (1993). S.A. Gurvitz and A.S. Rinat, Phys. Rev. C [47]{}, 2901 (1993); S.A. Gurvitz, Phys. Rev. C [52]{}, 1433 (1995). O.W. Greenberg, Phys. Rev. D [47]{}, 331 (1993). E. Pace, G. Salme and F.M. Lev, Phys. Rev. C [57]{}, 2655 (1998). O.W. Greenberg and D. Bhaumik, Phys. Rev. D [4]{}, 2048 (1971). O. Nachtmann, Nucl. Phys. [B63]{}, 237 (1973); Nucl. Phys. [B78]{}, 455 (1974). R. Barbieri, J. Ellis, M.K. Gaillard and G.G. Ross, Phys. Lett. [64B]{}, 171 (1976); R. Barbieri, J. Ellis, M.K. Gaillard and G.G. Ross, Nucl. Phys. [B117]{}, 50 (1976). This statement applies to the low energy effective theory. QCD matching effects make the rates dependent on the quark masses. X. Ji and P. Unrau, Phys. Rev. D [52]{}, 72 (1995); X. Ji and P. Unrau, Phys. Lett. B [333]{}, 228 (1994); X. Ji and W. Melnitchouk, Phys. Rev. D [56]{}, 1 (1997). A.De Rújula, H. Georgi and H.D. Politzer, Ann. Phys. (N.Y.) [103]{}, 315 (1975); Phys. Lett. [64]{} B, 428 (1976). J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B [247]{}, 399 (1990). J.D. Bjorken, in [Results and Perspectives in Particle Physics]{}, Proceedings of the $4^{th}$ Rencontre de Physique de la Valleé d’Aoste, La Thuile, Italy, 1990, edited by M. Greco (Editions Frontières, Gif-sur-Yvette, France, 1990); J.D. Bjorken, in [Proceedings of the XXVth International Conference on High Energy Physics]{}, Singapore, 1990, edited by K.K. Phua and Y. Yamaguchi (World Scientific, Singapore, 1992). N. Isgur and M. B. Wise, Phys. Rev. D [43]{}, 819 (1991). There has been some controversy lately over whether the formal suppression of duality-violation at order $1/m^2_Q$ is relevant. See N. Isgur, Phys. Lett. B [448]{}, 111 (1999); R. Lebed and N. Uraltsev, Phys. Rev. D [62]{}, 094011 (2000); A. Le Yaouanc [et al.]{}, Phys. Rev. D [62*]{}, 074007 (2000) .
--- abstract: 'It is highly desirable and challenging for a wireless ad hoc network to have self-organization properties in order to achieve network wide characteristics. Studies have shown that Small World properties, primarily low average path length and high clustering coefficient, are desired properties for networks in general. However, due to the spatial nature of the wireless networks, achieving small world properties remains highly challenging. Studies also show that, wireless ad hoc networks with small world properties show a degree distribution that lies between geometric and power law. In this paper, we show that in a wireless ad hoc network with non-uniform node density with only local information, we can significantly reduce the average path length and retain the clustering coefficient. To achieve our goal, our algorithm first identifies logical regions using Lateral Inhibition technique, then identifies the nodes that beamform and finally the beam properties using Flocking. We use Lateral Inhibition and Flocking because they enable us to use local state information as opposed to other techniques. We support our work with simulation results and analysis, which show that a reduction of up to $40\%$ can be achieved for a high-density network. We also show the effect of $hopcount$ used to create regions on average path length, clustering coefficient and connectivity.' author: - \| \ \ \ bibliography: - 'biblio.bib' title: 'Achieving Small World Properties using Bio-Inspired Techniques in Wireless Networks' --- Introduction {#sec:sec1} ============ Decades of academic and industrial research in wireless networks [@Akyildiz] has led to the tremendous growth of wireless networks requiring researchers to address manageability and scalability issues. Due to these issues, most of the research work has been oriented towards autonomous wireless networks. The autonomous behavior of the wireless nodes made decentralized computing and cost efficient topology deployment possible [@Dressler]. It was also proved that self-organization of the network can lead to better performance. An attractive model to achieve better network performance is the Small World network. Small world networks are characterized by reduced Average Path Length ($APL$) and high Clustering Coefficient ($CC$). Here, the $APL$ is the mean of $hopcount$ between all pairs of nodes in the network. Consider a node, $v$, with $k$ neighbors. In the sub-graph of these $k+1$ nodes, the $CC$ is defined as the fraction of links that exist to the maximum number of links that could have existed in the sub-graph. Drawing inspiration from the experimental work of Stanley Milgram [@Milgram], Watts et al [@Watts] proposed a model that could achieve small world properties. In the model, Watts et al proposed, small world properties could be reached by randomly rewiring a few existing links within the network. Watts et al showed that the dynamics of these small world networks lie between that of a regular network and a random network [@Watts; @Wattsbook]. To prove the findings, however, Watts et al used a regular wired network and called the rewired links as shortcuts. Many complex real world networks such as internet, biological networks, food web and social networks also demonstrate small world properties [@Barabasi; @BarabasiAlbert; @Newman]. In real world networks where there is a non-uniform distribution of nodes, these real world networks were shown to exhibit the properties of scale-free networks marked by power law degree distribution. Section \[sub:subswn\] provides more details on small world networks. In a wireless ad hoc network, achieving small world properties can help us in many ways. Having a low $APL$ would increase the performance of the network in terms of communication [@Brust; @KleinbergSWP] (reduced traffic per unit area, reduced congestion and reduced signal interference), low latency and reduce the overall energy consumption in the network during the data communication. On the other hand, maintaining the $CC$ would ensure connectivity to the neighborhood and would make the network resilient [@Albert; @GuidoniLoureiro]. However, Watts’ model cannot be applied directly to wireless ad hoc networks because of the spatial nature of such networks. In wireless ad hoc networks, addition of a shortcut between any two nodes should depend on the distance between two nodes. Helmy in [@Helmy] first studied the effect of adding few distance-limited links in the network. He showed that, upon introduction of distance-limited links, wireless ad hoc networks show small world properties. He concluded that, when the shortcut lengths are $\frac{1}{4}$th of the network diameter, there is a maximum reduction in the $APL$. Thus, proving that realization of small world properties in a wireless ad hoc network depends crucially on the length of shortcuts created among nodes. Another important factor in the realization of small world properties is the choice of nodes among which shortcuts are to be created. One method to obtain these nodes is that of preferential attachment [@Simon; @BarabasiAlbert], typically observed in real world networks, wherein links are created to nodes with high structural importance. It was shown that, analogous to real world networks, using preferential attachment for creation of distance-limited links in a spatial network resulted in reduced network diameter [@BarthelemyMark; @Manna]. This was accompanied by high clustering coefficient and a shift in the node degree distribution towards power law. These results motivate us to say that, creation of links to nodes having high structural importance in the network can result in the desired small world characteristics. The creation of a wireless ad hoc network with the small world properties also depends on the manner in which distance-limited links are added. Such links can be added through different techniques like: 1) creating the directional beam using the same power as when the node was operating in the omnidirectional mode; 2) increasing the omnidirectional transmission range of the node; 3) introducing of few long wired links [@Sharma]; 4) introducing special nodes with higher omnidirectional transmission range deterministically in the network [@GuidoniLoureiro]; 5) using another antenna for beamforming in addition to the omnidirectional antenna. Talking about the self-organization characteristics of the nodes, only techniques one and two mentioned above qualify. However, even though other techniques help in achieving desired network characteristics, they lack self-organization capabilities. In addition, the second technique suffers from the problem of early death of the node due to increased energy consumption. Thus leaving us only the first technique. Achieving reorganization or rewiring in a wireless ad hoc network through the first technique is hard due to the spatial nature of the wireless ad hoc network. Finding the beam direction, the beam length and determining the new neighborhood are primary issues associated with rewiring in a wireless ad hoc network. Our previous study, [@Banerjee], proved that the use of distance-limited long links in wireless ad hoc network to achieve small world properties is beneficial, (Cf. Fig. \[fig:figinitial\]). Motivated by this, in this study, we investigate how we can increase connectivity, reduce the $APL$ and almost maintain the $CC$ in a non-uniformly distributed wireless ad hoc network. We thus propose an algorithm that achieves these goals by creating long-range directional beams between nodes that have low and high structural importance. The decentralized computing and self-organizing requirements of such an approach motivate us to draw inspirations from nature. We further propose that Lateral Inhibition [@Lawrence; @Nagpal; @NagpalMamei; @Afek] and Flocking [@Reynolds], in conjunction with the centrality concept of graph theory, can provide valuable insights in building a solution to our problem. ![Source: [@Banerjee], Effect of beamforming on the $APL$ and the $CC$ when the nodes are using 1) Sector model and 2) $ULA$ Model. The results obtained show a reduction in the $APL$ while almost no change in the $CC$ for the case when we use a realistic antenna model. On the other hand, for the theoretical model, the reduction in the $APL$ is relatively less while the reduction in $CC$ is considerably large. The number of nodes that beamform is shown with a probability value in the log scale. The results also show that the reduction in the $APL$ increases with an increase in the number of beamforming nodes, [@Watts; @Wattsbook]. Here, $APL(p_r)$ and $CC(p_r)$ are the $APL$ and the $CC$ of the network when $p_r\%$ of the nodes create long-range links. $p_r=0$ means no node is beamforming. Further, in the figure, we normalize $APL(p_r)$ and $CC(p_r)$ to account for the variation in the $APL$ and the $CC$.[]{data-label="fig:figinitial"}](figinitial){width="50.00000%"} We use Lateral Inhibition to create small logical regions within a network. The use of Lateral Inhibition not only reduces the message complexity but also enables us to apply the Flocking rule analogy successfully. We use analogy of Flocking rules to identify the nodes that beamform and the beam properties. According to the rules, explained later in section \[sub:subflocking\], it is important to identify stray nodes, align the nodes and move them towards the centroid of their neighborhood. Analogous to this, after region formation in a non-uniformly distributed wireless ad hoc network, we use Flocking rules to identify the beamforming nodes and direct the beams of these beamforming nodes towards the centroid of the region. The centroid node in the region has a high structural importance. Beamforming towards the centroid node of the region contributes towards reducing the $APL$ because the centroid node of the region is the most connected node and has the highest Closeness Centrality measure. Thus, beamforming towards the centroid node is the preferential attachment behavior of the beamforming node, thereby making centroid finding a prerequisite to Flocking. In a distributed system where nodes only have local information and lack $GPS$ facilities, exact centroid node identification of the region is challenging. We can only make an estimate to the centroid node location in the region. We, therefore, use the self-organizing virtual coordinate scheme combined with the centrality concepts to identify the centroid nodes. Thus, our algorithm design is such that it first identifies regions using Lateral Inhibition, then identifies the centroid nodes of the regions and then uses the analogy of flocking rules to identify the nodes that will beamform along with their beam properties. Section \[sec:sec3\] gives a formal description of our proposed algorithm. The organization of rest of this paper is as follows. Section \[sec:sec3\] presents the assumptions used for the proposed algorithm along with the algorithm specifications. Section \[sec:sec5\] presents the formal definitions. Section \[sec:sec6\] and \[sec:sec7\] discuss the simulation setup and the results respectively. For the readers who are unfamiliar with the concepts used in this paper, we provide a detailed description of the same in section \[sec:sec2\]. We finally conclude our work in the section \[sec:sec8\] after providing insights to some future research directions in section \[sec:sec9\]. Assumptions and Algorithm {#sec:sec3} ========================= Assumptions ----------- To address issues mentioned in the Introduction, we focus ourselves towards the deployment of homogenous and autonomous wireless ad hoc nodes with no central entity controlling the nodes. This type of deployment enables us to easily apply self-organizing features, achieve global consensus with very limited local information, make any eligible node the group leader, make the system highly fault tolerant, ease the topological maintenance, lower the deployment cost and extend to incorporate the mobility of the nodes in the future. Further, the nodes are set to have an omnidirectional transmission range $r$. We assume a non-uniform distribution of nodes generated using thinning process defined by Bettstetter et al [@BettstetterGyarmati]. The non-uniform distribution of nodes allows us to realize scenarios that are more realistic. The algorithm proposed by Bettstetter et al proceeds by removing nodes which have less than $\ell_{min}$ neighbors within a transmission range $r_{b}$ (ref. section \[sub:subnonunifdistrib\]). Further, we assume the deployment of the nodes on a 2-D plane of area $A$. As part of our network setup, our algorithm assumes each node to have an antenna consisting of $M$ isotropic elements. The use of single antenna element results into omnidirectional beam while use of more than one antenna element results into a long-range directional beam. A node, however, decides to use more than one antenna element using simple local rules mentioned later in this section. The nodes use beamforming only to transmit data but use omnidirectional beams for reception. We have used the Sector model [@Yu] to visualize our algorithm and have assumed transmission of data to be synchronous. Further, we assume that the nodes lack $GPS$ facilities and global network knowledge. To achieve our goal, it is thus first essential to know what information can be used by the nodes. We limit a node to use local information along with that of its one hop neighborhood. Determining single hop neighborhood to build the local information is thus essential for the correct operation of the algorithm. Various studies have proposed many neighborhood discovery mechanisms, eg. [@Vasudevan], and have carefully analyzed them. Therefore, for our approach, we assume that all the nodes have information about their neighborhood. Notation Meaning Notation Meaning --------------------- ------------------------------------------------------------- ----------------------- -------------------------------------------------- $A$ simulation area $g$ gradient $G$ network with set of vertices $V$ and set of $g_{max}$ maximum gradient edges $E$ $e\_bet_{v}$ Egocentric Betweenness of $v$ w.r.t. its $G_{i}$ region $G_{i}\|G_{i} \subset G$ with set of cluster vertices $V_{i}$ and set of edges $E_{i}$ $hops(v,w)$ $hopcount$ between node $v$ and $w$ $N$ number of regions formed $v_{i}(x, y)$ virtual coordinates of $v$ in the region $G_{i}$ $v$ node $\|v \in V$ $v_{i}(x^{}, y^{})$ updated virtual coordinates of $v$ in the $v_{i}$ node $v$ in region $G_{i}\|v_{i} \in V_{i}$ region $G_{i}$ $r$ transmission radius $\varepsilon$ error margin $r_b$ Bettstetter transmission radius $M$ max antenna elements available with $v$ $\rho$ average node density $m$ number of antenna elements used by $v$ $ID_{v}$ identification number of node $v$ to beamform $\| m\in [2,M]$ $L_{v}$ neighbor list of $v\|v\in V$ $RC_{v}$ set of centroid nodes reachable from $v$ $L_{v,i}$ neighbor list of $v$ in the region $G_{i}\|v\in V_{i}$ with their $hopcount$ that are within $g_{max}$ $\ell_{min}$ minimum number of neighbors used for hops from $v$ when $v$ is not beamforming creating a non-uniform distribution $RC_{v}^{}$ set of centroid nodes reachable from $v$ $deg_{v}$ size of $L_{v}$, i.e., degree of $v$ with their $hopcount$ when $v$ is beamforming $H$ set of all region heads $\theta$ beam direction, i.e., the sector $h_{i}$ head node of the region $G_{i}\|h_{i} \in H$ $B_{b}$ boresight direction $C$ set of all centroid nodes $B_{l}$ beam length $c_{i}$ centroid node of the region $G_{i}\|c_{i}\in C$ $B_{w}$ beam width $P$ set of all peripheral nodes $APL$ Average Path Length $P_{i}$ set of peripheral nodes in the region $G_{i}\|P_{i}\in P$ $CC$ Clustering Coefficient $\wp_{i}$ peripheral node $\|\wp_{i}\in P_{i}$ $ULA$ Uniform Linear Antenna Array $\varrho_{\wp_{i}}$ peripheral neighbor of $\wp_{i}\|\varrho_{\wp_{i}}\in P_{i}$ $GSCC$ Giant Strongly Connected Component $S$ Set of nodes neither in $C$ nor in $P$ $GIN$ Giant In Component It is also essential to address the self-organizing paradigms, [@Prehofer], to claim for the self-organizing behavior of the network. Prehofer et al’s [@Prehofer] paradigms state: designing local rules to achieve global properties, implicit coordination, minimizing the use of historic information about the state of the network and designing an algorithm that changes with environment parameters. Our algorithm uses only locally available information to determine the beamforming nodes, beam properties and the regions. The nodes implicitly coordinate with their neighbors to determine the node with the highest $hopcount$ from the centroid of the region. For a given region, the nodes also coordinate implicitly to determine the centroid node of that region. The current discussion focuses on a static network. In dynamic network scenarios, optimizing the extent of reconfiguration to deal with frequent changes in state information is likely to be a crucial factor. We leave this for future investigation but offer some insights in section \[sec:sec9\]. We further describe the system model and the algorithm in the following sections. System Model ------------ Given a network, $G(V,E)$, where $V$ is the set of vertices and $E$ is the set of edges, we visualize $G$ as a network consisting of $N$ logical regions, $\{G_{1},G_{2},\dots,G_{N}\}$, i.e., $G=\bigcup_{i=1}^{N}{G_{i}}$. Each region, $G_{i}$, consists of the set of nodes, $V_{i}\|V_{i}\subset V$ and $V=\bigcup_{i=1}^{N}{V_{i}}$, and set of edges, $E_{i}\|E_{i}\subset E$ and $E=\bigcup_{i=1}^{N}{E_{i}}$. All vertices in $G_{i}$ are located within $g$ hops of a head node, $h_{i}$. As a part of our algorithm, we use Lateral Inhibition to identify regions and regional heads. We characterize the set of vertices, $V$, into three sets. These are termed as the Peripheral node set, the Centroid node set and the Standard node set. We provide separate role to the nodes in these sets. The Peripheral nodes set ($P$) contains the nodes that beamform. The Centroid node set ($C$) contains the nodes towards which the nodes in the Peripheral node set beamform. We call the set of remaining nodes, $S=V-(P\bigcup C)$, as the Standard node set. Further, we call nodes in these sets as the peripheral nodes, the centroid nodes and the standard nodes respectively. Mathematically, Closeness centrality of a node, $v \in V$, in a graph $G$ is equal to $\frac{1}{\sum_{w \in V}{hops(v,w)}}$, where $hops(v,w)$ is the $hopcount$ between nodes $v$ and $w$. The node having maximum Closeness Centrality is the centroid of the graph and has a high structural importance. For the vertex sets defined above, nodes in the set $P$ have lowest value of Closeness Centrality, i.e., $\operatorname{arg\,max}\limits_{v\in V}\{\sum_{w \in V,v\neq w} hops(v,w)\}$. However, the nodes in the set $C$ have highest value of closeness centrality, i.e., $\operatorname{arg\,min}\limits_{v\in V}\{\sum_{w \in V,v\neq w} hops(v,w)$}. A node in $P$ beamforms towards a node in $C$ in order to minimize the distance to other nodes and reduce $APL$. The directional beam is modeled using Sector model, i.e., for a given directional beam length $B_l$, the corresponding beam width, $B_w$, is $$\label{eq:blbw} B_{w}=\frac{2\pi r^2}{B_{l}^{2}}$$ In realistic antenna model, as beam length of the directional antenna is dependent on the number of antenna elements used, $m$, the corresponding value of $B_{l}$ used is $B_{l}=mr$. Further, table \[table:table3\] lists the notations used in this paper. Algorithm --------- We divide our approach into two parts: - Use of Lateral Inhibition technique and self-organizing virtual coordinate scheme for the identification of regions and the centroid nodes of the regions, so that there are less message overheads and nodes can beamform towards the centroid node to achieve reduced $APL$. Section \[sub:s1\] provides more details. - Use of flocking rules to identify the nodes that beamform, to determine beam properties that realize small world properties and improve connectivity. Section \[sub:s2\] provides more details. We describe these parts in detail in the next sub sections. ### Region formation and Centroid finding {#sub:s1} The Closeness Centrality [@Freeman; @Freemanlc] identifies the structural importance of the node in the network. The node with the highest Closeness Centrality value is the most central node in the network. Through this node, the spread of the information to other nodes is quick. To determine the Closeness Centrality of the node, the node requires the knowledge of other nodes in the region as suggested by the definition of Closeness Centrality, (ref. section \[subsubsec:closenessc\]). This makes the Closeness Centrality a global measure. Storing information about all the nodes in the network can consume a lot of node’s memory. When there is lack of global information, gathering such information can also be time consuming and the message complexity could be high. To overcome these problems, we create small logical regions. The creation of regions not only reduces the message complexity of the network but also reduces the effect on the $APL$ due to the failure of a node, thereby making the network more manageable, efficient and tolerant to failures [@BrustRibeiro]. Some algorithms designed in this direction were centralized. The Base Station chose the region heads based on the energy and the position of the nodes. Other techniques use either the transmission power or the degree or the mobility, eg., $WACA$ [@BrustAndronache]. On the contrary to centralized approaches, some algorithms were either distributed, [@Heinzelman], or probabilistic [@Younis]. We thus divide this part into two, identification of regions using Lateral Inhibition and identification of centroid node in the region. As we only have local information, we use degree of the node in the Lateral Inhibition process. For Lateral Inhibition, we consider that a node $v_{i}$ broadcasts and stores a message containing following information: the identity of the head node to which $v_{i}$ is associated ($h_{i}$), its $hopcount$ from $h_{i}$ and the degree of $h_{i}$ ($deg_{h_{i}}$), where $v_{i}\in V_{i}$. Initially, all the nodes, $v\in V$, consider themselves as heads, i.e. $H=V$, and store their own information, i.e., $h_{i}=v$, $hopcount=0$ and $deg_{h_{i}}=deg_{v}$. Each node, $v\in V$, then broadcasts this information to its neighbors, $L_{v}$. Similarly, $v$ receives information from each of its neighbors and subsequently updates the information stored in it. Thus, a node replaces its stored values, if the stored degree, $deg_{h_{i}}$, is less than that of the received value and $hopcount+1$ is less than $g$, where $g$ is the gradient or the desired size of the regions. Further, if the stored and the received $deg_{h_{i}}$ are same, the node decides to update the stored information based on lower $hopcount$ value. If the $hopcount$ is also same, then the node randomly decides to update the stored information to received information. The node $v$ then broadcasts the updated information after incrementing the $hopcount$ by 1. Subsequently, $v$ removes itself from $H$, i.e., $H=H-\{v\}$, and inhibits itself from acting as the regional head. The process continues until all the nodes within $g$ hops from the maximum degree node reach a consensus about the head node. Due to $g$, the algorithm assigns same $h_{i}$ to all the nodes within $g$ hops of the head node. We call the nodes having same $h_{i}$ to belong to one region, $G_{i}$. The nodes lying at different $hopcount$ from the $h_{i}$ virtually creates a gradient of different hops around $h_{i}$, (Cf. Fig. \[subfig:gradient\]). In the end, the algorithm tags a node with no neighborhood as the head as it has remained uninhibited, (Cf. Fig. \[subfig:regions\]). The regions created differ from other Lateral Inhibition algorithms, [@Afek], in a way that our algorithm creates regions that are not limited to 1 hop, (Cf. Fig. \[subfig:regions\] and Fig. \[subfig:regionsafek\]). However, the Lateral Inhibition technique does not guarantee that the head nodes identified above have a high Closeness Centrality value and are the most central nodes, (Cf. Fig. \[fig:f1\]). We thus now describe the steps for the centroid node identification in a given region, $G_{i}$, created using Lateral Inhibition described earlier. Due to the global properties of the Closeness Centrality and unavailability of any $GPS$ facilities within the nodes, we take insights from existing algorithms on self-organizing virtual coordinate systems. In self-organizing virtual coordinate system, the nodes identify their own coordinates relative to their neighborhood in the network. We however, make use of self-organizing virtual coordinate system to calculate centroid of the region. Existing techniques on self-organizing virtual coordinate system include [@Capkun; @Caruso; @Leong; @Dabek; @Rao; @Priyantha; @Watteyne; @Awad]. These studies deploy various mechanisms to reach consensus. We use a method for achieving consensus on centroid location based on self-organizing virtual coordinate techniques that rely on averaging of local neighborhood values [@Rao; @Watteyne]. This allows us to limit the information required to a single hop, and thereby have minimum communication overheads. Thus, in our algorithm, all nodes $v_{i}\in V_{i}$ in $G_{i}$ assign themselves randomly selected virtual $xy$ coordinates, $v_{i}(x, y)$. The identity of the nodes in the virtual coordinate system, however, remains the same. The nodes then communicate to their neighbors in $G_{i}$ these coordinates, i.e., $L_{v,i}$. Using the coordinates of their local neighborhood, the nodes compute an average of the coordinates, $v_{i}(x^{}, y^{})$, and broadcast the average coordinates to their neighbors. The neighbors in turn use these coordinates to compute a new average. This process continues until all nodes in the region reach consensus of having same average $xy$ coordinates of the centroid. The self-organizing virtual coordinate technique reveals the location of the centroid node in the self-organizing virtual coordinate system but not the identity of the node that is to be termed as centroid. In order to identify the centroid node of the region, nodes use their initially assigned virtual coordinates and the newly found average $xy$ coordinates. Each node $v_{i}$ checks if $v_{i}(x, y)=v_{i}(x^{}, y^{})\pm\varepsilon$, where $\varepsilon$ is the error margin, and declares itself as the centroid. This process might result into multiple nodes declaring themselves as the centroid as two or more nodes can lie within the $\varepsilon$ range of $v_{i}(x^{}, y^{})$. To avoid this, a node also considers its Degree and Egocentric Betweenness. The nodes within $\varepsilon$ range of $v_{i}(x^{}, y^{})$ share this information among themselves. Subsequently, the node having maximum sum of Degree and Egocentric Betweenness declares itself as the centroid of the region. As the node has same identity in the self-organizing virtual coordinate system as in the real coordinate system, the centroid node in the self-organizing virtual coordinate system will also be the centroid in the real coordinate system. After the identification of the centroid nodes, the centroid nodes broadcast their information in the network. All nodes then update their stored head information to their respective $c_{i}$’s and the $hopcount$ to $hops(v_{i},c{i})$. This broadcasting of the centroid node information enables the nodes to build $RC_{v}$ for future use. $RC_{v}$ is the set of centroid nodes within $g_{max}$ hops of the node $v$, where $g_{max}>g$. Algorithm \[Algo1\] represents the algorithmic description of the region formation and the centroid identification process. The Fig. \[subfig:centroidnodes\] shows the centroid nodes for the regions identified in the Fig. \[subfig:regions\]. ### Beamforming {#sub:s2} In this part, we describe the steps involved in beamforming. According to the results of [@Helmy], it requires only a small fraction of nodes with long link capabilities to achieve small world properties. In a self-organizing environment where all nodes possess beamforming capabilities, it is essential to identify nodes that create long-range beams along with the direction and the width of the beam. Flocking provides us with valuable insights in determining the answers to these questions. We use insights from the Alignment rule of Flocking to identify the set $P$. Alignment in Flocking is the change in the direction of the node to match its neighbors, in other words the change in the orientation of the node. Further, Alignment rule is, the node has to decide to change the direction and has to find the new direction. We modify the Alignment rule and say that our Alignment rule is only limited to the decision of whether to create the beam or not. The Alignment rule we apply is, thus, to identify the set of peripheral nodes, $P_{i}$ in the region $G_{i}$. Our algorithm uses the $hopcount$ of the neighborhood nodes to decide whether or not the node is a peripheral node, $\wp_{i}$, of the region $G_{i}$. If all $L_{v,i}$ of the node $v_{i}$ have $hopcount$ less than or equal to the node’s $hopcount$ to the $c_{i}$, then the node declares itself as a peripheral node. i.e., for a given region $G_{i}$ with centroid $c_{i}$, $\wp_{i} \in P_{i} \iff hops(\wp_{i},c_{i})\geq hops(L_{\wp_{i}},c_{i})$. This implies that, a single unconnected node will become a peripheral node because it does not have any neighborhood. Further, we can also infer that two peripheral nodes can be neighbors of each other due to the equality in the condition. Let $U=uninhibited$; Let $I=inhibited$; Let $ID=identity\; of\; node$; $\backslash\backslash$ Region formation; set $v_{Status}=U$ set $v\_coordinates=v_{i}(x,y)$ Initially broadcast($ID_{v}, hopcount=0, deg_{v}$) $recv$=receive($ID, hopcount+1, degree$) $v_{Status}=I$ & broadcast($recv$) $\backslash\backslash$ Centroid finding; $v_{i}(x^{},y^{})$=Cent\_finding($v_{i}(x,y), L_{v,i}(x,y)$) compute $sum_{v_{i}}=sum(deg_{v_{i}}, e\_bet_{v_{i}})$ $c_{i}=v_{i}\|v_{i}=max\{sum_{v_{i}}\}$ $C=C+{v_{i}}$ formulate $RC_{v}$ The peripheral nodes randomly choose the number of antenna elements, $m \in [2,M]$, and use the above rules to beamform. Considering $B_{l}$ to be equal to $mr$ in a Sector model, by keeping constant power as used for omnidirectional beam, we can easily compute $B_{w}$ from eq. (\[eq:blbw\]) as $B_{w}=\frac{2\pi}{m^2}$. From this we infer that, to cover all the directions, minimum number of sectors that we need to consider is $m^2$. The dependency of $B_{l}$ and $B_{w}$ on $m$ affects the connectivity of the network. The Fig. \[fig:figsectormodel\](a) shows the variation in $B_{l}$ and $B_{w}$ when $m>1$. When $B_{l}$ is smaller, i.e., when we use less number of antenna elements, the probability of connecting to the neighbors is high as the beam is wider, (Cf. Fig. \[fig:figsectormodel\](b)). However, when $B_{l}$ is longer, i.e., when we use more antenna elements, the probability of connecting to a neighbor is low as the beam is narrower, (Cf. Fig. \[fig:figsectormodel\](c)). As the number of sectors increase exponentially with an increase in the number of antenna elements, there is an increase in the time taken to decide the best sector. Checking all the sectors formed for all $m\in[2,M]$ requires a test of $\frac{(M)(M+1)(2M+1)}{6}-1$ sectors. The complexity of such a test is $O(M^3)$. This results into more energy consumption at the node. To reduce this energy consumption and the complexity to $O(M^2)$, our algorithm randomly selects the number of antenna elements, $m\in[2,M]$, and only tests the corresponding set of $m^{2}$ sectors. Non-uniformity reduces the size of the giant component in the wireless ad hoc network. It is thus important for the nodes to find different network components and connect them using beamforming. Separation* rule of Flocking provides us insight towards this problem. Separation rule states that the nodes should maintain certain distance with their neighbors. Our algorithm applies similar analogy to address the connectivity issue. We say, in order to increase connectivity, nodes create beam in different directions from their peripheral neighbors. Consider $\varrho_{\wp_{i}} \in P_{i}$ as a peripheral neighbor of $\wp_{i}$ then for all $\varrho_{\wp_{i}}$’s, $\wp_{i}(B_{b}) \neq \varrho_{\wp_{i}}(B_{b})$ must hold. Here $B_{b}$ is the boresight direction. To make this decision, if $\varrho_{\wp_{i}}$ of a $\wp_{i}$ decides to create the beam in certain direction, $\varrho_{\wp_{i}}$ informs $\wp_{i}$ about the chosen direction before it actually creates the beam. $\wp_{i}$ then tries to create the beam in another direction. Further, $\wp_{i}$ gives preference to connect to the nodes in other region rather than that of its own. This increases the possibility of connecting to an isolated region. The Fig. \[fig:f2\] shows two node $w$ and $x$ which were initially neighbors of each other, create beams in different direction in order to increase connectivity. Nevertheless, we still have to address the best direction of the beam and the knowledge of whether a $\wp_{i}$ has a node within its 1 hop. We address these problems next in this section. To the above-mentioned problem, we use analogy of Cohesion rule of Flocking to determine the best direction of the beam. In Flocking, Cohesion rule states that a node should move towards the centroid of the neighborhood to remain connected to all of its neighbors. We apply this definition of Cohesion in our algorithm because we want to bind a peripheral node with other nodes in minimum hops. From the previous section, we already know that the centroid node has the highest Closeness Centrality value in a given region. Directing the peripheral node’s beam towards the centroid node would help reduce the average distance of the peripheral node to other nodes of the region in which the centroid node lies. Combining Separation and Cohesion rules as discussed above, we can say that, if the centroid node chosen by the peripheral node and the peripheral node itself were not connected initially, connecting them would help in increasing the connectivity, (Cf. Fig. \[fig:f2\]). On the other hand, if the centroid node chosen by the peripheral node was within some hops from the peripheral node, it will lead to the reduction in the $APL$. To account for choosing the correct centroid to connect, the peripheral node, $\wp_{i}$, builds $RC_{\wp_{i}}^{}$, a set of all centroid nodes reachable when it is beamforming. To determine $RC_{\wp_{i}}^{}$, the peripheral nodes sweep through all the sectors ($m^2$) created with the chosen number of antenna elements except the sectors in which $\varrho_{\wp_{i}}$’s have created the beam. If $RC_{\wp_{i}}^{}-RC_{\wp_{i}}\neq\emptyset$ and $\|RC_{\wp_{i}}^{}-RC_{\wp_{i}}\|>1$, i.e., $\wp_{i}$ identified two or more potential centroid nodes, assuming the $hopcount$ to these centroid nodes as $\infty$ the decision to connect to one of them is randomly made. However, if $RC_{\wp_{i}}^{}-RC_{\wp_{i}}=\emptyset$, i.e., no new centroid is found, the $\wp_{i}$ decides to connect to farthest centroid node in $RC_{\wp_{i}}$. As we know that $APL$ is dependent on $\sum_{v,w\in V,v\neq w}^{} hops(v,w)$ any reduction in this summation will lead to a reduced network path length. In order to have maximum reduction in the path length, the node should connect to the farthest centroid. If the farthest centroid node was the $c_{i}$, then $\wp_{i}$ beamforms towards it. However, this decision also depends on the $hopcount$ between $c_{i}$ and $\wp_{i}$. Creating the beam toward the centroid that is less than two hops away will only reduce the initial neighborhood but not the $APL$. In this case $\wp_{i}$ drops the decision of being the peripheral node and remains omnidirectional. The Fig. \[fig:f3\](a) and the Fig. \[fig:f3\](b) depicts the same. In the Fig. \[fig:f3\](a), node $x$ is 5 hops away from $y$ while it is 4 hops away from $z$ and 2 hops away from the centroid of the region in which $x$ lies. Thus, in order to have a reduced path length, node $x$ decides to create beam towards $y$. On the contrary, in the case when the node $x$ does not have the previously stored information about the centroid nodes $y$ and $z$, the node considers $hopcount$ to these centroid nodes as $\infty$ and randomly chooses one of them to connect to, (Cf. Fig. \[fig:f3\](b)). Whenever a peripheral node creates a beam towards a centroid node that is more than 1 hop away, asymmetric link may arise. This is due to the fact that the $B_{l}$ of peripheral node is $mr$ while $B_{l}$ of a centroid node is $r$, in other words, $\frac{B_l \; of \; Centoid}{B_l \; of \; peripheral}=\frac{1}{m}$. Due to this difference, peripheral nodes will not know if they got connected to the centroid of other region or not. We propose to solve this issue as, when a centroid node receives information about the node trying to connect to it, it just for one time instant, to acknowledge the reception, creates the beam back to the node. We do this after determining angle of incidence of the beam. This works well for both connected and unconnected components. Algorithm \[Algo2\] represents a brief algorithmic description of beamforming using Flocking rule analogy. The Fig. \[fig:beamform\] shows the new network created after running our algorithm on the network shown in the Fig. \[subfig:fignodedistbnonunif\]. $\backslash\backslash$ Alignment; $P_{i}=P_{i}+\{v_{i}\}$ $P=P+\{v_{i}\}$ $\backslash\backslash$ Separation; set $m$ $RC_{\wp_{i}}^{} = RC_{\wp_{i}}^{}+\{reachable\; centroid\; nodes\}$ $\backslash\backslash$ Cohesion; $h = h+{hops(\wp_{i},c)}$ $h = h+{hops(\wp_{i},c)}$ $P_{i}=P_{i}-\{\wp_{i}\}$ $beamtonode=\max\{h\}$ $\theta$ = Sector containing $beamtonode$ Formal Definitions {#sec:sec5} ================== \[defcentroid\] Assume a centroid $c_{i}$ of the region $G_{i}$, and a node $v_{c}$ in $V_{i}$ which has the highest Closeness Centrality, then $$\begin{aligned} \label{eq:remnodes} Closeness(v_{c}) &=& \operatorname{arg\,\bf{max}}_{\forall v_{i} \in V_{i}} \left[Closeness(v_{i})\right] \nonumber\\ Closeness(c_{i}) &\simeq& Closeness(v_{c}) \end{aligned}$$ \[defperiph\] The node $v_{i}$ with neighborhood $L_{v,i}$ of the region $G_{i}$ with centroid $c_{i}$ is a peripheral node $\iff hops(v_{i},c_{i}) \geq hops(L_{v,i},c_{i})$. \[defremainnodes\] The expected number of nodes remaining after applying the thinning processes, [@BettstetterGyarmati], on a uniformly distributed network is $$\label{eq:remnodes} E(n) = \rho A \left( 1-\frac{\Gamma (r_b,\rho r_b^{2}\pi)}{(r_b-1)!}\right)$$ where $E(n)$ is the expected number of nodes remaining after the thinning process is applied, $\rho$ is the initial node density in a given area $A$ and $\Gamma(r_b,\rho r_b^{2}\pi)$ is the incomplete gamma function. \[defdistcnode\] The separation between any two head nodes is between $(g, 2g+1]$ where $g$ is the $hopcount$ used to create the region, [@NagpalMamei]. \[proofdefdistcnode\] Consider a head node with a gradient $g$ around itself. All the nodes within $g$ hops from the head node will be in its region. A node which is more than $g$ hops away will lie in another region. If in the neighboring region, a head node does not have any gradient around it, then the distance between the two head nodes in hops will be $g+1$. On the other hand, if the neighboring region also has a gradient $g$ around it, then the distance between two head nodes in hops will be $2g+1$. \[lemone\] The number of regions is equal to number of centroid nodes and each region has exactly one centroid node. \[proofone\] Our algorithm computes the centroid of the region based on average of coordinates, Degree and Egocentric Betweenness of the node for each region. According to our algorithm, the nodes are termed as centroid if the node falls within $\varepsilon$ range of the centroid coordinate estimation algorithm and have maximum sum of Degree and Egocentric Betweenness. If still there are multiple nodes that are termed as centroid nodes, the nodes randomly decide for being the centroid and thus only one node is chosen as centroid. The value of $\varepsilon$ is thus an important factor in the estimation of the centroid node. Also, smaller $\varepsilon$ will tend to provide better estimation of the centroid nodes. As there is only one centroid node per region, the number of centroid nodes is equal to the number of regions. \[lemnotcent\] If a node is not a centroid node, it is connected to a centroid node. \[prooflemnotcent\] Our algorithm identifies regions and their centroid nodes. An identified region is always connected, i.e., all the nodes in the identified region are connected to each other. Further, there is one and only one centroid node in a region, ref. lemma \[lemone\]. Thus for a given region, all nodes that are not centroid are connected to the centroid node. \[lemuncon\] An unconnected node is both the centroid node as well as the peripheral node. \[prooflemuncon\] A single unconnected node does not have any neighborhood. It thus remains uninhibited at the end of the region formation phase and becomes the head. As it is lacking any neighborhood, the node does not have any gradient around itself and is the only node in the region. In this region, the average coordinates perfectly match the virtual coordinates of the node. Thus requiring no further computation to correctly identify the centroid node. This node is also the peripheral node as the condition of Definition \[defperiph\] holds true because of the unavailability of the neighborhood. \[numregions\] For a node distribution and fully connected network with average node density $\rho$ and total number of nodes $\|V\|$, then $\|C\|$ is bounded by $\frac{\|V\|}{\rho g^2r^2\pi}$ and $\frac{\|V\|}{\rho g^2r^2\sqrt{3}}$. \[proofnumregions\] From lemma \[defdistcnode\], the hop distance between two heads is bounded by $(g,2g+1]$. Case 1 (Lower Bound): When the heads are separated by $2g+1$ hops, the number of regions formed are less. The number of nodes in one region is $\rho g^2r^2\pi$. Thus, the total number of nodes in all the $N$ regions is $\|N\|\rho g^2r^2\pi$. As the total number of nodes are $\|V\|$, $\therefore \|N\|=\frac{\|V\|}{\rho g^2r^2\pi}$. From lemma \[lemone\] $\|C\|=\|N\|$, $\therefore \|C\|=\frac{\|V\|}{\rho g^2r^2\pi}$ Case 2 (Upper Bound): When all the heads are separated by $g+1$ hops, the number of regions formed are more. A head in such a case is connected to only 6 other heads. This can be visualized as a hexagon with vertex-vertex distance equal to $g+1$ and a node at the center of hexagon. Each of the vertex nodes are shared between 3 other hexagons. Thus, the total number of heads that are exclusive for the hexagon are $\frac{6}{3}$+1 = 3. In other words, there are 3 heads in an area of $\frac{6g^2r^2\sqrt{3}}{2}$. Thus, for the area=$\frac{\|V\|}{\rho}$, $\|C\|$=$\frac{\|V\|}{\rho g^2r^2\sqrt{3}}$. ![Percolation of the giant component for nodes distributed uniformly and non-uniformly. We use $r_b=30m$ and $\ell_{min}=5$ to achieve non-uniform distribution of nodes. There is a difference in the values of the size of the giant component at the same average density because the algorithm used to generate non-uniformity [@BettstetterGyarmati] tends to create clusters of nodes that might be unconnected. This leads to a network that is less connected than the uniformly deployed network. However, when the density increases the size of the clusters also increases.[]{data-label="fig:percolation"}](percolation){width="\columnwidth"} \[numregionsunconnected\] Consider a network with $j$ components ($j>1$), average density of the nodes as $\rho_{k}$ and number of nodes as $\|V_{k}^{j}\|$ for $k\in j$, $\|C\|$ is bounded by $\sum_{k=1}^{j} \frac{\|V_{k}^{j}\|}{\rho_{k} g^2r^2\pi}$ and $\|V\|$, where $\|V\|=\sum_{k=1}^{j} \|V_{k}^{j}\|$. \[proofnumregionsunconnected\] From lemma \[defdistcnode\], the hop distance between two heads is bounded by $(g,2g+1]$. Case 1 (Lower Bound): Consider $k^{th}$ component of the network. When the heads are separated by $2g+1$ hops, the number of regions formed is less. The number of nodes in one region is $\rho_{k} g^2r^2\pi$. Thus, the total number of nodes in all the regions in the component is $N_{k}\rho_{k} g^2r^2\pi$, where $N_{k}$ are the number of region in $k^{th}$ component. But as the total number of nodes were assumed to be $\|V_{k}^{j}\|$, $\therefore$ $N_{k}$=$\frac{\|V_{k}^{j}\|}{\rho_{k} g^2r^2\pi}$. Thus for all the components, the number of regions formed is $\|N\|=\sum_{k=1}^{j}N_{i} = \sum_{k=1}^{j} \frac{\|V_{k}^{j}\|}{\rho_{k} g^2r^2\pi}$. From lemma \[lemone\], $\|C\|=\|N\|$, $\therefore \|C\|=\sum_{k=1}^{j} \frac{\|V_{k}^{j}\|}{\rho_{k} g^2r^2\pi}$ Case 2 (Upper Bound): Upper bound to the number of regions arises when all nodes in the network are disconnected. Thus, all nodes in such a case will be uninhibited thereby becoming region heads. Thus $\|C\|=\|V\|$. ![Relationship between centroid nodes and the nodes having maximum Socio-Centric Betweenness.[]{data-label="fig:relation"}](relation){width="\columnwidth"} \[numpnodes\] For a node distribution and fully connected network, and using lemma \[numregions\], the number of peripheral nodes in the network is bounded by $\frac{\|V\|(2g+1)}{g^2}$ and $\frac{\|V\|(2g+1)\pi}{g^2\sqrt{3}}$. \[proofpnodes\] Peripheral nodes are the nodes lying in the outer most gradient of the region. Thus, the number of nodes in the $g^{th}$ gradient of a region = $\rho g^2r^2\pi - \rho (g-1)^2r^2\pi$ = $\rho (2g+1)r^2\pi$ Now using lemma \[numregions\], the number of peripheral nodes for all regions thus varies between $\frac{\|V\|(2g+1)}{g^2}$ and $\frac{\|V\|(2g+1)\pi}{g^2\sqrt{3}}$. \[numpnodesdisconnected\] For a node distribution and network with $j$ components ($j>1$), and using lemma \[numregionsunconnected\] and lemma \[lemuncon\], the number of peripheral nodes in the network is bounded by $\sum_{k=1}^{j} \frac{\|V_{k}^{j}\|(2g+1)}{g^2}$ and $\|V\|$. \[proofpnodesdisconnected\] Peripheral nodes are the nodes lying in the outer most gradient of the region. Thus, the number of nodes in $g^{th}$ gradient of a region in $k^{th}$ component = $\rho_{k} g^2r^2\pi - \rho_{k} (g-1)^2r^2\pi$ = $\rho_{k} (2g+1)r^2\pi$. Now using lemma \[numregionsunconnected\] and lemma \[lemuncon\], the number of peripheral nodes for all regions thus varies between $\sum_{k=1}^{j} \frac{\|V_{k}^{j}\|(2g+1)}{g^2}$ and $\|V\|$. Simulation setup {#sec:sec6} ================ We use a simulation area of $A=500m$x$500m$ to simulate our algorithm. $r_b$ and $\ell_{min}$ are set to $30m$ and $5$ respectively to achieve the non-uniform distribution of node throughout the simulation area. The non-uniform node distribution enables us to visualize the real world scenarios. The range of average density, $\rho$, of nodes per unit area is set to \[$1$x$10^{-3}$, $2.5$x$10^{-3}$\]. We make the choice of this range for $\rho$ after considering the percolation of the giant component for the non-uniform node deployment, (Cf. Fig. \[fig:percolation\]). Initially, each node operates in omnidirectional mode using $m=1$ antenna element with the omnidirectional radius as $r=30m$. We set the maximum number of antenna elements that the nodes are equipped with to $M=6$. The separation between two antenna elements computed using $WiFi$ frequency, $f=2.4Ghz$. Through our simulations, we explore the effect on connectivity, $APL$ and $CC$ by varying the node densities and the gradient. We use MATLAB to simulate our algorithm with a confidence interval of $95\%$. We average All the results over $50$ topologies. Results and Analysis {#sec:sec7} ==================== First, we prove the correctness of the centroid finding in the region. For this, we compute the relation between the nodes that have maximum Socio-Centric Betweenness and the centroid nodes in the region. If the centroid node has the highest Socio-Centric Betweenness in the region, then the algorithm found centroid node correctly, (Cf. Fig. \[fig:relation\]). This depends on the value of the gradient. Larger gradients decrease the Socio-Centric Betweenness rank of the centroid node in the region. As the gradient increases, more nodes are now associated to a region thereby increasing the possibility of occurrences of the bridge nodes (bridge nodes have high Socio-Centric Betweenness value). Thus, we also calculate the distance in hops between the centroid node and the maximum Socio-Centric Betweenness node. According to the results, (Cf. Fig. \[fig:relation\]), for a $g$, the percentage of centroid nodes that also have high Socio-Centric Betweenness is more and all the centroid nodes in the network are within $hopcount<g$. The Fig. \[fig:relation\] however shows that for any $g\in[3,10]$ more than $95\%$ of the time the centroid node is within $4$ hop distance to the maximum Socio-Centric Betweenness value node and it is within $1$ hop $60\%$ of the time. Further, we use $g\in[3,10]$ to obtain the results when the Sector model is used in a non-uniformly distributed network, (Cf. Fig. \[fig:unifbiobeamsec\]). The Fig. \[subfig:avlsec\] shows the effect of beamforming on the $APL$. The $APL$ obtained in omnidirectional case is initially less than that obtained for the directional cases because the density of the nodes in the component is low. When the algorithm induces directional beams, due to the inclusion of the nodes in other network components, there is an increase in the $APL$. The $APL$ for the directional case is less than that of the omnidirectional case when $\rho>210^{-3}$ due to the fact that the nodes connect to the centroid node of other regions in the different component as well as in the same component. The gradient affects the $APL$. The lower the value of the gradient is, higher is the number of nodes that beamform, (Cf. Fig. \[subfig:pnodesec\]), leading to more shortcuts and in turn more reduction in the $APL$. For $\rho=2.510^{-3}$ and $g=10$, there is a reduction of almost 40% in the $APL$ while there is a reduction of almost $55\%$ for $g=3$, (Cf. Fig. \[subfig:smsec\]). However, for $\rho=110^{-3}$ and $g\in[3,10]$ when most nodes are unconnected, there is an increase of $70\%$ in $APL$ due to the above-mentioned facts. The introduction of the long-range beams also causes the $CC$ to change, (Cf. Fig. \[subfig:ccsec\]). For very low-density networks, the $CC$ for the directional case is less because beamforming leads to loss in the initial neighborhood. However, for higher density networks, the $CC$ does not vary as much as the $APL$ (Cf. Fig. \[subfig:smsec\]). For $\rho=2.510^{-3}$, there is a reduction of $25\%$ and $38\%$ for $g=10$ and $g=3$ respectively. However, for $\rho=110^{-3}$ and any $g\in[3,10]$, the reduction in $CC$ is almost $40\%$. The $CC$ for directional case for $g<6$ and $\rho\in[110^{-3},2.510^{-3}]$ is almost constant. This implies that the directional network shows modularity where $CC$ is independent of $\|V\|$ and evolves towards hierarchical network [@Ravaszv]. However, when $g\in(6,10]$ the evolution towards hierarchical networks cannot be justified. The number of components in the network can define connectivity. In a very low-density omnidirectional network, the number of disconnected components is higher, (Cf. Fig. \[subfig:connectivitysec\]. The number of disconnected components increases to a certain maximum and then decreases as the density increases. This is because, for a high density, all nodes can find at least one neighborhood node within their reach. In addition, as the number of components decreases, the connectivity increases. For the directional case however, as nodes beamform towards different components with the objective of increasing connectivity, the number of disconnected components is less than that of the omnidirectional case. The size of the giant component can also explain the connectivity of the network. For the directed graphs however, [@Dorogovtsev] defined the giant component using the Giant Strongly Connected Component ($GSCC$) and the Giant In-Component ($GIN$). Thus, we calculate the size of $GSCC$ and $GIN$. We further show the difference between the size of the giant component for omnidirectional network, $GSCC$ and $GIN$. As stated in [@Dorogovtsev] that $GSCC \subset GIN$, we also observe that $GIN$ is a bigger set and contains more nodes than $GSCC$. $GIN$ reaches percolation very early, (Cf. Fig. \[fig:percGSCCGIN\]). Comparing the size of the $GSCC$ of directional network with the giant component of the omnidirectional network, (Cf. Fig. \[fig:percolation\]), we see that the size of $GSCC$ varies between \[$0.84, 0.94$\] for $\rho=210^{-3}$ for different values of the gradient while the size of giant component for the omnidirectional network is $0.41$. Thus, we observe an increase of almost $2.1$ times. The Fig. \[fig:perccomp\] shows an increase of almost $2.2$ times when we compare of size of the $GSCC$ and the $GIN$ for $g=6$ with the giant component of the omnidirectional network. The number of centroid nodes ($\|C\|$) depends on the value of the gradient, (Cf. Fig. \[subfig:cnodesec\]). For a low-density network, the value of the gradient does not matter while as the density increases the value of the gradient affects the number of regions formed. As the gradient increases, more nodes inhibit leading to less number of regions. The difference between the number of regions formed for $g=3$ and $g=10$ is of $40$ for $\rho=2.510^{-3}$ while the difference for $\rho=110^{-3}$ is very less. The value of the gradient used also affects the number of peripheral nodes ($\|P\|$) identified, (Cf. Fig. \[subfig:pnodesec\]). For a low gradient value, as there are more regions, more nodes are included in $P$ because of the reduced neighborhood with respect to the region. However, when the value of the gradient is more, $\|P\|$ is less because there are more nodes in the region and the nodes have relatively more neighbors to check before making the decision of beamforming. $\|P\|$ greatly affects the number of unidirectional paths. However, it has an adverse effect on the $CC$. As the number of peripheral nodes increases, unidirectional paths between the nodes also increases leading to more loss in the $CC$. For $\rho=110^{-3}$ and $g\in[3,10]$, the difference between the number of peripheral nodes is almost negligible. For $\rho=2.510^{-3}$, however, the number of peripheral nodes varies by more than $120$ as the regions formed for lower gradient are more. Our algorithm affects the $APL$ and the $CC$ of the network when we use $ULA$ model, (Cf. Fig. \[fig:unifbiobeamULA\]). On the other hand, it does not affect $\|P\|$ and $\|C\|$. No dependency of the $ULA$ model on $\|P\|$ and $\|C\|$ is rightly justified because these sets are built when the network was omnidirectional, (Cf. Fig. \[subfig:pnodeULA\], \[subfig:cnodeULA\]). However, there is a reduction of almost $60\%$ and $68\%$ in the $APL$ for higher gradient value and for low gradient value respectively. On the other hand, there is no considerable reduction in the $CC$. The reduction in the $CC$ is only between $19\%$ to $22\%$. Due to variation in $B_{w}$ for different $B_{b}$ in $ULA$ model (Cf. Fig. \[fig:gainpatern\]), the values obtained for the $APL$, the $CC$ and connectivity are different from that of the Sector model. From the Fig. \[fig:smpcomp\] we observe that, for higher density networks, the change in the $APL$ for the $ULA$ model is more than that of the Sector model while the $CC$ changes at a much lower rate. ![Variation in the size of the $GSCC$ and the $GIN$ for different density of nodes and $g \in [3,10]$.[]{data-label="fig:percGSCCGIN"}](percoulationGSCCGIN){width="\columnwidth"} ![Comparison of size of the $GSCC$ and the $GIN$ for directed network with that of omnidirectional network for $g=6$.[]{data-label="fig:perccomp"}](percolationComp){width="\columnwidth"} ![Normalized $APL$ and the $CC$ for $N=625$ showing the effects of the gradient for both the Sector and the $ULA$ model.[]{data-label="fig:smpcomp"}](smpropertiesComp){width="\columnwidth"} Until now, we have shown that small world properties are achieved and connectivity be increased in a non-uniformly deployed network. However, it is also important to show the complexity of the algorithm. Due to the storage of three required data values in the region formation phase, neighborhood information and the knowledge about being the peripheral node for both itself and its neighbors is needed. Thus the required memory size is of the order O(3(d+r)+d+1)* where $d$ is the size of the neighborhood and $r$ is the size of reachable centroid nodes. For high-density network, reaching consensus in the region formation and the centroid finding phase is time consuming. However, for a low-density network, the algorithm reaches this consensus quickly. Useful concepts and Related Work {#sec:sec2} ================================ In this section, we define useful concepts giving an overview of the related work. We first define small world concepts in the section \[sub:subswn\] which form the basis of our research. The need of having long range links for achieving small world properties lead us to discuss beamforming in the section \[sub:subamb\]. We then define Lateral Inhibition in the section \[sub:subli\] and Flocking in the section \[sub:subflocking\]. The definitions of centrality concepts are discussed in the section \[sub:subcentral\]. Further, we discuss non-uniform deployment in the section \[sub:subnonunifdistrib\]. Small World Network {#sub:subswn} ------------------- Inspired by Stanley Milgram’s [@Milgram] experiment of “six degrees of separation", Watts et al [@Watts] suggested a model for the creation of small world network. Watts et al in [@Watts; @Wattsbook] showed that rewiring edges of a regular network with a probability $p_r$ results into reduction in the $APL$ of the network while there is very little change in the $CC$. Starting by choosing a random vertex and one of its edge to the vertex’s 1 hop neighbor with $p_r$, Watts et al reconnected the edge to a random vertex in the remaining network. Watts et al then considered all other vertices for rewiring. The process of rewiring continued with the edges now connecting the two hop neighbors. This process continued until all the edges were considered. $p_r$ highly affected the rewiring process. Probability $p_r=0$ meant that no rewiring while $p_r=1$ meant complete rewiring of the graph. Using $p_{r}=1$ resulted into complete randomness in the network. The small world model motivated many research studies, [@Helmy; @Barabasi; @AlbertBarabasi; @BarabasiAlbert], and many models were proposed. Newman,[@NewmanReview; @Newman], compiled a comprehensive list of the models on small world. Mostly, the researchers studied two kinds of network structures, one without network growth while another with the network growth. Researchers analyzed the scaling and performance issues for the growing networks [@Barabasi; @BarabasiAlbert]. Barabasi et al in [@AlbertBarabasi; @BarabasiAlbert] showed that small world properties also exists in a growing network and there is a preferential attachment of the nodes giving rise to “rich gets richer" property. Barabasi et al showed that the real world networks possess these properties. This led to the behavioral analysis of the networks. On the contrary, assuming spatial wireless ad hoc network without growth, Helmy [@Helmy] performed the small world analysis and showed that rewiring of links does not change the structure of the network. Two other results shown in [@Helmy] are significant in the context of this paper. First, the $APL$ is reduced at a greater rate when shortcuts are 25% to 40% in length of the network diameter. Second, the rate of the $APL$ reduction is more when there are only 0.2% to 2% shortcut links. The reduction rate stabilizes when there are more than 2% shortcut links. Antenna Model and Beamforming {#sub:subamb} ----------------------------- Authors of [@Bettstetter; @Balanis] provided an extensive study of antenna models and defined antenna gain using radiation intensity $u(\theta,\phi)$ where angle $\theta$ is angle with the $z$-axis and $\phi$ with the $xy$-plane as $$\label{eq:1} g(\theta ,\phi)=\frac{u(\theta ,\phi )}{\frac{1}{4\pi} \int_{0}^{2\pi }\int_{0}^{\pi }u(\theta ,\phi ) sin\theta d\theta d\phi}$$ Considering $m$ antenna elements and isotropic radiators with same phase shift between them, researchers defined two basic antenna models Uniform Linear Array antenna model ($ULA$), (Cf. Fig. \[fig:figULAUCA\]), and Uniform Circular Array antenna model ($UCA$). When $m=1$, there is no superimposition of the radiation. This leads to a beam with omnidirectional characteristics. However, when $m>1$, there is a constructive and destructive superimposition of the radiation due to the phase shift between the antenna elements. This leads to a beam with directional characteristics. ![Source: [@Bettstetter], Arrangement of $m=8$ antenna elements in $ULA$ model.[]{data-label="fig:figULAUCA"}](figULA){width="30.00000%"} The gain pattern for the $ULA$ antenna model is only dependent on the number of antenna elements. It has no dependency on the boresight direction ($B_{b}$, the direction of maximum radiation intensity, Cf. Fig. \[fig:gainpatern\]). On the other hand, for the $UCA$ antenna model, gain pattern is dependent on both the number of antenna elements and $B_{b}$. However, in wireless ad hoc networks, beamforming using $UCA$ model has been well studied. Classical beamforming techniques using $UCA$ model include Random Direction Beamforming ($RDB$) [@Bettstetter; @Vilzmann; @VilzmannBettstetterHartmann] and beamforming based on the angle of incidence and packet flow. Bettstetter et al [@Bettstetter] studied the use of $RDB$ with the path probability to improve the connectivity in the wireless networks. Vilzmann et al [@VilzmannWidmer] derived low complexity techniques for beamforming and proposed Maximum Node Degree Beamforming ($MNDB$). In $MNDB$ the nodes directed their beams towards the node that had maximum degree. The authors found that $MNDB$ leads to less number of inter-cluster connections but had more intra-cluster connections. To overcome this drawback, the authors proposed Two-hop Node Degree Beamforming ($TNDB$). In $TNDB$ the nodes directed their beams towards the node that had maximum two-hop neighborhood. The authors showed that $TNDB$ outperforms both $RDB$ and $MNDB$. Other works on beamforming include [@VilzmannWidmer; @Kiese; @Yu; @Li]. However, most of these studies were concentrated on nodes that were uniformly distributed at random in the given area but very few among them talk about non-uniform distribution of the nodes. Considering all nodes use directional beams, [@Bettstetter; @Vilzmann; @VilzmannWidmer; @Kiese; @Yu; @Li] addressed connectivity very well but do not discuss the impact on the $APL$ and the $CC$. Table \[table:table1\] illustrates a comparison between these studies. On the other hand, studies related to the small world properties lack connectivity analysis for the non-uniformly distributed network. Table \[table:table2\] illustrates comparisons between various studies performed in the direction of achieving small world properties in wireless ad hoc networks and our model. ------------------- ----------------------- ----------------------------- -------------------- ----------------- ----------------- Parameter\\ Vilzmann Widmer Kiese Yu Li Reference et al [@Vilzmann] et al [@VilzmannWidmer] et al [@Kiese] et al [@Yu] et al [@Li] Transmission mode Directional Directional Directional Directional Both Reception mode Directional Directional Directional Omnidirectional Both Mobility No Yes No No No Beam width Depends on Constant Constant Optional Constant, beam direction switched beam antenna Beam direction Random Optional Optional Optional Random Antenna model $UCA$ $UCA$ $UCA$ modeled Sector Keyhole as keyhole Node distribution Uniform Uniform and Non-Uniform Not specified Uniform Non-Uniform ------------------- ----------------------- ----------------------------- -------------------- ----------------- ----------------- -------------------- ----------------- ----------------------- ------------------------------ -------------------- --------------------- -------------------- Parameter\\ Our Model Banerjee Guidoni Helmy Sharma Verma Reference et al [@Banerjee] et al [@GuidoniLoureiro] et al [@Helmy] et al [@Sharma] et al [@Verma] Shortcut Creation Rewiring Rewiring Addition Addition Addition Addition Node distribution Non-Uniform Uniform Uniform Uniform Uniform Uniform External No No High range - Wired Two radios infrastructure Sensor for each node Global knowledge No No Yes Yes Yes Yes Density of nodes Low High High High - Low Shortcut Edge Directed Directed Undirected Undirected Undirected Directed Shortcut direction Towards Longest Random, Random Random Random centroid of Traffic Flow towards sink other region path Shortcut length Function of Function of Constant Limited Constant Constant antenna node density elements Shortcut width Depends on Depends on Constant - - Constant Shortcut Length Shortcut Length Prob. of Shortcut $(0,1]$ based Based on $\in (0,1]$ $\in (0,1]$ function of $\in (0,1]$ creation on model centrality network size parameters values Path length, Path length, Path length, Path length, Path length, Path length, Clust. Coeff. Connectivity Clust. Coeff. Clust. Coeff. Energy Clust. Coeff., Connectivity degree -------------------- ----------------- ----------------------- ------------------------------ -------------------- --------------------- -------------------- Lateral Inhibition {#sub:subli} ------------------ Lateral Inhibition is a process by which cells of animal tissues, based on the properties of neighbor cells, decide whether to perform a task or not. Lateral Inhibition ensures that the cells that perform the tasks are equidistant from each other. This helps in producing regular patterns throughout the surface. Lawrence [@Lawrence] modeled Lateral Inhibition as, when a cell performs a task, it inhibits its neighbors within $h$ hops from performing that task thereby resulting into equally spaced uninhibited cells. Lateral Inhibition thus creates clusters where the cluster heads are uninhibited nodes distributed over an area. Nagpal et al [@Nagpal; @NagpalMamei] described a simple algorithm to achieve Lateral Inhibition. In the algorithm, the cells assign themselves a random number. Each cell starts to count backwards. If before reaching $0$, a node receives an inhibition signal from the neighboring cell, the cell stops counting otherwise sends out an inhibition signal to all its neighbors. Nagpal et al [@Nagpal; @NagpalMamei] showed that the $hopcount$ used to create the cluster greatly affects the number of clusters formed. Recent studies revealed that Lateral Inhibition can be achieved in an optimal way [@Afek]. Inspired by the tissue of the fruit fly, Afek et al [@Afek] modeled distributed Lateral Inhibition using local information and requiring only two exchange mechanisms. These exchange mechanisms are, first, broadcasting a single control bit to the neighbors with certain probability and second, if the node receives no message from the neighbors, it sends out a control bit to inhibit its neighbors. As a variation to Nagpal et al’s algorithm, the algorithm used a probabilistic approach that varied over time in an increasing manner to perform Lateral Inhibition. The runtime complexity of the algorithm was of the order $O(\log^2\|V\|)$ where $\|V\|$ was the number of nodes in the system. Due to single bit exchange messages over single hop, the algorithm had a low message complexity. Flocking {#sub:subflocking} -------- Flocking, [@Reynolds], was first modeled by Reynolds in order to simulate the birds’ behavior. In nature, flocking is observed in many other social living organisms like cattle, fishes and humans. Reynolds, while modeling Flocking, termed each social entity as a boid and formulated three very simple rules, (a) Alignment (b) Separation and (c) Cohesion. Reynolds defined Alignment rule as the direction matching of a boid with its neighbors. He defined Separation rule as the collision avoidance with neighborhood boids and Cohesion rule as the tendency of a boid to remain as close to its neighbors as possible and not stray. The Fig. \[fig:figflocking\](a), shows that the boid orients itself in the direction in which its neighbors were moving. The Fig. \[fig:figflocking\](b), shows that the boid has to move away from the neighbors in order to avoid collision while the Fig. \[fig:figflocking\](c) shows that the boid moves towards the centroid of the neighbors in order to remain close to its neighborhood. Couzin in [@Couzin] formulated mathematical explanation of these rules. Due to the motion of a boid, velocity and displacement were associated with the boid. Alignment rule was modeled using the direction of a boid while Separation and Cohesion were modeled using both velocity and the displacement. Recent studies have revealed the use of Flocking in solving various problems in wireless ad hoc networks. Antoniou et al [@Antoniou] used Flocking to provide efficient congestion control mechanism by computing the congestion at the neighbor nodes while [@Kadrovach] used the Separation rule for the efficient placement of nodes to maximize the coverage area. Centrality {#sub:subcentral} ---------- Decades of research on network and graph theory has led researchers to derive many fundamental concepts related to the importance of a node in the network. The concept of centrality was one such concept that was developed and used to address the topological characteristics of the network nodes. Proposed definitions of centrality measures include those that use global parameters as well as those that only use local information. Some examples of global centrality measures are Socio-Centric Betweenness [@Freeman; @Freemanlc] and Closeness Centrality [@Freeman] while Degree Centrality [@Freeman] and Egocentric Betweenness Centrality [@Everett; @Daly] are examples of the local centrality measure. ### Socio-Centric Betweenness Centrality {#subsubsec:scbc} The Socio-Centric Betweenness Centrality, [@Freeman; @Freemanlc], is the measure of the number of shortest paths passing through the node thereby expressing the most important node in the network and through which most of the communication takes place. The Socio-Centric Betweenness is a frequency measure and requires the global network knowledge. Usually nodes with high degree and those that are acting as the bridge nodes tend to have relatively high Socio-Centric Betweenness. Mathematically the Socio-Centric Betweenness of a node $v$ is $$BC_v=\sum_{}\frac{sp(v)}{sp}$$ where $sp(v)$ is the number of shortest paths between any two nodes that pass through $v$ while $sp$ is the total number of shortest paths in the network. ### Egocentric Betweenness Centrality {#subsubsec:ebc} Aiming to compute the Betweenness centrality using local properties, [@Everett; @Daly] proposed the Egocentric Betweenness Centrality measure. Everett in [@Everett] computed the Egocentric Betweenness using upper diagonal adjacency matrix $A_{v}$. $A_{v}$ is created considering 1 hop neighborhood of the node $v$. Consider $I$ to be the identity matrix, then the sum of the inverse of all non-zero elements in $A_{v}^{2}$ along $[I-A_{v}]$ is the Egocentric Betweenness of the node. Marsden in [@Marsden] performed an empirical study to find the relation between the two types of Betweenness, the Socio-Centric and the Egocentric Betweenness, and found that the Egocentric Betweenness is strongly correlated to the Socio-Centric Betweenness and it can be used when global network information is lacking. ### Closeness Centrality {#subsubsec:closenessc} The Closeness Centrality [@Freeman] on the other hand is the measure of how fast a node can transfer data to all the nodes. The Closeness Centrality is the fraction of shortest distance between a node to all other nodes in the network. Assuming $sd(v,w)$ be the shortest distance between node $v$ and $w$, the Closeness Centrality of $v$ is $$C_{v}= \frac{1}{\sum_{w\neq v,w\in V} {sd(v,w)}}$$ A node with the highest Closeness Centrality value is the centroid of the network. As all the centrality measures convey different information, it is not necessary that a node having high value for one centrality measure also have high values for the others. Many other types of centralities, such as, Bridging Centrality, Eigen Vector Centrality and Spectral Centrality also exist. We refrain ourselves from describing them in detail. However, Katsaros, [@Katsaros], provided a brief survey on these centrality measures. Non-Uniform distribution of nodes {#sub:subnonunifdistrib} --------------------------------- Many non-uniform deployment strategies have been proposed, [@Weijen; @LeBoudec; @Hu; @Aitsaadi; @Riihijarvi; @BettstetterGyarmati]. We take insights from Bettstetter et al, [@BettstetterGyarmati], node deployment strategy. Bettstetter et al proposed the use of thinning process to generate a non-uniform node deployment. The authors started with uniform distribution of nodes in a given region, then pruned the nodes based on two factors, transmission radius, $r_b$, and the number of neighbor nodes, $\ell_{min}$. If the node had at least $\ell_{min}$ neighbors within $r_b$, the node was not removed else it was removed. Schilcher et al, [@SchilcherGyarmati], formulated and measured the degree of non-uniformity of this pruned network. Schilcher et al divided the region into smaller sub-regions and estimated the number of nodes in the sub-region. The estimated value was then used to calculate the non-uniformity index, $hIndex$. The Fig. \[subfig:figVincentnodedistbnonunif\] shows the deployment achieved when the thinning process is applied to the deployment shown by the Fig. \[subfig:figVincentnodedistbunif\]. The Fig. \[subfig:figVincentdensity\] shows the density distribution of nodes using kernel method. Future Work {#sec:sec9} =========== A Number of extensions to our algorithm can be visualized. Identifying the optimal gradient size to choose for the determination of minimal peripheral set of nodes is one way of extending our work. We are currently working on how we can apply game theory to successfully find the minimal peripheral set. We believe that by applying game theory nodes can determine what the suitable gradient size is and can reduce asymmetric links further. We would also like to extend our algorithm to support dynamic environment and asynchronous operation. Dynamic environments are likely to result in frequent changes to the state of the node. Any change in the state of the node would require reconfiguring in the network using the proposed algorithm. Information available at the neighborhood nodes would be helpful in learning about the previous configuration. This learning could be docitive [@Giupponi], meaning, partial learning from the neighborhood states could make nodes infer about the previous good configuration so that reconfiguration can be done easily and quickly. This will also help us to address the unaddressed paradigms of [@Prehofer]. Further, we would like to address network lifetime of the network when implementing our algorithm. Conclusion {#sec:sec8} ========== In this paper, we have presented an algorithm for achieving small world properties using beamforming and bio-inspired techniques in a wireless ad hoc network. Our algorithm works using locally available information and does not require the knowledge of the network. We have also removed the possibility of requirement of any external infrastructure for achieving our goal. Through our algorithm, we have shown how isolated communities can collaborate and connect with each other to achieve better and faster communication. Bio-Inspired techniques like Lateral Inhibition helped us to form communities within the network for the reduced message complexity while the Flocking analogy helped us to determine beam properties. Our results show that for both theoretical and realistic antenna models and relatively high-density networks, there is a reduction in the $APL$ by almost $40\%$ to $68\%$ for $g\in[3,10]$. On the other hand, reduction in the $CC$ is between $19\%$ to $38\%$. Our results also show improvement in the connectivity. The increase in the size of the $GSCC$ for the non-uniformly distributed directional network is around $10\%$ for high density network while it is around $61\%$ for relatively low density networks.
--- abstract: 'In the presence of a scattering potential, electron transport in a quantum wire is known to be dramatically modified by backward scattering and unaffected by forward scattering processes. We show that the scenario is quite different in Quantum Spin Hall effect edge states coupled at a constriction. The helical nature of these states leads to the appearance of a forward scattering spin channel that is absent in other Luttinger liquid realizations. Suitably applied ac gate voltages can thus operate on the spin of electrons tunneling across the constriction, and induce in the dc tunneling current a cusp pattern that represents the signature of the edge state electronic interaction.' author: - Fabrizio Dolcini title: Signature of interaction in dc transport of ac gated Quantum Spin Hall edge states --- The nanotechnological advances of last two decades have allowed to achieve various solid state realizations of one-dimensional (1D) electronic systems, such as semiconductor wires [@QW], carbon nanotubes [@SWNT], and Quantum Hall edge states[@QHE], where electron-electron correlations lead to a Luttinger liquid (LL) behavior [@LL]. One of the most striking features that distinguishes a LL from a non-interacting system is its unconventional electron transport. This can already be seen from the simple case of the current flowing through an interacting wire in the presence of one single localized impurity, modeled as a delta-like scattering potential $\lambda \, \delta(x)$. Indeed, a well known result of LL theory is that, no matter how weak the scatterer strength $\lambda$ is, the current-voltage characteristics exhibits a zero-bias anomaly with an interaction dependent power-law at low voltage bias [@kane-fisher; @glazman]. Such peculiar behavior originates from the interplay between electron-electron interaction and single-particle backward scattering (BS) processes occurring at the impurity. In general, besides BS, an impurity also gives rise to forward scattering (FS) processes, which are known to have no effect on the current of a quantum wire, though (see Ref. [@vondelft]). Thus, while BS has been widely discussed [@LL; @kane-fisher; @glazman; @vondelft], impurity FS terms are often omitted in models of quantum wires, with the underlying understanding that such terms never affect transport. In this paper we show that the situation can be quite different in the recently discovered edge states of Quantum Spin Hall effect (QSHE) systems [@QSHE]. These 1D electronic states, flowing at the edges of HgTe/CdTe quantum wells, are characterized by a tight connection between the direction of motion and spin orientation, and represent a new type of LLs, called [helical]{} Luttinger liquids [@hLL]. Here we shall show that in such systems a different type of FS emerges, which is absent in other 1D realizations, and which strongly affects transport via the spin channel. In particular we shall demonstrate that such effect can lead to a cusp pattern in the current-voltage characteristics that represents the signature of electron-electron interaction in the QSHE edge states.\ ![\[fig-setup\] (color on-line) Helical edge states flow in a QSHE bar (circled arrows denote spin orientation). Two ac gate voltages $V_{g,T}$ and $V_{g,B}$ applied with opposite phase across a QPC affect the spin of the tunneling electrons and, in the presence of electron-electron interaction, modify the dc current. ](Fig1){width="0.8\linewidth"} In order to illustrate this phenomenon, we first briefly recall the usual case of a quantum wire. We denote by $\mathcal{H}=\mathcal{H}_0+\mathcal{H}_{imp}^{FS}+\mathcal{H}_{imp}^{BS}$ the Hamiltonian of a quantum wire with an impurity, where $\mathcal{H}_0$ is the LL Hamiltonian of the clean interacting wire, $\mathcal{H}^{BS}_{imp}= \lambda \, \sum_{\sigma} (\Psi^\dagger_{R \sigma} \Psi^{}_{L \sigma} +\Psi^\dagger_{L \sigma} \Psi^{}_{R \sigma}) \, \, \|_{x=0}$ describes the impurity BS term, and $\mathcal{H}^{FS}_{imp}= \lambda \, \sum_{\sigma} (\Psi^\dagger_{R \sigma} \Psi^{}_{R \sigma}+\Psi^\dagger_{L \sigma} \Psi^{}_{L \sigma}) \, \|_{x=0} \, \,$ the impurity FS term. Here $\Psi_{r \sigma}$ ($r=R/L=\pm$) denotes the right(left) moving component of the electron field operator with the spin orientation $\sigma =\uparrow,\downarrow$. A handwaving argument to see that impurity FS has no effect on transport is to observe that, since $\mathcal{H}^{FS}_{imp}$ couples to the sum of right and left movers, it cannot affect their difference, i.e. the current. At a more formal level one expresses the electron fields $\Psi_{r \sigma}$ using the Bosonization identity [@LL], i.e. $\Psi_{r \sigma}(x)= (2 \pi a)^{-1/2} \kappa_{r \sigma} \, \exp{[ i r \sqrt{\frac{\pi}{2}} (\Phi_{c}+\sigma \Phi_{s}+r (\Theta_{c}+\sigma \Theta_{s}))(x)]}$, where $a$ denotes the underlying lattice spacing and $\kappa_{r \sigma}$ the Klein factors ensuring the anticommutation of different species. The bosonic fields $\Phi_{c(s)}(x)$ describe the charge and spin degrees of freedom, respectively, and $\Theta_{c(s)}(x)$ their dual fields, fulfilling the commutation relations $[ \Phi_{\nu}(x), \Theta_{\nu^\prime}(y)]= i \, \delta_{\nu, \nu^\prime} \, \mbox{sgn}(x-y)/2$, with $\nu=c,s$. Introducing now non-local fields $\xi_{ \nu \pm}(x) \doteq (\Phi_{\nu}(x) +\Theta_\nu(x)\pm (\Phi_{\nu}(-x)-\Theta_\nu(-x)))/2$, it is possible to show [@vondelft] that the impurity terms depend on $\xi_{\nu \pm}$ in the following way $$\begin{aligned} \mathcal{H}^{BS}_{imp}& =& \mathcal{H}^{BS}_{imp}(\xi_{c+},\xi_{s+}) \label{BS-bos1} \\ \mathcal{H}^{FS}_{imp}& =& \mathcal{H}^{FS}_{imp}(\xi_{c-}) \quad, \label{FS-bos1} \end{aligned}$$ whereas the term $\mathcal{H}_0$ splits into a sum $\mathcal{H}_0 = \sum_{\nu=c,s} \sum_{\alpha=\pm} \mathcal{H}_{0 \nu \alpha}(\xi_{\nu \alpha})$ of independent terms ($[\xi_{\nu +}(x), \xi_{\nu^\prime -}(y)]=0$). Since the term $\mathcal{H}^{FS}_{imp}$ is linear in $\xi_{c -}$, and the term $\mathcal{H}_{0 c -}$ is quadratic in $\xi_{c-}$, the former term can be gauged away simply by a unitary transformation [@vondelft]. Importantly, in doing that, a crucial point is that $\mathcal{H}^{FS}_{imp}$ and $\mathcal{H}^{BS}_{imp}$ depend on [different]{} fields, as shown in Eqs.(\[BS-bos1\])-(\[FS-bos1\]), so that in a quantum wire FS and BS terms are independent. Notice that such argument also holds for a time-dependent impurity, so that previous studies analyzing such situation have focussed on time-dependent BS terms [@feldman-gefen; @chamon; @komnik].\ Let us now discuss the case of the edge states of a QSHE system. The helical properties imply that, at each boundary of a HgTe/CdTe quantum well, a Kramers pair of counter-propagating states appears, so that at the -say- top boundary right- (left-) moving electrons are characterized by spin-$\uparrow$ (spin-$\downarrow$) only. The opposite occurs at the bottom boundary, as shown in Fig.\[fig-setup\], and each boundary carries both charge and spin. Including both intra- and inter-edge electron-electron interaction, the system of the two boundaries can be described by a LL Hamiltonian [@hLL] $$\mathcal{H}_0 = \frac{\hbar}{2} \sum_{\nu=c,s} \int dx \left[ v_\nu K_\nu (\partial_x \Theta_\nu)^2+ \frac{v_\nu}{K_\nu} (\partial_x \Phi_\nu)^2 \right]$$ where $v_c=v_s=v$ and $K_c=K_s^{-1}=K$ are interaction dependent parameters, with $v K^{\pm 1}=v_F (1+(2U_1\pm U_2)/2 \pi \hbar v_F)$. Here $v_F$ denotes the Fermi velocity, and $U_1$ and $U_2$ the intra- and inter-edge interaction. The sample is assumed infinitely long in the longitudinal direction $x$. The peculiarity of helical nature is encoded in the fact that, differently from quantum wires where the spin channel is essentially non interacting ($K_s =1$) due to SU(2) symmetry, in the QSHE edge states the spin channel is characterized by an effective attractive interaction $K_s >1$ [@hLL; @liu]. Scattering from impurities along one boundary is prevented from topological protection. Nevertheless, by etching the quantum well over a short region to form a Quantum Point Contact (QPC), the two boundaries are brought close to each other \[see Fig.\[fig-setup\]\], inducing inter-boundary tunneling [@hLL; @liu; @mio]. Due to the helical nature of the edges, an electron flowing along a given direction can only tunnel to the other boundary by reversing its group velocity [@nota]. Tunneling thus plays the same role as the BS in a quantum wire, and is described by a term $$\begin{aligned} \mathcal{H}^{}_{tun} = \displaystyle \left. \frac{\hbar v_{\rm F} \gamma}{2\pi a} \! \! \! \sum_{m=\pm, \sigma=\uparrow, \downarrow} \! \! \! \! \! \! m \, \kappa_{L \sigma} \kappa_{R \sigma} \, \, e^{i m \sqrt{2 \pi} (\xi_{c+} + \sigma \xi_{s+})}\right\|_{x=0} \label{Htun-p} \end{aligned}$$ where $\gamma$ is the dimensionless tunneling amplitude. Indeed Eq.(\[Htun-p\]) has the same form as (\[BS-bos1\]), and we shall utilize the expression ‘backward scattering’ (‘BS’) to emphasize such analogy. Furthermore, with the voltages $V_{g,T}$ and $V_{g,B}$ of two gates located at the two sides of the constriction, and coupled to the edge states over a lengthscale $l_g$ \[see Fig.\[fig-setup\]\], one generates local FS potential terms, namely $V_{g,T} (\rho_{R\uparrow}+\rho_{L\downarrow})$ for the top boundary, and $V_{g,B} (\rho_{R\downarrow}+\rho_{L\uparrow})$ for the bottom boundary, where $\rho_{r \sigma}=e \, \Psi^\dagger_{r \sigma} \Psi^{}_{r \sigma}$. Despite these similarities, important differences emerge with respect to the case of a quantum wire. Indeed, due to the space separation between the edge states, the two gates voltages couple differently to the edges, so that here two independently tunable FS terms appear. This difference can be highlighted by making the (unnecessary for the final result) assumption that the electron fields are not significantly varying along $l_g$. Then, utilizing the non-local charge and spin fields $\xi_{\nu\pm}$, one obtains $$\begin{aligned} \mathcal{H}^{}_{gate} &=& \sqrt{\frac{2}{\pi}} \left. e\, l_g \left( V_{g,c} \, \partial_x \xi_{c-}+ V_{g,s} \, \partial_x\xi_{s+} \right) \right\|_{x=0} \label{Hgate} \end{aligned}$$ where $V_{g,c/s}=(V_{g,T}\pm V_{g,B})/2$. The first term of the r.h.s. of Eq.(\[Hgate\]), controlled by $V_{g,c}$, is the usual charge density coupling also present in the term (\[FS-bos1\]) of an impurity in a quantum wire. As observed above, it has no effect on dc transport. However, Eq.(\[Hgate\]) exhibits an additional FS channel, which represents a coupling to the spin current and is controlled by spin gate $V_{g,s}$. Notably, such field $\xi_{s+}$ appears also in the ‘BS’ term (\[Htun-p\]), so that for the spin channel FS and BS processes are not independent, and the former cannot be simply gauged away. Thus, although the FS term [alone]{} cannot induce a dc current directly, in QSHE edge states it can in principle operate indirectly, by affecting the BS term. Indeed, generalizing now (\[Hgate\]) to arbitrary space and time-dependent profile $V_{g,s}=V_{g,s}(x,t)$ around the constriction, one can show that the additional spin FS term leads to a shift $\xi_{s+}\rightarrow \xi_{s+}+\xi_{s+}^0$ in the exponent of tunneling term (\[Htun-p\]), where $$\begin{aligned} \xi_{s+}^0(t) &=& -\frac{1}{\sqrt{2 \pi}} \frac{e}{\hbar v} \int V_{g,s}(x^\prime,t-\frac{\|x^\prime\|}{v}) \,dx^\prime \quad. \label{xis+0}\end{aligned}$$ This represents a $V_{g,s}$-dependent renormalization of the phase of the tunneling amplitude $\gamma$. Since only phase differences at the tunneling point matter, the spin FS term can affect dc transport in the presence of a time-dependent spin gate voltage $V_{g,s}(x^\prime,t)$.\ Thus, the above outlined difference between the two terms of Eq.(\[Hgate\]) also finds, mutatis mutandis, an interpretation in the context of photon-assisted tunneling [@aguado], where one studies the effect of an ac gate voltage on the dc current. For an energy independent scatterer (as a single impurity or tunneling term is) an ac gate voltage is known to yield no dc effect [@butti]. This holds for the conventional coupling to the charge degree of freedom. However, an electron tunneling across the QPC transfers not only charge but also spin. Helical QSHE edge states offer the possibility to ‘photon-assist’ tunneling via the spin-channel, through the second term on the r.h.s. of Eq.(\[Hgate\]). It is worth noticing, at this point, that also in QHE systems the edge states are space separated [@QHE] and, when a gate voltage difference $V_{g,T}-V_{g,B}$ is applied across a QPC in a Hall bar, an additional FS term arises w.r.t. quantum wires. However, since QHE the edge states are chiral, such FS term is the charge current. It thus breaks time-reversal symmetry (TRS) and still involves the usual degree of freedom, the charge, which exhibits repulsive interaction ($K_c \le 1$). In contrast, since in QSHE the edge states are helical, the additional coupling preserves TRS and involves the spin channel, which is characterized by an attractive interaction $K_s=K^{-1} \ge 1$. In this respect, the QSHE edge states exhibit an unconventional effect that cannot be addressed in other systems. ![\[Fig2\] (color on-line) The tunneling current across the QPC as a function of the dc bias voltage frequency, normalized to the value $G_{tun}^0=G_{tun}\|_{\omega=0}$. The cusps at integer values of the spin gate frequency $\omega$ are a signature of electron-electron interaction (here $K=0.7$). Different curves refer to different values of the spin gate parameter: $z=1$ (black thick line), $z=3$ (red thin line), and $z=4$ (blue dashed line). Inset: For non-interacting electrons, the two non equilibrium distributions of the electrons incoming to the QPC. ](Fig2){width="0.9\linewidth"} Let us now specify the conditions for this effect to occur. We shall henceforth consider a purely spin gate voltage configuration, i.e. $V_{g,T}=-V_{g,B}=V_{g,s}$, and assume for definiteness $eV_{g,s}(x,t)=W_{s}(x) A_s(t)$, where $W_s(x)=e V_{g,s}^0$ for $\|x\|<l_g/2$ and 0 otherwise, and $A_s(t)=\sin(\omega t)$. In terms of the setup in Fig.\[fig-setup\], such spin gate voltage corresponds to $V_{g,T}$ and $V_{g,B}$ oscillating with amplitude $V_{g,s}^0$, frequency $\omega$ and opposite phase. In this case one has $\xi_{s+}^0(t) = - (2 \pi)^{-1/2} 4 e V_{g,s}^0 \sin(\omega l_g/4v) \sin(\omega t) /\hbar \omega$. We now analyze how such ac gate voltage affects the dc current that flows through the four terminal setup when electrochemical potentials $\mu_{T/B}^S$ ($\mu_{T/B}^D$) are applied to the Top and Bottom source (drain) metallic electrodes, i.e. on the left (right) hand side of the sample. For simplicity, we focus here on the situation where the electrochemical potential of the electrodes are set in charge-bias configuration, i.e. $\mu^S_T=\mu_B^S=-\mu_T^D=-\mu_B^D=eV/2$. The current operator in a Top/Bottom electrode consists of a charge current and a spin density contribution, $\hat{I}_{T/B}= \hat{I}^{(c)}\pm \hat{I}^{(s)}$, given by $\hat{I}^{(c)}= e vK (2 \pi)^{-1/2} \partial_x\Theta_c$ and $\hat{I}^{(s)}=e vK (2 \pi)^{-1/2} \partial_x \Phi_s$, respectively. The average current $I_{T/B} \doteq \langle \hat{I}_{T/B} \rangle$ can be evaluated with the Keldysh technique [@keldysh]. To leading order in the tunneling amplitude $\gamma$, the contribution $I^{(c)}(x,t) \doteq \langle \hat{I}^{(c)}(x,t)\rangle$ at a point $x$ and time $t$ reads $$\begin{aligned} \lefteqn{I^{(c)}_{\gamma^2}(x,t) = \displaystyle - \frac{2 \pi i}{e\hbar} \left( \frac{\hbar v_F \gamma}{2 \pi a} \right)^2 \int_{t_1 \ge t_2} \hspace{-0.5cm} dt_1 dt_2 \, \, \, \sigma_{0c}(x,0;t_x- t_1) } \hspace{1.5cm} & & \nonumber \\ & & \hspace{-1cm} \times \left( \sum_{s=\pm} s \, \prod_{\nu=c,s} e^{ 2 \pi \mathcal{C}_{\nu +}(s(t_1-t_2)) } \right) \sin \left[ \frac{(t_1-t_2)e V}{\hbar} \right] \, \, \nonumber \\ & & \hspace{-1cm} \times \cos \left[ \sqrt{2 \pi} (\xi^0_{s+}(t_1) - \xi^0_{s+}(t_2)) \right] \, \, \, \label{Ic_gamp2_fin-inf-pre-pre}\end{aligned}$$ where $\sigma_{0c}(x,y;t)=(2K e^2/h) \, \theta(t) \sum_{p=\pm}\delta(t +p \|x-y\|/v)$ represents the charge conductivity, $\mathcal{C}_{\nu +}(t) \doteq \langle \xi_{\nu +}(0,t) \xi_{\nu +}(0,0)\rangle-\langle \xi^2_{\nu +}(0,0)\rangle =-K_\nu \ln[(t_a+i t)^2/t_a^2]/4\pi$ is the unperturbed correlation function of the charge and spin fields at the tunneling point, and $t_a=a/v_F$ is a small cut-off timescale. The last line of Eq.(\[Ic\_gamp2\_fin-inf-pre-pre\]) encodes the phase differences arising from the spin FS. After obtaining a similar expression for $I^{(s)}_{\gamma^2}$, one can evaluate the currents, which consist of a dc and an ac components, $I_{T/B}(x,t)=I_{dc}+I_{T/B,ac}(x,t)$, where $I_{dc}$ is independent of $x$ and $t$. In particular, $I_{dc}$ can be written as $I_{dc}=I_{0}+I_{\gamma^2}$, where $I_0=e^2 V/h$ is the current in the absence of the QPC, and the $I_{\gamma^2}$ represents the dc current tunneling across the constriction, which depends on the dc bias voltage $V$ and includes the effect of the ac gate voltage. Computing the tunneling conductance $G_{tun} \doteq dI_{\gamma^2}/dV$ we obtain $$\begin{aligned} \lefteqn{\displaystyle G_{tun} = - \frac{K e^2}{\hbar^3} \left( \frac{\hbar v_F \gamma}{2 \pi a } \right)^2 \! t_a^{2K^} \sin[\pi (1-2K^)] } & & \label{PAT} \\ & & \displaystyle \times \Gamma[1-2K^] (2K^-1) \sum_{n \in \mathbb Z} J^2_{\|n\|}(z) \, \, \| \omega_{SD}-n\omega\|^{2(K^-1)} \nonumber\end{aligned}$$ where $K^=(K_c+K_s)/2$ is the effective interaction strength, $J_n$ is the Bessel function, $\omega_{SD}=e V/\hbar$ the frequency related to the dc bias voltage, and $z \doteq 4eV_{g,s}^0 \, \sin( \frac{\omega l_g}{4v} )/\hbar \omega$ a spin gate amplitude parameter.\ Eq.(\[PAT\]) describes how the FS processes arising from the ac spin gate voltage affect the tunneling conductance. Formally, it is reminiscent of expressions obtained in the context of photon-assisted tunneling [@butti]. Here, however, it is the spin degree of freedom of the tunneling electron to be affected. Notice that Eq.(\[PAT\]) is intrinsically gauge invariant, for it only depends on energy differences ($z \propto V_{g,T}-V_{g,B}$). The behavior of $G_{tun}$, shown in Fig.\[Fig2\], consists of a pattern of cusps, located at values of $\omega_{SD}$ integer multiples of the ac spin gate frequency. The weight $J^2_{\|n\|}(z)$ of each cusp is a non-monotonous function of the spin gate amplitude $V_{g,s}^0$, and the cusp exponent is interaction dependent. We observe that, because the spin channel is attractive ($K_s \ge 1$), the effective interaction parameter is $K^* \ge 1$, and one obtains cusps, and not divergences. Importantly, in the non-interacting limit, $K^* \rightarrow 1$, the exponent of the singularities vanishes, and the Bessel rule $\sum_{n \in \mathbb Z} J^2_{\|n\|}(z) \equiv 1$ leads $G_{tun} $ to be independent of the spin-gate parameters $V_{g,s}^0$ and $\omega$. This can be understood considering the electrons incoming towards the tunneling point. A time-dependent spin gate voltage locally changes the distributions of the incoming electrons into strongly non-equilibrium distributions $f_{S/D}(E) \rightarrow \sum_{n \in \mathbb Z} J^2_{\|n\|}(z) f(E\mp eV/2-n \hbar \omega)$, where all the harmonics $\hbar \omega$ related to the gate ac frequency appear \[see inset in Fig.\[Fig2\]\]. For a non interacting system the tunneling current is given by $I_{\gamma^2} \sim \int \! R \, [f_S(E)-f_D(E)] dE$, where $R \propto \gamma^2$ is the energy independent ‘reflection coefficient’ induced by the tunneling term. Thus, although $f_S$ and $f_D$ are affected by the ac gate voltage, the integral is not. In contrast, when electron-electron interaction is included, $R$ acquires an effective energy dependence, with a singular behavior at the Fermi energy [@glazman]. The cusps thus emerge whenever the energy difference between the incoming states vanishes, providing a hallmark of electron interaction in QSHE edge states. We observe that previous works concerning time-dependent impurities [@feldman-gefen; @chamon; @komnik] have discussed the effect of a time-dependent [magnitude]{} of the BS term, in systems where FS cannot affect dc transport. In contrast, here we have shown that in QSHE edge states an ac FS term induces a time-dependence in the [phase]{} of the ‘BS’, even when the magnitude BS term is time-independent. This difference implies important conceptual consequences. First, even for a monocromatic ac gate voltage, all harmonics $n \omega$ appear in the tunneling current, thus broadening to a lower frequency range the possibility to observe resonances with the dc voltage ($\omega=\omega_{SD}/n$). Second, it provides an additional parameter, $V_{g,s}^0$, to modify $G_{tun}$ in a non-monotonous way. In conclusion, we have shown that two gate voltages applied at a QPC of a QSHE system lead to two types of FS processes: The first one corresponds to the usual charge gate coupling and has no effect on dc transport, just like the FS term of a single impurity in a quantum wire. The second one, stemming from the helical nature of the QSHE edge states, consists of a spin gate coupling, and does affect dc transport. Indeed an ac spin gate leads to the cusp pattern in the dc tunneling conductance Eq.(\[PAT\]), shown in Fig.\[Fig2\]. We have also demonstrated that such pattern is a signature of electron-electron interaction, for it disappears in the non-interacting case. The obtained tunneling current is reminiscent of the photon-assisted tunneling. In QSHE, however, it is the spin degree of freedom of the tunneling electron to be controlled by the ac gate voltages. This feature, combined with the attractive interaction of the spin sector, represents a unique unconventional effect that is not present in other 1D systems. For typical values $l_g \sim 100 {\rm nm}$, $v \sim 5 \cdot 10^{5} {\rm m/s}$, and for $\omega \sim 1{\rm GHz}$, $V_{g,s} \sim {\rm m eV}$ the cusps are located at bias voltages $eV \sim n \, \mu{\rm eV}$ ($n \in \mathbb Z$), and the observation of the effect is within reach of experiments.\ We greatly acknowledge fruitful discussions with B. Trauzettel, C. Brüne, F. Taddei, and financial support by the VIGONI Program of Ateneo Italo-Tedesco. [77]{} A. Yacoby et [al.]{}, Phys. Rev. Lett. [77]{}, 4612 (1996); O. M. Auslaender et [al.]{}, Science [308]{}, 88 (2005). M. Bockrath et [al.]{}; Nature [397]{}, 598 (1999); N. Y. Kim et [al.]{}, Phys. Rev. Lett. [99]{}, 036802 (2007). A. M. Chang, Rev. Mod. Phys. [75]{}, 1449 (2003); X. G. Wen, Phys. Rev. B [41]{}, 12838 (1990). T. Giamarchi, [Quantum Physics in One Dimension]{} (Oxford University Press, Oxford, 2004); V. V. Desphande, et [al.]{}, Nature [464]{}, 209 (2010). M. P. A. Fisher, and L. I. Glazman, in [Mesoscopic-Electron-Transport]{}, Ed. by L. L. Sohn et [al.]{}, Proc. NATO ASI Series, Kluwer Ac. Publ.Dordrecht, (1997). C. L. Kane, and M. P. A. Fisher, Phys. Rev. B [46]{}, 15233 (1992). J. von Delft, and H Sch[ö]{}ller, Ann. Phys. (Leipzig), [4]{}, 225 (1998). B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science [314]{}, 1757 (2006); M. König, et [al.]{} Science [318]{}, 766 (2006). C. Wu, B. A. Bernevig, and S-C. Zhang, Phys. Rev. Lett. [96]{}, 106401 (2006); C.Y. Hou, E-A. Kim, and C. Chamon, Phys. Rev. Lett. [102]{}, 076602 (2009); J. C. Y. Teo and C. L. Kane, Phys. Rev. B [79]{}, 235321 (2009). D.E. Feldman and Y. Gefen, Phys. Rev. B [67]{}, 115337 (2003). P. Sharma, and C. Chamon, Phys. Rev. Lett. [87]{}, 096401 (2001); Phys. Rev. B [68]{}, 035321 (2003). T. L. Schmidt, and A. Komnik, Phys. Rev. B [76]{}, 035113 (2007). C. X. Liu, J. C. Budich, P. Recher, and B. Trauzettel, Phys. Rev. B [83]{}, 035407 (2011). F. Dolcini, Phys. Rev B [83]{}, 165304 (2011). Here we limit our study to spin-preserving tunneling; see Refs.[@liu; @mio] for spin-flip effects. G. Platero, and R. Aguado, Phys. Rep. [395]{}, 1 (2004). M. H. Pedersen, and M. Büttiker, Phys. Rev. B [58]{}, 12993 (1998). L. V. Keldysh, Zh. Eksp. Teor. Fiz. [47]{}, 1515 (1964) \[Sov. Phys. JETP [20]{}, 1018 (1965)\].
--- abstract: \| Observations of the neutral Hydrogen () 21-cm signal hold the potential of allowing us to map out the cosmological large scale structures (LSS) across the entire post-reionization era ($z \leq 6$). Several experiments are planned to map the LSS over a large range of redshifts and angular scales, many of these targeting the Baryon Acoustic Oscillations. It is important to model the distribution in order to correctly predict the expected signal, and more so to correctly interpret the results after the signal is detected. In this paper we have carried out semi-numerical simulations to model the distribution and study the power spectrum $P_{\HI}(k,z)$ across the redshift range $1 \le z \le 6$. We have modelled the bias as a complex quantity $\tilde{b}(k,z)$ whose modulus squared $b^2(k,z)$ relates $P_{\HI}(k,z)$ to the matter power spectrum $P(k,z)$, and whose real part $b_r(k,z)$ quantifies the cross-correlation between the and the matter distribution. We study the $z$ and $k$ dependence of the bias, and present polynomial fits which can be used to predict the bias across $0 \le z \le6$ and $0.01 \le k \le 10 \, {\rm Mpc}^{-1}$. We also present results for the stochasticity $r=b_r/b$ which is important for cross-correlation studies. author: - \| Debanjan Sarkar$^{1}$[^1],Somnath Bharadwaj$^{1, 2}$[^2], Anathpindika S.$^{2}$\ $^1$Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India\ $^2$Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India title: 'Modelling the post-reionization neutral Hydrogen () bias' --- methods: statistical, cosmology: theory, large scale structures, diffuse radiation Introduction ============ Since its predictions by H. van der Hulst in 1944, the neutral hydrogen () 21-cm line has become a work horse for observational cosmology. One of the direct applications of the 21-cm emission is to measure the rotation curve of galaxies [e.g. see @begum-chengalur-karachentsev05 and references therein], which is one of the most direct probes of dark matter. The 21-cm emission is also a very reliable probe of the content of the galaxies for the nearby universe. Surveys like the HI Parkes All-Sky Survey [HIPASS; @zwaan-meyer05], the HI Jodrell All-Sky Survey [HIJASS; @lang-boyce-kilborn2003], the Blind Ultra-Deep HI Environmental Survey [BUDHIES; @jaffe-poggianti2012] and the Arecibo Fast Legacy ALFA Survey [ALFALFA; @martin-giovanelli12] aim to measure the 21-cm emission from individual galaxies at very low redshifts ($z<<1$) to quantify the distribution in terms of the mass function and the density parameter $\Omega_{HI}$. These studies also help us in understanding the effects of different environments in which resides. This method fails at higher redshifts where we cannot identify individual galaxies. Here the cumulative flux of the 21-cm radiation from high redshift emitters appears as a diffused background radiation. Measurements of the intensity fluctuations in this diffused background provide us a three dimensional probe of the large scale structures over a large redshift range in the post-reionization era ($z \lesssim 6$) [@bharadwaj-nath-sethi01; @bharadwaj01; @bharadwaj-pandey03; @bharadwaj-srikant04]. The advantage of studying the 21-cm emission in the post-reionization era lies in the fact that the modulation of the signal due to complicated ionizing fields is less and the 21-cm power spectrum is directly proportional to the matter power spectrum which enhances its usefulness as a probe of cosmology [@wyithe-loeb09]. This technique provides an independent estimate of various cosmological parameters [@loeb-wyithe08; @bharadwaj-sethi-saini09]. The Baryon Acoustic Oscillations (BAO) are embedded in the power spectrum of 21-cm intensity fluctuations at all redshifts and the comoving scale of BAO can be used as a standard ruler to constrain the evolution of the equation of state for dark energy [@wyithe-loeb-geil08; @chang08; @seo-dodelson10]. Several existing and upcoming experiments are planned to map this radiation at various redshifts. A number of methods also have been proposed or implemented to recover the informations from the signal faithfully. @lah-chengalur-briggs07 [@lah-pracy-chengalur09] used Giant Metrewave Radio Telescope (GMRT) observations at $z\sim0.4$ to co-add the 21-cm signals (“stacking") from galaxies with known redshifts in order to increase the signal to noise ratio and infer the average mass of the galaxies. This technique has been extended a little to $z\sim0.8$ by studying the cross-correlation between 21-cm intensity maps and the large scale structures traced by optically selected galaxies to constrain the amplitude of the fluctuations [@chang-pen-bandura10; @masui-switzer-banavar13]. @ghosh-bharadwaj-ali-chengalur11b [@ghosh-bharadwaj-ali-chengalur11a] devised a method to characterize and subtract the foreground contaminations in order to recover the signal and used $610$ MHz ($z=1.32$) GMRT observations to set an upper limit on the amplitude of the 21-cm signal. @kanekar-sethi-dwarkanath16 extended the signal stacking technique further to $z \sim 1.3$ using GMRT observations and obtained an upper limit on the average 21-cm flux density. A number of 21-cm intensity mapping experiments like Baryon Acoustic Oscillation Broadband and Broad-beam [BAOBAB; @pober-parsons13], BAO from Integrated Neutral Gas Observations [BINGO; @battye-brown12], Canadian Hydrogen Intensity Mapping Experiment [CHIME; @bandura14], the Tianlai project [@tianlai-chen12], Square Kilometre Array 1-MID/SUR [SKA1-MID/SUR; @bull-camera15-BAO-with-SKA] have been planned to cover the intermediate redshift range $z \sim 0.5 \,- \, 2.5$ where their primary goal is to measure the scale of BAO, particularly around the onset of acceleration at $z \sim 1$. Recent studies suggest that observations of 21-cm fluctuations on small scales, with SKA1, can constrain the sum of the neutrino masses [@villaescusa-navarro-bull-Viel15; @pal-guha-sarkar16]. Observations with SKA1-MID can also test different scalar field dark energy models [@hossain-thakur-guhasarkar-sen16]. @ali-bharadwaj14 and @bharadwaj-sarkar-ali15 present theoretical estimates for intensity mapping at $z \sim 3.35$ with the Ooty Wide Field Array (OWFA), while @chatterjee16 and @santos-bull-alonso15 present similar estimates for the upcoming uGMRT and SKA2 respectively. The main observable of the 21-cm intensity mapping experiments is the 21-cm brightness temperature fluctuation power spectrum $P_{T}(k,z)$. This can be interpreted in terms of the power spectrum $P_{\HI}(k,z)$ as, $$P_{T}(k,z)=\bar{T}_{\HI}^2(z)P_{\HI}(k,z) \,, \label{eq:1}$$ where $$\bar{T}_{\HI}(z) \simeq 4.0 {\rm mK} (1+z)^2 \left(\frac{\Omega_{gas}(z)}{10^{-3}}\right) \left(\frac{\Omega_{b}h^2}{0.02}\right) \left(\frac{H_0}{H(z)}\right) \, \label{eq:2}$$ is the mean brightness temperature of the 21-cm emission [@ali-bharadwaj14]. Here, $\Omega_{gas}(z)$ is the density parameter for the neutral gas which can be expressed in terms of the density parameter $\Omega_{\HI}(z)$ through a suitable conversion, all other symbols have their usual meaning. We can interpret the power spectrum $P_{\HI}(k,z)$ in terms of the matter power spectrum $P(k,z)$ as $$P_{\HI}(k,z)=b^2(k,z)P(k,z)\,, \label{eq:3}$$ under the assumption that the traces the underlying matter distribution with a linear bias $b(k,z)$ which quantifies the clustering of the relative to that of the total matter distribution. It is clear that we will need independent estimates of both $\Omega_{\HI}(z)$ and $b(k,z)$ in order to interpret the observable $P_{T}(k,z)$ in terms of the underlying matter power spectrum $P(k,z)$. Further, the amplitude of the expected signal $P_{T}(k,z)$ is very sensitive to both $\Omega_{\HI}(z)$ and $b(k,z)$, and it is crucial to have prior estimates of these parameters in order to make precise predictions for the upcoming experiments [@hamsa-tirthankar-refregier15] . Several measurements of $\Omega_{\HI}(z)$ have been carried out both at low and high redshifts. At low redshifts ($z \sim 1$ and lower) we have measurements of $\Omega_{\HI}$ from galaxy surveys [@zwaan-meyer05; @martin-papastergis-giovanellihaynes10; @delhaize-meyer13], Damped Lyman-$\alpha$ Systems (DLAs) observations [@rao-turnshek06; @meiring-tripp-prochaska11] and stacking [@lah-chengalur-briggs-colless07; @rhee-zwaan-briggs13]. At high redshifts ($1 < z < 6$), measurements of $\Omega_{\HI}$ come from the studies of the Damped Lyman-$\alpha$ Systems (DLAs) [@prochaska-wolfe09; @noterdaeme-petitjean-carithers12]. Earlier observations indicated the content of the universe to remain almost constant with $\Omega_{\HI} \sim 10^{-3}$ over the entire redshift range $z<6$ [@lanzetta-wolfe95; @storrielomb96; @rao-turnshek00; @peroux-mcmahon03]. However, some recent studies [@sanchez-ramirez-ellison-prochaska15] indicate that $\Omega_{\HI}$ evolves significantly from $z\sim2 \,{\rm to}\, z\sim5$, although the redshift evolution of $\Omega_{\HI}$ is debatable in the intermediate redshift range, $z=0.1 \, - \, 1.6$ [@sanchez-ramirez-ellison-prochaska15]. A combination of low redshift data with high redshift observations shows that $\Omega_{\HI}$ decreases almost by a factor of $4$ between $z=5$ to $z=0$ [@sanchez-ramirez-ellison-prochaska15; @crighton-murphy-prochaska15]. @martin-giovanelli12 have used selected galaxies to estimate the bias $b(k)$ at $z\sim 0.06$. Intensity mapping experiments have measured the product $\Omega_{\HI} \, b \, r$ [@chang-pen-bandura10; @masui-switzer-banavar13] by studying the cross-correlation of the intensity with optical surveys ($r$ here is the cross-correlation coefficient or “stochasticity”) while @switzer-masui-bandura-calin13 have measured the combination $\Omega_{\HI} \, b$, all these measurements being at $z <1$. We do not, at present, have any observational constraint on the bias $b(k,z)$ at redshifts $z>1$. It is therefore important to model $b(k,z)$ as an useful input for the future 21-cm intensity mapping experiments. @marin-gnedin10 have developed an analytic framework for calculating the large scale bias $b(k,z)$ and studying its redshift evolution using a relation between the mass $M_{\HI}$ and the halo mass $M_h$ motivated by observations of the $z=0$ mass function. Analytic techniques, however are limited in incorporating the effects of nonlinear clustering. In an alternative approach, @bagla10 have proposed a semi-numerical technique which utilizes a prescription to populate in the halos identified from dark matter only simulations. The same approach has also been used by @khandai-sethi-dimatteo11 and @tapomoy-mitra-majumder12 to study the power spectrum and the related bias. @navarro-viel-datta-choudhury14 have used high-resolution hydrodynamical N-body simulations along with three different prescriptions for distributing the . @seehars-paranjape15 have proposed a semi-numerical model for simulating large maps of the intensity distribution at $z<1$. The analytic and semi-numerical studies carried out till date are all limited in that each study is restricted to a few discrete redshifts. In a recent paper @hamsa-tirthankar-refregier15 have compiled all the available results for the bias and interpolated the values to cover the redshift range $z=0 \,- \, 3.4$. Their study is restricted to large scales where it is reasonable to consider a constant $k$ independent bias $b(z)$. We do not, at present, have a comprehensive study which uses a single technique to study the bias over a large $z$ and $k$ range. In this work, we study : (i) the evolution of the power-spectrum $P_{\HI}(k,z)$ across the redshift range $1 \le z \le 6$ by using N-Body simulations coupled with the third distribution model of @bagla10, (ii) the redshift variation of the complex bias $\tilde{b}(k,z)$ whose modulus squared, $b^{2}(k,z)$, relates $P_{\HI}(k,z)$ to the matter power-spectrum $P(k,z)$, and whose real component $b_r(k,z)$ quantifies the cross-correlation between the and the total matter distribution, and (iii) the spatial(rather, $k$) dependence of the bias and present polynomial fits which can be used to predict its variation over a large $z$ and $k$ range. We note that the entire analysis of this paper is restricted to real space [i.e.]{} it does not incorporate redshift space distortion arising due to the peculiar velocities. We plan to address the effect of peculiar velocities in future. An outline of the paper follows. In section \[sec:simulations\], we briefly describe the method of simulating the distribution. In section \[sec:results\], we present the results of our simulations. Section \[subsec:fitting\] contains the details of the polynomial fitting for the joint $k$ and $z$ dependence of the biases. The values of the fitting parameters are tabulated in Appendix \[sec:appendix\]. We finally summarize all the findings and discuss a few current results on the basis of our simulations in section \[sec:summary\]. We use the fitting formula of @eisenstein-hu99 for the $\Lambda$-CDM transfer function to generate the initial matter power spectrum. The cosmological parameter values used are as given in @planck-collaboration13. Simulating the distribution {#sec:simulations} =========================== We follow three main steps to simulate the post-reionization 21-cm signal. In the first step we use a Cosmological N-body code to simulate the matter distribution at the desired redshift $z$. Here we have used a Particle Mesh (PM) N-body code developed by @bharadwaj-srikant04. This ‘gravity only’ code treats the entire matter content as dark matter and ignores the baryonic physics. The simulations use $[1,072]^3$ particles in a $[2,144]^3$ regular cubic grid of spacing $0.07 \, {\rm Mpc}$ with a total simulation volume (comoving) of $[150.08 \, {\rm Mpc}]^3$. The simulation particles all have mass $10^{8} \, M_{\odot}$ each. We have used the standard linear $\Lambda$-CDM power spectrum to set the initial conditions at $z=125$, and the N-body code was used to evolve the particle positions and velocities to the redshift $z$ at which we desire to simulate the signal. We have considered $z$ values in the interval $\Delta z =0.5$ in the range $z=1$ to $z=6$. In the next step we employ the Friends-of-Friends (FoF) algorithm [@davis85] to identify collapsed halos in the particle distribution produced as output by the N-body simulations. For the FoF algorithm we have used a linking length of $l_f=0.2$ in units of the mean inter-particle separation, and furthermore, we require a halo to have at least ten particles. This sets $10^{9} \, M_{\odot}$ as the minimum halo mass that is resolved by our simulation. We also verify that the mass distribution of halos so detected are in good agreement with the theoretical halo mass function [@jenkins-frenk01; @sheth-tormen02] in the mass range $10^9 \leq M \leq 10^{13} \, M_{\odot}$. Our halo mass range is well in keeping with a recent study [@kim-wyithe-baugh-lagos16] which shows that at $z \ge 0.5$ a dark matter halo mass resolution better than $\sim 10^{10} \, h^{-1} \, M_{\odot}$ is required to predict 21-cm brightness fluctuations that are well converged. The observations of quasar (QSO) absorption spectra suggest that the diffuse Inter Galactic Medium (IGM) is highly ionized at redshifts $z \le 6$ [@becker-fan01; @fan-carilli06; @fan-strauss06]. This redshift range where the hydrogen neutral fraction has a value $x_{\rm \HI}<10^{-4}$ is referred to as the post-reionization era. Here the bulk of the resides within dense clumps (column density $N_{\HI} \geq 2\times10^{20}{\rm cm^{-2}}$) which are seen as the Damped Lyman-$\alpha$ systems (DLAs) found in the QSO absorption spectra [@wolfe05]. Observations indicate that the DLAs contain almost $\sim 80 \,\%$ of the total , [@storrie-lomb00; @prochaska-herbertfort-wolfe05; @zafar-peroux-popping-milliard13] and they are the source of the 21-cm radiation in the post-reionization era. While the origin of the high-$z$ DLAs is still not very well understood, it is found (eg. [@haehnelt00]) that it is possible to correctly reproduce many of the observed DLAs properties if it is assumed that the DLAs are associated with galaxies. From the cross-correlation study between DLAs and Lyman Break Galaxies (LBGs) at $z\sim 3$, @cooke-wolfe-gawiser06 showed that the halos with mass in the range $10^{9}<M_h<10^{12}\,M_{\odot}$ can host the DLAs. In this work we assume that in the post-reionization era is entirely contained within dark matter halos which also host the galaxies. In the third step of our simulation we populate the halos identified by the FoF algorithm with . Here we assume that the mass $M_{\HI}$ contained within a halo depends only on the halo mass $M_h$, independent of the environment of the halo. At the outset, we expect the mass to increase with the size of the halo. However, observations at low $z$ indicate that we do not expect the large halos, beyond a certain upper cut-off halo mass $M_{\rm max}$, to contain a significant amount of . For example, very little is found in the large galaxies which typically are ellipticals and in the clusters of galaxies [eg. see @serra-oosterloo-morganti12 and references therein]. Further, we also do not expect the very small halos, beyond a certain lower cut-off halo mass $M_{\rm min}$, to contain significant mass. The amount of gas contained in small halos $(M_h <M_{\rm min} )$ is inadequate for it to be self shielded against the ionizing radiation. Based on these considerations, @bagla10 have introduced several schemes for populating simulated halos to simulate the post-reionization distribution. In our work we have implemented one of the schemes proposed by @bagla10 to populate the halos. This uses an approximate relation between the virialized halo mass and the circular velocity as a function of redshift $$M_h \simeq 10^{10}\left(\frac{v_{\rm circ}}{60{\rm km/s}}\right)^3 \left(\frac{1+z}{4}\right)^{-\frac{3}{2}}M_{\odot} \,. \label{eq:5}$$ It is assumed that the neutral gas in the halos will be able to shield itself from the ionizing radiation only if the halo’s circular velocity exceeds $v_{\rm circ}\sim30$ km/s, which sets the lower mass limit of the halos $M_{\rm min}$. The upper mass cutoff $M_{\rm max}$ is decided by taking the upper limit of the circular velocity $v_{\rm circ}\sim200$ km/s, beyond which the content falls off. @pontzen08 have shown that halos more massive than $10^{11}$ $M_{\odot}$ do not contain a significant amount of neutral gas. \[c\]\[c\]\[1.4\]\[0\][Mpc]{} \[c\]\[c\]\[1.2\]\[0\] \[c\]\[c\]\[1.2\]\[0\] \[c\]\[c\]\[1.2\]\[0\] \[c\]\[c\]\[1.8\]\[0\] \[c\]\[c\]\[1.8\]\[0\] \[c\]\[c\]\[1.8\]\[0\] In our work we have used the third scheme proposed by @bagla10 where the mass in a halo is related to $M_h$ as $$M_{\rm \HI}(M_h) = \left\{ \begin{array}{l l} f_3\frac{M_h}{1+\left(\frac{M_h}{M_{\rm max}}\right)} & \quad \text{if $M_{\rm min}\leqslant M_h$} \\ 0 & \quad \text{otherwise}\\ \end{array} \right. \,. \label{eq:6}$$ According to this scheme only halos with mass greater than $M_{\rm min}$ host . The mass of a halo increases proportionally with the halo mass $M_{h}$ for $M_h \ll M_{\rm max}$. However, the mass saturates as $M_h \sim M_{\rm max}$, and $ M_{\rm \HI}$ attains a constant upper limit $ M_{\rm \HI} =f_3 M_{\rm max}$ for $M_h \gg M_{\rm max}$. The free parameter $f_3$ determines the total amount of in the simulation volume, and its value is tuned so that it produces the desired value of the density parameter $\Omega_{\rm \HI}\sim10^{-3}$. Our entire work here deals with the dimensionless density contrast $\delta \rho_{\rm HI}/ \bar{\rho}_{\rm HI}$ which is insensitive to the choice of $f_3$. We have run five statistically independent realizations of the simulation. These five independent realizations were used to estimate the mean value and the variance for all the results presented in this paper. The simulations require a large computation time, particularly the FoF which takes $\sim 10$ days for a single realization on our computers and this restricts us to run only five independent realizations. The computation time increases at lower redshifts, and we have restricted our simulations to $z \ge 1$. As mentioned earlier, our simulations have a halo mass resolution of $M_h = 10^9M_{\odot}$, but eq. \[eq:6\] shows that the mass of the smallest possible halo that retain falls as $M_{\rm min} \propto (1+z)^{-\frac{3}{2}}$ and so, $M_{\rm min}= 10^{9} \, M_{\odot}$ at $z$=3.5, [i.e.]{}, the minimum resolvable halo mass, $M_{\rm min}$, falls below our mass resolution of 10$^{9}$ M$_{\odot}$ at $z > 3.5$. At these redshifts therefore, our simulations cannot detect halos less massive than this threshold and which according to the model proposed by @bagla10, are also likely to host some . To study the effects of these missing low mass halos we have run a high resolution simulation (referred to as HRS) with $[2,144]^3$ particles in a $[4,288]^3$ regular cubic grid of spacing $0.035 \, {\rm Mpc}$, the total simulation volume remaining the same as earlier. The lower mass limit for the halo mass is $10^{8.1} \, M_{\odot}$ in the HRS, well below $M_{\rm min}$ in the entire redshift range. The HRS requires considerably larger computational resources compared to the other simulations, and we have run only a single realization for which we have compared the results with those from the earlier lower resolution simulations. Results {#sec:results} ======= Figure \[fig:density\_dist\] provides a visual impression of how the matter, the halos and the are distributed at different stages of the evolution. We show this by plotting the density contrasts $\delta(\x,t)=\delta\rho(\x,t)/\bar{\rho}(t)$ at three different redshifts, viz. $6,\, 3 \, \rm{and} \, 1$. It can be seen that the cosmic web is clearly visible in all three components even at the highest redshift $z=6$, though it is somewhat diffused for the matter at this redshift. Observe that the basic skeleton of the cosmic web is nearly the same for all the three components, and the basic skeleton does not change significantly with redshift. We see that for all the three components the cosmic web become more prominent with decreasing redshift. Considering the matter first, the density contrast grows with decreasing redshifts due to gravitational clustering. The halos are preferentially located at the matter density peaks, and it is evident that the halos have a higher density contrast. We see that the structures in the halo distribution are more prominent compared to the matter, particularly at high redshifts. The closely follows the halo distribution at $z=6$. However, in contrast to the matter and halo distribution, the distribution shows a much weaker evolution with $z$. It is possible to understand this in terms of the model for populating the halos with HI. We know that the halo masses increase as gravitational clustering proceeds. According to our population model, however, the mass remains fixed once the halo mass exceeds a critical value. \[c\]\[c\]\[1.2\]\[0\][${\Delta^{2}_{\HI}(k)}$]{} \[c\]\[c\]\[1.2\]\[0\][${\Delta^{2}(k)}$]{} \[c\]\[c\]\[1.2\]\[0\][$\Delta^{2}_{c}(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$k\,{\rm Mpc}^{-1}$]{} \[c\]\[c\]\[1.2\]\[0\][$z=1$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$2$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$3$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$4$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$5$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$6$]{} \[c\]\[c\]\[1\]\[0\][$10^{-2}$]{} \[c\]\[c\]\[1\]\[0\] \[c\]\[c\]\[1\]\[0\] We quantify the matter and the distributions with the respective power spectra $P(k)$ and $P_{\HI}(k)$. We also quantify the cross correlation between the matter and the through the cross-correlation power spectrum $P_{c}(k)$. For all the three power spectra we consider the dimensionless quantity $\Delta^{2}(k)=k^{3}P(k)/2 \pi^{2}$, respectively shown in the three panels of Figure \[fig:powspecs\] for different values of the redshift $z\in[1,6]$. The five independent realizations of the simulation each provides a statistically independent estimate of the power spectrum. We have used these to quantify the mean and the standard deviation which we show in the figures. For clarity of presentation, the $\pm \, 1\, \sigma$ confidence interval is shown for $z=3$ only. The left panel of Figure \[fig:powspecs\] shows $\Delta^{2}(k)$ as a function of $k$ at different redshifts. The matter distribution, whose evolution is well understood (e.g. Chapter 15 of @peacock99) serves as the reference against which we compare the distribution at different stages of its evolution. It is evident that $\Delta^{2}(k)$ increases proportional to the square of the growing mode leaving the shape of the power spectrum unchanged at small $k$ or large length-scales where the predictions of linear theory hold (e.g. Chapter 16 of @peacock99). At small scales, where nonlinear clustering is important, the shape of $\Delta^{2}(k)$ changes with evolution and the growth is more than what is predicted by linear theory. Note that the different power spectra shown in this paper have all been calculated using a grid whose spacing is double of that used for the simulations. The turn over seen in $\Delta^{2}(k)$ at $k \,\sim \,10 \, {\rm Mpc}^{-1}$ is an artefact introduced by the smoothing at this grid scale. We have restricted the entire analysis of this paper to the range $k \le 10 \, {\rm Mpc}^{-1}$. The central panel of Figure \[fig:powspecs\] shows ${\Delta^{2}_{\HI}(k)}$ as a function of $k$ at different redshifts. We can clearly see that the evolution of $\Delta^{2}(k)$ and ${\Delta^{2}_{\HI}(k)}$ are quite different. At small $k$, we find that ${\Delta^{2}_{\HI}(k)}$ shows almost no evolution over the entire redshift range. We find this behaviour over the entire region where the matter exhibits linear gravitational clustering. We find that ${\Delta^{2}_{\HI}(k)}$ grows to some extent at $k>2 \, {\rm Mpc}^{-1}$ where non-linear effects are important. This growth, however, is smaller than the growth of the matter power spectrum. Further, we also see that the shape of ${\Delta^{2}_{\HI}(k)}$ closely resembles $\Delta^{2}(k)$ at small $k$, however the two differ at large $k$, and these differences are easily noticeable at $k>2 \, {\rm Mpc}^{-1}$. The right panel of Figure \[fig:powspecs\] shows $\Delta^{2}_{c}(k)$ as a function of $k$ at different redshifts. We see that the evolution of $\Delta^{2}_{c}(k)$ is intermediate to that of $\Delta^{2}(k)$ and ${\Delta^{2}_{\HI}(k)}$, it grows faster than ${\Delta^{2}_{\HI}(k)}$ but not as fast as $\Delta^{2}(k)$. All three power spectra have the same shape at small $k$, indicating that the traces the matter at large length-scales. At large $k$ the shape of $\Delta^{2}_{c}(k)$, however, differs from both $\Delta^{2}(k)$ and ${\Delta^{2}_{\HI}(k)}$ indicating differences in the small-scale clustering of the and the matter. Redshift surveys of large scale structures and numerical simulations reveal that the galaxies trace underlying matter over-densities with a possible bias [@bbks-bardeen-bond-kaiser-szalay86; @mo-white96; @dekel-lahav99]. In the post-reionization era, the association of the with the halos implies that the follows the matter density field with some bias. The bias function relates the density contrast to that of the matter. Here we assume that a linear relation holds between the Fourier components of the and the matter density contrasts $$\Delta_{\HI}(\k) = \tilde{b}(\k)\, \Delta(\k) \label{eq:7}$$ where, $\tilde{b}(\k)$ is the linear bias function or simply bias, which can, in general, be complex. The complex bias allows for the possibility that the Fourier modes $\Delta_{\HI}(\k)$ and $ \Delta(\k)$ can differ in both the amplitude and also the phase. The ratio of the respective power spectra $$b(k)=\sqrt{\frac{P_{\HI}(k)}{P(k)}} \,. \label{eq:8}$$ allows us to quantify $b(k)$ which is the modulus of the complex bias $\tilde{b}(k)$, and the ratio $$b_r(k)=\frac{P_{c}(k)}{P(k)} \,. \label{eq:9}$$ allows us to quantify $b_r(k)$ which is the real part of the complex bias $\tilde{b}(k)$. With both $b(k)$ and $b_r(k)$ at hand, we can reconstruct the entire complex bias $\tilde{b}(k)$. One is mainly interested in the modulus $b(k)$ which allows us to interpret the power spectrum in terms of the underlying matter power spectrum. However, the real part of the bias $b_r(k)$ is the relevant quantity if one is considering the cross-correlation of the with either the matter distribution or with some other tracer of the matter distribution like Lyman-$\alpha$ forest [@tapomoy-somnath-trc-kanan11; @tapomoy-kanan15] or galaxy surveys [@chang-pen-bandura10; @masui-switzer-banavar13; @cohn-white-chang-holder15]. \[c\]\[c\]\[1.2\]\[0\][$b(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$b(k)$ and $b_{r}(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$b(z)$]{} \[c\]\[c\]\[1\]\[0\][$b(k)\:\:\:$]{} \[c\]\[c\]\[1\]\[0\][$b_r(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$z$]{} \[c\]\[c\]\[1.2\]\[0\][$k\, {\rm Mpc}^{-1}$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$5$]{} \[c\]\[c\]\[1.2\]\[0\][$\quad\:\:\:$$6$]{} \[c\]\[c\]\[1.2\]\[0\][$\:$$z=4$]{} \[c\]\[c\]\[1\]\[0\][Fractional difference in $\%$]{} \[c\]\[c\]\[1.2\]\[0\][$z=1$]{} \[c\]\[c\]\[1.2\]\[0\][$\:\:\:\quad2$]{} \[c\]\[c\]\[1.2\]\[0\][$\:\:\:\quad3$]{} \[c\]\[c\]\[1.2\]\[0\][$\:\:\:\quad4$]{} \[c\]\[c\]\[1.2\]\[0\][$\:\:\:\quad5$]{} \[c\]\[c\]\[1.2\]\[0\][$\:\:\:\quad6$]{} \[fig:bias\] The left panel of Figure \[fig:bias\] shows the behaviour of $b(k)$, the modulus of $\tilde{b}(k)$, as a function of $k$ at six different redshifts. We also show the $5 \, \sigma$ confidence interval at three different redshifts. The relatively small errors indicate that the results reported here are statistically representative values. We see that the value of $b(k)$ decreases with decreasing redshift. Further, the $k$ dependence is also seen to evolve with redshift. In all cases, we have a constant $k$ independent bias at small $k$ and the bias shoots up rapidly with $k$ at large $k$ ($\ge 4 \, {\rm Mpc}^{-1}$). However, for high redshifts $(z \ge 3)$ , $b(k)$ increases monotonically with $k$ whereas we see a dip in the values of $b(k)$ at $k \sim 2\, {\rm Mpc}^{-1}$ for $z < 3$. Interestingly, the $k$ range where we have a constant $k$ independent bias is maximum at the intermediate redshift $z = 3$ where it extends to $k \le 1 \, {\rm Mpc}^{-1}$, and it is minimum ($k \le 0.2 \, {\rm Mpc}^{-1}$) at the highest and lowest redshifts ($z=6,1$) whereas it covers $k \le 0.4 \, {\rm Mpc}^{-1}$ at the other redshifts $(z=2,4,5)$. The central panel of Figure \[fig:bias\] shows both $b(k)$ and $b_r(k)$ which is the real part of $\tilde{b}(k)$. The two quantities $b(k)$ and $b_r(k)$ show similar $k$ dependence. Both $b(k)$ and $b_r(k)$ will be equal if the bias $\tilde{b}({k})$ is a real quantity. We see that this is true at small $k$ where both quantities have nearly constant values independent of $k$. However, we find a $k$ independent bias $b_r(k)$ for a smaller range of $k$, in comparison to $b(k)$. The two quantities $b(k)$ and $b_r(k)$ differ at larger $k$, and the differences increase with increasing $k$. The difference between $b(k)$ and $b_r(k)$ is seen to increase with decreasing redshift. Also the $k$ value where these differences become significant shifts to smaller $k$ with decreasing redshift. \[c\]\[c\]\[1.2\]\[0\][$b(k)$ and $b_{r}(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$z$]{} \[c\]\[c\]\[1\]\[0\][$k=0.065\, \rm{Mpc}^{-1}$]{} \[c\]\[c\]\[0.8\]\[0\][$k=0.001\, \rm{Mpc}^{-1}$$\qquad\qquad\qquad\qquad\qquad$]{} \[c\]\[c\]\[1\]\[0\][$k=0.45\, \rm{Mpc}^{-1}$]{} \[c\]\[c\]\[1\]\[0\][$k=2.2\, \rm{Mpc}^{-1}$]{} \[c\]\[c\]\[0.8\]\[0\][$b(k)$ fitting$\quad\quad\;$]{} \[c\]\[c\]\[0.8\]\[0\][$b_{r}(k)$ fitting$\quad\quad\quad$]{} \[c\]\[c\]\[0.8\]\[0\][$b(k)$ simulations]{} \[c\]\[c\]\[0.8\]\[0\][$b_{r}(k)$ simulations]{} As already mentioned in Section \[sec:simulations\], for $z>3.5$, $M_{\rm min}$ (eq. \[eq:5\]) has a value that is smaller than $10^{9} \, M_{\odot}$ which is the smallest mass halo resolved by our simulations. Imposing a fixed lower halo mass limit of $10^{9} \, M_{\odot}$ will, in principle, change the bias $\tilde{b}(k)$ in comparison to the actual predictions of the halo population model proposed by @bagla10, and we have run higher resolution simulation in order to quantify this. It is computationally expensive to run several realizations of simulations with a smaller mass resolution, so we have run a single realization with a halo mass resolution of $10^{8.1} \, M_{\odot}$ which is well below $M_{\rm min}$ over the entire redshift range of our interest. The right panel of Figure \[fig:bias\] shows the percentage difference in the values of $b(k)$ computed using the low and the high resolution simulations. We find that the difference is minimum for $z=4$ and maximum for $z=6$ where we expect a larger contribution from the smaller halos. For $k<1.0\, \rm{Mpc}^{-1}$, the difference is $5 \, - \, 8\%$ at $z=4$ and $8 \, - \, 13\%$ at $z=6$. Beyond $1.0\, \rm{Mpc}^{-1}$, the difference increases but it is well within $20\%$ for redshifts $4\, \rm{and} \,5$ and less than $30 \%$ for redshift $6$. These differences are relatively small given our current lack of knowledge about how the is distributed at the redshifts of interest. It is therefore well justified to use the simulations with a fixed lower mass limit of $10^{9} \, M_{\odot}$ for the entire redshift range considered in this paper. Figure \[fig:diff\_bias\] shows the redshift evolution of $b(z)$ and $b_{r}(z)$ at three representative $k$ values. At $k=0.065 \, {\rm Mpc}^{-1}$ (left panel) which is in the linear regime we cannot make out the difference between $b(z)$ and $b_{r}(z)$ and this indicates that $\tilde{b}(z)$ is purely real. We find that the bias $b(z)$ has a value that is slightly less than unity at $z=1$ indicating that the is slightly anti-biased at this redshift. The bias increases nearly linearly with $z$ and it has a value $b(z) \approx 3$ at $z=6$. At $k=0.45 \, {\rm Mpc}^{-1}$ (central panel) where we have the transition from the linear to the non-linear regime we find that $b(z)$ is slightly larger than $b_r(z)$. Both $b(z)$ and $b_r(z)$ show a $z$ dependence very similar to that in the linear regime. At $k=2.2 \, {\rm Mpc}^{-1}$ (right panel) which is in the non-linear regime we find that $b_{r}(z)$ is appreciably smaller than $b(z)$, and the difference is nearly constant over the entire $z$ range. This indicates that the bias $\tilde{b}(z)$ is complex in the non-linear regime. Further, we see that the relative contribution from the imaginary part of $\tilde{b}(z)$ increases with decreasing $z$. The value of $b_{r}(z)$ is less than unity for $z \le 2$, whereas this is so only in the range $z \le 1.5$ for $b(z)$. The redshift dependence of the bias is much steeper as compared to the linear regime, and we have a larger value of $b(z) \approx 5$ at $z=6$. We find a nearly parabolic $z$ dependence in the no-linear regime as compared to the approximately linear redshift dependence found at smaller $k$. At all the three $k$ values we have fitted the redshift evolution of $b(z)$ and $b_{r}(z)$ with a quadratic polynomial of the form $b_{0} + b_{1}z + b_{2}z^{2}$. We find that the polynomials give a very good fit to the redshift evolution of the simulated data (Figure \[fig:diff\_bias\]). We also find that the quadratic term $b_2$ is much larger at $k=2.2 \, {\rm Mpc}^{-1}$ as compared to the two smaller $k$ values. Based on these results, we have carried out a joint fitting of the $k$ and $z$ dependence of the bias, the details of which are presented in the next section. Fitting the bias {#subsec:fitting} ---------------- \[c\]\[c\]\[1.2\]\[0\][$b(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$b_{r}(k)$]{} \[c\]\[c\]\[1.2\]\[0\][$k\, {\rm Mpc}^{-1}$]{} \[c\]\[c\]\[1.2\]\[-90\][$b_1\quad \quad$]{} \[c\]\[c\]\[1.2\]\[-90\][$b_0\quad \quad$]{} \[c\]\[c\]\[1.2\]\[-90\][$b_2\quad \;$]{} \[c\]\[c\]\[1.2\]\[-90\][$b_3\quad \;$]{} \[c\]\[c\]\[1.2\]\[-90\][$b_4\quad$]{} \[c\]\[c\]\[1.2\]\[0\][$z$]{} \[c\]\[c\]\[0.7\]\[0\][$b_m\:\:$]{} \[c\]\[c\]\[0.7\]\[0\][$b_{m0}$]{} \[c\]\[c\]\[1\]\[0\] \[c\]\[c\]\[1\]\[0\] \[c\]\[c\]\[1\]\[0\] \[c\]\[c\]\[1\]\[0\] We have carried out polynomial fitting for the $k$ dependence of the bias (Figure \[fig:bias\]). The fit was carried out for redshifts in the range $z=1$ to $6$ at an interval of $\Delta z=0.5$. We find that a linear function of the form $b(k) = b_0 + b_1 k$ gives a good fit to the simulated data for $z \ge 4$. However, a higher order polynomial is required at lower redshifts particularly because of the dip around $k \sim 2 \, {\rm Mpc}^{-1}$. We have used a quartic polynomial of the form $$b(k)=b_0 + b_1 k + b_2 k^2 + b_3 k^3 + b_4 k^4\,. \label{eqn:8}$$ which gives a good fit in the $k$ range $k \leq 10{\rm Mpc}^{-1}$ at all the redshifts that we have considered. The fit was carried out for both $b(k)$ and $b_r(k)$, and we have retained the subscript $r$ for the different fitting coefficients of $b_r(k)$. The left panel of Figure \[fig:fit\_bias\] shows how the $5$ fitting coefficients $b_0,...,b_4$ vary with redshift. The value of the coefficient $b_0$ corresponds to the scale independent bias which is seen to hold at small $k$ values. We find that $b_0$ and $b_{r0}$ are indistinguishable over the entire redshift range, indicating that the bias is real at small $k$ values. We also find that $b_0$ increases nearly linearly with $z$, consistent with the behaviour seen in the left panel of Figure \[fig:diff\_bias\]. The coefficients $b_1,...,b_4$ introduce a scale dependence in the bias, and these coefficients have progressively smaller values. We find that the redshift dependence of all the five coefficients can be well fit by quadratic polynomials of the form $$b_m(z)=c(m,0)+c(m,1) z + c(m,2) z^2, \label{eq:fit}$$ the fits also being shown in the left panel of Figure \[fig:fit\_bias\]. The fitting coefficients $c(m,n)$ allow us to interpolate the bias $b(k,z)$ at different values of $z$ and $k$ in the ranges $[1,6]$ and $[0.04,10]$ respectively. The fitting coefficients $c(m,n)$ and the $1 \, \sigma$ errors in these coefficients $\Delta c(m,n)$ are tabulated in the Appendix \[sec:appendix\]. The central panel of Figure \[fig:fit\_bias\] shows the fit along with the simulated data. We see that the fit reproduces the simulated data to a good level of accuracy over the entire $z$ and $k$ range of the fit. A similar fitting procedure was also carried out for $b_r$. The fitting coefficients $c_r(m,n)$ and the $1\, \sigma$ errors in these coefficients $\Delta c_r(m,n)$ are tabulated in the Appendix \[sec:appendix\]. The right panel of Figure \[fig:fit\_bias\] shows that the fit matches the simulated $b_r$ values to a good level of accuracy. \[c\]\[c\]\[1.2\]\[0\][$k\, {\rm Mpc}^{-1}$]{} \[c\]\[c\]\[1.2\]\[0\][$z$]{} \[c\]\[c\]\[1.3\]\[0\][$b(k,z)$]{} \[c\]\[c\]\[1.3\]\[0\][$b_r(k,z)$]{} \[c\]\[c\]\[1\]\[0\] \[c\]\[c\]\[1\]\[0\] Figure \[fig:cont\] provides a visual impression of how the bias varies jointly with $k$ and $z$. Here we have extrapolated our fit to cover a somewhat larger $k$ range ($[0.01,10] \, {\rm Mpc}^{-1}$) and $z$ range ($[0,6]$). We find a scale independent bias for $k \le 0.1 \, {\rm Mpc}^{-1} $ across the entire $z$ range. Further, we see that the biases $b(k,z)$ and $b_r(k,z)$ both decrease monotonically with decreasing $z$. We also see that the and the matter become anti-correlated where $b_r$ has a negative value for the $k$ range $k \sim 1-2 \, {\rm Mpc}^{-1}$ around $z \sim 0$. \[c\]\[c\]\[1.2\]\[0\][$k\, {\rm Mpc}^{-1}$]{} \[c\]\[c\]\[1.2\]\[0\][$z$]{} \[c\]\[c\]\[1\]\[0\] \[c\]\[c\]\[1\]\[0\] The cross-correlation between the and the matter can also be quantified using the stochasticity [@dekel-lahav99] $r=b_r/b$. By definition $\mid r \mid \leq 1$, values $r \sim 1$ indicate a strong correlation, $r \sim 0$ corresponds to a situation when the two are uncorrelated and $r < 0$ indicates anti-correlation. Figure \[fig:stocpar\] shows how the stochasticity $r$ varies jointly as a function of $z$ and $k$. We see that $r=1$ for $k \le 0.1 \, {\rm Mpc}^{-1} $ where the bias also is scale independent and real across the entire $z$ range. The $k$ value below which $r$ is unity increases with increasing redshift, with $k \sim 0.1 \, {\rm Mpc}^{-1} $ for $z=0$ and $k \sim 3 \, {\rm Mpc}^{-1} $ for $z=6$. The and the matter are highly correlated $(r > 0.8)$ across nearly the entire $k$ range for $z \ge 2$. We also find that $r$ has a negative value at $k \sim 1 \, - \, 2\, {\rm Mpc}^{-1}$ around $z \sim 0$, indicative of an anti-correlation. Summary and Discussion {#sec:summary} ====================== In this paper we have used semi-numerical simulations (the third scheme of @bagla10) to model the distribution and study the evolution of $P_{\HI}(k,z)$ in the post-reionization era. The simulations span the redshift range $1 \le z \le 6$ at an interval $\Delta z =0.5$. We have modelled the bias as a complex quantity $\tilde{b}(k,z)$ whose modulus $b(k,z)$ (squared) relates $P_{\HI}(k,z)$ to $P(k,z)$, and whose real part $b_r(k,z)$ quantifies the cross-correlation between the and the total matter distribution. While there are several earlier works which have studied the bias $b(k,z)$ at a few discrete redshifts (summarized in @hamsa-tirthankar-refregier15), this is the first attempt to model the post-reionization distribution across a large $z$ and $k$ range $(0.04 \le k \le 20 \, {\rm Mpc}^{-1})$ using a single simulation technique. We find that the assumption of a scale-independent bias $b(k,z)=b_0(z)$ holds at small $k$ (eq. \[eqn:8\]). The value of $b_0(z)$ increases nearly linearly with $z$, with a value that is slightly less than unity at $z=1$ and $b_0(z) \approx 3$ at $z=6$. The $k$ range where we have a constant $k$ independent bias is maximum at the intermediate redshift $z = 3$ where it extends to $k \le 1 \, {\rm Mpc}^{-1}$, and it is minimum ($k \le 0.2 \, {\rm Mpc}^{-1}$) at the highest and lowest redshifts ($z=6,1$) whereas it covers $k \le 0.4 \, {\rm Mpc}^{-1}$ at the other redshifts. The bias is scale dependent at larger $k$ values where non-linear effects become important. We find that a polynomial fit (eq. \[eq:fit\]) provides a good description of the joint $z$ and $k$ dependence of $b(k,z)$ (and also $b_r(k,z)$). The coefficients of the fit are presented in Appendix \[sec:appendix\], and Fig. \[fig:cont\] provides a comprehensive picture of the bias across the entire $k$ and $z$ range, all the way to $z=0$ where the results have been extrapolated from the fit. Our results which are based on a PM N-body code are qualitatively consistent with the earlier work of @bagla10 who have used a high resolution Tree-PM N-body code to calculate the bias at three different redshifts ($z=1.3, 3.4$ and $5.1$). The present work is also consistent with @tapomoy-mitra-majumder12 who have used a technique similar to ours to compute the bias across $z = 1.5 \, - \, 4$, and @hamsa-tirthankar-refregier15 who have applied the minimum variance interpolation technique to the different bias values collated from literature to predict the redshift evolution of the scale independent bias in the range $z = 0 \, - \, 3.4$. The analytic model of @marin-gnedin10 predicts the distribution to be anti-biased at low redshifts $(z\, \le \, 1)$. They also found that the bias decreases further for $k \ge 0.1 {\rm Mpc}^{-1}$. These predictions are consistent with observations at $z\, \sim \,0.06$ [@martin-giovanelli12] which suggest that rich galaxies are very weakly clustered and mildly anti-biased at large scales, but become severely anti-biased on smaller scales. The predictions of our simulations which are restricted to $z \ge 1$ are consistent with the findings of @marin-gnedin10. We find that the is mildly anti-biased at large scales at $z = 1$, and the bias drops further for $k \ge 0.1 {\rm Mpc}^{-1}$ (Fig \[fig:bias\]). We have also extrapolated our results to $z \sim 0$ (Fig \[fig:cont\]) where the predictions are found to be qualitatively consistent with the measurements of @martin-giovanelli12. In our analysis the real part $b_r(k,z)$ of the complex bias $\tilde{b}(k,z)$ quantifies the cross-correlation between the and the total matter, and the bias $\tilde{b}(k,z)$ is completely real if the two are perfectly correlated. The same issue is also quantified using the stochasticity $r=b_r(k,z)/b(k,z)$. We see that $b_r$ closely matches $b$ at small $k$ ($<0.1 \,{\rm Mpc}^{-1}$) where we have a scale independent bias across the entire $z$ range. The complex nature of the bias becomes important at larger $k$. Our results are summarized in Fig. \[fig:stocpar\] which shows $r$ across the entire $z$ and $k$ range. We find that the and the matter are well correlated $(r > 0.8)$ across nearly the entire $k$ range for $z \ge 2$. We also find that $r$ has a negative value at $k \sim 1 \, - \, 2\, {\rm Mpc}^{-1}$ around $z \sim 0$, indicative of an anti-correlation. The measurements of @chang-pen-bandura10 constrain the product $\Omega_{\HI} \,b \,r=(5.5 \pm 1.5) \times 10^{-4}$ at $z\sim 0.8$. From our analysis, we find that on large scales the product $b \,r\equiv b_r=0.79$ at $z=0.8$ which implies $\Omega_{\HI}=(6.96 \pm 1.89) \times 10^{-4}$. Again, @masui-switzer-banavar13 constrain the product $\Omega_{\HI}\, b\, r=(4.3 \pm 1.1) \times 10^{-4}$ at $z\sim 0.8$ using measurements in the $k$ range $0.05 \,{\rm Mpc}^{-1}<k<0.8 \,{\rm Mpc}^{-1}$ where our work predicts $br$ to vary from $0.83$ to $0.39$. The corresponding $\Omega_{\HI}$ varies between $(5.2 \pm 1.3) \times 10^{-4}$ to $(1.1 \pm 0.33) \times 10^{-3}$, which is a significant variation. On the other hand, @switzer-masui-bandura-calin13 constrain the product $\Omega_{\HI}\, b=6.2^{+2.4}_{-1.5}\times10^{-4} $ at $z\sim 0.8$ which implies $\Omega_{\HI}= 7.5^{+2.9}_{-1.8}\times10^{-4}$ if we consider $b=0.83$ from our analysis. The above estimates of $\Omega_{\HI}$ are consistent with the measurement $\Omega_{\HI}=7.41 \pm 2.71 \times 10^{-4}$ at $z\,\sim \, 0.609$ [@rao-turnshek06]. We note that our simulations are restricted to $z \ge 1$, and the results were extrapolated to $z=0.8$ for the discussion presented in this paragraph. @khandai-sethi-dimatteo11 have carried out simulations which were specifically designed to interpret the results of @chang-pen-bandura10, and they have predicted $b=0.55\ -\ 0.65$ and $r=0.9 \ - \ 0.95$ at $z=0.8$. We note that the bias value predicted by @khandai-sethi-dimatteo11 is considerably smaller than our prediction, and they predict $\Omega_{\HI}=11.2 \pm 3.0 \times 10^{-4}$ which also is larger than the measurements of @rao-turnshek06. We finally reiterate that it is important to model the distribution in order to correctly predict the signal for upcoming 21-cm intensity mapping experiments. Further, such modelling is also important to correctly interpret the outcome of the future observations. In the present work we have implemented a simple population scheme which incorporates the salient features of our present understanding [ie.]{} the resides in halos which also host the galaxies. This however ignores various complicated astrophysical processes which could possibly play a role in shaping the distribution. Further, the entire analysis has been restricted to real space, and the effects of redshift space distortion have not been taken into account. We plan to address these issues in future work. Acknowledgement {#acknowledgement .unnumbered} =============== Debanjan Sarkar wants to thank Rajesh Mondal for his help with simulations. Anathpindika, S., acknowledges support from the grant YSS/2014/000304 of the SERB, Department of Science & Technology, Government of India. The authors are grateful to J. S. Bagla, Nishikanta Khandai, Tapomoy Guha Sarkar and Kanan K. Datta for useful discussions. [80]{} natexlab\#1[\#1]{} S. S., [Bharadwaj]{} S., 2014, Journal of Astrophysics and Astronomy, 35, 157 J. S., [Khandai]{} N., [Datta]{} K. K., 2010, , 407, 567 K. [et al.]{}, 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9145, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, p. 22 J. M., [Bond]{} J. R., [Kaiser]{} N., [Szalay]{} A. S., 1986, , 304, 15 R. A. [et al.]{}, 2012, ArXiv e-prints R. H. [et al.]{}, 2001, , 122, 2850 A., [Chengalur]{} J. N., [Karachentsev]{} I. D., 2005, , 433, L1 S., [Nath]{} B. B., [Sethi]{} S. K., 2001, Journal of Astrophysics and Astronomy, 22, 21 S., [Pandey]{} S. K., 2003, Journal of Astrophysics and Astronomy, 24, 23 S., [Sarkar]{} A. K., [Ali]{} S. S., 2015, Journal of Astrophysics and Astronomy, 36, 385 S., [Sethi]{} S. K., 2001, Journal of Astrophysics and Astronomy, 22, 293 S., [Sethi]{} S. K., [Saini]{} T. D., 2009, , 79, 083538 S., [Srikant]{} P. S., 2004, Journal of Astrophysics and Astronomy, 25, 67 P., [Camera]{} S., [Raccanelli]{} A., [Blake]{} C., [Ferreira]{} P., [Santos]{} M., [Schwarz]{} D. J., 2015, Advancing Astrophysics with the Square Kilometre Array (AASKA14), 24 T.-C., [Pen]{} U.-L., [Bandura]{} K., [Peterson]{} J. B., 2010, , 466, 463 T.-C., [Pen]{} U.-L., [Peterson]{} J. B., [McDonald]{} P., 2008, Physical Review Letters, 100, 091303 S., [Sarkar]{} D., [Sarkar]{} A., [Bharadwaj]{} S., 2016, in preparation X., 2012, International Journal of Modern Physics Conference Series, 12, 256 J. D., [White]{} M., [Chang]{} T.-C., [Holder]{} G., [Padmanabhan]{} N., [Dor[é]{}]{} O., 2015, ArXiv e-prints J., [Wolfe]{} A. M., [Gawiser]{} E., [Prochaska]{} J. X., 2006, , 652, 994 N. H. M. [et al.]{}, 2015, , 452, 217 M., [Efstathiou]{} G., [Frenk]{} C. S., [White]{} S. D. M., 1985, , 292, 371 A., [Lahav]{} O., 1999, , 520, 24 J., [Meyer]{} M. J., [Staveley-Smith]{} L., [Boyle]{} B. J., 2013, , 433, 1398 D. J., [Hu]{} W., 1999, , 511, 5 X., [Carilli]{} C. L., [Keating]{} B., 2006, , 44, 415 X. [et al.]{}, 2006, , 132, 117 A., [Bharadwaj]{} S., [Ali]{} S. S., [Chengalur]{} J. N., 2011, , 411, 2426 A., [Bharadwaj]{} S., [Ali]{} S. S., [Chengalur]{} J. N., 2011, , 418, 2584 T., [Bharadwaj]{} S., [Choudhury]{} T. R., [Datta]{} K. K., 2011, , 410, 1130 T., [Datta]{} K. K., 2015, , 8, 001 T., [Mitra]{} S., [Majumdar]{} S., [Choudhury]{} T. R., 2012, , 421, 3570 M. G., [Steinmetz]{} M., [Rauch]{} M., 2000, , 534, 594 A., [Thakur]{} S., [Guha Sarkar]{} T., [Sen]{} A. A., 2016, ArXiv e-prints Y. L., [Poggianti]{} B. M., [Verheijen]{} M. A. W., [Deshev]{} B. Z., [van Gorkom]{} J. H., 2012, , 756, L28 A., [Frenk]{} C. S., [White]{} S. D. M., [Colberg]{} J. M., [Cole]{} S., [Evrard]{} A. E., [Couchman]{} H. M. P., [Yoshida]{} N., 2001, , 321, 372 N., [Sethi]{} S., [Dwarakanath]{} K. S., 2016, ArXiv e-prints N., [Sethi]{} S. K., [Di Matteo]{} T., [Croft]{} R. A. C., [Springel]{} V., [Jana]{} A., [Gardner]{} J. P., 2011, , 415, 2580 H.-S., [Wyithe]{} J. S. B., [Baugh]{} C. M., [Lagos]{} C. d. P., [Power]{} C., [Park]{} J., 2016, ArXiv e-prints P. [et al.]{}, 2007, , 376, 1357 P. [et al.]{}, 2007, , 376, 1357 P. [et al.]{}, 2009, , 399, 1447 R. H. [et al.]{}, 2003, , 342, 738 K. M., [Wolfe]{} A. M., [Turnshek]{} D. A., 1995, , 440, 435 A., [Wyithe]{} J. S. B., 2008, Physical Review Letters, 100, 161301 F. A., [Gnedin]{} N. Y., [Seo]{} H.-J., [Vallinotto]{} A., 2010, , 718, 972 A. M., [Giovanelli]{} R., [Haynes]{} M. P., [Guzzo]{} L., 2012, , 750, 38 A. M., [Papastergis]{} E., [Giovanelli]{} R., [Haynes]{} M. P., [Springob]{} C. M., [Stierwalt]{} S., 2010, , 723, 1359 K. W. [et al.]{}, 2013, , 763, L20 J. D. [et al.]{}, 2011, , 732, 35 H. J., [White]{} S. D. M., 1996, , 282, 347 P. [et al.]{}, 2012, , 547, L1 H., [Choudhury]{} T. R., [Refregier]{} A., 2015, , 447, 3745 A. K., [Guha Sarkar]{} T., 2016, ArXiv e-prints J. A., 1999, [Cosmological Physics]{} C., [McMahon]{} R. G., [Storrie-Lombardi]{} L. J., [Irwin]{} M. J., 2003, , 346, 1103 , 2014, , 571, A16 J. C. [et al.]{}, 2013, , 145, 65 A. [et al.]{}, 2008, , 390, 1349 J. X., [Herbert-Fort]{} S., [Wolfe]{} A. M., 2005, , 635, 123 J. X., [Wolfe]{} A. M., 2009, , 696, 1543 S. M., [Turnshek]{} D. A., 2000, , 130, 1 S. M., [Turnshek]{} D. A., [Nestor]{} D. B., 2006, , 636, 610 J., [Zwaan]{} M. A., [Briggs]{} F. H., [Chengalur]{} J. N., [Lah]{} P., [Oosterloo]{} T., [van der Hulst]{} T., 2013, , 435, 2693 R. [et al.]{}, 2015, ArXiv e-prints M. G. [et al.]{}, 2015, ArXiv e-prints S., [Paranjape]{} A., [Witzemann]{} A., [Refregier]{} A., [Amara]{} A., [Akeret]{} J., 2015, ArXiv e-prints H.-J., [Dodelson]{} S., [Marriner]{} J., [Mcginnis]{} D., [Stebbins]{} A., [Stoughton]{} C., [Vallinotto]{} A., 2010, , 721, 164 P. [et al.]{}, 2012, , 422, 1835 R. K., [Tormen]{} G., 2002, , 329, 61 L. J., [McMahon]{} R. G., [Irwin]{} M. J., 1996, , 283, L79 L. J., [Wolfe]{} A. M., 2000, , 543, 552 E. R. [et al.]{}, 2013, , 434, L46 F., [Bull]{} P., [Viel]{} M., 2015, , 814, 146 F., [Viel]{} M., [Datta]{} K. K., [Choudhury]{} T. R., 2014, , 9, 50 A. M., [Gawiser]{} E., [Prochaska]{} J. X., 2005, , 43, 861 J. S. B., [Loeb]{} A., 2009, , 397, 1926 J. S. B., [Loeb]{} A., [Geil]{} P. M., 2008, , 383, 1195 T., [P[é]{}roux]{} C., [Popping]{} A., [Milliard]{} B., [Deharveng]{} J.-M., [Frank]{} S., 2013, , 556, A141 M. A., [Meyer]{} M. J., [Staveley-Smith]{} L., [Webster]{} R. L., 2005, , 359, L30 {#sec:appendix} We have fitted the joint $k$ and $z$ dependence of the biases $b(k,z)$ and $b_r(k,z)$ using polynomial of the form $$b(k,z)=\sum\limits_{m=0}^{4} \sum\limits_{n=0}^{2} c(m,n) k^m z^n\,{\rm and} \label{eq:10}$$ $$b_r(k,z)=\sum\limits_{m=0}^{4} \sum\limits_{n=0}^{2} c_r(m,n) k^m z^n\, \label{eq:11}$$ The best fit values of the fitting coefficients $c(m,n)$ and $c_r(m,n)$, and the $1 \, \sigma$ uncertainties in fitting $\Delta c(m,n)$ and $\Delta c_r(m,n)$ respectively, are given below. $c(m,n)\times10^{-2}$= $\Delta c(m,n)\times10^{-3}$= $c_r(m,n)\times10^{-2}$= $\Delta c_r(m,n)\times10^{-3}$= [^1]: debanjan@cts.iitkgp.ernet.in [^2]: somnath@phy.iitkgp.ernet.in
--- abstract: \| When analyzing probabilistic computations, a powerful approach is to first find a martingale—an expression on the program variables whose expectation remains invariant—and then apply the optional stopping theorem in order to infer properties at termination time. One of the main challenges, then, is to systematically find martingales. We propose a novel procedure to synthesize martingale expressions from an arbitrary initial expression. Contrary to state-of-the-art approaches, we do not rely on constraint solving. Instead, we use a symbolic construction based on Doob’s decomposition. This procedure can produce very complex martingales, expressed in terms of conditional expectations. We show how to automatically generate and simplify these martingales, as well as how to apply the optional stopping theorem to infer properties at termination time. This last step typically involves some simplification steps, and is usually done manually in current approaches. We implement our techniques in a prototype tool and demonstrate our process on several classical examples. Some of them go beyond the capability of current semi-automatic approaches. author: - Gilles Barthe - Thomas Espitau - 'Luis Mar[í]{}a Ferrer[ ]{}Fioriti' - Justin Hsu bibliography: - 'header.bib' - 'main.bib' - 'literature.bib' title: \| Synthesizing Probabilistic Invariants\ via Doob’s Decomposition --- Introduction ============ Probabilistic computations are a key tool in modern computer science. They are ubiquitous in machine learning, privacy-preserving data mining, cryptography, and many other fields. They are also a common and flexible tool to model a broad range of complex real-world systems. Not surprisingly, probabilistic computations have been extensively studied from a formal verification perspective. However, their verification is particularly challenging. In order to understand the difficulty, consider the standard way to infer properties about the final state of a non-probabilistic program using a strong invariant (an assertion which is preserved throughout program execution) and a proof of termination. This proof principle is not easily adapted to the probabilistic case. First, probabilistic programs are interpreted as distribution transformers [@Kozen81] rather than state transformers. Accordingly, assertions (including strong invariants) must be interpreted over distributions. Second, the notion of termination is different for probabilistic programs. We are usually not interested in proving that all executions are finite, but merely that the probability of termination is $1$, a slightly weaker notion. Under this notion, there may be no finite bound on the number of steps over all possible executions. So, we cannot use induction to transfer local properties to the end of the program—more complex limiting arguments are needed. We can avoid some of these obstacles by looking at the average behavior of a program. That is, we can analyze numerical expressions (over program variables) whose average value is preserved. These expressions are known as martingales, and have several technical advantages. First, martingales are easy to manipulate symbolically and can be checked locally. Second, the average value of martingale is preserved at termination, even if the control-flow of the program is probabilistic. This fact follows from the optional stopping theorem (OST), a powerful result in martingale theory. While martingales are quite useful, they can be quite non-obvious. Accordingly, recent investigation has turned to automatically synthesizing martingales. State-of-the-art frameworks are based on constraint solving, and require the user to provide either a template expression [@KatoenMMM10; @ChakarovS13] or a limit on the search space [@ChenHWZ15; @ChakarovS14]. The main advantage of such approaches is that they are generally complete—they find all possible martingales in the search space. However, they have their drawbacks: a slightly wrong template can produce no invariant at all, and a lot of search space may be needed to arrive at the martingale. We propose a framework that complements current approaches—we rely on purely symbolic methods instead of solving constraints or searching. We require the user to provide a “seed” expression, from which we always generate a martingale. Our approach uses Doob’s decomposition theorem, which gives a symbolic method to construct a martingale from any sequence of random values. Once we have the martingale, we can apply optional stopping to reason about the martingale at loop termination. While the martingale and final fact may be quite complex, we can use non-probabilistic invariants and symbolic manipulations to automatically simplify them. We demonstrate our techniques in a prototype implementation, implementing Doob’s decomposition and the Optional Stopping Theorem. Although these proof principles have been long known to probability theory, we are the first to incorporate them into an automated program analysis. Given basic invariants and hints, our prototype generates martingales and facts for a selection of examples. Mathematical preliminaries ========================== We briefly introduce some definitions from probability theory required for our technical development. We lack the space to discuss the definitions in-depth, but we will explain informally what the various concepts mean in our setting. Interested readers can find a more detailed presentation in any standard probability theory textbook (e.g., @Williams91). First, we will need some basic concepts from probability theory. Let $\Omega$ be the set of outcomes. - A sigma algebra is a set $\SigmaAlgebra$ of subsets of $\Omega$, closed under complements and countable unions, and countable intersections. - A probability measure is a countably additive mapping $\Prob: \SigmaAlgebra \to [0, 1]$ such that $\Prob(\Omega) = 1$. - A probability space is a triple $(\Omega, \SigmaAlgebra, \Prob)$. Next, we can formally define stochastic processes. These constructions are technical but completely standard. Let $(\Omega, \SigmaAlgebra, \Prob)$ be a probability space. - A (real) random variable is a function $X : \Omega \to \Reals$. $X$ is $\SigmaAlgebra$-measurable if $X^{-1}((a, b]) \in \SigmaAlgebra$ for every $a, b \in \Reals$. - A filtration is a sequence $\{ \mathcal{F}_i \}_{i \in \Nats}$ of sigma algebras such that $\mathcal{F}_i \subseteq \mathcal{F}$ and $\mathcal{F}_{i - 1} \subseteq \mathcal{F}_i$ for every $i > 0$. When there is a fixed filtration on $\mathcal{F}$, we will often abuse notation and write $\mathcal{F}$ for the filtration. - A stochastic process adapted to filtration $\mathcal{F}$ is a sequence of random variables $\{ X_i \}_{i \in \Nats}$ such that each $X_i$ is $\mathcal{F}_i$-measurable. Intuitively, we can think of $\Omega$ as a set where each element represents a possible outcome of the samples. In our setting, grouping samples according to the loop iteration gives a natural choice for the filtration: we can take $\mathcal{F}_i$ to be the set of events that are defined by samples in iteration $i$ or before. A stochastic process $X$ is adapted to this filtration if $X_i$ is defined in terms of samples from iteration $i$ or before. Sampled variables at step $i$ are independent of previous steps, so they are not $\mathcal{F}_{i - 1}$-measurable. #### Expectation. {#expectation. .unnumbered} To define martingales, we need to introduce expected values and conditional expectations. The expected value of a random variable is defined as $$\Exp{X} \triangleq \int_\Omega X \cdot \dif \Prob$$ where $\int$ is the Lebesgue integral [@Williams91]. We say that a random variable is integrable if $\Exp{\|X\|}$ is finite. Given a integrable random variable $X$ and a sigma algebra $\SigmaAlgebra[G]$, a conditional expectation of $X$ with respect to $\SigmaAlgebra[G]$ is a random variable $Y$ such that $Y$ is $\SigmaAlgebra[G]$-measurable, and $\Exp{X \cdot \indFn{A}} = \Exp{Y \cdot \indFn{A}}$ for all events $A \in \SigmaAlgebra[G]$. (Recall that the indicator function $\indFn{A}$ of an event $A$ maps $\omega \in A$ to $1$, and all other elements to $0$.) Since one can show that this $Y$ is essentially unique, we denote it by $\CondExp{X}{\SigmaAlgebra[G]}$. #### Moments. {#moments. .unnumbered} Our method relies on computing higher-order moments. Suppose $X$ is a random variable with distribution $d$. If $X$ takes numeric values, the $k$th moment of $d$ is defined as $$G(d)_k \triangleq \Exp{ X^k }$$ for $k \in \Nats$. If $X$ ranges over tuples, the correlations of $d$ are defined as $$G(d, \{ a, b \})_{p, q} \triangleq \Exp{ \pi_a(X)^p \cdot \pi_b(X)^q } ,$$ for $p, q \in \Nats$, and similarly for products of three or more projections. Here, the projection $\pi_i(X)$ for $X$ a tuple-valued random variable is the marginal distribution of the $i$th coordinate of the tuple. #### Martingales. {#martingales. .unnumbered} A martingale is a stochastic process with a special property: the average value of the current step is equal to the value of the previous step. Let $\{ X_i \}$ be a stochastic process adapted to filtration $\{ \mathcal{F}_i \}$. We say that $X$ is a martingale with respect to $\mathcal{F}$ if it satisfies the property $$\CondExp{ X_i }{ \mathcal{F}_{i - 1} } = X_{ i - 1 } .$$ For a simple example, consider a symmetric random walk on the integers. Let $X \in \Ints$ denote the current position of the walk. At each step, we flip a fair coin: if heads, we increase the position by $1$, otherwise we decrease the position by $1$. The sequence of positions $X_0, X_1, \dots$ forms a martingale since the average position at time $i$ is simply the position at time $i - 1$: $$\CondExp{ X_i }{ \mathcal{F}_{i - 1} } = X_{i - 1} .$$ #### Doob’s decomposition. {#doobs-decomposition. .unnumbered} One important result in martingale theory is Doob’s decomposition. Informally, it establishes that any integrable random process can be written uniquely as a sum of a martingale and a predictable process. For our purposes, it gives a constructive and purely symbolic method to extract a martingale from any arbitrary random process. \[thm:doob\] Let $X = \{ X_i \}_{i \in \Nats}$ be a stochastic process adapted to filtration $\{ \mathcal{F}_i \}_{i \in \Nats}$ where each $X_i$ has finite expected value. Then, the following process is a martingale: $$M_i = \begin{cases} X_0 &: i = 0 \\ X_0 + \sum_{j = 1}^i X_j - \CondExp{ X_j }{ \mathcal{F}_{j - 1} } &: i > 0 \end{cases}$$ If $X$ is already a martingale, then $M = X$. We will think of the stochastic process $X$ as a seed process which generates the martingale. While the definition of the martingale involves conditional expectations, we will soon see how to automatically simplify these expectations. #### Optional stopping theorem. {#optional-stopping-theorem. .unnumbered} For any martingale $M$, it is not hard to show that the expected value of $M$ remains invariant at each time step. That is, for any fixed value $n \in \Nats$, we have $$\Exp{ M_n } = \Exp{ M_0 } .$$ The optional stopping theorem extends this equality to situations where $n$ itself may be random, possibly even a function of the martingale. Let $(\Omega, \SigmaAlgebra, \Prob)$ be a probability space with filtration $\{ \mathcal{F}_i \}_{i \in \Nats}$. A random variable $\tau : \Omega \to \Nats$ is a stopping time if the subset $\{ w \in \Omega \mid \tau(w) \leq i \}$ is a member of $\mathcal{F}_i$ for each $i \in \Nats$. Returning to our random walk example, the first time that the position is farther than $100$ from the origin is a stopping time since this time depends only on past samples. In contrast, the last time that a position is farther than $100$ from the origin is not a stopping time, since this time depends on future samples. More generally, the iteration count when we exit a probabilistic loop is a stopping time since termination is a function of past samples only. If we have a stopping time and a few mild conditions, we can apply the optional stopping theorem.[^1] \[thm:OST\] Let $\tau$ be a stopping time, and let $M$ be a martingale. If the expected value of $\tau$ is finite, and if $\|M_i - M_{i-1}\| \leq C$ for all $i > 0$ and some constant $C$, then $$\Exp{ M_\tau } = \Exp{ M_0 } .$$ To see this theorem in action, consider the random walk martingale $S$ and take the stopping time $\tau$ to be the first time that $\|S\| \geq 100$. It is possible to show that $\tau$ has finite expected value, and clearly $\|S_i - S_{i - 1}\| \leq 1$. So, the optional stopping theorem gives $$\Exp{ S_\tau } = \Exp{ S_0 } = 0 .$$ Since we know that the position is $\pm 100$ at time $\tau$, this immediately shows that the probability of hitting $+100$ is equal to the probability of hitting $-100$. This intuitive fact can be awkward to prove using standard probabilistic invariants, but falls out directly from a martingale analysis. Overview of method {#sec:tool} ================== Now that we have seen the key technical ingredients of our approach, let us see how to combine these tools into an automated program analysis. We will take an imperative program specifying a stochastic process and a seed expression, and we will automatically synthesize a martingale and an assertion that holds at termination. We proceed in three stages: extracting a polynomial representing the stochastic process in the program, applying Doob’s decomposition to the polynomial representation, and applying optional stopping. We perform symbolic manipulations to simplify the martingale and final fact. #### Programs. {#programs. .unnumbered} We consider programs of the form:[^2] $$I; \WWhile{e}{(S; B)}$$ where $I$ and $B$ are sequences of deterministic assignments (of the form $\Ass{\Var}{\Expr}$), and $S$ is a sequence of probabilistic samplings (of the form $\Rand{\SVar}{\DExpr}$). Note that we separate sample variables $s \in \SVar$, which are the target of random samplings, from process variables $x \in \Var$, which are the target of deterministic assignments. This distinction will be important for our simplifications: we know the moments and correlations of sample variables, while we have less information for process variables. We require that programs assign to sample variables before assigning to process variables in each loop iteration; this restriction is essentially without loss of generality. We take $\DExpr$ to be a set of standard distributions over the integers or over finite tuples of integers, to model joint distributions. For instance, we often consider the distribution $\mathsf{Bern}(1/2, \{-1, 1\}) \in \DExpr$ that returns $-1$ and $+1$ with equal probability. We assume that all distributions in $\DExpr$ have bounded support; all moments and correlations of the primitive distributions are finite. We will also assume that distributions do not depend on the program state. The set $\Expr$ of expressions is mostly standard, with a few notational conveniences for defining stochastic processes: $$\begin{array}{r@{\ \ }l@{\quad}l} \Expr ::= & \Var \mid \SVar \mid \Var[-n] & \mbox{process/sample/history variables} \\ \mid& \Ints & \mbox{constants}\\ % \mid& \indFn{\Expr} & \mbox{indicator}\\ \mid& \pi_a(\Expr) & \mbox{projections}\\ \mid& \Expr + \Expr \mid \Expr \cdot \Expr & \mbox{arithmetic}\\ \mid& \Expr < \Expr \mid \Expr \land \Expr \mid \neg \Expr & \mbox{guards}\\ \end{array}$$ History variables $\Var[-n]$ are indexed by a positive integer $n$ and are used inside loops. The variable $x[-n]$ refers to the value of $x$ assigned $n$ iterations before the current iteration. If the loop has run for fewer than $n$ iterations, $x[-n]$ is specified by the initialization step of our programs: $$I \triangleq \Ass{x_1[-n_1]}{e_1} ; \cdots ; \Ass{x_k[-n_k]}{e_k} .$$ #### Extracting the polynomial. {#extracting-the-polynomial. .unnumbered} For programs in our fragment, each variable assigned in the loop determines a stochastic process: $x$ is the most recent value, $x[-1]$ is the previous value, etc. In the first stage of our analysis, we extract polynomial representations of each stochastic process from the input program. We focus on the variables that are mutated in $B$—each of these variables determines a stochastic process. To keep the notation light, we will explain our process for first-order stochastic processes: we only use history variables $x[-1]$ from the past iteration. We will also suppose that there is just one process variable and one sample variable, and only samples from the current iteration are used. Since our expression language only has addition and multiplication as operators, we can represent the program variable $x$ as a polynomial of other program variables: $$\label{eq:p-poly} x = P_x(x[-1], s)$$ Next, we pass to a symbolic representation in terms of (mathematical) random variables. To variable $x$, we associate the random variable $\{ X_i \}_{i \in \Nats}$ modeling the corresponding stochastic process, and likewise for the sample variable $s$. By convention, $i = 0$ corresponds to the initialization step, and $i > 0$ corresponds to the stochastic process during the loop. In other words, $$\Ass{x[0]}{0}; \WWhile{e}{\Rand{s}{d}; \Ass{x}{x[-1] + s}}$$ desugars to $$\Ass{x[0]}{0}; \Ass{i}{0}; \WWhile{e}{\Ass{i}{i + 1}; \Rand{s}{d}; \Ass{x[i]}{x[i-1] + s}}$$ in a language with arrays instead of history variables. Then, the program variable $x[i]$ corresponds to the random variable $X_i$. Then, and the initial conditions specified by the command $I$ give an inductive definition for the stochastic process: $$\label{eq:p-poly-rv} X_i = P_x(X_{i - 1}, S_i) .$$ #### Applying Doob’s decomposition. {#applying-doobs-decomposition. .unnumbered} The second stage of our analysis performs Doob’s decomposition on the symbolic representation of the process. We know that the seed expression $e$ must be a polynomial, so we can form the associated stochastic process $\{ E_i \}_{i \in \Nats}$ by replacing program variables by their associated random variable: $$\label{eq:p-poly-seed} E_i = P_e(X_i, S_i) .$$ (Recall that the initial conditions $X_0$ and $S_0$, which define $E_0$, are specified by the initialization portion $I$ of the program.) Then, Doob’s decomposition produces the martingale: $$M_i = \begin{cases} E_0 &: i = 0 \\ E_0 + \sum_{j = 1}^i E_j - \CondExp{ E_j }{ \mathcal{F}_{j - 1} } &: i > 0 . \end{cases}$$ To simplify the conditional expectation, we unfold $E_j$ via and unroll the processes $X_i$ by one step with . $$\begin{aligned} \CondExp{ c \cdot f + c' \cdot g }{ - } &\mapsto c \cdot \CondExp{ f }{ - } + c' \cdot \CondExp{ g }{ - } \end{aligned}$$ ------------------------------------------------------------------------ $$\begin{aligned} \CondExp{ X_{i - n} \cdot f }{ \mathcal{F}_{i - 1} } &\mapsto X_{i - n} \cdot \CondExp{ f }{ \mathcal{F}_{i - 1} } \tag{$n > 0$} \\ \CondExp{ S_i \cdot S_i' }{ \mathcal{F}_{i - 1} } &\mapsto \CondExp{ S_i }{ \mathcal{F}_{i - 1} } \cdot \CondExp{ S_i' }{ \mathcal{F}_{i - 1} } \tag{$S \neq S'$} \end{aligned}$$ ------------------------------------------------------------------------ $$\begin{aligned} \CondExp{ S^k_i }{ \mathcal{F}_{i - 1} } &\mapsto G(d)_k \tag{$S \sim d$} \\ \CondExp{ \pi_a(S_i)^p \cdot \pi_b(S_i)^q }{ \mathcal{F}_{i - 1} } &\mapsto G(d_{a, b})_{p, q} \tag{$S \sim d$} \end{aligned}$$ Now, we apply our simplification rules; we present a selection in . The rules are divided into three groups (from top): linearity of expectation, conditional independence, and distribution information. The first two groups reflect basic facts about expectations and conditional expectations. The last group encodes the moments and correlations of the primitive distributions. We can pre-compute these quantities for each primitive distribution $d$ and store the results in a table. By the form of , the simplification removes all expectations and we can give an explicit definition for the martingale: $$\label{eq:q-poly} M_i = \begin{cases} E_0 &: i = 0 \\ Q_e(X_{i - 1}, \dots, X_0, S_i, \dots, S_1) &: i > 0 , \end{cases}$$ where $Q_e$ is a polynomial. #### Applying optional stopping. {#applying-optional-stopping. .unnumbered} With the martingale in hand, the last stage of our analysis applies the optional stopping theorem. To meet the technical conditions of the theorem, we need two properties of the loop: - The expected number of iterations must be finite. - The martingale must have bounded increments. These side conditions are non-probabilistic assertions that can already be handled using existing techniques. For instance, the first condition follows from the existence of a bounded variant [@HartSP83]: an integer expression $v$ such that - $0 \leq v < K$; - $v = 0$ implies the guard is false; and - the probability that $v$ decreases is strictly bigger than $\epsilon$ throughout the loop, for $\epsilon$ and $K$ positive constants. However in general, finding a bounded variant may be difficult; proving finite expected stopping time is an open area of research which we do not address here. The second condition is also easy to check. For one possible approach, one can replace stochastic sampling by non-deterministic choice over the support of the distribution, and verify that the seed expression $e$ is bounded using standard techniques [@CousotH78; @Mine06; @MouraB08]. This suffices to show that the martingale $M_i$ has bounded increments. To see why, suppose that the seed expression is always bounded by a constant $C$. By Doob’s decomposition, we have $$\begin{aligned} \| M_i - M_{i - 1} \| &= \left\|\left(\sum_{j = 1}^i E_j - \CondExp{ E_j }{ \mathcal{F}_{j - 1} }\right) - \left(\sum_{j = 1}^{i - 1} E_j - \CondExp{ E_j }{ \mathcal{F}_{j - 1} }\right) \right\| \\ &= \left\|E_i - \CondExp{ E_i }{ \mathcal{F}_{j - 1} }\right\| \leq 2C ,\end{aligned}$$ so the martingale has bounded increments. Thus, we can apply the optional stopping theorem to to conclude: $$E_0 = \Exp{M_0} =\Exp{M_\tau} = \Exp{Q_e(X_{\tau - 1}, \dots, X_0, S_\tau, \dots, S_1)}$$ Unlike the simplification step after applying Doob’s decomposition, we may not be able to eliminate all expected values. For instance, there may be expected values of $X$ at times before $\tau$. However, if we have additional invariants about the loop, we can often simplify the fact with basic symbolic transformations. #### Implementation. {#implementation. .unnumbered} ![Tool pipeline[]{data-label="fig:pipeline"}](pipeline){width="\textwidth"} We have implemented our process in a Python prototype using the `sympy` library for handling polynomial and summation manipulations [@Joyner:2012]. shows the entire pipeline. There are three parts of the input: the program describing a stochastic process, the seed expression, and hint facts. The output is a probabilistic formula that holds at termination. The most challenging part of our analysis is the last stage: applying OST. First, we need to meet the side conditions of the optional stopping theorem: finite expected iteration count and bounded increments. Our prototype does not verify these side conditions automatically since available termination tools are either not fully automatic [@EsparzaGK12] or can only synthesize linear ranking supermartingales [@ChakarovS13; @ChatterjeeFNH16] that are insufficient for the majority of our case studies[^3]. Furthermore, the final fact typically cannot be simplified without some basic information about the program state at loop termination. We include this information as hints. Hints are first-order formulae over the program variables and the loop counter (represented by the special variable $t$), and are used as auxiliary facts during the final simplification step. Hints can be verified using standard program verification tools since they are non-probabilistic. In our examples, we manually translate the hints and the program into the input language of EasyCrypt [@BartheGHB11], and perform the verifications there. Automating the translation poses no technical difficulty and is left for future work. We note that the performance of the tool is perfectly reasonable for the examples considered in the next section. For instance, it handles the “ABRACADABRA” example in less than 2 seconds on a modern laptop. Examples ======== Now, we demonstrate our approach on several classic examples of stochastic processes. In each case, we describe the code, the seed expression, and any hints needed for our tool to automatically derive the final simplified fact. #### Geometric distribution. {#geometric-distribution. .unnumbered} As a first application, we consider a program that generates a draw from the geometric distribution by running a sequence of coin flips. x[0] := 0; while (z $\neq$ 0) do z ~~ Bern(p, {1, 0}); x := x[-1] + z; end Here, $\lstt{Bern(p, \{1, 0\})}$ is the distribution that returns $1$ with probability $p > 0$, and $0$ otherwise. The program simply keeps drawing until we sample $0$, storing the number of times we sample $1$ in $x$. We wish to apply our technique to the seed expression $x$. First, we can extract the polynomial equation: $$X_i = X_{i - 1} + Z_i .$$ Applying Doob’s decomposition, our tool constructs and reduces the martingale: $$M_i = \begin{cases} X_0 &: i = 0 \\ X_i - p \cdot i &: i > 0 . \end{cases}$$ To apply optional stopping, we first need to check that the stopping time $\tau$ is integrable. This follows by taking $z$ as a bounded variant—it remains in $\{ 0, 1 \}$ and decreases with with probability $p > 0$. Also, the martingale $M_i$ has bounded increments: $\|M_i - M_{i - 1}\|$ should be bounded by a constant. But this is clear since we can use a loop invariant to show that $\|X_i - X_{i - 1}\| \leq 1$, and the increment is $$\|M_i - M_{i - 1}\| = \|X_i - X_{i - 1} - p\| \leq \|X_i - X_{i - 1}\| + p \leq 1 + p .$$ So, optional stopping shows that $$0 = \Exp{ X_\tau - p \cdot \tau } .$$ With the hint $x = t - 1$—which holds at termination—our tool replaces $X_\tau$ by $\tau - 1$ and automatically derives the expected running time: $$\begin{aligned} 0 &= \Exp{ \tau - 1 - p \cdot \tau } \\ \Exp{ \tau } &= 1/(1-p) .\end{aligned}$$ #### Gambler’s ruin. {#gamblers-ruin. .unnumbered} Our second example is the classic Gambler’s ruin process. The program models a game where a player starts with $a > 0$ dollars and keeps tossing a fair coin. The player wins one dollar for each head and loses one dollar for each tail. The game ends either when the player runs out of money, or reaches his target of $b > a$ dollars. We can encode this process as follows: x[0] := a; while (0 < x < b) do z ~~ Bern(1/2, {-1, 1}); x := x + z; end We will synthesize two different martingales from this program, which will yield complementary information once we apply optional stopping. For our first martingale, we use $x$ as the seed expression. Our tool synthesizes the martingale $$M_i = \begin{cases} X_0 &: i = 0 \\ X_i &: i > 0 . \end{cases}$$ So in fact, $x$ is already a martingale. To apply optional stopping, we first note that $x$ is a bounded variant: it remains in $(0, b)$ and decreases with probability $1/2$ at each iteration. Since the seed expression $x$ is bounded, the martingale $M_i$ has bounded increments. Thus, optional stopping yields $$a = \Exp{ X_0 } = \Exp{ X_\tau } .$$ If we give the hint $$\label{eq:ruin-inv} (x = 0) \lor (x = b)$$ at termination, our prototype automatically derives $$\begin{aligned} a &= \Exp{ X_\tau \cdot \indFn{X_\tau = 0 \lor X_\tau = b} } \\ &= \Exp{ X_\tau \cdot \indFn{ X_\tau = 0 } } + \Exp{ X_\tau \cdot \indFn{ X_\tau = b } } \\ &= 0 \cdot \Pr{}{ X_\tau = 0 } + b \cdot \Pr{}{ X_\tau = b } = b \cdot \Pr{}{ X_\tau = b } ,\end{aligned}$$ so the probability of exiting at $b$ is exactly $a/b$. Now, let us take a look at a different martingale generated by the seed expression $x^2$. Our prototype synthesizes the following martingale: $$M_i' = \begin{cases} X_0^2 &: i = 0 \\ X_i^2 - i &: i > 0 \end{cases}$$ Again, we can apply optional stopping: $x$ is a bounded variant, and the seed expression $x^2$ remains bounded in $(0, b^2)$. So, we get $$a^2 = \Exp{ M_0 } = \Exp{ X_\tau^2 - \tau } .$$ By using the same hint , our prototype automatically derives $$\begin{aligned} a^2 &= \Exp{ X_\tau^2 \cdot \indFn{X_\tau = 0 \lor X_\tau = b} } - \Exp{ \tau } \\ &= \Exp{ X_\tau^2 \cdot \indFn{ X_\tau = 0 } } + \Exp{ X_\tau^2 \cdot \indFn{ X_\tau = b } } - \Exp{ \tau } \\ &= 0 \cdot \Pr{}{ X_\tau = 0 } + b^2 \cdot \Pr{}{ X_\tau = b } - \Exp{ \tau } = b^2 \cdot \Pr{}{ X_\tau = b } - \Exp{ \tau } .\end{aligned}$$ Since we already know that $\Pr{}{ X_\tau = b } = a/b$ from the first martingale $\{ M_i \}_{i \in \Nats}$, this implies that the expected running time of the Gambler’s ruin process is $$\Exp{ \tau } = a(b - a) .$$ #### Gambler’s ruin with momentum. {#gamblers-ruin-with-momentum. .unnumbered} Our techniques extend naturally to stochastic processes that depend on variables beyond the previous iteration. To demonstrate, we’ll consider a variant of Gambler’s ruin process with momentum: besides just the coin flip, the gambler will also gain profit equal to the difference between the previous two dollar amounts. Concretely, we consider the following process: x[0] := a; x[1] := a; while (0 < x < b) do z ~~ Bern(p, {-1, 1}); x := x[-1] + (x[-1] - x[-2]) + z; end Note that we must now provide the initial conditions for two steps, since the process is second-order recurrent. Given seed expression $x$, our tool synthesizes the following martingale: $$M_i = \begin{cases} X_0 &: i = 0 \\ X_0 + X_i - X_{i - 1} &: i > 0 \end{cases}$$ Identical to the Gambler’s ruin process, we can verify the side conditions and apply optional stopping, yielding $$a = \Exp{ M_0 } = \Exp{ M_\tau } = \Exp{ X_0 + X_\tau - X_{\tau - 1} } .$$ Unfolding $X_0 = a$ and simplifying, our tool derives the fact $$\Exp{ X_\tau } = \Exp{ X_{\tau - 1} } .$$ We are not aware of existing techniques that can prove this kind of fact—reasoning about the expected value of a variable in the iteration just prior to termination. #### Abracadabra. {#abracadabra. .unnumbered} Our final example is a classic application of martingale reasoning. In this process, a monkey randomly selects a character at each time step, stopping when he has typed a special string, say “ABRACADABRA”. We model this process as follows: match$_0$[0] := 1; match$_1$[0] := 0; ... match$_{11}$[0] := 0; while (match$_{11}$ == 0) do s ~~ UnifMatches; match$_{11}$ := match$_{10}$[-1] * $\pi_{11}$(s); match$_{10}$ := match$_{9}$[-1] * $\pi_{10}$(s); ... match$_{1}$ := match$_{0}$[-1] * $\pi_1$(s); end Here, $\lstt{UnifMatches}$ is a distribution over tuples that represents a uniform $c$ draw from the letters, where the $k$th entry is $1$ if the $c$ matches the $k$th word and $0$ if not. The variables $\lstt{match}_i$ record whether the $i$ most recent letters match the first $i$ letters of target word; $\lstt{match}_0$ is always $1$, since we always match $0$ letters. Now, we will apply Doob’s decomposition. Letting $L$ be the number of possible letters and taking the seed expression $$e \triangleq 1 + L \cdot \lstt{match}_{1} + \cdots + L^{11} \cdot \lstt{match}_{11} ,$$ our tool synthesizes the following martingale: $$M_i = \begin{cases} 1 + L \cdot X^{(1)}_i + \cdots + L^{11} \cdot X^{(11)}_i &: i = 0 \\ \sum_{j = 1}^i \left( L \cdot X^{(1)}_j + \cdots + L^{11} \cdot X^{(11)}_j - L^0 \cdot X^{(0)}_{j - 1} + \cdots + L^{10} \cdot X^{(10)}_{j - 1} \right) &: i > 0 , \end{cases}$$ where $X^{(j)}$ is the stochastic process corresponding to $\lstt{match}_j$. The dependence on $L$ is from the expectations of projections of , which are each $1/L$—the probability of a uniformly random letter matching any fixed letter. To apply the optional stopping theorem, note that the seed expression $e$ is bounded in $(0, L^{12})$, and $1 + L + \cdots + L^{11} - e$ serves as a bounded variant: take the highest index $j$ such that $\lstt{match}_j = 1$, and there is probability $1/L$ that we increase the match to get $\lstt{match}_{j + 1} = 1$, decreasing the variant. So, we have $$1 = \Exp{ M_0 } = \sum_{j = 1}^\tau \Exp{ L \cdot X^{(1)}_j + \cdots + L^{11} \cdot X^{(11)}_j } - \Exp{ L^0 \cdot X^{(0)}_{j - 1} + \cdots + L^{10} \cdot X^{(10)}_{j - 1} } .$$ Our tool simplifies and uses the hints $X^{(11)}_j = 0$ and $X^{(0)}_j = 1$ for $j < \tau$ to derive $$1 = L^0 \cdot \Exp{ X^{(0)}_{\tau} } + \cdots + L^{11} \cdot \Exp{ X^{(11)}_{\tau} } - \Exp{ \tau } .$$ For the target string “ABRACADABRA”, we use hints $$\begin{aligned} (\lstt{match}_{11} = 1) &\implies (\lstt{match}_{4} = 1) \notag \\ (\lstt{match}_{11} = 1) &\implies (\lstt{match}_{1} = 1) \notag \\ (\lstt{match}_{11} = 1) &\implies (\lstt{match}_{0} = 1) \notag \\ (\lstt{match}_{11} = 1) &\implies (\lstt{match}_{j} = 0) \tag{for $j \neq 0, 1, 4, 11$} .\end{aligned}$$ For example, if $\lstt{match}_{11}$ is set then the full string is matching “ABRACADABRA”, so the most recently seen four characters are “ABRA”. This matches the first four letters of “ABRACADABRA”, so $\lstt{match}_4$ is also set. The hint can be proved from a standard loop invariant. Our tool derives the expected running time: $$\Exp{ \tau } = L^{1} + L^{4} + L^{11} .$$ Benchmarks. {#benchmarks. .unnumbered} ----------- To give an idea of the efficiency of our tool, we present some benchmarks for our examples in . We measured timing on a recent laptop with a 2.4 GHz Intel Core processor with 8 GB of RAM. We did not optimize for the performance; we expect that the running time could be greatly improved with some tuning. Example Running time (s) -------------------------------------------------------- ------------------ <span style="font-variant:small-caps;">geom</span> 0.14 <span style="font-variant:small-caps;">gamble</span> 0.11 <span style="font-variant:small-caps;">gamble2</span> 0.17 <span style="font-variant:small-caps;">miniabra</span> 0.87 <span style="font-variant:small-caps;">fullabra</span> 3.58 : Preliminary benchmarks.[]{data-label="fig:bench"} The example <span style="font-variant:small-caps;">miniabra</span> is a smaller version of the ABRACADABRA example, where the alphabet is just $\{ 0, 1 \}$, and we stop when we sample the sequence $111$; <span style="font-variant:small-caps;">fullabra</span> is the full ABRACADABRA example. While there is a growing body of work related to martingale techniques for program analysis (see the following section), it is not obvious how to compare benchmarks. Existing work focuses on searching for martingale expressions within some search space; this is a rather different challenge than synthesizing a single martingale from a programmer-provided seed expression. In particular, if the seed expression happens to already be a martingale by some lucky guess, our tool will simply return the seed expression after checking that it is indeed a martingale. Related work {#sec:related} ============ #### Martingales. {#martingales.-1 .unnumbered} Martingale theory is a central tool in fields like statistics, applied mathematics, control theory, and finance. When analyzing randomized algorithms, martingales can show tight bounds on tail events [@MotwaniR95]. In the verification community, martingales are used as invariant arguments, and as variants arguments to prove almost sure termination [@BournezG05; @ChakarovS13; @FerrerH15; @ChatterjeeFNH16]. Recently, martingale approaches were extended to prove more complex properties. @ChakarovVS16 propose proof rules for proving persistence and recurrence properties. @DimitrovaFHM16 develop a deductive proof system for PCTL$^$, with proof rules based on martingales and supermartingales. #### Probabilistic Hoare logic. {#probabilistic-hoare-logic. .unnumbered} @McIverMorgan05 propose a Hoare-like logic that is quite similar to our approach of using martingales and OST. Their approach is based on weakest pre-expectations, which are an extension of Dijkstra’s weakest preconditions [@Dijkstra75] based on “backward” conditional expectations. Their probabilistic invariants are similar to submartingales, as the expected value of the invariant at the beginning of the execution lower bounds the expected value of the invariant at termination. Their proof rule also requires an additional constraint to ensure soundness, but it requires a limiting argument that is more difficult to automate compared to our bounded increment condition. We could relax our condition using a weaker version of OST that generalizes their condition [@Williams91]. Another substantial difference with our approach is that their logic supports non-deterministic choices—ours does not. It is not obvious how we can extend our synthesis approach to the non-probabilistic case as we heavily rely on the concept of filtration, not applicable in the presence of non-determinism. #### Probabilistic model checking. {#probabilistic-model-checking. .unnumbered} In the last twenty years, model checking technology for probabilistic models have made impressive strides [@KwiatkowskaNP11; @CiesinskiB06; @KatoenZHHJ11] (@BaierKatoenBook provide overview). The main advantage of model checking is that it requires nothing from the user; our technique requires a seed expression. However, model checking techniques suffer from the state explosion problem—the time and memory consumption of the model checking algorithm depends on the number of reachable states of the program. Our approach can be used to verify infinite and parametric programs without any performance penalty, as we work purely symbolically. For example, a probabilistic model checker can find the expected running time of the gambler’s ruin process for concrete values of $a$ and $b$ but they cannot deduce the solution for the general case, unlike our technique. #### Invariant synthesis. {#invariant-synthesis. .unnumbered} There are several approaches for synthesizing probabilistic invariants. @KatoenMMM10 propose the first complete method for the synthesis of McIver and Morgan’s probabilistic linear invariants. It is an extension of the constraint solving approach by @ColonSS03 for the synthesis of (non-probabilistic) linear invariants. @ChakarovS13 later extended this work to martingales and ranking supermartingales. @ChakarovS14 propose a new notion of probabilistic invariants that generalizes the notion of supermatingales. They give a synthesis approach based on abstract interpretation, but it is not clear how their techniques can prove properties at termination time. @ChenHWZ15 propose a synthesis tool for verifying Hoare triples in the McIver and Morgan logic, using a combination of Lagrange’s interpolation, sampling, and counterexample guided search. One of the novelties is that they can synthesize non-linear invariants. The main disadvantages is that one must manually check the soundness condition, and one must provide a pre-expectation. For instance, we can apply the method of @ChenHWZ15 to the gambler’s ruin process only if we already know that the expected running time is $a(b - a)$. In contrast, we can deduce $\Exp{\tau} = a(b-a)$ knowing only that $\Exp{\tau}$ is finite. #### Expected running time. {#expected-running-time. .unnumbered} As the termination time of a probabilistic program is a random quantity, it is natural to measure its performance using the average running time. Rough bounds can be obtained from martingale-based termination proofs [@FerrerH15]. Recently, @ChatterjeeFNH16 showed that arbitrary approximations can be obtained from such proofs when the program is linear. They use Azuma’s inequality to obtain a tail distribution of the running time, and later they model check a finite unrolling of the loop. @Monniaux01 propose a similar approach that uses abstract interpretation to obtain the tail distribution of the running time. @KaminskiKMO16 extend Nielson’s proof system [@Nielson87] to bound the expected running time of probabilistic programs. #### Recurrence analysis. {#recurrence-analysis. .unnumbered} Our synthesis approach is similar to the use of recurrences relations for the synthesis of non-probabilistic invariants [@AmmarguellatH90; @RodriguezK07; @Kovacs08]. The main idea is to find syntactic or semantic recurrences relations, and later simplify them using known closed forms to obtain loop invariants. In essence, we apply algebraic identities to simplify the complex martingales from Doob’s decomposition. The difference is that our simplifications are more complex as we cannot always obtain a closed form but a simpler summation. However, we obtain the same closed form when we apply Doob’s decomposition to inductive variables. Another difference is that we rely on the syntactic criteria to identify which values are predictable and which values are random. Conclusion ========== We proposed a novel method for automatically synthesizing martingales expressions for stochastic processes. The basic idea is to transform any initial expression supplied by the user into a martingale using Doob’s decomposition theorem. Our method complements the state-of-the-art synthesis approaches based on constraint solving. On one hand, we always output a martingale expression, we are able to synthesize non-inductive martingales, and since we do not rely on quantifier elimination, we can synthesize polynomial expression of very high degree. On the other hand, we do not provide any completeness result, and the shape of martingale is difficult to predict. We considered several classical case studies from the literature, combining our synthesis method with the optional stopping theorem and non-probabilistic invariants to infer properties at termination time in a fully automatic fashion. Future work includes extending our approach to programs with arrays and improving the tool with automated procedures for checking side-conditions. It would also be interesting to consider richer programs, say distributions with parameters that depend on program state. Another possible direction would be improving the simplification procedures; possibly, the tool could produce simpler facts. Experimenting with more advanced computer-algebra systems and designing simplification heuristics specialized to handling the conditional expectations synthesized by Doob’s decomposition are both promising future directions. It would also be interesting to integrate our method as a special tool in systems for interactive reasoning about probabilistic computations. #### Acknowledgments. {#acknowledgments. .unnumbered} We thank the anonymous reviewers for their helpful comments. This work was partially supported by NSF grants TWC-1513694 and CNS-1065060, and by a grant from the Simons Foundation ($\#360368$ to Justin Hsu). [^1]: A basic version of the optional stopping theorem will suffice for our purposes, but there are alternative versions that don’t require finite expected stopping time and bounded increments. [^2]: We focus on programs for which our method achieves full automation. For instance, we exclude conditional statements because it is difficult to fully automate the resulting simplifications. We note however that there are standard transformations for eliminating conditionals; one such transformation is if-conversion*, a well-known compiler optimization [@Allen:1983]. [^3]: Although most of the ranking supermatingales needed in our case studies are non-linear, the bounded variants are always linear.
--- abstract: 'The two-component Fermi gas is the simplest fermion system displaying superfluidity, and as such is relevant to topics ranging from superconductivity to QCD. Ultracold atomic gases provide an exceptionally clean realisation of this system, where interatomic interactions and atom spin populations are both independently tuneable. Here we show that the finite temperature phase diagram contains a region of phase separation between the superfluid and normal states that touches the boundary of second-order superfluid transitions at a tricritical point, reminiscent of the phase diagram of $^3$He-$^4$He mixtures. A variation of interaction strength then results in a line of tricritical points that terminates at zero temperature on the molecular Bose-Einstein condensate (BEC) side. On this basis, we argue that tricritical points are fundamental to understanding experiments on polarised atomic Fermi gases.' author: - 'M. M. Parish' - 'F. M. Marchetti' - 'A. Lamacraft' - 'B. D. Simons' title: Finite temperature phase diagram of a polarised Fermi condensate --- Over the past decade, experimental progress in the field of cold atomic gases has resulted in unprecedented control over pairing phenomena in two-component Fermi gases. The ability to vary the effective interaction between atoms using magnetically tuned Feshbach resonances has already permitted the experimental investigation of the crossover from a BEC of diatomic molecules to the Bardeen-Cooper-Schrieffer (BCS) limit of weakly-bound Cooper pairs of fermionic atoms [@regal2004; @zwierlein2004; @chin2004; @bourdel2004; @kinast2004; @zwierlein2005]. A natural extension of these studies is an exploration of the Fermi gas with imbalanced spin populations, especially since this system has a far richer phase diagram than the equal spin case. As well as exhibiting a quantum phase transition between the superfluid and normal states, the polarized Fermi gas has been predicted to possess exotic superfluid phases such as the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [@Fulde1964; @larkin1965], where the pairing of fermions occurs at finite centre-of-mass momentum, and the deformed Fermi surface state [@sedrakian2005]. The exact nature of the superfluid states for the polarised Fermi gas is still the subject of considerable debate. However, atomic gases provide an ideal testing ground for this system, since the particle numbers can be varied independently from all other experimental parameters, and pioneering experiments have recently been performed [@zwierlein2006; @partridge2006; @zwierlein2006_2; @shin2006]. Contrast atomic gases with the case of superconductors, where the magnetic field used to generate a spin imbalance (via the Zeeman effect) also couples to orbital degrees of freedom. In this work, we elucidate the finite temperature phase diagram of a polarised Fermi gas. While much insight has been gained from previous theoretical studies [@bedaque2003; @carlson2005; @pao2005; @son2005; @Mizushima2005; @sheehy2006; @mannarelli2006; @pieri2005; @liu2006; @hu2006; @chien2006; @gu2006; @martikainen2006; @iskin2006; @desilva2006; @haque2006; @yi2006; @kinnunen2006], so far a key ingredient of the phase diagram has been overlooked: the tricritical point, at which the phase transition between superfluid and normal states switches from first to second order. By determining the behaviour of the tricritical point as a function of interaction strength, we can completely characterise the topology of the phase diagram without recourse to an extensive numerical treatment. Specifically, we shall focus on the uniform, infinite system, and concern ourselves almost exclusively with the phase boundary between the normal and homogeneous superfluid states. We will, however, discuss the ramifications of the inferred phase diagram for the trapped system. Formalism ========= Experiments to date exploit wide Feshbach resonances and are thus well described by the simplest single-channel Hamiltonian, where the two fermion species interact via an attractive contact potential $$\begin{gathered} \label{eq:ham} \hat{H} - \mu_{\uparrow} \hat{n}_{\uparrow} - \mu_{\downarrow} \hat{n}_{\downarrow} = \sum_{{{\mathbf k}}} \sum_{\sigma = \uparrow,\downarrow} \left(\epsilon_{{{\mathbf k}}} - \mu_{\sigma}\right) c_{{{\mathbf k}} \sigma}^\dag c_{{{\mathbf k}} \sigma} \\ + \frac{g}{V} \sum_{{{\mathbf k}},{{\mathbf k}}',{{\mathbf q}}} c_{{{\mathbf k}}+{{\mathbf q}}/2 \uparrow}^\dag c_{-{{\mathbf k}}+{{\mathbf q}}/2 \downarrow}^\dag c_{-{{\mathbf k}}'+{{\mathbf q}}/2 \downarrow} c_{{{\mathbf k}}'+{{\mathbf q}}/2 \uparrow}\; .\end{gathered}$$ Here, $\epsilon_{{{\mathbf k}}}={{{\mathbf k}}}^2/2m_{f}$ (we set $\hbar=1$ and $k_B=1$), $V$ is the volume, and we define the chemical potential $\mu$ and ‘Zeeman’ field $h$ such that $\mu_{\uparrow} = \mu + h$ and $\mu_{\downarrow} = \mu - h$. At present, only pairing between different hyperfine species of the same atom has been explored experimentally, so we restrict ourselves to a single mass $m_f$. The interaction strength $g$ is expressed in terms of the s-wave scattering length $a$ using the prescription: $$\frac{m_f}{4\pi a} = \frac{1}{g} + \frac{1}{V}\sum_{{{\mathbf k}}} \frac{1}{2\epsilon_{{{\mathbf k}}}} \; .$$ We also derive the Fermi momentum using the average density $n/2 \equiv (n_{\uparrow}+n_{\downarrow})/2$, so that $k_F = (3\pi^2 n)^{1/3}$. Throughout our calculations, we will keep $n$ fixed. The full phase diagram is parameterised by just a few observables: the temperature $T\equiv 1/\beta$, the interaction strength $1/k_Fa$, and the density difference or ‘magnetisation’ $m\equiv n_{\uparrow}-n_{\downarrow}$. To determine the position of the phase boundaries, we must minimise the mean-field free energy density $$\begin{gathered} \label{eq:energy} \Omega^{0} = -\frac{\Delta^2}{g} + \frac{1}{V}\sum_{{{\mathbf k}}} (\xi_{{{\mathbf k}}} - E_{{{\mathbf k}}}) \\ -\frac{1}{\beta V}\sum_{{{\mathbf k}}} \left[\ln\left(1+e^{-\beta(E_{{{\mathbf k}}}-h)}\right) + \ln\left(1+e^{-\beta(E_{{{\mathbf k}}}+h)}\right) \right],\end{gathered}$$ with respect to the BCS order parameter $\Delta$, where $\xi_{{{\mathbf k}}}=\epsilon_{{{\mathbf k}}}-\mu$ and $E_{{{\mathbf k}}}=\sqrt{\xi_{{{\mathbf k}}}^2+\Delta^2}$. Such a mean-field analysis provides a reasonable description of the zero temperature phase diagram, but at finite temperature, it neglects the contribution of non-condensed pairs to both the density $n = -\partial \Omega/\partial \mu$ and magnetisation $m = -\partial \Omega/\partial h$. This contribution is necessary to approach the transition temperature of an ideal Bose gas in the molecular limit, and can be included in the non-condensed phase ($\Delta=0$) through the Nozières-Schmitt-Rink (NSR) fluctuation correction to the energy [@nozieres1985] $$\label{fluct} \left.\Omega^{1}\right\|_{\Delta = 0} = \frac{1}{\beta V} \sum_{{{\mathbf q}},i\omega}\ln\Gamma^{-1}({{\mathbf q}},i\omega) \; ,$$ with $$\begin{gathered} \Gamma^{-1}({{\mathbf q}},i\omega) = -\frac{1}{g} \\ - \frac{1}{2V}\sum_{{{\mathbf p}}} \frac{\tanh{[\frac{\beta}{2}(\xi_{{{\mathbf p}}} + h)] + \tanh[\frac{\beta}{2}(\xi_{{{\mathbf p}}+{{\mathbf q}}} - h)]}}{i\omega + \xi_{{{\mathbf p}}} + \xi_{{{\mathbf p}}+{{\mathbf q}}}} \; .\end{gathered}$$ This gives an estimate of the effect of pair fluctuations on the second order phase boundary (but not the first order boundary, where $\Delta \neq 0$). ![The zero temperature phase diagram within mean-field theory for both Zeeman field $h/\varepsilon_F$ and magnetisation $m/n$ (inset) versus interaction $1/k_Fa$. There are four different phases: the normal (N) state, the phase-separated (PS) state, the ordinary superfluid (SF) and the magnetised superfluid (SF$_{\textrm{M}}$). Above the line $h/\varepsilon_F = 2^{-1/3}$, the normal state is completely polarised ($m/n=1$). The red and black lines enclosing the PS state are both first-order phase boundaries, while the SF$_{\textrm{M}}$-N transition is second-order, and the SF-SF$_{\textrm{M}}$ transition (green line) is at least third-order. The tricritical point is represented by orange circles at $1/k_Fa=2.368$ with $h/\varepsilon_F = 6.876$ or $m/n=1$. \[fig:zeroT\]](fT0_phd_paper){width="45.00000%"} Phase diagram for the uniform case ================================== Considerable insight can be gained by first examining the zero temperature mean-field phase diagram, as shown in Fig. \[fig:zeroT\]. The general structure parallels that of the two-channel case found in Ref. [@sheehy2006]. Since there is a gap in the quasiparticle excitation spectrum $E_{{{\mathbf k}}}$ of the unpolarised superfluid, the superfluid ground state will remain unchanged for $h<\mathrm{min}_{{\mathbf{k}}} E_{{\mathbf{k}}}$. We see that the $m=0$ superfluid line in the inset of Fig. \[fig:zeroT\] corresponds to an area in the $h/\varepsilon_F$ versus $1/k_Fa$ diagram, which expands as $1/k_Fa$ increases. A key feature of the strong coupling side is that for $1/k_F a\gtrsim 1$ the superfluid state is able to sustain a finite population of majority quasiparticles. This “gapless” [@pao2005; @son2005] superfluid phase is only stable for $\mu<0$ and it thus possesses only one Fermi surface. In the extreme BEC limit, this state is straightforwardly understood as an almost ideal mixture of bosonic pairs and fermionic quasiparticles. However, as we move towards unitarity, the bosons and fermions begin to interact more strongly, leading eventually to a first-order phase transition to the normal state. Here, a system with fixed $m$ will undergo phase separation into normal and superfluid regions if $m_N<m<m_S$, where $m_{N,S}$ denotes the magnetisation in the normal and superfluid phases at $h_c$, the critical field for the first-order transition. In the BCS limit ($\mu = \varepsilon_F$), $h_c=\Delta/\sqrt{2}$ which is less than the quasiparticle gap, so the superfluid state is unmagnetised $m_S=0$, and phase separation occurs for arbitrarily low magnetisation, consistent with Ref. [@bedaque2003]. For the moment we neglect the FFLO state, but will return to this point later. A crucial observation is that the line $m/n=1$ to the right of the region of phase separation can be thought of as a continuous zero temperature transition at which the condensate is totally depleted. It is thus natural to identify the point on $m/n=1$ where phase separation starts as a tricritical point. Indeed a Landau expansion of the free energy both confirms this and identifies the tricritical point at $1/k_Fa=2.368$. ![Finite temperature phase diagram as a function of magnetisation $m/n$ and interaction $1/k_Fa$. The plane at temperature $T=0$ is the phase diagram in Fig. \[fig:zeroT\]. The yellow line represents the locus of tricritical points calculated in the mean-field approximation, while the orange tricritical line corresponds to mean-field theory plus pair fluctuations. The fluctuation correction breaks down in the unitarity regime $-1 < 1/k_Fa < 1$, and is thus shown as a dotted line. The slice at $1/k_Fa = -1$ is based on a mean-field calculation and it shows the region of phase separation terminating in a tricritical point (yellow circle) at finite temperature, followed by a second-order phase transition from the superfluid to normal state. Note that the boundary between the FFLO and normal states (blue line) defines a small region of FFLO phase confined to the BCS side of the crossover, as explained in the text.[]{data-label="fig:tricritical"}](new_Fig2){width="45.00000%"} ![Finite temperature phase diagram for the two-channel model of a narrow Feshbach resonance, where the coupling between open and closed channels is weak: $\gamma = 0.1$. The effective interaction is parameterised by the detuning $\delta/\varepsilon_F$. The colour scheme for tricritical lines is the same as in Fig. \[fig:tricritical\].[]{data-label="fig:tricritical_2chan"}](ftricritical_2chan2){width="45.00000%"} With this background, we now turn to the analysis of the fate of the tricritical point when temperature is finite, beginning with the mean-field description. It is well known that there exists a finite temperature tricritical point in the BCS limit $1/k_Fa\to -\infty$, which is a natural consequence of having a first-order transition from the superfluid to normal state at $T=0$ and a second-order transition at $m=0$. First studied by Sarma in the context of superconductivity in the presence of a magnetic field [@sarma1963], the BCS tricritical point is located at $(T_{\textrm{crit}}/\Delta,h_{\textrm{crit}}/\Delta) = (0.3188,0.6061)$ [@casalbuoni2004], where $\Delta = \frac{8}{e^2}\varepsilon_F\exp\left[-\pi/2\|k_F a\|\right]$ (i.e. at weak coupling all energies scale with $\Delta$). This corresponds to a magnetisation $m=2\nu(\varepsilon_F)h_{\textrm{crit}}$, where $\nu(\varepsilon_F) = m_f^{3/2} \sqrt{\varepsilon_F}/ \sqrt{2}\pi^2$ is the Fermi surface density of states. To investigate how the BCS tricritical point is related to the one at zero temperature, we must develop a perturbative expansion of Eq. (\[eq:energy\]) for small $\Delta$ and general $1/k_Fa$. Doing so, one finds (Fig. \[fig:tricritical\]) that the tricritical point at $m/n=1$ is connected to that in the BCS limit by a line of tricritical points that passes through a maximum somewhere in the ‘unitarity’ regime $-1<1/k_Fa<1$. Moreover, for any given value of $1/k_Fa\leq 2.368$, the $(T/\varepsilon_F,m/n)$ phase diagram is highly reminiscent of the $^3$He-$^4$He system, with $m/n$ playing the role of the fraction of $^3$He. This is not surprising, as the finite $m$ system corresponds in general to a mixture of bosonic pairs and fermionic quasiparticles. Note that even the gapped superfluid can be magnetised at finite temperature due to thermal excitation of quasiparticles. Of course, at $m=0$ the transition into the superfluid state is second order at any point in the BCS-BEC crossover. It is interesting to examine how the FFLO phase fits in with the basic topology of the phase diagram. In the BCS limit, we already know that the point where the FFLO-normal phase boundary meets the normal-superfluid boundary asymptotes to the tricritical point [@casalbuoni2004]. Assuming that the transition from the FFLO state to the normal state is second-order (although Ref. [@combescot2004] found it to be weakly first order, this will make a relatively small difference), and performing a mean-field analysis, we find that the FFLO point of intersection leaves the finite temperature tricritical point with increasing interaction (see Fig. \[fig:tricritical\]), leading eventually to the extinction of the FFLO phase at $k_Fa=-0.35$. Note that although this treatment is somewhat approximate, as we have taken the SF-FFLO boundary to be the same as the SF-N boundary in the absence of FFLO, the point of intersection will coincide with that derived from a complete mean-field analysis. Moreover, despite all our assumptions, we expect the detachment of the point of intersection from the tricritical point and the eventual disappearance of FFLO to be robust features, since in the BEC regime we essentially have a mixture of bosons and fermions. The inclusion of the fluctuation contribution Eq. (\[fluct\]) is crucial for recovering the extreme BEC limit, where it is clear that the (second-order) transition temperature asymptotes to $T_{\mathrm{BEC}}(m)=T_{\mathrm{BEC}}\left(1-m/n\right)^{2/3}$ (with $T_{\mathrm{BEC}}\sim 0.218 \varepsilon_F$), the ideal BEC temperature of a gas of bosons of density $n_\downarrow=(n-m)/2$ and mass $2m_f$. More importantly, we find that fluctuations shift the mean-field tricritical line to lower temperatures and magnetisations on the BEC side, while leaving the tricritical points on the BCS side largely unchanged, as expected. However, in a broad region around unitarity, we find that the approximation underlying Eq. (\[fluct\]) generally leads to non-monotonic behavior of $m(h)$, with $m(h>0)<0$ for small $h$. We interpret this behaviour as a breakdown of the NSR treatment, yielding an unphysical compressibility matrix $-\partial^2\Omega/\partial \mu_{\sigma}\partial \mu_{\sigma'}$ that is not positive semi-definite. To address this problem, we note that the NSR scheme is a controlled approximation when we introduce resonant scattering with a finite width, with the width being a small fraction of the Fermi energy [@andreev2004]. The simplest such description is provided by the two-channel model [@timmermans2001; @holland2001]. The two-channel description of scattering depends upon two parameters: a detuning $\delta/\varepsilon_F$ describing the distance from the resonance, and a width $\gamma$ of the resonance measured in units of the Fermi energy. The one-channel description is recovered in the $\gamma\to\infty$ limit, while the treatment of Gaussian fluctuations is essentially perturbative in $\gamma$, with $\Gamma^{-1}$ in Eq. (\[fluct\]) being replaced with $\frac{{\mathbf{q}}^2}{4m}-i\omega_m+\gamma\Gamma^{-1}({\mathbf{q}},i\omega_{m})$, so in this case the NSR treatment is expected to be accurate. The resulting phase diagram is shown in Figure \[fig:tricritical\_2chan\]. The zero temperature phase diagram coincides with the result of Ref. [@sheehy2006]. With fluctuations accounted for, and for sufficiently small $\gamma$, we now find a well-behaved line of tricritical points spanning the crossover region. We expect that the true phase boundary at $\gamma\to\infty$ is qualitatively similar. ![Phase diagram at $1/k_Fa=0$ in the $\mu/h$-$T/h$ plane. The red and black lines are first- and second-order phase boundaries, respectively. The arrows at constant $T/h$ represent the trajectories followed when going from the centre to the edges of a trapped gas. The two trajectories correspond to two different magnetisations of the gas: one greater and one less than the tricritical point $h_{\textrm{crit}}$.[]{data-label="fig:tri_trap"}](ftri_trap){width="42.00000%"} Implications for experiment =========================== We now discuss the consequences of our results for trapped gases studied in experiment. Modeling the trapped gas by the local density approximation (LDA), the spatial dependence of the density induced by the trapping potential $V({{\mathbf r}})$ is accounted for by a spatially-varying chemical potential $\mu({{\mathbf r}}) = \mu - V({{\mathbf r}})$, with $h$ kept constant. In the $\mu/h$-$T/h$ plane, we thus move on a horizontal line (see Fig. \[fig:tri\_trap\]). At sufficiently low temperatures, a trapped gas will consist of a superfluid core surrounded by the normal state. The transition between normal and superfluid states in the trap can be either second or first order, depending on whether $T/h$ is above or below the tricritical point. Moreover, as long as the temperature is non-zero, we can always find a sufficiently small $h$ so that $T/h$ lies above the tricritical point. This leads us to a key point: if a trapped gas at a given temperature and magnetisation has a first-order transition between its normal and superfluid phases, then we will always cross the tricritical point by decreasing the magnetisation at fixed temperature. We emphasise that there are qualitative differences between first and second order transitions in a trap: the former yields a discontinuity in the density and magnetization at the phase interface, resulting in a form of phase separation as seen in recent experiments [@zwierlein2006; @partridge2006; @zwierlein2006_2; @shin2006], while the latter possesses a density that varies smoothly in space. Therefore, the magnetisation and temperature at which a tricritical point is crossed should be detectable experimentally. In fact, a critical magnetisation for the onset of phase separation in a trap has been observed experimentally [@partridge2006], and a calculation by Chevy supports the idea that this coincides with crossing a tricritical point [@chevy2006]. In addition, the order of the transition will have an impact on experiments that use phase separation as a signature of superfluidity [@zwierlein2006_2]. The presence of a first-order transition in the trap can be even more pronounced if the density discontinuities result in a breakdown of LDA. Experiments on highly elongated traps already provide evidence for such a breakdown [@partridge2006], and one requires the addition of surface energy terms at the phase interface to successfully model the trapped density profiles [@desilva2006_2]. An outstanding issue is the experimental detection of the gapless SF$_{\textrm{M}}$ phase. While optically probing the momentum distribution of the minority species is one promising method for detecting SF$_{\textrm{M}}$ [@yi2006_3], another possibility is to study density correlations using, for example, shot noise experiments as suggested in Ref. [@altman2004]. A simple mean-field calculation gives (for the uniform system): $$\begin{aligned} C_{\uparrow\downarrow}({{\mathbf k_1}},{{\mathbf k_2}}) & \equiv \langle \hat{n}_{\uparrow}({{\mathbf k_1}}) \hat{n}_{\downarrow}({{\mathbf k_2}})\rangle - \langle \hat{n}_{\uparrow}({{\mathbf k_1}}) \rangle \langle \hat{n}_{\downarrow}({{\mathbf k_2}})\rangle \\ & = \delta_{{{\mathbf k_1}},-{{\mathbf k_2}}} \frac{\Delta^2}{4E_{{{\mathbf k_1}}}^2} [1-f(E_{{{\mathbf k_1}}}+h)-f(E_{{{\mathbf k_1}}}-h)]^2\end{aligned}$$ where $f(E)$ is the Fermi-Dirac distribution. At $T=0$, the result is a ‘hole’ in the correlation function for momenta less than the Fermi wavevector of the majority quasiparticles. Such a measurement would therefore constitute both a confirmation of the SF$_{\textrm{M}}$ phase and a vivid demonstration of the blocking effect of quasiparticles on $\left(+{\mathbf{k}},-{\mathbf{k}}\right)$ pairing. In conclusion, we have determined the structure of the finite temperature phase diagram of the two component Fermi gas, as a function of both interaction strength and population imbalance, finding a region of phase separation terminating in a tricritical point for general coupling in the BCS-BEC crossover. A secondary result of our work is the demonstration that the NSR scheme yields unphysical results in a broad region around unitarity. This is significant, as it is widely viewed as offering a smooth, albeit uncontrolled approximation throughout the crossover. We emphasize that there is no a priori reason to believe in the accuracy of the NSR scheme without introducing an additional parameter, as we have done here. The Ginzburg criterion governing the smallness of fluctuation corrections is satisfied in both the BCS limit where it takes the form $(T_c/\varepsilon_F)^2\ll 1$, and in the BEC limit where $k_F a\ll 1$ is the relevant criterion. But at unitarity the shift in the transition temperature relative to the mean field value will be of order $\varepsilon_F$. At the same time the upper critical dimension at the tricritical point is three, so we may expect that our results there will be little changed. Finally, we have argued that these tricritical points play an important role in experiments on trapped Fermi gases (see, also, the subsequent related work on trapped gases at unitarity by Gubbels et al. [@gubbels2006]). Indeed, a recent comprehensive study of the temperature dependence of the phase-separated state at unitarity has yielded experimental results consistent with the phase diagram outlined here [@partridge2006_2]. We are grateful to P. B. Littlewood for stimulating discussions, and J. Keeling for help with the numerics. This work has been supported by EPSRC. [44]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , **, (). , , , , , , , (). , , , , , , , , (). , , , , , , , , , , (). , , , , , , (). , , , , , , (). , , (). , , (). , , , , , (). , , , , , (). , , , , , , (). , , , , , (), . , , , , , , (). , , , , (). , , (). , , , , (). , , (). , , , , (). , , (). , , , . , , (). , , (). , , (). , , , , , (), . , , , . , , (). , . , , (). , , (). , , (). , , , , (). , , (). , , (). , , (). , , (). , , , , (). , , , , , (). , , , , , (). , , (). , , (). , , (). , , , **, (). , , , . , , , , , , .
--- abstract: 'Model-based process design of ion-exchange simulated moving bed (<span style="font-variant:small-caps;">iex-smb</span>) chromatography for center-cut separation of proteins is studied. Use of nonlinear binding models that describe more accurate adsorption behaviours of macro-molecules could make it impossible to utilize triangle theory to obtain operating parameters. Moreover, triangle theory provides no rules to design salt profiles in <span style="font-variant:small-caps;">iex-smb</span>. In the modelling study here, proteins (, ribonuclease, cytochrome and lysozyme) on the chromatographic columns packed with strong cation-exchanger SP Sepharose FF is used as an example system. The general rate model with steric mass-action kinetics was used; two closed-loop <span style="font-variant:small-caps;">iex-smb</span> network schemes were investigated (, cascade and eight-zone schemes). Performance of the <span style="font-variant:small-caps;">iex-smb</span> schemes was examined with respect to multi-objective indicators (, purity and yield) and productivity, and compared to a single column batch system with the same amount of resin utilized. A multi-objective sampling algorithm, Markov Chain Monte Carlo (<span style="font-variant:small-caps;">mcmc</span>), was used to generate samples for constructing the Pareto optimal fronts. <span style="font-variant:small-caps;">mcmc</span> serves on the sampling purpose, which is interested in sampling the Pareto optimal points as well as those near Pareto optimal. Pareto fronts of the three schemes provide the full information of trade-off between the conflicting indicators of purity and yield. The results indicate the cascade <span style="font-variant:small-caps;">iex-smb</span> scheme and the integrated eight-zone <span style="font-variant:small-caps;">iex-smb</span> scheme have the similar performance that both outperforms the single column batch system.' author: - 'Qiao-Le He' - Zhaoxi Sun - 'Liming Zhao[^1]' bibliography: - '/home/kim/Bazinga/Templates/references.bib' date: 'Oct. 10, 2019' title: 'Model-based process design of a ternary protein separation using multi-step gradient ion-exchange SMB chromatography' --- Introduction ============ Protein purification and separation are major concerns in the downstream processing of pharmaceutical industries [@carta2010protein; @scopes2013protein]. They require a series of processes aimed at isolating single or multiple types of proteins from complex intermediates of process productions. Choosing a purification method, depending on the purpose for which the protein is needed, is a critical issue. Chromatography is a prevailing separation technology. Simulated moving bed (<span style="font-variant:small-caps;">smb</span>) chromatography [@broughton1961continuous] is an excellent alternative to the single column batch chromatography, because of its continuous counter-current operation, its potential to enhance productivities and to reduce solvent consumption [@seidel2008new; @rajendran2009simulated]. According to the position of the columns relative to the ports (, feed, raffinate, desorbent, extract), the process is divided into four zones (, I, II, III, IV), each has a specific functionality in the separations (see Fig. \[fig:4-zone\]). ![Schematic of four-zone <span style="font-variant:small-caps;">smb</span> chromatography[]{data-label="fig:4-zone"}](scheme_binary){width="45.00000%"} Multicolumn continuous chromatography is also commonly used in the protein separation and purification. For instance, there are sequential multicolumn chromatography (<span style="font-variant:small-caps;">smcc</span>) [@ng2014design], periodic counter-current packed bed chromatography (<span style="font-variant:small-caps;">pcc</span>) [@pollock2013optimising], gradient with steady state recycle (<span style="font-variant:small-caps;">gssr</span>) process [@silva2010new], multicolumn solvent gradient purification (<span style="font-variant:small-caps;">mcsgp</span>) [@aumann2007continuous] and capture <span style="font-variant:small-caps;">smb</span> [@angarita2015twin]. However, the prominent features of <span style="font-variant:small-caps;">smb</span> processes are based on the prerequisite that optimal operating conditions can be determined, which can be very challenging in practice. Determination of operating conditions is predominantly based on classical and extended triangle theories [@storti1993robust], which are originally derived from the ideal chromatographic model with linear isotherm. Powerful criteria to design processes with nonlinear isotherms (, Langmuir and Bi-Langmuir) have been elaborated in the past few years [@charton1995complete; @mazzotti1997optimal; @antos2001application]. Versatile results on triangle theory for the design of <span style="font-variant:small-caps;">smb</span> processes have been published [@nowak2012theoretical; @kim2016combined; @lim2004optimization; @kazi2012optimization; @bentley2013prediction; @bentley2014experimental; @sreedhar2014simulated; @toumi2007efficient; @silva2015modeling; @kiwala2016center]. Substances separated by <span style="font-variant:small-caps;">smb</span> processes have recently been evolved from monosaccharides to proteins and other macro-molecules. The adsorption behaviors of macro-molecules, as observed both in experiments [@clark2007new] and molecular dynamic simulations [@dismer2010structure; @liang2012adsorption; @lang2015comprehensive], are much more complex than that described by the linear isotherm, the Langmuir kinetics or the steric mass-action (<span style="font-variant:small-caps;">sma</span>) model [@brooks1992steric]. Macro-molecules can undergo conformation changes in the mobile phase, orientation changes on the functional surfaces of stationary phase and aggregation in the whole process. [@clark2007new] used an acoustically actuated resonant membrane sensor to monitor the changes of surface energy and found that the changes persist long after mass loading of the protein has reached the steady state. Therefore, more sophisticated adsorption models, such as the multi-state <span style="font-variant:small-caps;">sma</span> [@diedrich2017multi] based on the spreading model [@ghosh2013zonal; @ghosh2014zonal], have been proposed to describe dynamic protein-ligand interactions. Although these kinetics describe more accurately the adsorption behaviors by taking orientation changes into consideration, the strong nonlinearity of kinetics exerts tremendous difficulties in deriving analytical formulae to calculate flowrates, as it is in the triangle theory. Ion-exchange (<span style="font-variant:small-caps;">iex</span>) mechanism is commonly used to separate charged components in chromatography. In <span style="font-variant:small-caps;">iex</span> chromatography (both single column and <span style="font-variant:small-caps;">smb</span> processes), an auxiliary component acting as a modifier is used to change the electrostatic interaction force between functional groups and macro-molecules; sodium chloride is most frequently used. The higher the salt concentration is, the lower electrostatic force is, such that the binding affinity is decreased and bounded components are successively eluted [@lang2015comprehensive]. In the single column <span style="font-variant:small-caps;">iex</span> processes, this principle can be either implemented in an isocratic mode or a linear gradient mode. The linear gradient mode has frequently been applied to achieve better separation capability [@osberghaus2012optimizing]. While in the <span style="font-variant:small-caps;">iex-smb</span> processes, the isocratic mode (, the salt strength is identical in all zones) was originally applied. The gradient mode can be applied, by using a salt strength in the desorbent stream that is stronger than that in the feed (cf. Fig. \[fig:4-zone\]). Thus, high salt concentration in zone I and II (high salt region), low salt concentration in zone III and IV (low salt region). As shown in the studies, applying the gradient mode can further improve the separation performance [@antos2001application; @houwing2002effect; @houwing2003positioning; @li2007proteins]. The salt profiles of the gradient mode are referred to as two-step salt gradient in this work; and derivations of multi-step gradient to ternary separations are intuitive. However, the construction of the multi-step salt gradient imposes difficulty on the design of <span style="font-variant:small-caps;">iex-smb</span> processes. The difficulty can be detoured by using the open-loop multicolumn chromatography, where it is possible to apply linear salt gradients. The open-loop feature and possibility of linear gradient make the application potential of multicolumn continuous chromatography in the downstream processing unarguable [@faria2015instrumental]. As dealing with the model complexity induced by linear gradients is much easier than that by multi-step ones; moreover, the open-loop feature renders more degrees of freedom in the process designs. Conventional four-zone <span style="font-variant:small-caps;">smb</span> processes are tailored for binary separations. Ternary separations are requested in many applications; the selection of network configurations is application-specific. Ternary separations, also referred to as center-cut separations, can be achieved by cascading two four-zone <span style="font-variant:small-caps;">smb</span> units [@wooley1998nine; @nicolaos2001application; @nowak2012theoretical]. Cascade schemes require designing and operating two four-zone <span style="font-variant:small-caps;">smb</span> processes, which can be laborious and costly. Hence, integrated schemes have been developed using a single multi-port valve and fewer pumps than cascade schemes [@seidel2008new; @da2016evaluation]. Moreover, process design is simplified due to simultaneous switching of all columns. Due to the complexity of network configurations, in addition to the kinetic nonlinearity and multi-step gradient, a comprehensive column model, fast numerical solver and efficient optimizer are required. In this study, we shall present a model-based process design of <span style="font-variant:small-caps;">iex-smb</span> units for separating a protein mixture of ribonuclease, cytochrome and lysozyme, using cation-exchange columns packed with SP Sepharose FF beads. Two network configurations for the ternary separation will be used, that is, a cascade scheme and an eight-zone scheme. For comparison, a conventional single column batch system, with the same column geometry and resin utilization, will be studied. In this study, <span style="font-variant:small-caps;">smb</span> processes are modeled by weakly coupling individual models of the involved columns, , one large equation system is set up and sequentially solved [@he2018efficient]. The code has been published as open-source software, <span style="font-variant:small-caps;">cadet-smb</span>. The operating conditions of the network configurations will be optimized by a stochastic algorithm, Markov Chain Monte Carlo (<span style="font-variant:small-caps;">mcmc</span>), with respect to conflicting objectives of purity and yield. Pareto fronts are computed for illustrating the best compromises between the two conflicting performance indicators. Unlike multi-objective optimization algorithms (, non-dominated sorted genetic algorithm, strength Pareto evolutionary algorithms) that try to eliminate all the non-dominated points during optimization, <span style="font-variant:small-caps;">mcmc</span> serves on the sampling purpose, which is interested in sampling the Pareto optimal points as well as those near Pareto optimal. For samping purpose, <span style="font-variant:small-caps;">mcmc</span> not only accepts proposals with better objective value, but also accepts moves heading to non-dominated points with certain probability. Therefore, <span style="font-variant:small-caps;">mcmc</span> renders greater information for process design. Besides, uncertainties of parameters that is related to the robustness of process design can be examined, though the information is not involved in this study. As shown in the Pareto fonts, the performance indicators of the single column are dominated by that of both cascade and eight-zone schemes. The performances of the two <span style="font-variant:small-caps;">iex-smb</span> schemes are quite similar, though the cascade scheme has a slight advantage over the eight-zone scheme. Theory ====== This section introduces the transport model, binding kinetics, load-wash-elution mode of the single column, <span style="font-variant:small-caps;">smb</span> network connectivity, performance indicators and optimization algorithm. Transport model --------------- The transport behavior of proteins in the column is described by means of the general rate model (<span style="font-variant:small-caps;">grm</span>), which accounts for various levels of mass transfer resistance [@guiochon2006fundamentals]. <span style="font-variant:small-caps;">grm</span> considers convection and axial dispersion in the bulk liquid, as well as film mass transfer and pore diffusion in the porous beads: \[eq:GRM\] $$\begin{aligned} \frac{\partial c_i^j}{\partial t} & = -u_{\text{int}}^j \frac{\partial c_i^j}{\partial z} + D_{\text{ax}}^j \frac{\partial^2 c_i^j}{\partial z^2} - \frac{1-\varepsilon_c}{\varepsilon_c} \frac{3}{r_p} k_{f,i}^j \left(c_i^j - c_{p,i}^j(r\!=\!r_p)\right) \\ \frac{\partial c_{p,i}^j}{\partial t} & = D_{p,i}^j \left(\frac{\partial^2 c_{p,i}^j}{\partial r^2} + \frac{2}{r}\frac{\partial c_{p,i}^j}{\partial r}\right) - \frac{1-\varepsilon_p}{\varepsilon_p}\frac{\partial q_i^j}{\partial t} \end{aligned}$$ where $c_i^j$, $c_{p,i}^j$ and $q_i^j$ denote the interstitial, stagnant and stationary phase concentrations of component $i\in \{1,\dots,M\}$ in column $j \in \{1, \dots, N\}$. Furthermore, $z \in [0,L]$ denotes the axial position where $L$ is the column length, $r \in [0,r_p]$ is the radial position, $r_p$ is the particle radius, $t$ time, $\varepsilon_c$ column porosity, $\varepsilon_p$ particle porosity, $u_{\text{int}}^j$ interstitial velocity, $D_{\text{ax}}^j$ is the axial dispersion coefficient, $D_{p,i}^j$ the effective pore diffusion coefficient, and $k_{f,i}^j$ the film mass transfer coefficient. At the column inlet and outlet, Danckwerts boundary conditions [@Barber1998Boundary] are applied: $$\begin{cases} \left. \dfrac{\partial c_i^j}{\partial z} \right\|_{z=0} &= \dfrac{u_{\text{int}}^j}{D_{\text{ax}}^j} \left(c_i^j(z\!=\!0) - c_{\text{in},i}^j \right) \\ \left. \dfrac{\partial c_i^j}{\partial z} \right\|_{z=L} &= 0 \end{cases} \label{eq:Danckwerts_column}$$ where $c_{\text{in},i}^j$ is the inlet concentration of component $i$ in column $j$; the calculation of inlet concentration in <span style="font-variant:small-caps;">smb</span> chromatography is deferred to Eq. . The boundary conditions at the particle surface and center are given by: $$\begin{cases} \left. \dfrac{\partial c_{p,i}^j}{\partial r} \right\|_{r=r_p} &= \dfrac{k_{f,i}^j}{\varepsilon_p D_{p,i}^j} \left(c_i^j - c_{p,i}^j (r\!=\!r_p)\right) \\ \left. \dfrac{\partial c_{p,i}^j}{\partial r} \right\|_{r= 0} &= 0 \end{cases} \label{eq:Danckwerts_surface}$$ Binding kinetics ---------------- The steric mass-action (<span style="font-variant:small-caps;">sma</span>) model has been widely reported for predicting and describing nonlinear ion-exchange adsorption of proteins; the relationship between the stagnant liquid $c_{p,i}^j$ and the stationary phase $q_i^j$ is described as follows: $$\frac{\dd q_i^j}{\dd t} = k_{a,i}\, c_{p,i}^j\, (\bar{q}_0^j)^{\nu_i} - k_{d,i}\, q_i^j\, (c_{p,0}^j)^{\nu_i} \label{eq:sma}$$ where $k_{a,i}$ and $k_{d,i}$ denote adsorption and desorption coefficients, $\nu_i$ is the characteristic charge of the adsorbing molecules, and $\sigma_i$ the shielding factor. $c_0$, $c_{p,0}$, $q_0$ denote the salt concentration in the three different phases. Moreover, $\bar{q}_0$ denotes counter ions that are not shielded and available for protein binding, $$\bar{q}_0 = q_0 - \sum_{i=1}^M \sigma_i q_i$$ It can be expressed with electro-neutrality condition, $$\bar{q}_0 = \Lambda - \sum_{i=1}^{M} (\nu_i + \sigma_i) q_i$$ where $\Lambda$ is the ionic capacity. Single column batch system -------------------------- There are various possibilities to operate a single column in batch manner. The specific one described here is used for comparison with <span style="font-variant:small-caps;">smb</span> processes; we refer you the protocol to [@osberghaus2012optimizing]. Apart from regeneration steps, an operating cycle of the single column could consist of three phases: load, wash and elution, see Fig. \[fig:batch\_salt\_scheme\]. In phase I, the column is equilibrated with the running buffer, then the protein solution is injected to the column for a $t_{\text{load}}$ time. It is then followed by a wash step (II), that is, pumping through the column with running buffer for a $t_{\text{wash}}$ time. Afterwards, gradient modes (involving zero gradient) are set to elute the bounded protein for a $t_{\text{elute}}$ time. Elution (III) can be implemented by multi-step solvent gradients. As depicted in Fig. \[fig:batch\_salt\_scheme\] as an example, there are two parts (, IIIa, IIIb) with linear gradients but different slopes ($m_1$, $m_2$), and eventually an isocratic part (IIIc). Multi-step gradients are used, because in center-cut separations peak overlaps at the front and at the back of the target component need to be minimized. With the regeneration of the column (IV), it comprises an operating cycle of the single. (6,4) (0.5,0.5)[(0,1)[3]{}]{} (0.5,0.5)[(1,0)[6]{}]{} (0,3.4)[$c_i \quad [\si{\mole\per\cubic\metre]}$]{} (6.1,0)[$t_{\text{tot}}$]{} (0.5,1.5)[(1,0)[1.5]{}]{} (1,0.5)[(0,1)[1]{}]{} (0.8,0)[$t_{\text{load}}$]{} (0.65,0.8)[I]{} (2,0.5)[(0,1)[1.5]{}]{} (1.8,0)[$t_{\text{wash}}$]{} (1.35,0.8)[II]{} (1.2,1.9)[$c_{\text{init},0}$]{} (2,2)[(5,1)[2]{}]{} (2,2)[(1,0)[1.3]{}]{} (2.4,2.1)[$m_1$]{} (4,0.5)[(0,1)[1.9]{}]{} (4,2.4)[(1,0)[1.3]{}]{} (4,2.4)[(2,1)[1.5]{}]{} (4.4,2.5)[$m_2$]{} (3.6,0)[$t_{\text{elute}}^1$]{} (2.5,0.8)[IIIa]{} (4.2,0.8)[IIIb]{} (5.52,0.8)[IIIc]{} (6.5,0.8)[IV]{} (5.5,0.5)[(0,1)[2.64]{}]{} (5.5,3.15)[(1,0)[0.7]{}]{} (5.1,0)[$t_{\text{elute}}^2$]{} (6.2,0.5)[(0,1)[2.64]{}]{} SMB network connectivity ------------------------ In <span style="font-variant:small-caps;">smb</span> chromatography two adjacent columns, $j$ and $j\!+\!1$, are connected via a node $j$, see Fig. \[fig:adjacent\]. A circular <span style="font-variant:small-caps;">smb</span> loop is closed when two column indices point to the same physical column (, by identifying column $j=N\!+\!1$ with column $j=1$), see Fig. \[fig:4-zone\]. In this work, node $j$ is located at the downstream side of column $j$ and upstream of column $j\!+\!1$. At each node only one or none of the feed (F), desorbent (D), raffinate (R), or extract (E) streams exists at a time. The inlet concentration of component $i$ in column $j\!+\!1$ is calculated from mass balance of the node $j$: ![Schematic of network connection between two adjacent columns.[]{data-label="fig:adjacent"}](adjacent_column){width="40.00000%"} $$\begin{aligned} c_{\text{in},i}^{j+1} = \frac{c_{\text{out},i}^j Q^j + \delta_i^j}{Q^{j+1}} \label{eq:node_balance}\end{aligned}$$ where $c_{\text{out},i}^j = c_i^j(t, z\!=\!L)$ denotes the outlet concentration of component $i$ in column $j$, $Q^j = \varepsilon_c u_{\text{int}}^j \pi d_c^2/4$ the volumetric flowrates and $d_c$ the column diameter. Meanwhile, $\delta^j$ determines the current role of node $j$ (, F, D, R, E or none): $$\begin{aligned} \delta^j_i = \left\{ \begin{array}{l@{\quad \quad}l} \phantom{-}c_{\text{in,i}}^F Q^F & \text{feed} \\ \phantom{-}c_{\text{in,i}}^D Q^D & \text{desorbent} \\ -c_{\text{out,i}}^j Q^R & \text{raffinate} \\ -c_{\text{out,i}}^j Q^E & \text{extract} \\ \phantom{-}0 & \text{none} \end{array} \right. \label{eq:delta}\end{aligned}$$ where $c_{\text{in}}^F$ and $c_{\text{in}}^D$ are the component concentrations at feed and desorbent ports, and $Q^F$, $Q^D$, $Q^R$, $Q^E$ the volumetric flowrates at the feed, desorbent, raffinate, and extract ports, respectively. $\delta^j = 0$ indicates nodes that are currently not connected to a port (, in the interior of a zone); this occurs when more columns than zones are present, such as eight columns in a four-zone scheme. As there might be multiple feed, desorbent, raffinate, and extract ports in <span style="font-variant:small-caps;">smb</span> processes (, the cascade scheme and the integrated eight-zone scheme), an extension of the indexing scheme is required. Column shifting is implemented by periodically permuting $\delta$ each switching time $t_s$. Performance indicators ---------------------- Purity, yield and productivity are commonly used for evaluating the performance of chromatographic processes. All indicators can be defined within one collection time $t_c$ [@bochenek2013evaluating]. In the case of <span style="font-variant:small-caps;">smb</span> systems, $t_c$ is the switching time $t_s$, while in single column systems it is the length of pooling time interval $t_p$. In this study, performance indicators are all defined in terms of component, $i \in \{1, \dots, M\}$, withdrawn at a <span style="font-variant:small-caps;">smb</span> node, $j \in \{E, R, \dots\}$, or collected at the outlet of single column systems, $j \in \{B\}$, within the pooling interval. The definitions of performance indicators are all based on concentration integrals $A_{\text{out},i}^j$ of component $i$ at node $j$, $$A_{\text{out},i}^j = \int_{t=\tau}^{\tau + t_c} c_i^j(t, z\!=\!L)\, \dd t$$ In <span style="font-variant:small-caps;">smb</span>, $\tau$ is the starting time of one switching interval and all performance indicators are calculated when the system is upon cyclic steady state (<span style="font-variant:small-caps;">css</span>); it is the starting point of the pooling time interval in single column. The purity, $\mathtt{Pu}_{i}^j$, is the concentration integral of component $i$ relative to the integral sum of all components, Eq. . The yield, $\mathtt{Y}_i^j$, is the ratio of the withdrawn mass and the feed mass, Eq. . $$\mathtt{Pu}_{i}^j = \frac{A_{\text{out},i}^j }{\sum\limits_{k=1}^{M} A_{\text{out},k}^j} \label{eq:purity}$$ $$\mathtt{Y}_i^j = \frac{Q^j A_{\text{out},i}^j}{Q^F c_{\text{in},i}^F\, t_{\text{load}}} \label{eq:yield}$$ In <span style="font-variant:small-caps;">smb</span>, when the solution stream is continuously fed via the feed node and the extracts and raffinates are continuously withdrawn, $t_{\text{load}} = t_s = t_c$ and $Q^F$ differs with $Q^j$. While in the single column, the flowrates are the same at the inlet node and outlet node, $Q^j = Q^F$; and $t_{\text{load}} \neq t_p = t_c$. The productivity, $\mathtt{Pr}_{i}^{j}$, is the withdrawn mass of the component $i$ per collection time relative to the total volume of the utilized packed bed in all columns, $$\mathtt{Pr}_{i}^{j} = \frac{Q^j A_{\text{out},i}^j}{t_c\, (1-\varepsilon_c) V_c\, N}$$ In single column systems, $N = 1$. According to the definitions of $\mathtt{Pr}_i^j$ and $\mathtt{Y}_i^j$, both indicators increase with increasing amounts of the product collected, , $A_{\text{out},i}^j$. However, $\mathtt{Pr}_i^j$ can also be improved by reducing the collection time $t_c$. Optimization ------------ Generally, operating conditions of both <span style="font-variant:small-caps;">smb</span> and single column processes are systematically optimized by numerical algorithms such that the above performance indicators are all maximized. In studies of center-cut separations, the middle component $i$ is of interest. To specifically account for a vector of the conflicting performance indicators of $[\mathtt{Pu}_i^j, \mathtt{Y}_i^j]$ (referred to as $\mathcal{PI}$ hereafter) in this study, multi-objective optimization is applied. A set of objectives can be combined into a single objective by adding each objective a pre-multiplied weight (the weighted method [@marler2010weighted]), or keeping just one of the objectives and with the rest of the objectives constrained (the $\varepsilon$-constraint method [@mavrotas2009effective]). The latter method is used in this work, that is, maximizing the yield of the target component of the processes with the purity constrained to be larger than a threshold $\varepsilon_i^j$: $$\begin{array}[ ]{r l} \min & f(\theta) = - \mathtt{Y}_i^{j} \\ \text{s.t.} & \begin{cases} c_i(\theta) : \mathtt{Pu}^j_i - \varepsilon_i^j \geqslant 0 \\ \theta_{\min} \leqslant \theta \leqslant \theta_{\max} \end{cases} \end{array} \label{eq:eps_constraint}$$ where $\theta$ is a vector of optimized parameters with boundary limitations of $[\theta_{\min}, \theta_{\max}]$. The inequality $c_i(\theta)$ is lumped into the objective function using penalty terms in this study, such that it can be solved as a series of unconstrained minimization problems with increasing penalty factors, $\sigma_k$, $$\min \mathcal{H}(\theta; \sigma_k) = f(\theta) + \sigma_k g(\theta) \label{eq:objective}$$ In Eq. , the penalty function is chosen as $g(\theta) = \norm{ \min\{0, c(\theta)\} }^2$. Different types of methods can be applied to solve the minimization of $\mathcal{H}(\theta;\sigma_k)$, such as deterministic methods and heuristic methods. Additionally, stochastic methods can be chosen. In order to use a stochastic method, the minimization of $\mathcal{H}(\theta;\sigma_k)$ is further formulated to maximum likelihood estimation: $$\text{arg}\max_{\theta}\, \mathcal{L}(\theta) \label{eq:mle}$$ where the likelihood is defined as an exponential function of $\mathcal{H}(\theta;\sigma_k)$: $$\mathcal{L}(\theta) \overset{\text{def}}= \exp\left[ - \frac{1}{2} \mathcal{H}(\theta; \sigma_k) \right] \label{eq:likelihood}$$ Numerical solution ------------------ All numerical simulations were computed on an Intel(R) Xeon(R) system with 16 CPU cores (64 threads) running at . The mathematical models described above for each column of <span style="font-variant:small-caps;">smb</span> processes are weakly coupled together and then iteratively solved. The open-source code has been published on Github, <https://github.com/modsim/CADET-SMB.git>. <span style="font-variant:small-caps;">cadet-smb</span> repeatedly invokes <span style="font-variant:small-caps;">cadet</span> kernel to solve each individual column model, with default parameter settings. <span style="font-variant:small-caps;">cadet</span> is also an open-source software published on Github, <https://github.com/modsim/CADET.git>. The axial column dimension is discretized into $N_z = 40$ cells, while the radical bead is discretized into $N_r = 10$ cells. The resulting system of ordinary differential equations is solved using an absolute tolerance of $\num{1e-10}$, relative tolerance of $\num{1e-6}$, an initial step size of $\num{1e-14}$ and a maximal step size of $\num{5e6}$. A stochastic multi-objective sampling algorithm, <span style="font-variant:small-caps;">mcmc</span>, is applied in this study to optimize the operating conditions. Specifically, the Metropolis-Hastings algorithm, incorporating with delayed rejection, adjusted Metropolis and Gibbs sampling, has been published as open-source software on Github, <https://github.com/modsim/CADET-MCMC.git>. For sampling purpose, <span style="font-variant:small-caps;">mcmc</span> not only accepts proposals with better objective values, but also accepts moves heading to non-dominated points with certain probability. Samples are collected until the Geweke convergence criterion [@geweke1991evaluating] is smaller than $\num{1e-4}$ or the desired sample size is reached. The Pareto fronts in this study describe two-dimensional trade-offs between purity and yield. Thus, non-dominated stable sort method of Pareto front were is applied to generate the frontiers [@duh2012learning]. Case ==== A protein mixture of ribonuclease, cytochrome and lysozyme is of interest in this study, $i\in \{\text{RNase, cyt, lyz}\}$. It is a prototype example used in the academic field for modelling purposes [@osberghaus2012determination; @osberghaus2012optimizing]. The fractionation of this protein mixture is also referred to as center-cut separation, that is, the center component, cyt, is targeted and the other two components are regarded as impurities in separations. In the modelling study here, we have used the proteins on chromatographic columns packed with strong cation-exchanger SP Sepharose FF as an example system (see the column geometry in Tab. \[tab:literature\_data\]). [c c l l S]{} Catalogue & Symbol & Description & [Value]{} & [Unit]{}\ & $L$ & column length & $1.4\times10^{-2}$ &\ & $d_c$ & column diameter & $1\times10^{-2}$ &\ & $d_p$ & particle diameter & $9.00\times10^{-5}$ &\ & $\varepsilon_b$ & column porosity & 0.37 &\ & $\varepsilon_p$ & particle porosity & 0.75 &\ & $D_{\text{ax}}$ & axial dispersion & $5.75\times10^{-8}$ &\ & $D_{p}$ & pore diffusion & $6.07\times10^{-11}$ &\ & $k_f$ & film mass transfer & $6.90\times10^{-6}$ &\ & $\Lambda$ & ionic capacity & 1200 &\ & $k_{\text{eq},i}$ & equilibrium constant & $[7.70,\, 1.59,\, 35.5]\times10^{-3}$ &\ & $\nu_i$ & characteristic charges & $[3.70,\, 5.29,\, 4.70]$ &\ & $\sigma_i$ & steric factors & $[10.0,\, 10.6,\, 11.83]$ &\ Simplified models (, <span style="font-variant:small-caps;">tdm</span> and <span style="font-variant:small-caps;">edm</span>) that rely on an assumption of equal concentrations in the particle pores and in the mobile phase, can not been applied here; because mass transfer dynamics of macro-molecules in ca. beads can be rate limiting [@lodi2017ion]. Hence, the <span style="font-variant:small-caps;">grm</span> model is used to describe the mass transfer in porous beads. In addition, the <span style="font-variant:small-caps;">sma</span> model inherently considers the impact of salt on binding affinity, the hindrance effect of macro-molecules. The mathematical modelling of the prototype example has been presented in [@teske2006competitive] and [@puttmann2013fast]; the transport and binding parameters from experiments are shown in Tab. \[tab:literature\_data\]. The geometry, transport and binding parameters are directly used in this study; and the operating conditions will be optimized. In the pharmaceutical manufacturing usually at least $99\%$ purity of cyt is needed. Therefore, taking expensive computation cost in <span style="font-variant:small-caps;">iex-smb</span> simulations to generate Pareto optimal points with lower purities is also a trade-off. In this study, the $\varepsilon_{\text{cyt}}^j$ in Eq. \[eq:eps\_constraint\] is set to $99\%$; thus, only the Pareto optimal fragment with purity higher than $99\%$ is concerned in <span style="font-variant:small-caps;">iex-smb</span>. This implementation reduces the heavy computational burden of multi-objective sampling, as much fewer samples are requested to construct the frontiers. Pareto optimal fragment can be further extended by repeatedly solving Eq. with varying values of $\varepsilon_{\text{cyt}}^j$. However, in the single column the whole Pareto optimal front is concerned as the computational cost is not expensive. This can be one of the advantages of the $\varepsilon$-constraint method. Single column ------------- In the single column system, the equilibration, load-wash-elution and regeneration phases are performed. The interstitial velocity is ; the retention time for a non-retained component is . concentration of RNase, cyt and lyz is injected in the load phase. In both the load and wash steps, a salt concentration of is applied. The operating time intervals for the load and wash phases are and . The shapes of the multi-step elution phase, that can be characterized by the bilinear gradients (, $m_1$ and $m_2$), the operating time intervals, and the initial salt concentration (, $c_{\text{init},0}$), are optimized. Since the symbols as illustrated in Fig. \[fig:batch\_salt\_scheme\] are the elapsed times, the operating time intervals are calculated (, $\Delta t_1 = t_{\text{elute}}^1 - 50$ and $\Delta t_2 = t_{\text{elute}}^2 - t_{\text{elute}}^1$). Thus, the optimized operating parameters are $\theta = \{\Delta t_1, \Delta t_2, m_1, m_2, c_{\text{init},0}\}$. SMB --- Depending on initial states, <span style="font-variant:small-caps;">smb</span> processes often undergoes a ramp-up phase and eventually enter into a <span style="font-variant:small-caps;">css</span>. In this study, the system is assumed to be upon <span style="font-variant:small-caps;">css</span> when a difference between two iterations $(k, k\!+\!1)$ falls below a predefined tolerance error, $e_t$, see Eq. . $$\max_{j \in \{R,E\}} \sum_{i=1}^M \left( \int_0^{t_s} \left\| c_{i,k}^j(t, z=L) - c_{i,k+1}^j(t, z=L) \right\|^n \,\mathrm{d}t \right)^\frac{1}{n} \leqslant e_t \label{eq:difference_method}$$ In this study, the columns of <span style="font-variant:small-caps;">iex-smb</span> are initially empty except for the concentration of bound salt ions $q_0$ that is set to the ionic capacity $\Lambda$, in order to satisfy the electro-neutrality condition. Additionally, the initial salt concentrations in column $j$, $c_0^j(t\!=\!0, z)$, $c_{p,0}^j(t\!=\!0, z)$, $q_0^j(t\!=\!0, z)$, are set equal to the salt concentrations in the upstream inlet nodes. Taking the four-zone scheme as an example (cf. Fig. \[fig:4-zone\]), the salt concentrations of the mobile and stationary phases ($c_0^j$ and $c_{p,0}^j$) in zone III and IV are set to the value of $c_0^F$; while $c_0^j$ and $c_{p,0}^j$ in zone I and II are set to the value of $c_0^D$: $$\begin{cases} c_0^j(t\!=\!0, z) = c_{p,0}^j(t\!=\!0, z) = c_0^D \qquad j \in \{\text{I}, \text{II}\} \\ c_0^j(t\!=\!0, z) = c_{p,0}^j(t\!=\!0, z) = c_0^F \qquad j \in \{\text{III}, \text{IV}\} \\ q_0^j(t\!=\!0, z) = \Lambda \qquad j \in\{\text{I}, \text{II}, \text{III}, \text{IV}\} \end{cases} \label{eq:salt_initial}$$ In contrast to the single column system, the chosen initial state of <span style="font-variant:small-caps;">smb</span> processes is irrelevant to the performance indicators calculated upon <span style="font-variant:small-caps;">css</span>. A cascade of two four-zone units (named as $U_1$ and $U_2$ in the following content) and an integrated eight-zone scheme are applied, see Fig. \[fig:ternary\]. In both schemes, three single columns in each zone and 24 columns at all are designed. In this study, the volumetric flowrate of zone I is defined as recycle flowrate, $Q^{\text{rec}}$. In each sub-unit of the cascade scheme, the optimized operating parameters are the switching time (same for both units as they are synchronously switched), recycle flowrate, the inlet and outlet flowrates, and the salt concentrations at the inlet ports, $\theta = \{t_s, [Q^{\text{rec}}]_{U_1}, [Q^D]_{U_1},\allowbreak [Q^E]_{U_1}, [Q^F]_{U_1}, [c_0^F]_{U_1},\allowbreak [c_0^D]_{U_1}, [Q^{\text{rec}}]_{U_2}, [Q^D]_{U_2},\allowbreak [Q^E]_{U_2}, [Q^F]_{U_2},\allowbreak [c_0^F]_{U_2}, [c_0^D]_{U_2}\}$. While in the eight-zone scheme, the switch time, the recycle flowrate, the inlet and outlet flowrates, and salt concentrations at inlet ports are optimized, that is, $\theta = \{t_s, Q^{\text{rec}}, Q^{D1},\allowbreak Q^{E1}, Q^{F1}, Q^{R1},\allowbreak Q^{D2}, Q^{E2}, Q^{F2}, c_0^{F1},\allowbreak c_0^{F2}, c_0^{D1}, c_0^{D2}\}$. The flowrates $Q^R$ and $Q^{R2}$ are calculated from flowrate balance. [0.9]{}[![Schematics of the cascade scheme with two four-zone <span style="font-variant:small-caps;">iex-smb</span> units (top) and the integrated eight-zone scheme (bottom) applied in this study for ternary separations. In each zone, there can be multiple columns.[]{data-label="fig:ternary"}](scheme_cascade "fig:"){width="\textwidth"}]{} [0.6]{}[![Schematics of the cascade scheme with two four-zone <span style="font-variant:small-caps;">iex-smb</span> units (top) and the integrated eight-zone scheme (bottom) applied in this study for ternary separations. In each zone, there can be multiple columns.[]{data-label="fig:ternary"}](scheme_ternary_8 "fig:"){width="\textwidth"}]{} Results and discussion ====================== Single column ------------- The gradient shapes of the single column process are first optimized in order to have a comparison with the <span style="font-variant:small-caps;">iex-smb</span> processes. Searching domain of the parameters, $\theta$, is listed in Tab. \[tab:bounds\_batch\]. The intervals are based on literature values with additional safety margins. The maximal sampling length here is $\num{1.2e4}$ and the burn-in length is set to $50\%$ of the samples. [c l S S S]{} & & &\ & & [min]{} & [max]{} &\ $\Delta t_1$ & elution interval one & 500 & 8000 &\ $\Delta t_2$ & elution interval two & 1000 & 8000 &\ $m_1$ & elution gradient one & 1.0e-3 & 1.0e-2 &\ $m_2$ & elution gradient two & 1.0e-3 & 20 &\ $c_{\text{init},0}$ & initial salt conc. & 20 & 200 &\ Fig. \[fig:pareto\] shows the Pareto optimal front between purity and yield, as computed by Eq. and Eq. . At a rather low purity requirement of $85\%$, a yield of 0.9 can be achieved. However, at purity of $98\%$, the yield drops dramatically to 0.1. The Pareto front provides full information of the single column process; the corresponding operating conditions, that render the Pareto optima on demand in application, can be chosen on purpose. ![Pareto optimal front of the yield and purity performance indicators in the single column system.[]{data-label="fig:pareto"}](pareto){width="55.00000%"} [0.6]{}[![Chromatograms of the single column system (solid lines) and the corresponding multi-step elution gradients (dashed lines).[]{data-label="fig:batch_demo"}](chroma_batch_demo1 "fig:"){width="\textwidth"}]{} [0.6]{}[![Chromatograms of the single column system (solid lines) and the corresponding multi-step elution gradients (dashed lines).[]{data-label="fig:batch_demo"}](chroma_batch "fig:"){width="\textwidth"}]{} [0.6]{}[![Chromatograms of the single column system (solid lines) and the corresponding multi-step elution gradients (dashed lines).[]{data-label="fig:batch_demo"}](chroma_batch_demo3 "fig:"){width="\textwidth"}]{} Three characteristic points (, $a, b, c$) on the above Pareto front are exemplified and then compared. In Fig. \[fig:batch\_demo\], the corresponding chromatograms and multi-step salt gradients are shown; the grey areas indicate the pooling time intervals, $t_p$, of the target components. The calculated performance indicators of the three characteristic points are as follows: $\mathcal{PI}_{\text{cyt}}^B = [\mathtt{Pu}_{\text{cyt}}^B, \mathtt{Y}_{\text{cyt}}^B]$ is $[90.0\%, 0.85]$, and the productivity is $\mathtt{Pr}_{\text{cyt}}^B = \SI{4.94e-3}{\mole\per\cubic\metre\per\second}$ of the point $a$ (see Fig. \[fig:batch\_demo1\]). While they are $[95.59\%, 0.71]$ and of the point $b$ (see Fig. \[fig:batch\]), $[97.02\%, 0.49]$ and of the point $c$ (see Fig. \[fig:batch\_demo3\]). The respective operating parameters are listed in Tab. \[tab:est\_batch\]. [c S S S S]{} $\theta$ & $a$ & $b$ & $c$\ $\Delta t_1$ & 2.81e3 & 3.12e3 & 6.26e2 &\ $\Delta t_2$ & 3.53e3 & 1.56e3 & 3.68e3 &\ $m_1$ & 1.28e-3 & 4.29e-3 & 6.47e-3 &\ $m_2$ & 1.11 & 1.32e-2 & 1.62e-2 &\ $c_{\text{init},0}$ & 77.2 & 71.7 & 51.0 &\ The pooling time length $t_p = \SI{5808}{\second}$ in Fig. \[fig:batch\] is much bigger than that in Fig. \[fig:batch\_demo1\] (, ) and Fig. \[fig:batch\_demo3\] (, ). In this study, pooling time intervals were calculated from that the concentrations of other two components (, RNase and lyz) are lower than a predefined threshold, $\mu = \num{7.5e-5}$. Based on the same operating parameters, if we set $\mu$ to a larger value, higher yield and lower purity will be obtained. Therefore, the selection is also a trade-off, which is illustrated in a Pareto front in Fig. \[fig:pareto\_eps\], where $\mu$ was decreased from to with equivalent gap of $\num{2.5e-5}$ using operating parameters listed in the $b$ column of Tab. \[tab:est\_batch\]. ![Impact of $\epsilon$ values on the trade-off relationship of purity and yield. $\mu$ is decreased from $\num{1e-4}$ to $\num{2.5e-5}$ with equivalent gap of $\num{2.5e-5}$.[]{data-label="fig:pareto_eps"}](pareto_eps){width="50.00000%"} Shorter pooling time interval means the cyt peak is more stiffly concentrated; it can result in high yield but larger overlap areas (thus low purity) as a conflict, see Fig. \[fig:batch\_demo1\]. When pooling time interval is longer, the cyt peak is more spread and it renders relatively lower yield but higher purity, see Fig. \[fig:batch\]. As observed in Fig. \[fig:batch\_demo\], the peak of cyt always overlaps with the other two peaks, rather than thoroughly separated. In Fig. \[fig:batch\] and \[fig:batch\_demo3\], even when lyz is hardly washed out, it begins to be eluted out with small concentration values quite early. Additionally, as lyz is almost not eluted out in Fig. \[fig:batch\] and \[fig:batch\_demo3\], a strip step is needed before the regeneration step, in order to make the column reusable. The overlaps may denotes that cyt can not be collected with $100\%$ purity with the single column batch system. Cascade scheme -------------- Numerical optimization of the <span style="font-variant:small-caps;">iex-smb</span> schemes require suitable initial values. To this end, empirical rules have been developed with big efforts to achieve the center-cut separation first. With the operating conditions listed in the empirical column of Tab. \[tab:cascade\_op\], lyz is collected at the R port of $U_1$ with $[\mathcal{PI}_{\text{lyz}}^R]_{U_1} = [98.23\%, 0.99]$ (see Fig. \[fig:cascade\_unit1\_chroma\]). In $U_2$, cyt spreads towards the R port of $U_2$, resulting low yield at the E port (, $[\mathtt{Y}_{\text{cyt}}^E]_{U_2} = 0.33$) and low purity at the R port (, $[\mathtt{Pu}_{\text{RNase}}^R]_{U_2} = 58.39\%$) (the chromatogram is not shown). Based on the initial values with safety margins, searching domain as shown in Tab. \[tab:bounds\_cascade\] is used for numerical optimization. [c l S S S]{} & & &\ & & [min]{} & [max]{} &\ $Q^{\text{rec}}$ & recycle flowrate & 2.06e-8 & 3.06e-8 &\ $Q^{F}$ & feed one flowrate & 0.84e-8 & 1.84e-8 &\ $Q^{D}$ & desorbent one flowrate & 0.69e-8 & 1.69e-8 &\ $Q^{E}$ & extract one flowrate & 0.35e-8 & 1.25e-8 &\ $c^{F}_0$ & feed two salt conc. & 180 & 220 &\ $c^{D}_0$ & desorbent two salt conc. & 220 & 260 &\ [0.49]{}[![Left: Chromatograms of the first sub-unit in the cascade <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from the empirical design (left). Right: Salt profiles, $c_0(t, z)$, along the columns of the first sub-unit upon the <span style="font-variant:small-caps;">css</span> ($t = \tau$)](cascade_chroma_unit1 "fig:"){width="\textwidth"}]{} [0.49]{}[![Left: Chromatograms of the first sub-unit in the cascade <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from the empirical design (left). Right: Salt profiles, $c_0(t, z)$, along the columns of the first sub-unit upon the <span style="font-variant:small-caps;">css</span> ($t = \tau$)](cascade_salt_unit1 "fig:"){width="\textwidth"}]{} . \[fig:cascade\_unit1\_emp\] ![Pareto front of the purity and yield performance indicators of the cascade scheme.[]{data-label="fig:pareto_cascade"}](pareto_cascade){width="50.00000%"} <span style="font-variant:small-caps;">smb</span> processes often undergo a long ramp-up phase first and eventually enter into a <span style="font-variant:small-caps;">css</span>. Elaborately choosing the initial state of columns can also accelerate the convergence [@bentley2014experimental]. In this case, a total of ca. $k = 108$ switches was required for each <span style="font-variant:small-caps;">iex-smb</span> simulation to fall below the tolerance error of $e_t = \num{1e-5}$, see Eq. . Thus, gaining points for generating Pareto fronts in <span style="font-variant:small-caps;">smb</span> chromatographic processes is computational expensive. [@li2014using] have proposed computationally cheap surrogate models for efficient optimization of <span style="font-variant:small-caps;">smb</span> chromatography. In this study, only the operating parameters of $U_2$ are optimized; the operating conditions of $U_1$ are taken directly from the empirical design. Moreover, only the Pareto optimal fragment with purity higher than $99\%$ is concerned, such that fewer points are requested. The maximal sampling length of <span style="font-variant:small-caps;">mcmc</span> is $\num{300}$, and the burn-in length is $\num{50}$. The Pareto front of the cascade scheme is illustrated in Fig. \[fig:pareto\_cascade\]. As seen from the Pareto fronts, the cascade scheme outperforms the single column system in both purity and yield indicators. Three characteristic points (, $a, b, c$), ranging yields from high to low, on the Pareto front are compared. The corresponding operating conditions are listed in Tab. \[tab:cascade\_op\]; and the resulting chromatograms are shown in Fig. \[fig:cascade\_chroma\_op\]. The combined axial concentration profiles in all columns are displayed at multiples of the switching time. The performance vector $[\mathcal{PI}_{\text{cyt}}^E]_{U_2} = [\mathtt{Pu}^E_{\text{cyt}}, \mathtt{Y}_{\text{cyt}}^E]_{U_2}$ of the point $a$ is $[99.27\%, 0.96]$ (cf. Fig. \[fig:cascade\_chroma\_1\]), $[99.81\%, 0.95]$ of the point $b$ (cf. Fig. \[fig:cascade\_chroma\_2\]) and $[100\%, 0.39]$ of the point $c$ (cf. Fig. \[fig:cascade\_chroma\_3\]). After numerical optimization, the performance indicators tremendously increase from that of the empirical design, $[97.92\%, 0.33]$. The main difference between the chromatograms of $a$ and $b$ is that less RNase keeps retained to port E in $b$, resulting in higher purity at port E. The productivities of cyt, $[\mathtt{Pr}_{\text{cyt}}^E]_{U_2}$, are $[\num{3.13e-4}, \num{3.15e-4},\allowbreak \num{1.30e-4}]\, \si{\mole\per\cubic\metre\per\second}$, respectively. It indicates that productivity is high when the concentration values in the shadow areas at port E are high and flat. Though RNase and lyz are treated as impurities in the center-cut separation, they can be withdrawn with high performance indicators in this study. For instance, $[\mathcal{PI}_{\text{RNase}}^R]_{U_2} = [96.14\%, 0.97]$ of the point $a$, $[95.45\%, 0.98]$ of the point $b$. [0.6]{}[![Chromatograms of the second sub-unit in the cascade <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from numerical optimization.[]{data-label="fig:cascade_chroma_op"}](cascade_chroma_unit2_1 "fig:"){width="\textwidth"}]{} [0.6]{}[![Chromatograms of the second sub-unit in the cascade <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from numerical optimization.[]{data-label="fig:cascade_chroma_op"}](cascade_chroma_unit2_op "fig:"){width="\textwidth"}]{} [0.6]{}[![Chromatograms of the second sub-unit in the cascade <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from numerical optimization.[]{data-label="fig:cascade_chroma_op"}](cascade_chroma_unit2_3 "fig:"){width="\textwidth"}]{} The salt profiles, $c_0(t, z)$, along the two <span style="font-variant:small-caps;">iex-smb</span> units at time upon <span style="font-variant:small-caps;">css</span>, $t = \tau$, are depicted in Fig. \[fig:cascade\_salt\]. In $U_1$, the salt concentrations at inlet ports, $[c_0^D, c_0^F]_{U_1}$, used for constructing the two solvent gradients are $[290, 420]\, \si{\mole\per\cubic\metre}$ (cf. Fig. \[fig:cascade\_unit1\_salt\]). The salt concentrations at the inlet ports of $U_2$, $[c_0^D, c_0^F]_{U_2}$, are $[206.56, 243.53]\, \si{\mole\per\cubic\metre}$ of the point $a$; $[203.92, 244.93]\,\allowbreak \si{\mole\per\cubic\metre}$ of the point $b$ and $[208, 243]\, \si{\mole\per\cubic\metre}$ of the point $c$. ![Salt profiles, $c_0(t, z)$, along the columns of the cascade scheme upon the <span style="font-variant:small-caps;">css</span> ($t = \tau$) from the numerical optimization.[]{data-label="fig:cascade_salt"}](cascade_salt_unit2_op){width="70.00000%"} The salt gap of solvent between zones I,II and zones III,IV of $U_1$ (cf. Fig. \[fig:cascade\_unit1\_salt\]) facilitates to separate lyz from the mixture; RNase and cyt are still not rather separated in $U_2$. Because of the salt profile in zones III and IV (, ca. ), RNase and cyt have approximately the same electrostatic interaction forces with the stationary phase, causing rather similar desorption rates off the stationary phase. In order to separate RNase and cyt in $U_2$, the salt concentrations at the inlet ports should be reduced, leading one component (, cyt) to be strongly bounded. This can be achieved by diluting the bypass stream of $U_2$ with pure buffer (in the cascade scheme, the flow between the raffinate stream of $U_1$ and feed stream of $U_2$ is defined as a bypass stream): $$\left[ c_{\text{in},i}^F \right]_{U_2} = \frac{\left[ Q^R c_{\text{out},i}^R \right]_{U_1}}{\left[Q^R\right]_{U_1} + Q^{\text{dilute}}} \qquad i \in \{0, \dots, M\} \label{eq:dilute}$$ In numerical optimization, $\left[Q^F\right]_{U_2}$ and $\left[Q^R\right]_{U_1}$ are optimized, $Q^{\text{dilute}} = \left[Q^F\right]_{U_2} - \left[Q^R\right]_{U_1}$ and $[c_{\text{in},i}^F]_{U_2}$ are subsequently changed. As seen in Fig. \[fig:cascade\_salt\], the salt concentration in $U_2$ is decreased to ca. in zones I,II and in zones III,IV. With these salt concentrations, RNase and cyt can be separated in $U_2$. With the dilution, the protein concentrations are decreased as a side effect. Therefore, the peaks in $U_2$ are not as concentrated as have observed in $U_1$; they distribute over several zones, which makes the process design of the $U_2$ challenging. Eight-zone scheme ----------------- As in the design of the cascade scheme, the initial values were obtained from empirical designs with manual efforts. The searching domain for the numerical optimization is based on the initial values with safety margins, see Tab. \[tab:bounds\]. The dilution as described in the design of the cascade scheme was applied to the bypass stream (, the connection between the raffinate-I of the first sub-unit and the feed-II of the second sub-unit). With the operating conditions listed in the empirical column of Tab. \[tab:8-zone\], cyt spreads from zone VI to zone VII, RNase from zone V to VI, such that $\mathtt{Pu}_{\text{RNase}}^{R2} = 73.23\%$ and $\mathtt{Pu}_{\text{cyt}}^{E2} = 87.50\%$ (the chromatogram is not shown). [c l S S S]{} & & &\ & & [min]{} & [max]{} &\ $t_s$ & switch time & 90 & 110 &\ $Q^{\text{rec}}$ & recycle flowrate & 2.0e-8 & 2.5e-8 &\ $Q^{D1}$ & desorbent one flowrate & 0.94e-8 & 1.14e-8 &\ $Q^{E1}$ & extract one flowrate & 0.99e-8 & 1.19e-8 &\ $Q^{F1}$ & feed one flowrate & 0.45e-8 & 0.65e-8 &\ $Q^{R1}$ & raffinate one flowrate & 0.50e-8 & 0.70e-8 &\ $Q^{D2}$ & desorbent two flowrate & 1.30e-8 & 1.55e-8 &\ $Q^{E2}$ & extract two flowrate & 1.09e-8 & 1.29e-8 &\ $Q^{F2}$ & feed two flowrate & 0.65e-8 & 0.85e-8 &\ $c^{F1}_0$ & feed one salt conc. & 270 & 310 &\ $c^{D1}_0$ & desorbent one salt conc. & 410 & 450 &\ $c^{F2}_0$ & feed two salt conc. & 170 & 220 &\ $c^{D2}_0$ & desorbent two salt conc. & 240 & 260 &\ In this case, a total of ca. $k = 148$ switches was required for each eight-zone simulation to fall below the tolerance error of $e_t = \num{1e-5}$. The convergence time from an initial state to the <span style="font-variant:small-caps;">css</span>, thus the total computational time, in the eight-zone scheme is longer than that in the cascade scheme. The maximal sampling length of <span style="font-variant:small-caps;">mcmc</span> is $\num{300}$, and the burn-in length is $\num{50}$. The Pareto optimal front with purity larger than $99\%$ is illustrated in Fig. \[fig:pareto\_8zone\]; the Pareto fronts of single column and cascade systems are superimposed for comparison. The shadow areas of light blue show the sampling region. ![Pareto front of the purity and yield performance indicators of the eight-zone scheme.[]{data-label="fig:pareto_8zone"}](pareto_8zone){width="70.00000%"} By using the multi-objective sampling algorithm, <span style="font-variant:small-caps;">mcmc</span>, not only the Pareto front solutions but also the points near Pareto front can be studied. In this study, three points (, $a, b, c$; the point $c$ is on the Pareto front), ranging purities from low to high, were chosen to illustrate the impact of the composition of R1 stream on the performance of the second sub-unit. The corresponding operating conditions are listed in Tab. \[tab:8-zone\]; and the chromatograms upon <span style="font-variant:small-caps;">css</span> are shown in Fig. \[fig:8-zone\_chroma\_opt\]. The combined axial concentration profiles in all columns are displayed at multiples of the switching time. In Fig. \[fig:8-zone\_chroma\_opt\_1\], cyt is collected at the E2 node with $\mathcal{PI}_{\text{cyt}}^{E2} = [97.18\%, 0.95]$, while $[98.98\%, 0.90]$ in Fig. \[fig:8-zone\_chroma\_opt\_2\] and $[99.27\%, 0.97]$ in Fig. \[fig:8-zone\_chroma\_opt\_3\]. As seen from Fig. \[fig:8-zone\_chroma\_opt\], the higher concentration of the cyt composition in the R1 stream, the higher of the yield and productivity indicators at the E2 port. To be specific, $\mathtt{Y}_{\text{cyt}}^{E2} = 0.95$ and $\mathtt{Pr}_{\text{cyt}}^{E2} = \SI{3.10}{\mole\per\cubic\metre\per\second}$ of the point $a$; $\mathtt{Y}_{\text{cyt}}^{E2} = 0.90$ and $\mathtt{Pr}_{\text{cyt}}^{E2} = \SI{2.93}{\mole\per\cubic\metre\per\second}$ of the point $b$, and $\mathtt{Y}_{\text{cyt}}^{E2} = 0.97$ and $\mathtt{Pr}_{\text{cyt}}^{E2} = \SI{3.25}{\mole\per\cubic\metre\per\second}$ of the point $c$. Although in the center-cut separation, the other two components (, RNase and lyz) can be viewed as impurities, they can rather be withdrawn with high performance indicators. Specifically, for RNase at the R2 node it is $\mathcal{PI}_{\text{RNase}}^{R2} = [94.86\%, 0.97]$ of the point $a$, $[91.80\%, 0.99]$ of the point $b$ and $[98.29\%, 0.99]$ of the point $c$; for lyz at the E1 node it is $\mathcal{PI}_{\text{lyz}}^{E1} = [99.75\%, 1.00]$ of the point $a$, $[99.94\%, 1.00]$ of the point $b$ and $[99.31\%, 1.00]$ of the point $c$. [0.7]{}[![Chromatograms of the eight-zone <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from numerical optimization.[]{data-label="fig:8-zone_chroma_opt"}](8-zone_chroma_opt_1 "fig:"){width="\textwidth"}]{} [0.7]{}[![Chromatograms of the eight-zone <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from numerical optimization.[]{data-label="fig:8-zone_chroma_opt"}](8-zone_chroma_opt_2 "fig:"){width="\textwidth"}]{} [0.7]{}[![Chromatograms of the eight-zone <span style="font-variant:small-caps;">iex-smb</span> upon <span style="font-variant:small-caps;">css</span> from numerical optimization.[]{data-label="fig:8-zone_chroma_opt"}](8-zone_chroma_opt "fig:"){width="\textwidth"}]{} The salt profile, $c_0(t, z)$, along the eight-zone <span style="font-variant:small-caps;">iex-smb</span> unit at time upon <span style="font-variant:small-caps;">css</span>, $t = \tau$, are attached in Fig. \[fig:8-zone\_salt\]. The salt concentrations at inlet ports, $[c_0^{D1}, c_0^{F1},\allowbreak c_0^{D2}, c_0^{F2}]$, are $[442,270,\allowbreak 240, 211]\, \si{\mole\per\cubic\metre}$ of the point $a$; $[440, 270,\allowbreak 240, 209]\, \si{\mole\per\cubic\metre}$ of the point $b$ and $[443, 272,\allowbreak 240, 217]\,\allowbreak \si{\mole\per\cubic\metre}$ of the point $c$. Separating lyz at port E1 is facilitated with the salt gap between zones I,II and zones III,IV; RNase and cyt are both weekly bounded in zones III,IV and to be separated in the second sub-unit. Meanwhile, the bypass stream was diluted as described by Eq. \[eq:dilute\]. As seen in Fig. \[fig:8-zone\_salt\], the salt concentration is decreased to ca. in zones V,VI and in zones III,IV. With these salt concentrations, RNase is weakly bounded and cyt is strongly bounded in the second sub-unit. As seen from Fig. \[fig:8-zone\_salt\], the deviation of the salt profile along the columns of <span style="font-variant:small-caps;">iex-smb</span> is small, but the performance indicators varied. This indicates process design of <span style="font-variant:small-caps;">iex-smb</span> processes is sensitive to operating conditions. ![The actual salt profiles, $c_0(t=\tau, z)$, along the columns of the eight-zone scheme upon the <span style="font-variant:small-caps;">css</span> by controlling solvent gradients at inlet ports.[]{data-label="fig:8-zone_salt"}](8-zone_salt_opt){width="70.00000%"} Eight-zone schemes have fewer degrees of freedom than cascade schemes. Hence, $\mathtt{Pu}_{\text{cyt}}^{E2}$ decreases from $99.81\%$ of the cascade scheme to $97.18\%$, based on the same yield, 0.95. Moreover, in the cascade scheme, almost pure cyt can be collected, which can not be achieved with the eight-zone scheme. However, the productivity of the eight-zone scheme can be higher than that of the cascade scheme. The productivity of the cascade scheme, $\SI{3.13e-4}{\mole\per\cubic\metre\per\second}$ ($\mathcal{PI}_{\text{cyt}}^{E2} = [99.27\%, 0.96$\]), is slightly lower than that of the eight-zone scheme, $\SI{3.25e-4}{\mole\per\cubic\metre\per\second}$ ($\mathcal{PI}_{\text{cyt}}^{E2} = [99.27\%, 0.97]$). This suggests that the stationary phase is more efficiently utilized with integrated <span style="font-variant:small-caps;">smb</span> processes. Three columns in each zone was chosen and used in both the cascade and eight-zone schemes. In the empirical designs, neither one column nor two columns in each zone could lead to good separation performance. Having more columns in each zone and less switching time can approximate the true moving bed chromatography. However, introducing more columns into <span style="font-variant:small-caps;">iex-smb</span> system makes challenges for computational resource. Three columns in each zone is a good compromise. [c l S S S S S c]{} & & & &\ & & [$U_1$]{} & [$U_2$]{} & $a$ & $b$ & $c$ &\ $Q^F$ & feed flowrate & 0.55e-8 & 0.75e-8 & 1.48e-8 & 1.37e-8 & 1.56e-8 &\ $Q^R$ & raffinate flowrate & 0.60e-8 & 0.90e-8 & 1.49e-8 & 1.36e-8 & 1.63e-8 &\ $Q^D$ & desorbent flowrate & 1.14e-8 & 1.34e-8 & 1.24e-8 & 1.22e-8 & 1.24e-8 &\ $Q^E$ & extract flowrate & 1.09e-8 & 1.19e-8 & 1.23e-8 & 1.22e-8 & 1.17e-8 &\ $Q^{\text{I}}$ & zone I flowrate & 2.21e-8 & 2.56e-8 & 2.96e-8 & 2.90e-8 & 3.01e-8 &\ $Q^{\text{II}}$ & zone II flowrate & 1.12e-8 & 1.37e-8 & 1.73e-8 & 1.63e-8 & 1.84e-8 &\ $Q^{\text{III}}$ & zone III flowrate & 1.67e-8 & 2.12e-8 & 3.21e-8 & 3.04e-8 & 3.40e-8 &\ $Q^{\text{IV}}$ & zone IV flowrate & 1.07e-8 & 1.22e-8 & 1.72e-8 & 1.69e-8 & 1.77e-8 &\ $c^F_0$ & feed salt conc. & 290 & 200 & 206.56 & 203.92 & 208 &\ $c^D_0$ & desorbent salt conc. & 420 & 240 & 243.53 & 244.93 & 243 &\ $t_s$ & switching time & &\ [c l S S S S S]{} & & &\ & & [Empirical]{} & $a$ & $b$ & $c$ &\ $Q^{F1}$ & feed one flowrate & 0.477e-8 & 0.544e-8 & 0.544e-8 & 0.554e-8 &\ $Q^{R1}$ & raffinate one flowrate & 0.518e-8 & 0.586e-8 & 0.574e-8 & 0.558e-8 &\ $Q^{D1}$ & desorbent one flowrate & 1.086e-8 & 1.072e-8 & 1.082e-8 & 1.089e-8 &\ $Q^{E1}$ & extract one flowrate & 1.056e-8 & 1.023e-8 & 1.020e-8 & 1.045e-8 &\ $Q^{\text{I}}$ & zone I flowrate & 2.166e-8 & 2.000e-8 & 2.021e-8 & 2.000e-8 &\ $Q^{\text{II}}$ & zone II flowrate & 1.110e-8 & 0.977e-8 & 1.001e-8 & 0.955e-8 &\ $Q^{\text{III}}$ & zone III flowrate & 1.587e-8 & 1.521e-8 & 1.540e-8 & 1.508e-8 &\ $Q^{\text{IV}}$ & zone IV flowrate & 1.069e-8 & 0.935e-8 & 0.966e-8 & 0.950e-8 &\ $Q^{F2}$ & feed two flowrate & 0.753e-8 & 0.826e-8 & 0.837e-8 & 0.843e-8 &\ $Q^{R2}$ & raffinate two flowrate & 1.009e-8 & 1.122e-8 & 1.164e-8 & 1.189e-8 &\ $Q^{D2}$ & desorbent two flowrate & 1.510e-8 & 1.391e-8 & 1.408e-8 & 1.408e-8 &\ $Q^{E2}$ & extract two flowrate & 1.243e-8 & 1.102e-8 & 1.109e-8 & 1.101e-8 &\ $Q^{\text{V}}$ & zone V flowrate & 2.579e-8 & 2.327e-8 & 2.374e-8 & 2.358e-8 &\ $Q^{\text{VI}}$ & zone VI flowrate & 1.336e-8 & 1.225e-8 & 1.265e-8 & 1.257e-8 &\ $Q^{\text{VII}}$ & zone VII flowrate & 2.089e-8 & 2.051e-8 & 2.102e-8 & 2.100e-8 &\ $Q^{\text{VII}}$ & zone VIII flowrate & 1.080e-8 & 0.928e-8 & 0.939e-8 & 0.911e-8 &\ $c^{F1}_0$ & feed one salt conc. & 295.1 & 270.0 & 270.2 & 270.5 &\ $c^{D1}_0$ & desorbent one salt conc. & 422.0 & 442.0 & 440.3 & 442.7 &\ $c^{F2}_0$ & feed two salt conc. & 183.1 & 211.0 & 209.4 & 216.6 &\ $c^{D2}_0$ & desorbent two salt conc. & 240.7 & 240.0 & 240.0 & 240.5 &\ $t_s$ & switching time & 105.9 & 109.1 & 109.8 & 109.7 &\ Conclusions =========== We have presented model-based process designs of multi-step salt gradient single column system and <span style="font-variant:small-caps;">iex-smb</span> processes for separating ribonuclease, cytochrome and lysozyme on strong cation-exchanger SP Sepharose FF. The general rate model with steric mass-action binding kinetics was used in the column modelling. Two network configurations of closed-loop <span style="font-variant:small-caps;">iex-smb</span> have been studied, , one cascade scheme of two four-zone <span style="font-variant:small-caps;">iex-smb</span> sub-units and one integrated eight-zone scheme; the single column batch system has been studied for comparison. The multi-objective sampling algorithm, Markov Chain Monte Carlo (<span style="font-variant:small-caps;">mcmc</span>), has been used to generate samples for approximating the Pareto optimal fronts. This study shows that it is possible to design closed-loop <span style="font-variant:small-caps;">iex-smb</span> units for collecting the target, cytochrome. In the numerical optimizations of <span style="font-variant:small-caps;">iex-smb</span>, initial values are required. Empirical rules have been developed in this study to obtain the initial values, and the searching domains of numerical optimization are based on the initial values with safety margins. <span style="font-variant:small-caps;">mcmc</span> serves on the sampling purpose, which is interested in sampling Pareto optimal points as well as those near Pareto optimal. With $\varepsilon$-constraint method of the multi-objective optimization, Pareto optimal fragments can be concerned to reduce expensive computational cost in large-scale <span style="font-variant:small-caps;">iex-smb</span> processes. The Pareto fronts show the information of the trade-off relationship between the conflicting indicators (, purity and yield). Although in center-cut separations the other two components (, ribonuclease and lysozyme) are treated as impurities, they can be collected with rather high performance indicators. As illustrated by the Pareto fronts, the cascade scheme and the eight-zone scheme have approximately the same performance, that both outperforms the single column system with respect to the conflicting performance indicators, , purity and yield. In addition, the cascade scheme can produce almost pure target component with high yield. As seen from the productivity indicators, the stationary phase is more utilized in the integrated eight-zone scheme. However, single column system does not undergo a time-consuming ramp-up phase and eventually enters into a cyclic steady state, like it is in <span style="font-variant:small-caps;">smb</span> systems. It is also recommended starting a single column batch system prior to complex multi-column and <span style="font-variant:small-caps;">smb</span> systems. Dilution steps were applied in the bypass streams of both <span style="font-variant:small-caps;">iex-smb</span> schemes to reduce the salt concentrations of the outlet streams of the first sub-unit, leading cytochrome to be strongly bounded in the second sub-unit. Without the dilution steps, the salt concentrations are too high such that ribonuclease and cytochrome are both weekly bounded to the stationary phase of the second sub-unit. Though it is beneficial to implement dilution in the cascade and eight-zone schemes for center-cut separations, it consumes more running buffer on one hand, and the cytochrome product needs to be more concentrated in the subsequent downstream processes on the other hand. There are many options for modeling mass transfer in a column, , the equilibrium-dispersive model (<span style="font-variant:small-caps;">edm</span>), the transport-dispersive model (<span style="font-variant:small-caps;">tdm</span>) and the general rate model (<span style="font-variant:small-caps;">grm</span>) that can be combined with a multitude of binding kinetics. The software package provided in this study is flexible in choosing between various modelling options for the columns and hold-up volumes. <span style="font-variant:small-caps;">smb</span> processes do not necessarily need equivalent column numbers in each zone. Different column numbers in each zone can be applied, resulting the optimization problem to mixed-integer programming. The computational burden of <span style="font-variant:small-caps;">smb</span> processes in numerical optimizations can be reduced by optimizing initial states of the involved columns, or by using computationally cheap surrogate models. 3-dimensional Pareto optimal fronts can also be further investigated, taking buffer consumption, productivity or processing time into account. Acknowledgements ================ This work is partially supported by the China Scholarship Council (CSC, no. 201408310124) and partially by the National Key Research and Development Program of China (NKRDP, grant. 2017YFB0309302). [^1]: Corresponding author. [zhaoliming@ecust.edu.cn](zhaoliming@ecust.edu.cn)
--- abstract: 'Four-qubit Smolin bound entangled state[@Smolin] has a distinct feature: the state is not distillable when every qubit is seperated from each other; but it makes two separated qubit entangled if the other qubits group together. Here the feature is applied to quantum secret sharing, a QSS protocol similar to Ekert 91 protocol of QKD is proposed. The security problem, disadvantage and advantageof this protocol are disscused.' address: \| $^{a}$School for Information and Optoelectronic Science and Engineering,\ South China Normal University,\ GuangZhou,510006, PR China\ $^{b}$Laboratory of Photonic Information Technology, South China Normal\ University, GuangZhou,510006, PR China author: - \| Yafei Yu$^{\thanks{% corresponding author, e-mail: yfyuks@hotmail.com}a,b}$, Yi Xu$^{a}$and Jin Liu$^{a}$ title: '[A quantum secret sharing scheme]{} [among three parties ultilizing four-qubit Smolin bound entangled state]{}' --- Introduction ------------ In the past years, with the development of quantum key distribution[@Charlie; @Ekert], quantum secret sharing (QSS)[@Hillery] attracts much attention in both the theoretical and experimental aspects of quantum communication. QSS is a protocol to split a message to several parts so that no subset of parts is sufficient to read the message but the entire set is. For example, suppose that Alice wants to send a secret to two distant parties, Bob and Charlie. One of them, Bob or Charlie, is not entirely trusted by Alice, but she knows if they who can come together and carry it out for her, but she doesn’ttt know whether they are honest or whether there is an eavesdropper in the channel. She cannot simply send a message to both by a classical channel, because if the two of them coexist, the honest one will keep the dishonest one from doing any damage. Instead of giving entire secret messages to either of them, it may be desirable for Alice to split the secret messages into two encrypted parts and send each one a part so that neither individual is able to obtain all of the original information unless they collaborate. To achieve this end, classical cryptography can use a technique called secret sharing[@Schneier]. QSS is the generation of this concept to quantum scenario. Up to now, there are many kinds quantum secret sharing protocols with and without entanglement. The first QSS protocol has been proposed by Hillery et al[@Hillery], in which three-qubit GHZ (Greenberger-Horne-Zeilinger) entangled states is employed to allow information splitting and eavesdropper protection simultaneously. Moreover, Koashi and Imoto considered the correlation of the two-qubit Bell state in their quantum secret sharing scheme[@Koashi]. Then Bagherinezhad and Karimipour introduced a work for quantum secret sharing which utilizes the reusable GHZ states as secure carriers[@Bagherinezhad]. Entanglement swapping is another method used to realize QSS. Karimipour et al. [@Karimipour] proposed d-level secret sharing via entanglement swapping between a generalized cat states for d-level systems and a generalized Bell states. Other improved versions of QSS based on entanglement swapping were also presented[@Cabello; @Juhui; @Lee; @Zhan-jun; @Zhang]. Product states are alternative resources for realizing QSS. Li-Yi Hsu et al [@product; @state] suggested QSS schemes with a particular set of orthogonal product states in which an unknown quantum state cannot be determined only by LOCC(local operation and classical communication) if the order of the local measurements is private. A BB84-like QSS Scheme was given by the Group Guang-can Guo[@Guo-Ping; @Guo], which security is based on the quantum no-cloning theory. Recently, the experimental demonstrations of QSS by GHZ states[@Yu-Ao; @Chen] and by a single qubit[@Schmid] were reported, respectively. In this work, we use a bound entangled state as quantum resource to accomplish QSS task. Bound entanglement (BE)[@horodecki] is a kind of entanglement in multi-parties system which cannot be distilled to pure entangled form only by local operations and classical communication (LOCC). But for some bound entangled states, collective operation between some subparties induces distillable entanglement shared between the others. Four-party Smolin entangled state [@Smolin] is such state: when two parties come together, they can by LOCC enable the other two parties to have some pure entanglement. The Smolin state can be expressed as: $$\begin{aligned} \rho &=&\frac{1}{4}(\left\| \Phi ^{+}\right\rangle _{12}\left\langle \Phi ^{+}\right\| \otimes \left\| \Phi ^{+}\right\rangle _{34}\left\langle \Phi ^{+}\right\| +\left\| \Phi ^{-}\right\rangle _{12}\left\langle \Phi ^{-}\right\| \otimes \left\| \Phi ^{-}\right\rangle _{34}\left\langle \Phi ^{-}\right\| \nonumber \\ &&+\left\| \Psi ^{+}\right\rangle _{12}\left\langle \Psi ^{+}\right\| \otimes \left\| \Psi ^{+}\right\rangle _{34}\left\langle \Psi ^{+}\right\| +\left\| \Psi ^{-}\right\rangle _{12}\left\langle \Psi ^{-}\right\| \otimes \left\| \Psi ^{-}\right\rangle _{34}\left\langle \Psi ^{-}\right\| )\end{aligned}$$ where we use the usual notation for the maximally entangled states of two qubit.(Bell state) $$\left\| \Phi ^{\pm }\right\rangle =\frac{1}{\sqrt{2}}(\left\| 00\right\rangle \pm \left\| 11\right\rangle )$$ $$\left\| \Psi ^{\pm }\right\rangle =\frac{1}{\sqrt{2}}(\left\| 01\right\rangle \pm \left\| 10\right\rangle )$$ In other word, if 1 and 2 particle come together and do the nonlocal Bell measurement on their systems, they can determine reliably which Bell state they have since the four Bell states are orthogonal, and 3, 4 have the same one with 1,2. The feature of Smolin state is helpful for QSS scheme[@Smolin; @Augusiak; @Augusiak2]. Our work is organized as follows. In Sec. II we present a QSS scheme utilizing four party bound entangled states (Smolin states for short), and analyze the security in the sense of the violation to Bell inequality. In Sec.III the scheme is compared with others QSS protocol, and the conclusions are given. Quantum Secret Sharing With Smolin States ----------------------------------------- To achieve the purpose of sharing secret, the scheme is divided into three stages. The first stage is for preparation and distribution of Smolin states. Alice produces a series of Smolin states and sends two qubit in every event to Bob and Charlie. Secondly, for checking the security of quantum channel, Alice chooses randomly some Smolin states from the series. The three man determine whether the correlation in the Smolin states violate the Bell inequality or not. Then if the channel is secure, after Alice operates qubits in hand, Bob and Charlie share Alices’ private classical information which can be revealed only by Bob and Charlie’s collaboration. In the first phase, Alice prepares N copies of four-qubit bound entangled Smolin states each of which has a corresponding record number. For each Smolin state qubits 1, 2 are in the possession of Alice, and qubits 3, 4 are send to Bob and Charlie, respectively. Once Bob and Charlie receive one qubit, they publicly announce the facts. As a result, Alice has qubits 1, 2 of each Smolin state, Bob has qubit 3 and Charlie qubit 4. And the shared Smolin states between the three parties will serve as carriers of private classical information. In the phase of examining the security of quantum channel, Alice chooses randomly M copies from the Smolin states, and inform Bob and Charlie which their record numbers are. Then Alice projects qubit 1 into the basis $\frac{% \left\| x\right\rangle \pm \left\| y\right\rangle }{\sqrt{2}}$ and qubit 2 into $\left\| x\right\rangle ,\left\| y\right\rangle $. Bob and Charlie measure their qubits 3, 4 in the base of $\left\| x\right\rangle ,\left\| y\right\rangle $ and send their results back to Alice, respectively. Collecting all results of measurements on four qubits, Alice is able to calculate the value of correlation function $E$ of four qubits in Smolin state. For four qubits, a two-setting Bell-inequality similar to standard CHSH [@CHCS] inequality for two particles is given as [@Augusiak]: $$\left\| E(1,1,1,1)+E(1,1,1,2)+E(2,2,2,1)-E(2,2,2,2)\right\| \leq 2.$$ But for four-qubit Smolin entangled state, the correlation function $E_{QM}$ satisfies $$\left\| E_{QM}(1,1,1,1)+E_{QM}(1,1,1,2)+E_{QM}(2,2,2,1)-E_{QM}(2,2,2,2)\right\| =2% \sqrt{2}.$$ which gives violation, and it is proved that the above violation to two-setting Bell inequality is maximal [@Augusiak]. Therefore, if the qubits are not directly or indirectly disturbed, Alice gets the Equation (5), and the quantum channel is secure and can be used for private communication. In the phase of transferring information, Alice first measures the qubits 1,2 of the rest (N-M) copies of Smolin states in Bell basis $\left\{ \left\| \Phi ^{\pm }\right\rangle ,\left\| \Psi ^{\pm }\right\rangle \right\} $, and encode the results obtained, for example as $\left\| \Phi ^{+}\right\rangle =00,\left\| \Phi ^{+}\right\rangle =01,\left\| \Psi ^{+}\right\rangle =10,\left\| \Psi ^{-}\right\rangle =11.$ Hence, after the measurement, Alice creates a random and private bit string which, simultaneously, is send to and shared between Bob and Charlie when finishing the measurement is announced.. To reveal the bit string, Bob and Charlie have to come together and determine that in which one of four Bell states their qubits are, then read out Alice’s secure bit-string information. Actually and very importantly, in this phase, the key is created, send and shared just at the same time. As a result, the task of QSS is achieved. Then let’s go back to consider how to detect possible eavesdropping attacks.. If there is Eve in the channel who wants to extract out Alice’s private information. As discussed in Ref.[@Ekert91], Eve cannot elicit any information from the qubits while in transit from Alice to Bob and Charlie, because the private information is created only after Alice’s measurements on qubits 1,2 and announcement for finishing it. So Eve has to intercept and clone the qubits 3,4 send to Bob and Charlie, and distribute one copy between them. However the intervention of Eve will introduce noise to the original Smolin state. Now the modified Smolin state is expressed in a general form: $$\rho ^{noisy}=\frac{1-p}{16}I^{\otimes 4}+p\rho$$ where I stands for identity on one-qubit space, $p$ is scaling parameter which $p\leq \frac{2}{3}$ because of the intervention of Eve. The corresponding correlation function $E_{QM}$ is amended as follows: $$\begin{array}{c} E_{QM}(1,1,1,1)(\rho ^{noisy}(p))+E_{QM}(1,1,1,2)(\rho ^{noisy}(p)) \\ +E_{QM}(2,2,2,1)(\rho ^{noisy}(p))-E_{QM}(2,2,2,2)(\rho ^{noisy}(p))=2\sqrt{2% }p \end{array}$$ For $p\leq \frac{2}{3}$, the value of the Eq.\[7\] do not violate two-setting Bell-inequality of four qubits. Therefore, the existence of Eve will be detected while Alice tests the violation of the correlation function $E$ to Bell-inequality. There may be another method for Eve to choose. Eve makes Bell measurement on qubits 3,4, prepares two copies same as the result, and dispenses one to Bob and Charlie. It makes that both of Eve and Alice have the same Bell states as shared between Bob and Charlie. Eve’s trick can also be detected if Alice simply tests whether the correlation function between qubit 1,2 violate Bell-inequality of two qubits or not. Then the security for the present quantum secret sharing is guaranteed. Conclusion ---------- This investigation introduces a quantum secret sharing scheme using four-qubit Smolin bound entangled state as private channel. The idea is very similar to Ekert’s 91 protocol about quantum key distribution [@Ekert91]. By testing the violation of the four-qubit correlation function to two-setting Bell-inequality, the existence of eavesdropper can be detected. Comparing with the QSS protocol via GHZ state in which a half of qubits must be discarded because of incorrect direction for measurement, the efficiency of the present scheme can reach $100\%$ if the quantum channel is secure. Another QSS scheme with the help of bound entangled states is discussed in Ref.[@Augusiak2]. However, its efficiency can only approach 50% in principle. On other hand, because each Bell state carries two bits hidden classical information, for every copy of Smolin state Alice can send 2 bits shared by Bob and Charlie. But the generalization of this quantum secret sharing scheme to multi-parties is not good as expected. In each Smolin state, the number of qubits in hand of Alice is equal to the number of parties sharing secret information. With the growing of the parties, the dimension of collective measurement is increasing, while collective measurement on multi-qubit system is more difficult to accomplish. Acknowledgments --------------- This work was financially supported by National Natural Science Foundation of China under Grant. No. 10404007. References ---------- John A. Smolin, Phys. Rev. A 63, 032306(2001). C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers (IEEE, New York, 1984), pp. 175-179. A. Ekert, Phys. Rev. Lett. 67, 661 (1991). B. Schneier, Applied Cryptography, Wiley, New York, 1996;See also J. Gruska, Foundations of Computing, Thomson Computer Press, London, 1997. V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, Phys. Rev. A 65, 042320 (2002). M. Hillery, V. Bužek, and A. Berthiaume Phys. Rev. A 59,1829 (1999). M. Koashi and N. Imoto, Phys. Rev. Lett. 79, 2383 (1997). A. Cabello, e-print quant-ph/0009025. Juhui Lee, Soojoon Lee, Jaewan Kim, and Sung Dahm Oh, Phys. Rev. A 70, 032305 (2004) Zhan-jun Zhang and Zhong-xiao Man, Phys. Rev. A 72, 022303 (2005). Yu-Ao Chen, An-Ning Zhang, Zhi Zhao, Xiao-Qi Zhou, Chao-Yang Lu, Cheng-Zhi Peng, Tao Yang, and Jian-Wei Pan, Phys. Rev. Lett. 95, 200502(2005). Christian Schmid, Pavel Trojek, Mohamed Bourennane, Christian Kurtsiefer, Marek zukowski, and Harald Weinfurter, Phys. Rev. Lett. 95, 230505(2005). S. Bagherinezhad and V. Karimipour, Phys. Rev. A 67,044302 (2003). Li-Yi Hsu,Che-Ming Li Phys. Rev. A 71, 022321(2005). Guo-Ping Guo , Guang-Can Guo,Phy. Let. A 310 (2003) 247-251. M. Horodecki, P. Horodecki and R.Horodecki, Phys. Rev. Lett. 80, 5239(1998). R. Augusiak, P. Horodecki,quant-ph/0405187 (2004). Remigiusz Augusiak and Pawe Horodecki, Phys. Rev. A 73, 012318 (2006) M. Žukowski,Č.Brukner,Phys Rev Lett,88.210401 (2002) J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt,Phys. Rev. Lett. 23, 880 (1969); J. F. Clauser, M. A.Horne, Phys. Rev. D 10, 529 (1974). A K Ekert, Phys Rev Lett ,1991 ,67 :661.
--- abstract: \| Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$. For each $d\, <\,\eta(X)$, where $\eta(X)$ is the gonality of $X$, the symmetric product $\text{Sym}^d(X)$ embeds into $\text{Pic}^d(X)$ by sending an effective divisor of degree $d$ to the corresponding holomorphic line bundle. Therefore, the restriction of the flat Kähler metric on $\text{Pic}^d(X)$ is a Kähler metric on $\text{Sym}^d(X)$. We investigate this Kähler metric on $\text{Sym}^d(X)$. In particular, we estimate it’s Bergman kernel. We also prove that any holomorphic automorphism of $\text{Sym}^d(X)$ is an isometry. address: - 'Department of Mathematics,University of Hyderabad, Prof. C. R. Rao Road, Hyderabad 500046, India' - 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India' - 'Department of Mathematics,University of Hyderabad, Prof. C. R. Rao Road, Hyderabad 500046, India' - 'Department of Mathematics,University of Hyderabad, Prof. C. R. Rao Road, Hyderabad 500046, India' author: - Anilatmaja Aryasomayajula - Indranil Biswas - 'Archana S. Morye' - Tathagata Sengupta title: 'On the Kähler metrics over $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$' --- Introduction ============ Symmetric products of Riemann surfaces were studied by Macdonald [@Ma]; he explicitly computed their cohomologies. Interests on these varieties revived when it was realized that they constitute examples of vortex moduli spaces [@Br1], [@Br2], [@Ga]. One of the questions was to compute the volume, which was resolved in a series of papers [@Na], [@MN], [@Pe]; see also [@Ba] for Kähler structure on vortex moduli spaces. Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$, and let $\eta(X)$ denote the gonality of $X$ (this means that $X$ admits a nonconstant holomorphic map to ${\mathbb C}{\mathbb P}^1$ of degree $\eta(X)$ and it does not have any smaller degree nonconstant holomorphic map to ${\mathbb C}{\mathbb P}^1$). Take any integer $1\, \leq\, d\, <\, \eta(X)$. Let $$\varphi\, :\, \text{Sym}^d(X)\,\longrightarrow\, \text{Pic}^d(X)$$ be the map from the symmetric product that sends any $\{x_1,\, \ldots\, ,x_d\}$ to the holomorphic line bundle ${\mathcal O}_X(x_1+\cdots\, +x_d)$. We prove that $\varphi$ is an embedding. The natural inner product on $H^0(X,\, K_X)$, where $K_X\,\longrightarrow\, X$ is the holomorphic cotangent bundle, produces a flat Kähler metric on $\text{Pic}^d(X)$. It is natural to construct a metric on $\text{Sym}^d(X)$ by pulling back the flat metric using the embedding $\varphi$; see [@Ri], [@MR] (especially [@MR p. 1137, (1.2)], [@MR § 7]). Our aim here is to study this metric on $\text{Sym}^d(X)$. We prove that any holomorphic automorphism of $\text{Sym}^d(X)$ is in fact an isometry. Our main result is estimation of the Bergman kernel of the metric. Classically, the Bergman kernel which is the reproducing kernel for $L^2$-holomorphic functions has been extensively studied in complex analysis. The generalization of the Bergman kernel to complex manifolds as the kernel for the projection onto the space of harmonic $(p, q)$-forms with $L^2$-coefficients carries the information on the algebraic and geometric structures of the underlying manifolds. Using results from [@jkf] and [@jl], we derive the following estimate for $\operatorname{\mathcal{B}_{{\it{X}}}}(z)$, the Bergman kernel associated to the Riemann surface $X$: $$\operatorname{\mathcal{B}_{{\it{X}}}}(z)\leq \frac{48}{\pi}+\frac{4}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)},$$ where $\operatorname{{\it{r_{_X}}}}$ denotes the injectivity radius of $X$. We also study the above estimate for admissible sequences of compact hyperbolic Riemann surfaces. Our estimates are optimal, and these estimates continue to hold true for any compact hyperbolic Riemann surface. Comparison of Kähler metrics ============================ In this section, we introduce the hyperbolic and canonical metrics defined on a compact hyperbolic Riemann surface. Furthermore, we introduce the Bergman kernel, and derive estimates for it. We then extend these estimates to admissible sequences of compact hyperbolic Riemann surfaces. Canonical and hyperbolic metrics {#se2.1} -------------------------------- Let $X$ be a compact, connected Riemann surface of genus $g$, with $g\,>\, 1$. Let $$\begin{aligned} \mathbb{H}\,:=\,\lbrace z\,=\,x+\sqrt{-1}y\,\in\, \mathbb{C}\,\mid\,y\,>\,0 \rbrace \end{aligned}$$ be the upper half-plane. Using the uniformization theorem $X$ can be realized as the quotient space $\Gamma\backslash\mathbb{H}$, where $\Gamma\,\subset\, \mathrm{PSL}_{2}(\mathbb{R})$ is a torsionfree cocompact Fuchsian subgroup acting on $\mathbb{H}$, via fractional linear transformations. Locally, we identify $X$ with its universal cover $\mathbb{H}$ using the covering map $\mathbb{H}\,\longrightarrow\, X$. The holomorphic cotangent bundle on $X$ will be denoted by $K_X$. Let $$\mathrm{Jac}(X)\,=\,\text{Pic}^{0}(X)$$ be the Jacobian variety that parametrizes all the (holomorphic) isomorphism classes of topologically trivial holomorphic line bundles on $X$. It is equipped with a flat Kähler metric $g_J$ given by the Hermitian structure on $H^0(X,\, K_X)$ defined by $$\label{m0} (\alpha\, , \beta)\, \longmapsto\, \frac{\sqrt{-1}}{2}\int_X \alpha\wedge \overline{\beta}\, .$$ Fix a base point $x_0\, \in\, X$. Let $${\rm AJ}_X\, :\, X\, \longrightarrow\, \mathrm{Jac}(X)$$ be the Abel-Jacobi map that sends any $x\,\in\, X$ to the holomorphic line bundle on $X$ of degree zero given by the divisor $x-x_0$. It is a holomorphic embedding of $X$. The pulled back Kähler metric ${\rm AJ}^_X g_J$ on $X$ is called the [canonical metric]{}. The $(1,1)$-form on $X$ associated to the canonical metric is denoted by $\operatorname{\mu_{{\it{X}}}^{can}}$. The canonical metric has the following alternate description. Let $S_{2}(\Gamma)$ denote the $\mathbb{C}$-vector space of cusp forms of weight-$2$ with respect to $\Gamma$. Let $\lbrace f_{1}\, ,\ldots\, ,f_{g}\rbrace $ denote an orthonormal basis of $S_{2}(\Gamma)$ with respect to the Petersson inner product. Then, the $(1,1)$-form $\operatorname{\mu_{{\it{X}}}^{can}}(z)$ corresponding to the canonical metric of $X$ is given by $$\label{candefn1} \operatorname{\mu_{{\it{X}}}^{can}}(z)\,:=\, \frac{\sqrt{-1}}{2g} \sum_{j=1}^{g}\left\|f_{j}(z)\right\|^{2}dz\wedge d\overline{z}\, .$$ The volume of $X$ with respect to the canonical metric is one. Consider the hyperbolic metric of $X$, which is compatible with the complex structure on $X$ and has constant negative curvature $-1$. We denote by $\operatorname{\mu_{{\it{X}}}^{hyp}}$ the $(1,1)$–form on $X$ corresponding to it. The hyperbolic form on $\mathbb H$ is given by $$\frac{\sqrt{-1}}{2}\cdot\frac{dz\wedge d\overline{z}}{{\operatorname{Im}(z)}^{2}}\, .$$ So on $X$, the form $\operatorname{\mu_{{\it{X}}}^{hyp}}(z)$ is given by $$\operatorname{\mu_{{\it{X}}}^{hyp}}(z)\,:=\, \frac{\sqrt{-1}}{2}\cdot\frac{dz\wedge d\overline{z}}{{\operatorname{Im}(z)}^{2}},$$ for $z\in X$. The total volume $\operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X)$ of $X$ with respect to the hyperbolic metric $\operatorname{\mu_{{\it{X}}}^{hyp}}$ is given by the formula $$\operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X) \,=\, 4\pi\big(g -1 \big)\, .$$ Let $$\operatorname{\mu^{shyp}_{{\it{X}}}}(z)\,:= \,\frac{\operatorname{\mu_{{\it{X}}}^{hyp}}(z)}{ \operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X)}$$ denote the rescaled hyperbolic metric on $X$, which is normalized in such a way that the volume of $X$ is one. Estimates of the Bergman kernel {#subsec2.2} ------------------------------- For any $z\,\in\, X$, the Bergman kernel $\operatorname{\mathcal{B}_{{\it{X}}}}$ associated to the Riemann surface $X$ is given by the following formula $$\begin{aligned} \operatorname{\mathcal{B}_{{\it{X}}}}(z)\,:=\,\sum_{j=1}^{g}y^{2}\left\|f_{j}(z)\right\|^{2}\, ,\end{aligned}$$ where $y\,=\, {\rm Im}\, z$. The injectivity radius $\operatorname{{\it{r_{_X}}}}$ of $X$ is defined as $$\begin{aligned} \operatorname{{\it{r_{_X}}}}\,:=\, \inf\big\lbrace{d_{\mathbb{H}}(z,\gamma z)\,\mid\, z\,\in\, \mathbb{H}, \, \gamma\in\Gamma\backslash\lbrace \mathrm{id}\rbrace \big\rbrace}\, ,\end{aligned}$$ where $d_{\mathbb{H}}(z,\gamma z)$ denotes the hyperbolic distance between $z$ and $\gamma z$. Let $f$ be any positive, smooth, real valued decreasing function defined on $\mathbb{R}_{\geq 0}$. From [@jl Lemma 4], for any $\delta\,>\, \operatorname{{\it{r_{_X}}}}\slash 2$, and assuming that all the involved integrals exist, we have the following inequality $$\int_{0}^{\infty}f(\rho)dN_{\Gamma}(z_{1},z_{2};\rho)\,\leq\, \int_{0}^{\delta}f(\rho)dN_{\Gamma}(z_{1},z_{2};\rho)$$ $$\label{jlinequality} +\, f(\delta)\frac{\sinh(\operatorname{{\it{r_{_X}}}}\slash 2)\sinh(\delta)}{\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)} + \frac{1}{2\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}\int_{\delta}^{\infty}f(\rho) \sinh(\rho+\operatorname{{\it{r_{_X}}}}\slash 2)d\rho\, ,$$ where $$\begin{aligned} N_{\Gamma}(z_{1},z_{2};\rho)\,:=\,\mathrm{card}\,\big\lbrace \gamma\,\mid\,\gamma\in\Gamma,\,d_{\mathbb{H}}(z_1, \gamma z_2)\leq \rho \big\rbrace.\end{aligned}$$ Notice that the above injectivity radius $\operatorname{{\it{r_{_X}}}}$ is twice the injectivity radius defined in [@jl]. \[boundbk\] For any $z\in X$, the following estimate holds: $$\operatorname{\mathcal{B}_{{\it{X}}}}(z)\,\leq\, B_X\, :=\, \frac{48}{\pi}+\frac{4}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}\, .$$ Substituting $k\,=\,1$ in inequality (13) of [@jkf], we arrive at $$\begin{aligned} \label{boundbkeqn1} \operatorname{\mathcal{B}_{{\it{X}}}}(z)\,\leq\, \frac{\sqrt{2}}{3\pi}\sum_{\gamma\in\Gamma}\frac{1}{\cosh^2(\rho_{\gamma,z}\slash 2)}\int_{\rho_{\gamma,z}}^{\infty} \frac{ue^{-u\slash 2}}{\sqrt{\cosh(u)-\cosh(\rho_{\gamma,z})}}du\, ,\end{aligned}$$ where $\rho_{\gamma,z}\,=\,d_{\mathbb{H}}(z,\gamma z)$. Using the fact that $u\,\leq\,\sinh(u)$ for all $u\,\geq\, 0$, $$\begin{aligned} \int_{\rho_{\gamma,z}}^{\infty} \frac{ue^{-u\slash 2}}{\sqrt{\cosh(u)-\cosh(\rho_{\gamma,z})}}du \,\leq\, \int_{\rho_{\gamma,z}}^{\infty}\frac{ue^{-u\slash 2}}{\sqrt{\cosh(u)-1}}du\notag\\ =\,\int_{\rho_{\gamma,z}}^{\infty}\frac{ue^{-u\slash 2}}{\sqrt{2\sinh^{2}(u\slash2)}}du \,\leq\, \sqrt{2}\int_{\rho_{\gamma,z}}^{\infty}e^{-u\slash 2}du\,=\, 2\sqrt{2}e^{-\rho_{\gamma,z}}\, .\label{boundbkeqn2}\end{aligned}$$ Combining and , and using the fact that the inequality $\cosh(u)\,\geq\, e^{u}\slash 2$ holds for all $u\,\geq\, 0$, it follows that $$\begin{aligned} \operatorname{\mathcal{B}_{{\it{X}}}}(z)\,\leq\, \frac{4}{3\pi}\sum_{\gamma\in\Gamma} \frac{e^{-\rho_{\gamma,z}}}{\cosh^2(\rho_{\gamma,z} \slash 2)}\,\leq\, \frac{16}{3\pi}\sum_{\gamma\in\Gamma} \frac{e^{-\rho_{\gamma,z}}}{e^{\rho_{\gamma,z}}}\,=\,\frac{16}{3\pi}\int_{0}^{\infty} e^{-2\rho}dN_{\Gamma}(z,\gamma z;\rho)\, .\end{aligned}$$ As $e^{-2\rho}$ is a monotonically decreasing function in $\rho\,\in\, \mathbb{R}_{\geq 0}$, using we compute $$\operatorname{\mathcal{B}_{{\it{X}}}}(z)\,\,\,\leq\,\, \, \frac{16}{3\pi}\int_{0}^{\frac{3\operatorname{{\it{r_{_X}}}}}{4}}e^{-2\rho}dN_{\Gamma}(z,\gamma z;\rho)$$ $$\label{boundbkeqn3} + \frac{16e^{-\frac{3\operatorname{{\it{r_{_X}}}}}{2}}\sinh(\operatorname{{\it{r_{_X}}}}\slash 2)\sinh(3\operatorname{{\it{r_{_X}}}}\slash 4)}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)} +\frac{8}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}\int_{\frac{3\operatorname{{\it{r_{_X}}}}}{4}}^{\infty} e^{-2\rho}\sinh\bigl(\rho+\frac{\operatorname{{\it{r_{_X}}}}}{2}\bigr)d\rho\,.$$ From the definition of the injectivity radius $\operatorname{{\it{r_{_X}}}}$ we have $$\label{boundbkeqn4} \frac{16}{3\pi}\int_{0}^{\frac{3\operatorname{{\it{r_{_X}}}}}{4}}e^{-2\rho}dN_{\Gamma}(z,\gamma z;\rho) \,=\, \frac{16}{3\pi}\,.$$ Using the fact that $\sinh(u)$ is a monotone increasing function and that the inequality $\cosh(u)\,\leq \, e^{u}$ holds for all $u\,\geq\, 0$, we have the following estimate for the second term on the right-hand side of inequality in : $$\frac{16e^{-\frac{3\operatorname{{\it{r_{_X}}}}}{2}}\sinh(\operatorname{{\it{r_{_X}}}}\slash 2)\sinh(3\operatorname{{\it{r_{_X}}}}\slash 4)}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)} \,\leq\, \frac{16e^{-\frac{3\operatorname{{\it{r_{_X}}}}}{2}}\sinh(\operatorname{{\it{r_{_X}}}}\slash 2)\sinh(\operatorname{{\it{r_{_X}}}})}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}$$ $$\label{boundbkeqn5} \leq\,\frac{128e^{-\frac{3\operatorname{{\it{r_{_X}}}}}{2}}\cosh^2(\operatorname{{\it{r_{_X}}}}\slash 4)\cosh(\operatorname{{\it{r_{_X}}}}\slash 2)}{3\pi}\, \leq \,\frac{128e^{-\frac{\operatorname{{\it{r_{_X}}}}}{2}}}{3\pi}\,\leq\, \frac{128}{3\pi}\,.$$ Using the fact that $$\sinh(u)\,\leq\, e^{u}\slash 2$$ for all $u\,\geq\, 0$, we derive the following estimate for the third term on the right-hand side of the inequality in : $$\frac{8}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}\int_{\frac{3\operatorname{{\it{r_{_X}}}}}{4}}^{\infty}e^{-2\rho} \sinh\bigl(\rho+\frac{\operatorname{{\it{r_{_X}}}}}{2}\bigr)d\rho$$ $$\label{boundbkeqn6} \leq\, \frac{4e^{\frac{\operatorname{{\it{r_{_X}}}}}{2}}}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}\int_{\frac{3\operatorname{{\it{r_{_X}}}}}{4}}^{\infty}e^{-\rho}d\rho \,=\,\frac{4e^{-\frac{\operatorname{{\it{r_{_X}}}}}{4}}}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)} \,\leq\, \frac{4}{3\pi\sinh^{2}(\operatorname{{\it{r_{_X}}}}\slash 4)}\, .$$ Now the theorem follows from , , and . Let $\lbrace X_{N}\rbrace_{N\in\mathcal{N}}$, indexed by $\mathcal{N}\,\subseteq\, \mathbb{N}$, be a set of compact hyperbolic Riemann surfaces. We say that the sequence is admissible* if it is one of the following two types: 1. If $\mathcal{N}\,=\,\mathbb{N}$ and $N\,\in\,\mathcal{N}$, then $X_{N+1}$ is a finite degree unramified cover of $X_{N}$. 2. Let $\mathcal{N} \,\subset\, \mathbb{N}$ be such that for each $N \,\in \, \mathcal{N}$, the modular curves $X_{0}(N)$, $X_{1}(N)$, $X(N)$, have genus $g\,>\,1$. We consider families of modular curves $\lbrace X_{N} \rbrace_{N \,\in \,\mathcal{N}}$ given by $$\lbrace X_{0}(N) \rbrace_{N \in \mathcal{N}},\,\,\, \lbrace X_{1}(N) \rbrace_{N \,\in\, \mathcal{N}},\,\,\, \lbrace X(N) \rbrace_{N \in \mathcal{N}}\, .$$ See [@jkcomp p. 695–696, Definition 5.1]. Let $q_{_{\mathcal{N}}}\,\in\,\mathcal{N}$ be the minimal element of the indexing set $\mathcal{N}$. So in Case (1), we have $q_{_{\mathcal{N}}}\,=\,0$, while in Case (2), the integer $q_{_{\mathcal{N}}}$ is the smallest prime in $\mathcal{N}$. \[boundbkcor\] Let $\lbrace X_{N}\rbrace_{N\in\mathcal{N}}$ be an admissible sequence of compact hyperbolic Riemann surfaces. Then, for all $N\,\in\,\mathcal{N}$, the Bergman kernel $\operatorname{\mathcal{B}_{{\it{X_N}}}}(z)$ is bounded by a constant which depends only on the Riemann surface $X_{q_{_{\mathcal{N}}}}$. From Theorem \[boundbk\], we have $$\begin{aligned} \label{corxeqn1} \operatorname{\mathcal{B}_{{\it{X_N}}}}(z)\, \leq\,B_{X_{N}}\,= O\left(\frac{1}{r^{2}_{X_{N}}}\right)\,.\end{aligned}$$ Recall that injectivity radius $\operatorname{{\it{r_{_{X_N}}}}}$ is equal to $\ell_{X_{N}}$, the length of the shortest geodesic on $X_N$. From assertion (a) in [@jkcomp Lemma 5.3] we know that for all $N\,\in\,\mathcal{N}$, the number $\frac{1}{\operatorname{{\it{r_{_{X_N}}}}}}$ is bounded by a number that depends only on the Riemann surface $X_{q_{_{\mathcal{N}}}}$. Therefore, the estimate completes the proof. In [@bergmanbounds], B.-Y. Chen and S. Fu have also derived a similar estimate for the Bergman kernel as in Corollary \[boundbkcor\]. However, their estimate is valid only for any compact hyperbolic Riemann surfaces with injectivity radius greater than or equal to $\log(3)$. Cartesian product $\operatorname{{\it{X^{d}}}}$ =============================================== In this section, we introduce the hyperbolic and canonical metrics defined over the $d$-fold Cartesian product $\operatorname{{\it{X^{d}}}}$ of $X$. We, then compute an estimate for the volume form associated to the canonical metric. Canonical and hyperbolic metrics {#se3.1} -------------------------------- Take $X$ as before. Let $\operatorname{{\it{X^{d}}}}$ denote the $d$-fold Cartesian product $X\times \cdots\times X$. For each $1\, \leq\, i\,\leq\, d$, let $$p_i\, :\, X^d\,\longrightarrow\, X$$ be the projection to the $i$-th factor. Define $$\operatorname{\mu_{{\it{X^d}}}^{hyp}}\,=\, \sum_{i=1}^d p^_i \operatorname{\mu_{{\it{X}}}^{hyp}}\ \ \text{ and }\ \ \operatorname{\mu_{{\it{X^d}}}^{shyp}}\,=\, \sum_{i=1}^d p^_i \operatorname{\mu^{shyp}_{{\it{X}}}}\, .$$ We denote by $\operatorname{\mu_{{\it{X^d}},vol}^{shyp}}$ the volume form associated to $\operatorname{\mu_{{\it{X^d}}}^{shyp}}$. Note that the total volume of $X^d$ with respect to $\operatorname{\mu_{{\it{X^d}},vol}^{shyp}}$ is $1$, because the total volume of $X$ with respect to $\operatorname{\mu^{shyp}_{{\it{X}}}}$ is $1$. With respect to a local coordinate $z\,=\,(z_1\, ,\ldots\, ,z_d)$ on $X^d$, where $z_i\,=\, x_i+\sqrt{-1}y_i$ are hyperbolic coordinates on $X$, the hyperbolic volume form is given by $$\begin{aligned} \operatorname{\mu_{{\it{X^d}},vol}^{shyp}}(z)\,=\,\frac{1}{(\operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X))^{d}}\bigwedge_{j=1}^{d}\frac{\sqrt{-1}}{2}\cdot \frac{dz_{j}\wedge d\overline{z}_{j}}{y_{j}^{2}}\,=\, \frac{1}{(4\pi(g-1))^{d}}\bigwedge_{j=1}^{d}\frac{\sqrt{-1}}{2}\cdot \frac{dz_{j}\wedge d\overline{z}_{j}}{y_{j}^{2}}\, .\end{aligned}$$ The gonality of $X$ is defined to be the smallest among all positive integers $m$ such that $X$ admits a nonconstant holomorphic map to ${\mathbb C}{\mathbb P}^1$ of degree $m$. The gonality of $X$ will be denoted by $\eta(X)$. So $\eta(X)\,=\,2$ if and only if $X$ is hyperelliptic. We assume that $d\, <\, \eta(X)$. Let $\text{Pic}^d(X)$ denote the component of the Picard group of $X$ that parametrizes all the holomorphic line bundles of degree $d$. Consider the holomorphic map $$\label{phi} \phi\, :\, X^d\, \longrightarrow\, \mathrm{Pic}^{d}(X)\, ,\ \ (x_1\, ,\ldots\, , x_d)\, \longmapsto\, {\mathcal O}_X(x_1+\cdots +x_d)\, .$$ Since $d\, <\, m$, it can be shown that the fibers of the above map $\phi$ are zero dimensional. Indeed, if $$\phi((x_1, \ldots,x_d))\,=\, \phi((y_1, \ldots,y_d))\, ,$$ the holomorphic line bundle ${\mathcal O}_X(x_1+\cdots +x_d)$ has two nonzero sections given by the two effective divisors $x_1+\cdots +x_d$ and $y_1+\cdots +y_d$. These two sections can’t be linearly independent because that would contradict the assumption on $d$ that it is strictly smaller than $\eta(X)$. Since two sections are constant multiples of each other, it follows that $(x_1, \ldots,x_d)$ and $(y_1, \ldots,y_d)$ differ by a permutation of $\{1\, ,\ldots\, ,d\}$. Therefore, we have the following: \[l1\] Any two points of $X^d$ lying in a fiber of the map $\phi$ differ by a permutation of $\{1\, ,\ldots\, ,d\}$. The variety $\text{Pic}^d(X)$ is a torsor for $\mathrm{Jac}(X)$, because any two holomorphic line bundles of degree $d$ differ by tensoring with a unique holomorphic line bundle of degree zero. Therefore, by fixing a point of $\text{Pic}^d(X)$ we may identify $\mathrm{Jac}(X)$ with $\text{Pic}^d(X)$. Using this identification, we get a Kähler metric on $\text{Pic}^d(X)$ given by the metric on $\mathrm{Jac}(X)$ constructed in . This metric on $\text{Pic}^d(X)$ will be denoted by $g_d$. We note that $g_d$ does not depend on the choice of the point in $\text{Pic}^d(X)$ used in identifying $\mathrm{Jac}(X)$ with $\text{Pic}^d(X)$. The pullback $\phi^g_d$ is the canonical metric on $X^d$, which we denote by $\operatorname{\mu_{{\it{X^d}}}^{can}}$. The canonical metric degenerates along the divisor where two or more coordinates coincide (where the action of the group of permutations of $\{1\, ,\ldots\, ,d\}$ is not free). In Remark \[re1\] we will see that this is precisely the locus where $\operatorname{\mu_{{\it{X^d}}}^{can}}$ degenerates. As in Section \[se2.1\], let $\lbrace f_{1}\, ,\ldots\, ,f_{g}\rbrace $ be an orthonormal basis of $S_{2}(\Gamma)$ with respect to the Petersson inner product. The (1,1)-form associated to the canonical metric $\operatorname{\mu_{{\it{X^d}}}^{can}}$ is given by $$\begin{aligned} \label{eqn2} \operatorname{\mu_{{\it{X^d}}}^{can}}\,=\,\frac{\sqrt{-1}}{2g^d}\sum_{j=1}^{g}\sum_{a,b=1}^{d}f_{j}(z_{a}) \overline{f_{j}(z_{b})}dz_{a}\wedge d\overline{z}_{b}\, .\end{aligned}$$ The volume form associated to the canonical metric $\operatorname{\mu_{{\it{X^d}}}^{can}}$ measures the total volume of $\operatorname{{\it{X^{d}}}}$ to be one. For any $z=(z_{1}\, ,\ldots\, ,z_{d})\,\in\,\operatorname{{\it{X^{d}}}}$, the Bergman kernel associated to $\operatorname{{\it{X^{d}}}}$ is given by the formula $$\begin{aligned} \operatorname{\mathcal{B}_{{\it{X^d}}}}(z)\,=\, \prod_{i=1}^{d}\operatorname{\mathcal{B}_{{\it{X}}}}(z_i,w_i)\, . \end{aligned}$$ Estimates of $\operatorname{\mu_{{\it{X^d}},vol}^{can}}$ -------------------------------------------------------- In this subsection, using the estimate for the Bergman kernel $\operatorname{\mathcal{B}_{{\it{X}}}}(z)$ derived in Theorem \[boundbk\], we estimate $\operatorname{\mu_{{\it{X^d}},vol}^{can}}$, the volume form associated to the canonical metric $\operatorname{\mu_{{\it{X^d}}}^{can}}$. \[boundvol\] For any $z\,\in\, \operatorname{{\it{X^{d}}}}$, the following inequality holds: $$\begin{aligned} \Bigg\|\frac{\operatorname{\mu_{{\it{X^d}},vol}^{can}}(z)}{\operatorname{\mu_{{\it{X^d}},vol}^{shyp}}(z)}\Bigg\| \,\leq\, (d!)^2\bigg(\frac{\operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X)B_X}{g^{d-1}}\bigg)^d\, .\end{aligned}$$ For any $z\,=\,(z_1\, ,\ldots\, ,z_d)\,\in\,\operatorname{{\it{X^{d}}}}$, the canonical volume form $\operatorname{\mu_{{\it{X^d}},vol}^{can}}$ is given by $$\operatorname{\mu_{{\it{X^d}},vol}^{can}}(z)\,=\,$$ $$\Bigg(\frac{\sqrt{-1}}{2g^d}\Bigg)^d\sum_{\substack{j_1,\ldots ,j_d \in\lbrace1,\ldots,g \rbrace\\ \sigma,\tau\in S_d}}f_{j_1}(z_{\sigma(1)})\overline{f_{j_1}(z_{\tau(1)})}\cdots f_{j_d}(z_{\sigma(d)})\overline{f_{j_d}(z_{\tau(d)})}\bigwedge_{k=1}^{d}dz_{\sigma(k)} \wedge d\overline{z}_{\tau(k)}\,=$$ $$\Bigg(\frac{\sqrt{-1}}{2g^d}\Bigg)^d \sum_{\substack{j_1,\ldots ,j_d\in\lbrace1,\ldots,g \rbrace\\ \sigma,\tau\in S_d}} \mathrm{sgn}(\sigma)\mathrm{sgn}(\tau)f_{j_1}(z_{\sigma(1)}) \overline{f_{j_1}(z_{\tau(1)})}\cdots f_{j_d}(z_{\sigma(d)})\overline{f_{j_d}(z_{\tau(d)})} \bigwedge_{k=1}^{d}dz_{k}\wedge d\overline{z}_{k}\, .$$ Using the above expression, we observe that $$\Bigg\|\frac{\operatorname{\mu_{{\it{X^d}},vol}^{can}}(z)}{\operatorname{\mu_{{\it{X^d}},vol}^{shyp}}(z)}\Bigg\|^2 =\Bigg(\frac{\operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X)}{g^d}\Bigg)^{2d}$$ $$\times \Bigg\|\bigg(\prod_{k=1}^{d}y_{k}^2\bigg)\cdot\sum_{\substack{j_1,\ldots ,j_d\in\lbrace1,\ldots,g \rbrace\\ \sigma,\tau\in S_d}} \mathrm{sgn}(\sigma)\mathrm{sgn}(\tau)f_{j_1}(z_{\sigma(1)}) \overline{f_{j_1}(z_{\tau(1)})}\cdots f_{j_d}(z_{\sigma(d)}) \overline{f_{j_d}(z_{\tau(d)})} \Bigg\|^{2}\, .$$ Since the number of terms in the above summation are $(d!)^2g^d$, we arrive at the inequality $$\Bigg\|\frac{\operatorname{\mu_{{\it{X^d}},vol}^{can}}(z)}{\operatorname{\mu_{{\it{X^d}},vol}^{shyp}}(z)}\Bigg\|^2\leq (d!)^4\Bigg(\frac{g\operatorname{\mathrm{vol_{\mathrm{hyp}}}}(X)}{g^{d}}\Bigg)^{2d}$$ $$\label{boundvoleqn1} \times \sup_{\substack{j_1,\ldots ,j_d\in\lbrace1,\ldots,g\rbrace\\ \sigma,\tau\in S_d,\,z\in\operatorname{{\it{X^{d}}}}}} \Bigg\|\bigg(\prod_{k=1}^{d}y_{k}^2\bigg)\cdot f_{j_1}(z_{\sigma(1)}) \overline{f_{j_1}(z_{\tau(1)})}\cdots f_{j_d}(z_{\sigma(d)}) \overline{f_{j_d}(z_{\tau(d)})} \Bigg\|^2\, .$$ From Theorem \[boundbk\], we derive $$\sup_{\substack{j_1,\ldots ,j_d\in\lbrace1,\ldots,g\rbrace\\ \sigma,\tau\in S_d,\,z\in\operatorname{{\it{X^{d}}}}}} \Bigg\|\bigg( \prod_{k=1}^{d}y_{k}^2\bigg)\cdot f_{j_1}(z_{\sigma(1)}) \overline{f_{j_1}(z_{\tau(1)})}\cdots f_{j_d}(z_{\sigma(d)})\overline{f_{j_d}(z_{\tau(d)})} \Bigg\|^2$$ $$\label{boundvoleqn2} \leq\, \sup_{z\in\operatorname{{\it{X^{d}}}}}\big(\operatorname{\mathcal{B}_{{\it{X^d}}}}(z)\big)^2\leq \big(B_{X}\big)^{2d}\, .$$ Combining the inequalities and , the proof is completed. Singularities of the canonical metric on the symmetric product ============================================================== As before, take $d\, <\,\eta(X)$. Let $S_d$ denote the group permutation of $\{1\, ,\ldots\, ,d\}$. It acts on $X^d$ by permuting the factors. Let $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$ denote the $d$-fold symmetric product of $X$. In other words, $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$ is the quotient of $X^d$ for the action of $S_d$. The metric $\operatorname{\mu_{{\it{\mathrm{Sym}^d(X)}}}^{can}}$ on $X^d$ is clearly invariant under the action of the group $S_d$. Let us denote the push-forward of the canonical metric $\operatorname{\mu_{{\it{X^d}}}^{can}}$ onto $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$. \[symmsing\] Consider the map $\phi\, :\, X^d\, \longrightarrow\, \mathrm{Pic}^{d}(X)$ in . It factors through the quotient $X^d\, \longrightarrow\, X^d/S_d \,=\, {\rm Sym}^d(X)$. The resulting map $${\rm Sym}^d(X)\, \longrightarrow\, \mathrm{Pic}^{d}(X)$$ is an embedding. If two elements $(x_1, \ldots,x_d)$ and $(y_1, \ldots,y_d)$ of $X^d$ lie in the same orbit for the action of $S_d$ on $X^d$, then the line bundles ${\mathcal O}_X(x_1+\cdots +x_d)$ and ${\mathcal O}_X(y_1+\cdots +y_d)$ are isomorphic. Hence $\phi$ descends to a morphism $$\label{vp} \varphi\,:\, \operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}\, \longrightarrow\, \mathrm{Pic}^{d}(X)\, .$$ From Lemma \[l1\] we know that $\varphi$ is injective. Therefore, it suffices to show that $\varphi$ is an immersion. Take any point $\underline{x}\, =\, \{x_1\, ,\ldots\, ,x_d\}\,\in\, \operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$. The divisor $\sum_{i=1}^d x_i$ will be denoted by $D$. Let $$0\, \longrightarrow\, {\mathcal O}_X(-D)\, \longrightarrow\, {\mathcal O}_X \, \longrightarrow\, Q'(\underline{x})\,:=\, {\mathcal O}_X/{\mathcal O}_X(-D) \, \longrightarrow\, 0$$ be the short exact sequence corresponding to the point $\underline{x}$. Tensoring it with the line bundle ${\mathcal O}_X(-D)^\,=\, {\mathcal O}_X(D)$ we get the short exact sequence $$0\, \longrightarrow\, End({\mathcal O}_X(-D))\,=\,{\mathcal O}_X \, \longrightarrow\, Hom({\mathcal O}_X(-D)\, ,{\mathcal O}_X)\,=\, {\mathcal O}_X(D)$$ $$\longrightarrow\, Q(\underline{x})\,:=\, Hom({\mathcal O}_X(-D)\, ,Q'(\underline{x}))\, \longrightarrow\, 0\, .$$ Let $$\label{e1} 0\, \longrightarrow\, H^0(X,\, {\mathcal O}_X)\,\stackrel{\alpha}{\longrightarrow}\, H^0(X,\, {\mathcal O}_X(D))\,\stackrel{\beta}{\longrightarrow}\, H^0(X,\, Q(\underline{x}))\,\stackrel{\gamma}{\longrightarrow}\, H^1(X,\, {\mathcal O}_X)$$ be the long exact sequence of cohomologies associated to this short exact sequence of sheaves. The holomorphic tangent space to $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$ at $\underline{x}$ is $$T_{\underline{x}}\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}\,=\, H^0(X,\, Q(\underline{x}))\, ,$$ and the tangent bundle of $\text{Pic}^d(X)$ is the trivial vector bundle with fiber $H^1(X,\, {\mathcal O}_X)$. The differential at $\underline{x}$ of the map $\varphi$ in $$(d\varphi)(\underline{x})\, :\, T_{\underline{x}}\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}\,=\, H^0(X,\, Q(\underline{x}))\,\longrightarrow\, T_{\varphi(\underline{x})}\text{Pic}^d(X)\,=\, H^1(X,\, {\mathcal O}_X)$$ satisfies the identity $$\label{e2} (d\varphi)(\underline{x})\,=\, \gamma\, ,$$ where $\gamma$ is the homomorphism in . Now, $H^0(X,\, {\mathcal O}_X)\,=\, \mathbb C$. In the proof of Lemma \[l1\] we saw that $$H^0(X,\, {\mathcal O}_X(D))\,=\, {\mathbb C}\, .$$ Hence the homomorphism $\alpha$ in is an isomorphism. Consequently, $\beta$ in the exact sequence is the zero homomorphism and $\gamma$ in is injective. Since $\gamma$ in is injective, from we conclude that $\varphi$ is an immersion. \[re1\] Since $\varphi$ is an embedding, the metric $\operatorname{\mu_{{\it{\mathrm{Sym}^d(X)}}}^{can}}$ on $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$ is nonsingular. Therefore, the metric $\operatorname{\mu_{{\it{X^d}}}^{can}}$ on $X^d$ is singular exactly on the divisor where the quotient map $X^d\, \longrightarrow\, \operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$ is ramified. We note that this ramification divisor consists of all points of $X^d$ such that the $d$ points of $X$ are not distinct. Automorphisms of $\text{Sym}^d(X)$ ================================== Consider the nonsingular Kähler metric $\operatorname{\mu_{{\it{\mathrm{Sym}^d(X)}}}^{can}}$ on $\operatorname{{\it{{\mathrm{Sym}^{d}(X)}}}}$ (see Remark \[re1\]). \[th3\] Let $T\, :\, {\rm Sym}^d(X)\, \longrightarrow\, {\rm Sym}^d(X)$ be any holomorphic automorphism. Then the pulled back Kähler form $T^\operatorname{\mu_{{\it{\mathrm{Sym}^d(X)}}}^{can}}$ coincides with $\operatorname{\mu_{{\it{\mathrm{Sym}^d(X)}}}^{can}}$. In particular, $T$ is a isometry for the metric $\operatorname{\mu_{{\it{\mathrm{Sym}^d(X)}}}^{can}}$. Since $\varphi$ (constructed in ) is the Albanese map for $\text{Sym}^d(X)$, there is a holomorphic automorphism $$\widehat{T}\, :\, \text{Pic}^d(X)\, \longrightarrow\,\text{Pic}^d(X)$$ such that $$\label{com} \varphi\circ T\,=\, \widehat{T}\circ\varphi\, .$$ From [@Fa] we know that $\widehat{T}$ preserves the polarization on $\text{Pic}^d(X)$. A theorem due to Weil says a holomorphic automorphism of $\mathrm{Jac}(X)\,=\, \text{Pic}^0(X)$ that preserves the polarization is generated by the following: - translations of $\text{Pic}^0(X)$, - automorphisms of $\text{Pic}^0(X)$ given by the holomorphic automorphisms of $X$, and - the inversion of $\text{Pic}^0(X)$ defined by $L\, \longmapsto\, L^$. (See [@We Hauptsatz, p. 35].) But all these three types of automorphisms of $\text{Pic}^0(X)$ are isometries for the flat Kähler form on $\text{Pic}^0(X)$ constructed in . From this it follows immediately that $\widehat{T}$ is an isometry for the flat Kähler form $g_d$ on $\text{Pic}^d(X)$ constructed in Section \[se3.1\]. Since $\widehat{T}$ is an isometry, from it follows immediately that $T^\omega_d\,=\, \omega_d$. Acknowledgements {#acknowledgements .unnumbered} ================ We thank the referee for pointing out a reference. The second-named author wishes to thank the University of Hyderabad for hospitality while the work was carried out. He is supported by a J. C. Bose Fellowship. [999]{} J. M. Baptista, On the $L^2$-metric of vortex moduli spaces, [Nuclear Phys. B]{} [844]{} (2011), 308–333. B.-Y. Chen and S. Fu, Stability of the Bergman kernel on a tower of coverings, Jour. Diff. Geom.* (to appear), arXiv:1202.4371. S. B. Bradlow, Vortices in holomorphic line bundles over closed Kähler manifolds, Commun. Math. Phys. 135 (1990) 1–17. S. B. Bradlow, Special metrics and stability for holomorphic bundles with global sections, Jour. Diff. Geom. 33 (1991), 169–213. N. Fakhruddin, Torelli’s theorem for high degree symmetric products of curves, arxiv:0208180v1. J. S. Friedman, J. Jorgenson, and J. Kramer, Uniform sup-norm bounds on average for cusp forms of higher weights, arXiv:1305.1348. O. García-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1994) 88–96. J. Jorgenson and J. Kramer, Bounds on canonical Green’s functions, [Compositio Math.]{} [142]{} (2006), 679–700. J. Jorgenson and R. Lundelius, Convergence of the heat kernel and the resolvent kernel on degenerating hyperbolic Riemann surfaces of finite volume, [Questiones Math.]{} [18]{} (1995), 345–363. I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319–343. N. S. Manton and S. M. Nasir, Volume of vortex moduli spaces, [Comm. Math. Phys.]{} [199]{} (1999), 591–604. N. S. Manton and N. Romão, Vortices and Jacobian varieties, [Jour. Geom. Phys.]{} [61]{} (2011), 1135–1155. S. M. Nasir, Vortices and flat connections, [Phys. Lett.]{} [419]{}, (1998), 253–257. T. Perutz, Symplectic fibrations and the abelian vortex equations, [Comm. Math. Phys.]{} [278]{} (2008), 289–306. N. A. Rink, Vortices and the Abel-Jacobi map, [Jour. Geom. Phys.]{} [76]{} (2014), 242–255. A. Weil, Zum beweis des Torelli satzes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II [2]{} (1957) 33–53.
--- author: - Luca Visinelli title: Cosmological perturbations for an inflaton field coupled to radiation --- Introduction ============ The mechanism of inflation [@kazanas; @starobinsky; @guth; @sato; @albrecht; @linde] addresses the problems of the flatness, homogeneity, and the lack of relic monopoles raised by the standard Big-Bang cosmology [@linde_book; @kolb_book; @weinberg_book], and provides an explanation to the inhomogeneities observed in the Cosmic Microwave Background Radiation (CMBR) [@mukhanov; @guth_pi; @hawking; @starobinsky1; @bardeen1]. It is usually postulated that during the inflationary period, the expansion of the Universe is driven by the potential energy of the so-called “inflaton” field. When the kinetic energy of the inflaton is no longer negligible with respect to its potential energy, the inflaton decays into lighter particles, which might be both hypothetical and Standard Model particles, and the Universe reheats and transits to a radiation-dominated epoch [@linde_hybrid; @kofman; @kofman1; @boyanovsky; @greene; @braden]. An alternative model postulates that the decay of the inflaton field occurs during the whole inflationary period, as in Warm Inflation (WI) scenarios ([@berera_fang; @berera; @berera_review; @bastero_review; @berera_spectral_index; @taylor_berera; @oliveira; @oliveira1; @delcampo; @hall; @bastero_gil; @moss_xiong; @moss; @graham; @delcampo2010; @bastero1; @moss1; @setare2013; @bastero2011; @bartrum], see also Refs. [@fang1980; @hosoya; @moss_ewi; @lonsdale; @yokoyama; @liddle_ewi]). Other models include multi-inflaton fields [@garcia_bellido; @randall; @tsujikawa; @parkinson; @kadota; @wands2007; @bassett], natural inflation [@natural_inflation; @natural_inflation1; @natural_inflation2; @mohanty; @visinelli_NWI], and chain inflation [@freese2005; @freese2005_1]; alternative mechanisms include the curvaton [@enqvist2002; @lyth2002; @moroi2002; @moroi2002_1]. The only constraint on the reheating temperature of the Universe ${T_{\rm RH}}\gtrsim 4{\rm ~MeV}$, valid in both the standard and WI models, comes from the requirement that a sufficiently large population of thermal neutrinos has to be produced [@kawasaki; @kawasaki1; @hannestad; @ichikawa], since a modification of the neutrino number density would spoil the observed abundance of light elements, the observed CMBR and matter power spectra [@ichikawa1; @bernardis], and the large-scale structure. As previously mentioned, inhomogeneities might have generated during inflation [@bardeen1980; @kodama; @mukhanov1992], either as quantum fluctuations of the inflaton (standard inflation scenarios [@mukhanov; @guth_pi; @hawking; @starobinsky1; @bardeen1]), or as thermal fluctuations (WI scenarios [@berera_fang; @berera]). These cosmological fluctuations later evolve into perturbations of the matter and radiation power spectra that are observed in the CMBR. The evolution of the fields participating inflation is described by a set of coupled Boltzmann equations, whose first-order perturbation describes the evolution of inhomogeneities. To some extent, these fields can be treated as fluids, with a pressure related to the energy density by an equation of state. The formalism for interacting fluids in an expanding Friedmann-Robertson-Walker (FRW) metric has been extensively discussed in Refs. [@kodama; @mukhanov1992; @malik; @malik1]. In this work, we present the equations describing the coupling of the inflaton to a relativistic field using the interacting fluids approach developed in Refs. [@malik; @malik1]. The equations for the bulk and the cosmological fluctuations are solved numerically and tested against the analytical approximation around the reheating period. The paper is organized as follows. In Sec. \[Equations for the background fields\], we provide a numerical solution to the basic equations for the background fields, while in Sec. \[Evolution of perturbations during inflation\] we study the time evolution of the perturbations in the inflaton and radiation fields. Sec. \[Stochastic perturbations\] is devoted to the modifications induced in the equation by an additional stochastic term, which is usually considered in WI scenarios. Results are summarized in Sec. \[Discussion and conclusions\]. Equations for the background fields {#Equations for the background fields} =================================== We consider a model in which the energy density of the inflaton field $\rho_\phi$ dominates the energy density of the Universe, decaying into relativistic matter with energy density $\rho_r$ at a rate $\Gamma$. We assume a FRW metric with cosmic time $t$ and scale factor $a = a(t)$, $$\label{line_element_0} ds^2 = -dt^2 + a^2(t)\,\delta_{ij}\,dx^i\,dx^j.$$ If radiation thermalizes on a time scale much shorter than $1/\Gamma$, the potential $U(\phi, T)$ [@bellini; @matsuda], that expresses the interaction between the inflaton and radiation, splits as $U(\phi,T) = U(\phi) + \rho_r$ [@berera; @moss]. Denoting the zero-th order (background) term in the perturbation with a bar over a quantity, we write the background expressions for the energy density and pressure fields as $$\begin{array}{l} \displaystyle \bar{\rho}_\phi = \frac{1}{2}\,\dot{\bar{\phi}}^2 + U(\bar{\phi}),\quad\quad \bar{p}_\phi = \frac{1}{2}\,\dot{\bar{\phi}}^2 - U(\bar{\phi}),\\ \displaystyle \bar{\rho}_r = \frac{\pi^2}{30}\,g_(T)\,T^4, \quad \quad \bar{p}_r = c_r^2\,\bar{\rho}_r, \end{array} \label{energy_pressure}$$ where $c_r^2 = 1/3$ describes the equation of state for relativistic matter. Under these conditions, Friedmann equations read [@weinberg_book] $$\begin{array}{l} \displaystyle H^2 = \frac{8\pi\,G}{3}\,\bar{\rho} = \frac{8\pi\,G}{3}\left(\frac{1}{2}\dot{\bar{\phi}}^2+U+\bar{\rho}_r\right),\\ \displaystyle\\ \displaystyle \dot{H} = -4\pi\,G\,\left(\bar{p} + \bar{\rho}\right) = - 4\pi\,G\,\left(\dot{\bar{\phi}}^2 + \frac{4}{3}\,\bar{\rho}_r\right), \end{array} \label{friedmann}$$ where $G$ is Newton’s gravitational constant, $H = \dot{a}/a$ is the Hubble rate, and where a dot indicates a derivation with respect to the cosmic time $t$. For each species $\alpha$, the continuity equation is [@kodama] $$\label{cons_energy_alpha} \begin{array}{l} \displaystyle \dot{\bar{\rho}}_\alpha + 3H\,\left(\bar{p}_\alpha + \bar{\rho}_\alpha\right) = \bar{Q}_\alpha,\\ \end{array}$$ where $\bar{Q}_\alpha$ is a source term describing the conversion between the species $\alpha$ and the other species accounted in the theory. To assure the conservation of total energy density in a co-moving volume, the sum of all source terms must satisfy $$\sum_\alpha\,\bar{Q}_\alpha = 0.$$ The conversion of the inflaton energy density into radiation energy density is thus described by the set of Boltzmann equations $$\label{cons_energy} \begin{array}{l} \displaystyle \dot{\bar{\rho}}_\phi + 3H\,\left(\bar{p}_\phi + \bar{\rho}_\phi\right) = -\bar{Q},\\ \displaystyle \dot{\bar{\rho}}_r + 3H\,\left(\bar{p}_r + \bar{\rho}_r\right) = \bar{Q}. \end{array}$$ At early times $t \ll 1/\Gamma$, the source term in the Boltzmann equation can be neglected, and the inflaton energy density is described by $$\label{no_source_phi} \frac{d\bar{\rho}_\phi}{da} + \frac{3}{a}\,\left(\bar{p}_\phi + \bar{\rho}_\phi\right) = 0.$$ For a pressure-less fluid $\bar{p}_\phi =0$, Eq. (\[no\_source\_phi\]) predicts $\bar{\rho}_\phi \propto a^{-3}$ [@erickcek]. In this paper instead, we use the expression $\bar{p}_\phi + \bar{\rho}_\phi = 2\epsilon\,\bar{\rho}_\phi/3$, where $\epsilon$ is the first slow-roll parameter defined in Eq. (\[slow\_roll\_parameters\]) below. With this expression, the energy density is constant during during $\phi$ domination as $\bar{\rho}_\phi \propto a^{-2\epsilon}$. Inserting the expressions for $\bar{\rho}_\phi$ and $\bar{p}_\phi$ given in Eq. (\[energy\_pressure\]), together with the source term $$\label{source_term} \bar{Q} = \Gamma\,\left(\bar{p}_\phi + \bar{\rho}_\phi\right),$$ into Eq. (\[cons\_energy\]), we obtain the evolution of the inflaton and radiation fields in a FRW metric, $$\label{eq_motion} \begin{array}{l} \displaystyle \ddot{\bar{\phi}} + \left(3H + \Gamma\right)\,\dot{\bar{\phi}} + U_\phi = 0,\\ \displaystyle \dot{\bar{\rho}}_r + 4H\,\bar{\rho}_r = \Gamma\,\dot{\bar{\phi}}^2, \end{array}$$ where $U_\phi = \partial U/\partial\bar{\phi}$. The second line in Eq. (\[eq\_motion\]) corresponds to the Boltzmann equation for the massive scalar field $\phi$ decaying into a massless field $\chi$ via an interaction proportional to $\phi\,\chi^2$ [@braden]. Here, we do not account for a microscopical model for the radiation field, which is treated as a fluid. The set in Eq. (\[eq\_motion\]) has been often presented various time in the literature. Examples include warm inflation [@oliveira; @oliveira1; @hall; @delcampo; @graham; @visinelli_NWI], the decay of the inflaton [@allahverdi; @allahverdi1] or the curvaton field [@gupta; @matarrese], or the early domination of a massive scalar field [@chung; @erickcek]. We solve the set of Eq. (\[eq\_motion\]) numerically, assuming that the inflaton potential be [@lyth] $$\label{potential} U(\phi) = \frac{\lambda}{4}\,\left(\phi^2 - \phi_0^2\right)^2,$$ where $\lambda$ is a dimensionless parameter describing the strength of the potential, and $\phi_0$ is the value of the inflaton field at which the potential reaches its minimum value. Initial conditions for $\dot{\phi}$ and $\rho_r$ are obtained by first solving the set of Eq. (\[eq\_motion\]) in the regime $\ddot{\phi} \ll H\,\dot{\phi}$ and $\dot{\rho}_r \ll H\,\rho_r$, starting from an initial configuration of the inflaton $\phi = \phi_i$. In Fig. \[figure:background\], we show the energy densities $\bar{\rho}_\phi$ (red solid line) and $\bar{\rho}_r$ (blue solid line) as a function of time $t$, for the values of the parameters given in Table \[table\_parameters\]. Also shown is the Hubble rate $H$, multiplied by the constant $\phi_0^2\,\Gamma$ in order to obtain the units of the energy density. [l]{} [Quantity]{}\ \ $\lambda=10^{-8}$\ $\Gamma = 10^{13}{\rm ~GeV}$\ $\phi_0 = 10^{17}{\rm ~GeV}$\ $\phi_i = 0.01\,\phi_0$\ \ ![The energy density for the inflaton field $\bar{\rho}_\phi$ (red line), the radiation field $\bar{\rho}_r$ (blue line), and the Hubble rate $\phi_0^2\,\Gamma H$, as a function of time $t$.[]{data-label="figure:background"}](PlotBackground.eps){width="15cm"} Defining the reheating time ${t_{\rm RH}}$ as the time at which $$\bar{\rho}_\phi({t_{\rm RH}}) = \bar{\rho}_r({t_{\rm RH}}),$$ the total energy density is dominated by the inflaton energy density for $t < {t_{\rm RH}}$ , with a constant value $\bar{\rho}_\phi \approx U(\phi_i)$. When $t \approx {t_{\rm RH}}$, the inflaton field starts oscillating around its equilibrium value $\phi_0$, so we write the expansion $\phi = \phi_0 + \varphi$, where $\varphi$ is a small perturbation. In this approximation, the inflaton potential is $U(\phi_0+\varphi) \approx \lambda\,\phi_0^2\,\varphi^2$, and Eq. (\[eq\_motion\]) reads $$\label{eq_motion_tRH} \begin{array}{l} \displaystyle \ddot{\varphi} + \left[\sqrt{24\pi\,G}\,\left(\lambda\,\phi_0^2\,\varphi^2 + \frac{\dot{\varphi}^2}{2} + \bar{\rho}_r\right)^{1/2} + \Gamma\right]\,\dot{\varphi} + 2\lambda\,\phi_0^2\,\varphi = 0,\\ \displaystyle \dot{\bar{\rho}}_r + 4\,\sqrt{\frac{8\pi\,G}{3}}\,\left(\lambda\,\phi_0^2\,\varphi^2 + \frac{\dot{\varphi}^2}{2} + \bar{\rho}_r\right)^{1/2}\,\bar{\rho}_r -\Gamma\,\dot{\varphi}^2 = 0. \end{array}$$ Keeping only the leading terms in $\varphi$ and $\dot{\varphi}$, we obtain $$\label{eq_motion_tRH1} \begin{array}{l} \displaystyle \ddot{\varphi} + \Gamma\,\dot{\varphi} + 2\lambda\,\phi_0^2\,\varphi = 0,\\ \displaystyle \dot{\bar{\rho}}_r + 4\,\sqrt{\frac{8\pi\,G}{3}}\,\bar{\rho}_r^{3/2} = 0, \end{array}$$ with solution $$\label{eq_motion_tRHsol} \begin{array}{l} \displaystyle \varphi(t) \propto e^{-\Gamma\,t/2}\,\cos\left(\frac{t}{2}\,\sqrt{\Gamma^2 - 8\lambda\,\phi_0^2} \right),\\ \displaystyle \bar{\rho}_r(t) \propto t^{-2}. \end{array}$$ The inflaton field approaches its equilibrium value $\phi_0$ with an exponential decay; for the sake of illustration, in Fig. \[figure:phi\] we show a detail of the value of the inflaton field around $t = {t_{\rm RH}}$. The dependence of the energy density of the relativistic field $\bar{\rho}_r(t) \propto t^{-2}$ is a standard result, which can also be inferred from the decoupled conservation equation, $$\label{eq_motion_rad} \displaystyle \dot{\bar{\rho}}_r + 4H\,\bar{\rho}_r \approx 0,$$ and the fact that $a(t) \propto t^{1/2}$ during the radiation epoch. From this result, it follows that the Hubble rate (the black dashed line in Fig. \[figure:background\]) is $H(t) = 1/2t$ when $t > {t_{\rm RH}}$. ![The behavior of the inflaton field $\phi(t)$ around the time ${t_{\rm RH}}$ at which the inflaton converts into relativistic degrees of freedom, as a function of time $t$.[]{data-label="figure:phi"}](PlotPhi.eps){width="15cm"} Evolution of perturbations during inflation {#Evolution of perturbations during inflation} =========================================== We consider the perturbations of the FRW background metric in Eq. (\[line\_element\_0\]), described by the metric in the longitudinal gauge [@bardeen1980; @mukhanov1992] $$\label{line_element} ds^2 = -\left(1 + 2\Psi\right)\,dt^2 + a^2(t)\,\delta_{ij}\,\left(1 - 2\Psi \right)\,dx^i\,dx^j,$$ where $\Psi$ parametrizes the perturbations in the metric. Eq. (\[line\_element\]) is valid whenever perturbations of the total energy-momentum tensor do not give rise to anisotropic stress. Following Eqs. (2.15) and (2.17) in Ref. [@malik] with our notation and in the case in which the shear is zero, the expression for the perturbations in the energy density and pressure fields of a generic species $\alpha$ are $$\delta \dot{\rho}_\alpha + 3H\,\left(\delta \rho_\alpha + \delta p_\alpha\right) - 3\left(\bar{\rho}_\alpha + \bar{p}_\alpha\right)\,\dot{\Psi} + a^{-2}\,\nabla^2\,\left(\bar{\rho}_\alpha + \bar{p}_\alpha\right)\,\delta u_\alpha - \bar{Q}_\alpha\,\Psi - \delta Q_\alpha = 0,\\ \label{perturbation_energy_malik}$$ and $$\delta p_\alpha + a^{-3}\,\left[a^3\,\left(\bar{\rho}_\alpha + \bar{p}_\alpha\right)\,\delta u_\alpha\right]^{\centerdot} + \left(\bar{\rho}_\alpha + \bar{p}_\alpha\right)\,\Psi = \bar{Q}_\alpha\,\delta u. \label{perturbation_pressure_malik}$$ Here, a dot indicates a derivation with respect to $t$, $\delta \rho_\alpha$ and $\delta p_\alpha$ are respectively the first-order perturbations in the energy density and pressure fields of the $\alpha$-fluid, $c^2_\alpha = \dot{p}_\alpha/\dot{\rho}_\alpha$, $\delta u$ is the total covariant velocity perturbation, and $\nabla^2$ is the co-moving spatial Laplacian. Eq. can be alternatively reformulated as $$\delta \dot{u}_\alpha + \left[\frac{\bar{Q}_\alpha}{\bar{\rho}_\alpha + \bar{p}_\alpha}\,\left(1 + c^2_\alpha\right)-3\,c^2_\alpha\,H\right]\,\delta u_\alpha + \Psi + \frac{1}{\bar{\rho}_\alpha + \bar{p}_\alpha}\,\left[\delta p_\alpha -\bar{Q}_\alpha\,\delta u\,\right] = 0. \label{perturbation_pressure_malik_old}$$ Defining the Fourier transform of a generic quantity $F({\bf x}, t)$ as $$\begin{array}{l} \displaystyle F_q \equiv F({\bf q}, t) = \frac{1}{(2\pi)^3}\,\int\,d^3x\,e^{-i\,{\bf q}\cdot{\bf x}}\,F({\bf x}, t), \end{array}$$ we take the Fourier transform of the equations for density and pressure perturbations, and set $\nabla^2 \to -q^2$. In momentum space, Eq. (\[perturbation\_energy\_malik\]) for the inflaton and the radiation fields reads $$\begin{split} \begin{array}{l} \displaystyle \delta \dot{\rho}_{\phi q} + 3H\,(\delta \rho_{\phi q} + \delta p_{\phi q}) + \left(\bar{\rho}_\phi+\bar{p}_\phi\right)\,\left[\Gamma\,\Psi_q - 3\dot{\Psi}_q - \frac{q^2}{a^2}\delta u_{\phi q}\right] = -\delta Q,\\ \displaystyle \delta \dot{\rho}_{r q} +3H({\delta \rho_{rq}}+ {\delta p_{rq}}) - \bar{\rho}_r\,\left[\frac{4}{3}\frac{q^2}{a^2}\,{\delta u_{rq}}+ 4\dot{\Psi}_q\right] - \Gamma\,\left(\bar{p}_\phi + \bar{\rho}_\phi\right)\, \Psi_q = \delta Q. \end{array} \label{perturbation_energy1} \end{split}$$ The first-order perturbation of the source terms in Eq. (\[source\_term\]) is $$\label{cold_perturbations} \delta Q = \Gamma\,\left(\delta p_\phi + \delta \rho_\phi\right),$$ while the term $\bar{Q}\,\Psi_q$ containing the perturbations in the gravitational field has already been included in Eq. (\[perturbation\_energy1\]) which, once the term $\delta Q$ in Eq. (\[cold\_perturbations\]) has been substituted, reads $$\begin{split} \begin{array}{l} \displaystyle \delta \dot{\rho}_{\phi q} + \left(3H+\Gamma\right)\,\left(\delta \rho_{\phi q} + \delta p_{\phi q}\right) + \left(\bar{\rho}_\phi+\bar{p}_\phi\right)\,\left[\Gamma\,\Psi_q - 3\dot{\Psi}_q - \frac{q^2}{a^2}\delta u_{\phi q}\right] = 0,\\ \displaystyle \delta \dot{\rho}_{r q} +3H\,({\delta \rho_{rq}}+{\delta p_{rq}}) - \bar{\rho}_r\,\left[\frac{4}{3}\frac{q^2}{a^2}\,{\delta u_{rq}}+ 4\dot{\Psi}_q\right] - \Gamma\,\left(\bar{p}_\phi + \bar{\rho}_\phi\right)\, \Psi_q = \Gamma\,\left(\delta \rho_{\phi q} + \delta p_{\phi q}\right). \end{array} \label{perturbation_energy2} \end{split}$$ For velocity perturbations, we rewrite Eq. (\[perturbation\_pressure\_malik\_old\]) for the inflaton and radiation fields as $$\label{perturbation_pressure1} \begin{array}{l} \displaystyle \delta \dot{u}_{\phi q} - \left(3H + \Gamma\right)\,c^2_\phi\,\delta u_{\phi q} + \Psi_q + \Gamma\,\left(\delta u_q - \delta u_{\phi q}\right) + \frac{\delta p_{\phi q}}{\bar{\rho}_\phi + \bar{p}_\phi} = 0,\\ \displaystyle\\ \displaystyle \frac{4}{3}\,\bar{\rho}_r\,\left(\delta \dot{u}_{r q} - H\,\delta u_{r q} + \Psi_q \right) + \Gamma\,\left(\bar{\rho}_\phi + \bar{p}_\phi\right)\,\left(\frac{4}{3}\,\delta u_{r q} - \delta u_q\right) + \delta p_{r q} = 0.\\ \end{array}$$ Perturbations in the gravitational field, described by the field $\Psi$ appearing in the metric in Eq. , are described by [@malik] $$\dot{\Psi}_q + H\,\Psi_q + 4\pi\,G \,\left[\frac{4}{3}\,\bar{\rho}_r\,\delta u_{r q} + \left(\bar{\rho}_\phi + \bar{p}_\phi\right)\, \delta u_{\phi q}\right] = 0. \label{perturbation_field1}$$ Following Weinberg [@weinberg_book], the perturbations in the energy density, pressure, and velocity of the inflation field are related to the perturbations in the scalar field $\delta \phi$ by $$\begin{array}{l} \displaystyle \delta \rho_{\phi q} = \dot{\bar{\phi}}\,\delta\dot{\phi}_q + U_\phi\,\delta\phi_q - \dot{\bar{\phi}}^2\,\Psi_q,\\ \displaystyle \delta p_{\phi q} = \dot{\bar{\phi}}\,\delta\dot{\phi}_q - U_\phi\,\delta\phi_q - \dot{\bar{\phi}}^2\,\Psi_q,\\ \displaystyle \delta u_{\phi q} = -\frac{\delta \phi_q}{\dot{\bar{\phi}}}. \end{array} \label{field_perturbations}$$ In addition, we assume that perturbations in the radiation fluid are adiabatic, setting ${\delta p_{rq}}= c_r^2\,{\delta \rho_{rq}}= {\delta \rho_{rq}}/3$. Combining the set in Eq. (\[field\_perturbations\]) with the expressions for the energy density and pressure fields in Eqs. (\[perturbation\_energy2\]), (\[perturbation\_pressure1\]) and (\[perturbation\_field1\]), we obtain $$\label{perturbation_eq_phi} \delta \ddot{\phi}_q + \left(3H +\Gamma\right)\,{\delta \dot{\phi}_q}+ \left(U_{\phi\phi} + \frac{q^2}{a^2}\right)\,{\delta \phi_q}= 4\,\dot{\bar{\phi}}\,\dot{\Psi}_q - \left(2\,U_\phi + \Gamma\,\dot{\bar{\phi}}\right)\,\Psi_q,$$ $$\label{perturbation_eq_r} \delta \dot{\rho}_{r q} +4H\,{\delta \rho_{rq}}- \frac{4}{3}\frac{q^2}{a^2}\,\bar{\rho}_r\,\delta u_{r q} = 4\bar{\rho}_r\,\dot{\Psi}_q + \Gamma\,\dot{\bar{\phi}}\,\left(2\delta\dot{\phi}_q - \dot{\bar{\phi}}\,\Psi_q\right),$$ $$\label{perturbation_eq_u} \frac{4}{3}\,\bar{\rho}_r\,\left(\delta \dot{u}_{r q} - H\,\delta u_{r q} + \Psi_q \right) + \Gamma\,\dot{\bar{\phi}}\,\left(\frac{4}{3}\,\dot{\bar{\phi}}\,\delta u_{r q} + \delta \phi_q\right) + \frac{1}{3}\,\delta \rho_{r q} = 0,$$ $$\label{perturbation_eq_psi} \dot{\Psi}_q + H\,\Psi_q + 4\pi\,G \,\left(\frac{4}{3} \bar{\rho}_r\,\delta u_{r q} - \dot{\bar{\phi}}\,\delta \phi_q\right) = 0.$$ In the literature, velocity perturbations in the radiation fluid have often been expressed in terms of the scalar potential velocity $v_r$, related to the covariant velocity perturbation used here by $\delta u_{r q} = a\,v_r/q$. When substituting for $v_r$, Eq. (\[perturbation\_eq\_u\]) reads $$\label{scalar_potential_r} \dot{v}_r + \frac{\Gamma\,\dot{\bar{\phi}}^2}{\bar{\rho}_r}\,v_r +\frac{q}{a}\left(\Psi_q + \frac{3\Gamma\,\dot{\bar{\phi}}}{4\bar{\rho}_r}\,\delta \phi_q + \frac{\delta \rho_{r q}}{4\bar{\rho}_r}\right) = 0.$$ In their Eq. (A20), Moss and Xiong [@moss_xiong] quote the same expression as our Eq. (\[scalar\_potential\_r\]), once set their $\alpha = \Psi_q$ and their $J = -\Gamma\,\dot{\bar{\phi}}\,\delta \phi_q$ as in their Eq. (A13). Oliveira and Joras [@oliveira] quote the same as our Eq. (\[scalar\_potential\_r\]) in the third line of their Eq. (A2); the second line of their Eq. (A2) agrees with our Eq. (\[perturbation\_eq\_phi\]) once set $\Gamma'=0$, while their first line in Eq. (A2) shows a term with an extra factor of three with respect to our Eq. (\[perturbation\_eq\_r\]). Our Eq. (\[perturbation\_eq\_phi\]) shows all terms on the RHS of the equation with the opposite sign with respect to Eq. (61) quoted in Hall [et al.]{} [@hall]. To obtain the initial conditions for $\delta \phi_q$ and $\Psi_q$, we perform a WKB expansion using the fact that, at early times, the term $q/a(t)$ is much larger than $H$, and radiation can be neglected. Introducing the conformal time $d\tau = dt/a(t)$, we write $$\displaystyle \delta \phi_q = f\,\exp\left(-i\,q\,\tau\right),\quad\hbox{and}\quad \Psi_q = g\,\exp\left(-i\,q\,\tau\right),$$ where $f(t)$ and $g(t)$ are slowly-varying functions. Substituting these expressions in Eqs. (\[perturbation\_eq\_phi\]) and (\[perturbation\_eq\_psi\]), and keeping the leading terms in $q/a$ only, we obtain $$\label{perturbation_eq1} \displaystyle f \sim a^{-3/2}, \quad \hbox{and} \quad g = \frac{4\pi\,i\,G\,\dot{\bar{\phi}}\,a}{q}\,f.$$ Initial conditions for $\delta \rho_{r q}$ and $\delta u_{r q}$ are obtained by solving Eqs. and for these variables, neglecting their time dependence and given the initial conditions for $\delta \phi_q$ and $\Psi_q$. We solve the set of Eqs. numerically using these initial conditions and the numerical solution to the background Eq. (\[eq\_motion\]). The scale factor $a(t)$ that appears in the set of equations is obtained as the numerical solution of the differential equation $\dot{a} = H\,a$, where the Hubble rate $H$ is defined by the Friedmann Eq. (\[friedmann\]). In Fig. \[figure:pertQ\], we show results for the numerical resolution of this set of equations with the values for the parameters as in Table \[table\_parameters\], for the values $q = 0.01$ (a), $q = 0.1$ (b), $q = 1$ (c). ![Perturbation in the energy density of the (red solid line) and radiation (blue solid line) and gravitational perturbations (black dashed line) in order of $\Gamma^2\phi_0^2$ obtained from the numerical resolution of the set of Eqs. , , , , with the parameters in Table \[table\_parameters\] and for $q = 0.01$ (top figure), $q=0.1$(middle figure), and $q=1$ (bottom figure).[]{data-label="figure:pertQ"}](PlotPert.eps){width="12cm"} After the reheating stage, the radiation perturbations dominate over the inflaton perturbations. The first-order perturbations in the radiation field oscillate with a frequency that increases with $q$. To explain this oscillatory behavior, we set $\Gamma = 0$, and we introduce the variable $\delta_{r q} = \delta \rho_{r q}/\bar{\rho}_r$, so that Eqs. , , and , when $t > {t_{\rm RH}}$ are simplified as $$\label{perturbation_eq_1} \begin{array}{l} \displaystyle \dot{\delta}_{r q} - \frac{4}{3}\frac{q}{a}\,v_{r q} - 4\dot{\Psi}_q = 0,\\ \displaystyle \dot{v}_r+\frac{q}{a}\left(\Psi_q + \frac{1}{4}\,\delta_{r q}\right) = 0,\\ \displaystyle \dot{\Psi}_q + H\,\Psi_q + 2H^2\,\frac{a}{q}\,v_{r q} = 0. \end{array}$$ Neglecting the perturbations in the gravitational potential $\Psi_q$, we find that both $\delta_{rq}$ and $v$ follow the same differential equation, $$\label{perturbation_eq_X} \displaystyle \ddot{x} + H\,\dot{x} + \frac{q^2}{3a^2}\,x = 0,$$ where $x(t)$ expresses either $\delta_{rq}$ or $v$. For a radiation-dominated cosmology, setting $$\label{perturbation_eq_solutionX} H = \frac{1}{2t}, \quad a = {a_{\rm RH}}\,\sqrt{\frac{t}{{t_{\rm RH}}}},\quad Q^2 = \frac{q^2}{3\,{a_{\rm RH}}^2},$$ this differential equation has solution $$x(t) = x_1\,\sin \left(2 Q\sqrt{\frac{t}{{t_{\rm RH}}}}\right) + x_2\,\cos \left(2 Q\sqrt{\frac{t}{{t_{\rm RH}}}}\right),$$ where $x_1$ and $x_2$ are arbitrary constants. In Fig. \[figure:pertvR\], we compare the solution to the expression in Eq. in a radiation-dominated Universe (black dashed line) for the case of the density perturbations (top) and velocity perturbations (bottom), with the full numerical solution obtained from Eqs. (blue solid line) and (green solid line), setting $q = 0.1$. Once the matching initial conditions have been chosen, the analytical solution captures the oscillatory behavior and the trend obtained with the numerical resolution. ![Top: Density perturbations in the radiation fluid $\delta \rho_{r q}$ obtained with the numerical resolution of Eq. (blue solid line) and with the analytical solution of Eq. with $x(t) = \delta_{rq}$ (black dashed line). Bottom: Velocity perturbations in the radiation fluid $\delta u_{r q}$ obtained with the numerical resolution of Eq. (green solid line) and with the analytical solution of Eq. with $x(t) = v$ (black dashed line).[]{data-label="figure:pertvR"}](FigureAnalytical.eps){width="12cm"} Effects of a viscosity term {#dissipative_terms} =========================== In the non-relativistic theory, the Navier-Stokes equation describes the evolution of the velocity field ${{\bf v}}$ for a fluid of pressure $p$ and density $\rho$. In formulas [@padmanabhan_book], $$\label{navier_stokes} \frac{d{{\bf v}}}{dt} = -\frac{1}{\rho}\,{{\bf \nabla}}\,p - {{\bf \nabla}}\cdot {{\bf \Pi}},$$ where the stress tensor ${{\bf \Pi}}$ introduces a dissipative behavior in the fluid motion, and satisfies $$\label{dissipative} {{\bf \nabla}}\cdot{{\bf \Pi}}= -\eta_s\,{{\bf \nabla}}^2\,{{\bf v}}- \left(\eta_b + \frac{\eta_s}{3}\right)\,{{\bf \nabla}}\,\left({{\bf \nabla}}\cdot{{\bf v}}\right).$$ In the last expression, the divergence of ${{\bf v}}$ can be written in terms of the identity $${{\bf \nabla}}\,\left({{\bf \nabla}}\cdot{{\bf v}}\right) = {{\bf \nabla}}^2\,{{\bf v}}+ {{\bf \nabla}}\times{{\bf \nabla}}\times{{\bf v}}.$$ Assuming that the vorticity of the fluid is zero, ${{\bf \nabla}}\times{{\bf v}}= 0$, which is a valid assumption for first-order cosmological perturbations [@kodama], allows us to write the velocity field as the gradient of a scalar field as ${{\bf v}}= {{\bf \nabla}}\,u$, where the scalar velocity potential $u$ is the non-relativistic analogue of the velocity potential $\delta u_\alpha$ introduced in Eq. . With this substitution, the Navier-Stokes Eq. reads $${{\bf \nabla}}\,\frac{du}{dt} = -\frac{1}{\rho}\,{{\bf \nabla}}\,p + \eta_0\,{{\bf \nabla}}\left({{\bf \nabla}}^2\, u\right),\quad\hbox{with}\quad \eta_0 = \frac{4}{3}\,\eta_s + \eta_b.$$ So far the reviewing. We now discuss how viscosity enter the equation for a relativistic field. In Eq. , viscosity acts as an effective pressure, once we substitute [@delcampo; @bastero2011; @bastero12a] $$\label{substitute_p} \bar{p}_r \to \bar{p}_r + \Pi_0,$$ where $\Pi_0 = -3H\,\eta_b$. This modification alters the set of coupled equations for the background as $$\label{eq_motion1} \begin{array}{l} \displaystyle \ddot{\bar{\phi}} + \left(3H + \Gamma\right)\,\dot{\bar{\phi}} + U_\phi = 0,\\ \displaystyle \dot{\bar{\rho}}_r + 3H\,\left(\frac{4}{3}\,\bar{\rho}_r + \Pi_0\right) = \Gamma\,\dot{\bar{\phi}}^2. \end{array}$$ We find that the inclusion of the viscous term in Eq. drastically affects the behavior of the solution for $\bar{\rho}_r$ when the scalar field has decayed at $t \gg 1/\Gamma$. In fact, in this regime the second line of Eq. yields to a constant value of the radiation energy density, $$\label{viscosity_radiation_rho} \bar{\rho}_r = \frac{27\,\pi\,G}{2}\,\eta_b^2,$$ where we have used the fact that for a radiation-dominated cosmology the Hubble rate is $H^2 = 8\pi\,G\,\bar{\rho}_r/3$. A possible solution to this problem is found by considering a time-dependent viscous term, for example the usual result $\bar{\rho}_r \sim 1/t^2$ when $t \gg 1/\Gamma$ is recovered by setting $\eta_b \sim 1/t$. These results are confirmed in Fig. \[figure:backgroundviscosity\], where we show the numerical resolution of Eq. , with the values in Table \[table\_parameters\] and with the choices for the viscosity coefficient as $\eta_b = 0$ (blue solid line), $\eta_b = \Gamma\,\phi_0^2$ (brown solid line), and $\eta_b = \Gamma\,\phi_0^2/(\Gamma\,t+1)$ (black dashed line). Notice that the blue solid line for $\bar{\rho}_r$ in Fig. \[figure:backgroundviscosity\] is the same as that obtained previously and shown in Fig. \[figure:background\]. Since $\eta_b$ enters the equation of motion as a negative pressure, its effect is that of enhancing the energy density of the radiation field. The viscosity term affects initial conditions, since they are obtained from Eq. once $\ddot{\phi}$ and $\dot{\bar{\rho}}_r$ have been neglected. In particular, the solution with $\eta_b = \Gamma\,\phi_0^2(\Gamma t+1)^{-1}$ behaves like the one with $\eta_b~=~\Gamma\,\phi_0^2$ for $t \ll 1/\Gamma$, and like the one with $\eta_b = 0$ for $t \gg 1/\Gamma$. ![The energy density in the radiation field $\bar{\rho}_r$, obtained from solving the set of Eqs. with three different expressions for the viscosity term. For the viscosity coefficient, we set $\eta_b~=~0$ (blue solid line), $\eta_b~=~\Gamma\,\phi_0^2$ (brown solid line), and $\eta_b~=~\Gamma\phi_0^2/(\Gamma t+1)$ (black dashed line).[]{data-label="figure:backgroundviscosity"}](PlotRViscosity.eps){width="15cm"} The effects of the dissipative terms into the relativistic expression for momentum and energy density perturbations have been extensively discussed in the literature [@delcampo; @delcampo2010; @bastero2011; @bastero1; @setare2013; @bastero12a]. Viscosities in the radiation fluid are included in Eq. by substituting the pressure as in Eq. , while pressure perturbations modify as $$\label{delta_Pi} \delta p_{rq} \to \delta p_{rq} + \delta \Pi_0.$$ This gives $$\delta \dot{\rho}_{r q} + 3H\,({\delta \rho_{rq}}+{\delta p_{rq}}+ \delta \Pi_0 ) - \left(\bar{\rho}_r+\bar{p}_r+\Pi_0\right)\,\left(\frac{q^2}{a^2}\,{\delta u_{rq}}+ 3\dot{\Psi}_q\right) = \Gamma\,\dot{\bar{\phi}}\,\left(2\delta\dot{\phi}_q - \dot{\bar{\phi}}\,\Psi_q\right). \label{perturbation_energy_viscosity}$$ In order to derive the momentum equation for $\delta u_{rq}$, we add to the matter-energy tensor $T_{\mu\nu}$ of the system the stress tensor $\Pi_{\mu\nu}$, whose non-zero components are [@bastero1] $$\label{def_viscosity} \Pi_{ij}^{(\alpha)} = -\eta_0\,\nabla_i\,\nabla_j \,\delta u_\alpha,$$ so that $\nabla_i \,\Pi_{ij}^{(\alpha)} = -\eta_0\,\nabla_j \,\nabla^2 \delta u_\alpha$. Eq. can also be derived from the definition in Eq. with the assumptions of zero vorticity. With the substitutions for the pressure term in Eq. and its perturbation in Eq. , the momentum Eq. reads $${\delta p_{rq}}+ \delta \Pi_0 + a^{-3}\,\left[a^3\,\left(\bar{\rho}_r + \bar{p}_r + \Pi_0\right)\,\delta u_{rq}\right]^{\centerdot} + \left(\bar{\rho}_r + \bar{p}_r + \Pi_0\right)\,\Psi_q + \eta_0\,\frac{q^2}{a^2}\,\delta u_{rq}= \bar{Q}_r\,\delta u_q. \label{perturbation_pressure_viscosity}$$ Finally, Eq. giving the perturbations in the gravitational field modifies as $$\dot{\Psi}_q + H\,\Psi_q + 4\pi\,G \,\left[\left(\frac{4}{3}\,\bar{\rho}_r + \Pi_0\right)\,\delta u_{r q} - \dot{\phi}\,\delta \phi_q\right] = 0. \label{perturbation_field_viscosity}$$ Summing up, Eqs. - in the case where the viscosity is included as $\Pi_0 = -3H\,\eta_b$, with perturbations $\delta \Pi_0 = -3\left(H\,\Psi_q + \dot{\Psi}_q\right)\,\eta_b$, read $$\label{perturbation_eq_phi_viscosity1} \delta \ddot{\phi}_q + \left(3H +\Gamma\right)\,{\delta \dot{\phi}_q}+ \left(U_{\phi\phi} + \frac{q^2}{a^2}\right)\,{\delta \phi_q}= 4\,\dot{\bar{\phi}}\,\dot{\Psi}_q - \left(2\,U_\phi + \Gamma\,\dot{\bar{\phi}}\right)\,\Psi_q,$$ $$\delta \dot{\rho}_{r q} + 4H\,{\delta \rho_{rq}}-9H^2\,\eta_b\,\Psi_q - \left(\frac{4}{3}\bar{\rho}_r - 3H\,\eta_b\right)\,\frac{q^2}{a^2}\,{\delta u_{rq}}= 4\,\bar{\rho}_r\,\dot{\Psi}_q + \Gamma\,\dot{\bar{\phi}}\,\left(2\delta\dot{\phi}_q - \dot{\bar{\phi}}\,\Psi_q\right), \label{perturbation_energy_viscosity1}$$ $$\frac{1}{3}{\delta \rho_{rq}}- 3\eta_b\,\left(2H\,\Psi_q + \dot{\Psi}_q\right) + \frac{4}{3}\,\bar{\rho}_r\,\Psi_q + a^{-3}\,\left[a^3\,\left(\frac{4}{3}\,\bar{\rho}_r -3H\eta_b\right)\,\delta u_{rq}\right]^{\centerdot} + \eta_0\,\frac{q^2}{a^2}\,\delta u_{rq} + \Gamma\,\dot{\bar{\phi}}\,\delta\phi_q = 0, \label{perturbation_pressure_viscosity1}$$ $$\dot{\Psi}_q + H\,\Psi_q + 4\pi\,G \,\left[\left(\frac{4}{3}\,\bar{\rho}_r -3H\,\eta_b \right)\,\delta u_{r q} - \dot{\phi}\,\delta \phi_q\right] = 0. \label{perturbation_field_viscosity1}$$ For completeness, we have included Eq. although it has not been modified from Eq. by the inclusion of viscosities. This set of equations describes the evolution of perturbations in the presence of viscous terms $\eta_b$ and $\eta_0$, and reduces to the results in the previous Section when $\eta_b = \eta_0 = 0$. We solve numerically the set of Eqs. -, with the background for the inflaton field and $\bar{\rho}_r$ given by the solution to Eq. . Results are shown in Fig. \[figure:eta\], where we fixed $q = 0.1$. For the viscosity coefficients we set $\eta_0 = \eta_b$ and we choose $\eta_b = 0$ (blue line), corresponding to the solution described in Sec. \[Evolution of perturbations during inflation\], $\eta_b = \Gamma\,\phi_0^2$ (brown line), which for the background gives the results in Eq. for large values of $t$, and $\eta_b = \Gamma\phi_0^2/(\Gamma t+1)$ (black dashed line), which allows us to retrieve the $\bar{\rho}_r \sim 1/t^2$ for large values of $t$. The presence of a constant viscosity term damps oscillations in all three fields shown. ![Perturbations in the gravitational field $\Psi_{q}$ (top figure), in the velocity perturbation $\delta u_{rq}$ (middle figure) and in the perturbations of the radiation energy density $\delta \rho_{rq}$ (bottom figure), obtained by solving the set of Eqs. -. For the viscosity coefficient, we set $\eta_b~=~0$ (blue solid line), $\eta_b~=~\Gamma\,\phi_0^2$ (brown solid line), and $\eta_b~=~\Gamma\phi_0^2/(\Gamma t+1)$ (black dashed line).[]{data-label="figure:eta"}](FigureDissipation.eps){width="12cm"} Recently, Bastero-Gil [et al.]{} [@bastero1] proposed a model which uses the formalism of Hwang and Noh [@hwang2002] for cosmological perturbations and Landau theory for the statistical fluctuations in the radiation fluid [@landau]. Contrary to our assumption in Eq. , where we indirectly assumed that the entropy density $s$ of the radiation fluid is a function of temperature only, in Ref. [@bastero1] the entropy density is a function of both $T$ and $\phi$. The work in Ref. [@bastero1] includes additional terms in the expression for the perturbations of the radiation energy density and velocity such as the viscosity coefficients introduced before $\eta_s$ and $\eta_b$, a possible perturbation in the dissipation coefficient $\delta\Gamma_q$, and stochastic sources ${\bf P}_q$, $\xi^{(r)}_q$, and $\xi^{(\phi)}_q$ (we discuss stochastic perturbations in Sec. \[Stochastic perturbations\] below). We have checked that imposing $\delta\Gamma_q = {{\bf P}_q}= \xi^{(r)}_q = \xi^{(\phi)}_q = s_{,\phi} = 0$ in Eqs. (3.20), (3.24), (3.27), (3.28), and (3.30) of Ref. [@bastero1] allows us to recover our set of Eqs. -. Slow-roll regime with a stochastic source {#Stochastic perturbations} ========================================= Inflation occurs if the potential $U(\phi)$ is sufficiently flat and much larger than all other forms of energy, so that the Hubble expansion rate $H$ is constant. During this period, which is known as the slow-roll regime of the inflaton field, higher derivatives in Eq. (\[eq\_motion\]) can be neglected, $$\label{slow_roll_conditions} \ddot{\phi} \ll H \,\dot{\phi},\quad\hbox{and}\quad \dot{\rho}_r \ll H\,\rho_r.$$ In this regime, the Friedmann Eq. (\[friedmann\]) and Eq. (\[eq\_motion\]) for the motion of the inflaton field read $$\begin{array}{l} \displaystyle H^2 \simeq \frac{8\pi G}{3}\,U,\\ \displaystyle \\ \displaystyle \dot{\phi} \simeq -\frac{U_\phi}{3H+\Gamma},\\ \displaystyle \\ \displaystyle \rho_r \simeq \frac{\Gamma\,\dot{\phi}^2}{4H}, \end{array} \label{eq_motion_slow_roll}$$ where we use the symbol “$\simeq$” for an equality that holds only in the slow-roll regime. Defining the slow-roll parameters, $$\label{slow_roll_parameters} \epsilon = \frac{1}{16\pi G}\left(\frac{U_\phi}{U}\right)^2, \quad \eta = \frac{1}{8\pi G}\,\frac{U_{\phi\phi}}{U},\quad\beta = \frac{1}{8\pi G}\left(\frac{\Gamma_{\phi}\,U_\phi}{\Gamma\,U}\right),$$ the condition in Eq. (\[slow\_roll\_conditions\]) is met if the slow-roll parameters satisfy [@visinelli_NWI; @taylor_berera; @hall; @moss_xiong; @moss] $$\label{slow_roll} \epsilon \ll 1+\frac{\Gamma}{3H},\quad \|\eta\| \ll 1+\frac{\Gamma}{3H},\quad \|\beta\| \ll 1+ \frac{\Gamma}{3H}.$$ We now discuss the evolution of perturbations during the slow-roll regime. In the following, we take into account the effects from both quantum and thermal fluctuations, the latter being the predominant source for perturbations in WI scenarios. For this, we introduce a stochastic noise $\xi({\bf x},t)$ in the source term for $\delta \rho_\phi$, according to the Schwinger-Keldysh approach to describe the quantum mechanical evolution of a system in a non-equilibrium state [@schwinger; @keldysh; @rammer]. This term has been discussed by Calzetta and Hu [@calzetta_hu] in the context of the Boltzmann equation, and by Berera and Fang [@berera_fang] in the setting of WI. Instead of $\delta Q_\phi$ in Eq. (\[cold\_perturbations\]), we thus consider the source term $$\label{stochastic_source} \delta Q_\phi = -\Gamma\,\left(\delta p_\phi + \delta \rho_\phi\right) - \dot{\bar{\phi}}\,\Gamma_{\rm eff}\,\xi_q,$$ where $\xi_q$ is the Fourier transform of the stochastic noise, which follows the statistical average $$\label{condition_stochastic1} \displaystyle \langle \xi_q\,\xi_{q'}\rangle = a(t)^{-3}\,\delta^{(3)}({\bf q + q'})\,\delta(t-t'),$$ and we have defined the parameter [@moss] $$\Gamma_{\rm eff} = \sqrt{2\,\left[\Gamma + H\right]\,T},$$ where the temperature of the radiation bath is defined through $\bar{\rho}_r$ in Eq. (\[energy\_pressure\]). Stochastic perturbations may arise from microscopical models for radiation, as discussed for example in Refs. [@bastero12b; @bastero14; @bastero13] in the context of supersymmetric models. In this work, we decided to treat the radiation density $\bar{\rho}_r$ as a fluid, and to introduce the stochastic source $\xi_q$ in a phenomenological way, without making assumptions on the underlying model for radiation. This will be the subject of a future work, along the line of Ref. [@visinelli_NWI], in which perturbations in models of axion-like particles are discussed. With the addition of the stochastic source perturbations in Eq. , the equation for $\delta \phi$ in Eq. modifies as $$\label{perturbation_eq_stochastic} \delta \ddot{\phi}_q + \left(3H +\Gamma\right)\,{\delta \dot{\phi}_q}+ \left(U_{\phi\phi} + \frac{q^2}{a^2}\right)\,{\delta \phi_q}- 4\,\dot{\bar{\phi}}\,\dot{\Psi}_q + \left(2\,U_\phi + \Gamma\,\dot{\bar{\phi}}\right)\,\Psi_q = -\Gamma_{\rm eff}\,\xi_q.$$ We have checked that this expression is recovered also by using the results in Ref. [@bastero1], once set to zero the perturbations on the dissipation term $\delta \Gamma_q = 0$ which we do not include. We assume that, during the slow-roll regime, the gravitational field perturbation varies slowly as [@polarski] $$\dot{\Psi} \ll H\,\Psi, \quad$$ so that, using Eq. (\[eq\_motion\_slow\_roll\]), perturbations for the inflaton field in Eq. (\[perturbation\_eq\_stochastic\]) are given by $$\label{perturbation_slow_roll} \delta \ddot{\phi}_q + \left(3H +\Gamma\right)\,{\delta \dot{\phi}_q}+ \left[U_{\phi\phi} + \frac{q^2}{a^2} - \frac{2H\,(3H + \Gamma)(6H+\Gamma)}{\Gamma}\,\frac{\bar{\rho}_r}{U}\right]\,{\delta \phi_q}\simeq -\Gamma_{\rm eff}\,\xi_q.$$ We redefine the latter term in the square brackets as $$- \frac{2H\,(3H + \Gamma)(6H+\Gamma)}{\Gamma}\,\frac{\bar{\rho}_r}{U} = \kappa\,\frac{U_\phi^2}{2U},$$ where we have parametrizes the strength of $\Gamma$ with respect to $H$, as $\kappa = 1$ when $H \gg \Gamma$ and $\kappa = 1/2$ when $H \ll \Gamma$. With this substitution, the first line of Eq. (\[perturbation\_slow\_roll\]) gives the dissipation equation presented in Ref. [@moss], except for the extra term containing $U_{\phi\phi}$ and $U_\phi^2$, $$\delta\ddot{\phi}_q + \left(3H +\Gamma\right)\,\delta \dot{\phi}_q + \left(U_{\phi\phi} + \kappa \,\frac{U_\phi^2}{2U} + \frac{q^2}{a^2}\right)\,\delta\phi_q \simeq -\Gamma_{\rm eff}\,\xi_q, \label{perturbation_slow_roll1}$$ In order to solve for $\phi_q$, we switch to the variable $$z = \frac{q}{H\,a(t)},$$ and, using the slow-roll Eqs. (\[slow\_roll\_parameters\]) in the form $U_\phi^2 = 3H^2\,U\,\epsilon/2$ and $U_{\phi\phi} = 3H^2\,\eta$, we define $$\nu = \frac{3}{2}\,(1+R),\quad \hbox{and}\quad \mu = \sqrt{\nu^2 - 3\,(\kappa\,\epsilon + \eta)}.$$ Eq. (\[perturbation\_slow\_roll1\]) is rewritten as $$\begin{array}{l} \displaystyle \frac{d^2\,\delta\phi}{dz^2} + \frac{1-2\nu}{z}\,\frac{d\,\delta \phi}{dz} + \left[1+\frac{3\,(\kappa\,\epsilon + \eta)}{z^2}\right]\,\delta \phi = \frac{\left[2(\Gamma+H)\,T\right]^{1/2}}{H^2\,z^2}\,\xi_q, \end{array} \label{diff_eq_perturbation_phi_moss1}$$ which is a second order stochastic differential equation, with solution for the field and its first derivative given respectively by $$\begin{array}{l} \displaystyle \delta \phi_q(z) = z^\nu\,\left(\delta\phi_1\,J_\mu(z) + \delta\phi_2\,Y_\mu(z)\right) -\frac{\pi}{2}\,\frac{\left[2(\Gamma+H)\,T\right]^{1/2}}{H^2}\,\int_z^{+\infty}\,G^{(1)}_\mu(z,z')\,\left(\frac{z}{z'}\right)^\nu\,\xi_q(z')\,\frac{dz'}{z'},\\ \displaystyle\\ \displaystyle \delta \phi'_q(z) = \delta\phi'_1+\delta\phi'_2 -\frac{\pi}{2}\,\frac{\left[2(\Gamma+H)\,T\right]^{1/2}}{H^2}\,\int_z^{+\infty}\,G^{(2)}_\mu(z,z')\,\left(\frac{z}{z'}\right)^\nu\,\xi_q(z')\,\frac{dz'}{z'}. \end{array}$$ Here, $J_\mu(z)$ and $Y_\mu(z)$ are the Bessel functions of respectively the first and second kind of order $\mu$, $\delta\phi_2$ and $\delta\phi_2$ are arbitrary constants, and we have introduced the functions $$\begin{array}{l} G^{(1)}_\mu(z,z') = J_\mu(z)\,Y_\mu(z')-J_\mu(z')\,Y_\mu(z),\\ G^{(2)}_\mu(z,z') = J_{\mu-1}(z)\,Y_\mu(z')-J_\mu(z')\,Y_{\mu-1}(z). \end{array}$$ This solution generalizes the result presented in Ref. [@moss] to the case $\mu \neq \nu$, and it is valid whenever the slow-roll parameters can be considered constant. Conclusions {#Discussion and conclusions} =========== We shall summarize the main points of this paper. In Sec. \[Equations for the background fields\], we have revised the Boltzmann equations for an inflaton field coupled to radiation, and we have solved the system of equation numerically in the case of a quartic inflaton potential $U(\phi)$, see Eq. . In Sec. \[Evolution of perturbations during inflation\], we have applied the interacting fluids formalism, based on the framework given in Refs. [@kodama; @mukhanov1992] and described in Refs. [@malik; @malik1], to derive the differential equations governing the cosmological fluctuations arising during the inflationary period in Eqs. -. These equations have been solved numerically for a quartic inflaton potential $U(\phi)$. We give an analytical expression for $\delta \phi$ around the reheating period in Eq. (\[perturbation\_eq\_X\]), which approximates the frequency of oscillation obtained with the numerical solution. We have considered the effects of a viscosity term $\eta_0$ in Sec. \[dissipative\_terms\], following the result of other work in the literature [@delcampo; @bastero12a; @bastero1]. We have obtained that a constant viscosity term predicts a constant value for the energy density of the radiation field in the post-inflationary period, see Eq. , and affects perturbations in the energy density and velocity of the fluid by damping oscillations. These occurrences can be avoided in theories which predict a time-depending $\eta_0$, as we have shown in Fig. \[figure:eta\] for a specific function of time. These results will be further explored in a future work on realistic microscopic models of the radiation field. In conclusion, we discuss perturbations in the presence of viscosities in Eqs. -, and we show numerical results in Fig. \[figure:eta\]. In Sec. \[Stochastic perturbations\], we added a stochastic source to the expression describing $\delta \phi_q$, which is the approach usually considered in warm inflation scenarios. In the slow-roll regime, the equation for the inflaton perturbations can be solved exactly, including the stochastic term. The solution extends the result obtained in Ref. [@moss] to the case in which the slow-roll parameters are not neglected in the differential equation. The author would like to thank Paolo Gondolo and Xuefang Sui for useful discussions on cosmological perturbations. [50]{} D. Kazanas, [Dynamics of the universe and spontaneous symmetry breaking]{}, Astrophys. J. [241]{}, L59 (1980). A. A. Starobinsky, [A new type of isotropic cosmological models without singularity]{}, Phys. Lett. B [91]{}, 99 (1980). A. H. Guth, [Inflationary universe: A possible solution to the horizon and flatness problems]{}, Phys. Rev. D [23]{}, 347 (1981). K. Sato, [Cosmological baryon-number domain structure and the first order phase transition of a vacuum]{}, Phys. Lett. B [99]{}, 66 (1981). A. Albrecht and P. J. Steinhardt, [Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking]{}, Phys. Rev. Lett. [48]{}, 1220 (1982). A. D. Linde, [A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems]{}, Phys. Lett. B [108]{}, 389 (1982); [Chaotic Inflation]{}, Phys. Lett. B [129]{}, 177 (1983). A. D. Linde, [Particle Physics and Inflationary Cosmology]{}, Harwood Academic, Switzerland (1990). E. W. Kolb and M. S. Turner, [The Early Universe]{}, Addison-Wesley (1990). S. Weinberg, [Cosmology]{}, Oxford University Press, USA (2008). V. F. Mukhanov and G. V. Chibisov, [Quantum fluctuations and a nonsingular Universe]{}, PisÕma Zh. Eksp. Teor. Fiz. [33]{}, 549 (1981) \[JETP Lett. [33]{}, 532 (1981)\]. A. H. Guth and S. Y. Pi, [Fluctuations in the New Inflationary Universe]{}, Phys. Rev. Lett. [49]{}, 1110 (1982). S. W. Hawking, [The development of irregularities in a single bubble inflationary universe]{}, Phys. Lett. B [115]{}, 295 (1982). A. A. Starobinsky, Phys. Lett. B [117]{}, 175 (1982). J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D [28]{}, 679 (1983). A. Linde, [Hybrid Inflation]{}, Phys. Rev. D [49]{}, 748 (1994) \[[astro-ph/9307.002](http://arxiv.org/abs/astro-ph/9307002)\]. L. Kofman, A. Linde, and A. Starobinsky, [Reheating after Inflation]{}, Phys. Rev. Lett. [73]{}, 3195 (1994) \[[hep-th/9405.187](http://arxiv.org/abs/hep-th/9405187)\]. L. Kofman, A. Linde, and A. Starobinsky, [Towards the Theory of Reheating After Inflation]{}, Phys. Rev. D [56]{}, 3258 (1997) \[[hep-ph/9704.452](http://arxiv.org/abs/hep-ph/9704452)\]. D. Boyanovsky, H. J. de Vega, R. Holman, and J. F. J. Salgado, [Analytic and Numerical Study of Preheating Dynamics]{}, Phys. Rev. D [54]{}, 7570 (1996) \[[hep-ph/9608.205](http://arxiv.org/abs/hep-ph/9608205)\]. P. B. Greene and L. Kofman, [Preheating of Fermions]{}, Phys. Lett. B [448]{}, 6 (1999) \[[hep-ph/9807.339](http://arxiv.org/abs/hep-ph/9807339)\]. J. Braden, L. Kofman, and N. Barnaby, [Reheating the Universe After Multi-Field Inflation]{}, JCAP [7]{}, 16 (2010) \[[hep-th/1005.2196](http://arxiv.org/abs/arXiv:1005.2196)\]. A. Berera and L. Z. Fang, [Thermally Induced Density Perturbations in the Inflation Era]{}, Phys. Rev. Lett. [74]{}, 1912 (1995) \[[astro-ph/9501.024](http://arxiv.org/abs/astro-ph/9501024)\]. A. Berera, [Warm Inflation]{}, Phys. Rev. Lett. [75]{}, 3218 (1995) \[[astro-ph/9509.049](http://arxiv.org/abs/astro-ph/9509049)\]. A. Berera, [Warm Inflation in the Adiabatic Regime- a Model, an Existence Proof for Inflationary Dynamics in Quantum Field Theory]{}, Nucl. Phys. B [595]{}, 666 (2000) \[[hep-ph/9904.409](http://arxiv.org/abs/hep-ph/9904409)\]. A. N. Taylor and A. Berera, [Perturbation Spectra in the Warm Inflationary Scenario]{}, Phys. Rev. D [62]{}, 083517 (2000) \[[astro-ph/0006.077](http://arxiv.org/abs/astro-ph/0006077)\]. H. P. De Oliveira, S. E. Joras, [On Perturbations in Warm Inflation]{}, Phys. Rev. D [64]{}, 063513 (2001) \[[gr-qc/0103.089](http://arxiv.org/abs/gr-qc/0103089)\]. H. P. De Oliveira, [Density perturbations in warm inflation and COBE normalization]{}, Phys. Lett. B [526]{}, 1 (2002) \[[gr-qc/0202.045](http://arxiv.org/abs/gr-qc/0202045v1)\]. L. M. H. Hall, I. G. Moss, A. Berera, [Scalar perturbation spectra from warm inflation]{}, Phys. Rev. D [69]{} 083525 (2004) \[[astro-ph/0305.015](http://arxiv.org/abs/astro-ph/0305015)\]. M. Bastero-Gil and A. Berera, [Determining the regimes of cold and warm inflation in the supersymmetric hybrid model]{}, Phys. Rev. D [71]{}, 063515 (2005) \[[hep-ph/0411.144](http://arxiv.org/abs/hep-ph/0411144)\]. S. del Campo, R. Herrera, D. Pavon, [Cosmological perturbations in warm inflationary models with viscous pressure]{}, Phys. Rev. D [75]{}, 083518 (2007) \[[astro-ph/0703.604](http://arxiv.org/abs/astro-ph/0703604v1)\]. I. G. Moss and C. Xiong, [Non-gaussianity in fluctuations from warm inflation]{}, JCAP [0704]{}, 007 (2007) \[[astro-ph/0701.302](http://arxiv.org/abs/astro-ph/0701302v2)\]. I. G. Moss and C. Xiong, [On the consistency of warm inflation]{}, JCAP [0811]{}, 023 (2008) \[[astro-ph/0808.0261](http://arxiv.org/abs/0808.0261)\]. A. Berera, I. G. Moss, and R. O. Ramos, [Warm Inflation and its Microphysical Basis]{}, Rept. Prog. Phys. [72]{}, 026901 (2009) \[[hep-ph/0808.1855](http://arxiv.org/abs/0808.1855)\]. M. Bastero-Gil and A. Berera, [Warm inflation model building]{}, Int. J. Mod. Phys. A [24]{}, 2207 (2009) \[[hep-ph/0902.0521](http://arxiv.org/abs/0902.0521)\]. C. Graham and I. G. Moss, [Density fluctuations from warm inflation]{}, JCAP [0907]{}, 013 (2009) \[[astro-ph/0905.3500](http://arxiv.org/abs/0905.3500)\]. S. del Campo, R. Herrera, D. Pavon, and J. R. Villanueva, [On the consistency of warm inflation in the presence of viscosity]{}, JCAP [1008]{}, 002 (2010) \[[astro-ph/1007.0103](http://arxiv.org/abs/1007.0103)\]. I. G. Moss and T. Yeomans, [Non-gaussianity in the strong regime of warm inflation]{}, \[[astro-ph/1102.2833](http://arxiv.org/abs/1102.2833)\] (2011). M. Bastero-Gil, A. Berera,, and R. O. Ramos, [Shear viscous effects on the primordial power spectrum from warm inflation]{}, \[[astro-ph/1106.0701](http://arxiv.org/abs/1106.0701v1)\] (2011). S. Bartrum, A. Berera, and J. G. Rosa, [Warming up for Planck]{}, JCAP [1306]{}, 025 (2013) \[[astro-ph/1303.3508](http://arxiv.org/abs/1303.3508)\]. M. R. Setare and V. Kamali, [Cosmological perturbations in warm-tachyon inflationary universe model with viscous pressure]{}, Phys. Lett. B [736]{}, 86 (2014). M. Bastero-Gil, A. Berera, I. G. Moss, and R. O. Ramos, [Cosmological fluctuations of a random field and radiation fluid]{}, \[[astro-ph/1401.1149](http://arxiv.org/abs/1401.1149)\] (2014). L. Z. Fang, [Entropy generation in the early universe by dissipative processes near the Higgs phase transition]{}, Phys. Lett. B [95]{} 154 (1980). A. Hosoya and M.-aki Sakagami, [Time development of Higgs field at finite temperature]{}, Phys. Rev. D [29]{}, 2228 (1984). I. G. Moss, [Primordial inflation with spontaneous symmetry breaking]{}, Phys. Lett. B [154]{}, 120 (1985). S. R. Lonsdale and I. G. Moss, [A superstring cosmological model]{}, Phys. Lett. B [189]{}, 12 (1987). J. Yokoyama and K. Maeda, [On the Dynamics of the Power Law Inflation Due to an Exponential Potential]{}, Phys. Lett. B [207]{}, 31 (1988). A. R. Liddle, [Power Law Inflation With Exponential Potentials]{}, Phys. Lett. B [220]{}, 502 (1989). J. Garcia-Bellido, A. D. Linde, and D. Wands, [Density Perturbations and Black Hole Formation in Hybrid Inflation]{}, Phys. Rev. D [54]{}, 6040 (1996) \[[astro-ph/9605.094](http://arxiv.org/abs/astro-ph/9605094)\]. L. Randall, M. Soljacic, and A. H. Guth, [Supernatural Inflation: Inflation from Supersymmetry with No (Very) Small Parameters]{}, Nucl. Phys. B [472]{}, 377 (1996) \[[hep-ph/9512.439](http://arxiv.org/abs/hep-ph/9512439)\]. S. Tsujikawa, D. Parkinson, and B. A. Bassett, [Correlation-consistency cartography of the double inflation landscape]{}, Phys. Rev. D [67]{}, 083516 (2003) \[[astro-ph/0210.322](http://arxiv.org/abs/astro-ph/0210322)\]. D. Parkinson, S. Tsujikawa, B. A. Bassett, L. Amendola, [Testing for double inflation with WMAP]{}, Phys. Rev. D [71]{}, 063524 (2005) \[[astro-ph/0409.071](http://arxiv.org/abs/astro-ph/0409071)\]. K. Kadota and E. D. Stewart, [Successful Modular Cosmology]{}, JHEP [0307]{}, 013 (2003) \[[hep-ph/0304.127](http://arxiv.org/abs/hep-ph/0304127)\]. D. Wands, [Multiple field inflation]{}, Lect. Notes Phys. [738]{}, 275 (2008) \[[astro-ph/0702.187](http://arxiv.org/abs/astro-ph/0702187v1)\]. B. A. Bassett, S. Tsujikawa, and D. Wands, [Inflation Dynamics and Reheating]{}, Rev. Mod. Phys. [78]{}, 537 (2006) \[[astro-ph/0507.632](http://arxiv.org/abs/astro-ph/0507632)\]. K. Freese, J. A. Frieman, A. V. Olinto, [Natural inflation with pseudo Nambu-Goldstone bosons]{}, Phys. Rev. Lett. [65]{}, 3233 (1990). F. Adams, J. R. Bond, K. Freese, J. Frieman, A. Olinto, [Natural inflation: Particle physics models, power-law spectra for large-scale structure, and constraints from the Cosmic Background Explorer]{}, Phys. Rev. D [47]{}, 426 (1993) \[[hep-ph/9207245](http://arxiv.org/abs/hep-ph/9207245)\]. C. Savage K. Freese and W. H. Kinney, [Natural Inflation: status after WMAP 3-year data]{}, Phys. Rev. D [74]{},123511 (2006) \[[hep-ph/0609.144](http://arxiv.org/abs/hep-ph/0609144)\]. S. Mohanty, A. Nautiyal, [Natural inflation at the GUT scale]{}, Phys. Rev. D [78]{}, 123515 (2008) \[[hep-ph/0807.0317](http://arxiv.org/abs/0807.0317)\]. L. Visinelli, [Natural Warm Inflation]{}, JCAP [1109]{}, 013 (2011) \[[astro-ph/1107.3523](http://arxiv.org/abs/1107.3523)\]. K. Freese and D. Spolyar, [Chain Inflation in the Landscape: “Bubble Bubble Toil and Trouble”]{}, JCAP [0507]{}, 007 (2005) \[[hep-ph/0412.145](http://arxiv.org/abs/hep-ph/0412145)\]. K. Freese, J. T. Liu, and D. Spolyar, [Inflating with the QCD Axion]{}, Phys. Rev. D [72]{}, 123521 (2005) \[[hep-ph/0502.177](http://arxiv.org/abs/hep-ph/0502177)\]. K. Enqvist and M. S. Sloth, [Adiabatic CMB perturbations in pre-big bang string cosmology]{}, Nucl. Phys. B [626]{}, 395 (2002) \[[hep-ph/0109.214](http://arxiv.org/abs/hep-ph/0109214)\]. D. H. Lyth and D. Wands, [Generating the curvature perturbation without an inflaton]{}, Phys. Lett. B [524]{}, 5 (2002) \[[hep-ph/0110.002](http://arxiv.org/abs/hep-ph/0110002)\]. T. Moroi and T. Takahashi, [Effects of Cosmological Moduli Fields on Cosmic Microwave Background]{}, Phys. Lett. B [522]{}, 215 (2001) \[Erratum-ibid. B [539]{}, 303 (2002)\] \[[hep-ph/0110.096](http://arxiv.org/abs/hep-ph/0110096)\]. T. Moroi and T. Takahashi, [Cosmic Density Perturbations from Late-Decaying Scalar Condensations]{}, Phys. Rev. D [66]{}, 063501 (2002) \[[hep-ph/0206026](http://arxiv.org/abs/hep-ph/0206026)\]. M. Kawasaki, K. Kohri, and N. Sugiyama, [Cosmological Constraints on Late-time Entropy Production]{}, Phys. Rev. Lett. [82]{}, 4168 (1999) \[[astro-ph/9811.437](http://arxiv.org/abs/astro-ph/9811437)\]. M. Kawasaki, K. Kohri, and N. Sugiyama, [MeV-scale Reheating Temperature and Thermalization of Neutrino Background]{}, Phys. Rev. D [62]{}, 023506 (2000) \[[astro-ph/0002.127](http://arxiv.org/abs/astro-ph/0002127)\]. S. Hannestad, [What is the lowest possible reheating temperature?]{}, Phys. Rev. D [70]{}, 043506 (2004) \[[astro-ph/0403.291](http://arxiv.org/abs/astro-ph/0403291)\]. K. Ichikawa, M. Kawasaki, and F. Takahashi, [The oscillation effects on thermalization of the neutrinos in the universe with low reheating temperature]{}, Phys. Rev. D [72]{}, 043522 (2005) \[[astro-ph/0505395](http://arxiv.org/abs/astro-ph/0505395)\]. K. Ichikawa, M. Kawasaki, and F. Takahashi, [Constraint on the Effective Number of Neutrino Species from the WMAP and SDSS LRG Power Spectra]{}, JCAP [5]{}, 7 (2007), \[[astro-ph/0611784](http://arxiv.org/abs/astro-ph/0611784)\]. F. de Bernardis, L. Pagano, and A. Melchiorri, Astroparticle Physics [30]{}, 192 (2008). J. Bardeen, [Gauge-invariant cosmological perturbations]{}, Phys. Rev. D [22]{}, 1882 (1980). H. Kodama and M. Sasaki, [Cosmological Perturbation Theory]{}, Prog. Theor. Phys. Suppl. [78]{} (1984). V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, [Theory of cosmological perturbations]{}, Phys. Rep. [215]{}, 203 (1992). K. A. Malik and D. Wands, [Adiabatic and entropy perturbations with interacting fluids and fields]{}, JCAP [02]{}, 007 (2005) \[[astro-ph/0411.703](http://arxiv.org/abs/astro-ph/0411703)\]. K. A. Malik, D. Wands, and C. Ungarelli, [Large-scale curvature and entropy perturbations for multiple interacting fluids]{}, Phys. Rev. D [67]{}, 063516 (2003) \[[astro-ph/0211.602](http://arxiv.org/abs/astro-ph/0211602)\]. M. Bellini, [Fresh inflation with increasing cosmological parameter]{}, Phys. Rev. D [67]{}, 027303 (2003) \[[gr-qc/0211.044](http://arxiv.org/abs/gr-qc/0211044)\]. T. Matsuda, [Aspects of warm-flat directions]{}, Int. J. Mod. Phys. A [25]{}, 4221 (2010) \[[hep-ph/0908.3059](http://arxiv.org/abs/0908.3059)\]. A. L. Erickcek, K. Sigurdson, [Reheating Effects in the Matter Power Spectrum and Implications for Substructure]{}, Phys. Rev. D [84]{}, 083503 (2011) \[[astro-ph/1106.0536](http://arxiv.org/abs/1106.0536v2)\]. R. Allahverdi and M. Drees, [Production of massive stable particles in inflaton decay]{}, Phys. Rev. Lett. [89]{}, 091302 (2002) \[[hep-ph/0203.118](http://arxiv.org/abs/hep-ph/0203118)\]. R. Allahverdi and M. Drees, [Thermalization after inflation and production of massive stable particles]{}, Phys. Rev. D [66]{}, 063513 (2002) \[[hep-ph/0205.246](http://arxiv.org/abs/hep-ph/0205246)\]. S. Gupta, K. A. Malik, and D. Wands, [Curvature and isocurvature perturbations in a three-fluid model of curvaton decay]{}, Phys. Rev. D [69]{}, 063513 (2004) \[[astro-ph/0311.562](http://arxiv.org/abs/astro-ph/0311562)\]. S. Matarrese and A. Riotto, [Large-scale curvature perturbations with spatial and time variation of the inflaton decay rate]{}, JCAP [0308]{}, 007 (2003) \[[astro-ph/0306.416](http://arxiv.org/abs/astro-ph/0306416v2)\]. D. J. H. Chung, E. W. Kolb, A. Riotto, [Production of massive particles during reheating]{}, Phys. Rev. D [60]{}, 063504 (1999) \[[hep-ph/9809.453](http://arxiv.org/abs/hep-ph/9809453)\]. D. H. Lyth and A. Riotto, Phys. Rept. [314]{}, 1 (1999) \[[hep-ph/9807.278](http://arxiv.org/abs/hep-ph/9807278)\]. T. Padmanabhan, [Theoretical Astrophysics, Volume I: Astrophysical Processes]{}, Cambridge University Press, Cambridge (2000). M. Bastero-Gil, A. Berera, R. Cerezo, R. O. Ramos, and G. S. Vicente, [Stability analysis for the background equations for inflation with dissipation and in a viscous radiation bath]{}, JCAP [1211]{}, 042 (2012) \[[astro-ph/1209.0712](http://arxiv.org/abs/1209.0712)\]. J.-c. Hwang and H. Noh, [Cosmological Perturbations with Multiple Fluids and Fields]{}, Class. Quant. Grav. [19]{}, 527 (2002) \[[astro-ph/0103.244](http://arxiv.org/abs/astro-ph/0103244)\]. L. D. Landau and E. M. Lifshitz, [Fluid Mechanics]{}, Addison-Wesley, Reading (MA) (1975). J. Schwinger, [Brownian motion of a quantum oscillator]{}, J. Math. Phys. [2]{}, 407 (1961). L. V. Keldysh, [Diagram Technique for Non-equilibrium Processes]{}, J. Exptl. Theoret. Phys. [47]{}, 1515 (1964) \[[http://www.jetp.ac.ru](http://www.jetp.ac.ru/cgi-bin/dn/e_020_04_1018.pdf)\]. J. Rammer and H. Smith, [Quantum field-theoretical methods in transport theory of metals]{}, Rev. Mod. Phys. [58]{}, 323 (1986). E. Calzetta and B. L. Hu, [Stochastic dynamics of correlations in quantum field theory: From Schwinger-Dyson to Boltzmann-Langevin equation]{}, Phys. Rev. D [61]{}, 025012 (2000) \[[hep-ph/9903.291](http://arxiv.org/abs/hep-ph/9903291v1)\]. D. Polarski and A. A. Starobinsky, Nucl. Phys. B [385]{}, 623 (1992). S. Weinberg, [Adiabatic Modes in Cosmology]{}, Phys. Rev. D [67]{}, 123504 (2003) \[[astro-ph/0302.326](http://arxiv.org/abs/astro-ph/0302326)\]. M. Bastero-Gil, A. Berera, R. O. Ramos, and J. G. Rosa, [General dissipation coefficient in low-temperature warm inflation]{}, JCAP [01]{}, 016 (2013) \[[hep-ph/1207.0445](http://arxiv.org/abs/1207.0445)\]. M. Bastero-Gil, A. Berera, R. O. Ramos, and J. G. Rosa, [Observational implications of mattergenesis during inflation]{}, JCAP 1410, 053 (2014) \[[astro-ph/1404.4976](http://arxiv.org/abs/1404.4976)\]. M. Bastero-Gil, A. Berera, T. P. Metcalf, J. G. Rosa, [Delaying the waterfall transition in warm hybrid inflation*]{}, \[[hep-ph/1312.2961](http://arxiv.org/abs/1312.2961)\] (2013).
--- abstract: \| Supply Shortage Outages are a major concern during peak demand for developing countries. In the Philippines, commercial loads have unused backup generation of up to 3000 MW, at the same time there are shortages of as much as 700 MW during peak demand. This gives utilities the incentive to implement Demand Response programs to minimize this shortage. But when considering Demand Response from a modeling perspective, social welfare through profit is always the major objective for program implementation. That isn’t always the case during an emergency situation as there can be a trade-off between grid resilience and cost of electricity. The question is how the Distribution Utility (DU) shall optimally allocate the unused generation to meet the shortage when this trade-off exists. We formulate a combined multi-objective optimal dispatch model where we can make a direct comparison between the least-cost and resilience objectives. We find that this trade-off is due to the monotonically increasing nature of energy cost functions. If the supply is larger than the demand, the DU can perform a least-cost approach in the optimal dispatch since maximizing the energy generated in this case can lead to multiple solutions. We also find in our simulation that in cases where the supply of energy from the customers is less than shortage quantity, the DU must prioritize maximizing the generated energy rather than minimizing cost. author: - - - title: 'Resilient Energy Allocation Model for Supply Shortage Outages' --- INTRODUCTION ============ There is potential to increase the reliability[^1] of electricity grids during Supply Shortage Outages (SSO). This can done through the use of the in-house distributed generation of commercial and industrial loads. In fact, in the Philippines, there is as much as 3000 MW worth of unused distributed power through these generators even with energy crisis of as much as 700MW [@intro]. Therefore, there is an incentive for the regulator to implement a Demand Side Management (DSM) program [@DSM] to allow distribution utilities (DU) to contract certain commercial establishments to either (1) reduce energy consumption, or (2) use their in-house generators during peak demand. We put our attention on the latter case. In the literature, managing energy demand through incentive contracts fall in the realm of Demand Response (DR) [@DR-berk]. The US Department of Energy defines Demand Response as “Changes in electric usage by end-use customers from their normal consumption patterns in response to changes in the price of electricity over time, or to incentive payments designed to induce lower electricity use at times of high wholesale market prices or when system reliability is jeopardized” [@DR_def]. Specifically, we are dealing with the Interruptible Load Program (ILP) [@DR-berk]. For most cases in DR, and DSM in general, profits, social welfare, and prices are put into the spotlight in the design of these programs [@DSM; @DR-berk]. In certain cases, such as emergency situations, this approach may leave the implementation of certain DR schemes ineffective in increasing resiliency. Most of the time during emergency situations, the amount of shortage in a distribution system would be too high for the PCs to be able to fully compensate the shortage. This is because generation is usually not part of the core competencies of the PCs. Resilience must then be explicitly considered because there may be a trade-off between the minimization of cost, and the minimization of the shortage. Therefore, this paper focuses on creating a mechanism that considers both the allocation of the amount of energy each PC must generate through a least-cost optimal dispatch, or by increasing grid resiliency. This is done through a modified optimal energy dispatch model for the distributed generators of the PCs [@elec-market]. The rest of the paper is organized as follows: in Section II, we review the different DR mechanism with the goal of optimizing social welfare through profit [@MWG]. In Section III, we develop the agent model and make assumptions on their cost functions. In Section IV and V, we formulate the optimal dispatch models, and analyze their properties. In Section VI, we show a simulation on the proposed model. We conclude this paper in Section VII. LITERATURE REVIEW ================= DR techniques are used to shift consumption patterns as a solution to reduce energy shortage [@DSM; @DR-berk]. The drawback to using DR is that these methods do not necessarily reduce energy consumption as shifting demand may lead to profiles with new peaks [@DSM]. To fix this problem, DR methods can also involve methods that induce consumers to generate their own energy [@DR-berk]. There are two kinds of DR strategies that are currently being used: incentive-based and time-based DR [@DR-berk]. Incentive-based DR schemes refer to programs where DUs give incentives to customers to influence consumption and generation, while time-based DR schemes refer to mechanisms where DUs modify prices throughout the day to influence consumption patterns [@DR-over]. Work has been done to discuss a competitive market model for demand response assuming firms are price-takers under a profit-maximization model [@RRL1; @RRL2]. Gadham and Ghose show how social welfare, as the sum of consumer and producer surplus, is affected when the equilibrium price change through a sensitivity analysis [@RRL3]. Chen and Low model social welfare maximization through a residential demand response scheme where each household operates different appliances [@RRL4]. Nijhuis, et al. implemented a demand response program at a franchise level using the elasticity of aggregated demand [@RRL5]. Also, we see that Sabounchi, et al. defined social welfare as the preferences of a consumer using the aggregate energy consumed of all devices in the household [@RRL6]. Chen, et al. created two market models for DR under the assumption of a perfect competition, and an oligopoly [@RRL7]. Also, Su, and Kirschen quantify how social welfare, defined as the sum of the utility functions, is affected as the number of participating customers (PC) increases [@RRL8]. Common among the incentive-based schemes is that DUs contract to affect consumption rather than explicitly contract to generate energy. Another issue with most DR mechanisms is that they mostly consider social welfare through profit optimization. In cases where there are SSOs, this objective may lead to the ineffective implementation of the program because of a possible trade-off between cost and resilience. With the Philippines, since there are a lot commercial establishments have their own distributed generators, it is more efficient to contract with the latter case. With that in mind, it is also not explicitly mentioned how the DU optimally allocates the amount of shortage each PC is required to generate. Therefore, we focus our attention on how the DU shall allocate the amount of shortage through a modified optimal dispatch [@elec-market]. Traditionally, optimal dispatch problems have the objective of minimizing the cost to buy a certain amount of power from the generation sector [@elec-market]. Barley and Winn study choosing an optimal energy mix with corresponding dispatch strategies for microgrids [@RRL9]. Traditionally, optimal dispatch problems are implemented for generators competing in the transmission system [@RRL10]. Also, it is standard to model dispatch problems as a deterministic model, but Viviani and Heydt model stochastic optimal dispatch problems where the parameters are random variables, and the agents are risk-averse [@RRL11]. Research has also been done by Baran and El-Markabi to model optimal dispatch in distribution systems with the objective of minimizing the total reactive power dispatched to regulate the voltage in distribution feeders [@RRL12]. Also, DallAnese, et al. implemented the optimal dispatch of photovoltaic systems in residential distribution systems [@RRL13]. Common among all the optimal dispatch models is the knowledge that the market is operating with a business as usual. Since we are dealing with SSOs, the objective of the DU in the dispatch may change drastically. Therefore in our case, we will study whether it is still efficient to implement a least-cost optimal dispatch when resilience can also be a big issue in cases when there are SSOs. AGENT MODEL =========== Agent Cost Function ------------------- In this section, we discuss the preliminary definitions and assumptions for the model. We begin by defining the costs for each PC. For a set of PCs $\{1,...,N\}$ where the PC $i \in \{1,...,N\}$, Let $\theta_i \in \Theta_i$ denote the unit price of generating $\bar{e}_i \in E_i \subseteq \mathbb{R}_{\geq 0}$ amount of energy for the PC. We define the cost of the PC $i$ to generate $\bar{e}_i$ to be $C_i: E_i \times \Theta_i \rightarrow \mathbb{R}_{\geq 0}$. In a standard optimal dispatch problem, this is the cost to produce $\bar{e}_i$ amount of energy for PC $i$. The goal of the DU is to minimize the overall cost to society as mandated by the system regulator. We state the assumed properties of the cost function model for each PC to generate his own electricity. We base this on the standard cost model [@MWG] used in microeconomics. \[cost-assumption\] The generator cost function $C_i(\bar{e}_i, \theta_i) \in C^2$ is continuous, twice differentiable[^2], and monotonically increasing in $\bar{e}_i$ and $\theta_i$, while convex in $\bar{e}_i$ and concave in $\theta_i$. This assumption is reasonable as most distributed generators do have a convex cost function [@elec-market]. Lastly, we make define the DU demand. The total DU demand is $\bar{e} \in \mathbb{R}_{\geq 0}$. This implies that the DU demand $\bar{e}$ is an external variable that is dependent on the nature of the SSO. In the economics literature, this means that the $\bar{e}$ is exogenous [@MWG]. Agent Generator Constraints --------------------------- In this section, we characterize the capacity of the generators being operated by the PC. In a market setting, it is sufficient to characterize the generator using an inequality constraint [@elec-market]. The PC has a maximum average power consumption of $P^{max}_i \in \mathbb{R}_{> 0}$. This can be justified because PCs generally have finite average power consumption. Since most ILPs last only a few hours, the amount of energy they would need to generate is finite. Next we need to assume the minimum power consumption each generator shall take. In the literature [@elec-market], it is usually assumed that each generator generates a minimum of $0W$. In our case, generating $0W$ is akin to not participating in the contract at all, and is undesirable when one signs up for ILP. The PC is always required to join the program and generate energy with a minimum average power consumption of $ P^{min}_i > 0$ where $P^{min}_i \in \mathbb{R}$ and $P^{max}_i > P^{min}_i$. Since ILP programs mostly have energies as an input instead of power [@ILP], the next remark combines both the previous assumptions. The PC satisfies the inequality constraint $0 < P^{min}_i \leq \frac{\bar{e}_i}{T} \leq P^{max}_i$ for the optimal dispatch problem. In this model, $\frac{\bar{e}_i}{T} = P_i$ where $P_i$ is assumed to be the average power consumed during the ILP. Cost-Based Optimal Dispatch Model ================================= In this Section, we discuss the least-cost based model for being able to do optimal energy dispatch. Market Clearing Constraint -------------------------- First, we state an assumption[^3] on the nature of the supply of the DGs. \[market-clearing-assumption\] The PCs can always supply as much as the DU demands. The previous assumption is only valid when the sum of the maximum generator constraints satisfy the following relationship: $$\label{market-clearing} T\sum_{i=1}^{N}P_i^{max} \geq \bar{e}.$$ This is a strong assumption because it is not always the case that the DGs of the PCs can supply enough to clear the shortage. We will see in the section V that if this property does not hold, the feasibility set in the optimization problem is empty. With this, the market clearing constraint [@MWG] is $$\label{market-clear} \sum_{i=1}^N\bar{e}_i = \bar{e}.$$ The market clearing constraint implies that any combination of $\bar{e}_i$ $\forall i$ can be used as long as it satisfies . We call this homogeneity since each $\bar{e}_i$ can be replaced by any $\bar{e}_j$ for $j \neq i$ [@MWG]. Cost Optimization Problem ------------------------- We base our optimization problem on a dispatch that minimizes the total cost to generate $\bar{e}$. Therefore, with the market clearing constraint and the generation constraints, the optimization problem is $$\label{standard_op_prob} \min_{[\bar{e}_1, \bar{e}_2, ..., \bar{e}_N]} \sum_{i=1}^{N}C_i(\bar{e}_i, \theta_i).$$ subject to: $$P^{min}_i \leq \frac{\bar{e}_i}{T} \leq P^{max}_i \text{ } \forall i$$ $$\sum_{i=1}^N\bar{e}_i = \bar{e}.$$ The next Lemma shows when the problem is feasible with a global optimal solution [@elec-market]. \[lemma1\] If $T\sum_{i=1}^{N}P_i^{max} \geq \bar{e}$ and Assumption \[cost-assumption\] is satisfied, then there exists a global optimal solution, $[\bar{e}_1, ..., \bar{e}_N]^T$, that satisfies . Intuitively, this means that the sum of the maximum capacities must be greater than the demand for there to be a solution. If the premise is not satisfied, then no combination of $[\bar{e}_1, ..., \bar{e}_N]^T$ will satisfy the market clearing constraint. We treat this case in the next section. Modified Optimal Dispatch Model for Emergency Situations ======================================================== In reality, such as in emergency situations due to SSOs, the amount of shortage is almost always too high such that the total amount of energy the PCs generate is not enough to cover the shortage. With that, In this section, we analyze this case where Lemma \[lemma1\] because Assumption \[market-clearing-assumption\] is not satisfied. More concisely, we consider the case where the DGs don’t have the capacity to clear the shortages. To preempt the results from this section, we show that the trade-off between resilience and cost is due to the inefficiency of the least-cost objective in minimizing the shortage. This is due to Assumption \[cost-assumption\]. With that, we form a multi-objective optimal dispatch problem that can characterize this trade-off. Market Clearing Constraint Breakdown ------------------------------------ The next Lemma formally states what happens when Assumption \[market-clearing-assumption\], which assures Lemma \[lemma1\], does not hold. \[lemma2\] If $T\sum_{i=1}^{N}P_i^{max} < \bar{e}$, then the feasible set that satisfies is empty. This just means that if the individual maximum capacities don’t add up to the shortage, then there is no way the market clears. This is an issue because Lemma \[lemma2\] implies there is no feasible solution to the optimal dispatch problem. We combine Lemma \[lemma1\] and Lemma \[lemma2\] to produce the following Theorem. \[theorem1\] $T\sum_{i=1}^{N}P_i^{max} \geq \bar{e}$ iff $\exists$ a $[\bar{e}_1, ..., \bar{e}_N]^T$ that satisfies and Assumption \[cost-assumption\]. Theorem \[theorem1\] provides a necessary and sufficient condition for the feasibility of . When considering emergency situations where supply of energy from the PCs can’t meet demand, as formalized in Theorem \[theorem1\], we quantify this insight by relaxing the market cleating constraint with $$\label{ineq-market-clearing} \sum_{i=1}^N\bar{e}_i \leq \bar{e}.$$ In this case where supply can’t meet demand, the aggregate energy supply the PCs produce will always be less than $\bar{e}$. Least-Cost Objective Trade-Off ------------------------------ As stated previously, we are dealing with the problem of minimizing the total cost of the $N$ PCs subject to the relaxed market clearing constraint. An issue here is that minimizing the cost that is monotonically increasing subject to leads to a trade-off with grid resiliency - a big objective of DR. \[theorem2\] If $\exists$ a $[\bar{e}_1, ..., \bar{e}_N]^T$ that satisfies while relaxing the market clearing constraint, then $\bar{e}_i = TP_i^{min}$. This is because with a monotonically increasing cost function found in Assumption \[cost-assumption\], $[\bar{e}_1, ..., \bar{e}_N]^T$ would minimize the total cost by dispatching the minimum amount of energy, $TP_i^{min}$, the PCs can generate while in the program. This result is troublesome as the ultimate goal of a DR scheme during SSOs is to minimize the total shortage. Using a least-cost objective would necessarily decrease the system’s resilience, and would render any incentive-based DR scheme for energy generation during emergency situations useless. Therefore, we consider adding to the least-cost objective the objective of minimizing the shortage in the sequel. Resilience Objective -------------------- Since the DU cannot tell PCs to generate sufficient energy, we add a second objective to maximize the amount of energy generated. This is equivalent to minimizing the shortage during the SSO. Therefore, we consider another objective $$\label{new-obj} \max_{[\bar{e}_1,...,\bar{e}_i]}\sum_{i=1}^N\bar{e}_i.$$ We characterize the solution to this problem in the next Remark. \[lemma3\] The optimization of with the same constraints as has multiple optimizers if the generator constraints for each PC $i$ are not all tight at the optimal solution of the new problem. Therefore, for cases where supply of energy is greater than demand, the DU still has to decide which optimal solution it will use for the allocation. In this case, we recommend the DU to use the standard least-cost dispatch. Multi-Objective Optimal Dispatch -------------------------------- Lastly, because of the trade-off between least cost and least shortage, DUs generally don’t want to focus a big chunk their resources in minimizing their energy due to a loss of revenue. Therefore, we formulate the modified optimal dispatch problem as a multi-objective convex optimization problem [@Convex-Optimization]. With the objectives and the constraints laid out, the generalized optimal dispatch problem, after regularization [@Convex-Optimization], is then $$\label{relaxed-optimization} \min_{[\bar{e}_1, ..., \bar{e}_i]}\lambda \sum_{i=1}^{N}C_i(\bar{e}_i, \theta_i) -(1-\lambda)\sum_{i=1}^N\bar{e}_i.$$ subject to: $$P^{min}_i \leq \frac{\bar{e}_i}{T} \leq P^{max}_i \text{ } \forall i$$ $$\sum_{i=1}^N\bar{e}_i \leq \bar{e}.$$ Since the model is convex [@Convex-Optimization], all we need to do is prove that there is a feasible set to state a necessary and sufficient condition for global optimality. \[theorem3\] The feasible set of is non-empty. Notice how the regularization of the problem differs slightly from the standard Tikhonov Regularization [@Convex-Optimization]. This has some analytic and interpretive significance. The solution to the optimal dispatch problem is going to be a Pareto Frontier [@Convex-Optimization] from $\lambda \in [0,1]$ instead of $\lambda \in \mathbb{R}_{\geq 0}$. The value of $\lambda$ then could be interpreted as the ratio of how much the DU should prioritize ones cost objective over the resilience objective. If $\lambda =0$ then the firm prioritizes resilience. The cost objective is prioritized when $\lambda = 1$. With this framework, firms can do a sensitivity analysis on $\lambda$ to see whether or not they should opt to prioritize minimizing the shortage or minimizing their costs. If we characterize the Pareto Frontier of the optimal solution, we are able to see the effectiveness of the program when one prioritizes one cost over resilience. APPLICATION AND SIMULATION RESULTS ================================== In this section, we provide a simple analysis of the claim above. Consider the quadratic fuel cost function . We consider three cases for this application: 1. $\lambda$ Sensitivity Analysis 2. Cost-Based Optimal Dispatch Assumed Cost Function --------------------- In general, any cost function that satisfies the properties found in any standard graduate microeconomic theory textbook [@MWG] will work, but as is usual for power engineering, we modify the standard quadratic fuel cost curve for the purpose of application [@elec-market]. Assuming the generator generates an average power $\frac{\bar{e}}{T}$, the fuel cost function is $$\label{fuel-cost} C_i(\bar{e}_i, \theta_i)=T[a_2\theta_i(\frac{\bar{e}_i}{T})^2 + a_1\theta_i(\frac{\bar{e}_i}{T}) + a_0\theta_i]$$ for some cost parameters $a_2, a_1, a_0 \in \mathbb{R}, a_2 > 0$, and length of shortage $T$. The quadratic component corresponds to all nonlinear variable costs present due to system nonlinearities such as starting and stopping the generator. The linear term is a variable cost that represents the cost of operating and maintaining the generator at steady-state. The $a_0$ term represents all fixed costs [@elec-market]. Optimal Dispatch Simulation Parameters -------------------------------------- We consider implementing the analysis of the optimal dispatch problems with 5 PCs. We set $P_i^{min}=30 kW$ $\forall i$. The maximum average generation for each PC to be $P^{max}=[60,100, 125, 85, 130]^T kW$. We consider an ILP where $T=1 hr$. We consider the generator parameters given in Table \[gen-par\] [@gen-parameters]. \[ht\] ------------------------ ---------- ----------- ---------- ------------ ---------- Parameter PC 1 PC 2 PC 3 PC 4 PC 5 \[0.5ex\] $a_0 \theta$ $96.6$ $96.6046$ $96.279$ $100.3937$ $95.856$ $a_1 \theta$ $7.588$ $7.5874$ $7.592$ $6.9761$ $7.374$ $a_2 \theta$ $0.0414$ $0.0414$ $0.042$ $0.0533$ $0.047$ \[1ex\] ------------------------ ---------- ----------- ---------- ------------ ---------- : Generator Parameters \[gen-par\] $\lambda$ Sensitivity Analysis ------------------------------ We consider $P_i^{min}=30kW$ $\forall i$, and $\bar{e}=700kWhr$ because $700 kWhr > 500kWhr = T\sum P_i^{max}$ In this case, we consider how the dispatch changes when the parameter $\lambda$ changes. Figure 1 shows how the dispatch is sensitive to $\lambda$. \[ht\] \[dispatch\_700\] ![PC Relaxed Optimal Dispatch Sensitivity With Respect to $\lambda$](dispatch_700.jpg "fig:"){width="\linewidth"} We observe that the dispatch remains constant until around $\lambda=0.05$. This implies that the effect of minimizing costs starts to dominate even when the DU prioritizes maximizing energy generated more. In fact, the cost objective totally dominates when $\lambda \geq 0.09$, which is less than $\lambda=0.5$ where the DU prioritizes the two objectives equally. Therefore, when an SSO occurs during an emergency situation, the DU must decide to prioritize maximizing energy generated more than minimizing the total cost to generate. Cost-Based Optimal Dispatch --------------------------- For cases where the supply of electricity is greater than the demand due to a shortage, we use Remark \[lemma3\] to justify using a Cost-Based Optimal Dispatch. We consider checking the sensitivity of the Strict Dispatch from $\bar{e}=150kWhr$ to $\bar{e}=500kWhr$. $150kWhr$ was chosen because the $\sum P_i^{min} = 150kW.$ Figure 2 shows how the optimal dispatch changes when the total shortage increases from $150kWhr$ to $500kWhr$. \[ht\] \[sensitivity\_DP\] ![PC Dispatch Sensitivity With Respect to $\bar{e}$](sensitivity_requested.jpg "fig:"){width="\linewidth"} It can be seen that the dispatch monotonically increases with respect to $\bar{e}$ until one PC’s full capacity is reached. We observe the behavior that when firm $j$ reaches its full capacity, the rate of change of the dispatch $\forall i \neq j$ increases. This implies that the DU demands more from the other PCs marginally when PC $i$ reaches full capacity. CONCLUSION AND FUTURE WORK ========================== In this paper, we develop an optimal dispatch model for allocating generation during supply shortage outages (SSO) in emergency situations. This is for the development of the Interruptible Load Program. We develop a model for optimal dispatch for SSOs by modifying the least-cost optimal dispatch found in standard literature [@elec-market]. Using the modified model, we have found that when the supply of energy provided by the contracted establishments is less than the shortage demand, there is a trade-off between the resilience of the dispatch with the cost. This is because using the least-cost approach may lead to the inefficient implementation of the program since grid resilience will increase minimally. Mathematically, this inefficiency is caused by the reasonable assumption that costs are monotonically increasing in the amount of energy. Characterizing this inefficiency is crucial for the interruptible load program as grid resilience should be of top priority during emergency situations. Though, for special cases where all the total number of PCs can cover the energy shortage, we recommend using a least-cost approach as it will not lead to the inefficient implementation of the mechanism. In this model, we have considered PCs that are taken to be price-takers. Also, in this model, we have assumed that the generation prices and valuations are known such that the PCs can’t cheat the system by raising prices. Lastly, the model breaks down in the case where the cost is not a linear function of price [@MWG]. In future studies, we consider energy with significant market power such that they can influence the price. We also consider the case where the energy valuations of each PC is unknown. This can be done by implementing through either a uniform or discriminatory price auction. Finally, we consider models where the costs are not a linear function of the unit price commonly found in contract models [@MWG]. APPENDIX {#appendix .unnumbered} ======== In this section, we provide the proofs for Lemma \[lemma1\], Lemma \[lemma2\], Theorem \[theorem2\], Lemma \[lemma3\], and Theorem \[theorem3\]. Theorem \[theorem1\] is not included as it is a direct consequence of Lemmas \[lemma1\] and \[lemma2\]. Let $X_i =[P_i^{min}, P_i^{max}] \in \mathbb{R}_{> 0}$. Since $P_i^{max} = \sup\{X_i\}$, $\sum_{i=1}^{N}TP_i^{max} \geq \bar{e}$, and $P_i^{max} \geq 0$, take $\bar{P}_i \geq 0$ $\forall i$ s.t. $\sum_{i=1}^{N}T(P_i^{max} - \bar{P}_i) = \bar{e}$. Since $P_i^{max} - \bar{P}_i \in X_i$, then $\exists$ $[x_1 \in X_1, x_2 \in X_2,...,x_N \in X_N]^T$ that satisfies . Additionally, since Assumption \[cost-assumption\] implies that the problem is convex [@Convex-Optimization], then the solution is a global optima. Let $X_i =[P_i^{min}, P_i^{max}] \subset \mathbb{R}_{> 0}$. Since $\sum_{i=1}^{N}TP_i^{max} < \bar{e}$, and $P_i^{max} = \sup\{X_i\}$, then any $x_i \in X_i$ $\forall i$ will lead to $\sum_{i=1}^{N}Tx_i < \sum_{i=1}^{N}TP_i^{max} < \bar{e}$. Therefore, $\nexists$ $[x_1 \in X_1, x_2 \in X_2,...,x_N \in X_N]^T$ that satisfies . Therefore, the feasible set of is empty. Let $X_i =[P_i^{min}, P_i^{max}] \in \mathbb{R}_{> 0}$. Since the cost function is monotonically increasing, $\arg\min_{\bar{e}} C_i(\bar{e}_i, \theta_i)$ subject to the minimimum generation constraint is $TP_i^{min}.$ Restating the contrapositive, we have “If the solution to replacing the objective with is unique, then the generator constraints for each PC $i$ are all tight in the new problem.” We prove this by contradiction. Let $\mathbf{e} = [\bar{e}_{11}, ..., \bar{e}_{1i}]$ be an optimal solution to the problem. Let $x \in \mathbb{R}_{\geq 0}$ where $\min\{\mathbf{e}\} < x < \max\{\mathbf{e}\}$ s.t. $\min\{\mathbf{e}\} + x$ is less than $TP^{max}_j$ for its corresponding $j \in \{1,...,N\}$. This is possible by the completeness of the reals. We can take the vector $\mathbf{e}^* = [\bar{e}_{11}, ..., \bar{e}_{1i}]$ s.t. $\max\{\mathbf{e}^\} = \max\{\mathbf{e}\} - x$, and $\min\{\mathbf{e}^\} = \min\{\mathbf{e}\} + x$. Since this is also an optimal solution, we have a contradiction on uniqueness - proving the contrapositive. $\perp$ Let $X_i =[P_i^{min}, P_i^{max}] \in \mathbb{R}_{> 0}$. Take $\frac{\bar{e}_i}{T}= \frac{P_i^{max}+P_i^{min}}{2}$. Thus, $\sum_{i=1}^{N}\frac{\bar{e}_i}{T}=\sum_{i=1}^{N}\frac{P_i^{max}+P_i^{min}}{2} \leq \sum_{i=1}^{N}P_i^{max}$. Therefore, since $\sum_{i=1}^{N}\frac{\bar{e}_i}{T}\leq \sum_{i=1}^{N}P_i^{max}$, then $\sum_{i=1}^{N}\bar{e}_i\leq \sum_{i=1}^{N}TP_i^{max}=\bar{e}$. Therefore, The feasible set is non-empty. ACKNOWLEDGMENT {#acknowledgment .unnumbered} ============== This work was done through the financial support from the Philippine Commission of Higher Education under the Philippine-California Advanced Research Institution program. [99]{} P. L. Viray, “Gov’t-owned companies expected to join Interruptible Load Program,” philstar.com. P. Palensky and D. Dietrich, “Demand Side Management: Demand Response, Intelligent Energy Systems, and Smart Loads,” IEEE Transactions on Industrial Informatics, vol. 7, no. 3, pp. 381–388, Aug. 2011. J. Dudley and M. Piette, “Solutions for Summer Electric Power Shortages: Demand Response and Its Application in Air Conditioning and Refrigerating Systems,” Refrigeration, Air Conditioning, amp; Electric Power Machinery, vol. 29, pp. 1–4, 2008. US Department of Energy, “Energy policy Act of 2005.” . D. R. Biggar and M. Hesamzadeh, The economics of electricity markets. Chichester, West Sussex, United Kingdom: Wiley, 2014. A. Mas-Colell, M. D. Whinston, and J. R. Green, Microeconomic theory. New York, NY: Oxford Univ. Press, 1995. M. H. Albadi and E. F. El-Saadany, “Demand Response in Electricity Markets: An Overview,” in 2007 IEEE Power Engineering Society General Meeting, 2007, pp. R. N. Boisvert and B. F. Neenan, “Social Welfare implications of demand response programs in competitiv e electricity markets,” Lawrence Berkeley National Laboratory, Aug. 2003. Q. Dong, L. Yu, W. Z. Song, L. Tong, and S. Tang, “Distributed Demand and Response Algorithm for Optimizing Social-Welfare in Smart Grid,” in 2012 IEEE 26th International Parallel and Distributed Processing Symposium, 2012, pp. 1228–1239. K. R. Gadham and T. Ghose, “Importance of social welfare point for the analysis of demand response,” in 2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI), 2016, pp. 182–185. N. Li, L. Chen, and S. H. Low, “Optimal demand response based on utility maximization in power networks,” in 2011 IEEE Power and Energy Society General Meeting, 2011, pp. 1–8. M. Nijhuis, M. Babar, M. Gibescu, and S. Cobben, “Demand Response: Social Welfare Maximization in an Unbundled Energy Market Case Study for the Low-Voltage Networks of a Distribution Network Operator in The Netherlands,” IEEE Transactions on Industry Applications, vol. 53, no. 1, pp. 32–38, Jan. 2017. M. Sabounchi, H. Khazaei, and S. K. H. Sani, “A new residential demand response management method based on a social welfare optimization framework,” in 2014 2nd International Conference on Electrical, Electronics and System Engineering (ICEESE), 2014, pp. 146–151. L. Chen, N. Li, S. H. Low, and J. C. Doyle, “Two Market Models for Demand Response in Power Networks,” in 2010 First IEEE International Conference on Smart Grid Communications, 2010, pp. 397–402. C. L. Su and D. Kirschen, “Quantifying the Effect of Demand Response on Electricity Markets,” IEEE Transactions on Power Systems, vol. 24, no. 3, pp. 1199–1207, Aug. 2009. C. Dennis Barley and C. Byron Winn, “Optimal dispatch strategy in remote hybrid power systems,” Solar Energy, vol. 58, no. 4, pp. 165–179, Oct. 1996. R. S. Fang and A. K. David, “Optimal dispatch under transmission contracts,” IEEE Transactions on Power Systems, vol. 14, no. 2, pp. 732–737, May 1999. G. L. Viviani and G. T. Heydt, “Stochastic Optimal Energy Dispatch,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 7, pp. 3221–3228, Jul. 1981. M. E. Baran and I. M. El-Markabi, “A Multiagent-Based Dispatching Scheme for Distributed Generators for Voltage Support on Distribution Feeders,” IEEE Transactions on Power Systems, vol. 22, no. 1, pp. 52–59, Feb. 2007. E. Dall’Anese, S. V. Dhople, and G. B. Giannakis, “Optimal Dispatch of Photovoltaic Inverters in Residential Distribution Systems,” IEEE Transactions on Sustainable Energy, vol. 5, no. 2, pp. 487–497, Apr. 2014. D. Thimsen, “Interruptible Power Rates and Their Role in Utility Distributed Resources Programs,” 20-Feb-2003. \[Online\] S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge, UK ; New York: Cambridge University Press, 2004. S. Sayah and A. Hamouda, “Novel application of differential evolution algorithm for estimating fuel cost function of thermal generating units,” in 2015 Third World Conference on Complex Systems (WCCS), 2015, pp. 1–6. [^1]: Or resiliency. [^2]: $C^2$ is the space of continuous functions with 2nd derivatives [^3]: Only valid in this section.
--- author: - Wieland Brendel - Matthias Bethge bibliography: - 'main.bib' date: 'Dated: ' title: 'Comment on Biologically inspired protection of deep networks from adversarial attacks' --- A recent paper [@1703.09202] suggests that Deep Neural Networks can be protected from gradient-based adversarial perturbations by driving the network activations into a highly saturated regime. Here we analyse such saturated networks and show that the attacks fail due to numerical limitations in the gradient computations. A simple stabilisation of the gradient estimates enables successful and efficient attacks. Thus, it has yet to be shown that the robustness observed in [@1703.09202] is not simply due to numerical limitations. Evaluating the robustness of neural networks is difficult. One core reason is the ambiguity between network robustness and deficiencies of the adversarial attack: a network might just appear robust because the core assumptions of the chosen attacker are not met. This is particularly obvious for gradient-based adversarial attacks which inadvertently rely on stable gradient estimates. Even in cases without gradients, however, many other adversarial attacks may still succeed, including attacks that do not use gradient information like [@clune; @papernot] or gradient-based attacks using estimates of the gradient. A recent paper [@1703.09202] suggests that highly saturated deep neural networks (DNNs) might be robust against gradient-based adversarial perturbations. We here show that the observed robustness is likely a side-effect of numerical limitations that arise in the high-saturation limit and which prevent stable computations of the gradient. These limitations can be lifted by a simple and stable estimate of the gradients. In a first step we tried to reproduce the results of [@1703.09202] by training a three-layer Multi-layer Perceptron (MLP) together with the proposed saturation penalty. This penalty pushes the hidden-layer activations of the network into the saturated parts of the non-linearity (zero and one for sigmoid, zero for ReLU). We report the classification accuracy for both the vanilla and the saturated network for normal and adversarial images in Table \[table:accuracy\]. In agreement with [@1703.09202] the performance of the MLP with and without penalty is almost identical, but the robustness to a simple adversarial attack with the fast-gradient sign method (FGSM) drastically increases for the latter. We also verified that the weight and activation distribution qualitatively match the results in [@1703.09202], Figure \[fig:parameterDistribution\]. ----------- ------- ------------ ------------- ------- ------------ ------------- Plain naive FGSM stable FGSM Plain naive FGSM stable FGSM Vanilla 98% 2,5% - 98,7% 0,2% - Saturated 97,2% 96,6% 1,7% 98,1% 98,0% 8,4% ----------- ------- ------------ ------------- ------- ------------ ------------- : \[table:accuracy\]Adversarial robustness of vanilla and saturated sigmoidal and ReLU MLPs. While the naive application of FGSM fails to generate suitable adversarial examples, a slightly modified FGSM based on a more stable gradient estimate is still highly successful. In highly saturated networks the gradients of the loss with respect to the input are either exactly zero or numerically unstable. Using these unreliable values from the saturated network directly without numerical stabilisation is likely to fail. In Figure \[fig:gradients\] we plot the distribution of elements of the gradients. In the saturated network more then 98,2% of the gradient elements are exactly zero, compared to none in the vanilla network. At the same time, all non-zero elements are sixteen orders of magnitude smaller the gradient elements of the vanilla network, suggesting that the residual gradients of the saturated network are due to rounding errors or are susceptible to numerical instabilities. For 97,9% of the images exactly all elements of the gradient are zero. In this case FGSM applies absolutely no perturbation to the corresponding image and the attack is inadvertently unsuccessful. For all other images the FGSM attack was successful in 62% of the cases. The overall success rate of FGSM is directly related to the number of zero-valued gradients. In Figure \[fig:gradient\_attack\] we plot the success of FGSM and the ratio of non-zero gradients as a function of the gain. For gains below $10^{-3}$ FGSM is highly successful (as evaluated on the saturated sigmoid MLP with gain 1). For larger gains, however, zero-valued gradients start to appear. Unsurprisingly, the success rate of FGSM strongly decreases with the number of zero-valued gradients. ![\[fig:gradients\]Histogram over the elements of the gradients of the input image with respect to the cross-entropy loss (the direction of the adversarial perturbation) for both the vanilla sigmoid MLP (left) and the saturated sigmoid MLP (right). In the saturated network more then 98% of the gradient elements are exactly zero while the rest is sixteen orders of magnitude smaller then in the vanilla network.](gradients){width="80.00000%"} It is straight-forward to attack saturated networks through a simple trick that allows more stable gradient estimates. To this end we note that in highly saturated networks a modest reduction in the gain of the sigmoids barely change the activations (they will still be close to zero and one) but can significantly increase the numerical stability of the gradient estimates. We then use this gradient to generate an adversarial example according to the usual FGSM procedure and use it as an input to the original saturated network. As can be seen in Table \[table:accuracy\], this simple modification of FGSM is highly successful in fooling the saturated network. For saturated ReLU MLPs we observed a saturation of the softmax and devised a similar attack by down-scaling the activations of the readout layer (Table \[table:accuracy\]). ![\[fig:gradient\_attack\] The success of the FGSM attack clearly reflects the ratio of non-zero gradients. Networks with different gain are only used to generate adversarial images using the FGSM method. The accuracy by which the adversarials fool the saturated network (gain = 1) is plotted in red. The success of FGSM is highly correlated with the ratio of non-zero gradients (black).](gradient_attack){width="80.00000%"} Taken together, we demonstrated that the robustness observed in [@1703.09202] likely originates from the numerical instabilities of gradient computations in highly saturated networks. A simple stabilisation of the gradient computations still allow standard gradient-based adversarial attack methods to succeed. While we cannot say with certainty that the same attack succeeds for the networks analysed in [@1703.09202] (without access to the source code we cannot exclude unintended differences in the implementation and the network parameters), it has yet to be shown that the robustness observed in [@1703.09202] does not originate from numerical limitations. More generally, our findings highlight the critical importance of choosing the most suitable methods to challenge the robustness of the network. [1]{} ![\[fig:parameterDistribution\] (a) Sigmoid MLP Weight and activation distribution for both the vanilla (top) and saturated (bottom) network. We observe a qualitatively similar increase in the kurtosis of the weights and the bimodality of the activations as in [@1703.09202]. (b) ReLU MLP Same as (a) but for ReLU nonlinearities. Similar to [@1703.09202] the activations are not bimodal as in the sigmoid MLP but feature a high kurtosis.](sigm_weights_activations "fig:"){width="1\linewidth"} [1]{} ![\[fig:parameterDistribution\] (a) Sigmoid MLP Weight and activation distribution for both the vanilla (top) and saturated (bottom) network. We observe a qualitatively similar increase in the kurtosis of the weights and the bimodality of the activations as in [@1703.09202]. (b) ReLU MLP Same as (a) but for ReLU nonlinearities. Similar to [@1703.09202] the activations are not bimodal as in the sigmoid MLP but feature a high kurtosis.](relu_weights_activations "fig:"){width="1\linewidth"}
--- author: - \| Pedro Ferreira [^1], and R. Santos [^2]\ Centro de Física Teórica e Computacional,\ Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal\ date: 'January, 2006' title: Contributions from flavour changing effective operators to the physics of the top quark at LHC --- Introduction ============ The Large Hadron Collider (LHC) is starting its operation in 2007. In the low luminosity run, production of around 8 million top quark pairs per year can be anticipated. That is why the LHC is considered the ideal laboratory to study the heaviest of all known particles. Recently [@nos; @nos2] we undertook a model-independent study of possible new physics effects on the phenomenology of the top quark. Following reference [@buch] we considered a set of dimension six effective operators and analyzed its impact on observable quantities related to the top quark, such as its width or the cross section for single top quark production at the LHC. Due to the large number of arbitrary coupling constants, we have excluded the ones with little or no impact on phenomena occurring at energy scales inferior to the LHC’s. This framework of effective lagrangians has been widely used to study the top particle [@whis; @saav; @fcnc; @sola; @liu; @4f]. Let us remark that our philosophy in [@nos; @nos2] was also somewhat different from that of most previous works in this field, in that we presented, whenever possible, analytical expressions. Our aim was, and is, to provide our experimental colleagues with formulae they can use directly in their Monte Carlo simulations. Effective operator formalism {#sec:eff} ============================ The effective operator approach is based on the assumption that, at a given energy scale $\Lambda$, physics effects beyond those predicted by the SM make themselves manifest. We describe this by assuming the lagrangean $${\cal L} \;\;=\;\; {\cal L}^{SM} \;+\; \frac{1}{\Lambda}\,{\cal L}^{(5)} \;+\; \frac{1}{\Lambda^2}\,{\cal L}^{(6)} \;+\; O\,\left(\frac{1}{\Lambda^3}\right) \;\;\; , \label{eq:l}$$ where ${\cal L}^{SM}$ is the SM lagrangean and ${\cal L}^{(5)}$ and ${\cal L}^{(6)}$ are all of the dimension 5 and 6 operators which, like ${\cal L}^{SM}$, are invariant under the gauge symmetries of the SM. The ${\cal L}^{(5)}$ terms break baryon and lepton number conservation, and are thus not usually considered. This leaves us with the ${\cal L}^{(6)}$ operators, some of which, after spontaneous symmetry breaking, generate dimension five terms. The list of dimension six operators is quite vast [@buch], therefore some sensible criteria of selection are needed. Underlying all our work is the desire to study a new possible type of physics, flavour changing strong interactions. The first criterion is to choose those ${\cal L}^{(6)}$ operators that have no sizeable impact on low energy physics (below the TeV scale, say). Another criterion was to only consider operators with a single top quark, since we will limit our studies to processes of single top production. Finally, we will restrict ourselves to operators with gluons, or four-fermion ones. No effective operators with electroweak gauge bosons will be considered. The gluon operators that survive these criteria are but two, which, in the notation of ref. [@buch], are written as $$\begin{aligned} {\cal O}_{uG} &= \;\;i\,\frac{\alpha_{ij}}{\Lambda^2}\,\left(\bar{u}^i_R\, \lambda^a\, \gamma^\mu\,D^\nu\,u^j_R\right)\,G^a_{\mu\nu} \nonumber \vspace{0.2cm} \\ {\cal O}_{uG\phi} &= \;\;\frac{\beta_{ij}}{\Lambda^2}\,\left(\bar{q}^i_L\, \lambda^a\, \sigma^{\mu\nu}\,u^j_R\right)\,\tilde{\phi}\,G^a_{\mu\nu} \;\;\; . \label{eq:op}\end{aligned}$$ $q_L$ and $u_R$ are spinors (a left quark doublet and up-quark right singlet of $SU(2)$, respectively), $\tilde{\phi}$ is the charge conjugate of the Higgs doublet and $G^a_{\mu\nu}$ is the gluon tensor. $\alpha_{ij}$ and $\beta_{ij}$ are complex dimensionless couplings, the $(i,j)$ being flavour indices. According to our criteria, one of these indices must belong to the third generation. After spontaneous symmetry breaking the neutral component of the field $\phi$ acquires a vev ($\phi_0\,\rightarrow\,\phi_0\,+\,v$, with $v\,=\, 246/\sqrt{2}$ GeV) and the second of these operators generates a dimension five term. The lagrangean for new physics thus becomes $$\begin{aligned} {\cal L}\;\; =&\;\;\; \alpha_{tu}\,{\cal O}_{tu}\;+\; \alpha_{ut}\,{\cal O}_{ut} \;+\; \beta_{tu}\,{\cal O}_{tu\phi}\;+\;\beta_{ut}\,{\cal O}_{ut\phi}\;+\; \mbox{h.c.} \nonumber \vspace{0.2cm} \\ =& \;\;\;\frac{i}{\Lambda^2}\,\left[\alpha_{tu}\,\left(\bar{t}_R\,\lambda^a\, \gamma^\mu \,D^\nu\,u_R\right)\;+\; \alpha_{ut}\,\left(\bar{u}_R\,\lambda^a\, \gamma^\mu\, D^\nu\,t_R\right)\right]\,G^a_{\mu\nu} \;\;\;+ \nonumber \vspace{0.2cm} \\ & \;\;\;\frac{v}{\Lambda^2}\,\left[\beta_{tu}\,\left(\bar{t}_L\,\lambda^a\, \sigma^{\mu\nu}\,u_R\right)\;+\; \beta_{ut}\,\left(\bar{u}_L\,\lambda^a\, \sigma^{\mu\nu}\,t_R\right)\right]\,G^a_{\mu\nu} \;\;+\;\; \mbox{h.c.} \;\;\;. \label{eq:lf}\end{aligned}$$ Several extensions of the SM, such as supersymmetry and two Higgs doublet models, may generate contributions to this type of operator [@chro]. The Feynman rules for these anomalous vertices are shown in figure , with quark momenta following the arrows and incoming gluon momenta. In ref. [@nos] we calculated the effect of these operators on the width of the quark top. They allow for the decay $t\,\rightarrow\,u\,g$ ($t\,\rightarrow \,c\,g$) (which is also possible in the SM, albeit at higher orders), and the corresponding width is given by $$\Gamma (t \rightarrow u g) = \frac{m^3_t}{12 \pi\Lambda^4}\,\Bigg\{ m^2_t \,\left\|\alpha_{ut} + \alpha^_{tu}\right\|^2 \,+\, 16 \,v^2\, \left(\left\| \beta_{tu} \right\|^2 + \left\| \beta_{ut} \right\|^2 \right) + 8\, v\, m_t\,\mbox{Im}\left[ (\alpha_{ut} + \alpha^_{tu}) \, \beta_{tu} \right] \Bigg\} \label{eq:wid}$$ and an analogous expression for $\Gamma (t \rightarrow c g)$. In this expression, and throughout the entire paper, we will consider all quark masses, except the top’s, equal to zero; the imprecision introduced by this approximation is extremely small, as we verified having performed the full calculations. Direct top production is also possible with these new vertices (meaning, the production of a top quark from partonic reactions such as $g\,u\, \rightarrow\,t$ or $g\,c\, \rightarrow\,t$), and the corresponding cross section at the LHC is given by $$\sigma(p\,p\,\rightarrow\,t)\;\;=\;\;\sum_{q\,=\,u,c}\,\Gamma (t\,\rightarrow\,q \,g)\, \frac{\pi^2}{m_t^2}\;\int^1_{m^2_t/E_{CM}^2}\frac{2\, m_t}{E_{CM}^2\, x_1} f_g (x_1)\, f_q (m^2_t/(E_{CM}^2\, x_1)) \, dx_1 \;\;\; . \label{eq:ppt}$$ In this expression $E_{CM}$ is the proton-proton center-of-mass energy (14 TeV at the LHC) and $f_g$ and $f_q$ are the parton density functions of the gluon and quark, respectively. Notice how both the top width and the cross section depend on $\Lambda^{-4}$. There are processes with a $\Lambda^{-2}$ dependence, namely the interference terms between the anomalous operators and the SM diagrams of single top quark production, via the exchange of a W gauge boson - processes like $u\,\bar{d}\,\rightarrow\,t\,\bar{d}$. They were studied in ref. [@nos] in detail, and we discovered that, due to a strong CKM suppression, the contributions from the anomalous vertices are extremely small. Now, the operators that compose the lagrangean are not, in fact, completely independent. If one performs integrations by parts and uses the fermionic equations of motion [@buch; @grz], one obtains the following relations between them: $$\begin{aligned} {\cal O}^{\dagger}_{ut} &= {\cal O}_{tu}\;-\;\frac{i}{2} (\Gamma^{\dagger}_u\, {\cal O}^{\dagger}_{u t \phi} \,+\, \Gamma_u \,{\cal O}_{t u \phi}) \nonumber \\ {\cal O}^{\dagger}_{ut} &= {\cal O}_{tu}\;-\;i\, g_s\, \bar{t}\, \gamma_{\mu}\, \gamma_R\, \lambda^a\,u\, \sum_i (\bar{u}^i\, \gamma^{\mu}\, \gamma_R\, \lambda_a u^i\,+\, \bar{d}^i\, \gamma^{\mu}\, \gamma_R\, \lambda_a\, d^i) \;\;\; , \label{eq:rel}\end{aligned}$$ where $\Gamma_u$ are the Yukawa couplings of the up quark and $g_s$ the strong coupling constant. In the second of these equations we see the appearance of four-fermion terms, indicating that they have to be taken into account in these studies. Equations then tell us that there are two relations between the several operators, which means that we are allowed to set two of the couplings to zero. A careful analysis of the operators listed in [@buch] leads us to consider three types of four-fermion operators: - [Type 1, $${\cal O}_{u_1}\;\;=\;\; \frac{g_s\,\gamma_{u_1}}{\Lambda^2} \left(\bar{t}\, \lambda^a\,\gamma^{\mu}\, \gamma_R\, u\right)\,\left(\bar{q} \, \lambda^a\,\gamma_{\mu}\, \gamma_R\, q\right)\;+\;\mbox{h.c.} \;\;\; ,$$ where $q$ is any given quark, other than the top;]{} - [Type 2, $${\cal O}_{u_2}\;\;=\;\; \frac{g_s\,\gamma_{u_2}}{\Lambda^2} \left[\left(\bar{t}\, \lambda^a\, \gamma_L\, u^\prime\right)\,\left( \bar{u}^{\prime\prime}\,\lambda^a\, \gamma_R\, u\right) \; + \; \left(\bar{t}\, \lambda^a\, \gamma_L\, d^\prime\right)\,\left(\bar{d}^{\prime\prime}\,\lambda^a \,\gamma_R\, u\right) \right] \;+\;\mbox{h.c.} \;\;\; ,$$ with down and up quarks from several possible generations, excluding the top once more;]{} - [Type 3, $${\cal O}_{u_3}\;\;=\;\; \frac{g_s\,\gamma_{u_3}}{\Lambda^2} \left[\left(\bar{t}\, \lambda^a\, \gamma_R\, u\right)\,\left( \bar{b}\,\lambda^a\, \gamma_R\, d^\prime\right) \; - \; \left(\bar{t}\, \lambda^a\, \gamma_R\, d^\prime\right)\,\left(\bar{b}\,\lambda^a\,\gamma_R\,u \right) \right] \;+\;\mbox{h.c.} \;\;\; , \label{eq:ga31}$$ and also, $$\frac{g_s\,\gamma_{u_3}^}{\Lambda^2} \left[\left(\bar{t}\, \lambda^a\, \gamma_L\, u\right)\,\left( \bar{d}^\prime\,\lambda^a\, \gamma_L\, d^{\prime\prime}\right) \; - \; \left( \bar{t}\, \lambda^a\, \gamma_L\, d\right)\,\left(\bar{d}^\prime\,\lambda^a\, \gamma_L\,u^{\prime\prime}\right) \right] \;+\;\mbox{h.c.} \;\;\; . \label{eq:ga32}$$ ]{} The $\gamma_u$’s are complex couplings. We of course consider identical operators for the case of flavour changing interactions with the $c$ quark. In the notation of ref. [@buch] these operators correspond, respectively, to $\bar{R}R\bar{R}R$, $\bar{L}R\bar{R}L$ and $\bar{L}R$ $\widetilde{(\bar{L} \, R)}$, in the octet configuration. We could have also considered the singlet operators but, since their spinorial structure is identical to these (lacking only the Gell-Mann matrices) we opted to leave them out. The presence of the $\lambda^a$ in these operators also signals their origin within the strong interaction sector, in line with our aim of studying strong flavour changing effects. For this reason, and for an easier comparison between the effects of the several operators, we included, in the definitions of the four-fermion terms above, an overall factor of $g_s$. Cross sections for $g\,g\,\rightarrow\,t\,\bar{u}$ and $g\,u\, \rightarrow\,g\,t$. Four-fermion channels. ========================================================================================================= The Feynman diagrams contributing to the partonic cross sections, $g\,g\,\rightarrow\,t\,\bar{u}$ and $g\,u\,\rightarrow\,g\,t$ are shown in figs. and respectively. Details of the calculations can be found in [@nos2]. If we assume that the branching ratio $BR(t\,\rightarrow\,b\,W)$ is approximately 100% and use $\Gamma (t\,\rightarrow\,b\,W)\,=\,1.42\, \left\| V_{tb}\right\|^2$ GeV (a value which includes QCD corrections) [@qcdc], we may express the partial widths as $\Gamma (t\,\rightarrow\,q\,g)\,=\,1.42\,\left\|V_{tb}\right\|^2\,BR(t\, \rightarrow\,q\,g)$. In terms of these branching ratios, and using the CTEQ6M structure functions [@cteq6] [^3] to perform the integration in the pdf’s, we obtain, for the total cross sections, the following results (expressed in picobarn): $$\begin{aligned} \sigma(p\,p\,\rightarrow\,g\,g\,\rightarrow\,t\,\bar{q}) &=\;\;\;\left[\,0.5\, BR (t\,\rightarrow\,u\,g)\;+\; 0.5\,BR(t\,\rightarrow\,c\,g) \right]\,\left\| V_{tb}\right\|^2\,10^4 \nonumber \vspace{0.5cm} \\ \sigma(p\,p\,\rightarrow\,g\,g\,\rightarrow\,\bar{t}\,q) &=\;\;\;\sigma(p\,p\, \rightarrow\,g\,g\,\rightarrow\,t\,\bar{q}) \nonumber \vspace{0.5cm} \\ \sigma(p\,p\,\rightarrow\,g\,q\,\rightarrow\,g\,t) &=\;\;\;\left[\,8.2\, BR (t\,\rightarrow\,u\,g)\;+\; 0.8\,BR(t\,\rightarrow\,c\,g) \right]\,\left\| V_{tb}\right\|^2\,10^4 \nonumber \vspace{0.5cm} \\ \sigma(p\,p\,\rightarrow\,g\,\bar{q}\,\rightarrow\,g\,\bar{t}) &=\;\;\;\left[\, 1.5\,BR(t\,\rightarrow\,u\,g)\;+\; 0.8\,BR(t\,\rightarrow\,c\,g) \right]\, \left\| V_{tb}\right\|^2\,10^4 \;\;\; . \label{eq:sigg}\end{aligned}$$ and for the direct top cross section we have, $$\begin{aligned} \sigma(p\,p\,\rightarrow\,g\,q\,\rightarrow\,t) &=\;\;\;\left[\,10.5\, BR (t\,\rightarrow\,u\,g)\;+\; 1.6\,BR(t\,\rightarrow\,c\,g) \right]\,\left\| V_{tb}\right\|^2\,10^4 \nonumber \vspace{0.5cm} \\ \sigma(p\,p\,\rightarrow\,g\,\bar{q}\,\rightarrow\,\bar{t}) &=\;\;\;\left[\, 2.7\, BR(t\,\rightarrow\,u\,g)\;+\; 1.6\,BR(t\,\rightarrow\,c\,g) \right]\, \left\| V_{tb}\right\|^2\,10^4 \;\;\; . \label{eq:sigd}\end{aligned}$$ The larger values of the coefficients affecting the up-quark branching ratios in eqs. and derive from the fact that the pdf for that quark is larger than the charm’s. The numerical integration has an error of less than one percent. Except for the direct top channel, all of these cross sections (as well as the four-fermion results we will soon present) are integrated with a cut on the transverse momentum ($p_T$) of the light parton in the final state of 15 GeV. This is to remove the collinear and soft singularities in the gluon-quark subprocesses to render finite partonic cross sections, for a finite $p_T$ cut eliminates both of those divergences in two-to-two scattering processes. In a realistic analysis including backgrounds, a higher $p_T$ cut might well be needed, to suppress background rates in order to observe the signal events. That study, however, is beyond the scope of this work. Observe how the direct channel cross section is larger than the others. Notice, however, that due to the kinematics of that channel, no $p_T$ cut was applied. When imposing such a cut on the decay products of the top quark produced in the direct channel, the corresponding cross section will certainly be reduced. It is quite remarkable that these cross sections are all proportional to the branching ratios for rare decays of the top. These are possible even within the SM, at higher orders. For instance, one expects the SM value of $BR(t\, \rightarrow\,c\,g)$ to be of about $10^{-12}$ [@chro; @juan], $BR(t\, \rightarrow\,u\,g)$ two orders of magnitude smaller. What this means is that, if whatever new physics lies beyond the SM has no sizeable impact on the flavour changing decays of the top quark, so that its branching ratios are not substantially different from their SM values, then one does not expect any excess of single top production at the LHC through these channels. On the other hand, if an excess of single top production is observed, even a small one, the expressions and tell us that $BR(t\,\rightarrow \,c\,g)$ and $BR(t\, \rightarrow\,u\,g)$ will have to be very different from their SM values. In fact, in models with two Higgs doublets or supersymmetry, one expects the branching ratios $BR(t\,\rightarrow\,c\,g)$ and $BR(t\, \rightarrow\,u\,g)$ to increase immensely [@chro; @juan], in some models becoming as large as $\sim 10^{-4}$. If that is the case, eqs. and predict a significant increase in the cross section for single top production at the LHC. This cross section is therefore a very sensitive observable to probe for new physics. A single top in the final state can also be produced through quark-quark or quark-antiquark scattering. The complete list of processes is $u\,u\,\rightarrow\,t\,u$, $u\,c\,\rightarrow\,t\,c$, $u\,\bar{u}\,\rightarrow\,t\,\bar{u}$, $u\,\bar{u}\,\rightarrow\,t\,\bar{c}$, $u\,\bar{c}\,\rightarrow\,t\,\bar{c}$, $d\,\bar{d}\,\rightarrow\,t\,\bar{u}$, $u\,d\,\rightarrow\,t\,d$ and $u\,\bar{d}\,\rightarrow\,t\,\bar{d}$. We have however excluded from this list, processes that are not consistent with our choice of gluonic operators, like, for instance, $s\,\bar{d}\,\rightarrow\,t\,\bar{u}$. In fig. we show the Feynman diagrams for the process $u\,u \,\rightarrow\,t\,u$ and the details of the calculation can again be found in [@nos2]. Results and discussion {#sec:conc} ====================== We can now gather all the results obtained in refs. [@nos; @nos2] for the cross sections of single top production. In terms of the couplings, the direct channel, eq. , gives us $$\sigma_{g\,u\,\rightarrow\,t} \;=\;\left\{ 321\, \left\|\alpha_{ut} + \alpha^_{tu}\right\|^2\,+\,5080\,\left(\left\| \beta_{tu} \right\|^2 + \left\| \beta_{ut} \right\|^2\right) + 2556\,\mbox{Im}\left[( \alpha_{ut} + \alpha^_{tu})\, \beta_{tu} \right] \right\}\,\frac{1}{\Lambda^4}\; \mbox{pb}\;\;\; ,$$ for the partonic channel $g\,u\,\rightarrow\,t$. For the gluon-gluon and gluon-quark channels, we have, from eqs. , $$\begin{aligned} \sigma_{g\,g\,\rightarrow\,t\bar{u}} &=\; \left\{\,14\, \left\|\alpha_{ut} + \alpha^_{tu}\right\|^2\,+\,221\,\left(\left\|\beta_{tu} \right\|^2 + \left\| \beta_{ut} \right\|^2\right) + 111\,\mbox{Im}\left[( \alpha_{ut} + \alpha^_{tu})\, \beta_{tu} \right] \right\}\,\frac{1}{\Lambda^4} \;\mbox{pb} \vspace{0.1cm}\nonumber \\ \sigma_{g\,u\,\rightarrow\,g\,t} &=\; \left\{\,250\, \left\|\alpha_{ut} + \alpha^_{tu}\right\|^2\,+\,3952\,\left(\left\|\beta_{tu} \right\|^2 + \left\| \beta_{ut} \right\|^2\right) + 1988\,\mbox{Im}\left[( \alpha_{ut} + \alpha^_{tu})\, \beta_{tu} \right] \right\}\, \frac{1}{\Lambda^4}\;\mbox{pb} \;\;\; .\vspace{-1cm}\end{aligned}$$ Finally, the four-fermion processes can all be gathered (after integration on the parton density functions, as before) in a single expression, $$\begin{aligned} \sigma^{(u)}_{4F} &=\;\left[\frac{}{}\,171\,\left\|\alpha_{ut} \right\|^2\,+\,179\,\left\|\alpha_{tu}\right\|^2\,-\,176\,\mbox{Re}(\alpha_{ut}\, \alpha_{tu})\,+\,331\,\mbox{Im}(\alpha_{ut}\,\beta_{tu})\,-\,362\,\mbox{Im}( \alpha_{tu}\,\beta_{tu}^)\right. \vspace{0.3cm}\nonumber \\ & \hspace{0.7cm}+\,689\,\left(\left\|\beta_{tu}\right\|^2 + \left\| \beta_{ut} \right\|^2\right)\,+\,177\,\mbox{Re}(\alpha_{ut}\,\gamma_{u_1})\,-\, 185\,\mbox{Re}(\alpha_{tu}\,\gamma^_{u_1})\,-\,16\,\mbox{Im}(\beta_{tu}\, \gamma^_{u_1})\vspace{0.6cm}\nonumber \\ & \hspace{0.7cm}-\,17\,\mbox{Re}(\alpha_{ut}\,\gamma_{u_2})\,+\,17\, \mbox{Re}(\alpha_{tu}\,\gamma^_{u_2})\,+\,0.1\,\mbox{Im}(\beta_{tu}\, \gamma^_{u_2}) \vspace{0.3cm}\nonumber \\ & \hspace{0.7cm}+\,\left. 525\,\left\|\gamma_{u_1}\right\|^2\,+\,94\, \left\| \gamma_{u_2}\right\|^2\,+\,88\, \left\|\gamma_{u_3}\right\|^2 \frac{}{}\right] \frac{1}{\Lambda^4}\;\mbox{pb} \;\;\; . \vspace{-1cm} \label{eq:sigtu}\end{aligned}$$ For the channels proceeding through the charm quark, we have analogous expressions, with different numeric values in most cases due to different parton content inside the proton. Within the four-fermion cross sections we show the results for the production of a bottom quark alongside the top, through the processes $u\,b\,\rightarrow\,t\,b$ and $u\, \bar{b}\,\rightarrow\,t\,\bar{b}$ (and analogous processes for the $c$ quark). They are given by $$\begin{aligned} \sigma^{(u)}_{t + b} &=\;\left[\frac{}{}\,8\,\left\|\alpha_{ut} \right\|^2\,+\,9\,\left\|\alpha_{tu}\right\|^2\,-\,2\,\mbox{Re}(\alpha_{ut}\, \alpha_{tu})\,+\,28\,\mbox{Im}(\alpha_{ut}\,\beta_{tu})\,-\,32\,\mbox{Im}( \alpha_{tu}\,\beta_{tu}^)\right. \vspace{0.3cm}\nonumber \\ & \hspace{0.7cm}+\,59\,\left(\left\|\beta_{tu}\right\|^2 + \left\| \beta_{ut} \right\|^2\right)\,+\,12\,\mbox{Re}(\alpha_{ut}\,\gamma_{u_1})\,-\, 13\,\mbox{Re}(\alpha_{tu}\,\gamma^_{u_1})\,-\,3\,\mbox{Im}(\beta_{tu}\, \gamma^_{u_1})\vspace{0.6cm}\nonumber \\ & \hspace{0.7cm}-\,2\,\mbox{Re}(\alpha_{ut}\,\gamma_{u_2})\,+\,2\, \mbox{Re}(\alpha_{tu}\,\gamma^_{u_2})\,+\,0.5\,\mbox{Im}(\beta_{tu}\, \gamma^_{u_2}) \vspace{0.3cm}\nonumber \\ & \hspace{0.7cm}+\,\left. 19\,\left\|\gamma_{u_1}\right\|^2\,+\,5\, \left\| \gamma_{u_2}\right\|^2\,+\,16\, \left\|\gamma_{u_3}\right\|^2 \frac{}{}\right] \frac{1}{\Lambda^4}\;\mbox{pb}\vspace{-1cm}\end{aligned}$$ and $$\begin{aligned} \sigma^{(c)}_{t + b} &=\;\left[\frac{}{}\,0.4\,\left\|\alpha_{ct} \right\|^2\,+\,0.6\,\left\|\alpha_{tc}\right\|^2\,+\,0.2\,\mbox{Re}(\alpha_{ct}\, \alpha_{tc})\,+\,2\,\mbox{Im}(\alpha_{ct}\,\beta_{tc})\,-\,3\,\mbox{Im}( \alpha_{tc}\,\beta_{tc}^)\right. \vspace{0.3cm}\nonumber \\ & \hspace{0.7cm}\left. +\,5\,\left(\left\|\beta_{tc}\right\|^2 + \left\| \beta_{ct} \right\|^2\right)\,+\,\left\|\gamma_{c_1}\right\|^2\,+\,0.2\, \left\| \gamma_{c_2}\right\|^2\,+\,0.6\, \left\|\gamma_{c_3}\right\|^2 \frac{}{} \right] \frac{1}{\Lambda^4}\;\mbox{pb}\end{aligned}$$ where the interference terms between the $\{\alpha\,,\,\beta\}$ and the $\gamma$ were left out because they were too small when compared with the remaining terms. Finally, by changing the pdf integrations, and using the second vertex in fig.1, we can also obtain the cross sections for anti-top production. We have thus far presented the complete expressions for the cross sections but, as was discussed earlier and is made manifest by equation , some of the operators we considered are not independent. In fact, eq. implies that we can choose two of the couplings $\{\alpha_{ut}\,,\,\alpha_{tu}\, ,\,\beta_{ut}\,,\,\beta_{tu}\,,\,\gamma_{u_1}\}$ to be equal to zero. Notice that $\gamma_{u_2}$ and $\gamma_{u_3}$ are not included in this choice, as the respective operators do not enter into equations . A similar conclusion may be drawn, of course, about the couplings $\{\alpha_{ct}\,,\, \alpha_{tc}\,,\,\beta_{ct}\,,\,\beta_{tc}\,,\,\gamma_{c_1}\}$. We choose to set $\beta_{tu}$ and $\gamma_{u_1}$ to zero, as this choice eliminates many of the interference terms of the cross sections. Summing all of the different contributions, we obtain, for the single top production cross section, the following results: $$\begin{aligned} \sigma^{(u)}_{single\;\, t} &=\;\left[\frac{}{}\,756\,\left\|\alpha_{ut} \right\|^2\,+\,764\,\left\|\alpha_{tu}\right\|^2\,+\,994\,\mbox{Re}(\alpha_{ut}\, \alpha_{tu})\,+\,9942\,\left\|\beta_{ut}\right\|^2 \right. \vspace{0.2cm} \nonumber \\ & \hspace{0.7cm}\left. -\,17\,\mbox{Re}(\alpha_{ut}\,\gamma_{u_2})\,+\,17\, \mbox{Re}(\alpha_{tu}\,\gamma^_{u_2})\,+\,94\, \left\| \gamma_{u_2}\right\|^2\,+\,88\, \left\|\gamma_{u_3}\right\|^2 \frac{}{}\right] \frac{1}{\Lambda^4}\;\mbox{pb} \;\;\; , \nonumber \vspace{0.3cm} \\ \sigma^{(c)}_{single\;\, t} &=\;\left[\frac{}{}\,109\,\left\|\alpha_{ct} \right\|^2\,+\,109\,\left\|\alpha_{tc}\right\|^2\,+\,166\,\mbox{Re}(\alpha_{ct}\, \alpha_{tc})\,+\,1514\,\left\|\beta_{ct}\right\|^2 \right. \vspace{0.3cm} \nonumber \\ & \hspace{0.7cm}\left. -\,3\,\mbox{Re}(\alpha_{ct}\,\gamma_{c_2})\,+\,3\, \mbox{Re}(\alpha_{tc}\,\gamma^_{c_2})\,+\,24\, \left\| \gamma_{c_2}\right\|^2\,+\,27\, \left\|\gamma_{c_3}\right\|^2 \frac{}{}\right] \frac{1}{\Lambda^4}\;\mbox{pb} \;\;\; . \label{eq:res}\end{aligned}$$ For anti-top production, $$\begin{aligned} \sigma^{(u)}_{single\;\, \bar{t}} &=\;\left[\frac{}{}\,174\,\left\|\alpha_{ut} \right\|^2\,+\,174\,\left\|\alpha_{tu}\right\|^2\,+\,265\,\mbox{Re}(\alpha_{ut}\, \alpha_{tu})\,+\,2422\,\left\|\beta_{ut}\right\|^2 \right. \vspace{0.3cm} \nonumber \\ & \hspace{0.7cm}\left. +3\,\mbox{Re}(\alpha_{ut}\,\gamma_{u_2})\,-\, \mbox{Re}(\alpha_{tu}\,\gamma^_{u_2})\,+\,26\, \left\| \gamma_{u_2}\right\|^2\,+\,35\, \left\|\gamma_{u_3}\right\|^2 \frac{}{}\right] \frac{1}{\Lambda^4}\;\mbox{pb} \;\;\; , \nonumber \vspace{0.3cm} \\ \sigma^{(c)}_{single\;\, \bar{t}} &=\left[\frac{}{}\,109\,\left\|\alpha_{ct} \right\|^2\,+\,109\,\left\|\alpha_{tc}\right\|^2\,+\,166\,\mbox{Re}(\alpha_{ct}\, \alpha_{tc})\,+\,1514\,\left\|\beta_{ct}\right\|^2 \right. \vspace{0.3cm} \nonumber \\ & \hspace{0.7cm}\left. +\,7\,\mbox{Re}(\alpha_{ct}\,\gamma_{c_2})\,-\,7\, \mbox{Re}(\alpha_{tc}\,\gamma^_{c_2})\,+\,29\, \left\| \gamma_{c_2}\right\|^2\,+\,29\, \left\|\gamma_{c_3}\right\|^2 \frac{}{}\right] \frac{1}{\Lambda^4}\;\mbox{pb} \;\;\; .\end{aligned}$$ There is an extensive literature on the subject of single top production [@topcr]. For the LHC, the SM prediction is usually considered to be $319.7\,\pm\,19.3$ pb [@singt]. Considering the large numbers we are obtaining in the expressions above - specially the coefficients of the $\beta$ couplings, though the others are not in any way negligible - we can see that even a small deviation from the SM framework will produce a potentially large effect in this cross section. It is indeed a good observable to test new physics, as it seems so sensible to its presence. Alternatively, if the cross section for single top production at the LHC is measured in the years to come and is found to be in complete agreement with the SM predicted value, then we will be able to set extremely stringent bounds on the couplings $\{\alpha\,,\, \beta\,,\,\gamma \}$ - on new physics in general - precisely for the same reasons. In conclusion, we have calculated the contributions from a large set of dimension six operators to cross sections of several processes of single top production at the LHC. All cross sections involving gluons in the initial or final states are proportional to branching ratios of rare top quark decays. This makes these processes extremely sensitive to new physics, since those branching ratios may vary by as much as eight orders of magnitude in the SM and extended models. The four-fermion operators we chose break this proportionality so that, even if the branching ratios of the top quark conform to those of the SM, we may still have an excess of single top production at the LHC, stemming from those same operators. One of the advantages of working in a fully gauge-invariant manner is the possibility of using the equations of motion to introduce relations between the operators and thus reduce the number of independent parameters. One possible further simplification, if one so wishes, would be to consider each generation’s couplings related by the SM CKM matrix elements, so that, for instance, $\alpha_{tu}\,=\,\alpha_{tc}\,\|V_{ub}/V_{cb}\|$. This should constitute a reasonable estimate of the difference in magnitude between each generations’ couplings. Finally, in this paper we presented both the total anomalous cross sections for single top production and those of the individual processes that contribute to it. If there is any experimental method - through kinematical cuts or jet analysis - to distinguish between each of the possible partonic channels (direct top production; gluon-quark fusion; gluon-gluon fusion; quark-quark scattering), the several expressions we presented here will allow a direct comparison between theory and experiment. At this point a thorough detector simulation of these processes is needed to establish under which conditions, if any, they might be observed at the LHC, and what precision one might expect to obtain on bounds on the couplings $\{\alpha\,,\,\beta\,,\, \gamma\}$. [99]{} P.M. Ferreira, O. Oliveira and R. Santos, [Phys. Rev.]{} [D73]{} (2006) 034011. P.M. Ferreira and R. Santos, [hep-ph/0601078]{}; to be published in Phys. Rev. D. W. Buchmüller and D. Wyler, [Nucl. Phys.]{} [B268]{} (1986) 621. E. Malkawi and T. Tait, [Phys. Rev.]{} [D54]{} (1996) 5758; T. Han, K. Whisnant, B.L. Young and X. Zhang, [Phys. Lett.]{} [B385]{} (1996) 311; T. Han, M. Hosch, K. Whisnant, B.L. Young and X. Zhang, [Phys. Rev.]{} [D55]{} (1997) 7241; K. Whisnant, J.M. Yang, B.L. Young and X. Zhang, [Phys. Rev.]{} [D56]{} (1997) 467; M. Hosch, K. Whisnant and B.L. Young, [Phys. Rev.]{} [D56]{} (1997) 5725; T. Han, M. Hosch, K. Whisnant, B.L. Young and X. Zhang, [Phys. Rev.]{} [D58]{} (1998) 073008; K. Hikasa, K. Whisnant, J.M. Yang and B.L. Young, [Phys. Rev.]{} [D58]{} (1998) 114003. F. del Àguila and J.A. Aguilar-Saavedra, [Phys. Rev.]{} [D67]{} (2003) 014009. T. Tait and C. P. Yuan, [Phys. Rev.]{} [D63]{}, (2001) 014018; D. O. Carlson, E. Malkawi, and C. P. Yuan, [Phys. Lett.]{} [B337]{}, (1994) 145; G. L. Kane, G. A. Ladinsky, and C. P. Yuan, [Phys. Rev.]{} [D45]{}, (1992) 124; T. G. Rizzo, [Phys. Rev.]{} [D53]{}, (1996) 6218; T. Tait and C. P. Yuan, [Phys. Rev.]{} [D55]{}, (1997) 7300; A. Datta and X. Zhang, [Phys. Rev.]{} [D55]{}, (1997) 2530; E. Boos, L. Dudko, and T. Ohl, [Eur. Phys. J.]{} [C11]{}, (1999) 473; D. Espriu and J. Manzano, [Phys. Rev.]{} [D65]{}, (2002) 073005. Gad Eilam, Mariana Frank and Ismail Turan, [hep-ph/0601253]{}; J. Guasch, W. Hollik, S. Peñaranda and J. Solà, [hep-ph/0601218]{}. J. J. Liu, C. S. Li, L. L. Yang and L. G. Jin, [Phys. Rev.]{} [D72]{} (2005) 074018; L.L. Yang, C. S. Li, Y. Gao and J. J. Liu, [hep-ph/0601180]{}. C.T. Hill and S.J. Parke, [Phys. Rev.]{} [D49]{} (1994) 4454; G. J. Gounaris, D. T. Papadamou, F. M. Renard, [Z. Phys.]{} [C76]{} (1997) 3 33. B. Grzadkowski, J.F. Gunion and P. Krawczyk, [Phys. Lett.]{} [B268]{} (1991) 106; G. Eilam, J.L. Hewett and A. Soni, [Phys. Rev.]{} [D44]{} (1991) 1473; T.P. Cheng and M. Sher, [Phys. Rev.]{} [D35]{} (1987) 3484; L.J. Hall and S. Weinberg, [Phys. Rev.]{} [D48]{} (1993) R979. B. Grzadkowski, Z. Hioki, K. Ohkuma and J. Wudka, [Nucl. Phys.]{} [B689]{} (2004) 108. V.M. Budnev, I.F. Ginzbirg, G.V. Meldon and V.G. Serfo, [Phys. Rep.]{} [15C]{} (1975) 183; R. Cutler and D. Sivers, [Phys. Rev.]{} [D17]{} (1978) 17. A. Denner and T. Sack, [Nucl. Phys.]{} [B358]{} (1991) 46; G. Eilam, R.R. Mendel, R. Migneron and A. Soni, [Phys. Rev. Lett.]{} [66]{} (1991) 3105; A. Czarnecki and K. Melnikov, [Nucl. Phys.]{} [B554]{} (1999) 520; K.G. Chetyrkin, R. Harlander, T. Seidensticker and M. Steinhauser, [Phys. Rev.]{} [D60]{} (1999) 114015; S.M. Oliveira, L. Brücher, R. Santos and A. Barroso, [Phys. Rev.]{} [D64]{} (2001) 017301. J. Pumplin [et al]{}, [JHEP]{} [0207]{} (2002) 012. A. Belyaev and E. Boos, [Phys. Rev.]{} [D63]{} (2001) 034012; Z. Sullivan, [Phys. Rev.]{} [D70]{} (2004) 114012. M.E. Luke and M.J. Savage, [Phys. Lett.]{} [B307]{} (1993) 387; D. Atwood, L. Reina and A. Soni, [Phys. Rev.]{} [D55]{} (1997) 3156; J.M. Yang, B.L. Young and X. Zhang, [Phys. Rev.]{} [D58]{} (1998) 055001; J. Guasch and J. Solà, [Nucl. Phys.]{} [B562]{} (1999) 3; D. Delepine and S. Khalil, [Phys. Lett.]{} [B599]{} (2004) 62; J.J. Liu, C.S. Li, L.L. Yang and L.G. Jin, [Phys. Lett.]{} [B599]{} (2004) 92; J.A. Aguilar-Saavedra, [Acta Phys. Pol.]{} [B35]{} (2004) 2695. S. S. Willenbrock and D. A. Dicus, [Phys. Rev.]{} [D34]{} (1986) 155; C. P. Yuan, [Phys. Rev.]{} [D41]{}, (1990) 42; R. K. Ellis and S. Parke, [Phys. Rev.]{} [D46]{}, (1992) 3785; D. O. Carlson and C. P. Yuan, [Phys. Lett.]{} [B306]{}, (1993) 386; A. P. Heinson, A. S. Belyaev and E. E. Boos, [Phys. Rev.]{} [D 56]{}, (1997) 3114; G. Bordes and B. van Eijk, [Nucl. Phys.]{} [B435]{}, (1995) 23; T. Stelzer, Z. Sullivan and S. Willenbrock, [Phys. Rev.]{} [D56]{} (1997) 5919; B. W. Harris, E. Laenen, L. Phaf, Z. Sullivan, and S. Weinzierl, [Phys. Rev.]{} [D66]{}, (2002) 054024; S. Cortese and R. Petronzio, [Phys. Lett.]{} [B253]{}, (1991) 494; T. Stelzer and S. Willenbrock, [Phys. Lett.]{} [B357]{}, (1995) 125; Martin C. Smith and S. Willenbrock, [Phys. Rev.]{} [D54]{} (1996) 6696; S. Mrenna and C. P. Yuan, [Phys. Lett.]{} [B416]{}, (1998) 200; B. W. Harris, E. Laenen, L. Phaf, Z. Sullivan, and S. Weinzierl, [Int. J. Mod. Phys.]{} [A16S1A]{}, (2001) 379; T. M. P. Tait, [Phys. Rev.]{} [D61]{}, (2000) 034001; S. Moretti, [Phys. Rev.]{} [D56]{}, (1997) 7427; S. Zhu, [Phys. Lett.]{} [B524]{} (2002) 283; [Erratum-ibid.]{} [B537]{} (2002) 351. T. Stelzer, Z. Sullivan, and S. Willenbrock, [Phys. Rev.]{} [D58]{}, (1998) 094021. [^1]: ferreira@cii.fc.ul.pt [^2]: rsantos@cii.fc.ul.pt [^3]: We used a factorization scale equal to the mass of the quark top, that being the characteristic scale of these reactions. This choice of $\mu_F$ produces smaller cross section values than, saying, choosing it equal to the partonic center-of-mass energy [@singt].
--- abstract: \| We present the results of a study that uses numerical simulations to interpret observations of tidally disturbed satellites around the Milky Way. When analysing the simulations from the viewpoint of an observer, we find a break in the slope of the star count and velocity dispersion profiles in our models at the location where unbound stars dominate. We conclude that ‘extra-tidal’ stars and enhanced velocity dispersions observed in the outskirts of Galactic satellites are due to contamination by stellar debris from the tidal interaction with the Milky Way. However, a significant bound population can exist beyond the break radius and we argue that it should not be identified with the tidal radius of the satellite. We also develop and test a method for determining the mass loss rate from a Galactic satellite using its extra-tidal population. We apply this method to observations of globular clusters and dwarf spheroidal satellites of the Milky Way, and conclude that a significant fraction of both satellite systems are likely be destroyed within the next Hubble time. Finally, we demonstrate that this mass loss estimate allows us to place some limits on the initial mass function (IMF) of stars in a cluster from the radial dependence of its present day mass function (PDMF). author: - \| Kathryn V. Johnston,$^1$ Steinn Sigurdsson,$^2$ and Lars Hernquist$^3$\ $^1$ Institute for Advanced Study, Princeton, NJ 08450\ $^2$ Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, England\ $^3$ Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064. title: Measuring mass loss rates from Galactic satellites --- epsf ==== == == == == \#1[[ \#1]{}]{} \#1[[ \#1]{}]{} @mathgroup@group @mathgroup@normal@group[eur]{}[m]{}[n]{} @mathgroup@bold@group[eur]{}[b]{}[n]{} @mathgroup@group @mathgroup@normal@group[msa]{}[m]{}[n]{} @mathgroup@bold@group[msa]{}[m]{}[n]{} =“019 =”016 =“040 =”336 ="33E == == == == \#1[[ \#1]{}]{} \#1[[ \#1]{}]{} == == == == globular clusters: general – stellar dynamics INTRODUCTION ============ Accurate estimates of mass loss rates from the satellite galaxies and globular clusters in orbit around the Milky Way offer the possibility of better understanding the dynamical history of the Galaxy. When integrated over the entire population, such measurements would tell us about the current accretion rate onto the Galaxy. This in turn would indicate whether the Galaxy has grown significantly through the gradual process of tidal stripping and disruption of its satellites. The direct inference of mass loss rates from individual objects would place new constraints on detailed analytic or numerical dynamical models of these systems. In addition, the measurements could be used to identify satellites which are likely to have well-populated streams of tidal debris associated with them. In particular, the formation, evolution and ultimate fate of the Galactic globular clusters is a long standing puzzle in astrophysics. Dynamically, globular clusters are clean and relatively simple systems, yet on close inspection they reveal a wealth of complex behaviour, including relaxation and evaporation from internal dynamical interactions and mass loss in response to the tidal field of the Milky Way [@me97; @el87]. Ideally we would like to be able to assess the importance of each of these processes for the evolution of individual globular clusters from observations. Of particular interest for mass-segregated, relaxed clusters is the differential mass loss of stars due to the different relative densities of low and high mass stars at various radii, and the consequence of this loss for the evolution of the present day mass function (PDMF) away from the initial mass function (IMF). Understanding the form of the IMF of the current globulars is relevant for theories of star formation and globular cluster formation [@fr77], and the related issue of how the Galaxy was assembled. The mass function of a stellar system $dN/dM$ is defined as the number of stars $dN$ in the mass interval $(M,M+dM)$. A common functional form chosen to approximate this distribution is $${dN \over dM} \propto M^{-(x+1)} \label{mf}$$ where $x$ is called the [mass function index]{}. The PDMF $x$ of a cluster has been found to be related to the Galactocentric radius $R_{\rm GC}$ and height above the Galactic plane $Z$ of the cluster (Capaccioli, Ortolani & Piotto 1991). Using simple semi-analytic models Capaccioli, Piotto & Stiavelli (1993) showed that these correlations could be reproduced by evolving a population of globular clusters which all have the same initial IMF in the tidal field of the Milky Way. Similarly, in a comparison of several globular clusters Piotto, Cool & King (1997) found the luminosity function of NGC 6397 to be much flatter than M15, M30 and M92, and concluded that this could be due to the extreme dynamical evolution implied by its orbit calculated from its proper motion [@d96]. In addition McClure et al. (1986) reported a dependence of the PDMF on metallicity, and Djorgovski, Piotto & Capaccioli (1993) successfully incorporated this together with the trends in $R_{\rm GC}$ and $Z$ into a trivariate analysis. In contrast, in a recent study of Pal 1, Rosenberg et al. (1997) found its mass function to be inconsistent with the trends in $R_{\rm GC}$, $Z$, and metallicity and concluded that this could either be due to evaporation of low mass stars, or an intrinsically different IMF. One step towards solving the puzzle of just how far the PDMF of a cluster differs from its IMF is determining how to interpret the signatures of tidal effects in observations of globular clusters and other satellites. In the past, the limiting radii of Galactic satellites (which are assumed to correspond to the tidal radii $r_{\rm tide}$ imposed by the Milky Way) have been interpreted using King’s (1962) tidal radius formula $$r_{\rm tide}=R_{\rm GC} \left({m_{\rm sat} \over 3 M_{\rm GC}}\right)^{1\over 3}, \label{rtide}$$ where $R_{\rm GC}$ is the distance of the satellite from the Galactic centre, $m_{\rm sat}$ is the mass of the satellite, and $M_{\rm GC}$ is the mass of the Galaxy enclosed within $R_{\rm GC}$. For the observed $r_{\rm tide}$ this formula has been used to: (i) find $m_{\rm sat}$ from the current $R_{\rm GC}$ [@fl83]; (ii) estimate the pericentre of the satellite’s orbit given the value for $m_{\rm sat}$ implied by its internal dynamics (Oh, Lin & Aarseth 1995; Irwin & Hatzidimitriou 1995 – hereafter IH85); and (iii) compare the Galaxy’s tidal field with the satellite’s internal field [@ih95]. Such analyses have been complicated by the detection of ‘extra-tidal’ stars around both dwarf spheroidal galaxies (IH95; Kuhn, Smith & Hawley 1996) and globular clusters (Grillmair et al. 1995 – hereafter G95; for a summary of current observations of both Galactic and extra-galactic globular clusters, see Grillmair 1998), and these morphological features have been shown to be consistent with those seen in simulations of tidally disrupting systems [@ola95; @g95]. Hut & Djorgovski (1992) also estimated destruction rates of globular clusters by finding the extent to which their distribution of half-mass relaxation times deviates from a power law and interpreting this difference in terms of evolution of the system. An alternative to these very direct interpretations of observations is to build models that include some representation of all the expected dynamical effects on globular clusters and integrate the cluster’s evolution over the lifetime of the Galaxy [@aho88; @ch96; @go97; @mw97; @v97]. The success of these semi-analytic methods (as with those in the previous paragraphs) rests upon the representation of the physics involved, and there is some disagreement about the level of sophistication required for accurate predictions – one example is the theory of tidal shocking which has received renewed attention [@w94; @ko95; @jhw98; @gho98]. However, all these studies predict that the Milky Way’s globular cluster system is currently undergoing significant evolution. In this paper, we return to the earlier philosophy of trying to understand how much we can learn directly from observations of tidal signatures. We approach the problem by using numerical simulations to assess how successfully such features can be interpreted. Our study differs from earlier ones in several ways: (i) we include a mass spectrum, and begin with mass segregated models; (ii) there is a one-one representation of stars in e.g. globular clusters; and (iii) the tidal field of a full Milky Way model is included rather than idealised as giving rise to a spherical tidal boundary or being represented by a one-component potential. In our analysis, we explicitly distinguish between the bound and unbound stars in a satellite, which allows us to identify the characteristics of the extra-tidal population. We use this analysis to develop and test methods for quantifying the mass loss rate and evolution of the mass function of clusters and satellites directly from current observations. We present the simulation and analysis methods used throughout the paper in §2. The simulations are used to provide an overview of the characteristics of evolution in a tidal field in §3. We analyse our models from the viewpoint of an observer §4. In §5, the interpretations developed in §4 are applied to observations and used to measure the rate of destruction of the Milky Way’s satellite system. We summarise our results in §6. METHODS ======= General approach ---------------- In our calculations, the satellite is represented as a discrete set of particles of different masses, one for each star expected in a cluster of the chosen mass and mass function. The particular choice of cluster models in this paper follows the parameters adopted by Chernoff & Weinberg (1990) (see also Sigurdsson & Phinney 1995). The initial particle masses, positions and velocities are taken from an explicit, maximal $N$ realisation of an isotropic Michie–King model of a cluster [@k62; @dc76; @gg79], as described in §2.2. The cluster’s evolution along an orbit in a three component rigid model of the Galaxy (described in §2.3) is followed using a self-consistent field (SCF) code to calculate the mutual interactions of stars in the cluster (see §2.4.1). The SCF approach cannot follow evolution due to close encounters of individual stars in the system, so we repeat one case with the effects of two-body encounters included via diffusion coefficients to illustrate their importance (see §2.4.2). This study is intended to isolate the response of a cluster to tidal effects with the dual purposes of identifying observable signatures of tidal interactions and providing a set of control models to compare to future studies which will include more of the physics that might influence the cluster’s evolution. In particular, we make the following simplifications: 1. The distribution of particle masses is chosen to represent an evolved stellar population with a turnoff at $\sim 0.8 M_{\odot}$. We assume that continued stellar evolution is slow compared to the dynamical timescales involved. This assumption will be relaxed in subsequent papers, with explicit stellar evolution done in tandem with the dynamical evolution. 2. The initial cluster models are truncated at some finite radius, $r_T$, corresponding to the ‘tidal radius’ of the King model. However, $r_T$ is not set to the limiting radius expected along each orbit, as a key purpose of the study is to investigate observable effects of tidal shocks and stripping of a cluster that is not in exact equilibrium with the tidal field. The chosen models and orbits cover a range of interaction strengths – from those that only survive a few orbits to those that could last for the lifetime of the Galaxy. The scenario implicit in these initial conditions is that a cluster is formed very compact, with most of its mass well inside its actual tidal radius. Mass loss during stellar evolution subsequent to the cluster’s formation leads to expansion of the cluster, until the tidal radius is reached. Here we assume that there is negligible loss of stars due to tidal effects until late in the cluster’s evolution, and that we can start with a relaxed, mass segregated cluster with an exact King model profile, and follow subsequent mass loss due to tidal effects. Clearly this is an approximation to the real physics, but in order to isolate the tidal effects we are exploring, the initial conditions of the cluster must be close to equilibrium or internal dynamical evolution will instead dominate. 3. As noted above, the effects of two body encounters on the internal dynamical evolution have not been included in the majority of our models. In this paper we restrict detailed discussions to effects that are insensitive to these simplifications. For example, in §4 we use our models to test how a cluster’s extra-tidal population can be used to determine its mass loss rate – we expect this to be valid because although the mass loss rate itself will depend on the internal dynamics of the system (which we do not model exactly), the characteristics of the extra-tidal population are determined only by the tidal potential (which we do model exactly). Conversely, we do not attempt to arrive at quantitative conclusions in our discussion of the evolution of clusters along different orbits (§3) since these would be influenced by all three of the above simplifications. Initial cluster models ---------------------- ------- ------------------ ---------------- --------- --------- ------------------- ------ --------------- ----------------- ----------- Model mass $r_{\rm half}$ $N$ $W_{0}$ \# density $x$ $T_{\rm dyn}$ $T_{\rm relax}$ orbits $10^5 M_{\odot}$ pc \#$/pc^3$ $10^6$ years $10^9$ years 0a 0.34284 10.6 142365 4 50 1.35 6.19 2.68 p1-3,d1-2 0b 2.73969 21.2 1136640 4 50 1.35 6.19 17.66 p1 0c 0.20614 6.24 29904 4 50 0.00 3.61 0.383 p1 0d 0.43807 13.4 269366 4 50 2.50 7.74 6.00 p1 1 1.22626 7.36 508669 6 1000 1.35 1.89 2.62 p1,p2,d1 2 2.86636 14.1 1186585 9 1000 1.35 3.28 9.92 p1 3 3.30330 4.62 1368230 12 1.0 $\times 10^5$ 1.35 0.573 1.98 p1,p2 ------- ------------------ ---------------- --------- --------- ------------------- ------ --------------- ----------------- ----------- Columns: (1) Model number; (2) mass; (3) half mass radius; (4) number of stars; (5) King model; (6) central number density; (7) mass function index; (8) internal dynamical time (see equation \[\[tdyn\]\]); (9) half-mass relaxation time; (10) orbits simulated. Four representative cluster models (labelled 0-3) are considered, with properties summarised in [Table \[modstab\]]{}. Model 0a was repeated with the same number density profile, but larger mass (Model 0b), and with different mass function indices (Models 0c and 0d). [Figure \[profifig\]]{} shows the number density and mass density as a function of spherical radius, and surface brightness as a function of projected radius for each initial distribution. Masses are converted to visual magnitudes and luminosities in this and subsequent plots using Bergbusch & VandenBerg’s (1992) isochrone for a 14 Gyr, \[Fe/H\]=-1.66 cluster. [ Figure \[cummfig\]]{} illustrates the level of mass-segregation by showing the cumulative fraction of stars within a given radius for each mass group in the models. The cluster is generated as a Monte Carlo realisation of the Michie–King distribution function, using 8 discrete mass groups and a truncated, evolved power law initial mass function as described in Sigurdsson & Phinney (1995). The structure of each cluster is determined from a set of input parameters: the depth of the central potential, $W_0$, the central number density, $n_0$, and the mean central dispersion, $\sigma $. Some of the observable structure parameters, such as the concentration, core radius and light profile, are derived quantities and depend on the mass function and mass–luminosity relation chosen. The particles representing the stars in mass group $\alpha (= 1\ldots 8)$, of mass $m_{\alpha}$, are assumed to have some initial distribution function, $g_{\alpha} (\varepsilon )$, where $\varepsilon=-\Phi-{1\over 2}{\bf v}^2$ is the energy and ${\bf v}$ is the velocity of the particle (both in the cluster centre–of–mass frame), and $\Phi$ is the cluster potential. The distribution function is then given by $$g_{\alpha} (\varepsilon ) = {{n_{0_{\alpha}}}\over {(2\pi \sigma_{\alpha})^{3/2}}} \Bigl [ e^{{{ \varepsilon }/{\sigma_{\alpha} }}} - 1 \Bigr ],$$ where $\sigma_{\alpha}$ is the core dispersion of mass group $\alpha $ and the $n_{0_{\alpha}}$ are the normalised densities of each mass group. The $n_{0_{\alpha}}$ are solved iteratively given the cluster structure parameters, according to the scheme described in Sigurdsson & Phinney (1995). ---------- -------------- --------------- -------------- -------------- --------------- -------------- -------------- --------------- -------------- $\alpha$ $x=0.0$ $x=1.35$ $x=2.50$ $m_{\alpha}$ $dm_{\alpha}$ $f_{\alpha}$ $m_{\alpha}$ $dm_{\alpha}$ $f_{\alpha}$ $m_{\alpha}$ $dm_{\alpha}$ $f_{\alpha}$ 1 0.12629 0.01847 1.00000 0.12346 0.23393 1.00000 0.12114 0.50252 1.00000 2 0.22461 0.04963 1.00000 0.21330 0.29045 1.00000 0.20449 0.33330 1.00000 3 0.34806 0.02601 1.00000 0.34600 0.08360 1.00000 0.34427 0.05484 1.00000 4 0.44265 0.03596 1.00000 0.43956 0.08355 1.00000 0.43695 0.04163 1.00000 5 0.57392 0.12309 0.34155 0.56679 0.13430 0.52623 0.56126 0.03946 0.67655 6 0.71009 0.16844 0.32948 0.70416 0.11552 0.58798 0.70078 0.02525 0.77595 7 0.99995 0.22745 0.00000 0.96585 0.04281 0.00000 0.93946 0.00264 0.00000 8 1.38462 0.35095 0.00000 1.36336 0.01583 0.00000 1.34231 0.00036 0.00000 ---------- -------------- --------------- -------------- -------------- --------------- -------------- -------------- --------------- -------------- Columns: (1) – mass group; (2), (5) and (8) – average mass $m_{\alpha}$; (3), (6) and (9) – mass fraction $dm_{\alpha}$; (4), (7) and (10) – fraction of luminous stars assigned to bin $\alpha$. For each mass function index $x$ considered in this paper, Table \[startab\] gives the average mass $m_{\alpha}$ of stars and total mass fraction $dm_{\alpha}$ assigned to bin $\alpha$, along with the fraction $f_{\rm \alpha}$ of stars in each bin that are luminous. Stars that have evolved beyond the main–sequence turnoff are assigned to remnant stellar classes. In particular, stars with initial masses between the turnoff and $4.7 M_{\odot}$ are assumed to have evolved to white dwarfs, and stars with initial masses between $8$ and $15 M_{\odot}$ are assumed to have formed $1.4 M_{\odot}$ neutron stars. This follows the prescription in Chernoff & Weinberg (1990). Future work will consider other prescriptions for assigning remnant masses as a function of initial stellar mass. The presence of a substantial population of intermediate mass white dwarfs changes the present day mass function from the simple power law of the initial mass function. In particular, there is a pronounced ‘bump’, an excess of stars with masses $\sim 0.5 M_{\odot}$ in the distribution function, which complicates interpretation of the local luminosity function in terms of the mass function, for intermediate mass stars. Each cluster is generated by a simple acceptance–rejection algorithm, selecting particle masses, positions and velocities from the distribution function. The masses are assigned in discrete mass bins, with the value of the mass in each bin being the mean mass of that interval, weighted by the evolved mass function. That is, the mean mass in each bin allows for the presence of stellar remnants in that bin. Future models may incorporate a continuous distribution of stellar masses. The positions are drawn from a radial grid (typically of about 200 points, spaced to sample the density profile efficiently). A particle selected to be at radius $r_i$ is assigned to some radius $r_i \pm \epsilon$ where $\epsilon$ is a uniform random variable on the grid space interval. The initial distribution is thus a set of thin stepped radial shells, with local density that deviates slightly from the true density profile. Phase mixing ensures that the model settles down to a smooth representation of the cluster on a dynamical timescale. The initial model has a virial ratio $\sim 1 \pm 1/\sqrt{N}$, and is stationary and stable. Cluster models were run in isolation using the SCF code (see §2.4.1 and Hernquist & Ostriker 1992): the initial mass segregation is robust and there is no spurious evolution. The mass segregation corresponds to thermal equilibrium for the cluster structure parameters (central potential and initial mass function) chosen. This is the equilibrium state expected for the internal distribution of masses within the cluster once the time scale for mass loss due to stellar evolution becomes longer than the core relaxation time, and provided that tidal effects have not yet had a strong effect on the cluster. There is also some evidence for primordial mass segregation in young clusters [@f98; @e98], possibly due to bias in the formation of massive stars towards high density regions, or due to rapid relaxation in the young cluster. Mass segregation continues on a relaxation time scale, and would lead to core collapse in the absence of other effects. Cluster orbits in the Galactic potential ---------------------------------------- A three-component model is used to represent the Galactic potential: $\Psi=\Psi_{\rm disk}+\Psi_{\rm spher}+\Psi_{\rm halo}$, in which the disk is represented by a Miyamoto-Nagai potential [@mn75], the spheroid by a Hernquist potential [@h90], and the halo by a logarithmic potential: $$\Psi_{disk}=-{GM_{\rm disk} \over \sqrt{R^{2}+(a+\sqrt{z^{2}+b^{2}})^{2}}}, \label{disk}$$ $$\Psi_{\rm spher}=-{GM_{\rm spher} \over r+c}, \label{bulge}$$ $$\Psi_{\rm halo}=v_{\rm halo}^2 \ln (r^{2}+d^{2}). \label{halo}$$ Here, $M_{\rm disk}=1.0 \times 10^{11}, M_{\rm spher}=3.4 \times 10^{10}, v_{\rm halo}= 128, a=6.5, b=0.26, c=0.7$, and $d=12.0$, where masses are in $M_{\odot}$, velocities are in km s$^{-1}$ and lengths are in kpc. This choice of parameters provides a nearly flat rotation curve between 1 and 30 kpc and a disk scale height of $0.2 $ kpc. The radial dependence of the z epicyclic frequency ($\kappa_z$) in the disk between radii at 3 and 20 kpc is similar to that of an exponential disk with a 4 kpc scale length. ------- ---------------- --------------- --------------- ------------------------- ---------------- ------------------------- ----------------- Orbit $r_{\rm peri}$ $r_{\rm apo}$ $T_{\rm orb}$ $A_{\rm disk}$ $T_{\rm disk}$ $A_{\rm bulge}$ $T_{\rm bulge}$ kpc kpc $10^7 years$ (km s$^{-1}$)/kpc/Myear $10^6 years$ (km s$^{-1}$)/kpc/Myear $10^6 years$ p1 1.1 3.5 8.0 30 1.0 20-30 3. p2 3.0 5.8 14.5 20 1.0 2-3 10. p3 9.5 12.5 32.0 5-10 1.0 0.5 20. d1 2.9 3.15 8.5 30 2.0 3-4 12. d2 4.6 5.4 14.5 30 2.2 1 20. ------- ---------------- --------------- --------------- ------------------------- ---------------- ------------------------- ----------------- Columns: (1) orbit; (2) closest approach; (3) apocentre; (4) azimuthal orbital time period; (5) defined in equation (\[a\]); (6) timescale for disk passage (see equation \[\[tdisk\]\]); (7) and (8) as columns (5) and (6) but for bulge passages. To contrast the evolution of the models in different tidal fields we choose orbits roughly 3kpc, 5kpc and 10kpc from the Galactic centre, three with polar orientations (p1, p2 and p3) and two near the Galactic disk (d1 and d2). Figures \[orbpfig\] and \[orbdfig\] show the paths of these orbits and Table \[orbstab\] gives their peri- and apo-galacticon and radial time periods. Hereafter we refer to the simulation representing the evolution of cluster Model $m$ along orbit $o$ as Model $(m,o)$. Integration methods ------------------- Individual particle trajectories are integrated using a leapfrog integration scheme, with accelerations calculated from $$\label{accel} {\bf \ddot r} = \nabla (\Psi ({\bf r})+\Phi({\bf r})) + {\bf a_{dyf}} + {\bf a_{kick}},$$ where the Milky Way is represented by the rigid potential $\Psi$ given in §2.2, and the cluster’s internal potential $\Phi$ is calculated from the particles’ masses and positions using the SCF code described in §2.4.1. The effect of accelerations due to two-body effects (i.e. terms ${\bf a_{dyf}} + {\bf a_{kick}}$) were included in only one simulation using the method described in §2.4.2. Dynamical friction of the cluster’s orbit was ignored. In all cases, the time step was smoothly varied along the orbit between $T_{\rm disk}/100$ and $T_{\rm dyn}/100$ to ensure that energy was conserved to better than 1 percent of the initial internal potential energy of the cluster. Here $$T_{\rm disk}=b/v_z \label{tdisk}$$ is the timescale for a disk passage, where $b$ is the vertical scale of the Galactic disk (see equation \[\[disk\]\]), and $v_z$ is the z-component of the satellite’s velocity as it crosses the disk plane at $Z=0$, and $$\label{tdyn} T_{\rm dyn}=\pi \sqrt{r_{\rm half}^3 \over 2 G m_{\rm sat}},$$ is the internal dynamical timescale for a system of total mass $m_{\rm sat}$ and half-mass radius $r_{\rm half}$ [@bt]. The timescales for the orbits and models are given in Tables \[orbstab\] and \[modstab\] respectively. ### Self-consistent field code The SCF code uses a bi-orthogonal basis function expansion to calculate the internal potential of the cluster from the individual particle positions and masses [@ho92]. The expansion is sensitive to large-scale fluctuations in the cluster’s potential but smooths local potential fluctuations arising from the discrete particles. Hence the SCF approach will underestimate relaxation when (as in our case) the number of particles in the simulation is equivalent to the number of stars in the system represented. The computations using this scheme (i.e. not including two-body encounters) were performed at the National Center for Supercomputing Applications (NCSA) with a version of the SCF code which had been parallelised to run on the Connection Machine 5 [@her95]. On average, each step required $4\times 10^{-4}$cpusec/particle/processor. ### Two-body relaxation calculation In order to explore the consequences of continuing mass segregation and two–body relaxation on the structure of the cluster, we incorporated a Fokker–Planck diffusion scheme into a version of the SCF code parallelised to run on the T3E [@ss97] at the Pittsburgh Supercomputing Center. The results of Model (0a,p3), run with and without diffusion are compared in in §3.3. The Fokker–Planck scheme calculates the first and second order velocity diffusion coefficients, $D(\Delta v_i), D(\Delta v_i\Delta v_j)$ [@bt]. The diffusion is directly incorporated into the explicit time evolution of the particles by including the last two terms in equation (\[accel\]), where ${\bf a_{dyf}} $ is the dynamical friction experienced by the star, and ${\bf a_{kick}}$ is the effective acceleration due to scattering by individual stars in the cluster (see Sigurdsson & Phinney 1995 for discussion). To calculate ${\bf a_{dyf}} $ and ${\bf a_{kick}}$, we choose an orthonormal basis local to each particle defined by the particle’s velocity ${\bf v}$ and position ${\bf x}$ in the cluster frame. This gives us three independent diffusion coefficients, $D(\Delta v_{\parallel })$, $D(\Delta v^2_{\parallel})$, and $D(\Delta v^2_{\perp})$. To calculate those we need the local density, which is obtained directly from the SCF expansion [@ho92], and a local velocity distribution. We approximate the velocity distribution as a Gaussian, and calculate it every $N_v \sim 100$ integration steps, by sampling the velocity distribution in radial bins, calculating the local dispersion, and using spline interpolation to obtain the dispersion as a function of radius. Provided the dynamical evolution of the cluster is slow compared to the dynamical time scale, the sparse updating of the dispersion profile is adequate, and provided the cluster is close to being relaxed, the approximation of taking the velocity distribution as an isotropic Gaussian is acceptable. The diffusion approximation incorporates a Coulomb logarithm term whose magnitude is uncertain to a magnitude comparable to other errors incurred by the approximations we make. Near the truncation radius, the interpolation of the dispersion must be positive definite or negative dispersion may be inferred. Outside the truncation radius the diffusion coefficients may be set to zero, or the local dispersion approximated by the Keplerian velocity due to the total enclosed mass. Given the diffusion coefficients, then ${\bf a_{dyf}} = D(\Delta v_{\parallel })$ and we model ${\bf a_{kick}}$ by random fluctuations in velocity, $\Delta {\bf v}$, where $ {\bf a_{kick}} = {{\Delta {\bf v}}\over {\Delta t}}$, with $$\begin{aligned} \Delta v^2_{\parallel } &= \varsigma_i ^2 (D(\Delta v^2_{\parallel } ) \Delta t ) \\ \Delta v^2_{\perp } &= \varsigma_i ^2 (D(\Delta v^2_{\perp }) \Delta t ),\end{aligned}$$ where $\varsigma_i $ is a random number with zero mean and unit standard deviation, chosen here from a normal distribution. This choice of diffusion coefficients provides a fast and surprisingly good approximation to two–body relaxation in clusters, and allows a fair representation of the effects of two–body relaxation on the structure of the cluster in the presence of other perturbing dynamical processes. For isolated clusters, this scheme provides a realisation of evolution towards core collapse on relaxation time scales, and can follow the evolution of the cluster over several orders of magnitude in central density. Ultimately though, with a finite number of expansion terms, the SCF code fails to resolve the resulting density cusp and core collapse to infinite density is not observed. Details of the scheme will be discussed in another paper. Analysis methods ---------------- ### Isolating the bound population of the cluster In parts of this study, identical analyses are performed on the particle data from the simulations both with and without the unbound particles, and the results are compared. Much of the discussion of the interpretation of observations rests upon the identification of stars that are cluster members. We define members of the cluster to be the maximum set of mutually bound particles. This set is found iteratively, starting from the set of all particles, by: (i) calculating the internal potential field of the particles in the set considered; (ii) finding the velocity of each particle with respect to the velocity of the minimum of this potential field; (iii) defining each particle’s internal kinetic energy to be the kinetic energy of this relative motion; (iv) labeling those particles whose internal kinetic energy is greater than their internal potential energy as ‘unbound’, removing them from the set considered and returning to step (i). Steps (i)-(iv) are repeated until no new particles in an iteration are labelled ‘unbound’. Using this method we find that the instantaneous mass bound to the cluster is a monotonically decreasing function of time, and the number of stars that later become bound again to the cluster once they have first been classified as unbound is negligible. ### Extrapolating mass functions Each particle in our simulations is assigned the average mass $m_\alpha$ of the mass bin $\alpha$ that it occupies. The first four mass bins contain only luminous matter, the next two contain a fraction $f_\alpha$ of luminous stars and the two most massive bins contain only stellar remnants (see Table \[startab\]). Hence, we can trivially find the mass function of observable stars in any sample at these average points by calculating $$\left({dN \over dm}\right)_\alpha= {f_\alpha N_\alpha \over \Delta m_\alpha}, \label{dndm}$$ where $\Delta m_\alpha$ is the width of the mass bin and $N_\alpha$ is the number of stars in the bin. Since mass functions are typically calculated from stars near the turnoff in a globular cluster, we estimate $x$ using only the two heaviest luminous mass bins to be $$x={LOG(N_6 f_6 \Delta m_5)-LOG(N_5 f_5 \Delta m_6) \over LOG(m_6)-LOG(m_5)}-1. \label{x}$$ GENERAL EVOLUTION IN A TIDAL FIELD ================================== Cluster heating --------------- We expect mass loss to occur predominantly where the tidal field of the Milky Way is strongest – i.e. at pericentric points along the orbit and during disk passages. In column 5 of Table \[orbstab\] we give values for $$A_{\rm disk}=d^2 \Phi_{\rm disk}/dz^2. \label{a}$$ For a globular cluster whose physical scale is given by its half mass radius $r_{\rm half}$, this can be used to estimate the specific energy change due to the disk passage in the impulsive regime, $$\Delta E_{\rm imp} = {1\over 2} \left(A_{\rm disk} T_{\rm disk}r_{\rm half}\right)^2, \label{eimp}$$ [@s58] which we shall subsequently refer to as the [shock strength]{}, where $T_{\rm disk}$ is the timescale for the disk passage (see equation \[\[tdisk\]\]). We expect this estimate to be appropriate when $T_{\rm dyn} \gg T_{\rm disk}$, and can otherwise use it as an upper limit on the average energy change. Columns 7 and 8 in Table \[orbstab\] give similar estimates for the importance of bulge shocking for these orbits (with $A_{\rm bulge}=d^2\Phi_{\rm bulge}/dR^2$), which is clearly competitive with disk shocking for orbit p1 (see Aguilar, Hut & Ostriker 1988 for a discussion of this effect). In our chosen Galactic potential, all these orbits lie within the core radius $d=12$kpc of the halo component (see equation \[\[halo\]\]), so its tidal field is roughly constant along the orbit and halo shocking is unimportant (in contrast, this is likely to be an important factor in the evolution of the Sagittarius dwarf spheroidal galaxy – see Johnston, Spergel & Hernquist 1995). To test the accuracy of the impulse approximation, we calculated the change in the cluster’s internal potential energy per unit mass ($\Delta \Phi$) and the impulsive estimate for this change ($\Delta E_{\rm imp}$ – from equations \[\[tdisk\]\], \[\[a\]\] and \[\[eimp\]\]) for each of the $N_{\rm disk}$ disk passages in each simulation. The points in [Figure \[dphifig\]]{} show the average values of these quantities in the simulations, in units of the cluster’s velocity dispersion, $\sigma^2$. The models in the bottom left hand corner of the plot are expected to be the most resilient to the Milky Way’s influence. There is a large spread in the response of the clusters’ potential energy for a given shock strength, $\Delta E_{\rm imp}$. The clusters on disk orbits (filled symbols) are shocked less noticeably than those on polar orbits (open symbols), because the timescales for the disk passages in the former case are longer, and the impulse approximation will only be valid in the outer region of the cluster. In general, Figure \[dphifig\] shows that while this crude estimate serves as a useful guide to the importance of shock heating for a cluster in a given orbit a much more detailed calculation of the interaction of the internal and external dynamics is needed for an accurate assessment of cluster evolution in general. More rigorous analytic estimates for tidal heating have been discussed extensively elsewhere in the literature [@w94; @ko95; @go97; @jhw98]. Differential mass loss ---------------------- [Figures \[masspfig\] and \[massdfig\]]{} follow the bound mass fraction remaining as a function of time for the first 0.5 Gyrs of evolution along the polar and disk orbits. On each of these figures we show the time of disk passages as vertical dashed lines, and the pericentric points along the orbits as vertical dotted lines. As noted in the previous section, the mass loss rate is greatest at these locations along the orbit. A comparison of these plots with Figure \[dphifig\] confirms the loose correlation between mass loss rate and shock strength. [Figure \[evolfig\]]{} compares the effect of the tidal field on each individual mass group in Model (1,p1) (left hand panels) with those in Model (0a,p3) (right hand panels). The solid lines are for the smallest and largest masses and the dotted lines are for the intermediate ones. The top panels illustrate differential mass loss due to mass segregation, as the most weakly bound (and therefore lower mass) stars are preferentially stripped. The lower two panels give a measure of the response in the spatial distributions - $r_{\rm lim}$ is calculated as the average radius of the outermost 1 percent of bound particles in each group, and $r_{\rm half}$ is calculated as the radius containing half the mass of each group. Again, the compressive disk shocks (and subsequent expansion due to the energy input) are clearly seen. The effect is more dramatic in the left hand panels since this model is more susceptible to the influence of tides (as demonstrated by its position in Figure \[dphifig\], and large mass loss rate). Similar characteristics are seen in all the models. [Figure \[m0abfig\]]{} repeats Figure \[evolfig\] for Models (0a,p1) (left hand panels) and (0b,p1) (right hand panels). These models had identical mass density profiles, but with different radial scales (and hence total masses). The comparison of the two demonstrates that the general characteristics of the response for models with the same density profiles subjected to the same tidal field are identical. Figures \[mevolfig\] – \[xevoldfig\] illustrate our intuitive understanding of tidal mass loss in a mass-segregated system. As an example, the solid lines in Figure \[mevolfig\] show the bound mass fraction for each mass group as a function of time for Model (0a,d1). The dashed lines show the prediction if the initial model were simply truncated in radius at the surface enclosing the same mass as is instantaneously bound to the system in the simulation. The dotted lines show the prediction if the initial model is instead truncated in binding energy. As might be anticipated, initial binding energy provides the more accurate of these simple guides to which stars are lost first from the system. This behaviour is seen in all our simulations, as shown in [Figures \[xevolpfig\] and \[xevoldfig\]]{}. The solid lines show the global mass function index $x$ as a function of time for each simulation, the dashed lines show the result of estimating $x$ by truncating the initial model in radius and the dotted lines show $x$ for truncation in energy. Note in general that the agreement is good, except when there has been substantial evolution of the system (e.g Models (0a,p1), (0c,p1)). In summary, the results in this section demonstrate that: (i) the impulse approximation provides a rough guide to the mass loss rate from a system; (ii) tidal mass loss along a given orbit is determined by the density profile of the cluster; and (iii) tidal shocking of a mass-segregated system leads to evolution of the mass function, which can approximately be predicted for a given mass lost by simply truncating the initial distribution in energy. Clearly these general trends fit in with theoretical interpretations of observed correlations of cluster properties with $R_{\rm GC}$ and $Z$ [@cop91; @cps93; @dpc93]. However, a quantitative comparison of our simulations with observations would require more detailed analytic models both of the tidal effects seen in the simulations and of other physical processes not included (one example is given in the next section) and is beyond the scope of the current study. Competition of tides and relaxation ----------------------------------- Figure \[mrelaxfig\] compares the mass loss rate for each group in Model (0a,p3) evolving over 8 Gyrs both with (dotted lines) and without (solid lines) diffusion included. The half-mass relaxation time for this cluster is $T_{relax}\sim 2.68$ Gyrs (see Table \[modstab\]). The evaporation time for a cluster is $\sim$ 100 $T_{\rm relax}$ [@me97], so we expect only a few percent change in bound mass fraction due to relaxation over the course of the simulation, as is seen in the figure. Relaxation contributes to mass loss by increasing the number of low mass stars lost and decreasing the number of high mass stars lost, with a net increase in mass loss. Hence, the evolution of the mass function will be modified from the purely tidal response described in the previous section. [Figure \[vrelaxfig\]]{} shows that the density and velocity properties of the model evolved with (solid squares) and without relaxation (open squares) are similar (the initial model is shown in stars). These figures demonstrate that our neglect of relaxation in the models considered in this paper will not substantially affect their evolution since a significant fraction of mass is lost due to tidal influences over a few relaxation times. In contrast, Pal 5 is an example of a cluster whose evolution is thought to be dominated by relaxation [@r98] and a natural extension to our work would be to consider models where the tidal and evaporation mass loss rates are comparable [@vh97]. INTERPRETING OBSERVATIONS OF TIDALLY DISRUPTING, MASS-SEGREGATED SYSTEMS ======================================================================== ‘Observing’ the simulations --------------------------- Observed properties of globular clusters are often interpreted as resulting from the tidal influence of the Milky Way. In this section, we ‘observe’ our models to confirm the validity of some common interpretations and explore other signatures of tidal disruption that future studies might be sensitive to. We focus our analysis on Model (0a,p3) as an example of a cluster with a steady mass loss rate that could survive for the lifetime of the Galaxy. We examine the observable properties of this model at the end of the simulation (after 8 Gyrs of evolution), and compare and contrast it with the same model at earlier times and with the other models in our sample. Since the Sun’s position along the Solar Circle in the simulations is arbitrary, rather than assuming a single viewpoint we examine each set of particle data along three perpendicular axes defined by the cluster’s instantaneous position and velocity: the $x$ axis lies along the vector joining the cluster to the Galactic centre; the $z$ axis is perpendicular to the cluster’s position and velocity vector (i.e. looking down on the orbital plane); and the $y$ axis is perpendicular to these two, along the orbit of the satellite (i.e. it coincides with the velocity vector when the Galactocentric radial velocity is zero). We also make the simplification that the lines of sight across the face of the cluster are perfectly parallel (i.e. the observer is at infinite distance from the cluster) and state distances in our projected coordinates in $pc$ rather than $arcmin$. This will not significantly affect our analysis of density profiles in §4.2, but can alter the measurement of cluster rotation, and we discuss this simplification further in §4.3. Star count profiles ------------------- ### Qualitative interpretation [Figure \[prof20fig\]]{} shows the number surface density for all stars (top panels), stars brighter than 4 magnitudes below the turnoff (middle panels) and stars down to the turnoff (bottom panels) at the end of the simulation of Model (0a,p3) with the three labelled viewpoints along the axes defined in §4.1. The closed symbols are for stars still bound to the satellite and the open symbols include unbound stars in the calculation (where the ‘bound’ and ‘unbound’ populations are separated following the method described in §2.5.1). In this Figure, there is a clear break in the slope of the profile defined by the open symbols, approximately corresponding to the point where the closed and open symbols become distinct and the fraction of unbound stars in an annulus becomes significant. Such ‘extra-tidal’ stars have been detected in several studies of dwarf spheroidal galaxies [@ih95; @ksh96] and globular clusters [@g95; @g98]. Our investigation confirms the common interpretation that these can be identified as stars escaping from the satellites. [Figure \[proffig\]]{} demonstrates that this is a general result by repeating the $x$-axis view for all stars in Model (0a,3p) at earlier times (left hand panels) and at random points for several of the other models (as labelled). ### Quantitative interpretation Tremaine (1993) pointed out that the change in the orbital frequency of a star torn from the satellite at radius $r_{\rm break}$ (the point at which the slope of the surface density profile changes) should approximately be given by $\Delta \Omega \sim r_{\rm break} d\Omega/dR$, where $\Omega(R)$ is the frequency of a circular orbit in the parent galaxy at radius $R$. Equivalently, the orbital energies in tidal debris from a satellite orbiting in a potential $\Phi$ will be spread over a characteristic range $\epsilon=r_{\rm tide} d\Phi/dR$. Johnston (1998) tested this simple physical argument by looking at the spread in energies in streamers seen in numerical simulations of tidal disruption and found that $\epsilon < \|\Delta E\| < 2\epsilon$. Hence, debris will spread over an angular distance comparable to the size of the cluster ($r_{\rm break}/R$) in a time $(r_{\rm break}/R)/2 \Delta \Omega \sim T_{orb}/\pi$, where $T_{\rm orb}$ is the azimuthal time period of the orbit. This suggests that we can estimate the average surface density of stars in an annulus between $r_{\rm break}$ and $r$ from the centre of the cluster to be $$\begin{aligned} \label{sigbar} \langle\Sigma_{\rm xt}(r)\rangle= {1 \over g(\theta)} \left[ {dm \over dt} {(r -r_{\rm break}) \over r_{\rm break}} {T_{\rm orb} \over \pi}\right] \\ \left/ \left[\pi (r^2-r_{\rm break}^2)\right]\right., \nonumber\end{aligned}$$ where $dm/dt$ is the mass loss rate from the cluster. The function $g$ depends on the angle $\theta$ of our line of sight with the plane perpendicular to the direction of motion of the satellite. Eventually, the debris spreads out along the satellite’s orbit so we can take $g(\theta)=\cos \theta$ as a crude approximation. In fact, mass tends to leave the satellite at the inner and outer Lagrange points along the vector joining the satellite to the centre of the galaxy, so the geometry of the streamers close to the satellite is rather more complicated than this. We can differentiate equation (\[sigbar\]) to find an approximate expression for the absolute surface density $$\label{sigma} \Sigma_{\rm xt}(r) = {1 \over g(\theta)} {dm \over dt} {T_{\rm orb} \over \pi} {1 \over 2 \pi r_{\rm break} r}.$$ This estimate is overlaid in dashed lines on the profiles shown in Figure \[proffig\] using $dm/dt$ averaged over each simulation, $T_{\rm orb}$ from Table \[orbstab\] and $r_{\rm break}$ indicated by the vertical dotted lines in each panel. The open squares and dashed line agree well in all cases except for Model (3,p1). In this model, the simulation was run for only a few orbital periods so the debris only had time to disperse a few $r_{\rm break}$ from the satellite and the streamers were not well populated. Equation (\[sigma\]) and Figure \[proffig\] demonstrate that we expect the extra-tidal population around a cluster to have $\Sigma_{\rm xt}(r) \propto r^{-1}$. In observations of real globulars, the surface density of extra-tidal stars has been found to fall as $\Sigma_{\rm xt}(r) \sim r^{-\gamma}$ with $-5<\gamma<-0.7$ (G95; Zaggia, Piotto & Capaccioli 1997). This suggests that the measured $\gamma$ might be used as a test of whether the globular is sufficiently obscured, either by tidal debris further along its own tidal tail or by the Galactic field, that the above interpretation is invalid. Note that these estimates are independent of the method of mass loss from the cluster since the dynamics once the star is lost is determined only by the external field, and hence should apply equally to mass lost by tidal stripping, shocking, or evaporation due to relaxation. ### Estimating mass loss rates using extra-tidal stars Equations (\[sigbar\]) and (\[sigma\]) suggest two ways of estimating the fractional mass loss rate $(df/dt)$ from a cluster using observations of extra-tidal stars. If the extra-tidal population is well defined out to a radius $r_{\rm xt}$ with $\Sigma_{\rm xt} \sim r^{-1}$ we can count the number of stars $n_{\rm break}$ within $r_{\rm break}$ (the point where there is a break in the slope of the surface density profile) and the number of extra-tidal stars $n_{\rm xt}$ between $r_{\rm break}$ and $r_{\rm xt}$ and use them to find $$\left({df \over dt}\right)_1 = g(\theta) {r_{\rm break}\over r_{\rm xt}-r_{\rm break}} {n_{\rm xt} \over n_{\rm break}} {\pi \over T_{\rm orb}}. \label{dfdt1}$$ Since the timescale for stars to diffuse beyond a few $r_{\rm break}$ is an orbital timescale, so long as $r_{\rm xt} > 2 r_{\rm break}$ this estimate should be sensitive to the average mass loss rate rather than the instantaneous one. The left hand panel of [Figure \[mlossfig\]]{} plots the known fractional mass loss rate, $df/dt$, for each of the models with fractional mass loss rates less than 1Gyr$^{-1}$ (see Figures \[masspfig\] and \[massdfig\]) against $(df/dt)_1$ calculated from equation (\[dfdt1\]). To find $(df/dt)_1$, we evaluated the ratio $n_{\rm xt}/n_{\rm break}$ for $r_{\rm xt}=2 r_{\rm break}$, $r_{\rm xt}= 5 r_{\rm break}$ and $r_{\rm xt}=10 r_{\rm break}$. This process was repeated looking along the $x$-axis (solid squares), $y$-axis (open squares) and $z$-axis (stars). The open triangles show the estimate made along $y$-axis if the viewing angle is not taken into account (i.e. $g(\theta)\equiv 1$). The solid line in the figure shows where the estimated and known mass loss rates agree, and the dotted lines are for factor of 2 discrepancies. The figure suggests that we can use this technique to estimate the mass loss rate from a Galactic satellite provided our viewpoint lies close to the plane perpendicular to the satellite’s velocity vector (equivalent to the $xz$-plane in our projection). Otherwise, we are likely to poorly estimate the rate of mass loss as the debris along the satellite’s orbit confuses the calculation. If the extra-tidal population is either not well defined or non-existent we can estimate $(df/dt)_2$ as an upper limit for the mass loss rate from $\Sigma_{\rm xt}(r_{\rm break})$, where $r_{\rm break}$ is taken to be the point either where the slope of the profile changes or the last point where the surface density is separable from the background. Then, from equation (\[sigma\]), $$\left({df \over dt}\right)_{2}=g(\theta) {\Sigma_{\rm xt}(r_{\rm break}) \over n_{\rm break}} {\pi \over T_{\rm orb}} 2 \pi r^2_{\rm break}, \label{dfdt2}$$ which is shown in the right hand panel of Figure \[mlossfig\]. As with the first case, the view perpendicular to the velocity vector provides the best estimate. Line of sight velocity distributions ------------------------------------ [Figure \[strip20fig\]]{} summarises the line-of-sight velocity analysis at the end of the simulation for Model (0a,p3) from the stated viewpoints and with the observer assumed to be at infinite distance from the cluster (i.e with the individual lines of sight across the cluster assumed to be parallel to the one to the cluster centre – the effect of this simplification is discussed in §4.3.1). For each viewpoint the average $v_{\rm los}$ and dispersion $\sigma_{\rm los}$ in the velocities of stars are calculated in bins lying in strips along the two perpendicular axes across the cluster. [Figure \[stripfig\]]{} demonstrates the general nature of these plots, repeating the $x$-axis view of Figure \[strip20fig\] at different orbital phases of this model and for several different models. The solid symbols show the analysis for the bound stars, and the open symbols shows the results for all stars. ### Average velocities and cluster rotation The average velocities are zero except for the viewpoints which are sensitive to the cluster’s orbital motion. In the panels corresponding to the $x$-axis view of Figures \[strip20fig\] and \[stripfig\], the dotted lines indicate the expected velocity gradient if the cluster were rigidly co-rotating with its orbit. The bound stars follow this line fairly closely, with some indication of a smaller gradient in velocities towards the centres of the clusters. The unbound stars appear to be rotating faster than this as they move to orbits with higher/lower angular velocities to form the leading/trailing streamers. In our analysis, two effects are clearly contributing to the measured rotation of the cluster – the intrinsic rotation of the cluster (in our simulations, roughly corresponding to co-rotation with the tidal field), and the velocity gradient in stripped material. Observers also have to contend with ‘perspective rotation’ [@ftw61; @mmm97; @d97] from the projection of the tangential velocity onto the non-parallel lines of sight across the cluster. If we were indeed viewing a rigidly co-rotating cluster from the centre of the Galaxy, the intrinsic and perspective rotation would cancel each other out and we would be sensitive only to tidal stripping, but if we were closer to/farther from the cluster than the centre of the Galaxy, we would be dominated by the perspective/intrinsic rotation. Unfortunately, we cannot assume that tidal torquing of a real satellite necessarily results in co-rotation. The rotation seen in our simulations is likely to be an artifact of the near-circular orbits we have chosen. In models, run along eccentric orbits, of the disruption of the Sagittarius dwarf galaxy, Johnston, Spergel & Hernquist (1995) found that the line of sight velocity gradient is dominated by perspective rotation, despite the fact that our viewpoint is farther from the satellite than the centre of the Galaxy, presumably because the satellite is not co-rotating. Velázquez & White (1995) made an estimate of 225km s$^{-1}$ for the tangential velocity of the Sagittarius dwarf galaxy by considering only the effect of perspective rotation and this roughly agrees with the observed proper motion of 250km s$^{-1}$ [@i97]. Rotation has been detected in several observational surveys of Galactic globular clusters. Merritt, Meylan & Mayor (1997) analysed the phase-space distribution of 469 stars in $\omega$Cen and found, after taking perspective rotation into account, it to be consistent with solid body rotation out to 11pc from the centre of the cluster, and decreasing beyond. Drukier et al. (1997), in an analysis of the velocities of 230 stars in the outskirts of M15, found some evidence for rotation. In both these studies, rotation was found to be less important towards the edge of the cluster, suggesting that in these cases it is intrinsic and not due to tidal torquing or stripping of the bound system. ### Velocity dispersions and tidal heating In most panels of Figures \[strip20fig\] and \[stripfig\] there is good agreement between $\sigma_{\rm los}$ calculated with and without the unbound stars. However, the observed results can be confusing when looking along the orbit ($y$-axis view panels of Figure \[strip20fig\]), as there is significant unbound material along the line of sight from the trailing and leading streamers. This problem may not be so severe in reality if the sample can be selected to exclude stars beyond a few tidal radii from the cluster (in the figure, all stars at the projected separation were included). An observer would also see an apparently enhanced velocity dispersion when looking at the outskirts of the cluster from the centre of the Galaxy. In particular, note that the dispersion in the unbound material roughly corresponds to the dispersion of the bound material within the point where stripping occurs. This suggests that satellites that are being more violently stripped will have larger dispersions in their debris trails, though not in excess of the maximum dispersion of the bound material. The Sagittarius dwarf spheroidal galaxy is an example of a system where it can be plausibly argued that this behaviour is observed: its highly distorted surface density contours suggest that it is likely to be surrounded by a cloud of unbound stars and its velocity dispersion is roughly constant along the entire length of its major axis [@i97]. In their analysis of the velocities in the outskirts of M15, Drukier et al. (1997) found that the dispersion of the stars decreased to a minimum at 7 arcmin, increasing again beyond this radius. They suggest that this deviation from the behaviour expected for an isolated cluster could be due to tidal heating, and our simulations confirm this interpretation. They comment that the radial position of this minimum is much smaller than the tidal radius found by G95 by fitting King Models to star count profiles. This is also consistent with our analyses – the star counts and velocity analyses become contaminated by unbound stars well within the outermost radius of the bound system. Hence the minimum in the velocity dispersion profile is a good indicator of where this contamination becomes important, but does not necessarily correspond to the edge of the system. Mass functions -------------- ### Measuring present day mass functions In [Figures \[pdmffig\] and \[pdmfm0o7fig\]]{} we mimic the observational analyses recently applied to the globular clusters NGC 1261 [@z98] and M55 [@zpc97]. The solid line in Figure \[pdmffig\] shows the evolution of the mass function index $x$ for Model (0a,p3). The points indicate what $x$ would be measured to be at different points along the orbit and at different positions in the cluster if viewed from the centre of the Galaxy – within the core radius (filled squares), between the core and half-light radii (open squares) and between the half-light and tidal radii (stars). Each of these points is calculated using several thousand stars. In Figure \[pdmfm0o7fig\] we plot the mass and luminosity functions at the beginning (solid lines), and the end (dotted lines) of the simulation. The dashed lines show the final analysis for the same Model but for the simulation that included relaxation effects. ### Placing limits on the initial mass function of a cluster Suppose we have estimated the fractional mass-loss rate from a cluster using its observed population of extra-tidal stars. If we also know the PDMF, both globally and locally in the outskirts of the cluster, we can place some limits on its IMF. We test this idea with our simulations by using the methods outlined in §4.1 to find $(df/dt)_1$. [Figure \[imffig\]]{} shows the final mass function $(dN/dM)_{\rm final}$ at the end of the five labelled simulations (dashed line) and the mass function for the 100 stars at the projected edge of each system $(dN/dM)_{\rm edge}$ (dotted line), where each mass function has been scaled by the known IMF ($(dN/dM)_0$, represented by the horizontal solid line). We then simply estimate the IMF (solid squares) to be, $$\left({dN \over dM} \right)_{0, \rm est}= \left({dN \over dM} \right)_{\rm final}+ T_{\rm sim} \left({df \over dt}\right)_{1} \left({dN \over dM} \right)_{\rm edge} \label{imf}$$ where $T_{\rm sim}$ is the duration of the simulation. (The open squares show the result if the PDMF of the outermost 1000 stars is used.) Several assumptions make this estimate uncertain: (i) $(df/dt)_{1}$ is taken from star counts, implicitly assuming that the mass function of the extra-tidal material is the same as the global mass function; (ii) we assume the mass-function of the stripped material to be constant with time; (iii) we neglect relaxation effects; and (iv) we assume that the satellite fills its tidal radius, and hence loses mass at a roughly constant rate due to either evaporation or tidal effects. The first approximation can be addressed if the luminosity function of the cluster is known as a function of radius, as is the case in our simulations – Figure \[imffig\] demonstrates that we can make reasonable predictions for the simulations even without this level of refinement; the second two assumptions will both lead to an underestimate of the evolution of IMF; and the last approximation will place some limits on how far back we might be able to ‘integrate’ the differential mass loss. Despite these uncertainties, this method provides a new approach – more directly based on observations rather than using complex dynamical models – to exploring the question of whether the IMF in globulars was universal, or environment dependent. DISCUSSION: APPLICATIONS TO OBSERVED SYSTEMS ============================================ Destruction rate of galactic satellites --------------------------------------- ### Dwarf spheroidal galaxies IH95 studied the morphology of eight of the dwarf spheroidal satellites of the Milky Way (all the ones currently known with the exception of Sagittarius) determined from star counts made using the APM facility at Cambridge. Extra-tidal stars are clearly seen in the number count profiles for six of these satellites, but in no case do they clearly follow $\Sigma_{\rm xt} \sim r^{-1}$. Hence, we use equation (\[dfdt2\]), to find an upper limit for the mass loss rate. Using the data from Table 3 of IH95 (kindly made available to us by M. Irwin), we subtract off the mean background stellar density from each of their bins and take $r_{\rm break}$ to be the smaller of the point at which the slope of the star count profile changes (by visual inspection of their figure 2) and the last measured surface density above the background. The orbital time period $T_{\rm orb}$ is calculated for a circular orbit in a logarithmic potential, with circular velocity $v_c=200$km s$^{-1}$ $$\label{torb} T_{\rm circ}={2 \pi R_{\rm GC} \over v_c}= 1.5 \left({R_{\rm GC} \over 50 {\rm kpc}}\right) {\rm Gyrs},$$ with $R_{\rm GC}$ taken to be the current distance of each satellite from the Galactic centre. In fact, if the time period of orbits is assumed to be independent of angular momentum (shown to be approximately true in Johnston 1998), then a satellite at $R_{\rm GC}$ with velocity $v$ in the range $0< v < 3 v_{\rm c}/2$ will have an orbital time period $T_{\rm orb}$ in the range $T_{\rm circ}(R_{\rm GC})/2 < T_{\rm orb} < 2 T_{\rm circ}(R_{\rm GC})$, so equation (\[torb\]) is expected to be good to within a factor of 2. Several of the dwarf spheroidal satellites do have proper motion measurements – with large errors. These could in principle be used to determine a specific time period, but since the mass loss estimate itself is expected to be uncertain to within a factor of 2, the approximation in equation (\[torb\]) is deemed to be sufficiently accurate. For those with proper motions we calculate the angle $\theta$ between our line of sight and the plane perpendicular to the velocity vector of the satellite in the Galactic rest frame. Otherwise, we assume the orbit is circular and calculate $\theta$ as the angle between our line of sight and the Galactocentric radius vector of the satellite. ------------ -------------- --------------- ----------- -------------- Name $R_{\rm GC}$ $T_{\rm orb}$ $\theta$ $(df/dt)_2$ (kpc) (Gyr) (degrees) (Gyr$^{-1}$) Carina 8.66E+01 2.60E+00 2.35E+01 $<$ 3.31E-01 Draco 7.20E+01 2.16E+00 4.50E+01 $<$ 2.18E-01 Fornax 1.22E+02 3.66E+00 2.94E+01 $<$ 6.17E-02 LeoI 2.02E+02 6.05E+00 5.61E+01 $<$ 5.92E-02 LeoII 2.10E+02 6.29E+00 3.66E+01 $<$ 1.44E-01 Sculptor 7.22E+01 2.16E+00 1.64E+01 $<$ 2.84E-01 Sextans 8.60E+01 2.58E+00 4.36E+01 $<$ 2.61E-01 Ursa Minor 6.59E+01 1.98E+00 1.30E+01 $<$ 3.22E-01 ------------ -------------- --------------- ----------- -------------- Columns: (1) name; (2) Galactocentric distance; (3) time period of circular orbit at that distance; (4) angle between line of sight and plane perpendicular to satellite’s velocity, calculated from proper motion measurements for Sculptor [@s95] and Ursa Minor [@s98]; (5) mass loss rate estimate from equation (\[dfdt2\]). All these quantities are shown in [Table \[dsphtab\]]{}. The estimates $(df/dt)_2$ range from a few percent (Fornax and Leo I) up to more than 30 percent in the next Gyr (Ursa Minor and Carina). They do not correlate well with previous calculations that attempted to determine the robustness of each object either from the ratio of the expected to observed tidal radius (where the former was calculated given a dynamical estimate of the satellite’s mass), or of the external tidal field to internal field (see IH95). Nevertheless, these mass loss rates clearly indicate that the satellite system of the Milky Way could easily be diminished by several members in the next 10 Gyrs, which in turn suggests that there may have been several more satellites of the Milky Way in the past. If these dSph do indeed have such large mass loss rates, why have we not detected tidal streamers? In the cases of Ursa Minor and Sculptor this question was addressed by Johnston (1998), using a semi-analytic technique. She found that if each was losing mass at the rate of 10 percent per Gyr the local number count densities along the streamers (i.e. not averaged over annuli centred in the cluster) would not exceed 1 percent of the background star counts predicted by the Bahcall-Soneira [@bs80] model of the Milky Way. Hence, even increasing these mass loss rates to the tabulated values would not make the streamers striking features in the sky. However, the estimates bolster the notion that these streamers might be discovered and traced over large angular extents using integrated star counts (such as the method of Great Circle Cell Counts proposed by Johnston, Hernquist & Bolte 1996) or with color and velocity information to distinguish them from the background. ### Globular clusters ------ -------------- --------------- ----------- ----------- -------------- -------------- -------------- NGC $R_{\rm GC}$ $T_{\rm orb}$ $\theta$ $\gamma$ $(df/dt)_1$ $(df/dt)_2$ G&O (kpc) (Gyr) (degrees) (Gyr$^{-1}$) (Gyr$^{-1}$) (Gyr$^{-1}$) 288 1.14E+01 3.43E-01 4.19E+00 -5.21E-01 5.26E-02 1.26E-02 1.10E-01 362 9.04E+00 2.71E-01 1.40E+01 -1.58E-01 6.22E-01 5.77E-01 3.54E-02 1904 1.81E+01 5.44E-01 8.66E+01 – – $<$ 8.51E-03 3.54E-02 2808 1.07E+01 3.20E-01 2.33E+01 – – $<$ 4.37E-02 1.61E-02 3201 8.85E+00 2.65E-01 4.95E+01 – – $<$ 4.50E-01 3.45E-02 4590 9.94E+00 2.98E-01 5.17E-01 – – $<$ 1.28E+00 8.22E-03 5824 2.60E+01 7.81E-01 7.94E+01 -1.33E+00 6.46E-02 1.18E-01 3.06E-03 6864 1.17E+01 3.52E-01 8.60E+01 – – $<$ 4.91E-02 1.89E-02 6934 1.17E+01 3.52E-01 4.33E+01 – – $<$ 4.18E-01 2.89E-02 6981 1.22E+01 3.65E-01 5.84E+01 – – $<$ 2.96E-01 1.76E-02 7078 1.01E+01 3.02E-01 5.06E+01 – – $<$ 7.00E-02 2.17E-02 7089 1.01E+01 3.02E-01 1.91E+01 -1.62E+00 1.29E-01 3.13E-01 5.76-03 ------ -------------- --------------- ----------- ----------- -------------- -------------- -------------- Columns: (1) name; (2) Galactocentric distance; (3) time period of circular orbit at that distance; (4) angle between line of sight and plane perpendicular to satellite’s velocity, calculated from proper motions in Dauphole et al. 1996; (5) slope of extra-tidal star surface density profile; (6) mass loss rate estimate from equation (\[dfdt1\]); (7) mass loss rate estimate from equation (\[dfdt2\]) (8) mass loss rate estimate from Gnedin & Ostriker (1997). [Table \[globstab\]]{} presents the results of mass loss estimation for the 12 globular clusters analysed by G95 (using their tables 3-14, kindly made available to us by C. Grillmair). Four of these clusters have well-defined tidal tails with slopes $\gamma \approx 1$ (given in column 5), and in these cases both $(df/dt)_1$ and $(df/dt)_2$ are calculated. In the other cases, only the latter estimate is made as an upper limit on the mass loss rate. The estimated limits for the mass loss rates range from a few to over 100 percent in the next Gyr, again implying that the Galaxy’s globular cluster system will evolve substantially in the next Hubble time. This provides observational support for the many purely theoretical studies that have reached the same conclusion using semi-analytic models [@aho88; @go97; @mw97; @v97; @ch96]. The last column shows the destruction rates predicted by Gnedin & Ostriker (1997). There is no striking correspondence between our ‘observational’ results and the purely ‘theoretical’ values. However, the upper limit we calculate is only in direct contradiction with the theoretical calculation in the cases of NGC 288 and NGC 1904. In the former case our viewing angle is favourable for making an accurate estimate, but in the latter the value of $\theta$ is sufficiently large that we might expect the result to be confused by the debris geometry. In general, our mass loss rates are higher, but Gnedin & Ostriker (1997) themselves point out that their destruction rates should be taken as a lower limit since the rates could be increased with the inclusion of other effects ignored in their study (such as a mass-spectrum). ### Future prospects The results in Tables \[dsphtab\] and \[globstab\] should be treated with some caution as in most cases the observed densities of extra-tidal stars do not follow $\Sigma(r) \propto r^{-1}$ predicted by the simple model which we use to interpret the data. However, the original profiles were made either using star counts directly, or with additional photometry to subtract off some of the background. Since we expect the tidal debris to typically have velocities within $\pm 10 {\rm \, km\, s^{-1}}$ of the satellite, a study of radial velocities around a cluster has the potential of refining these measurements considerably. Further in the future, the astrometric satellites SIM (the Space Interferometry Mission) and GAIA (the Galactic Astrometric Imaging Satellite) promise proper motion measurements with a few $\mu arcsec$ accuracy (or tangential velocities to better than 1 km s$^{-1}$ out to tens of kpc), and a similar advance in identifying debris. We have also, for the sake of simplicity, restricted our discussion to annularly averaged surface densities. Clearly, more information is contained in two-dimensional surface density maps [@g95]. However, these will be more sensitive to the orbital phase and the mass loss history of the satellite and would require more detailed analytic modeling to interpret. Determining the IMF from observations of the PDMF ------------------------------------------------- The method proposed in §4.5 uses observations of a cluster’s PDMF and extra-tidal stars to place some limits on the IMF. This is clearly a powerful tool for making progress towards understanding whether the IMF is in any sense ‘universal’. Of course it is non-trivial to find the global PDMF, the local mass function in the exterior, and to detect extra-tidal stars around a cluster. However, there are currently two examples in the literature where this has already been done – M15 and M55 (see G95; Piotto, Cool & King 1997; Zaggia et al. 1997). In the case of M15, Piotto, Cool & King (1997) argue that mass-segregation is important only in the centre of the cluster and that their local sample is a fair representation of the global PDMF. Unfortunately, in the absence of mass segregation our method would find no evolution of the IMF since it does not model differential mass-loss due to relaxation effects, but only steady stripping of the most weakly bound stars. (Since Piotto et al. 1997 make the same statement about NGC 6397, which has a very different $x$ from M15, this might imply that the differences can only be due to relaxation effects or an intrinsically different IMF – it is unclear whether this is a robust argument, or whether a low level of mass segregation farther out in the cluster might be sufficient to account for the differences through tidal shocking.) In the case of M55, Zaggia, Piotto & Capaccioli (1997) find some evidence for extra-tidal stars, but do not give the density of this material since there are several other effects that it could be attributed to. Despite these problems, these studies show that it is currently feasible to design future observations that can address these issues. CONCLUSIONS =========== We summarise our main conclusions as follows: 1. Observations of extra-tidal stars and enhanced velocity dispersions in the outskirts of clusters can be attributed to tidally stripped material. The star counts become dominated by unbound stars at the point where the slope of the surface density profile changes or the dispersion reaches a minimum. This should not be identified with the tidal radius of the cluster since the edge of the bound population can still lie significantly beyond this radius. 2. The mass loss rate from a Galactic satellite can be directly estimated from the population of extra-tidal stars within a few of its tidal radii, the orbital time period and our line of sight. 3. Using current observations we calculated the mass-loss rate from dwarf spheroidal satellites and globular clusters and found that both systems will undergo significant evolution in the next Hubble time. Future observations should be able to place much stronger limits on these destruction rates. 4. Mass-loss estimate together with measurements of the local and global PDMFs of a cluster can place limits on its IMF. Our review of the literature suggests that it is currently observationally feasible to carry out this program. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to Piet Hut, Douglas Heggie and the other occupants of E-building at the IAS for helpful discussions, and to the IAS, IoA Cambridge and Sterrewacht Leiden for hospitality. We thank Mike Irwin and Carl Grillmair for providing data used for our analysis. This work was supported in part by the National Center for Supercomputing Applications, the Pittsburgh Supercomputer Center, and the EPCC; NASA Grant NAGW–2422 and the NSF under grants AST 90–18526, ASC 93–18185, the Presidential Faculty Fellows Program; funds from the IAS, a PPARC theory grant, an EU Marie Curie Fellowship and the generous support of the British Council. Aguilar, L., Hut, P. & Ostriker, J.P., 1988, , 335, 720 Bahcall, J. N. & Soneira, R. M., 1980, , 44, 73 )[bv92]{} Bergbusch, P. A. & VandenBerg, D. A., 1992, , 81, 163 Binney, J. & Tremaine, S., 1987, Galactic Dynamics (Princeton University Press, Princeton) Capaccioli, M., Ortolani, S. & Piotto, G., 1991, , 244, 298 Capaccioli, M., Piotto, G. & Stiavelli, M., 1993, , 261, 819 Capriotti, E.R. & Hawley, S.L., 1996, , 464, 765 Chernoff, D.F. & Weinberg, M.D., 1990, , 351, 121 Da Costa, G.S. & Freeman, K., 1976, , 206, 128 Dauphole, B., Geffert, M., Colin, J., Ducourant, C., Odenkirchen, M. & Tucholke, H.J., 1996, , 313, 119 Djorgovski, S., Piotto, G. & Capaccioli, M., 1993, , 105, 2148 Drukier, G.A., Slavin, S.D., Cohn, H.N., Lugger, P.M., Berrington, R.C., Murphy, B.W., & Seitzer, P.O., 1997, , 115, 708 Elson, R.A.W., Hut, P. & Inagaki, S., 1987, , 25, 565 Elson, R.A.W., Sigurdsson, S., Davies, M.B., Hurley, J. & Gilmore, G., 1998, , in press Faber, S.M. & Lin, D.N.C., 1983, , 266, L17 Fall, M. & Rees, M.J., 1977, , 181, 37P Feast, M.W., Thackeray & Wesselink, A.J., 1961, , 122, 433 Fischer, P., Pryor, C., Murray, S., Mateo, M. & Richtler, T., , 1998, 115, 592 Gnedin, O. & Ostriker, J.P., 1997, , 488, 579 Gnedin, O., Hernquist, L. & Ostriker, J.P., 1998, , in press Grillmair, C.J., 1998, in Galactic Halos, ed. D. Zaritsky, A.S.P. Conf. Ser, 136, 45 Grillmair, C.J., Freeman, K.C., Irwin, M. & Quinn, P.J., 1995, , 109, 2553 Gunn, J.E. & Griffin, R.F., 1979, , 84, 752 Hernquist, L., 1990, , 356, 359 Hernquist, L. & Ostriker, J.P., 1992, , 386, 375 Hernquist, L., Sigurdsson, S. & Bryan, G.L., 1995, , 446, 717 Hut, P. & Djorgovski, S., 1992, Nature, 359, 806 Ibata, R. A., Wyse, R.F.G., Gilmore, G., Irwin, M. J. & Suntzeff, N.B., 1997, , 113, 634 Irwin, M. J. & Hatzidimitriou, D., 1995, , 277, 1354 (IH95) Johnston, K.V., 1998, , 495, 297 Johnston, K. V., Spergel, D. N. & Hernquist, L., 1995, , 451, 598 Johnston, K. V., Hernquist, L. & Bolte, M., 1996, , 465, 278 Johnston, K. V., Hernquist, L. & Weinberg, M.D., 1998, , submitted King, I. R., 1962, , 67, 471 Kuhn, J. R., Smith, H. A. & Hawley, S. L., 1996, , 469, L93 Kundic, T. & Ostriker, J.P., 1995, , 438, 702 McClure, R.D. et al., 1986, , 307, L49 Merritt, D., Meylan, G. & Mayor, M., 1997, , 114, 1074 Meylan, G. & Heggie, D.C., 1997, , 8, 1 Miyamoto, M. & Nagai, R., 1975, , 27, 533 Murali, C. & Weinberg, M. D. 1997, , 291, 717 Oh, K.S., Lin, D.N.C. & Aarseth, S.J., 1995, , 442, 142 Piotto, G., Cool, A.M. & King, I.R., 1997, , 113, 1354 Rosenberg, A., Saviane, I., Piotto, G., Aparicio, A. & Zaggia, S.R., 1998, , 115, 658 Sigurdsson, S. & Phinney, E.S., 1995, , 99, 609 Sigurdsson, S., He, B., Melhem, R. Hernquist, L., 1997, Computers in Physics, 11.4, 378 Spitzer, L., 1958, , 127, 17 Schweitzer, A. E., Cudworth, K. M., Majewski, S. R. & Suntzeff, N. B., 1995, , 110, 2747 Schweitzer, A. E., Cudworth, K. M. & Majewski, S. R., 1998, , submitted Tremaine, S., 1993 in Back to the Galaxy eds. S. S. Holt & F. Verter, (AIP Conf. Proc. : New York), p.599 Velázquez , H. & White, S.D.M., 1995, , 275, L23 Vesperini, E., 1997, , 287, 915 Vesperini, E. & Heggie, D.C., 1997, , 289, 898 Weinberg, M.D., 1994 , 108, 1403 Zaggia, S.R., Piotto, G. & Capaccioli, M., 1997, , 327, 1004 Zoccali, M., Piotto, G., Zaggia, S.R., & Capaccioli, M., 1998, , 331, 541
--- abstract: 'Approximate Bayesian computation (ABC) and synthetic likelihood (SL) techniques have enabled the use of Bayesian inference for models that may be simulated, but for which the likelihood cannot be evaluated pointwise at values of an unknown parameter $\theta$. The main idea in ABC and SL is to, for different values of $\theta$ (usually chosen using a Monte Carlo algorithm), build estimates of the likelihood based on simulations from the model conditional on $\theta$. The quality of these estimates determines the efficiency of an ABC/SL algorithm. In standard ABC/SL, the only means to improve an estimated likelihood at $\theta$ is to simulate more times from the model conditional on $\theta$, which is infeasible in cases where the simulator is computationally expensive. In this paper we describe how to use bootstrapping as a means for improving SL estimates whilst using fewer simulations from the model, and also investigate its use in ABC. Further, we investigate the use of the bag of little bootstraps as a means for applying this approach to large datasets, yielding Monte Carlo algorithms that accurately approximate posterior distributions whilst only simulating subsamples of the full data. Examples of the approach applied to i.i.d., temporal and spatial data are given.' author: - 'Richard G. Everitt' bibliography: - 'bootstrap\_paper.bib' title: Bootstrapped synthetic likelihood --- Introduction {#sec:Introduction} ============ This paper is concerned with performing Bayesian inference for parameter $\theta$ conditional on data $y$ (consisting of $N$ data points) using the prior $p\left(\theta\right)$ and likelihood $L_{\theta}\left(y\right)$, in situations where the likelihood cannot be evaluated pointwise at $\theta$ but where it is possible to simulate from $L_{\theta}$ for each $\theta$. Such a choice of $L_{\theta}$ is sometimes referred to as an “implicit” model; exact Bayesian inference is rarely possible in this setting. The most common approach to inference in this setting is approximate Bayesian computation (ABC): a technique for approximate Bayesian inference originally introduced in the population genetics literature [@Pritchard1999; @Beaumont2002], but which is now used for a wide range of applications including ecology [@VanderVaart2015], cosmology [@Akeret2015; @Jennings2017], epidemiology [@Kypraios2017] and econometrics [@Martin2017]. ABC has also been used for inference for models where that are intractable due to the presence of a partition function [@Grelaud2009; @Everitt2012], such as undirected graphical models. The key idea in ABC is to approximate the likelihood at each $\theta$ based on simulations from $L_{\theta}$. Usually a non-parametric kernel estimator of the likelihood is employed, with a bandwidth parameter $\epsilon$ used to trade off properties of the estimator: for $\epsilon=0$ the estimator has no bias, but a very high variance, with the bias increasing and the variance reducing if a larger $\epsilon$ is chosen. In cases where a sufficient (with respect to $\theta$) vector of statistics $S_{N}$ is available, ABC uses instead an approximation to the distribution $f_{\theta}$ of $S_{N}$ given $\theta$ in order to create an approximation with a lower variance. In practice it is usually only possible to choose a near-sufficient vector of “summary” statistics, introducing a further approximation. An alternative approach, the focus of this paper, is to use a Gaussian approximation [@Wood2010f] $\bar{f}_{\theta}$ on the summary statistic likelihood, with mean $\mu_{\theta}$ and covariance $\Sigma_{\theta}$. This approach is known as “synthetic likelihood” (SL), with $\bar{f}_{\theta}$ being estimated by $$\widehat{f}_{\theta}=\mathcal{N}\left(\cdot\mid\widehat{\mu}_{\theta},\widehat{\Sigma}_{\theta}\right),\label{eq:sl_llhd}$$ where $$\widehat{\mu}_{\theta}=\frac{1}{M}\sum_{m=1}^{M}S_{N}^{(m)},\label{eq:sl_mu}$$ $$\hat{\Sigma}_{\theta}=\hat{\mathbb{V}}\left[S_{N}^{(1:M)}\right]:=\frac{ss^{T}}{M-1},\label{eq:sl_sigma}$$ with $s=\left(S_{N}^{(1)}-\widehat{\mu}_{M,\theta},...,S_{N}^{(M)}-\widehat{\mu}_{M,\theta}\right)$ and $S_{N}^{(m)}$ is the summary statistic vector found from $x^{(m)}\sim L_{\theta}$ for $1\leq m\leq M$ for some $M$ (thus $\hat{\mathbb{V}}$ denotes taking the sample variance). The approximate likelihood $\widehat{f}_{\theta}$ is evaluated at the statistic vector $S(y)$ of the observed data $y$. Clearly this approach only provides a good approximation when the true summary statistic likelihood is approximately Gaussian, but in practice this occurs in a wide range of applications. this method in a setting where the summary statistics are regression coefficients (whose distribution is approximately Gaussian), and recommends transforming $S_{N}$ for cases where the Gaussian assumption is not appropriate the original parameterisation. The SL approximation, using the estimate $\widehat{f}_{\theta}$ may be used within an MCMC Monte Carlo methods where the likelihood is estimated rather than known exactly have been much studied in recent years. If $\widehat{f}_{\theta}$ were an unbiased estimate of $\bar{f}_{\theta}$, SL would result in an instance of a pseudo-marginal method [@Andrieu2009], of which ABC is also a special case. For a pseudo-marginal MCMC algorithm, the target distribution of $\theta$ is precisely the same as it is when the exact likelihood is used no matter the variance of $\widehat{f}_{\theta}$, although a larger variance results in higher variance estimates from the MCMC output [@Andrieu2014a]. However the estimate in equation \[eq:sl\_llhd\] is biased, thus Monte Carlo methods using $\widehat{f}_{\theta}$ are a particular case of “noisy” Monte Carlo methods [@Alquier2016], in which the exact target is not obtained. For such algorithms, under certain conditions, it is possible to show that the target of the noisy algorithm (using $\widehat{f}_{\theta}$) converges to the target of the corresponding exact algorithm (i.e. the “ideal” algorithm that uses $\bar{f}_{\theta}$). In the case of SL, we have such a result as $M\rightarrow\infty$. Thus in practice, an increased value for $M$ will result in reduced bias and variance of estimates from the SL-MCMC algorithm, although @Price2017 find empirically that $M$ does not usually need to be very large in order that the bias in SL-MCMC is low. @Price2017 also introduce an unbiased estimator of $\bar{f}_{\theta}$ yielding a variant of SL-MCMC that has exactly the correct target no matter the choice of $M$. However, empirically they find that the results are not significantly improved over standard SL-MCMC. @Price2017 [@Everitt2017] find that SL often outperforms ABC, and is easier to tune, even in some cases where the distribution of the summary statistics is clearly not Gaussian. However, SL can be expensive to implement for simulators that have a high computational cost, since the simulator needs to be run $M$ times for each $\theta$ that is visited. Even if the bias is often low for relatively small values of $M$, @Price2017 find empirically that for small $M$ the efficiency of SL-MCMC is poor since the variance of the likelihood estimator is prohibitively large. In some situations, it may be possible to exploit the embarrassingly parallel nature of SL and run the $M$ simulators in parallel. However, this is not always possible, in cases where we wish to use parallelism to explore multiple $\theta$ points simultaneously, or when a single run of the simulator itself requires parallel computing. In addition to running the simulator $M$ times, SL can be costly when the size $N$ of the data is large (sometimes known as “tall data”). In this paper we introduce an approach to using SL where only subsets of data of size $n\ll N$ are simulated. Using SL (or ABC) is appealing for tall data, since prior to running an inference algorithm the dimensionality of the data is reduced by taking a lower dimensional summary statistic vector. The method then never uses the full data; the only point in the algorithm that scales with $N$ is the simulation from $l_{\theta}$. In this paper we show that in some cases this requirement can be removed, since it is possible to accurately approximate the posterior using only simulations of size $n$. One striking result in the literature on Monte Carlo methods for tall data [@Bardenet2017] is that previous methods that use subsamples of size $n$ (all outside of the ABC/SL context), need to be run for $N/n$ times as long in order to give the same accuracy as an algorithm that runs on the full data (with the exception of @Pollock2016). We show empirically that our approach does not appear to have this requirement (as seen in section \[subsec:Ising-model\]). In summary, this paper investigates methods that provide likelihood estimates at (sometimes substantially) lower simulation cost, through reducing the number of simulations $M$ needed from the likelihood, and also in some cases the size of each simulation from $N$ to $n$. Methodology {#sec:Methodology} =========== This paper investigates using the bootstrap [@Efron1979] as a means for approximating $f_{\theta}$ using fewer simulations from the likelihood. Further, we consider the case where the likelihood is expensive due to its consisting of a large number $N$ of data points. In this case we investigate the use of the bag of little bootstraps (BLB) [@Kleiner2014] as a means for approximating $f_{\theta}$ that involves simulating only subsets of the full data. This section gives an overview of the paper, and describes its relationship to previous work. To estimate the synthetic likelihood, we must estimate the functions $\mu_{\theta}$ and $\Sigma_{\theta}$ of $\theta$. The standard SL approach is to estimate $\mu_{\theta}$ and $\Sigma_{\theta}$ independently for each $\theta$. @Meeds2014a present an alternative in which the variance of estimates is lowered by using a Gaussian process model of each function. For $\Sigma_{\theta}$, this requires introducing an approximation by modelling only the diagonal of the matrix. In this paper we remove this requirement by using bootstrap estimators of $\Sigma_{\theta}$ which we find empirically to have a lower variance than the raw estimates in equation \[eq:sl\_sigma\]. @An2016 uses the Graphical Lasso as an alternative approach to yield low variance estimates of $\Sigma_{\theta}$. Section \[subsec:Bootstrapped-synthetic-likelihoo\] describes how to use bootstrapping to estimate a synthetic likelihood, outlines some conditions under which this is possible, and describes how to implement the approach in a computationally efficient way when the approximation is used within a Monte Carlo algorithm. The use of the bootstrap in this context has not been considered previously, although resampling-based ideas have previously been used in the ABC literature [@Peters2010u; @Buzbas2015; @Vo2015; @Zhu2016a]. The only directly related work to this paper is that of @Buzbas2015, in which a small number of simulations from the likelihood are used to construct an approximate likelihood. A simulation from this approximate likelihood is given by a weighted resampling of the existing simulations. This approach may be seen as a relatively crude non-parametric estimator of the likelihood, which we may expect to be improved using a more sophisticated estimator, such as Gaussian processes [@Wilkinson2014] (also used outside of the ABC context in @Drovandi2015a) or neural density estimators [@Papamakarios2016]. Our approach differs in that we use a (parametric) conditional Gaussian model of the likelihood, and use bootstrapping to estimate its variance. The method is applicable in any model for which a bootstrapping procedure is available. We give suggestions for temporal and spatial models in section \[subsec:Temporal-and-spatial\]. In section \[subsec:Synthetic-likelihood-with\] we extend the method to cases where a single simulation requires simulating a large number of data points. Here the BLB is used, leading to a cost that is independent of the size of the data with little loss of accuracy. Again we describe how this method may be used efficiently within a Monte Carlo algorithm. Low variance estimators of $\mu_{\theta}$ cannot be found using the bootstrap. Therefore we use a variant on the regression ideas in @Meeds2014a to estimate $\mu_{\theta}$. Section \[subsec:Regression-for-estimates\] describes the approach in full, in which $\mu_{\theta}$ is estimated via regression and $\Sigma_{\theta}$ is approximated by bootstrap estimates. Empirical results for each approach are given in section \[sec:Empirical-results\], where we begin by studying a toy example with independent data in section \[subsec:Toy-example\] in order to establish the behaviour of the methods on an example where the ground truth is known. This is followed by temporal data (from the Lotka-Volterra model) in section \[subsec:Lotka-Volterra-model\], where the summary statistic vector is 9-dimensional, where the likelihood is difficult to estimate, and where the summary statistic distribution and posterior are not close to Gaussian. Finally we study spatial data (from the Ising model) in section \[subsec:Ising-model\], where we focus on using the BLB to obtain a good approximation to the true posterior in a tall data setting. Sections \[subsec:Toy-example\] and \[subsec:Ising-model\] both investigate empirically the impact of changing the quality of estimates of $\mu_{\theta}$ and $\Sigma_{\theta}$ on estimates of the SL. We then conclude with a discussion in section \[sec:Conclusions\]. Bootstrapped synthetic likelihood\[subsec:Bootstrapped-synthetic-likelihoo\] ---------------------------------------------------------------------------- In this section we introduce the bootstrap, and describe how it may be used within SL. Our notation is the same as in section \[sec:Introduction\], except that we are now more precise about the distributions of each quantity. The most common use of the the bootstrap is to estimate the variance $\mathbb{V}_{Q \left(P\right)}\left[\theta_{N}\left(P_{N}\right)\right]$ (or some other property) of the sampling distribution $Q$ of an estimator $\theta_{N}\left(P_{N}\right)$ of some population value $\theta\left(P\right)$ based on data $y$ (with empirical distribution $P_{N}$) from some unknown population distribution $P$. The useful result exploited by the bootstrap is that the variance $\mathbb{V}_{Q\left(P\right)}\left[\theta_{N}\left(P_{N}\right)\right]$ may be accurately approximated by $\mathbb{V}_{Q\left(P_{N}\right)}\left[\theta_{N}\left(P_{N}\right)\right]$: i.e. we may approximate the variance of $\theta_{N}$ by using the empirical distribution of the data in place of the true population distribution. ### Using the bootstrap to approximate $\Sigma_{\theta}$\[subsec:Using-the-bootstrap\] In the SL context we may exploit this idea since we wish to estimate the variance $\Sigma_{\theta}=\mathbb{V}_{f_{\theta}\left(L_{\theta}\right)}\left[S_{N}\right]$ (to plug into the SL approximation) of an estimator $S_{N} = S_{N}\left(L_{N,\theta}\right)$ of $S\left(L_{\theta}\right)$. Here $L_{N,\theta}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}$ is the empirical distribution of a sample $x$ from $L_{\theta}$. Using the bootstrap, we may approximate $\Sigma_{\theta}$ by $\varSigma_{\theta}=\mathbb{V}_{f_{\theta}\left(L_{N,\theta}\right)}\left[S_{N} \right]$. When using SL, it is possible to simulate multiple ($M$) samples from $L_{\theta}$, thus we introduce an additional superscript $m$ into all quantities that depend on a sample $x^{(m)}$ from $L_{\theta}$. To improve our approximation of $\Sigma_{\theta}$ we take the sample average of multiple approximations $\varSigma_{\theta}^{(m)}=\mathbb{V}_{f_{\theta}\left(L_{N,\theta}^{(m)}\right)}\left[S_{N}^{(m)}\right]$, yielding the approximation $\Sigma_{\theta}^{\text{boot}}=\frac{1}{M}\sum_{m=1}^{M}\varSigma_{\theta}^{(m)}$. Standard results on the consistency of bootstrap estimates give us that each $\varSigma_{\theta}^{(m)}\rightarrow\Sigma_{\theta}$ as $N\rightarrow\infty$ (see @Horowitz2001 for details). For any $N$, the average $\Sigma_{\theta}^{\text{boot}}$ yields a lower variance estimator than the individual $\varSigma_{\theta}^{(m)}$. The quantities $\left\{ \varSigma_{\theta}^{(m)}\right\} _{m=1}^{M}$ are not available analytically, but may be estimated via resampling the single simulation $x^{(m)}$ from $L_{\theta}$. $R$ resamples $\left\{ x^{(m,r)}\right\} _{r=1}^{R}$ from $L_{N,\theta}^{(m)}$ yield the Monte Carlo estimate $$\hat{\varSigma}_{\theta}^{(m)}=\hat{\mathbb{V}}_{f_{\theta}\left(L_{N,\theta}^{(m)}\right)}\left[S_{N}^{(m,1:R)}\right],$$ where $S_{N}^{(m,r)}\sim f_{\theta}\left(L_{N,\theta}^{(m)}\right)$ is the summary statistic vector found from $x^{(m,r)}$, with the sample variance $\hat{\mathbb{V}}_{f_{\theta}\left(L_{N,\theta}^{(m)}\right)}$ being taken over the $R$ summary statistic vectors. Let $\hat{\Sigma}_{\theta}^{\text{boot}}=\frac{1}{M}\sum_{m=1}^{M}\hat{\varSigma}_{\theta}^{(m)}$, where we for simplicity of notation we have omitted the dependence of $\hat{\Sigma}_{\theta}^{\text{boot}}$ on $M$ and $R$. Making computational savings through using bootstrapped SL (B-SL) compared to standard SL requires that resampling a single simulation $x$ is cheaper than simulating from the likelihood. This is the case for most applications, but we may make further computational savings when the B-SL approximation is used when $M$ is large, or when a large number of $\theta$ are used as is the case when this approximation is embedded within Monte Carlo methods. For i.i.d. data, a single resample involves sampling $N$ times without replacement from $\left\{ 1,...,N\right\} $, thus for $R$ resamples, $R\times N$ samples from $\left\{ 1,...,N\right\} $ are required; we denote such a sample by the matrix of indices $u=\left[u_{r,i}\right]_{r=1:R,i=1:N}$. We may make a computational saving by reusing this same index matrix for resampling every different simulation from the likelihood (i.e. for every value of $m$ for every $\theta$). Algorithm \[alg:boot\] summarises our proposed procedure for estimating $\Sigma_{\theta}$. We will observe empirically in section \[sec:Empirical-results\] that, compared to standard SL, this approach significantly reduces the variance of likelihood estimates without introducing noticeable bias. $x^{(m)} \sim L_{\theta}$ $x^{(m,r)} \sim L_{N,\theta}^{(m)}$ Find $S_{N}^{(m,r)}$ from $x^{(m,r)}$. $\hat{\varSigma}_{\theta}^{(m)}=\hat{\mathbb{V}}_{f_{N,\theta}\left(L_{N,\theta}^{(m)}\right)}\left[S_{N}^{(m,1:R)}\right]$. Calculate $\hat{\Sigma}_{\theta}^{\text{boot}}=\frac{1}{M}\sum_{m=1}^{M}\hat{\varSigma}_{\theta}^{(m)}$. ### Bootstrapped approximate Bayesian computation One may also consider using this approach in ABC, where the bootstrap is used to obtain lower variance estimates of the ABC likelihood. For each sample $x^{\left(m\right)}$, the ABC kernel (often the uniform kernel with bandwidth $\epsilon$) is evaluated at the summary statistic $S_{N}^{(m,r)}$ of each resample, and the average of these results is taken to give the estimated likelihood for sample $m$. The bootstrapped ABC (B-ABC) likelihood is then the average of the estimated likelihoods for each $m$. However, in this case we find that the bootstrapped estimates introduce a significant bias into likelihood estimates. Specifically, we find that a bootstrapped ABC (B-ABC) likelihood for some tolerance $\epsilon_{1}$ resembles the standard ABC likelihood for some $\epsilon_{2}>\epsilon_{1}$. It is not clear in general whether lower variance estimates would be achieved by using the standard ABC likelihood estimate for $\epsilon_{2}$, or by using a B-ABC likelihood estimate for $\epsilon_{1}$. Empirical results (section \[subsec:Lotka-Volterra-model\]) suggest that the bootstrapped approach can outperform the standard approach, but need not always be the case. A significant drawback of the bootstrapped approach is that the posterior will not converge to the true posterior as the tolerance decreases to zero. Additionally we note that there are alternative methods for achieving lower variance likelihood estimates in ABC, such as @Prangle2016. ### Temporal and spatial models\[subsec:Temporal-and-spatial\] B-SL may also be used in situations outside of i.i.d. data, in any situation where a bootstrapping procedure has been defined (the review papers @Horowitz2001 and @Kreiss2012 give an overview of the literature). Here we discuss the use of the block bootstrap, which was introduced for bootstrapping stationary time series by @Kunsch1989. Instead of resampling data independently, the block bootstrap instead resamples blocks of data. These blocks are chosen to be sufficiently large such that they retain the short range dependence structure of the data, so that a resampled time series constructed by concatenating resampled blocks has similar statistical properties to a real sample. Below we outline the scheme that is used in this paper; others are also possible (see @Kreiss2012 for a review). Suppose that $y_{1:N}$ is time indexed data, and that $x_{1:N}^{(m)}$ is a time series sampled from $l_{\theta}$. In the block bootstrap, using a block of length $B$ (for simplicity we consider the case where $B$ is a divisor of $N$), we first construct a set of overlapping blocks of indices of the variables $$\mathcal{B}=\left\{ (1:B),(2:B+1),...,(N-B+1:N)\right\} .\label{eq:index_set}$$ Then a resample $x_{1:N}^{(m,r)}$ from $x_{1:N}^{(m)}$ consists of $N/B$ concatenated blocks whose indices are sampled with replacement from $\mathcal{B}$. The summary statistics of $R$ resamples $\left\{ x_{1:N}^{(m,r)}\right\} _{r=1}^{R}$ may then be computed, followed by calculating the sample variance of these statistics over the $R$ resamples. This procedure is repeated for each sample $m$, with the approximation to $\Sigma_{\theta}$ taken to be the sample mean of the variance estimate for each $m$. As for the i.i.d. case, the resampling indices may be the same for every $m$ and every $\theta$: again the indices may be stored in a matrix $u=\left[u_{r,i}\right]_{r=1:R,i=1:N}$, where in this case each row is given by concatenating the blocks of indices sampled from $\mathcal{B}$. In this paper we also study the case of stationary spatial models, where the variables are arranged on a regular two-dimensional grid. In this case we use the analogous scheme, where each resample is constructed of randomly selected sub-tiles of the sample $x^{(m)}$. Again the indices of the sub-tiles may be the same for every $m$ and $\theta$. In both temporal and spatial models, for many summary statistics there is an additional computational saving that is possible when $R$ is large. We focus on the temporal case for simplicity. Suppose that the summary statistic for a resampled time series may be computed directly from corresponding statistics computed from its constituent blocks. For example, the sample mean $\frac{1}{N}\sum_{i=1}^{N}x_{i}^{(m)}$ of a time series $x_{1:N}$ is given by $\frac{1}{N}\sum_{b=1}^{N-B+1}n_{k}B\left(\frac{1}{B}\sum_{k=1}^{B}x_{s_{b}+k-1}^{(m)}\right),$ where $s_{b}$ is the index at the beginning of the $b$th block, the expression in the brackets is the sample mean of the $b$th block, and $n_{k}$ is the number of times block $k$ appears in the resample. In such a case, for each sampled time series $x^{(m)}$, rather than compute the statistic for each resampled time series, it may be cheaper to compute the statistic for each block then combine them. To see this, consider the case where the cost of computing the statistic is linear in the length $N$ of the time series. Here the cost of computing the statistic for each resampled time series is $O\left(RN\right)$, whereas when the statistics may be precomputed for each block the cost is $O\left(\left(B+R\right)\left(N-B+1\right)\right)$. Therefore as $R$ grows, the latter scheme offers a computational saving. Synthetic likelihood with the bag of little bootstraps\[subsec:Synthetic-likelihood-with\] ------------------------------------------------------------------------------------------ For i.i.d. or stationary models, in the case where $N$ is large, we may reduce the cost of B-SL through using the “bag of little bootstraps” introduced by @Kleiner2014. This approach avoids the simulation of data sets of size $N$, and instead constructs approximations based on subsamples of size $n$, where $n<N$. We begin by outlining the method mathematically, extending the notation of section \[subsec:Using-the-bootstrap\]. We use the notation $l_\theta$ to denote the likelihood of a dataset of size $n$ (as opposed to previously, where $L_\theta$ is used for data of size $N$). Recall that we wish to estimate the variance $\Sigma_{\theta}=\mathbb{V}_{f_{N,\theta}\left(L_{\theta}\right)}\left[S_{N}\right]$ of $S_{N}$. Using the bootstrap, we approximated $\Sigma_{\theta}$ by $\varSigma_{\theta}=\mathbb{V}_{f_{N,\theta}\left(L_{N,\theta}\right)}\left[S_{N}\right]$. The BLB instead uses the approximation $\varSigma_{n,\theta}=\mathbb{V}_{f_{N,\theta}\left(l_{n,\theta}\right)}\left[S_{N}\right]$ where $l_{n,\theta}=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$ is the empirical distribution of a sample $x$ of size $n$ from $l_{\theta}$. In this case, the empirical distribution used in place of $l_{\theta}$ only requires data of size $n$ to be simulated from the likelihood, but the resamples from $l_{n,\theta}$ are of size $N$. If $n=N$, this method is the same as the standard bootstrap, with $\varSigma_{\theta}=\varSigma_{N,\theta}$. The computational saving of the BLB arises since in practice, the resamples of size $N$ need not actually be constructed: all that needs to be stored are counts of the number of times each point in the subsample is used in the resample, then all of the calculations may be based on the subsample and these counts. As previously, we simulate multiple ($M$) samples (this time of size $n$) from $l_{\theta}$, and average over the approximations $\varSigma_{n,\theta}^{(m)}$ given by each sample yielding the approximation $\Sigma_{n,\theta}^{\text{blb}}=\frac{1}{M}\sum_{m=1}^{M}\varSigma_{n,\theta}^{(m)}$. This procedure differs slightly to the BLB as introduced in @Kleiner2014, where there is only a single dataset available. In such a case, the multiple simulations used are $M$ subsets (simulated without replacement) of the data. When used in SL (we refer to this approach as BLB-SL), we instead simulate each “subset” independently from the likelihood, enabling us to completely avoid the simulation of data of size $N$. For the standard BLB, $\Sigma_{n,\theta}^{\text{blb}}\rightarrow\Sigma_{\theta}$ as $N\rightarrow\infty$ for any sequence $n\rightarrow\infty$ and any fixed $M$, with convergence at the same rate as the bootstrap (see @Kleiner2014 for a precise statement of these results). Importantly these results hold for $n\ll N$, giving the promise of significant computational savings. The BLB may be combined with the block bootstrap to be used for stationary temporal and spatial data (previously considered for the temporal case in @Laptev2012). Focussing on the temporal case, for each $m$ we sample a time series of length $n$ from the likelihood. From this we construct blocks of time series, using the index set in equation \[eq:index\_set\] to index the blocks (using $B<n$). We may think of each resampled time series of length $N$ as a concatenation of $N/B$ blocks whose indices are sampled with replacement from $\mathcal{B}$, although in practice this concatenation need not actually be constructed: all that needs to be stored are the counts of the number of times each block is selected for each resample. As previously, the indices sampled from $\mathcal{B}$ may be the same for every $m$ and $\theta$. In addition, if the summary statistic for a resample may be computed directly from statistics of its constituent blocks, then for each $m$ we may compute the statistic for each block then combine them. The cost of this is $O\left(\left(B+R\right)\left(n-B+1\right)\right)$ which, crucially, is independent of $N$. For estimating the variance $\Sigma_{\theta}$ fir SL, the BLB potentially offers a significant advantage over the bootstrap, in that we need only simulate datasets of size $n$ rather than size $N$, whilst maintaining accuracy. The empirical investigations in sections \[subsec:Toy-example\] and \[subsec:Ising-model\] suggest that this potential is fulfilled in practice. Regression for estimates of the expectation\[subsec:Regression-for-estimates\] ------------------------------------------------------------------------------ The previous two sections focus exclusively on estimates of the variance $\Sigma_{\theta}$. However, to use SL, we also need an estimate of the mean $\mu_{\theta}$; in order that any computational saving is made through using the bootstrap or BLB to estimate $\Sigma_{\theta}$, the estimates of $\mu_{\theta}$ must not involve any further simulation from the likelihood. Bootstrapping does not lead to lower variance estimates of the mean, so other approaches are required. When using the bootstrap to estimate $\Sigma_{\theta}$, we simulate $M$ datasets of size $N$ from the likelihood. In this case we may simply use the standard estimator used in SL in equation \[eq:sl\_mu\]. When using the BLB, we simulate $M$ datasets of size $n$. For each dataset of size $n$, we may estimate $\mu_{\theta}$ by simulating a dataset of size $N$ from the corresponding empirical distribution $l_{n,\theta}^{(m)}$ then again use the estimator in equation \[eq:sl\_mu\]. Or alternatively, for statistics that satisfy a law of large numbers in the size of the data (including all of those used in sections \[subsec:Toy-example\] and \[subsec:Lotka-Volterra-model\]) it is sufficient (and introduces less variance) to calculate the statistic based on $n$ data points rather than $N$, then to simply use these raw estimates within equation \[eq:sl\_mu\]. Section \[subsec:Ising-model\] uses the same idea, except that in this case the statistic needs to be scaled by the size of the data, since it is a sum rather than an average. Estimates of $\mu_{\theta}$ constructed in this way, based on only $n$ data points, have a larger variance than those from $N$ data points. We expect this increased variance to increase the variance of SL estimates, but a further concern (borne out in practice) is that it also increases the bias of SL estimates. @Price2017 use an identity from @Ghurye1969 to correct for bias in the Gaussian estimator resulting from errors in the expectation estimate, but this identity is not applicable when using subsampling. In section \[sec:Empirical-results\] we observe empirically the bias introduced through using high variance estimates of $\mu_{\theta}$. The variance of these estimates may be reduced using regression, an idea used in several previous papers. @Meeds2014a [@Sherlock2017] describe MCMC schemes for estimating the likelihood as the MCMC algorithm runs. In both of these methods, at each new value of $\theta$ visited by the MCMC, random variables $x$ are simulated in order to estimate the likelihood (in @Meeds2014a these are the simulations from $l_{\theta}$ used in equations \[eq:sl\_mu\] and \[eq:sl\_sigma\]). The likelihood regression then makes use of the entire history of $\left(\theta,x\right)$ pairs built up as the MCMC runs. One weakness of the approach in @Sherlock2017 is that since the tails of the posterior are visited infrequently, the regression estimates of the likelihood in these regions has a higher variance. @Meeds2014a address this issue by, where needed, actively acquiring additional points in order to lower the variance of the regression estimate. Also related is @Moores2015 in which a preprocessing stage is used to train a regression, which in this case is used to smooth out the effect of using a finite value of $M$ in SL. In the current paper we instead use the approach in @Everitt2017c, in which a history of $\left(\theta,x\right)$ pairs is built up as an SMC algorithm runs, where low variance regression estimates in the tails are naturally obtained through beginning the SMC with a heavy tailed proposal. For simplicity we use a local linear regression within this SMC algorithm. In contrast to @Meeds2014a, we only perform this regression on $\mu_{\theta}$, since we use the bootstrap for approximating $\Sigma_{\theta}$. As remarked at the beginning of section \[sec:Methodology\], this removes a restriction of the @Meeds2014a approach, where the regression on $\Sigma_{\theta}$ necessitates diagonalising the covariance matrix. @Everitt2017c uses marginal SMC [@DelMoral2006c] to infer the posterior distribution. This algorithm maintains a population of weighted particles $\left\{ \left(\theta_{t}^{(p)},w_{t}^{(p)}\right)\right\} _{p=1}^{P}$ that for each $t$ approximate a target distribution $\pi_{t}\left(\theta\right)=p\left(\theta\right)\hat{f}_{\theta}^{\nu_{t}}\left(y\right)$, where $\hat{f}_{\theta}^{\nu_{t}}$ is an estimated likelihood raised to a power and $\nu_{t}$ moves from 0 to 1 as $t$ increases. At each target the kernel $K_{t}$ is used to move each particle, then update $$\tilde{w}_{t}^{(p)}=\frac{p\left(\theta_{t}^{(p)}\right)\hat{f}_{\theta_{t}^{(p)}}^{\nu_{t}}\left(y\right)}{\sum_{r=1}^{P}w_{t-1}^{(r)}K_{t}\left(\theta_{t}^{(p)}\mid\theta_{t-1}^{(r)}\right)}\label{eq:pmc_weight}$$ is used at target $t$ to calculate unnormalised weights $\tilde{w}_{t}^{(p)}$ for the particles, which are normalised to give $w_{t}^{(p)}$. The particles are then resampled. This approach has the advantage over other SMC methods when using estimated likelihoods that bias in the likelihood estimates does not accumulate as the algorithm runs. It is particularly appropriate for the case where $\hat{f}_{\theta}$ is an estimated synthetic likelihood, since the estimates $\hat{\mu}_{\theta}$ at each $\theta$ may be stored, and used to fit regression models to improve estimates of $\mu_{\theta}$ at future iterations. The annealed sequence of distributions, which are heavier tailed than the posterior in earlier iterations, leads to a useful set of estimates to use in the regression. @Everitt2017c uses a similar approach for doubly intractable distributions, and describes how (similar to @Sherlock2017), for any particle $\theta_{t}^{(p)}$, to use a KD-tree [@Bentley1975] to efficiently locate nearby previously values of $\theta$ that may be used in the regression. Algorithm \[alg:SMC\] outlines our approach in full, in which a local linear regression is used to estimate $\mu_{\theta_{t}^{(p)}}$. In this algorithm, each simulated $x$ has an additional superscript: the first refers to the index of the particle; the second to the index of the data simulated from $l_{\theta}$; the third (if present) to the index of the resample. As discussed in @Everitt2017c, this method is limited to applications where $\theta$ is of low to moderate dimension. $\theta^{(p)}_0 \sim p\left( \cdot \right)$ $x^{(p,m)}_{0} \sim f\left( \cdot \mid \theta^{(p)}_0 \right)$ $t=0$. $\theta^{(p)}_{t+1} \sim K_{t+1}\left( \cdot \mid \theta^{(p)}_{t} \right)$ $x^{(p)}_{t+1} \sim l_{\theta^{(p)}_{t+1}}\left( \cdot \right)$ Find the approximation $\mu^{\text{pred}}_{\theta^{(p)}_{t}}$ to $\mu_{\theta^{(p)}_{t}}$: the predicted value at $\theta^{(p)}_{t}$ from the local linear regression of the raw estimates of $\mu_{\theta}$ on $\theta$, fitted to the nearest $L$ points to $\theta^{(p)}_{t}$. Find the variance approximation $\Sigma_{n,\theta^{(p)}_{t}}^{\text{blb}}$ using BLB. $\tilde{w}_{t+1}^{(i)}=\frac{p\left(\theta_{t+1}^{(p)}\right) \left( \mathcal{N}\left( S(y) \mid \mu^{\text{pred}}_{\theta^{(p)}_{t}},\Sigma_{n,\theta^{(p)}_{t}}^{\text{blb}}\right) \right)^{\nu_{t+1}} }{ \sum_{q=1}^{P}w_{t}^{(q)}K_{t+1}\left(\theta_{t+1}^{(p)}\mid\theta_{t}^{(q)}\right)}$ Normalise $\left\{ \tilde{w}_{t+1} \right\}_{i=1}^N$ to give normalised weights $\left\{ w_{t+1} \right\}_{i=1}^N$. Resample. Empirical results\[sec:Empirical-results\] ========================================== This section analyses the new methods empirically, through their application to i.i.d., temporal and spatial data. Toy example\[subsec:Toy-example\] --------------------------------- In this section we explore the behaviour of B-SL, BLB-SL and ABC on a toy model. We examine the bias and variance of likelihood estimates, and the efficiency of Monte Carlo estimates when these approximate likelihoods are used in MCMC algorithms. In this section regression is not used to improve estimates of $\mu_{\theta}$. Let data $y$ be $10^{5}$ points simulated from a univariate Gaussian distribution with mean 0 and precision 0.25. Choosing a $\Gamma\left(1,1\right)$ prior on the precision $\tau$, we study the performance of the proposed methods in estimating statistics of the posterior on the precision $\tau$. In this example, the posterior is known by conjugacy. In addition, the sample standard deviation is a sufficient statistic, and its distribution conditional on $\tau$ is known analytically. We may therefore estimate the error of estimates of statistics of the posterior, and also of the likelihood approximations. We ran MCMC algorithms with likelihoods given by SL, B-SL, BLB-SL, ABC and B-ABC. All bootstrap algorithms used 100 resamples. For the SL approaches, in order to distinguish the error resulting from using a bootstrapped estimate of the variance from the error resulting from using an estimated mean (which may have a high variance, particularly in the BLB approaches), we run every SL approach with both the true mean of the summary statistic for each $\tau$, and the estimated mean. The proposal was taken to be normal with standard deviation 0.002. Each MCMC chain was started from the true posterior mean. For our simulated $y$, the posterior mean and standard deviation are respectively, to 3 s.f., $0.252$ and $1.13\times10^{-3}$. We ran 40 MCMC algorithms for each approximate likelihood, and report estimates of the the bias, standard deviation, root mean squared error (RMSE) of posterior mean and standard deviation estimates from the MCMC output. In this toy example the likelihood is relatively easy to estimate, and we find that for several of the approaches $M=10$ results in MCMC algorithms that have similar autocorrelation to an MCMC algorithm targeting the true posterior. Figure \[fig:Estimated-bias,-standard\] compares the efficiency of posterior mean and standard deviation estimates from MCMC using SL, B-SL, ABC and B-ABC estimates. The ABC algorithms used a Gaussian distribution as the kernel in the ABC likelihood estimator, with standard deviation $\epsilon=0.001$. We observe that standard SL performs poorly for $M=2$, with the bias in estimates of the true summary statistic likelihood leading to bias in the posterior mean and s.d.. The error in SL relative to ABC decreases as $M$ increases. For a comparison of standard SL and ABC on a more challenging problem (where there is a clear advantage to using SL), we refer the reader to section \[subsec:Lotka-Volterra-model\] (and also to @Price2017). We now compare the performance of standard ABC and SL with their bootstrapped versions. B-ABC has the advantage over ABC that for small values of $M$ (when the likelihood variance is highest) it results in chains of lower autocorrelation: for $M=1$, the mean estimated integrated autocorrelation time (IAT) for the ABC chain is $\sim26$, compared to $\sim9$ for B-ABC. However, as $M$ grows and the ABC estimates of the likelihood improve, any advantage of B-ABC is negated, particularly since it results in an overestimation of the posterior uncertainty, clearly seen in figure \[fig:Estimated-bias-of\]. B-SL exhibits improved performance over SL in almost every case (and can be implemented for $M=1$, where SL cannot). These comparisons are revisited in section \[subsec:Lotka-Volterra-model\] on a more challenging example. The comparison of SL and B-SL with the case where the true mean is used in place of the estimated mean suggests the potential of an approach that combines the bootstrap estimates of $\Sigma_{\theta}$ with improved estimates of $\mu_{\theta}$. We see that B-SL with the true value of $\mu_{\theta}$ outperforms all other approaches: the results are comparable to using an MCMC with the true likelihood. Figure \[fig:Estimated-bias,-standard-1\] compares the efficiency of posterior mean and standard deviation estimates from MCMC using different BLB-SL approximations. We observe that the subsampling has two large effects (both of which are increased by decreasing the size of the subsample): the posterior s.d. is overestimated (see figure \[fig:Estimated-bias-of-1\]); and the variance of the estimates is increased (due to an increased autocorrelation in the chains). However, we observe that both of these effects are reduced dramatically by using the true value of $\mu_{\theta}$ in the SL estimates. In this situation, the autocorrelation in the MCMC chains and the errors in posterior estimates are similar between B-SL and BLB-SL, no matter the size of the subsample (for $n=10,000$, $1000$ or $100$). This suggests great potential for the BLB approach, as long as accurate estimates of $\mu_{\theta}$ may be obtained through other means. Section \[subsec:Ising-model\] illustrates that the regression approach suggested in section \[subsec:Regression-for-estimates\] provides a way of achieving this. Lotka-Volterra model\[subsec:Lotka-Volterra-model\] --------------------------------------------------- ### Introduction {#introduction} The Lotka-Volterra model is well-studied in the ABC literature. The model is a stochastic Markov jump process that describes how the number of individuals in two populations (one of predators, the other of prey) change over time. We use the form of the model in @Wilkinsona, in which $X$ represent the number of predators and $Y$ the number of prey. The following reactions may take place: - A prey may be born, with rate $\theta_{1}Y$, increasing $Y$ by one. - The predator-prey interaction in which $X$ increases by one and $Y$ decreases by one, with rate $\theta_{2}XY$. - A predator may die, with rate $\theta_{3}X$, decreasing $X$ by one. Figure \[fig:Data-simulated-from\] shows the simulated data $y$ studied in this section: it consists of two oscillating time series: one giving the size of the predator population, the other of the prey. The simulation starts with initial populations $X=50$ and $Y=100$, and including the initial values has 32 measurements for each series, with the values of $X$ and $Y$ being recorded every 2 time units. The model may be simulated exactly using the Gillespie algorithm [@Gillespie1977], but it is not possible to evaluate its likelihood. We followed the ABC approaches in @Wilkinsona [@Papamakarios2016], using as summary statistics a 9-dimensional vector composed of the mean, log variance and first two autocorrelations of each time series, together with the cross-correlation between them (scaled by dividing by the summary statistic vector $T$ of the observed data). This model has a number of properties that provide a challenge to our proposed approach: 1. The simulations from the model give temporal data, which requires the use of the block bootstrap (as described in section \[subsec:Temporal-and-spatial\]). 2. The simulations from the model cannot always be considered to be stationary time series, since sometimes (more commonly for inappropriate parameters) the sizes of the populations decreases to zero, or diverges towards infinity. We do not treat such simulations differently to any other simulation, thus our results help to illustrate how robust our approach is to this situation. 3. The 9-dimensional summary statistics allow us to illustrate the performance of our bootstrapping approach for estimating a covariance matrix. 4. The distribution of the summary statistics is not close to being Gaussian, and there are complex dependencies between the statistics (see figure \[fig:Scatter-plots-of\]). The former point allows us to examine the performance of SL when the Gaussian assumption is not satisfied (as previously studied in @Price2017; the latter suggests that the approach of @Meeds2014a may not be appropriate, since they assume a diagonal covariance matrix. ![Scatter plots and kernel density estimates from draws from the distribution of the summary statistics conditional on the true parameters, and estimated correlations between the statistics.\[fig:Scatter-plots-of\]](ss) We compared the output of MCMC algorithms employing the approximate likelihoods given by SL, B-SL, ABC and B-ABC. We ran each method with several different choices of $M$, and the bootstrap algorithms used $R=100$ resamples. The algorithms were run for $5\times10^{4}$ iterations, and were initialised at the parameters $\theta_{1}=1$, $\theta_{2}=0.005$ and $\theta_{3}=0.6$ for which the data $y$ was simulated. The MCMC proposal was a multivariate Gaussian with diagonal covariance matrix whose diagonal is $\left(0.2^{2},0.001^{2},0.2^{2}\right)$, these values being determined using pilot runs. Our prior followed @Wilkinsona, being uniform in the $\log$ domain $$p\left(\log\left(\theta\right)\right)\propto\prod_{i=1}^{3}\mathcal{U}\left(\log\left(\theta_{i}\right)\mid\text{lower}=-6,\text{upper}=2\right).$$ The block bootstrap used blocks of length 8, so that each bootstrap resample consists of 4 blocks. Each statistic of each resample is calculated by combining the corresponding statistic of each of the constituent blocks in the obvious way. ### Results Table \[tab:The-estimated-integrated\] shows the mean (over the three parameters) estimated IAT of each sampler, and figure \[fig:SL-and-ABC\] (b-f) shows kernel density estimates of the marginal posterior distributions based on the MCMC samples. In comparing the results from standard SL to standard ABC, we see that SL usually results in more efficient MCMC samplers, and we do not see any clear indications that the Gaussian assumption made in SL is problematic. The bootstrapped algorithms do not appear to be adversely affected by any of the challenges described in the previous section, giving similar posterior distributions to the standard approaches. Further, the autocorrelation properties of the MCMC chains from the bootstrapped algorithms are improved over their standard counterparts. This provides further evidence for the observation made in section \[subsec:Toy-example\] that the bootstrapped methods are useful when the corresponding standard estimates of the likelihood have a high variance. Bootstrapped SL consistently exhibits the best performance: figure \[fig:Density-estimates-for\] shows that the low IAT found for SL when $M=5$ is not representative of the performance of the algorithm, since the sampler is only exploring a region in the tails of the posterior. Algorithm $M=1$ $M=2$ $M=5$ $M=10$ $M=50$ ---------------------- ------- ------- ------- -------- -------- SL N/A 4210 326 B-SL 504 ABC $\epsilon=0.1$ 6006 2969 B-ABC $\epsilon=0.1$ 5443 2874 1409 496 199 ABC $\epsilon=0.2$ 2923 2223 1416 427 : The estimated integrated autocorrelation time (to 0 d.p.) of each chain.\[tab:The-estimated-integrated\] Ising model\[subsec:Ising-model\] --------------------------------- Undirected graphical models, or Markov random fields (MRFs), have previously been studied using ABC in a number papers, beginning with @Grelaud2009. Such models have the form $$L_{\theta}\left(y\right)=\frac{\gamma_{\theta}\left(y\right)}{Z\left(\theta\right)},$$ where $\gamma_{\theta}\left(y\right)$ is tractable, but the partition function $Z\left(\theta\right)$ cannot, in practice, be evaluated pointwise. @Moller2006 [@Murray2006] pioneered the approach of estimating $L_{\theta}$ at each $\theta$ using importance sampling (known as auxiliary variable methods), embedded in an MCMC algorithm to perform Bayesian inference on $\theta$. ABC may be used as an alternative, but the likelihood estimates are typically high variance in comparison with those from the @Moller2006 approach (@Everitt2017c establishes a connection between the two approaches). However, when using a latent MRF model, ABC can be competitive with auxiliary variable methods [@Everitt2017b]. Also, SL provides a lower variance alternative to ABC that can be competitive with auxiliary variable methods [@Moores2015; @Everitt2017]. All of these previous approaches require simulating from $L_{\theta}$, which for most MRFs needs to be done approximately by using a run of MCMC with $L_{\theta}\left(\cdot\right)$ as the target distribution [@Caimo2011]. The use of MCMC introduces an approximation, which is small as long the chain is run long enough to have essentially forgotten its initial condition [@Everitt2012]. In this section, we focus on the Ising model. This is a pairwise Markov random field model on binary variables, each taking values in $\left\{ -1,1\right\} $. Its distribution is given by $$L_{\theta}\left(y\right)\propto\exp\left(\theta\sum_{\left(i,j\right)\in\mathbf{N}}y_{i}y_{j}\right),$$ where $\theta_{x}\in\mathbb{R}$, $x_{i}^{h}$ denotes the $i$th random variable in $x^{h}$ and where $\mathbf{N}$ is a set that defines pairs of nodes that are neighbours. We consider the case where the neighbourhood structure is given by a regular 2-dimensional grid, using a first order model (so that variables horizontally and vertically adjacent are neighbours) and toroidal boundary conditions, and use a Gibbs sampler to simulate from $L_{\theta}\left(\cdot\right)$. The mixing properties of the Gibbs sampler on Ising models are well understood, and indicate a limitation of all of the approaches to inference outlined above: namely that the approaches do not scale to large MRFs. The mixing time of Gibbs samplers on Ising models on 2-d grids is at best polynomial in the number of rows in the grid [@Lubetzky2012]. Therefore, as the size of the grid grows, in addition to the number of single variable updates growing linearly in the size of the grid, we expect to need to run the Gibbs sampler for more iterations. In this section we study data from a $1,000\times1,000$ Ising model (so that $N=10^{6}$), generated with $\theta=0.3$, and compare results from the exchange algorithm (an auxiliary variable MCMC approach introduced in @Murray2006) and BLB-SL. In all cases, the Gibbs sampler for simulating from the likelihood is burned in for 10 iterations. The exchange algorithm was initialised at $\theta=0.298$ and run for $1000$ iterations, using a normal proposal with standard deviation $0.001$. For BLB-SL, the spatial block bootstrap was used (as described in section \[subsec:Temporal-and-spatial\]), with the size of the subsample being either $100\times100$ or $50\times50$ (so that $n=10^{4}$ or $2,500$) and the block size being $50\times50$ or $25\times25$ (so that $B=2,500$ or $625$). The (sufficient) statistic $S\left(y\right)=\sum_{\left(i,j\right)\in\mathbf{N}}y_{i}y_{j}$ was used, and we took $M=1$. The SMC algorithm from section \[subsec:Regression-for-estimates\] was used, with $P=1,000$ particles and $T=10$ target distributions, with $\nu_{t}=\left(t/T\right)^{2}$. For each $\theta_{t}^{(p)}$ (the $p$th particle at the $t$th target) a sample $x_{t}^{(p)}$ of size $\sqrt{n} \times \sqrt{n}$ is simulated from $l_{\theta}\left(\cdot\right)$. $\mu_{\theta}$ and $\Sigma_{\theta}$ was then approximated as follows. - Calculate $\left(N/n\right)S_n$ (the statistic rescaled from a grid of size $n$ to a grid of size $N$) as a “raw” estimate of $\mu_{\theta}$. Find the $C$ closest $\theta$ values to $\theta_{t}^{(p)}$ (including $\theta_{t}^{(p)}$ itself) and perform a linear regression of $\left(N/n\right)S_n$ on $\theta_{t}^{(p)}$. Then, from the regression use the predicted value of the response at $\theta_{t}^{(p)}$ as the estimate of $\mu_{\theta}$. - Compose $R=100$ resamples from $x_{t}^{(p)}$ by using the following procedure - Take (overlapping) blocks of size $\sqrt{B}\times \sqrt{B}$ from $x_{t}^{(p)}$ as described in the spatial block bootstrap: there are $(1+\sqrt{B}) \times (1+\sqrt{B})$ of these blocks in total. Compute the statistic for each block, denoting it by $\mathcal{S}_b$ for the $b$th block. - Randomly compose a resample of size $\sqrt{N} \times \sqrt{N}$ by piecing together $N/B$ blocks. The indices $\mathcal{B}$ of the blocks used in the resample may be the same for all $p$ and $t$, thus can be generated in a pre-processing step. - Compute the statistic for each resample, using for the $r$th resample $$S_{N}^{(r)}=\left(\frac{N}{N-\frac{N}{\sqrt{B}}}\right)\sum_{b\in\mathcal{B}} \mathcal{S}_b,$$ where the rescaling accounts for the absence of edges between the blocks. The sample variance of the $\left\{ S_{N}^{(r)}\right\} _{r=1}^{R}$ is then our approximation of $\Sigma_{\theta}$ from the block-BLB. Figure \[fig:Density-estimates-from\] shows the estimated posterior distributions from the exchange algorithm, and runs of the BLB-SL SMC method for different values of $C$ and $n$. We observe that the posterior distribution from the exchange algorithm is very well approximated by the posterior distribution from BLB-SL SMC when $C=100$ and $n=10,000$, with the posterior standard deviation being overestimated for smaller $C$. This is due to the increasing variance estimates of the SL mean as $C$ decreases, as was previously observed in section \[subsec:Toy-example\]. In this example, the combination (through regression) of $C=100$ raw estimates of $\mu_{\theta}$ is sufficient to reduce the variance sufficiently that an accurate posterior results. This variance reduction, without the introduction of significant bias, is possible since the assumptions made in the regression are appropriate. Figure \[fig:Raw-estimates-of\] illustrates shows the raw estimates of $\mu_{\theta}$ against $\theta$ in the region of the posterior, together with the predicted regression with $C=100$ for each point. We also find that using $C=200$ and $n=2,500$ yields a fairly accurate posterior, indicating that our approach can be accurate even when the subsampling ratio $N/n$ is quite large ($400$ in this case). Figure \[fig:The-ESS-over\] shows the effective sample size (ESS) over the iterations of the SMC sampler for different values of $C$. As in section \[subsec:Toy-example\] we observe that the efficiency of the Monte Carlo method in which the BLB-SL estimate is embedded decreases when the estimates of the mean have higher variance. For comparison, the ESS for the exchange algorithm was estimated at $98$ using the `LaplacesDemon` package in R (whilst recalling that the ESS is defined differently for importance sampling and MCMC algorithms). Conclusions\[sec:Conclusions\] ============================== This paper introduces methodology for improving the performance of SL in cases where a bootstrap may be used to estimate the variance of the chosen statistics. Further, it provides a method for using SL in “tall data” settings, where subsampling may be used such that the cost of the sampling algorithm depends on the size $n$ of the subsets rather than $N$, the size of the full data. In summary - In situations in which the likelihood is difficult to estimate, bootstrap approximations of the variance of a statistic result in lower variance likelihood estimates, thus improve the efficiency of Monte Carlo methods that use SL. Further, using the bootstrap only results in a small bias. The same is true to an extent when using bootstrapped estimates of the ABC likelihood. - Using the BLB to estimate the variance of a statistic has a similar performance to using the bootstrap, paving the way for using subsampling to estimate SLs. However, estimates of the mean of the statistic using subsamples are too high variance to result in either accurate approximations of the true posterior distribution, or low variance Monte Carlo algorithms. - When using the BLB, regression estimates of the statistic mean may be used in order to reduce the variance sufficiently that an accurate approximation to the true posterior is obtained, even for large values of $N/n$. In this paper a local linear regression was used, but in other models different regression techniques such as Gaussian processes may be more appropriate. The methods in this paper should be of use whenever a bootstrap method is available to estimate the variance of the chosen statistics, with the BLB being applicable to stationary models. It is possible that, since it only requires the simulation of subsamples of size $n$, BLB-SL may be useful in big data settings where ABC/SL would not usually be applied. One further remark about the big data setting is that often in such cases sophisticated Monte Carlo methods are not required since the posterior is approximately Gaussian. This is not necessarily the case in many models where ABC/SL might be used, since parameters are often non-identifiable, leading to complex posterior distributions no matter how much data is observed (see the differential equation models in @Maybank2017, for example).
--- abstract: \| Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph \[Gagie et al., TCS 2017\]—which extends naturally the concept of prefix sorting to labeled graphs—we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. We proceed by proving several results related to Wheeler automata: 1. We show that every Wheeler NFA (WNFA) with $n$ states admits an equivalent Wheeler DFA (WDFA) with at most $2n-1-\|\Sigma\|$ states ($\Sigma$ being the alphabet) that can be computed in $O(n^3)$ time. This is in sharp contrast with general NFAs (where the blow-up could be exponential). 2. We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a $O(n\log n)$-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton’s size. 3. We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in $O(n\log n)$ time in the general case. 4. We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time. Our contributions imply new results of independent interest. Contributions (i-iii) extend the universe of known regular languages for which membership can be tested efficiently \[Backurs and Indyk, FOCS 2016\] and provide a new class of NFAs for which the minimization problem can be approximated within constant factor in polynomial time. In general, the NFA minimization problem does not admit a polynomial-time $o(n)$-approximation unless P=PSPACE. Contribution (iv) is a big step towards a complete solution to the well-studied problem of indexing graphs for linear-time pattern matching queries: our algorithm provides a provably minimum-size solution for the deterministic-acyclic case. author: - Jarno Alanko - 'Giovanna D’Agostino' - Alberto Policriti - 'Nicola Prezza[^1]' bibliography: - 'recognition.bib' title: 'Regular Languages meet Prefix Sorting[^2]' --- Introduction ============ Prefix-sorting is the process of ordering the positions of a string in the co-lexicographic order of their corresponding prefixes[^3]. Once this step has been performed, several problems on strings become much easier to solve: for example, substrings can be located efficiently in the string without the need to read all of its characters. Given the versatility of this tool, it is natural trying to generalize it to more complex objects such as edge-labeled trees and graphs. For example, a procedure for lexicographically-sorting the states of a finite-state automaton could be useful to speed up subsequent membership queries in its accepting language or its substring/suffix closure; as shown by Backurs and Indyk [@backurs2016regular], membership and pattern matching problems on regular languages are hard in the general case. The newborn theory of Wheeler graphs [@gagie2017wheeler] provides such a generalization. Intuitively, a labeled graph is Wheeler if and only if its nodes can be co-lexicographically sorted in a total order, i.e. pairwise-distinct nodes are ordered according to (i) their incoming labels or (when the labels are equal), according to (ii) their predecessors. As a consequence, Wheeler graphs admit indexes for linear-time exact pattern matching queries (also known as path queries). Wheeler graphs generalize several lexicographically-sorted structures studied throughout the past decades: indexes on strings [@manber1993suffix; @weiner1973linear; @ferragina2005indexing], sets of strings [@mantaci2007extension], trees [@ferragina2009compressing], de Bruijn graphs [@de1946combinatorial], variable-order de Bruijn graphs [@siren2014indexing], wavelet trees [@grossi2003high], wavelet matrices [@claude2015wavelet]. These efforts are part of a more general wave of interest (dating as far back as 27 years ago [@manber1992approximate]) towards techniques aimed at solving pattern matching on labeled graphs [@equi2019complexity; @siren2014indexing; @gagie2017wheeler; @ferragina2009compressing; @jain2019complexity; @vaddadi2017sequence; @rautiainen2017aligning; @amir2000pattern; @equi2019complexity2; @manber1992approximate]. As discussed above, existing graph-indexing solutions can only deal with simple labeled graphs. The problem of indexing general (or even just acyclic) graphs with a solution of provably-minimum size remains unsolved. Unfortunately, not all graphs are Wheeler and, as Gibney and Thankachan [@gibney2019hardness] have recently shown, the problem of recognizing (and sorting) them turns out to be NP-complete even when the graph is acyclic (this includes, in particular, acyclic NFAs). Even worse, not all regular languages admit an accepting Wheeler finite automaton: the set of Wheeler languages is a proper superset of the finite languages and a proper subset of the regular languages [@gagie2017wheeler]. Even when an index is not used, exact pattern matching on graphs is hard: Equi et al. [@equi2019complexity; @equi2019complexity2] have recently shown that any solution to the problem requires at least quadratic time (under the Orthogonal Vectors hypothesis), even on acyclic DFAs. In particular, this implies that converting an acyclic DFA into an equivalent Wheeler DFA cannot be done in less than quadratic time in the worst case. The remaining open questions, therefore, are: what are the properties of Wheeler languages? which class of automata admits polynomial-time prefix-sorting procedures? Can we efficiently build the smallest (that is, with the minimum number of states) prefix-sortable finite-state automaton that accepts a given regular language? These questions are also of practical relevance: as shown in [@siren2014indexing], acyclic DFAs recognizing pan-genomes (i.e. known variations in the reference genome of a population) can be turned into equivalent WDFAs of the same expected asymptotic size. While the authors do not find the minimum such automaton, their theoretical analysis (as well as experimental evaluation) suggests that the graph-indexing problem is tractable in some real-case scenarios. Our Contributions ----------------- In this paper we provide the following contributions: 1. We show that Wheeler languages are the natural version of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of intervals, each formed by elements from a single Myhill-Nerode equivalence class. In regular languages, those intervals are replaced with general sets. 2. We show that every Wheeler NFA (WNFA) with $n$ states admits an equivalent Wheeler DFA (WDFA) with at most $2n-1-\|\Sigma\|$ states ($\Sigma$ being the alphabet) that can be computed in $O(n^3)$ time. This is in sharp contrast with general NFAs (where the blow-up could be exponential). 3. Let $d$-NFA denote the class of NFAs with at most $d$ equally-labeled edges leaving any state. We show that the problem of recognizing and sorting Wheeler $d$-NFAs is in P for $d\leq 2$. A recent result from Gibney and Thankachan [@gibney2019hardness] shows that the problem is NP-complete for $d\geq 5$. Our result almost completes the picture, the remaining open cases being $d=3$ and $d=4$. 4. We provide an online incremental algorithm that, when fed with an acyclic Wheeler DFA’s nodes in any topological order, can dynamically compute the co-lexicographic rank of each new incoming node among those already processed with just logarithmic delay. 5. We improve the running time of (4) to linear in the offline setting for arbitrary WDFAs. 6. Given a Wheeler DFA $\mathcal A$ of size $n$, we show how to compute, in $O(n\log n)$ time, the smallest Wheeler DFA recognizing the same language as $\mathcal A$. If $\mathcal A$ is acyclic, running time drops to $O(n)$. 7. Given any acyclic DFA $\mathcal A$ of size $n$, we show how to compute, in $O(n+m\log m)$ time, the smallest Wheeler DFA $\mathcal A'$, of size $m$, recognizing the same language as $\mathcal A$. The paper is structured in a top-down fashion to make it accessible also to non-experts. We provide the main theorems and proof sketches within the first ten pages. We start in Section \[sec:WL\] with results [1]{} and [2]{}: a Myhill-Nerode theorem for Wheeler languages and a linear conversion from WNFAs to WDFAs. Result [1]{} shows that Wheeler languages are precisely those admitting a “finite interplay” between the co-lexicographic ordering and the Myhill-Nerode equivalence relation (i.e. the relation characterizing general DFAs). Result [2]{} implies that the WNFA minimization problem admits a polynomial-time $2$-approximation. We remark that the NFA minimization problem is notoriously hard (even to approximate within $o(n)$-factor) in the general case [@approx_min_NFA; @gruber2007computational; @Malcher:2004:MFA:1046130.1046139]. In Section \[sec:sorting\] we describe results [3-5]{}: polynomial-time algorithms for recognizing and sorting Wheeler $2$-NFAs. These results generalize to labeled graphs existing prefix-sorting algorithms on strings [@navarro2014optimal] and labeled trees [@ferragina2009compressing] that have been previously described in the literature. Combined with contribution [2]{}, our algorithms can be used to index any Wheeler NFA at the price of a moderate linear blow-up in the number of states. This result expands the universe of known regular languages for which membership and pattern matching problems can be solved efficiently [@backurs2016regular]. Contributions [6]{} and [7]{} (Section \[sec:minimization\]) combine our sorting algorithms [3-5]{} with DFA minimization techniques to solve the following problem: to compute, given a finite language represented either explicitly by a set of strings or implicitly by an acyclic DFA, the smallest accepting WDFA. Note that this can be interpreted as a technique to index arbitrary deterministic acyclic graphs using the smallest prefix-sortable equivalent automaton. While we do not provide a lower bound stating that the space of our index is minimum, we note that all known fast indexes on strings, sets of strings, trees, and variable-order de Bruijn graphs are Wheeler graphs [@gagie2017wheeler]. In this sense, any index on acyclic graphs improving our solution would probably require techniques radically different than those developed in the last decades to solve the indexing problem (virtually any known full-text index uses prefix sorting, including those based on LZ77 [@KN13], run-length BWT [@gagie2018optimal], and grammars [@CNspire12]). Definitions {#sec:definitions} ----------- We start by giving a definition of finite-state automata that captures, to some extent, the amount of nondeterminism of the automaton. A $d$-NFA is a nondeterministic finite state automaton that has at most $d$ transitions with the same label leaving each state. Note that $1$-NFAs correspond to DFAs, while $\infty$-NFAs correspond to NFAs. \[def:d-NFA\] A $d$-NFA is a quintuple $\mathcal A = (V,E,F,s,\Sigma)$, where $V$ is a set of states (or vertices), $\Sigma$ is the alphabet (or set of labels), $E\subseteq V\times V \times \Sigma$ is a set of directed labeled edges, $F\subseteq V$ is a set of accepting states, and $s \in V$ is a start state (or source). We moreover require that $s\in V$ is the only node with in-degree zero and that for each $u\in V$ and $a\in\Sigma$, $\|\{(u,v,a)\in E\}\| \leq d$. We denote $\sigma = \|\Sigma\|$. The notation $\mathcal L(\mathcal A)$ indicates the language accepted by $\mathcal A$, i.e. the set of all strings labeling paths from $s$ to an accepting state. We assume that each state either is $s$ or is reachable from $s$. Otherwise, any state that cannot be reached from $s$ can be removed without changing $\mathcal L(\mathcal A)$. Note that we allow states with incomplete transition function, i.e. such that the set of labels of their outgoing edges does not coincide with $\Sigma$. If state $s$ misses outgoing label $a$, then any computation following label $a$ from $s$ is considered as non-accepting. In a standard NFA definition, this would be equivalent to having an outgoing edge labeled $a$ to a universal non-accepting node (a sink). We call a $d$-NFA acyclic when the graph $(V,E)$ does not have cycles. We say that a $d$-NFA is input-consistent if, for every $v\in V$, all incoming edges of $v$ have the same label. If the $d$-NFA is input-consistent, we indicate with $\lambda(v)$, $v\in V$, the label of the incoming edges of $v$. For the source, we take $\lambda(s) = \# \notin \Sigma$. We will assume that characters in $\Sigma$ are totally ordered by $\prec$ and that $\#$ is minimum. We extend $\prec$ to $\Sigma^$ co-lexicographically, still denoting it by $\prec$. On DFAs, we denote by $succ_a(u)$, with $u\in V$ and $\ a\in\Sigma$, the unique successor of $v$ by label $a$, when it exists. We define the size* of an automaton to be the number of its edges. The notion of Wheeler graph generalizes in a natural way the concept of co-lexicographic sorting to labeled graphs: \[def\_WG\] A triple $G=(V, E, \Sigma)$, where $V$ is a set of vertices and $E \subseteq V \times V \times \Sigma$ is a set of labeled edges, is called a Wheeler graph if there is a total ordering $<$ on $V$ such that vertices with in-degree $0$ precede those with positive in-degree and for any two edges $(u_1,v_1,a_1), (u_2, v_2,a_2)$ we have (i) $a_1 \prec a_2 \rightarrow v_1 < v_2$, and (ii) $\left(a_1=a_2 \right) \wedge \left( u_1 < u_2 \right) \rightarrow v_1 \leq v_2$. Note that the above definition generalizes naturally the concept of prefix-sorting from strings to graphs: two nodes (resp. string prefixes) can be ordered either looking at their incoming labels (resp. last characters) or, if the labels are equal, by looking at their predecessors (resp. previous prefixes). Considering that, differently from strings and trees, a graph’s node can have multiple predecessors, it should be clear that there could exist graphs whose nodes cannot be sorted due to conflicting predecessors: not all labeled graphs enjoy the Wheeler properties. We call a total order of nodes satisfying Definition \[def\_WG\] a Wheeler order of the nodes and we write WDFA,WNFA as a shortcut for Wheeler DFA, Wheeler NFA. By property (i), input-consistence is a necessary condition for a graph to be Wheeler. An important property of Wheeler graphs is path coherence: \[def:path coherence\] An edge-labeled directed graph $G$ is path coherent if there is a total order of the nodes such that for any consecutive range $[i,j]$ of nodes and string $\alpha$, the nodes reachable from those in $[i,j]$ in $\|\alpha\|$ steps by following edges whose labels form $\alpha$ when concatenated, themselves form a consecutive range. A Wheeler graph is path coherent with respect to any Wheeler order of the nodes [@gagie2017wheeler]. Wheeler Languages {#sec:WL} ================= In this section we collect our basic results on regular languages accepted by automata whose transition relation is a Wheeler graph: Wheeler languages. Let $\Sigma$ be a finite set. A language $\mathcal L$ is [Wheeler]{} if $\mathcal L= \mathcal L(\mathcal A)$ for a Wheeler NFA $\mathcal A$. Let us begin with some basic notation. Given a language $\mathcal L \subseteq \Sigma^$ we denote by $ \text{\em Pref}(\mathcal L), \text{\em Suff}(\mathcal L), $ and $ \text{\em Fact}(\mathcal L)$ the set of prefixes, suffixes, and factors of strings in $\mathcal L$, respectively. More formally: $ \text{\em Pref}(\mathcal L) = \{\alpha: \exists \beta \in \Sigma^~ \alpha \beta\in \mathcal L\}$, $\text{\em Suff}(\mathcal L) = \{\beta: \exists \alpha \in \Sigma^~ \alpha \beta\in \mathcal L\}$, $\text{\em Fact}(\mathcal L) = \{\alpha: \exists \beta, \gamma \in \Sigma^~ \gamma \alpha \beta\in \mathcal L\}$. Given two states $u,v$ of an NFA $\mathcal{A}$, we denote by $u \rightsquigarrow v$ a path from $u$ to $v$ in $\mathcal{A}$. If $\mathcal A=(V,E,F,s,\Sigma)$ is an NFA, $ u\in V $, and $\alpha\in \text{\em Pref}(\mathcal L(\mathcal A))$, we define: 1. $V_\alpha=\{ v ~\|~ \alpha \text{ labels } s \rightsquigarrow v \}$, 2. $ \text{\em Pref}(\mathcal L(\mathcal A))_u:=\{\alpha \in \text{\em Pref}(\mathcal L(\mathcal A)): \alpha \text{ labels } s \rightsquigarrow u\} $ . Clearly, from the above definition it follows that (i) $V_{\alpha} \subseteq V $, (ii) $ \text{\em Pref}(\mathcal L(\mathcal A))_{u}\subseteq \text{\em Pref}(\mathcal L(\mathcal A)) $, and (iii) $u \in V_{\alpha} \text{ if and only if } \alpha \in \text{\em Pref}(\mathcal L(\mathcal A))_{u}$. The prefix/suffix property introduced below is going to be crucial to determine the Wheeler ordering among states—when such an ordering exists. Consider a linear order $(L, <)$. 1. An [interval]{} in $(L, <)$ is a $I\subseteq L$ such that $(\forall x,x'\in I)(\forall y \in L) (x< y< x' \rightarrow y\in I)$. 2. Given $I,J$ intervals in $(L, <)$ and $I\subseteq J$, then: - $I$ is a [prefix]{} of $J$ if $(\forall x \in I )( \forall y \in J\setminus I) ( x< y)$; - $I$ is a [suffix]{} of $J$ if $(\forall y \in J \setminus I )( \forall x \in I )( y< x)$. 3. A family $\mathcal C$ of non-empty intervals in $(L, <)$ is said to have the [prefix/suffix property]{} if, for all $I,J \in \mathcal C$ such that $I \subseteq J$, $I$ is either a prefix or a suffix of $J$. The following lemma will allow us to bound (linearly) the blow-up of the number of states taking place when moving from a WNFA to a WDFA. \[3n\] Let $(L, <)$ be a finite linear order of cardinality $\|L\|=n$, let $\mathcal C$ be a [prefix/suffix]{} family of non-empty intervals in $(L, <)$. Then: 1. $\|\mathcal C\| \leq 2n-1$. 2. The upper bound is tight: for every $n$, there exists a prefix/suffix family of size $2n-1$. Let $\mathcal C$ be a family of non-empty intervals of a linear order $(L, <)$ having the prefix/suffix property. Let $<^{i}$ (or simply $ < $) the binary relation over $\mathcal C$ defined by $$\begin{aligned} I<^{i} J & \text{ if and only if } ( \exists x \in I)( \forall y\in J )( x< y )\lor (\exists y \in J)( \forall x \in I )( x< y). \end{aligned}$$ The following lemma is easily proved. \[convex+order\] Let $\mathcal C$ be a family of non-empty intervals of a linear order $(L, <)$ having the prefix/suffix property, then $ (\mathcal C,<^{i}) $ is a linear order. Note that whenever the linear order $(L, <)$ is finite, any non-empty interval $I$ has minimum $m_I$ and maximum $M_I$. In this special case, the above order $<^{i}$ can be equivalently described on a family having the prefix/suffix property, by: $ I <^{i} J \text{ if and only if } (m_I < m_J) \lor [ (m_I=m_J) \wedge (M_I < M_J)].$ We now have the basics to start our study of Wheeler languages. In this section, we use $V,E,F,s,\Sigma,<$ to denote the set of states, edges, final states, initial state, alphabet, and Wheeler order of a generic WNFA. The key property of path-coherence will be re-proved below—in Lemma \[convex\_sets\]—, together with what we may call a sort of its “dual”, that is, the the set of strings read while reaching a given state is an interval. More precisely, if $\mathcal A$ is a WNFA, $ u\in V $, and $\alpha\in \Sigma^$, we have that $ V_{\alpha}$ is an interval in $ (V,<) $ ($ I_{\alpha} $, from now on), $ \text{\em Pref}(\mathcal L(\mathcal A))_{u}$ is an interval in $(\text{\em Pref}(\mathcal L(\mathcal A)),\prec)$ ($ I_{u} $, from now on), and $$\alpha \in I_u \text{ if and only if } u \in I_\alpha.$$ Preliminary to our result are the following lemmas, exploiting the interval-structure of both Wheeler languages and automata. \[prec\_versus\_minus\] If $\mathcal A$ is a WNFA, $u,v\in V$ are states, and $\alpha, \beta \in \text{Pref}(\mathcal L(\mathcal A))$, then: 1. if $\alpha\in I_u, \beta\in I_v$, and $\{\alpha,\beta\}\not\subseteq I_v\cap I_u$, then $\alpha\prec \beta \text{ if and only if } u<v$; 2. if $u\in I_\alpha, v\in I_\beta $, and $\{u,v\}\not\subseteq I_\beta \cap I_\alpha$, then $\alpha\prec \beta \text{ if and only if } u<v$. Let $ I_{V} =\{I_u:~ u\in V\}$ and $I_ {\text{\em Pref}(\mathcal L(\mathcal A))} =\{I_\alpha :~\alpha\in \text{\em Pref}(\mathcal L(\mathcal A)) \}$. \[convex\_sets\] If $\mathcal A$ is a WNFA and $\mathcal L = \mathcal L(\mathcal A)$, then: 1. for all $u\in V$, the set $I_u$ is an interval in $( \text{Pref}(\mathcal L(\mathcal A)), \prec)$; 2. $ I_{V}$ is a prefix/suffix family of intervals in $( \text{Pref}(\mathcal L(\mathcal A)), \prec)$; 3. for all $\alpha \in \text{Pref}(\mathcal L(\mathcal A))$, the set $I_\alpha$ is an interval in $(V,<)$; 4. $I_ {\text{Pref}(\mathcal L(\mathcal A))}$ is a prefix/suffix family of intervals in $(V,<)$. From Lemma \[convex+order\] it follows that both $(I_{v}, \prec^{i})$ and $(I_ {\text{\em Pref}(\mathcal L(\mathcal A))}, <^{i})$ are linear orders. The link between such orders is made explicit below. \[compare\] Consider $ I_{u},I_{v}\in I_{V} $ and $ I_{\alpha}, I_{\beta} \in I_ {\text{Pref}(\mathcal L(\mathcal A))}$. 1. $ I_u \prec^{i} I_v $ implies that $ u < v $ and $ u < v $ implies that $ I_u \preceq^{i} I_v $ 2. $ I_{\alpha} <^{i} I_{\beta} $ implies that $ \alpha \prec \beta $ and $ \alpha \prec \beta $ implies that $ I_{\alpha} \leq^{i} I_{\beta} $ If $\mathcal A$ is a WNFA we can prove that the following interval construction—which is the analogous of the power-set construction for NFAs—allows determinization. If $\mathcal A$ is a WNFA we define its (Wheeler) determinization* as the automaton ${\mathcal A^{d}}=(V^{d}, E^{d},F^{d} s^{d},<^{d}, \Sigma)$, where: - $V^{d}=I_{\text{ Pref}(\mathcal L(\mathcal A))}$; - $s^{d}= I_{\epsilon}=\{s\}$ - $F^{d}=\{I_\alpha ~\|~ \alpha \in \mathcal L(\mathcal A)\}$; - $E^{d}$ is the set of triples $(I_\alpha,I_{\alpha e}, e)$, for all $e\in \Sigma$ and $\alpha e\in \text{Pref}(\mathcal L(\mathcal A))$; - $<^{d}=<^{i}$. The bound proved in Lemma \[3n\] can be sligthly improved in the special case of prefix/suffix families corresponding to WNFA intervals. \[Wdeterminization\] If $\mathcal A$ is a WNFA with $n$ states over an alphabet $\Sigma$, then $\mathcal A^{d}$ is a WDFA with at most $2n-1-\|\Sigma\|$ states, and $ \mathcal L(\mathcal A^{d}) = \mathcal L(\mathcal A)$. \[ComputeWdeterminization\] If $\mathcal A$ is a WNFA with $n$ states, then $\mathcal A^{d}$ can be computed in $O(n^3)$ time. Lemma \[Wdeterminization\] above— saying that we can restrict the automata recognizing Wheeler Languages to deterministic ones without an exponential blow up—marks a difference between the standard and the Wheeler case for regular languages and can be seen as the first step in the study of Wheeler Languages. Further differences can be observed. For example, the reader can check that the language $ \mathcal L(\mathcal A)=\mathcal L= b^{+}a $ is accepted by incomplete WDFAs only. The subsequent step to take in a theory of Wheeler Languages is a Myhill-Nerode like theorem for this class. To this end, we define: \[right\_inv\] Given a language $\mathcal L\subseteq \Sigma^$, an equivalence relation $\sim$ over $\text{\em Pref}(\mathcal L)$ is: - [right invariant]{}, when for all $\alpha, \beta \in \text{\em Pref}(\mathcal L)$ and $\gamma\in \Sigma^$: if $ \alpha \sim \beta$ and $\alpha\gamma \in \text{\em Pref}(\mathcal L)$, then $\beta\gamma \in \text{\em Pref}(\mathcal L)$ and $ \alpha\gamma \sim \beta\gamma$; - [convex]{} if $\sim$-classes are intervals of $(\text{\em Pref}(\mathcal L),\prec)$; - [input consistent]{} if all words belonging to the same $\sim$-class end with the same letter. Consider a language $\mathcal L\subseteq \Sigma^$. The Myhill-Nerode equivalence $\equiv_{\mathcal L}$ among words in $ Pref(L)$ is defined as $$\begin{aligned} \alpha \equiv_{\mathcal L} \beta \text{ if and only if } & (\forall \gamma \in \Sigma^{}) ( \alpha \gamma \in {\mathcal L} \Leftrightarrow \beta \gamma \in {\mathcal L} ). \end{aligned}$$ The input consistent, convex refinement of $ \equiv_{\mathcal L} $ is defined as follows. For all $\alpha, \beta \in \text{Pref}(\mathcal L)$: $$\begin{aligned} \alpha\equiv_{\mathcal L}^{c}\beta \text{ if and only if } & \alpha \equiv_{\mathcal L} \beta \wedge ~end(\alpha)=end(\beta) \wedge (\forall \gamma \in \text{\em Pref}(\mathcal L)) (\alpha\prec \gamma \prec \beta \rightarrow \gamma \equiv_{\mathcal L} \alpha),\end{aligned}$$ where $end(\alpha)$ is the final character of $\alpha$ when $\alpha\neq\epsilon$, and $ \epsilon $ otherwise. Using the above results in this section we can prove: \[Myhill-Nerode\] Given a language $\mathcal L\subseteq \Sigma^$, the following are equivalent: 1. $\mathcal L$ is a Wheeler language (i.e. $\mathcal L$ is recognized by a WNFA). 2. $\equiv_{\mathcal L}^{c}$ has finite index. 3. $\mathcal L$ is a union of classes of a convex, input consistent, right invariant equivalence over $\text{Pref}(\mathcal L)$ of finite index. 4. $\mathcal L$ is recognized by a WDFA. This theorem and other results on Wheeler Languages are going to be part of a companion paper of this one. Sorting Wheeler Finite Automata {#sec:sorting} =============================== In this section we provide efficient algorithms to sort a relevant sub-class of the Wheeler automata. Combined with the results of the previous section, our algorithms can be used to index any* WNFA. We start with a reduction from the problem of recognizing Wheeler $2$-NFAs to 2-SAT. The reduction introduces only a polynomial number of boolean variables and can be computed in polynomial time; since 2-SAT is in P, this implies that Wheeler $2$-NFA recognition is in P. \[thm:2NFA in P\] Let $\mathcal A=(V,E,F,s,\Sigma)$ be a $2$-NFA. In $O(\|E\|^2)$ time we can: 1. Decide whether $\mathcal A$ is a Wheeler graph, and 2. If $\mathcal A$ is a Wheeler graph, return a node ordering satisfying the Wheeler graph definition. It is easy to express the Wheeler properties (i)-(ii) and antisymmetry/connex of the Wheeler order with 2-SAT clauses. Transitivity, however, requires 3-SAT clauses on general graphs. The core of the full proof in Appendix \[proof thm 2NFA in P\] is to show that, on $2$-NFAs, transitivity automatically “propagates” from the source to all nodes and does not require additional clauses. Gibney and Thankachan [@gibney2019hardness] have recently shown that the problem of recognizing Wheeler $d$-NFAs is NP-complete for $d\geq 5$. Theorem \[thm:2NFA in P\] almost completes the picture, the remaining open cases being $d=3$ and $d=4$. We note that Theorem \[thm:2NFA in P\] combined with our determinization result of Section \[sec:WL\] does not break the problem’s NP-completeness: in principle, our determinization algorithm could turn a non-Wheeler NFA into a WDFA. We now describe more efficient algorithms for the deterministic case. The first, Theorem \[thm:n log n\], is an online algorithm that solves the problem considered in Theorem \[thm:2NFA in P\] in $O(\|E\|\log\|V\|)$ time when the graph is an acyclic DFA. The algorithm is online in the following sense. We assume that the nodes, together with their incoming labeled edges, are provided to the algorithm in any valid topological ordering. At any step, we maintain a prefix-sorted list of the current nodes, which is updated when a new node is added. When a new node $v$ arrives together with its incoming labeled edges $(u_1,v,a), \dots, (u_k,v,a)$, then $u_1,\dots, u_k$ have already been seen in the past node sequence and can be used to decide the co-lexicographic rank of $v$. If $v$ falsifies the Wheeler properties, we detect this event, report it, and stop the computation. Our algorithm is an extension of an existing one that builds online the Burrows-Wheeler transform of a string [@navarro2014optimal]. In Section \[sec:minimization\] we will modify this algorithm so that, instead of failing on non-Wheeler graphs, it computes the smallest Wheeler DFA equivalent to the input acyclic DFA. First, note that Lemma \[prec\_versus\_minus\] implies that Wheeler DFAs admit a unique admissible ordering (this follows from the fact that, on WDFAs, $\{\alpha,\beta\}\not\subseteq I_v\cap I_u$ always holds): \[lem:clusters\] Let $\mathcal A$ be a Wheeler DFA, $<$ be the node ordering satisfying the Wheeler properties, and $\prec$ be the co-lexicographic order among strings. For any two nodes $u \neq v$, the following holds: $\alpha_u \prec \alpha_v$ for all string pairs $\alpha_u, \alpha_v$ labeling paths $s \rightsquigarrow u$ and $s\rightsquigarrow v$ if and only if $u<v$. Corollary \[lem:clusters\] has two important consequences: on DFAs, (i) we can use any paths connecting $s$ with two nodes $u\neq v$ to decide their co-lexicographic order, and (ii) if it exists, the total ordering of the nodes is unique. The corollary is crucial in proving the following (as well as others) result: \[thm:n log n\] Let $\mathcal A=(V,E,F,s,\Sigma)$ be an acyclic DFA. There exists an algorithm that either prefix-sorts the nodes of $\mathcal A$ or returns $\mathtt{FAIL}$ if such an ordering does not exist online with $O(\log\|V\|)$ delay per input edge. All details of our algorithm (description, pseudocode and data structures) and the proof of its correctness are reported in in Appendices \[app: Sorting WDAGs Online\] and \[sec:data structures\]. To conclude the section, we show that in the offline setting we can improve upon the previous result. We first need the following lemma (see Appendix \[proof lemma check range consistency\] for the full proof): \[lem: check range consistency\] Given an input-consistent edge-labeled graph $G=(V,E,\Sigma)$ and a permutation of $V$ sorted by a total order $<$ on $V$, we can check whether $<$ satisfies the Wheeler properties in optimal $O(\|V\|+\|E\|)$ time. \[thm:DFA linear\] Let $\mathcal A=(V,E,F,s,\Sigma)$ be a DFA. In $O(\|V\|+\|E\|)$ time we can: 1. Decide whether $\mathcal A$ is a Wheeler graph, and 2. If $\mathcal A$ is a Wheeler graph, return a node ordering satisfying the Wheeler graph definition. By Corollary \[lem:clusters\], if $\mathcal A$ is a Wheeler graph then we can use the strings labeling any paths $s\rightsquigarrow u$ and $s\rightsquigarrow v$ to decide the order of any two nodes $u$ and $v$. We build a spanning tree of $\mathcal A$ rooted in $s$ and prefix-sort it using [@ferragina2009compressing Thm 2]. Finally, we verify correctness using Lemma \[lem: check range consistency\]. We note that the above strategy cannot be used to sort Wheeler NFAs, since the spanning tree could connect $s$ with several distinct nodes using the same labeled path: this would prevent us to find the order of those nodes using the spanning tree as support. Wheeler DFA Minimization {#sec:minimization} ======================== We are now ready to use the algorithms of the previous sections to prove our main algorithmic results: (i) a minimization algorithm for WDFAs (Theorem \[thm: min DFA\]) and (ii) a near-optimal algorithm generating the minimum acyclic WDFA equivalent to any input acyclic DFA (Theorem \[thm: ADFA -> WADFA\]). Let $\equiv$ be an equivalence relation over the states $V$ of an automaton $\mathcal A = (V,E,F,s,\Sigma)$. The quotient automaton is defined as $\mathcal A/_\equiv = (V/_\equiv, E/_\equiv, F/_\equiv,[s]_{\equiv},\Sigma)$, where $E/_\equiv = \{([u]_\equiv, [v]_\equiv, c)\ :\ (u,v,c)\in E\}$. The symbol $\approx$ denotes the Myhill-Nerode equivalence among states [@nerode1958linear]: $u \approx v$, with $u,v\in V$, if and only if, for any string $\alpha$, we reach a final state by following the path labeled $\alpha$ from $u$ if and only if the same holds for $v$. Note that this is the “state” version of the relation $\equiv_{\mathcal L}$ given in Section \[sec:WL\] (which instead is defined among strings). The goal of any DFA-minimization algorithm is to find $\approx$, which is the, provably existing and unique, coarsest (i.e. largest classes) equivalence relation stable with respect to the initial partition in final/non-final states. To abbreviate, we will simply say “coarsest equivalence relation” instead of “coarsest equivalence relation stable with respect to an initial partition”. In our case, assuming that $\mathcal A$ is Wheeler, we want to find the (unique as proved below) coarsest equivalence relation $\equiv_w$ finer than $\approx$, such that $\mathcal A/_{\equiv_w}$ is Wheeler. Our Algorithm \[alg:minimize\] achieves precisely this goal: we start with $\approx$ and then refine it preserving stability with respect to characters, while also ensuring that the resulting equivalence classes can be ordered consistently with the Wheeler constraints. Again, it can be proved that $\equiv_w$ is the “state” version of the relation $\equiv_{\mathcal L}^c$ given in Section \[sec:WL\]. For the purposes of the following results, we do not need to prove the connection between the two relations and we keep a distinct notation to avoid confusion. We show (formal proof in Appendix \[proof thm: min DFA\]): \[thm: min DFA\] Let $\mathcal A$ be a WDFA. The automaton $\mathcal{A}/_{\equiv_w}$ returned by Algorithm \[alg:minimize\] is the minimum WDFA recognizing $\mathcal L(\mathcal A)$. 1. Compute the Myhill-Nerode equivalence $\approx$ among states of $\mathcal A$. 2. Prefix-sort $\mathcal A$’s states, obtaining the ordering $v_1 < \dots < v_n$. 3. Compute a new relation $\equiv_{w}$ defined as follows. Insert in the same equivalence class all maximal runs $v_i < v_{i+1} < \dots < v_{i+t}$ such that: 1. $v_i \approx v_{i+1} \approx \dots \approx v_{i+t}$ 2. $\lambda(v_i) = \lambda(v_{i+1}) = \dots = \lambda(v_{i+t})$. 4. Return $\mathcal{A}/_{\equiv_w}$. Note that uniqueness of the minimum WDFA follows from Corollary \[lem:clusters\] (uniqueness of the Wheeler order) and Algorithm \[alg:minimize\]. Note also that, in the automaton output by Algorithm \[alg:minimize\], adjacent states in co-lexicographic order are distinct by the relation $\approx$ unless their incoming labels are different (in which case they might be equivalent). It follows that if a sorted Wheeler DFA does not have this property, then it is not minimum (otherwise Algorithm \[alg:minimize\] would collapse some of its states). Conversely, If a Wheeler DFA has this property, then Algorithm \[alg:minimize\] does not collapse any state, i.e. the automaton is already of minimum size. We therefore obtain the following characterization: \[thm:characterization minimum\] Let $\mathcal A$ be a Wheeler DFA, let $v_1 < v_2 < \dots < v_t$ be its co-lexicographically ordered states, and let $\approx$ be the Myhill-Nerode equivalence among them. $\mathcal A$ is the minimum Wheeler DFA recognizing $\mathcal L(\mathcal A)$ if and only if the following holds: for every $1 \leq i < t$, if $v_i \approx v_{i+1}$ then $\lambda(v_i) \neq \lambda(v_{i+1})$. Theorem \[thm: min DFA\] implies the following corollaries. \[thm: min DFA1\] Given a WDFA $\mathcal A$ of size $n$, in $O(n\log n)$ time we can build the minimum WDFA recognizing $\mathcal L(\mathcal A)$. We run Algorithm \[alg:minimize\] computing $\approx$ with Hopcroft’s algorithm [@hopcroft1971n] ($O(n\log n)$ time), and prefix-sorting $\mathcal A$ with Theorem \[thm:DFA linear\] ($O(n)$ time). Note that we can check $u\approx v$ in constant time by representing the equivalence relation as a vector $EQ[v] = [v]_{\approx}$, where we choose $V=\{1, \dots, \|V\|\}$ and where $[v]_{\approx}$ is any representative of the equivalence class of $v$ (e.g., the smallest one, which we can identify in linear time by radix-sorting equivalent states). Then, $u\approx v$ if and only if $EQ[u]=EQ[v]$. Using this structure, the runs of Algorithm \[alg:minimize\] can easily be identified in linear time. \[thm: min ADFA\] Given an acyclic WDFA $\mathcal A$ of size $n$, in $O(n)$ time we can build the minimum acyclic WDFA recognizing $\mathcal L(\mathcal A)$. We run Algorithm \[alg:minimize\] computing $\approx$ with Revuz’s algorithm [@revuz1992minimisation] ($O(n)$ time), prefix-sorting $\mathcal A$ with Theorem \[thm:DFA linear\] ($O(n)$ time), and testing $u\approx v$ in constant time as done in Corollary \[thm: min DFA1\]. Note that Corollary \[thm: min ADFA\] implies that we can, in optimal linear time, build the minimum WDFA $\mathcal A/_{\equiv_w}$ recognizing any input finite language $\mathcal L$ represented as a set of strings: we build the tree DFA accepting $\mathcal L$ and apply Corollary \[thm: min ADFA\]. The corollary can be applied since trees are always Wheeler [@ferragina2009compressing; @gagie2017wheeler]. In the next subsection we treat the (more interesting) case where $\mathcal L$ is represented by a DFA. Note that this result could already be achieved by unraveling the DFA into a tree and minimizing it using Corollary \[thm: min ADFA\]. However, the intermediate tree could be exponentially larger than the output. Acyclic DFAs to Smallest Equivalent WDFAs {#sec:ADFA->WADFA} ----------------------------------------- We show how to build the smallest acyclic Wheeler DFA equivalent to any acyclic DFA in output-sensitive time. Let $\mathcal A = (V,E,F,s,\Sigma)$ be an acyclic DFA. We first minimize $\mathcal A$ using Revuz’s algorithm [@revuz1992minimisation] and obtain the equivalent minimum acyclic DFA $\mathcal A_1 = \mathcal A/_\approx = (V_1,E_1,F_1,s_1,\Sigma)$. Let us denote $\|V_1\|=t$. The idea is to run a modified version of the online Algorithm \[alg:step\] on $\mathcal A_1$. The difference is that now we will solve (not just detect) violations to the Wheeler properties without changing the accepting language. The next step is to topologically-sort $\mathcal A_1$’s states (e.g. using Kahn’s algorithm [@kahn1962topological]). At this point, we modify $\mathcal A_1$ in $t$ steps by processing its states in topological order. This defines a sequence of automata $\mathcal A_1, \mathcal A_1, \dots, \mathcal A_t$. At each step, the states of $\mathcal A_i$ are partitioned in two sets: - those not yet processed: $N_i = \{ v_{i+1}, v_{i+2}, \dots, v_{t} \}$, and - the remaining states $V_i - N_i$, sorted by a total ordering $<$ in a sequence $\mathtt{LEX}_i$. At the beginning, $N_1 = \{v_2, \dots, v_t\}$ and $\mathtt{LEX}_1 = s$. Note that $N_t = \emptyset$ (i.e. at the end we will have processed all states). At each step $i$, we maintain the following invariants: 1. $\mathcal L(\mathcal A_i) = \mathcal L(\mathcal A_1)$. 2. States in $\mathtt{LEX}_i$ are sorted by a total order $<$ that does not violate the Wheeler properties among states in $\mathtt{LEX}_i$ itself: in Definition \[def\_WG\], we require $u_1,u_2,v_1,v_2 \in \mathtt{LEX}_i$. 3. for each $j=1, \dots, \|\mathtt{LEX}_i\|-1$, if $\mathtt{LEX}_i[j] \approx \mathtt{LEX}_i[j+1]$ then $\lambda(\mathtt{LEX}_i[j]) \neq \lambda(\mathtt{LEX}_i[j+1])$. Invariant [1]{} implies $\mathcal L(\mathcal A_t) = \mathcal L(\mathcal A)$. Since $N_t=\emptyset$ and $\mathtt{LEX}_t$ contains all $\mathcal A_t$’s states, invariant [2]{} implies that $\mathcal A_t$ is Wheeler (note that intermediate automata $\mathcal A_i$, with $1 < i < t$ might be non-Wheeler). Finally, invariant [3]{} and Theorem \[thm:characterization minimum\] imply that $\mathcal A_t$ is the minimum WDFA accepting $\mathcal L(\mathcal A_t)$. As a result, $\mathcal A_t = \mathcal A/_{\equiv_w}$. We describe all the details of our algorithm in Appendix \[app:DFA->WDFA\] for space constraints; here we give an overview of the procedure. The idea is to process states in topological order as done in Theorem \[thm:n log n\]. This time, however, we also solve inconsistencies of type 1 and 2 among nodes in $\mathtt{LEX}_{i} \cup \{v_{i+1}\}$ by splitting nodes in $\approx$-equivalent copies. Here, splitting means creating two or more copies of a state $v$ in such a way that (i) each copy duplicates all $v$’s outgoing edges, (ii) $v$’s incoming edges are distributed (not duplicated) among the copies, and (iii) each copy is a final state if and only if $v$ is a final state. Our splitting process creates $\approx$-equivalent nodes, therefore the accepted language never changes (invariant [1]{} stays true). Moreover, since the states of $\mathcal A$ have already been collapsed by the equivalence $\approx$, after inserting nodes (or their copies) in $\mathtt{LEX}_{i}$ we never create runs of length greater than one of $\approx$-equivalent states with equal incoming labels (invariant [3]{} stays true). As a result, we incrementally build the minimum WDFA $\mathcal A/_{\equiv_w}$ recognizing $\mathcal L(\mathcal A)$. Since our algorithm never deletes edges, the running time is bounded by the output’s size (which could nevertheless be much larger — or smaller — than $\mathcal A$). In Appendix \[app:DFA->WDFA\] we show: \[thm: ADFA -> WADFA\] Given an acyclic DFA $\mathcal A$ of size $n$, we can build and prefix-sort the minimum acyclic WDFA, of size $m$, recognizing $\mathcal L(\mathcal A)$ in $O(n + m\log m)$ time. Theorem \[thm: ADFA -> WADFA\] solves the problem of indexing deterministic DAGs for linear-time pattern matching queries in nearly-optimal time with a solution of minimum size. Note that the hardness result of Equi et al. [@equi2019complexity2] implies that, under the Orthogonal Vectors hypothesis, in the worst case the minimum WDFA has size $\Omega(n^{2-\epsilon})$ for any constant $\epsilon>0$. We can do better: in Appendix \[sec:worst case blow up\] we show that, in the worst case, the minimum WDFA can be of size $\Omega(2^{n/4})$. Indexing Wheeler Automata {#sec:indexing} ========================= We show that any Wheeler NFA can be efficiently indexed in order to support fast membership queries in its accepting language or in its substring/suffix closure. Let $\mathcal A$ be any Wheeler NFA. We first remove all states that do not lead to a final state. This preserves the accepted language, the total ordering, and the Wheeler properties. We then use our algorithms to convert the automaton to a WDFA, prefix-sort it in polynomial time, and build a (generalized) FM-index on the graph as described in [@gagie2017wheeler]. We mark in a bitvector $B[1..\|V\|]$ supporting constant-time rank queries [@jacobson1988succinct] all accepting states of the Wheeler NFA in our array $\mathtt{LEX}$ containing the states in co-lexicographic order. To check membership of a word $w$, we search the word $\#w$ and get a range $\mathtt{LEX}[L,R]$ of all states reachable from the root by a path labeled $w$. At this point, $w$ is accepted if and only if $B[L,R]$ contains at least one bit set (constant time using rank on $B$). Note that this procedure works in $O(w\log\sigma)$ time also if the original automaton is nondeterministic (this, in general, is not possible for general NFAs). If we search for $w$ instead of $\#w$, then we get the range of states reachable by a path labeled $uw$, for any $u\in\Sigma^$. This range is non-empty if and only if $w$ belongs to the substring closure of $\mathcal L(\mathcal A)$. Finally, if we search a word $w$ and get a range $\mathtt{LEX}[L,R]$, then $w$ is in the suffix closure of $\mathcal L(\mathcal A)$ if and only if $B[L,R]$ contains at least one bit set. Conclusions and Future Extensions ================================= In this paper, we have initiated the study of Wheeler languages, that is, regular languages that can be indexed via prefix-sorting techniques. On our way, we provided new results of independent interest: (i) we provided a new class of NFAs for which the minimization problem can be approximated up to multiplicative factor 2 in polynomial time and that admit fast membership and pattern matching algorithms, and (ii) we solved the problem of indexing finite languages with prefix-sortable DFAs of minimum size. Our work leaves several intriguing lines of research (some of which will be explored in future extensions of this paper). First of all, is the problem of recognizing Wheeler languages (encoded, e.g. as regular expressions) decidable? We believe that the answer to this question is positive: the Wheelerness* of a regular language seems to translate into particular constraints (that can be verified in bounded time) on the topology of its minimum accepting DFA. Once a regular language has been classified as Wheeler, can we build the minimum accepting Wheeler DFA? Also in this case, we believe that the task can be solved by iterating conflict-resolution (Section \[sec:ADFA->WADFA\]) from the minimum DFA until the process converges to the minimum Wheeler DFA. Proofs of Section \[sec:WL\] ============================ (of Lemma \[3n\]) Let us order the elements of $L$ by the relation $<$, and let us denote by $L[i]$ the $i$-th element in the ordering. The notation $L[i,j]$, with $j\geq i$, denotes the interval $\{L[k]:i\leq k \leq j\}$. In particular, $L[1,n] = L$. We say that an interval $I\in \mathcal C$ is maximal if $I$ is not the prefix nor the suffix of any other interval in $\mathcal C$. We say that an interval $I\in \mathcal C$ is prefix (resp. suffix) if $I$ is the proper prefix (resp. suffix) of a maximal interval of $\mathcal C$. Note that, by this definition, intervals of $\mathcal C$ are either maximal or prefix/suffix. Note also that there could be elements of $\mathcal C$ being both prefix and suffix intervals. \(1) We first prove that (1.i) $\mathcal C$ contains at most $n$ prefix intervals, then (1.ii) slightly improve this bound to $n-1$, and finally (1.iii) show that the sum between the number of maximal and suffix intervals is at most $n$. To prove (1.i), we show that every $L[j]$, $1\leq j \leq n$, can be the largest element of at most one prefix interval. In turn, this is shown by considering the prefixes of any two pairwise distinct maximal intervals of $\mathcal C$. Consider two distinct maximal intervals $I=L[i,j]$ and $J=L[i',j']$. If $I$ and $J$ do not overlap (i.e. $j < i'$ or $j' < i$), then the property is trivially true: if $I'$ and $J'$ are prefixes of $I$ and $J$, respectively, then $\max(I') \neq \max(J')$. Consider now the case where $I$ and $J$ overlap. Without loss of generality, we can assume $i < i' \leq j < j'$ (the strict inequalities follow from the fact that, by maximality, it cannot be $i=i'$ or $j=j'$). Assume, for contradiction, that $\mathcal C$ contains two intervals $L[i,j'']$ and $L[i',j'']$ such that $i' \leq j'' < j$, i.e. $L[i,j'']$ and $L[i',j'']$ are (proper) prefixes of $I$ and $J$, respectively, that share their largest element $j''$. Then, we have $i < i' \leq j'' < j$: interval $L[i',j'']$ is strictly contained inside $L[i,j]\in\mathcal C$ (i.e. $L[i',j''] \subset L[i,j]$) and it is not a prefix nor a suffix of it. This is forbidden by the definition of prefix/suffix family. From this contradiction we deduce that any two distinct prefix intervals $I', J' \in \mathcal C$ satisfy $\max(I') \neq \max(J')$, which implies that $\mathcal C$ contains at most $n$ prefix intervals. To improve the above bound to $n-1$ and prove (1.ii), consider the rightmost maximal interval $I=L[i,j]$, i.e. the one having largest $j$. We show that $j$ cannot be the maximum element of any prefix interval. Assume, for contradiction, that such a prefix interval $K=L[i',j]$ exists. Then, the corresponding maximal interval $J = L[i',j']$ of which $K$ is a proper prefix satisfies $j'>j$. This contradicts the fact that $I$ is the rightmost maximal interval. The next step is to prove (1.iii), i.e. that the sum between the number of maximal and suffix intervals is at most $n$. We proceed by induction on the number $M$ of maximal intervals. If $M=1$, then the unique maximal interval $I=L[i,j]$ contains at most $j-i$ suffix intervals. In total, $\mathcal C$ contains at most $1 + (j-i) \leq n$ maximal and suffix intervals. For $M>1$, consider the maximal interval $I = L[i,j]$ with minimum $j$ (call it the “leftmost”). Now, consider the immediate maximal “successor” $J = L[i',j']$ of $I$, i.e. the maximal interval with the smallest endpoint $j'\geq j$. Clearly, such $j'$ satisfies $j'>j$, otherwise $J$ would be a suffix of $I$ (contradicting maximality of $J$). Note that it must also be the case that $i'>i$: if $i=i'$, then $I$ would be a prefix of $J$ (contradicting maximality of $I$); on the other hand, if $i'<i$ then $I$ would be strictly contained in $J$, contradicting the definition of prefix/suffix family. We are left with two cases: \(a) $i \leq j < i' \leq j'$. In this case, $I$ and $J$ are disjoint. As seen above, $I$ contributes to at most one maximal interval ($I$ itself) and $j-i$ suffix intervals. In total, $I$ contributes to at most $j-i+1$ maximal and suffix intervals. We are left to count the number of maximal and suffix intervals in the remaining portion of the linear order $L[i',...,n]$. Note that there are no other intervals to be considered: if $L[i'', j'']$ is a maximal interval in $\mathcal C$, different from $I,J$, then $j''>j'$ and hence $i''>i'$ or $L[i'',j'']$ would contain $J$. The portion $L[i',...,n]$ contains $M-1$ maximal intervals, so we can apply the inductive hypothesis and obtain that this segment contains at most $n-i'+1$ maximal and suffix intervals. In total, we have that $L[1,...,n]$ contains at most $(j-i+1) + (n-i'+1)$ maximal and suffix intervals. Since $i'>j$ and $i\geq 1$, this quantity is at most $n$. \(b) $i < i' \leq j < j'$. Denote by $k = i'-i$ the number of $L$’s elements belonging to $I \setminus J$. Then, $\mathcal C$ can contain at most $k$ proper suffixes of $I$: $L[i+1,j]$, $L[i+2,j]$, ..., $L[i', j]$. All other suffixes of $I$ are strictly contained inside $J$, and cannot belong to $\mathcal C$ due to the prefix/suffix property. Actually, one of those suffixes, $L[i',j]$, is a prefix of $J$ so it has already been counted above in points (1.i) and (1.ii). We are left with $k-1$ suffixes to take into account, plus the maximal interval $I$ itself: in total, $k = i'-i$ maximal and suffix intervals. As noted above, all remaining maximal and suffix intervals of $\mathcal C$ to take into account are those contained in $L[i',n]$. Since $L[i',n]$ contains $M-1$ maximal intervals, we can apply the inductive hypothesis and deduce that it contains at most $n-i'+1$ maximal and suffix intervals. In total, $L[1,n]$ contains therefore at most $(i'-i) + (n-i'+1) \leq n$ maximal and suffix intervals. This concludes the proof of the upper bound $\|\mathcal C\| \leq 2n-1$. \(2) Consider the prefix/suffix family containing just one maximal interval and all its proper prefixes and suffixes: $\mathcal C = \{ L[1,n], L[1,1], \dots, L[1,n-1], L[2,n], \dots, L[n,n]\}$. This family satisfies $\|\mathcal C\| = 2n-1$. (of Lemma \[convex+order\]) We just prove transitivity when $I<^{i} J$ and $J<^{i} K$ are witnessed by $x_0\in I$ satisfying $ (\forall y\in J) ( x_0< y )$, and $ z_0 \in K$ satisfying $ (\forall y \in J) ( y< z_0)$, respectively (the other cases are similar). We claim that $z_0>x$, for all $x\in I$. Suppose, for contradiction, that there exists $x_1\in I$ with $z_0\leq x_1$; then, from $x_0<y<z_0\leq x_1$ for all $y\in J$ and the fact that $I$ is an interval, it follows that $z_0\in I$, $J\subseteq I$, so that, by prefix/suffix property of $\mathcal C$, $J$ is either a prefix or a suffix of $I$. Since $ x_0< y$ for all $y\in J$, we see that $J$ must be a suffix of $I$ and this, knowing that $z_0\in I$, implies $z_0\in J$. A contradiction. In the following proofs, we always refer to a WNFA $\mathcal A=(V,E,F,s, \Sigma,<)$. (of Lemma \[prec\_versus\_minus\]) 1. Suppose $\alpha\in I_u, \beta\in I_v$ and $\{\alpha,\beta\}\not\subseteq I_v\cap I_u$. From this we have that $\alpha\in I_u\setminus I_v$ or $\beta\in I_v\setminus I_u$, hence $u\neq v$ and $\alpha\neq \beta$ follows. If $u=s$ or $v=s$, either $ \alpha $ or $ \beta $ is the empty string $ \epsilon $ and the result follows easily. Hence, we suppose $u\neq s \neq v$ and (hence) $\alpha\neq \epsilon \neq \beta$. To see the left-to-right implication, assume $\alpha \prec \beta$: we prove that $u<v$ by induction on the maximum betwewn $\|\alpha\|$ and $\|\beta\|$. If $\|\alpha\| =\|\beta\|=1$, then the property follows from the Wheeler-(i). If $\max(\|\alpha\|, \|\beta\|)>1$ and $ \alpha $ and $ \beta $ end with different letters, then again the property follows from Wheeler-(i). Hence, we are just left with the case in which $\alpha=\alpha' e$ and $\beta=\beta' e$, with $e\in \Sigma$. If $\alpha\prec \beta$, then $\alpha'\prec\beta'$. Consider states $u',v'$ such that $\alpha' \in I_{u'}, \beta'\in I_{v'}$, and $(u',u,e), (v',v,e)\in E$. Then $\alpha'\in I_{u'}\setminus I_{v'}$ or $\beta'\in I_{v'}\setminus I_{u'}$ because otherwise we would have $\alpha'\in I_{v'}$ and $\beta'\in I_{u'}$ which imply respectively $\alpha\in I_v$ and $\beta\in I_u$. By induction we have $u'<v'$ and therefore, by Wheeler-(ii), $u\leq v$. From $u\neq v$ it follows $u<v$. Conversely, for the right-to-left implication, suppose $u<v$. Since $\alpha \neq \beta$, if it were $\beta\prec \alpha$ then, by the above, we would have $v<u$: a contradiction. Hence, $\alpha\prec \beta $ holds. 2. Recall that, by definition, $ \alpha \in I_u \text{ if and only if } u \in I_\alpha$ and $ \beta \in I_v \text{ if and only if } v \in I_\beta$. Hence, the hypothesis that $u\in I_\alpha, v\in I_\beta$ and $\{u,v\} \not\subseteq I_\beta \cap I_\alpha$, is equivalent to say that $ \alpha\in I_u, \beta\in I_v$ and $\{\alpha, \beta\} \not\subseteq I_v \cap I_u$. Therefore, (2) follows from (1). (of Lemma \[convex\_sets\]) 1. Suppose $\alpha\prec \beta\prec\gamma$ with $\alpha, \gamma\in I_u$ and $\beta \in \text{\em Pref}(\mathcal L(\mathcal A))$; we want to prove that $\beta \in I_u$. From $\beta\in \text{\em Pref}(\mathcal L(\mathcal A))$ it follows that there exists a state $v$ such that $\beta \in I_v$. Suppose, for contradiction, that $\beta\not \in I_u$. Then $\beta \in I_v\setminus I_u$ and from $\alpha\prec \beta$ and Lemma \[prec\_versus\_minus\], it follows $u<v$. Similarly, applying again Lemma \[prec\_versus\_minus\], from $\beta\prec \gamma$ we have $v<u$, which is a contradiction. 2. Suppose, for contradiction, that $I_u, I_v\in I_{V}$ are such that $I_u\subsetneq I_v$ and $I_u$ is neither a prefix nor a suffix of $I_v$. In these hypotheses there must exist $\alpha, \alpha' \in I_v\setminus I_u$ and $\beta \in I_u$ such that $\alpha \prec \beta \prec \alpha'$. Lemma \[prec\_versus\_minus\] implies $v<u<v$, which is a contradiction. Points $(3), (4)$ follow similarly from Lemma \[prec\_versus\_minus\]. (of Lemma \[Wdeterminization\]) The verification that $ \mathcal L(\mathcal A^{d}) = \mathcal L(\mathcal A)$ follows the same lines of the proof in the classical regular case. We prove that $<^{d}$ is a Wheeler order on the states of the automaton $\mathcal A^{d}$. By Lemma \[convex\_sets\], the set $V^{d}=I_{\text{\em Pref}(\mathcal L(\mathcal A))}$ of states of $\mathcal A^{d}$ is a prefix/suffix family of intervals, so that, by Lemma \[convex+order\], $<^{d}$ is a linear order on $V^{d}$. Next, we check the Wheeler properties. The only vertex with in-degree $0$ is $I_\epsilon$, and it clearly precedes those with positive in-degree. For any two edges $(I_\alpha,I_{\alpha a_1}, a_1)$, $(I_\beta,I_{\beta a_2}, a_2)$ we have: - if $a_1 \prec a_2$ then $\alpha a_1\prec \beta a_{2}$, and from Lemma \[compare\] it follows $I_{\alpha a_1}\leq^d I_{\beta a_2}$. Moreover, by the input consistency of $\mathcal A$, states in $I_{\alpha a_1}$ are $a_1$-states, while states in $I_{\beta a_2}$ are $a_2$-states; hence $I_{\alpha a_1}\neq I_{\beta a_2}$, so that $I_{\alpha a_1}<^d I_{\beta a_2}$ follows. - If $a=a_{1}=a_{2}$ and $I_\alpha < I_\beta$, from Lemma \[compare\] it follows $\alpha\prec \beta$, so that $\alpha a \prec \beta a$ and, using again Lemma \[compare\], we obtain $I=I_{\alpha a}\leq^i I=I_{\beta a}$. Finally, we prove that $\|V^d\|\leq 2n-1-\|\Sigma\|$. By the Wheeler properties, we know that the only interval in $I_{\text{\em Pref}(\mathcal L(\mathcal A))}$ containing the initial state $s$ of the automaton $\mathcal A$ is $\{s\}$ and that the remaining intervals can be partitioned into $\|\Sigma\|$-classes, by looking at the letter labelling incoming edges. If $\Sigma=\{a_1, \ldots, a_k\}$, and, for every $i=1,\ldots,k$, we denote by $m_i$ the number of states of the automaton $\mathcal A$ whose incoming edges are labeled $a_i$, we have $\sum_{i=1}^k m_i=n-1$. Using Lemma \[3n\] we see that the intervals in $V^d$ composed by $a_i$ states are at most $2m_i-1$, so that the total number of intervals in $V^d$ is at most $$1+ \sum_{i=1}^k (2m_i-1)=1+ 2( \sum_{i=1}m_i )-k= 1+2(n-1)-k=2n-1-k= 2n-1-\|\Sigma\|.$$ We apply the standard powerset construction algorithm starting from the original WNFA $\mathcal A$. By Lemma \[Wdeterminization\], the powerset algorithm does not generate more than $2n-1-\|\Sigma\|$ distinct sets of states. Remember that such algorithm starts from the set $\{s\}$ containing the NFA’s source and simulates a visit of the final DFA, whose states are represented as sets of states of the original NFA. At each step, the successor with label $a\in\Sigma$ of a set $K$ is computed by calculating all the $a$-successors of states in $K$, and taking their union. In the worst case, $\|K\| = O(n)$ and each state in $K$ has $O(n)$ $a$-successors. After having obtained the $a$-successor $K'$ of $K$, we need to check if $K'$ had already been visited. Since $K'$’s cardinality is at most $n$, this operation takes $O(n)$ time using a standard dictionary (e.g. a hash table). Overall, we spend $O(n^2)$ time to simulate an edge traversal of the final DFA. By Lemma \[Wdeterminization\], we visit at most $O(n)$ distinct sets of states. Overall, the powerset algorithm’s complexity is $O(n^3)$. (of Theorem \[Myhill-Nerode\]) - \(1) $\Rightarrow$ (2) If $\mathcal A$ is a Wheeler NFA such that $\mathcal L= \mathcal L(\mathcal A)$, consider the following equivalence relation $\sim_{\mathcal A}$ over $ \text{\em Pref}(\mathcal L)$: $$\alpha \sim_{\mathcal A} \beta ~~\Leftrightarrow I_\alpha=I_\beta.$$ Using the fact that the $I_\alpha$ are intervals (see Lemma \[convex\_sets\]), and other properties of Wheeler automata, one can easily prove that the equivalence $\sim_{\mathcal A}$ is a refinement of $\equiv_{\mathcal L}^{c}$, so that each $\equiv_{\mathcal L}^{c}$-class is a union of $\sim_{\mathcal A}$-classes. Moreover, the equivalence $\sim_{\mathcal A}$ has finite index, bounded by the number of intervals $I_\alpha$, hence $\equiv_{\mathcal L}^{c}$ has finite index as well. - \(2) $\Rightarrow$ (3) We prove that the relation $\equiv_{\mathcal L}^{c}$ is a convex, input consistent, right invariant equivalence, and that $\mathcal L$ is a union of $\equiv_{\mathcal L}^{c}$-classes; this last property is true simply because $\mathcal L$ is a union of $\equiv_{\mathcal L}$-classes and $\equiv_{\mathcal L}^{c}$ is a refinement of $ \equiv_{\mathcal L}$. The fact that $\equiv_{\mathcal L}^{c}$ is convex and input consistent follows directly from its definition. We prove that $\equiv_{\mathcal L}^{c}$ is right invariant. Suppose $\alpha , \alpha' , \gamma\in \text{\em Pref}(\mathcal L)$ and $\alpha\equiv_{\mathcal L}^{c} \alpha'$. Note that if $\alpha \gamma \in \text{\em Pref}(\mathcal L)$ then there exists $\nu \in \Sigma^$ such that $\alpha \gamma \nu \in \mathcal L$, so that $\alpha' \gamma \in \text{\em Pref}(\mathcal L)$ follows from $\alpha\equiv_{\mathcal L} \alpha'$. Hence, we are left to prove that $\alpha\gamma \equiv_{\mathcal L} ^{c} \alpha'\gamma$. We easily prove the following: - $\alpha \gamma \equiv_{\mathcal L} \alpha'\gamma$ (it follows from $\alpha\equiv_{\mathcal L}\alpha'$). - If $\alpha \gamma \prec \beta' \prec \alpha' \gamma$, for $\beta'\in \text{\em Pref}(\mathcal L)$, then $\beta' \equiv_{\mathcal L} \alpha \gamma$: from $\alpha \gamma \prec \beta' \prec \alpha' \gamma$ it follows that $\beta'=\beta \gamma$, for some $\beta\in \text{\em Pref}(\mathcal L)$, and $\alpha \prec \beta \prec \alpha'$. Since $\alpha, \alpha'$ belong to the same $\equiv_{\mathcal L}^{c} $ class, then $\beta\equiv_{\mathcal L} \alpha$, and $\beta \gamma \equiv_{\mathcal L} \alpha \gamma$ follows from the right invariance of $\equiv_{\mathcal L} $. Since $\alpha \gamma, \beta \gamma $ end with the same letter, the previous points imply that $\alpha\gamma \equiv_{\mathcal L} ^{c} \alpha'\gamma$ and $\equiv_{\mathcal L}^{c} $ is right invariant. - \(3) $\Rightarrow$ (4) Suppose $\mathcal L$ is a union of classes of a convex, input consistent, right invariant equivalence relation $ \sim $ of finite index. We build a WDFA ${\mathcal A}_\sim=(V_{\sim}, E_{\sim}, F_{\sim}, s_{\sim}, \Sigma, <_{\sim} )$ such that $\mathcal L=\mathcal L(\mathcal A)$ as follows: - $ V_{\sim}=\{[\alpha]_{\sim}~\|~\alpha \in \text{\em Pref}(\mathcal L)\}$; - $ s_{\sim}=[\epsilon]_{\sim} $ (note that, by input consistency, $[\epsilon]_{\sim}=\{\epsilon\}$); - $ (I,J,e) \in E_{\sim}$ if and only if $ Ie \cap \text{\em Pref}(\mathcal L) \neq \emptyset $ and $ Ie \subseteq J $, where $Ie=\{\alpha e ~\|~ \alpha \in I\}$ (note that $ J $, if existing, is unique by right invariancy); - $ F_{\sim}= \{I ~\|~I \subseteq \mathcal L\}$; - $<_{\sim} = \prec^{i}$ (being pairwise disjoint and convex, the classes in $V_{\sim}$ form a prefix/suffix family of intervals of $( \text{\em Pref}(\mathcal L), \prec)$). Note that all words in $\text{\em Pref}(\mathcal L)$ label a computation in ${\mathcal A}_\sim$. We claim that, for all $\sim$-class $I$ and $\alpha\in\text{\em Pref}(\mathcal L)$: $$\alpha\in I ~~\Leftrightarrow ~~ s_\sim \rightsquigarrow I \text{ in } \mathcal{A}_{\sim} \text{ reading } \alpha.$$ We prove the implication from right to left by induction on the length of $\alpha\in \text{\em Pref}(\mathcal L)$. If $\alpha=\epsilon$ then the claim follows from the definition of $s_\sim$. If $\alpha=\alpha' e\in \text{\em Pref}(\mathcal L)$ with $e\in \Sigma$, then $\alpha'\in \text{\em Pref}(\mathcal L)$. Then, if $K\in V_{\sim}$ is such that $s_\sim \rightsquigarrow K$ reading $\alpha'$ in $\mathcal{A}_{\sim}$, by induction we know that $\alpha'\in K$. Since $\alpha =\alpha' e\in Ke $, we have $ Ke\cap \text{\em Pref}(\mathcal L) \neq \emptyset$; by right invariance of $\sim$ there exists a unique $J$ such that $Ke\subseteq J$. From $\alpha=\alpha'e\in Ke\subseteq J$ it follows $\alpha\in J$, and also $J=I$, because ${\mathcal A}_\sim$ is a deterministic automaton and $s_\sim \rightsquigarrow I$, $s_\sim \rightsquigarrow J$, both by reading $\alpha$. In order to prove the implication from left to right of the claim, suppose $\alpha \in I$, and $J\in V_\sim$ is such that $s_\sim \rightsquigarrow J \text{ in } \mathcal{A}_{\sim}$ reading $\alpha$. Then, by the first part of the proof of the claim we obtain $\alpha \in J$; since $J$ and $I$ are equivalence classes and $\alpha \in I \cap J$, it follows that $I=J$ and $s_\sim \rightsquigarrow I \text{ in } \mathcal{A}_{\sim}$ reading $\alpha$. From the above claim and the definition of $ F_{\sim} $, it easily follows that $\mathcal L$ is the language recognised by ${\mathcal A}_{\sim}$. We conclude by checking that ${\mathcal A}_{\sim}$ is Wheeler, proving the two Wheeler properties (i) and (ii) with respect to the linear order $(V_\sim, <_{\sim})$. To see Wheeler-(i) assume $e\prec e' $ with $e,e'\in\Sigma$. Consider $I,J \in V_{\sim}$ such that $(I,H,e)\in E_{\sim}$ and $(J,K,e')\in E_{\sim}$. We want to prove that $H<_{\sim} K$ (i.e. $H\prec^{i} K$). By definition of $E_{\sim}$, in our hypotheses there are $\alpha\in I$, $ \alpha'\in J$ with $\alpha e \in H$ and $ \alpha' e' \in K$. From $e\prec e'$ it follows $\alpha e \prec \alpha' e'$ and hence $H \preceq^{i} K$. To conclude observe that $H \prec^{i} K$ since all words in $H$ end with $e$, while all words in $K$ end with $e'$. To see Wheeler-(ii) assume $I<_{\sim} J$ (i.e. $I \prec^{i} J$), $e\in \Sigma$, $(I,H,e)\in E_{\sim}$, and $(J,K,e)\in E_{\sim}$. In these hypotheses there are $\alpha\in I$, $\alpha'\in J $, with $\alpha e \in H$ and $ \alpha' e \in K$. From $I \prec^{i} J$ and the fact that different classes are disjoint it follows $\alpha \prec \alpha'$; therefore, $\alpha e \prec \alpha' e$ and hence $ H \preceq^{i} K $. This ends the proof of the implication $(3) \Rightarrow (4)$. - \(4) $\Rightarrow$ (1) Trivial. Proofs of Section \[sec:sorting\] ================================= (of Theorem \[thm:2NFA in P\]) \[proof thm 2NFA in P\] We can assume, without loss of generality, that $\mathcal A$ is input-consistent, since checking this property takes linear time. If $\mathcal A$ is not input-consistent, then it is not Wheeler. We show a reduction of problem [1]{} to 2-SAT, which can be solved in linear time using Aspvall, Plass, and Tarjan’s (APT) algorithm based on strongly connected components computation. The reduction introduces $O(\|V\|^2)$ variables and $O(\|E\|^2)$ clauses, hence the final running time will be $O(\|E\|^2)$. Moreover, since a satisfying assignment to our boolean variables will be sufficient to define a total order of the nodes, APT will essentially solve also problem [2]{}. For every pair $u\neq v$ of nodes we introduce a variable $x_{u<v}$ which, if true, indicates that $u$ must precede $v$ in the ordering. We now describe a 2-SAT CNF formula whose clauses are divided in two types: clauses of the former type ensure that the Wheeler graph property is satisfied, while clauses of the second type ensure that the order of nodes induced by the variables is total. The following formulas ensure that the Wheeler properties are satisfied: - For each $u,v$, if $\lambda(u)\prec \lambda(v)$ then we add the unary clause $x_{u<v}$. - For each $u\neq v$, if $\lambda(u)=\lambda(v)=a$, then for every pair $u'\neq v'$ such that $(u',u,a)\in E$ and $(v',v,a)\in E$ we add the clause $x_{u'<v'}\rightarrow x_{u<v}$. There are at most $\|V\|^2\leq \|E\|^2$ clauses of type (a) and at most $\|E\|^2$ clauses of type (b). The following formulas guarantee that the order is total. Note that we omit transitivity which, on a general graph, would require a 3-literals clause $(x_{u<v} \wedge x_{v<w}) \rightarrow x_{u<w}$ for each triple $u,v,w$. We will show that, if the graph is an input-consistent $2$-NFA, then transitivity is satisfied “for free”. - Antisymmetry. For every pair $u\neq v$, add the clause $x_{u<v} \rightarrow \neg x_{v<u}$. - Completeness.* For every pair $u\neq v$, add the clause $x_{u<v}\ \vee\ x_{v<u}$. There are at most $O(\|V\|^2) = O(\|E\|^2)$ clauses of types (1) and (2). We now show that on input-consistent $2$-NFAs transitivity propagates from the source to all nodes. Consider a variable assignment that satisfies clauses (a),(b),(1), and (2) (if $\mathcal A$ is a Wheeler $2$-NFA, then such an assignment exists by definition). Assume, moreover, that $x_{u<v}$ and $x_{v<w}$ are set true by the assignment, for three pairwise distinct nodes $u,v,w$. We want to show that also $x_{u<w}$ must be true. Consider a directed shortest-path tree $\mathcal T$ with root $s$ of $\mathcal A$. Since we assume that each state is reachable from $s$, $\mathcal T$ must exist and must contain all nodes of $\mathcal A$. Let $d_v$ be the length of a shortest directed path connecting $s$ to $v$. By definition of $\mathcal T$, the path connecting $s$ to $v$ in $\mathcal T$ has length $d_v$, with $d_s=0$. We proceed by induction on $k=\max\{ d_u,d_v,d_w \}$. The case $k=0$ is trivial, since there are no triples of pairwise distinct nodes in $\{u\ :\ d_u\leq 0\}$ (this set contains just $s$). Take now a general $k>0$. We consider two main cases: \(i) $\|\{\lambda(u),\lambda(v),\lambda(w)\}\|>1$. Then, since $x_{u<v}$ and $x_{v<w}$, for some $a<b<c\in\Sigma$ either: (i.1) $\lambda(u)=a,\ \lambda(v)=b,\ \lambda(w)=c$, or (i.2) $\lambda(u)=a,\ \lambda(v)=a,\ \lambda(w)=b$, or (i.3) $\lambda(u)=a,\ \lambda(v)=b,\ \lambda(w)=b$. Any other choice would force one of the variables $x_{v<u}, x_{w<v}$ to be true (by an (a)-clause), forcing a contradiction by a (1)-clause. In all cases (i.1)-(i.3) we have that $\lambda(u)<\lambda(w)$, therefore $x_{u<w}$ must be true by (a). \(ii) $\lambda(u) = \lambda(v) = \lambda(w) = a$ for some $a\in\Sigma$ (note that $a\neq \#$ since the NFA has only one source and $u,v,w$ are distinct by assumption). Let $u',v',w'$ be the parents of $u,v,w$, respectively, in $\mathcal T$. Note that $u',v',w'$ cannot be the same vertex, since $u,v,w$ are distinct and every node has at most two outgoing edges with the same label. We therefore consider two sub-cases. (ii.1) $\|\{u',v',w'\}\| = 2$. We first show that $u'=w'\neq v'$ generates a contradiction. Since $x_{u<v}$ and $x_{v<w}$ are true and $u'\neq v'$ and $v'\neq w'$ hold, $x_{u'<v'}$ and $x_{v'<w'}$ must be true: otherwise, by (b), would imply that $x_{v<u}$ and $x_{w<v}$ are true, which generates a contradiction. Now, $u'=w'$ means that $x_{v'<w'}$ and $x_{v'<u'}$ have the same truth value; since $x_{u'<v'}$ and $x_{v'<u'}$ cannot be both true, we have a contradiction. We are therefore left with the case $u'=v'\neq w'$ ($u'\neq v' = w'$ is symmetric). Remember that we assumed $x_{u<v}$ and $x_{v<w}$ are true. Hence, $x_{v'<w'}$ must be true: otherwise, by (b), the truth of $x_{w'<v'}$ would imply that $x_{w<v}$ is true, which generates a contradiction. Since $x_{v'<w'} = x_{u'<w'}$ is true, by (b) we conclude that also $x_{u<w}$ must be true. (ii.2) $u',v',w'$ are pairwise distinct. We show that $x_{u'<v'}$ and $x_{v'<w'}$ must be true. Suppose, for contradiction, that $x_{u'<v'}$ is false (the proof is symmetric for $x_{v'<w'}$). Then, by (2), $x_{v'<u'}$ is true. But then, by (b) it must be the case that $x_{v<u}$ is true. Since we are assuming that $x_{u<v}$ is true, this introduces a contradiction by (1). Therefore, we conclude that $x_{u'<v'}$ and $x_{v'<w'}$ are true for the (pairwise distinct) parents $u',v',w'$ of $u,v,w$ in $\mathcal T$. Now, by definition of the shortest-path tree $\mathcal T$ it must be the case that $d_{u'} = d_{u}-1$, $d_{v'} = d_{v}-1$, and $d_{w'} = d_{w}-1$ as $u',v',w'$ are the parents of $u,v,w$ in $\mathcal T$. As a consequence, $\max\{d_{u'},d_{v'},d_{w'}\} = k-1$. We can therefore apply the inductive hypothesis and conclude that $x_{u'<w'}$ is true. But then, by (b) we conclude that $x_{u<w}$ must also be true. From the above proof correctness follows: if $\mathcal A$ is an input-consistent $2$-NFA and there exists a truth assignment satisfying the formula, then the assignment induces a total ordering of the nodes satisfying the Wheeler properties. Conversely, the algorithm is clearly complete: if $\mathcal A$ is a Wheeler $2$-NFA, then there exists a total ordering of the nodes satisfying the Wheeler properties. This defines a truth assignment of the variables that satisfies our 2-SAT formula. We note that it is tempting to try to generalize the above solution to general NFAs by simulating arbitrary degree-$d$ nondeterminism using binary trees: a node with $d$ equally-labeled outgoing edges could be expanded to a binary tree with $d$ leaves (bringing down the degree of nondeterminism to $2$). Unfortunately, while this solution works for transitivity (which is successfully propagated from the source), it could make the graph non-Wheeler: the topology of those trees cannot be arbitrary and must satisfy the co-lexicographic ordering of the nodes, i.e. the solution we are trying to compute. Sorting WDFAs Online {#app: Sorting WDAGs Online} -------------------- Algorithm \[alg:sort\] initializes all variables used by our procedure and implements Kahn’s topological-sorting algorithm [@kahn1962topological]. Every time a new node is appended to the topological ordering, we call Algorithm \[alg:step\]—our actual online algorithm—to update also the co-lexicographic ordering. This step also checks if the new node and its incoming edges falsify the Wheeler properties. We use the following structures (indices start from 1): - $\mathtt{LEX}$ is a dynamic sequence of distinct nodes $v_1, \dots, v_k\in V$ supporting the following operations: 1. $\mathtt{LEX[i]}$ returns $v_i$. 2. $\mathtt{LEX^{-1}[v]}$, with $v\in \mathtt{LEX}$, returns the index $i$ such that $\mathtt{LEX}[i]=v$. 3. $\mathtt{LEX.insert(v,i)}$ inserts node $v$ between $\mathtt{LEX[i-1]}$ and $\mathtt{LEX[i]}$. If $i=1$, $v$ is inserted at the beginning of the sequence. This operation increases the sequence’s length by one.\ - $\mathtt{IN}$ and $\mathtt{OUT}$ are dynamic sequences of strings, i.e. sequences $\alpha_1, \dots, \alpha_k$, where $\alpha_i\in\Sigma^$ (note that $\alpha_i$ could be the empty string $\epsilon$). To make our pseudocode more readable, we index $\mathtt{IN}$ and $\mathtt{OUT}$ by nodes of $\mathtt{LEX}$ (these three arrays will be synchronized). Let $\mathtt{T} = \alpha_1, \alpha_2,\dots, \alpha_k$, with $\mathtt{T\in \{IN,OUT\}}$. Both arrays support the following operation: 4. $\mathtt{T.insert(\alpha,v)}$, where $\alpha\in\Sigma^$ and $v\in \mathtt{LEX}$: insert $\alpha$ between $\alpha_{\mathtt{LEX}^{-1}[v]-1}$ and $\alpha_{\mathtt{LEX}^{-1}[v]}$. If $\mathtt{LEX^{-1}[v]}=1$, then $\alpha$ is inserted at the beginning of $\mathtt{T}$. This operation increases the sequence’s length by one. \ Sequence $\mathtt{OUT}$ supports these additional operations: 5. $\mathtt{OUT}[v]$, with $v\in \mathtt{LEX}$, returns $\alpha_{\mathtt{LEX}^{-1}[v]}$. 6. $\mathtt{OUT.append(\alpha,v)}$, where $\alpha\in\Sigma^$ and $v\in \mathtt{LEX}$: append the string $\alpha$ at the end of the string $\mathtt{OUT[v]}$, i.e. replace $\mathtt{OUT[v]} \leftarrow \mathtt{OUT[v]}\cdot \alpha$. Note that this operation does not increase $\mathtt{OUT}$’s length. 7. $\mathtt{OUT.rank(c,u)}$, with $u\in \mathtt{LEX}$ and $c\in \Sigma$: return the number of characters equal to $c$ in all strings $\mathtt{OUT[v]}$, with $v = \mathtt{LEX[1]}, \mathtt{LEX[2]}, \dots, \mathtt{LEX[LEX^{-1}[u]]}$. 8. $\mathtt{OUT.reserve(u,v,c)}$, with $u,v\in \mathtt{LEX}$ and $c\in \Sigma$: from the moment this operation is called, the sequence $\alpha_{\mathtt{LEX^{-1}[u]}}, \dots, \alpha_{\mathtt{LEX^{-1}[v]}}$ is marked with label $c$. Note that inserting new elements inside $\alpha_{\mathtt{LEX^{-1}[u]}}, \dots, \alpha_{\mathtt{LEX^{-1}[v]}}$ will increase the length of the reserved sequence. 9. $\mathtt{OUT.is\_reserved(v,c)}$, with $v\in \mathtt{LEX}$ and $c\in \Sigma$: return $\mathtt{TRUE}$ iff $\alpha_{\mathtt{LEX}^{-1}(v)}$ falls inside a sequence that has been marked (reserved) with character $c$. In our algorithm, sequence $\mathtt{IN}$ will always be partitioned in at most $t\leq \sigma+1$ sub-sequences $\mathtt{IN} = \alpha^{c_1}_1, \dots, \alpha^{c_1}_{k_{c_1}}, \alpha^{c_2}_1, \dots, \alpha^{c_2}_{k_{c_2}}, \dots, \alpha^{c_t}_1, \dots, \alpha^{c_t}_{k_{c_t}}$, where each $\alpha^{c}_i$ contains only character $c$ and $c_1 \prec c_2 \prec \dots \prec c_t$. We define an additional operation on $\mathtt{IN}$:\ 10. $\mathtt{IN.start(c)}$, with $c\in \Sigma$, returns the largest integer $j\geq 1$ such that all characters in $\mathtt{IN}[v]$ are strictly smaller than $c$, for all $v = \mathtt{LEX}[1], \dots, \mathtt{LEX}[j-1]$. Figure \[fig:inconsistencies\] shows how our dynamic structures evolve while processing states in topological order. In Appendix \[sec:data structures\] we discuss data structures implementing the above operations in $O(\log k)$ time, $k$ being the sequence’s length. Intuitively, these three dynamic sequences have the following meaning: $\mathtt{LEX}$ will contain the co-lexicographically-ordered sequence of nodes. $\mathtt{IN}[v]$ and $\mathtt{OUT}[v]$, with $v\in\mathtt{LEX}$, will contain the labels of the incoming and outgoing edges of $v$, respectively. To keep the three sequences synchronized, when inserting $v$ in $\mathtt{LEX}$ we will also need to update the other two sequences so that $\mathtt{IN}[v] = c^t$, where $t$ is the number of incoming edges, labeled $c$, of $v$, and $\mathtt{OUT}[v] = \epsilon$, since $v$ does not have yet outgoing edges. $\mathtt{OUT}[v]$ will (possibly) be updated later, when new nodes adjacent to $v$ will arrive in the topological order. Our representation is equivalent to that used in [@siren2014indexing] to represent the GCSA data structure. Intuitively, $\mathtt{OUT}$ is a generalized version of the well-known Burrows-Wheeler transform (except that we sort prefixes in co-lexicographic order instead of suffixes in lexicographic order). If the graph is a path (i.e. a string) then $\mathtt{OUT}$ is precisely the BWT of the reversed path. We proceed with a discussion of the pseudocode. In Lines \[line:init in\]-\[line:init OUT\] of Algorithm \[alg:sort\] we initialize all variables and data structures. Let $u\in V$. The variable $\mathtt{u.in}$ memorizes the number of incoming edges in $u$; we will use this counter to implement Kahn’s topological sorting procedure. $\mathtt{u.label}$ is the label of all incoming edges of $u$, or $\#$ if $u=s$. $\mathtt{IN}, \mathtt{LEX}$, and $\mathtt{OUT}$ are initialized as empty dynamic sequences. Lines \[line: init S\]-\[line:cycle\] implement Kahn’s topological sorting algorithm [@kahn1962topological]. Each time a new node $u$ is appended to the order, we call our online procedure $\mathtt{update(u)}$, implemented in Algorithm \[alg:step\]. Algorithm \[alg:step\] works as follows. Assume that we have already sorted $v_1, \dots, v_k$, that $\mathtt{LEX}$ contains the nodes’ permutation reflecting their co-lexicographic order, and that $\mathtt{IN[v_i]}$ and $\mathtt{OUT[v_i]}$ contain the incoming and outgoing labels for each $i=1, \dots, k$ in the sub-graph induced by $v_1, \dots, v_k$. When a new node $u$ arrives in topological order, all its $t$ predecessors are in $\mathtt{LEX}$. Let $b = \mathtt{u.label}$ be the incoming label of $u$. We find the co-lexicographically smallest $v_{min}$ and largest $v_{max}$ predecessors of $u$ (using function $\mathtt{LEX^{-1}}$ on all $u$’s predecessors). In our pseudocode, if $u=s$ then $v_{min} = v_{max} = \mathtt{NULL}$. To keep the Wheeler properties true, note that there cannot be $b$’s in the range $\mathtt{OUT[v_{min}..v_{max}]}$: if there are, since we will append $b$ to $\mathtt{OUT[v_{min}]}$ and $\mathtt{OUT[v_{max}]}$, there will be three nodes $v_{min} < v' < v_{max}$ such that $(v_{min},u,b), (v',u',b), (v_{max},u, b) \in E$ for some $u'$. Then, by Wheeler property (ii), this would imply that $u < u' < u$, a contradiction. We therefore check this event using function $\mathtt{contains}$ (note: this function can be easily implemented using two calls to $\mathtt{rank}$). If $b$’s are present, then the graph is no longer Wheeler: such an event is shown in Figure \[fig:inconsistencies\], left-hand side (where $u = v_5$). Otherwise, the number $j$ of $b$’s before $v_{min}$ (which is equal to the number of $b$’s before $v_{max}$) tells us the co-lexicographic rank $i$ of $u$ (similarly to the standard string-BWT, we obtain this number by adding $j$ to the starting position of $b$’s in $\mathtt{IN}$), and we can mark (reserve) range $\mathtt{OUT[v_{min}..v_{max}]}$ with letter $b$ using function $\mathtt{reserve}$. Such an event is shown in Figure \[fig:inconsistencies\], left-hand side, when inserting, e.g., node $v_3$. At this point, we may have an additional inconsistency falsifying the Wheeler properties in the case that one of the predecessors $v_i$ of $u$ falls inside a reserved range for $b$ (reserved by a node other than $u$): this happens, for example, when inserting $v_6$ in Figure \[fig:inconsistencies\], right-hand side. This check requires calling function $\mathtt{is\_reserved}$. If all tests succeed, we insert $u$ in position $i$ of $\mathtt{LEX}$ and we update $\mathtt{IN}$ and $\mathtt{OUT}$ by inserting $b^t$ at the $i$-th position in $\mathtt{IN}$ (i.e. the position corresponding to $u$) and by appending $b$ at the end of each $\mathtt{OUT[v_i]}$ for each predecessor $v_i$ of $u$. $\mathtt{s.label \leftarrow \#}$; $\mathtt{IN} \leftarrow \mathtt{new\_dyn\_sequence(\Sigma^)}$ $\mathtt{LEX} \leftarrow \mathtt{new\_dyn\_sequence(V)}$ $\mathtt{OUT} \leftarrow \mathtt{new\_dyn\_sequence(\Sigma^)}$\[line:init OUT\] $S \leftarrow \{s\}$\[line: init S\] $\mathtt{LEX}$ $\mathtt{v_{min} \leftarrow min\_pred(u)}$ $\mathtt{v_{max} \leftarrow max\_pred(u)}$ $\mathtt{IN.insert(u.label}^p,u)$ $\mathtt{OUT.insert(\epsilon,u)}$ ![Left: Inconsistency of type 1. The five tables show how arrays $\mathtt{IN}$, $\mathtt{LEX}$, and $\mathtt{OUT}$ evolve during insertions of nodes $v_1, \dots, v_5$ in topological order. Up to node $v_4$, the Directed Acyclic Graph (DAG) is Wheeler. When inserting node $v_3$, we successfully reserve the interval $[v_1,v_2]$ with label ’a’ (shown in red). From this point, no ’a’s can be inserted inside the reserved interval. When inserting node $v_5$ (with incoming label ’b’), the co-lexicographically smallest and largest predecessors of $v_5$ are $v_3$ and $v_4$, respectively. This means we have to reserve the interval $[v_3,v_4]$ with label ’b’ (shown in blue dashed line); however, this is not possible since there already is a ’b’ (highlighted in blue) in $\mathtt{OUT}[v_3,\dots,v_4]$. Right: Inconsistency of type 2. Up to node $v_5$, the DAG is Wheeler. Note that we successfully reserve two intervals: $[v_1,v_2]$ (with letter ’a’, red interval), and $[v_3,v_4]$ (with letter ’b’, blue interval). When inserting node $v_4$ we do not need to reserve any interval since the node has only one predecessor. The confict arises when inserting node $v_6$ (with incoming label ’b’). Since $v_5$ is a predecessor of $v_6$, we need to append ’b’ in $\mathtt{OUT[v_5]}$. However, this ’b’ (underlined in the picture) falls inside a reserved interval for ’b’ (in blue). []{data-label="fig:inconsistencies"}](DAG-both-crop) Data Structure Details {#sec:data structures} ---------------------- In this section we show how to implement operations [1]{}-[10]{} used by Algorithms \[alg:sort\] and \[alg:step\] using state-of-the-art data structures. At the core of $\mathtt{LEX}$ and $\mathtt{OUT}$ stands the dynamic sequence representation of Navarro and Nekrich [@navarro2014optimal]. This structure supports insertions, access, rank, and select in $O(\log n)$ worst-case time, $n$ being the sequence’s length. The space usage is bounded by $nH_0 + o(n\log\sigma) + O(\sigma\log n)$ bits, where $H_0$ is the zero-th order entropy of the sequence. Sequence $\mathtt{IN}$ will instead be represented using a dynamic partial sum data structure, e.g. a balanced binary tree or a Fenwick tree [@fenwick1994new], and a dynamic bitvector. All details follow. Sequence $\mathtt{LEX}$ is stored with Navarro and Nekrich’s dynamic sequence representation [@navarro2014optimal]. Operations [1]{}-[3]{} are directly supported on the representation. Operation [2]{} is simply a $select_{v}(1)$ (i.e. the position of the first $v$). Sequence $\mathtt{OUT}$ is stored using a dynamic sequence $\mathtt{out}$ and a bitvector, both represented with Navarro and Nekrich’s dynamic sequence. The idea is to store all the strings $\mathtt{OUT}[1], \dots, \mathtt{OUT}[\|\mathtt{OUT}\|]$ concatenated in a single sequence $\mathtt{out}$, and mark the beginning of those strings with a bit set in a dynamic bitvector $\mathtt{B_{out}[1..n]}$, were $n=\|\mathtt{out}\|$. Clearly, operations [4]{}-[7]{} on $\mathtt{OUT}$ can be simulated with a constant number of operations (insert, access, rank, select) on $\mathtt{out}$ and $\mathtt{B_{out}}$. Operations [8]{}-[9]{} require an additional dynamic sequence of parentheses $\mathtt{PAR}[1..n]$ on alphabet $\{ \mathtt{(}_c\ :\ c\in\Sigma\} \cup \{ \mathtt{)}_c\ :\ c\in\Sigma\} \cup \{ \square \}$. Every time a new character is inserted at position $i$ in $\mathtt{out}$, we also insert $\square$ at position $i$ in $\mathtt{PAR}$. When $\mathtt{OUT.reserve(u,v,c)}$ is called (i.e. operation [8]{}), let $i_v$ and $i_u$ be the positions in $\mathtt{out}$ corresponding to the two occurrences of character $c$ in $\mathtt{OUT[u]}$ and $\mathtt{OUT[u]}$, respectively (remember that the automaton is deterministic, so these positions are unique). These positions can easily be computed in $O(\log n)$ time using select* and rank operations on $\mathtt{out}$ and $\mathtt{B_{out}}$. Then, we replace $\mathtt{PAR}[i_u]$ and $\mathtt{PAR}[i_v]$ with characters $\mathtt{(_c}$ and $\mathtt{)_c}$, respectively (replacing a character requires a deletion followed by an insertion). Note that reserved intervals for a fixed character do not overlap, so this parentheses representation permits to unambiguously reconstruct the structure of the intervals. At this point, operation [9]{} is implemented as follows. Let $i_v$ be the position in $\mathtt{out}$ corresponding to the first character in $\mathtt{OUT}[v]$. This position can be computed in $O(\log n)$ time with two select operations on $\mathtt{LEX}$ and $\mathtt{B_{out}}$. Then, $\mathtt{OUT.is\_reserved(v,c)}$ returns true if and only if $\mathtt{PAR.rank_{(_c}(i_v)}>\mathtt{PAR.rank_{)_c}(i_v)}$, i.e. if we did not close all opening parentheses $\mathtt{{(_c}}$ before position $i_v$ (note that does not make any difference if $i_v$ is the first or last position in $\mathtt{OUT}[v]$, since when we call this operation $\mathtt{OUT}[v]$ does not contain characters equal to $c$). To conclude, $\mathtt{IN}$ is represented with a dynamic bitvector $\mathtt{B_{IN}[1..n]}$ and a partial sum $\mathtt{PS[1..\sigma+1]}$ supporting the following operations in $O(\log\sigma)$ time: - partial sum: $\mathtt{PS.ps(i)} = \sum_{j=1}^i\mathtt{PS[j]}$. - update: $\mathtt{PS[i]} \leftarrow \mathtt{PS[i]}+\delta$. Fenwick trees [@fenwick1994new] support the above operations within this time bound. Bitvector $\mathtt{B_{IN}[1..n]}$ contains the bit sequence $110^{t_{v_2}-1} 10^{t_{v3}-1} \dots 10^{t_{v_k}-1}$, where $t_{v_i}$ is the number of predecessors of $v_i$ in the current sequence $\mathtt{LEX} = v_1, \dots, v_k$ of sorted nodes (note that $v_1$ is always the source $s$). Assume, for simplicity, that $\Sigma = [1,\sigma+1]$, where $\#$ corresponds to $1$ (this is not restrictive, as we can map the alphabet to this range at the beginning of the computation). At the beginning, the partial sum is initialized as $\mathtt{IN[c]=0}$ for all $c$. Operation [10]{}, $\mathtt{IN.start(c)}$, with $c\neq \#$, is implemented as $\mathtt{PS.ps(c-1)+1}$. If $c=\#$, the operation returns $1$. Operation [4]{}, $\mathtt{IN.insert(c^p,u)}$, is implemented as $\mathtt{PS[c]} \leftarrow \mathtt{PS[c]}+p$ followed by $\mathtt{B_{IN}.insert(10^{p-1},i_u)}$ (i.e. $p$ calls to insert on the dynamic bitvector at position $i_u$), where $i_u$ is the position of the $j$-th bit set in $\mathtt{B_{IN}}$ (a select operation) and $j = \mathtt{LEX^{-1}[u]}$, or $i_u=n+1$ if $\mathtt{B_{IN}}$ has $j-1$ bits set (note that, when we call $\mathtt{IN.insert(c^p,u)}$, node $u$ has already been inserted in $\mathtt{LEX}$). Proof of Theorem \[thm:n log n\] {#proof thm n log n} -------------------------------- In Appendix \[sec:data structures\] we show that all operations on the dynamic sequences can be implemented in logarithmic time. Correctness follows from the fact that we always check that the Wheeler properties are maintained true. To prove completeness, note that at each step we place $u$ between two nodes $v_1$ and $v_2$ in array $\mathtt{LEX}$ only if the smallest $u$’s predecessor is larger than the largest $v_1$’s predecessor, and if the largest $u$’s predecessor is smaller than the smallest $v_2$’s predecessor. This is the only possible choice we can make in order to satisfy $w_{v_1} \prec w_{u} \prec w_{v_2}$ for all strings labeling paths $s \rightsquigarrow v_1$, $s \rightsquigarrow u$, and $s \rightsquigarrow v_2$ and to obtain, by Corollary \[lem:clusters\], the only possible correct ordering of the nodes. It follows that, if the new node $v$ does not falsify the Wheeler properties, then we are computing its co-lexicographic rank correctly. (of Lemma \[lem: check range consistency\]) \[proof lemma check range consistency\] First, we sort edges by label, with ties broken by origin, and further ties broken by destination. This can be achieved in time $O(\|E\| + \|V\|)$ by radix sorting the edges represented as triples $(a, u, v)$, where $a$ is the label, and $u$ and $v$ respectively are the ranks of the source and destination nodes in the given order $<$. Let $L$ denote the sorted list of edges. We claim that the given order $<$ satisfies the Wheeler properties (Definition \[def\_WG\]) if and only if for all pairs of consecutive edges $(a_i, u_i, v_i), (a_{i+1}, u_{i+1}, v_{i+1})$ in $L$, we have $(a_i = a_{i+1}) \rightarrow v_i \leq v_{i+1}$ and $(a_i \neq a_{i+1}) \rightarrow v_i < v_{i+1}$. Clearly this can be checked in time $O(\|E\|)$ with one scan over $L$. We now argue the correctness of this algorithm. Wheeler property (ii) is equivalent to the condition that when all edges labeled by some character $a \in \Sigma$ are sorted by source with ties broken by destination, the sequence of destinations is monotonically increasing, which is expressed by the condition $(a_i = a_{i+1}) \rightarrow v_i \leq v_{i+1}$. Wheeler property (i) is equivalent to the condition that for all pairs of characters $a,b \in \Sigma$ such that $b$ is a successor of $a$ in the order of $\Sigma$, denoting by $v_a$ the largest node with an incoming $a$-edge, and by $v_b$ the smallest node with an incoming $b$-edge, we have $v_a < v_b$. If Wheeler property (ii) holds, then destinations $v_a$ and $v_b$ are consecutive in $L$ because the list is sorted primarily by label and destinations are monotonically increasing for each label. Hence checking for $(a_i \neq a_{i+1}) \rightarrow v_i < v_{i+1}$ verifies Wheeler property (i) given that Wheeler property (ii) holds. (of Theorem \[thm:DFA linear\]) \[proof thm: DFA linear\] In $O(\|V\|+\|E\|)$ time we build a directed spanning tree $\mathcal T$ of $\mathcal A$ with root $s$ (e.g. its directed shortest-path tree with root $s$). Note that this is always possible since we assume that all states are reachable from $s$. By Corollary \[lem:clusters\], if $\mathcal A$ is a Wheeler graph then we can use the strings that label any two paths $s\rightsquigarrow u$ and $s\rightsquigarrow v$ to decide the order of any two nodes $u$ and $v$. We can therefore sort $V$ according to the paths spelled by $\mathcal T$; by Corollary \[lem:clusters\], if $\mathcal A$ is Wheeler then we obtain the correct (unique) ordering. To prefix-sort $\mathcal T$, we compute its XBW transform [^4] in $O(\|V\|)$ time [@ferragina2009compressing Thm 2]. The array containing the lexicographically-sorted nodes (i.e. the prefix array of $\mathcal T$) can easily be obtained from the XBW transform using, e.g. the partial rank counters defined in the proof of Lemma \[lem: check range consistency\] to navigate the tree (this is analogous to repeatedly applying function LF on the BWT in order to obtain the suffix array). At this point, we check that the resulting node order satisfies the Wheeler properties using Lemma \[lem: check range consistency\]. If this is this case, then the above-computed prefix array contains the prefix-sorted nodes of $\mathcal A$. Proofs of Section \[sec:minimization\] ======================================= (of Theorem \[thm: min DFA\])\[proof thm: min DFA\] Let $\mathcal A=(V,E,F,s,\Sigma)$. Consider the (possibly infinite) deterministic automaton $\mathcal T$ that is a tree and that is equivalent to $\mathcal A$ in the following sense: $\mathcal T$ is the (unique) tree obtained by “unraveling” $\mathcal A$, i.e. the tree containing all words in $\mathcal L(\mathcal A)$ such that each path labeled with such a word leads to an accepting state. Clearly, $\mathcal T$ is a (possibly infinite) deterministic automaton recognizing $\mathcal L(\mathcal A)$: a string $\alpha$ leads to a final state in $\mathcal A$ if and only if it does in $\mathcal T$. Let $L^u = \{u^1, u^2, \dots, u^{k_u}\}$ be the (possibly infinite) set of nodes of $\mathcal T$ reached by following, from its root, all the paths labeled $\alpha$ for each $\alpha$ labeling a path $s \rightsquigarrow u$ connecting $s$ with $u$ in $\mathcal A$. Note that each state $u$ of $\mathcal A$ can be identified by the set $L^u$ of states of $\mathcal T$; this allows us to extend $\equiv_w$ to the states of $\mathcal T$ as follows: $u^i \equiv_w u^j$ for all $u^i,u^j\in L^u$, $u\in V$, and $u^i \equiv_w v^j$ for $u^i\in L^u$, $v^j\in L^v$ if and only if $u \equiv_w v$. Consider now the process of minimizing $\mathcal T$ by collapsing states in equivalence classes in such a way that (i) the quotient automaton is finite, (ii) the accepting language of the quotient DFA is the same as that of $\mathcal T$ and (iii) the quotient DFA is Wheeler. By the existence of $\mathcal A$, there exists such a partition (not necessarily the coarsest): the one putting $u^i$ and $u^j$ in the same equivalence class if and only if $u^i, u^j \in L^u$, for some $u\in V$ (in this case, $\mathcal A$ itself is the resulting quotient automaton). Call $\equiv$ the relation among states of $\mathcal T$ yielding the smallest such WDFA $\mathcal A/_\equiv$. By definition, $\mathcal A/_\equiv$ is the smallest WDFA recognizing $\mathcal L(\mathcal A)$. Our claim is that $\equiv\ =\ \equiv_w$, i.e. that Algorithm \[alg:minimize\] returns this automaton. We observe that: 1. $u^i \approx u^j$ for any $u^i,u^j\in L^u$ and all $u\in V$. Otherwise, assume for a contradiction that there exists a string $\alpha$ leading to an accepting state from $u^i$ but not from $u^j$. By construction of $\mathcal T$, $u^i$ and $u^j$ are $\approx$-equivalent to $u$: this leads to a contradiction, since the state reached from $u$ with label $\alpha$ cannot be both accepting and not accepting. 2. Since $\mathcal A$ is a Wheeler DFA, Corollary \[lem:clusters\] applied to $\mathcal A$ tells us that, for any two nodes $u<v\in V$, all strings labeling paths from the root of $\mathcal T$ to nodes in $L^u$ are co-lexicographically smaller than those labeling paths from the root of $\mathcal T$ to nodes in $L^v$. We express this fact using the notation $L^u < L^v$. 3. Since $\mathcal A$ is Wheeler, then each $u\in V$ has only one distinct incoming label and $\lambda(u^j) = \lambda(u)$ for all $u^j\in L^u$. By the above properties, $u^i \equiv u^j$ for all $u^i,u^j\in L^u$, $u\in V$. To see this, note that, by property [1]{}, those states are all equivalent by relation $\approx$. Moreover, properties [2-3]{} combined with Corollary \[lem:clusters\] imply that, by grouping states in each $L^u$, we cannot break any Wheeler property. It follows that $\equiv$ must group those states, being the coarsest partition finer than $\approx$ with these two properties. Let us indicate with $L^u \equiv L^v$ the fact that $u^i \equiv v^j$ for all $u^i\in L^u,\ v^j\in L^v$. Suppose now, for a contradiction, that there exist $L^u < L^v < L^w$ with $L^u \equiv L^w \not\equiv L^v$. Then, by Corollary \[lem:clusters\], $L^u < L^v$ implies that, in the quotient automaton, states $[L^w]_\equiv = [L^u]_\equiv$ and $[L^v]_\equiv$ are reachable from the source by two paths $\alpha$ and $\beta$, respectively, with $\alpha \prec \beta$. Conversely, $L^v < L^w$ implies that states $[L^v]_\equiv$ and $[L^w]_\equiv$ are reachable from the source by two paths $\alpha'$ and $\beta'$, respectively, with $\alpha' \prec \beta'$. Then, by Corollary \[lem:clusters\] we cannot define a total order on $\mathcal A/_\equiv$’s states, i.e. $\mathcal A/_\equiv$ is not Wheeler. By all the above observations, we conclude that $\equiv$ must (i) group only equivalent states by $\approx$, (ii) group only states with the same incoming label, (iii) group all states inside each $L^u$, and (iv) group only states in adjacent sets $L^u$, $L^v$ in the co-lexicographic order. By its definition, the relation $\equiv_w$ induces the coarsest partition that satisfies (i)-(iv), therefore we conclude that $\equiv\ =\ \equiv_w$. Converting DFAs to minimum WDFAs {#app:DFA->WDFA} -------------------------------- We describe an online step of our algorithm. Assume we successfully built $\mathcal A_i$, with $i<t$, and we are about to process $v_{i+1}$ in order to build $\mathcal A_{i+1}$. Let $\{c_1, \dots, c_k\}$ be the labels of incoming $v_{i+1}$’s edges. We first replace (split) $v_{i+1}$ by $k$ equivalent states $v_{i+1}^{c_1} \approx \dots \approx v_{i+1}^{c_k}$: each $v_{i+1}^{c_k}$ (i) is accepting if and only if $v_{i+1}$ is accepting, (ii) keeps only the incoming edges of $v_{i+1}$ labeled $c_i$, and (iii) it duplicates all its outgoing edges: we replace each $(v_{i+1},u,c)$ with the edges $(v_{i+1}^{c_1},u,c), \dots, (v_{i+1}^{c_k},u,c)$. Note that all the newly-created edges must be present in the final automaton $\mathcal A_t$ since the states $v_{i+1}^{c_1}, \dots, v_{i+1}^{c_k}$ cannot be collapsed back by $\equiv_w$ (as they have different incoming labels); it follows that in this step we are not creating more edges than necessary. We now insert separately $v_{i+1}^{c_1}, \dots, v_{i+1}^{c_k}$ in $\mathtt{LEX}_i$ in any order as follows. The procedure is the same for all those vertices, therefore we may simply assume we are about to process a node $v$ with all incoming edges labeled with the same character $a$. Let $u_1 < \dots < u_k$ be the predecessors of $v$ in the graph; note that those nodes must belong to $\mathtt{LEX}_i$ (since we are processing states in topological order), therefore their order $<$ is well-defined. We now must detect and solve inconsistencies of type 1 and 2 as defined in the proof of Theorem \[thm:n log n\] (see also Figure \[fig:inconsistencies\]). We start with inconsistencies of type 1: there already are nodes $w_i \notin \{u_1, \dots, u_k\}$ with outgoing edges labeled $a$ inside the range $[u_1, u_k]$. This breaks the sequence $u_1 < \dots < u_k$ into $q$ sub-intervals $[u_{i_j}, u'_{i_j}]$, $j=1, \dots, q$, that do not contain nodes with outgoing label $a$ different than those in $\{u_1, \dots, u_k\}$. The range has therefore the following form, where we denote with $w_i$ and $w'_i$ all nodes not in $\{u_1, \dots, u_k\}$ with outgoing edges labeled $a$ and we highlight in bold the runs $[u_{i_j}, u'_{i_j}]$: $$w_1 < \mathbf{u_{i_1} \leq \dots \leq u'_{i_1}} < w_2 \leq \dots \leq w'_2 < \mathbf{u_{i_2} \leq \dots \leq u'_{i_2}} < \dots < \mathbf{u_{i_q} \leq \dots \leq u'_{i_q}} < w_{q+1} \;,$$ where $u_{i_1} = u_1$, $u'_{i_{q}} = u_k$, and $w_1<u_1,\ w_{q+1}>u_k$ are the rightmost and leftmost states with an outgoing edge labeled $a$, respectively (if they exist). The top part of Figure \[fig:make WDAG\] depicts this situation, where $k=4$ and $u_1, \dots, u_4$ are clustered in $q=3$ runs: $w_1 < \mathbf{u_1} < w_2 < w_3 < \mathbf{u_2 < u_3} < w_4 < \mathbf{u_4} < w_5$. We solve the inconsistencies of type 1 by splitting $v$ in (i.e. replacing it with) $q$ equivalent nodes: $v_1 \approx \dots \approx v_q$. Each $v_j$ is final if and only if $v$ is final, duplicates all $v$’s outgoing edges (as seen above), and keeps only incoming edges from $v$’s predecessors inside the corresponding run $[u_{i_j}, u'_{i_j}]$. This is depicted in the bottom part of Figure \[fig:make WDAG\]: $v$ has been split into the three equivalent nodes $v_1 \approx v_2 \approx v_3$. Inconsistencies of type 2 are solved similarly by splitting $a$-successors of $w_1, \dots, w_{q+1}$ that belong to $\mathtt{LEX}_i$ when necessary. Let $\mathtt{LEX}_i \cap \{succ_a(w_1), \dots, succ_a(w_{q+1})\} = \{z_1 < \dots < z_{q'}\}$ be the $a$-successors of $w_1, \dots, w_{q+1}$ in $\mathtt{LEX}_i$. Note that it might be the case that $q'<q+1$. Note also that some of the nodes $z_i$ might belong to $\{u_1, \dots, u_k\} \cup \{w_1, \dots, w_{q+1}\}$. We have an inconsistency of type 2 (among nodes in $\mathtt{LEX}_i$) whenever $succ_a(w_i) = succ_a(w_{i+1}) = z_e$, for some $1\leq e \leq q'$, and there exist some $u_j$ such that $w_i < u_j < w_{i+1}$. In this case, we split $z_e = succ_a(w_i) = succ_a(w_{i+1})$ in two equivalent nodes $z_e' \approx z_e''$ ordered as $z_e' < succ_a(u_j) < z_e''$. This cannot contradict the Wheeler properties (even if $z_i\in \{u_1, \dots, u_k\} \cup \{w_1, \dots, w_{q+1}\}$), since $succ_a(u_j)$ is one of the copies of $v$ (or $v$ itself if $v$ has not been splitted in the previous step) and has therefore no successors in the current automaton. The process of fixing inconsistencies of type 2 is shown in Figure \[fig:make WDAG\]: nodes $w_3$ and $w_4$ are separated by $u_2, u_3$ as $w_3 < u_2 < u_3 < w_4$. In this case, $succ_a(w_3) = succ_a(w_4) = z_3$, and we split $z_3$ in the two equivalent nodes $z'_3$ and $z''_3$. Note also that we only need to check those $w_i$ that immediately precede or follow a predecessor of $v$ (i.e. $w_1, w_2, w_2', \dots , w_{q+1}$): those nodes are at most $O(k)$, where $k$ is the number of $v$’s predecessors. As shown in Figure \[fig:make WDAG\] (bottom), after solving the inconsistencies of type 1 and 2 the nodes in $\mathtt{LEX}_{i+1}$ are again range-consistent: the $a$-successors of any (sorted) range of nodes form themselves a (sorted) range. Moreover, the splitting process defines unambiguously a total ordering of the new nodes among those already in $\mathtt{LEX}_i$, which can be therefore updated to $\mathtt{LEX}_{i+1}$ by inserting those nodes at the right place: to insert a node $v'$ in $\mathtt{LEX}_i$, let $u'$ be its $a$-predecessor: $succ_a(u') = v'$. Let moreover $u'' < u'$ be the rightmost node preceding $u'$ (in $\mathtt{LEX}_i$) having an outgoing edge labeled $a$, and let $v''$ be its $a$-successor: $succ_a(u'') = v''$. By range-consistency, node $v'$ has to be inserted immediately after $v''$ in $\mathtt{LEX}_i$. If such a node $u''$ does not exist (i.e. $u'$ is the leftmost node in $\mathtt{LEX}_i$ having an outgoing edge labeled $a$), then $v'$ has to be inserted in $\mathtt{LEX}_i$ so that it becomes the first node with incoming edges labeled $a$ (i.e. in the position immediately following the rightmost node $v''$ with incoming label $a'$, where $a'$ is the lexicographically-largest character such that $a'\prec a$, or at the first position in $\mathtt{LEX}_i$ if such a character $a'$ does not exist). This shows that invariant [2]{} is maintained: the Wheeler properties are kept true among nodes in $\mathtt{LEX}_{i+1}$. It is also clear that we do not insert $\approx$-equivalent adjacent states with the same incoming label (see Figure \[fig:make WDAG\]: by construction, the newly-inserted nodes $v_1, z'_3, v_2, z''_3, v_3$ are non-equivalent to their neighbors), i.e. invariant [3]{} is maintained. Finally, the accepted language does not change since the splitting process generates $\approx$-equivalent nodes: also invariant [1]{} stays true. Note that the minimization process on the original acyclic DFA $\mathcal A$ takes linear time. After that, we only insert edges/nodes in the minimum output WDFA: never delete. It follows that the number of performed operations is equal to the output’s size. The final automaton could be either smaller or exponentially-larger than $\mathcal A$. We note that all the discussed operations can be easily implemented in logarithmic time using the data structures discussed in Section \[sec:data structures\]: finding the $q$ runs of states $[u_{i_j}, u'_{i_j}]$, as well as finding the $O(k)$ states $w_i$, requires executing a constant number of rank operations on sequence $\mathtt{OUT}$ and start operations on $\mathtt{IN}$ for each predecessor of $v$. Nodes can be inserted at the right position in sequence $\mathtt{LEX}$ exactly as done in Algorithm \[alg:step\] (by also updating $\mathtt{IN}$ and $\mathtt{OUT}$). Finally, the graph can be dynamically updated (i.e. splitting nodes) and queried (i.e. navigation) by keeping it as a dynamic adjacency list: since we can spend logarithmic time per edge, we can store the graph as a self-balancing tree associating nodes to their predecessors and successors (also kept as self-balancing trees). This structure supports all updates and queries on the graph in logarithmic time. It follows that the overall procedure terminates in $O(n+m\log m)$ time, $n$ and $m$ being the input and output’s sizes, respectively. ![Inconsistency resolution. Nodes are ordered left-to-right by the total ordering $<$ (except $v$ in the top part of the figure). Top: we are trying to insert $v$ in $\mathtt{LEX}_i$, but this violates the Wheeler properties (edges’ destinations are not ordered as the sources, no matter where we insert $v$). Bottom: we solve the inconsistencies by splitting $v$ in three equivalent nodes $v_1 \approx v_2 \approx v_3$ and $z_3$ in two equivalent nodes $z_3' \approx z_3''$. Note that (i) the splitting procedure induces naturally an ordering of the nodes that satisfies the Wheeler properties, and (ii) after splitting, no two adjacent states with the same incoming label are equivalent by $\approx$. By Theorem \[thm:characterization minimum\], this is the minimum way of splitting nodes. For simplicity, in the figure nodes $z_1,\dots, z_4$ do not coincide with any node $w_1, \dots, w_5$ or $u_1, \dots, u_4$. This may not necessarily be the case. In our full proof in Appendix \[app:DFA->WDFA\] we show that our procedure is correct even when this happens.[]{data-label="fig:make WDAG"}](WDAG-crop) Worst case blowup from an acyclic DFA to the minimum equivalent WDFA {#sec:worst case blow up} ==================================================================== In this section we show that the running time of the conflict resolution algorithm in Section $\ref{sec:ADFA->WADFA}$ is exponential in the worst case, i.e. there exists a family of regular languages where the size of the smallest WDFA is exponential in the size of the smallest DFA. We now show that one such family the sequence of languages $L_1,L_2,\ldots$, where $L_m = \{ c \alpha e \; \| \; \alpha \in \{a,b\}^m \} \cup \{ d \alpha f \; \| \; \alpha \in \{a,b\}^m \}$. For an example, Figure \[fig:dfa\_wdfa\_worst\_case2\] shows a DFA and the smallest WDFA for the language $L_3$. In general, we can build a DFA for $L_m$ by generalizing the construction in the figure: the source node has outgoing edges labeled with $c$ and $d$, followed by simple linear size “universal gadgets” capable of generating all binary strings of length $m$, with one gadget followed by an $e$ and the other by an $f$. The two sink states are the only accepting states. The smallest WDFA for $L_m$ is an unraveling of the described DFA, such that all paths up to (but not including) the sinks end up in distinct nodes, i.e. the universal gadgets are replaced by full binary trees (see Figure \[fig:dfa\_wdfa\_worst\_case2\]). It is easy to see that the automaton is Wheeler as the only nodes that have multiple incoming paths are the sinks, and the sinks have unique labels. To prove that this is the minimal WDFA, we need to check the condition of Theorem \[thm:characterization minimum\], i.e. that all colexicographically consecutive pairs of nodes with the same incoming label are Myhill-Nerode inequivalent. As labels $c,d,e$ and $f$ occur only once, it is enough to focus on nodes that have label $a$ or $b$. Let $B_1, B_2, B_{2^{m+1}-1}$ be the colexicographically sorted sequence of all possible binary strings with lengths $1 \leq \|B_i\| \leq m$ from the alphabet $\{a,b\}$. Observe that the nodes with incoming label $a$ and $b$ correspond to path labels of the form $c B_i$ and $dB_i$ for all $1 \leq i \leq 2^{m+1}-1$. The colexicographically sorted order of these path labels is: $$c B_1 < d B_1 < c B_2 < d B_2 < \ldots < c B_{2^{m+1}-1} < d B_{2^{m+1}-1}$$ Here we can see that all consecutive pairs have a different first character, and therefore they lead to a different sink in the construction, and hence they are not Myhill-Nerode equivalent. Therefore the automaton is the minimum WDFA. The DFA has $n = 4m + 5$ states and the WDFA has $1 + 2^{m+2} = 1 + 2^{(n-5)/4 + 2}$ states, so we obtain the following result: The minimal WDFA equivalent to an acyclic DFA with $n$ states has $\Omega(2^{n/4})$ states in the worst case. =\[scale=0.4\] (T21) at (-4,0+2) [$e$]{}; (T18) at (-5,-1+2) [$a$]{}; (T17) at (-5,1+2) [$b$]{}; (T16) at (-6,-1+2) [$a$]{}; (T15) at (-6,1+2) [$b$]{}; (T14) at (-7,-1+2) [$a$]{}; (T13) at (-7,1+2) [$b$]{}; (T12) at (-8,-0+2) [$c$]{}; (T10) at (-4,-4+2) [$f$]{}; (T7) at (-5,-5+2) [$a$]{}; (T6) at (-5,-3+2) [$b$]{}; (T5) at (-6,-5+2) [$a$]{}; (T4) at (-6,-3+2) [$b$]{}; (T3) at (-7,-5+2) [$a$]{}; (T2) at (-7,-3+2) [$b$]{}; (T1) at (-8,-4+2) [$d$]{}; (T0) at (-9,-2+2) ; (T0) edge node (T1); (T1) edge node (T2); (T1) edge node (T3); (T2) edge node (T4); (T2) edge node (T5); (T3) edge node (T4); (T3) edge node (T5); (T4) edge node (T6); (T4) edge node (T7); (T5) edge node (T6); (T5) edge node (T7); (T6) edge node (T10); (T6) edge node (T10); (T7) edge node (T10); (T7) edge node (T10); (T0) edge node (T12); (T12) edge node (T13); (T12) edge node (T14); (T13) edge node (T15); (T13) edge node (T16); (T14) edge node (T15); (T14) edge node (T16); (T15) edge node (T17); (T15) edge node (T18); (T16) edge node (T17); (T16) edge node (T18); (T17) edge node (T21); (T17) edge node (T21); (T18) edge node (T21); (T18) edge node (T21); (S0) at (1-3,0) ; (S11) at (2.5-3,1/2) [$c$]{}; (S21) at (4-3,1/2) [$a$]{}; (S22) at (4-3,5/2) [$b$]{}; (S31) at (5.5-3,1/2) [$a$]{}; (S32) at (5.5-3,3/2) [$b$]{}; (S33) at (5.5-3,5/2) [$a$]{}; (S34) at (5.5-3,7/2) [$b$]{}; (S41) at (7.0-3,1/2) [$a$]{}; (S42) at (7.0-3,2/2) [$b$]{}; (S43) at (7.0-3,3/2) [$a$]{}; (S44) at (7.0-3,4/2) [$b$]{}; (S45) at (7.0-3,5/2) [$a$]{}; (S46) at (7.0-3,6/2) [$b$]{}; (S47) at (7.0-3,7/2) [$a$]{}; (S48) at (7.0-3,8/2) [$b$]{}; (Se) at (8.5-3,4.5/2) [$e$]{}; (D11) at (2.5-3,-1/2) [$d$]{}; (D21) at (4-3,-1/2) [$a$]{}; (D22) at (4-3,-5/2) [$b$]{}; (D31) at (5.5-3,-1/2) [$a$]{}; (D32) at (5.5-3,-3/2) [$b$]{}; (D33) at (5.5-3,-5/2) [$a$]{}; (D34) at (5.5-3,-7/2) [$b$]{}; (D41) at (7.0-3,-1/2) [$a$]{}; (D42) at (7.0-3,-2/2) [$b$]{}; (D43) at (7.0-3,-3/2) [$a$]{}; (D44) at (7.0-3,-4/2) [$b$]{}; (D45) at (7.0-3,-5/2) [$a$]{}; (D46) at (7.0-3,-6/2) [$b$]{}; (D47) at (7.0-3,-7/2) [$a$]{}; (D48) at (7.0-3,-8/2) [$b$]{}; (Df) at (8.5-3,-4.5/2) [$f$]{}; (S0) edge node (S11); (S11) edge node (S21); (S11) edge node (S22); (S21) edge node (S31); (S21) edge node (S32); (S22) edge node (S33); (S22) edge node (S34); (S31) edge node (S41); (S31) edge node (S42); (S32) edge node (S43); (S32) edge node (S44); (S33) edge node (S45); (S33) edge node (S46); (S34) edge node (S47); (S34) edge node (S48); (S41) edge node (Se); (S42) edge node (Se); (S43) edge node (Se); (S44) edge node (Se); (S45) edge node (Se); (S46) edge node (Se); (S47) edge node (Se); (S48) edge node (Se); (S0) edge node (D11); (D11) edge node (D21); (D11) edge node (D22); (D21) edge node (D31); (D21) edge node (D32); (D22) edge node (D33); (D22) edge node (D34); (D31) edge node (D41); (D31) edge node (D42); (D32) edge node (D43); (D32) edge node (D44); (D33) edge node (D45); (D33) edge node (D46); (D34) edge node (D47); (D34) edge node (D48); (D41) edge node (Df); (D42) edge node (Df); (D43) edge node (Df); (D44) edge node (Df); (D45) edge node (Df); (D46) edge node (Df); (D47) edge node (Df); (D48) edge node (Df); [^1]: Corresponding author. Supported by the project MIUR-SIR CMACBioSeq (“Combinatorial methods for analysis and compression of biological sequences”) grant n. RBSI146R5L. [^2]: We wish to thank Travis Gagie for introducing us to the problem and for stimulating discussions. [^3]: Usually, the lexicographic order is used to sort string suffixes. In this paper, we use the symmetric co-lexicographic order of the string’s prefixes, and extend the concept to labeled graphs. [^4]: note: this requires mapping the labels of $\mathcal T$ to alphabet $\Sigma' \subseteq [1,\|V\|]$ while preserving their lexicographic ordering. Since we assume that the original alphabet’s size does not exceed $\|E\|^{O(1)} = \|V\|^{O(1)}$, this step can be performed in linear time by radix-sorting the labels.
--- abstract: 'In the present paper, we introduced the extended bicomplex plane $\overline{\mathbb{T}}$, its geometric model: the bicomplex Riemann sphere, and the bicomplex chordal metric that enables us to talk about the convergence of the sequences of bicomplex meromorphic functions. Hence the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains emerges. Besides obtaining a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bicomplex meromorphic functions are also obtained.' author: - \| Kuldeep Singh Charak\ Department of Mathematics, University of Jammu\ Jammu-180 006, India\ E-mail: kscharak7@rediffmail.com - \| Dominic Rochon\ Département de mathématiques et d’informatique\ Université du Québec à Trois-Rivières\ C.P. 500 Trois-Rivières, Québec, Canada G9A 5H7\ E-mail: Dominic.Rochon@UQTR.CA - \| Narinder Sharma\ Department of Mathematics, University of Jammu\ Jammu-180 006, India\ E-mail: narinder25sharma@sify.com title: Normal Families of Bicomplex Meromorphic Functions --- [^1] [^2] Introduction ============ The concept of normality of a family of bicomplex holomorphic functions was introduced in [@Sharma], and we now intend to study the same property for a family of bicomplex meromorphic functions. A family $\boldsymbol F$ of meromorphic functions on a domain $D\subset\mathbb{C}$ is said to be normal in $D$ if every sequence in $\boldsymbol F$ contains a subsequence which converges uniformly on compact subsets of $D$; the limit function is either meromorphic in $D$ or identically equal to $\infty$. Of course, the convergence in this situation is with respect to the chordal metric on the Riemann sphere $\overline {\mathbb C} =\mathbb C \cup \{\infty\}$ (cf. [@Schiff]). Unfortunately, the one complex variable case doesn’t admit any simple generalization to extend facts to the bicomplex case. In order to discuss the convergence of sequences of bicomplex meromorphic functions on bicomplex plane domains, we introduce the extended bicomplex plane $\overline{\mathbb{T}}$, its geometric model viz., the bicomplex Riemann sphere, the bicomplex chordal metric on the bicomplex Riemann sphere, and the idea of convergence on $\overline{\mathbb{T}}$. In turn, these developments facilitate the introduction of the concept of the normality of a family of bicomplex meromorphic functions on bicomplex domains. This forms the content of Section 3. In Section 4 of the paper, after introducing the concept of normality of a family of bicomplex meromorphic functions, a normality criterion for such families, the bicomplex analog of the Montel theorem for meromorphic functions and the Fundamental Normality Tests for families of bicomplex holomorphic functions and bicomplex meromorphic functions are also obtained. Preliminaries ============= As in [@Shapiro] (see also [@Charak] and [@Sharma]), the algebra of bicomplex numbers $$\mathbb{T}:=\{z_1+z_2{\ensuremath{{\bf i_2}}}\ \|\ z_1, z_2 \in \mathbb{C}({\ensuremath{{\bf i_1}}}) \} \label{enstetra}$$ is the space isomorphic to $\mathbb{R}^{4}$ via the map $$z_1+z_2{\ensuremath{{\bf i_2}}}=x_0+x_1{\ensuremath{{\bf i_1}}}+x_2{\ensuremath{{\bf i_2}}}+x_3{\ensuremath{{\bf j}}}\rightarrow (x_0,x_1,x_2,x_3)\in\mathbb{R}^{4},$$ and the multiplication is defined using the following rules: $${\ensuremath{{\bf i_1}}}^2={\ensuremath{{\bf i_2}}}^2=-1,\qquad{\ensuremath{{\bf i_1}}}{\ensuremath{{\bf i_2}}}={\ensuremath{{\bf i_2}}}{\ensuremath{{\bf i_1}}}={\ensuremath{{\bf j}}}\quad\mbox{ so that }\quad{\ensuremath{{\bf j}}}^2=1.$$ Note that we define ${\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i}}}_k):=\{x+y{\ensuremath{{\bf i}}}_k\ \|\ {\ensuremath{{\bf i}}}_k^2= -1$ and $x,y\in {\ensuremath{\mathbb{R}}}\}$ for $k=1,2$. Hence, it is easy to see that the multiplication of two bicomplex numbers is commutative. In fact, the bicomplex numbers $$\mathbb{T}\cong {\rm Cl}_{\Bbb{C}}(1,0) \cong {\rm Cl}_{\Bbb{C}}(0,1)$$ are unique among the complex Clifford algebras (see [@Brackx; @Delanghe] and [@Ryan]) in that they are commutative but not division algebra. Also, since the map $z_1+z_2{\ensuremath{{\bf i_2}}}\rightarrow (z_1,z_2)$ gives a natural isomorphism between the $\mathbb{C}$-vector spaces $\mathbb{T}$ and $\mathbb{C}^{2}$, we have $\mathbb{T}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}$. That is, we can view the algebra $\mathbb{T}$ as the complexified $\mathbb{C}({\ensuremath{{\bf i_1}}})$ exactly the way $\mathbb{C}$ is complexified $\mathbb{R}$. In particular, in the equation (\[enstetra\]), if we put $z_1=x$ and $z_2=y{\ensuremath{{\bf i_1}}}$ with $x,y \in \mathbb{R}$, then we obtain the following subalgebra of hyperbolic numbers, also called duplex numbers (see, e.g. [@Shapiro], [@Sobczyk]): $$\mathbb{D}:=\{x+y{\ensuremath{{\bf j}}}\ \|\ {\ensuremath{{\bf j}}}^2=1,\ x,y\in \mathbb{R}\}\cong {\rm Cl}_{\Bbb{R}}(0,1).$$ The two projection maps $\mathcal{P}_1,\mathcal{P}_2:\mathbb{T}\longrightarrow\mathbb{C}(\bold{i_1})$ defined by $$\label{projections} \mathcal{P}_1(z_1+z_2\bold{i_2})=z_1-z_2\bold{i_1}\qquad\mbox{ and }\qquad\mathcal{P}_2(z_1+z_2\bold{i_2})=z_1+z_2\bold{i_1},$$ are used extensively in the sequel. The complex (square) norm $CN(w)$ of the bicomplex number $w$ is the complex number ${z_1}^2 +{z_2}^2$; writing $w^=z_1-z_2{\ensuremath{{\bf i_2}}}$, we see that $CN(w)=ww^$. Then a bicomplex number $w=z_1+z_2{\ensuremath{{\bf i_2}}}$ is invertible if and only if $CN(w)\neq 0$. Precisely, $$w^{-1}= \displaystyle \frac{w^}{CN(w)}.$$ The set of units in the algebra $\mathbb{T}$ forms a multiplicative group which we shall denote by $\mathbb{T}_$ (see [@Baird]). Unlike the algebra $\mathbb{C}$, the bicomplex algebra $\mathbb{T}$ has zero divisors given by $$\mathcal{NC}=\{w \in \mathbb{T}: CN(w)=0\}=\{z(1\pm{\ensuremath{{\bf j}}})\|\ z\in \mathbb{C}({\ensuremath{{\bf i_1}}})\},$$ which we may call the [null-cone]{}. Note that, using orthogonal idempotents $$\bold{e_1}=\frac{1+\bold{j}}{2},\qquad \bold{e_2}=\frac{1-\bold{j}}{2},\ \text{ in }\ \mathcal{NC},$$ each bicomplex number $w=z_1+z_2{\ensuremath{{\bf i_2}}}\in \mathbb{T}$ can be expressed uniquely as $$w=\mathcal{P}_1(w){\ensuremath{{\bf e_1}}}+\mathcal{P}_2(w){\ensuremath{{\bf e_2}}},$$ where $\mathcal{P}_1$ and $\mathcal{P}_2$ are projection maps defined in $(\ref{projections})$. This representation of $\mathbb{T}$ as $\mathbb{C}\oplus\mathbb{C}$ helps to do addition, multiplication and division term-by-term. With this representation we can directly express $\|w\|_j$ as $$\|w\|_j := \|\mathcal{P}_1(w)\|{\ensuremath{{\bf e_1}}}+\|\mathcal{P}_2(w)\|{\ensuremath{{\bf e_2}}}$$ and will be referred to as the $\bold{j}$*-modulus* of $w=z_1+z_2{\ensuremath{{\bf i_2}}}\in\mathbb{T}$ (see [@Shapiro]). Let $X_1$ and $X_2$ be subsets of $\mathbb{C}(\bold{i_1})$. Then the following set $$X_{1}\times_e X_{2}:=\{w=z_1+z_2\bold{i_2}\in\mathbb{T}~:~\mathcal{P}_1(w)\in X_1\ \text{ and }\ \mathcal{P}_2(w)\in X_2\}$$ is called a $\mathbb{T}$*-cartesian set* determined by $X_1$ and $X_2$, where $\mathcal{P}_1$ and $\mathcal{P}_2$ are projections as defined in . It is easy to see that if $X_1$ and $X_2$ are domains (open and connected) of $\mathbb{C}(\bold{i_1})$ then $X_1\times_e X_2$ is also a domain of $\mathbb{T}$. We define the “discus" with center $a=a_1+a_2{\ensuremath{{\bf i_2}}}$ of radius $r_1$ and $r_2$ of $\mathbb{T}$ as follows [@Price]: $$\begin{aligned} D(a;r_1,r_2)&=& B^{1}(a_1-a_2{\ensuremath{{\bf i_1}}},r_1)\times_{e}B^{1}(a_1+a_2{\ensuremath{{\bf i_1}}},r_2)\\ &=& \{w_1\bold{e_1}+w_2\bold{e_2}: \|w_1-(a_1-a_2{\ensuremath{{\bf i_1}}})\|<r_1,\|w_2-(a_1+a_2{\ensuremath{{\bf i_1}}})\|<r_2\},\end{aligned}$$ where $B^{n}(z,r)$ is an open ball with center $z\in\mathbb{C}^n(\bold{i_1})$ and radius $r>0$. In the particular case where $r=r_1=r_2$, $D(a;r,r)$ will be called the $\mathbb{T}$-disc with center $a$ and radius $r$. In particular, we define $$\overline{D}(a;r_1,r_2):=\overline{B^{1}(a_1-a_2{\ensuremath{{\bf i_1}}},r_1)}\times_{e}\overline{B^{1}(a_1+a_2{\ensuremath{{\bf i_1}}},r_2)}\subset \overline{D(a;r_1,r_2)}.$$ We remark that $D(0;r,r)$ is, in fact, the Lie Ball (see [@Avanissian]) of radius $r$ in $\mathbb{T}$. Further, the projections as defined in , help to understand bicomplex holomorphic functions in terms of the following Ringleb’s Decomposition Lemma [@Riley]. \[theo5\] Let $\Omega\subset\mathbb{T}$ be an open set. A function $f:\Omega\longrightarrow\mathbb{T}$ is $\mathbb{T}$-holomorphic on $\Omega$ if and only if the two natural functions $f_{e1}:\mathcal{P}_1(\Omega)\longrightarrow\mathbb{C}(\bold{i_1})$ and $f_{e2}:\mathcal{P}_2(\Omega)\longrightarrow\mathbb{C}(\bold{i_1})$ are holomorphic, and $$f(w)=f_{e1}(\mathcal{P}_1(w)){\ensuremath{{\bf e_1}}}+f_{e2}(\mathcal{P}_2(w)){\ensuremath{{\bf e_2}}},\ \forall\mbox{ }w=z_1+z_2\bold{i_2}\in \Omega,$$ The Ringleb’s Lemma for bicomplex meromorphic functions is as follows [@Charak]. \[mero1\] Let $\Omega\subset\mathbb{T}$ be an open set. A function $f:\Omega\longrightarrow\mathbb{T}$ is bicomplex meromorphic on $\Omega$ if and only if the two natural functions $f_{e1}:\mathcal{P}_1(\Omega)\longrightarrow\mathbb{C}(\bold{i_1})$ and $f_{e2}:\mathcal{P}_2(\Omega)\longrightarrow\mathbb{C}(\bold{i_1})$ are meromorphic, and $$f(w)=f_{e1}(\mathcal{P}_1(w)){\ensuremath{{\bf e_1}}}+f_{e2}(\mathcal{P}_2(w)){\ensuremath{{\bf e_2}}},\ \forall\mbox{ }w=z_1+z_2\bold{i_2}\in \Omega.$$ Let $f:\Omega\longrightarrow\mathbb{T}$ be a bicomplex meromorphic function on the open set $\Omega\subset\mathbb{T}$, and let $f_{e1}:\mathcal{P}_1(\Omega)\longrightarrow\mathbb{C}({\ensuremath{{\bf i_1}}})$ and $f_{e2}:\mathcal{P}_2(\Omega)\longrightarrow\mathbb{C}({\ensuremath{{\bf i_1}}})$ be the natural maps. Then we say that $w=\mathcal{P}_1(w){\ensuremath{{\bf e_1}}}+\mathcal{P}_2(w){\ensuremath{{\bf e_2}}}\in \Omega$ is a (strong) *pole* for the bicomplex meromorphic function $$f(w)=f_{e1}\mathcal{P}_1(w){\ensuremath{{\bf e_1}}}+f_{e1}\mathcal{P}_2(w){\ensuremath{{\bf e_2}}}$$ if $\mathcal{P}_1(w)$ (and) or $\mathcal{P}_2(w)$ is a pole for $f_{e1}$ or $f_{e2}$, respectively. \[pole\] Poles of bicomplex meromorphic functions are not isolated singularities. It is also easy to obtain the following characterization of poles. Let $f:X\longrightarrow\mathbb{T}$ be a bicomplex meromorphic function on the open set $\Omega\subset\mathbb{T}$. If $w_0\in \Omega$ then $w_0$ is a pole of $f$ if and only if $$\mathop {\lim }\limits_{w \to w_0 } \left\| {f(w)} \right\| = \infty .$$ A classical example of bicomplex meromorphic function is the bicomplex Riemann zeta function introduced by Rochon in [@Rochon1]. The Extended Bicomplex Plane $\overline{\mathbb{T}}$ ==================================================== Since the range of bicomplex meromorphic function lies beyond the bicomplex plane, we need the [extended bicomplex plane]{} to study the bicomplex meromorphic functions. Further, it would help to study the limit points of unbounded sets in bicomplex plane. We obtain this extended bicomplex plane by using extended $\mathbb {\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})$-plane. For, we may consider the set $$\begin{aligned} \overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})} \times_{e} \overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})} &=& \left({\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}}) \cup \{ \infty \} \right)\times_{e}\left( {\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})\cup \{ \infty \}\right)\\ &=& \left({\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}}) \times_{e} {\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})\right) \cup \left({\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}}) \times_{e} \{ \infty \} \right)\cup \left(\{ \infty \} \times_{e} {\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})\right) \cup \{ \infty \}\\ &=& {\mathbb{T}} \cup {I_{\infty}},\end{aligned}$$ writing $I_{\infty}$ for the set $\left({{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}}) \times_{e} \{ \infty \}}\right) \cup \left({\{ \infty \} \times_{e} {\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}\right) \cup {\{ \infty \}}$. Clearly, any unbounded sequence in $\mathbb{T}$ will have a limit point in $I_{\infty}$. The set $\overline{\mathbb{T}}=\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})} \times_{e} \overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ is called the extended bicomplex plane. That is, $$\overline{\mathbb{T}} = \mathbb{T} \cup I_{\infty},\quad\mbox{ with }\quad I_{\infty} = \{w \in \overline{\mathbb{T}} : \left\\|w\right\\| = \infty \}.$$ It is of significant importance to observe that formation of the extended bicomplex plane $\overline{\mathbb{T}}$ requires us to add an infinity set viz. $I_{\infty}$, which we may call the bicomplex infinity set. We need some definitions in order to give a characterization of this set. An element $w \in I_{\infty}$ is said to be a $\mathcal{P}_1$-infinity ($\mathcal{P}_2$-infinity) element if $\mathcal{P}_{1}(w)= \infty \ (\mathcal{P}_{2}(w)= \infty)$ and $\mathcal{P}_{2}(w)\neq \infty \ (\mathcal{P}_{1}(w)\neq \infty).$ The set of all $\mathcal{P}_1$-infinity elements is called the $I_1$*-infinity set. It is denoted by $I_{1,\infty}$. Therefore, $$I_{1,\infty} = \{w \in \overline{\mathbb{T}} : \mathcal{P}_{1}(w)= \infty ,\\\ \mathcal{P}_{2}(w)\neq \infty \}.$$ Similarly we can define the $I_2$-infinity set* as: $$I_{2,\infty} = \{w \in \overline{\mathbb{T}} : \mathcal{P}_{1}(w) \neq \infty ,\\\ \mathcal{P}_{2}(w)= \infty \}.$$ An element $w \in \overline{\mathbb{T}}$ is said to be a $\mathcal{P}_1$-zero ($\mathcal{P}_2$-zero) element if $\mathcal{P}_{1}(w)= 0 \ (\mathcal{P}_{2}(w)= 0)$ and $\mathcal{P}_{2}(w)\neq 0 \ (\mathcal{P}_{1}(w)\neq 0).$ The set of all $\mathcal{P}_1$-zero elements is called the $I_1$*-zero set; it is denoted by $I_{1,0}$. That is, $I_{1,0} = \{w \in \overline{\mathbb{T}}: \mathcal{P}_{1}(w)= 0 ,\mathcal{P}_{2}(w)\neq 0 \}$. Similarly, we may define the $I_2$-zero set* as the set $\{w \in \overline{\mathbb{T}}: \mathcal{P}_{1}(w) \neq 0 ,\mathcal{P}_{2}(w)= 0 \}$. We now construct the following two new sets: $$I^{-}_{\infty} = I_{1,\infty} \cup I_{2,\infty},\qquad I^{-}_{0} = I_{1,0} \cup I_{2,0},$$ so that $I_{\infty} = I^{-}_{\infty} \cup \{ \infty \}$ and $\mathcal{NC} = I^{-}_{0} \cup \{0\}$. With these definitions, each element in the null-cone has an inverse in $I_{\infty}$ and vice versa. One can easily check that the elements of the set $I^{-}_{\infty}$ do not satisfy all the properties as satisfied by the ${\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})$-infinity but the element $\infty = \infty {\ensuremath{{\bf e_1}}}+ \infty {\ensuremath{{\bf e_2}}}$ does. We may call the set $I^{-}_{\infty}$, the weak bicomplex infinity set and the element $ \infty = \infty {\ensuremath{{\bf e_1}}}+ \infty {\ensuremath{{\bf e_2}}}$, the strong infinity. This nature of the set $ I_{\infty}$ generates the idea of weak and strong poles for bicomplex meromorphic functions (see [@Charak]). Now, in order to work in the extended bicomplex plane, it is desirable to have a geometric model wherein the elements of $\overline{\mathbb{T}}$ have a concrete representative so as to treat the points of $I_{\infty}$ as good as any other point of $\overline{\mathbb{T}}$. To obtain such a model, one can use the usual stereographic projections of $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ as two components in the idempotent decomposition to get a one-to-one and onto correspondence between the points of $S \times S$, where $S$ is the unit sphere in $\mathbb{R}^{3}$, and $\overline{\mathbb{T}}$. Hence, we can visualize the extended bicomplex plane directly in $\mathbb{R}^{6}=\mathbb{R}^{3} \times \mathbb {R}^{3}$. With this representation, we call $\overline{\mathbb{T}} $ the bicomplex Riemann sphere. Observe that what is done above is basically a compactification of $\mathbb{C}^2$, using bicomplex setting. That is, suitable points at infinity are added to $\mathbb{T}$ to get the extended bicomplex plane $\overline{\mathbb{T}}$. In higher dimensions such compactifications are well known under the name, conformal compactifications. In fact, such compactifications are obtained as homogeneous spaces of Lie groups (see [@Baird] and [@Baston]). The Chordal Metric on $\overline{\mathbb{T}}$ --------------------------------------------- To initiate a study of normal families of bicomplex meromorphic functions, we first have to extend the chordal distance to the extended bicomplex plane in such a way that facilitates introduction of notions like convergence of sequences and continuity of bicomplex meromorphic functions. The chordal metric on $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ can be used to define a distance on $\overline{\mathbb T}.$ If $\chi : \overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})} \times \overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}\longrightarrow \mathbb{R}$ be the chordal metric on $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$. Then the mapping $ \mathbb \chi_e : \overline{\mathbb{T}}\times \overline{\mathbb{T}}\longrightarrow \mathbb R \ \ \text{defined as:}$ $$\mathbb \chi_e\left({z,w}\right) =\sqrt{\frac{\chi^{2}(\mathcal{P}_{1}(z),\mathcal{P}_{1}(w)) + \chi^{2}(\mathcal{P}_{2}(z),\mathcal{P}_{2}(w))}{2}}$$ is a metric on $\overline{\mathbb{T}}.$ It is easy to verify that $\forall z,w\in\overline{\mathbb{T}}$ we have: $$\mathbb \chi_e\left({z,w}\right)\geq 0;$$ $$\mathbb \chi_e\left({z,w}\right) = 0 \ \ \text{iff } z = w;$$ $$\mathbb \chi_e\left({z,w}\right)= \mathbb \chi_e\left({w,z}\right).$$ Now, we show that $\mathbb \chi$ also satisfies the triangle inequality. Let $ z, \ w \ ,\ v \in \overline{\mathbb{T}} .$ We have to show that $$\mathbb \chi_e\left({z,w}\right)\leq \mathbb \chi_e\left({z,v}\right)+ \mathbb \chi_e\left({v,w}\right).$$\ For this, $\mathbb \chi_e\left({z,w}\right) = \sqrt{\displaystyle\frac{\chi^{2}\left({\mathcal{P}_{1}(z),\mathcal{P}_{1}(w)}\right) + \chi^{2}\left({\mathcal{P}_{2}(z),\mathcal{P}_{2}(w)}\right)}{2}}$\ $\leq\sqrt{\frac { \left\{{\chi\left({\mathcal{P}_{1}(z),\mathcal{P}_{1}(v)}\right) + \chi\left({\mathcal{P}_{1}(v),\mathcal{P}_{1}(w)}\right)}\right\} ^{2} + \left\{{\chi\left({\mathcal{P}_{2}(z),\mathcal{P}_{2}(v)}\right) + \chi\left({\mathcal{P}_{2}(v),\mathcal{P}_{2}(w)}\right) }\right\}^{2} }{2}}.$ Now, using Minkowski’s inequality in the above inequality, we obtain that $$\leq \sqrt{\frac {\chi^{2}\left({\mathcal{P}_{1}(z),\mathcal{P}_{1}(v)}\right) + \chi^{2}\left({\mathcal{P}_{2}(z),\mathcal{P}_{2}(v)}\right)}{2}}$$ $$+ \sqrt{\frac {\chi^{2}\left({\mathcal{P}_{1}(v),\mathcal{P}_{1}(w)}\right) + \chi^{2}\left({\mathcal{P}_{2}(v),\mathcal{P}_{2}(w)}\right)}{2}}$$ $$= \mathbb \chi_e\left({z,v}\right)+ \mathbb \chi_e\left({v,w}\right).$$ Hence, $\mathbb \chi_e$ is a metric on $\overline{\mathbb{T}}$. We call this metric $\mathbb \chi_e$ on $\overline{\mathbb{T}}$ the bicomplex chordal metric. The virtue of the bicomplex chordal metric is that it allows $w\in I_{\infty}$ to be treated like any other point. Hence, we are able now to analyse the behavior of the bicomplex meromorphic functions in the extended bicomplex plane, especially on the set $I_{\infty}$. As for the ${\ensuremath{{\bf j}}}-modulus$, let us define $${\chi_{{\ensuremath{{\bf j}}}}}(z, w):=\chi(\mathcal{P}_1(z),\mathcal{P}_1(w)){\ensuremath{{\bf e_1}}}+\chi(\mathcal{P}_2(z), \mathcal{P}_2(w)){\ensuremath{{\bf e_2}}}$$ in the extended hyperbolic numbers. Then $${\mathrm{Re}}({\chi_{{\ensuremath{{\bf j}}}}}^{2}(z, w))={\chi_{e}}^2(z, w)$$ and thus we have $$\chi_{e}(z, w)=\sqrt{{\mathrm{Re}}({\chi_{{\ensuremath{{\bf j}}}}}^{2}(z, w))}$$ where $$\mathbb \chi_{{\ensuremath{{\bf j}}}}(z,w)=\frac{\left\|z-w\right\|_{{\ensuremath{{\bf j}}}}}{\sqrt{1+\left\|z\right\|_{{\ensuremath{{\bf j}}}}}\sqrt{1+\left\|w\right\|_{{\ensuremath{{\bf j}}}}}}\mbox{ if } z,w\in\mathbb{T}.$$ Some of the important properties of the bicomplex chordal metric are discussed in the following results. If $ z = z_{1}{\ensuremath{{\bf e_1}}}+ z_{2}{\ensuremath{{\bf e_2}}}\ \text{and} \ w = w_{1}{\ensuremath{{\bf e_1}}}+ w_{2}{\ensuremath{{\bf e_2}}}$ are any two elements in the extended bicomplex plane and $\mathbb \chi_e$ is the bicomplex chordal metric on $\overline{\mathbb{T}}$. Then, $ 1.\ \mathbb \chi_e(z,w) \leq 1 $; $ 2.\ \mathbb \chi_e(0,\infty) = 1 $; $ 3.\ \mathbb \chi_e(z,w)= \frac{1}{\sqrt{2}}{\chi(z_{1},\infty)}\ \text{if } \mathcal{P}_{2}(z)=\mathcal{P}_{2}(w)= 0 \ \text{and } \mathcal{P}_{1}(w)= \infty$; $ 4.\ \mathbb \chi_e(z,w)= \frac{1}{\sqrt{2}}\chi(z_{1},w_{1}) \ \text{if } \mathcal{P}_{2}(z)=\mathcal{P}_{2}(w)= \infty$; $ 5.\ \mathbb \chi_e(z,\infty)= \frac{1}{\sqrt{2}}\chi(z_{2},\infty)\ \text{if } \mathcal{P}_{1}(z)=\infty$; $ 6.\ \mathbb \chi_e(z,w) = \mathbb \chi_e(z^{-1},w^{-1})$; $ 7.\ \mathbb \chi_e(z,w)=\chi(z,w)$ if $z,w\in\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$; $ 8.\ \mathbb \chi_e(z,w) \leq \left\\|z-w\right\\|$ if $z,w\in\mathbb{T}$; $9.\ \mathbb \chi_e(z,w)$ is a continuous function on $\mathbb{T}$. The following implication $$\left\\|z\right\\|\leq\left\\|w\right\\|\Longrightarrow\mathbb\chi(0,z)\leq\mathbb\chi(0,w),\ z,w\in\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$$ need not be true in case of the bicomplex chordal metric $\mathbb \chi_e$ on $\overline{\mathbb{T}}$. To support our argument we give the following examples. Let $$z = (1+2\bold{i_{1}}){\ensuremath{{\bf e_1}}}+(2+3\bold{i_{1}}){\ensuremath{{\bf e_2}}}\ \text{and} \ \ w = (1+\bold{i_{1}}){\ensuremath{{\bf e_1}}}+(3+3\bold{i_{1}}){\ensuremath{{\bf e_2}}}.$$ Then, $\left\\|z\right\\| \leq \left\\|w\right\\|$, but $\mathbb\chi_e(0,z) = \sqrt{0.88}$ and $\mathbb\chi_e(0,w) = \sqrt{0.80}$ implies that $\chi_e(0,z) > \mathbb\chi_e(0,w)$. Let $$z = (4+\bold{i_{1}}){\ensuremath{{\bf e_1}}}+(2+3\bold{i_{1}}){\ensuremath{{\bf e_2}}}\ \text{and } \\ w = (1+2\bold{i_{1}}){\ensuremath{{\bf e_1}}}+(3+4\bold{i_{1}}){\ensuremath{{\bf e_2}}}.$$ Then, $\left\\|z\right\\| = \left\\|w\right\\|$, but $\mathbb\chi_e(0,z) = \sqrt{0.93}$ and $\mathbb\chi_e(0,w) = \sqrt{0.89}$ implies that $\chi_e(0,z) > \mathbb\chi_e(0,w)$. However, we can prove the following result. Let $z,w\in\mathbb{T}$. If $\left\\|z\right\\| \leq \left\\|w\right\\|$ then $$\chi_e(0,z) \leq \chi_e(0,\sqrt{2}\left\\|w\right\\|).$$ By definition, $$\mathbb \chi_e(0,z) = \sqrt { \frac{\chi^{2}(0,\mathcal{P}_{1}(z)) + \chi^{2}(0,\mathcal{P}_{2}(z))}{2}}$$ $$= \sqrt{ \frac{1}{2}\left\{ \frac{\left\|\mathcal{P}_{1}(z)\right\|^2}{1+\left\|\mathcal{P}_{1}(z)\right\|^2} + \frac{\left\|\mathcal{P}_{2}(z)\right\|^2}{1+\left\|\mathcal{P}_{2}(z)\right\|^2}\right\}}.$$ Since, $$\left\|\mathcal{P}_{i}(z)\right\|\leq \sqrt{2}\left\\|z\right\\|\leq \sqrt{2}\left\\|w\right\\|\mbox{ for }i=1,2$$ then $$\chi(0,\mathcal{P}_{i}(z))=\chi(0,\left\|\mathcal{P}_{i}(z)\right\|) \leq \chi(0,\sqrt{2}\left\\|w\right\\|)\mbox{ for }i=1,2.$$ Hence, $$\chi_e(0,z) \leq \chi_e(0,\sqrt{2}\left\\|w\right\\|).$$ Convergence in $\overline{\mathbb{T}}$ -------------------------------------- A sequence of functions $\{f_n\}$ converges bispherically uniformly to a function $f$ on a set $E\subset\mathbb{T}$ if, for any $\epsilon>0$, there is a number $n_0$ such that $n\geq n_0$ implies $$\chi_e(f_n(w),f(w))<\epsilon,$$ for all $w\in E$. Note that if $\{f_n\}$ converges uniformly to $f$ on $E\subset\mathbb{T}$, then it also converges spherically uniformly to $f$ on $E$. The converse holds if the limit function is bounded. $ \mathbb \chi_e(z,w) \geq\displaystyle\frac{\left\\|z-w\right\\|}{\sqrt{1+2\left\\|z\right\\|^2}\sqrt{1+2\left\\|w\right\\|^2}}$, if $z,w\in\mathbb{T}.$ \[lemmaineq\] We shall establish the validity of the inequality in this lemma by obtaining an equivalent inequality that holds trivially. For $z,w\in\mathbb{T},$ put $\mathcal{P}_1(z)=a, \ \mathcal{P}_2(z)=b, \ \mathcal{P}_1(w)=c, \text{ and } \mathcal{P}_2(w)=d.$ Then\ $\mathbb \chi_e(z,w) \geq\displaystyle\frac{\left\\|z-w\right\\|}{\sqrt{1+2\left\\|z\right\\|^2}\sqrt{1+2\left\\|w\right\\|^2}}$\ $\Leftrightarrow \mathbb \chi_e^2(z,w) \geq\displaystyle\frac{{\left\\|z-w\right\\|}^2}{(1+2\left\\|z\right\\|^2)(1+2\left\\|w\right\\|^2)}$\ $\Leftrightarrow \chi^2(a,c)+ \chi^2(b,d) \geq\displaystyle\frac{\left\|a- c\right\|^2 + \left\|b- d\right\|^2}{(1 + \left\|a\right\|^2 + \left\|b\right\|^2 )(1 + \left\|c\right\|^2 + \left\|d\right\|^2)}$\ $\Leftrightarrow\displaystyle\frac{\left\|a- c\right\|^2}{(1 + \left\|a\right\|^2)(1 + \left\|c\right\|^2)} + \frac{\left\|b- d\right\|^2}{(1 + \left\|b\right\|^2)(1 + \left\|d\right\|^2)}$\ $\geq\displaystyle\frac{\left\|a- c\right\|^2 }{(1 + \left\|a\right\|^2 + \left\|b\right\|^2 )(1 + \left\|c\right\|^2 + \left\|d\right\|^2)} + \frac{\left\|b-d\right\|^2}{(1 + \left\|a\right\|^2 + \left\|b\right\|^2 )(1 + \left\|c\right\|^2 + \left\|d\right\|^2)}$\ $\Leftrightarrow \left\|a- c\right\|^2 \left[\displaystyle\frac{1}{(1 + \left\|a\right\|^2)(1 + \left\|c\right\|^2)} - \frac{1}{(1 + \left\|a\right\|^2 + \left\|b\right\|^2 )(1 + \left\|c\right\|^2 + \left\|d\right\|^2)}\right]$\ $\geq \left\|b- d\right\|^2 \left[\displaystyle\frac{1}{(1 + \left\|a\right\|^2 + \left\|b\right\|^2 )(1 + \left\|c\right\|^2 + \left\|d\right\|^2)} - \frac{1}{(1 + \left\|b\right\|^2)(1 + \left\|d\right\|^2)}\right]$\ $\Leftrightarrow\displaystyle\frac{\left\|a- c\right\|^2 \left[ \left\|d\right\|^2 + \left\|b\right\|^2 + \left\|a\right\|^2 \left\|d\right\|^2 + \left\|b\right\|^2 \left\|c\right\|^2 + \left\|b\right\|^2 \left\|d\right\|^2\right]}{(1 + \left\|a\right\|^2)(1 + \left\|c\right\|^2)}$\ $\geq\displaystyle\frac{\left\|b- d\right\|^2 \left[-\left\{ \left\|c\right\|^2 + \left\|a\right\|^2 + \left\|a\right\|^2 \left\|c\right\|^2 + \left\|a\right\|^2 \left\|d\right\|^2 + \left\|b\right\|^2 \left\|c\right\|^2\right\}\right]}{(1 + \left\|b\right\|^{2})(1 + \left\|d\right\|^2)}. $\ The left hand side of the last inequality is a positive real number where the right hand side is a negative real number and hence holds trivially. If the sequence $\{f_n\}$ converges bispherically uniformly to a bounded function $f$ on $E\subset \mathbb{T}$, then $\{f_n\}$ converges uniformly to $f$ on $E$. Since $\{f_n\}$ converges bispherically uniformly to a bounded function $f$ on $E\subset \mathbb{T}$, for every $\epsilon >0$ there is $n_0$ such that for all $n\geq n_0$, we have $$\chi_e(f_n(w),f(w))<\epsilon.$$ Now from this inequality and by the definition of the bicomplex chordal metric it follows that $\mathcal{P}_i(f_n(w))$ converges uniformly to $\mathcal{P}_i(f(w))$ on $\mathcal{P}_i(E)$, $i=1,2.$ Further, since $f$ is bounded on $E$, $\mathcal{P}_i(f(w))$ is bounded on $\mathcal{P}_i(E)$, $i=1,2,$ and hence $\mathcal{P}_i(f_n(w))$ is bounded on $\mathcal{P}_i(E)$, $i=1,2,$ for all but finitely many $n$. This implies that there is a positive constant $L$ such that $$\left\\|f_n(w)\right\\|<L \ \forall n\geq n_0,$$ on $\mathcal{P}_1(E)\times_e \mathcal{P}_2(E) \supseteq E.$ Now by Lemma \[lemmaineq\], we have $$\left\\|f_n(w)-f(w)\right\\|\leq \sqrt{1+2\left\\|f_n(w)\right\\|^2}\sqrt{1+2\left\\|f(w)\right\\|^2}\chi_e(f_n(w),f(w))$$ for all $n\geq n_0$ and for all $w\in E.$ But $f$ is bounded on $E$ and $\{f_n\}$ is bounded on $E$ for all $n\geq n_0$, so it follows from the last inequality that $\{f_n\}$ converges uniformly to $f$ on $E.$ The notion of continuity with respect to the bicomplex chordal metric is given in the following definition. A function $f$ is bispherically continuous at a point $w_0\in \mathbb{T}$ if, given $\epsilon>0$, there exists $\delta>0$ such that $$\chi_e(f(w),f(w_0))<\epsilon,$$ whenever $\left\\|w-w_0\right\\|<\delta$. In the case of bicomplex meromorphic functions we have the following result. If $f(w)$ is a bicomplex meromorphic function in a domain $E\subset \mathbb{T}$, then $f$ is bispherically continuous in $E$. Since $f(w)$ is a bicomplex meromorphic function on $E$, then there exist meromorphic functions (see Thm. \[mero1\]) $f_{e1}:E_1\longrightarrow\mathbb{C}(\bold{i_1})$ and $f_{e2}:E_2\longrightarrow\mathbb{C}(\bold{i_1})$ with $E_1=\mathcal{P}_1(E)$ and $E_2=\mathcal{P}_2(E)$ such that $$f(z_1+z_2\bold{i_2})=f_{e1}(z_1-z_2\bold{i_1})\bold{e_1}+f_{e2}(z_1+z_2\bold{i_1})\bold{e_2} \mbox{ }\forall \mbox{ }z_1+z_2\bold{i_2}\in E.$$ If $f$ is $\mathbb{T}$-holomorphic at $w_0\in E$, then $f_{ei}$ is holomorphic on $E_i$ for $i=1,2$. Hence, it is bispherically continuous on $E$ since $$\chi_e(f(w),f(w_0))\leq\left\\|f(w)-f(w_0)\right\\|. \label{meroza}$$ If $w_0$ is a strong pole, then $\frac{1}{f_{e1}}$ and $\frac{1}{f_{e2}}$ is continuous at $\mathcal{P}_1(w_0)$ and $\mathcal{P}_2(w_0)$ respectively. Moreover, noting that $$\mathbb \chi_e(f(w),f(w_0)) = \mathbb \chi_e(\frac{1}{f(w)},\frac{1}{f(w_0)})$$ $$= \sqrt{\displaystyle\frac{\chi^{2}\left(\displaystyle\frac{1}{f_{e1}(\mathcal{P}_1(w))},\displaystyle\frac{1}{f_{e1}(\mathcal{P}_1(w_0))}\right) + \chi^{2}\left(\displaystyle\frac{1}{f_{e2}(\mathcal{P}_2(w))},\displaystyle\frac{1}{f_{e2}(\mathcal{P}_2(w_0))}\right)}{2}},$$ the result follows as in the preceding case. If $w_0$ is a weak pole, then $\frac{1}{f_{e1}}$ or $\frac{1}{f_{e2}}$ is continuous at $\mathcal{P}_1(w_0)$ or $\mathcal{P}_2(w_0)$ respectively. Suppose, without loss of generality, that $\frac{1}{f_{e1}}$ is continuous at $\mathcal{P}_1(w_0)$ with $f_{e2}$ continuous at $\mathcal{P}_2(w_0)$. Then, $\mathbb \chi_e(f(w),f(w_0))=$ $$\sqrt{\displaystyle\frac{\chi^{2}\left(\displaystyle\frac{1}{f_{e1}(\mathcal{P}_1(w))},\displaystyle\frac{1}{f_{e1}(\mathcal{P}_1(w_0))}\right) + \chi^{2}(f_{e2}(\mathcal{P}_2(w),f_{e2}(\mathcal{P}_2(w_0)))}{2}},$$ and the result follows using the Equation in the complex plane (in $\bold{i_1}$). A family $\boldsymbol F$ of bicomplex functions defined on a domain $\Omega\subset\mathbb{T}$ is said to be bispherically equicontinuous at a point $w_{0}\in\Omega$ if for each $\epsilon > 0, \ \exists\delta = \delta(\epsilon,w_{0})$ such that $$\mathbb \chi_e \left(f(w),f(w_{0})\right)< \epsilon \ \ \text{whenever}\ \left\\|w-w_{0}\right\\|< \delta \, \ \forall f \in \boldsymbol F.$$ Moreover, $\boldsymbol F$ is bispherically equicontinuous on a subset $E\subset\Omega$ if it is bispherically equicontinuous at each point of $E$. Since $$\mathbb \chi_e\left(f(w),f(w_{0})\right) \leq \left\\|f(w)-f(w_{0})\right\\|,$$ we see that equicontinuity with respect of the euclidean metric implies bispherical equicontinuity. Normal Families of Bicomplex Meromorphic Functions ================================================== Basic results ------------- A family $\boldsymbol F$ of bicomplex meromorphic functions in a domain $\Omega\subset\mathbb{T}$ is normal in $\Omega$ if every sequence $\{f_{n}\}\subset\boldsymbol F$ contains a subsequence which converges bispherically uniformly on compact subsets of $\Omega$. That the limit function is either bicomplex meromorphic in $\Omega$ or in the set $I^{-}_{\infty}$ or identically $\infty$ is a consequence of Corollary \[meroinft\]. That the limit can actually be identically $\infty$ is given by the following example. Let $f_n(w)=\frac{n}{w}$, $n=1,2,3,\ldots,$ on the Lie Ball $D(0;r,r)$. Then each $f_n$ is bicomplex meromorphic and $\{f_n\}$ converges bispherically uniformly to $\infty$ in $D(0;r,r)$. A family $\boldsymbol F$ of bicomplex meromorphic functions is normal in a domain $\Omega$ with respect to the bicomplex chordal metric if and only if the family of meromorphic functions $F_{ei}=\mathcal{P}_i(\boldsymbol F)$ is normal in $\mathcal{P}_i(\Omega)$ for $i=1,2$ with respect to the chordal metric. \[promero\] Suppose that $\boldsymbol F$ is normal in $\Omega$ with respect to the bicomplex chordal metric. Let $\{(f_n)_1\}$ be a sequence in $\boldsymbol F_{e1}=\mathcal{P}_{1}(\boldsymbol F)$. We want to prove, without loss of generality, that the family of meromorphic functions $\{(f_n)_1\}$ contains a subsequence which converges spherically locally uniformly on $\mathcal{P}_1(\Omega)$. By definition, we can find a sequence $\{f_n\}$ in $\boldsymbol F$ such that $\{\mathcal{P}_1(f_n)\}=\{(f_n)_1\}$. Moreover, for any $z_0\in\mathcal{P}_1(\Omega)$, we can find a $w_0\in\Omega$ such that $\mathcal{P}_1(w_0)=z_0$. Now, consider a closed $\mathbb{T}$-disk $\overline{D}(w_0;r,r)$ in $\Omega$. By hypotheses, the sequence $\{f_{n}\}$ contains a subsequence $\{f_{n_k}\}$ which converges bispherically uniformly on $\overline{D}(w_0;r,r)$. Hence, $\mathcal{P}_1(f_{n_k})=(f_{n_{k}})_1$ converges spherically uniformly on $\overline{B^1(z_0,r)}\subset\mathcal{P}_1(\Omega)$. Conversely, suppose that $\boldsymbol F_{ei}=\mathcal{P}_{i}(\boldsymbol F)$ is normal on $\mathcal{P}_{i}(\Omega)=\Omega_i$ for $i=1,2.$ We want to show that $\boldsymbol F$ is normal in $\Omega$ with respect to the bicomplex chordal metric. Let $\{f_n\}$ be any sequence in $\boldsymbol F$ and $K$ be any compact subset of $\Omega$. Then $\{\mathcal{P}_1(f_n)\}=\{(f_n)_1\}$ is a sequence in $\boldsymbol F_{e1}=\mathcal{P}_{1}(\boldsymbol F).$ Since $\boldsymbol F_{e1}=\mathcal{P}_{1}(\boldsymbol F)$ is normal in $\mathcal{P}_1(\Omega)$ then $\{(f_n)_1\}$ has a subsequence $\{(f_{n_k})_1 \}$ which converges spherically uniformly on $\mathcal{P}_1(K)$ to a $\overline{\mathbb{C}({\ensuremath{{\bf i_1}}})}$-function. Now, consider $\{f_{n_k}\}$ in $\boldsymbol F$. Then $\{\mathcal{P}_2(f_{n_k})\}=\{(f_{n_k})_2\}$ is a sequence in $\boldsymbol F_{e2}=\mathcal{P}_{2}(\boldsymbol F)$. Since $\boldsymbol F_{e2}=\mathcal{P}_{2}(\boldsymbol F)$ is normal in $\mathcal{P}_1(\Omega)$ then $\{(f_{n_k})_2\}$ has a subsequence $\{(f_{n_{k_l}})_2 \}$ which converges spherically uniformly on $\mathcal{P}_2(K)$ to a $\overline{\mathbb{C}({\ensuremath{{\bf i_1}}})}$-function. This implies that $\{(f_{n_{k_l}})_1{\ensuremath{{\bf e_1}}}+(f_{n_{k_l}})_2{\ensuremath{{\bf e_2}}}\}$ is a subsequence of $\{f_n\}$ which converges bispherically uniformly on $\mathcal{P}_1(K)\times_{e}\mathcal{P}_2(K)\supseteq K$ to a $\overline{\mathbb{T}}$-function showing that $\boldsymbol F$ is normal in $\Omega$ with respect to the bicomplex chordal metric. Since the limit function of a locally convergent sequence of meromorphic functions is either meromorphic or identically equal to $\infty$, we have automatically the following result as a direct consequence of Theorems \[mero1\] and \[promero\]. Let $\{f_{n}\}$ be a sequence of bicomplex meromorphic functions on $\Omega$ which converges bispherically uniformly on compact subsets to $f$. Then $f$ is either a bicomplex meromorphic function on $\Omega$ or in the set $I^{-}_{\infty}$ or identically $\infty$. \[meroinft\] Moreover, from the fact that a family of analytic functions is normal with respect to the usual metric if and only if the family is normal with respect to the chordal metric (see [@Schiff], Cor. 3.1.7) and from the characterization of the notion of normality for a family of bicomplex holomorphic functions (see [@Sharma], Thm. 8), we obtain the following result as a consequence of Theorems \[theo5\] and \[promero\]. A family $\boldsymbol F$ of $\mathbb{T}$-holomorphic functions is normal in a domain $\Omega$ with respect to the euclidian metric if and only if $\boldsymbol F$ is normal in $\Omega$ with respect to the bicomplex chordal metric. Bicomplex Montel Theorem ------------------------ In this subsection, we will give a proof of a bicomplex version of the Montel theorem for a family of bicomplex meromorphic functions. We start with the following results. If $\{f_{n}\}$ is the sequence of bispherically continuous functions which converges bispherically uniformly to a function $f$ on a compact subset $E\subset\mathbb{T}$. Then $f$ is uniformly bispherically continuous on $E$ and the functions $\{f_{n}\}$ are bispherically equicontinuous on $E$. \[biequi\] The proof is same, with necessary changes, as that of one complex variable analogue (see [@Schiff], Prop. 1.6.2). The bicomplex Riemann sphere is a compact metric space. \[biequi2\] We will prove that $\overline{\mathbb{T}}$ is sequentially compact. Let $\{w_n\}$ be a sequence in $\overline{\mathbb{T}}$. We have that $\{\mathcal{P}_i(w_n)\}$ is a sequence in $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ for $i=1,2$. Since the Riemann sphere is the one-point compactification of the complex plane, $\{\mathcal{P}_1(w_n)\}$ has a spherically convergent subsequence $\{\mathcal{P}_1(w_{n_k})\}$ in $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ and $\{\mathcal{P}_2(w_{n_k})\}$ has also a spherically convergent subsequence $\{\mathcal{P}_2(w_{{n_k}_l})\}$ in $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ such that $\{\mathcal{P}_i(w_{{n_k}_l})\}$ converges spherically in $\overline{{\ensuremath{\mathbb{C}}}({\ensuremath{{\bf i_1}}})}$ for $i=1,2$. Hence, $\{w_{{n_k}_l}\}$ converges bispherically in $\overline{\mathbb{T}}$. As for one complex variable, in discussing the normality of a family of bicomplex meromorphic functions, the concept of local boundedness is not entirely relevant. However, bispherical equicontinuity can be substituted in the following counterpart of Montel’s theorem. A family $\boldsymbol F$ of bicomplex meromorphic functions in a bicomplex domain $\Omega\subset\mathbb{T}$ is normal if and only if $\boldsymbol F$ is bispherically equicontinuous in $\Omega$. Suppose $\boldsymbol F$ is normal but not bispherically equicontinuous in $\Omega$. Then there is a point $w_{0} \in \Omega$, some $\epsilon > 0 $, a sequence $\{w_{n}\}\longrightarrow w_{0}$ and a sequence $\{f_{n}\} \subset \boldsymbol F$ such that $$\mathbb \chi_e\left(f_n(w_{0}),f_n(w_{n})\right) > \epsilon, \ \ n = 1,2,3\ldots. \label{eps}$$ Since $\boldsymbol F$ is normal, so $\{f_{n}\}$ has a subsequence $\{f_{n_{k}}\}$ converging bisherically uniformly on compact subsets of $\Omega$ and in particular on a compact subset containing $\{w_{n}\}$. By the Lemma \[biequi\], this implies that $\{f_{n_{k}}\}$ is bispherically equicontinuous at $w_{0}$. This is a contradiction with the Equation . Therefore $\boldsymbol F$ is bispherically equicontinuous. Conversely, let $\boldsymbol F$ be a bispherically equicontinuous family of bicomplex meromorphic functions defined on $\Omega$. To show that $\boldsymbol F$ is normal in $\Omega$ we need to extract a locally bispherically uniformly convergent subsequence from every sequence in $\boldsymbol F.$ Let $\{f_{n}\}$ be any sequence in $\boldsymbol F$ and let $E$ be a countable dense subset of $\Omega$, for example we can take $E\bigcap \Omega$ where $E=\{w_{n}=w_{1,n}\bold{e_1}+w_{2,n}\bold{e_2}: w_{j,n}=x_{j,n}+i_{1}y_{j,n}\text{ where } x_{j,n}, y_{j,n} \in \mathbb Q, j=1,2 \}.$ Take any sequence $\{f_{n}\}\subset \boldsymbol F$ and consider the sequence of bicomplex numbers $\{f_{n}(w_1)\}$. Since the bicomplex Riemann sphere is a compact metric space (see Lemma \[biequi2\]), $\{f_{n}(w_{1})\}$ has a subsequence $\{f_{n,1}\}$ converging bispherically at $w_1$. Next, consider the sequence $\{f_{n,1}(w_{2})\}$, we can also find a subsequence $\{f_{n,2}\}$ of $\{f_{n,1}\}$ such that $\{f_{n,2}(w_{2})\}$ converges bispherically at $w_2$. Since $\{f_{n,2}\}$ is a subsequence of $\{f_{n,1}\},$ $\{f_{n,2}(w_{1})\}$ also converges bispherically at $w_1$. Therefore, $\{f_{n,2}\}$ converges bispherically at $w_1$ and $w_2$. Continuing this process, for each $k\geq 1$ we obtain a subsequence $\{f_{n,k}\}$ that converges bispherically at $w_1, w_2, \ldots, w_k$ and $\{f_{n,k}\} \subset \{f_{n,k-1}\}$. Now by Cantor’s diagonal process we define a sequence $\{g_n\}$ as $$g_{n}(w)=f_{n,n}(w), \ \ \ n\in \mathbb N.$$ Hence, $\{g_n(w_k)\}$ is a subsequence of the bispherically convergent sequence $\{f_{n,k}(w_k) \}_{n\geq k}$ and hence converges for each $w_k \in E$. Now, by hypothesis, $\{g_n \}$ is bispherically equicontinuous on every compact subset of $\Omega$. So for every $\epsilon>0$ and every compact subset $K$ of $\Omega$ there is a $\delta>0$ such that $$\chi_e (g_n(w),g_n(w^{\prime}))< \frac{\epsilon}{3},\ \\\ \forall n\in\mathbb N \text{ and } \forall \ w, w^{\prime} \in K \text{ with } \left\\|w-w^{\prime} \right\\|< \delta. \label{eq02}$$ Since $K$ is compact, we can cover it by a finite subcover, say $$K\subset\bigcup^p_{j=1}\{B^2(\varsigma_{j},\frac {\delta}{2}): \varsigma_{j} \in E\}.$$ Since $\varsigma_{j}\in E$, $\{g_{n}(\varsigma _{j})\}$ converges for each $j:1\leq j \leq p$ which further implies that $\{g_{n}(\varsigma _{j})\}$ is a Cauchy sequence. That is, there is a positive integer $n_0$ such that $$\chi_e(g_{n}(\varsigma _{j}),g_{m}(\varsigma _{j})) <\frac{\epsilon}{3},\mbox{ } \forall \ m, n \geq n_{0}, \ (1\leq j \leq p). \label{eq03}$$ Finally, for any $w\in K$, $w\in B^2(\varsigma_{j},\frac {\delta}{2})$ for some $1\leq j_0 \leq p$. Thus, from Equations and , we have $$\chi_e(g_{n}(w), g_{m}(w)) \leq \chi_e(g_{n}(w), g_{n}(\varsigma _{j_0}))+\chi_e (g_{n}(\varsigma _{j_0}), g_{m}(\varsigma _{j_0}))+\chi_e(g_{n}(\varsigma _{j_0}), g_{m}(w))$$ $$< \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3}, \\\ \forall m,n\geq n_0.$$ Therefore, by construction, $\ \{g_{n}\}$ is locally bispherically uniformly Cauchy and hence converges locally bispherically uniformly on $\Omega$. Fundamental Normality Test -------------------------- Finally, we prove the bicomplex version of the Fundamental Normality Test for meromorphic functions. First, we prove it for bicomplex holomorphic functions. Let $\boldsymbol F$ be a family of bicomplex holomorphic functions in a domain $\Omega\subset \mathbb {T}$. Suppose there are $\alpha, \ \beta \in \mathbb{T}$ such that\ (a) $\alpha - \beta$ is invertible, and\ (b) $S\cap \mathcal{R}(f) =\varnothing, \ \forall f\in \boldsymbol F,$ where $S=\{w\in \mathbb{T}:w-\alpha \in \mathcal{NC}\}\cup \{w\in \mathbb{T}:w-\beta \in \mathcal{NC}\}$ and $\mathcal{R}(f)$ denotes the range of $f.$\ Then $\boldsymbol F$ is a normal family in $\Omega.$ \[FNTH\] Conditions $(a)$ and $(b)$ of the hypothesis imply that for each $f\in \boldsymbol F$ the projection $P_i(f)$ does not assume $P_i(\alpha)$ and $P_i(\beta)$, where $P_i(\alpha)\neq P_i(\beta)$, for $i=1,2$. Then by the fundamental normality test for holomorphic functions (see [@Schiff]), it follows that $P_i(\boldsymbol F)$ is normal in $P_i(\Omega)$ for $i=1,2.$ Now by Theorem 11 of [@Sharma] we conclude that $\boldsymbol F$ is normal in $\Omega.$ Following [@Rochon2], one can easily obtain a bicomplex version of the Picard’s Little Theorem for meromorphic functions. Let $f$ be a bicomplex meromorphic function in $\mathbb {T}$. Suppose there exist $\alpha, \ \beta, \ \gamma \in \mathbb {T}$ such that\ (a) $\alpha - \beta, \ \beta - \gamma, \ \gamma -\alpha$ are invertible, and\ (b) $S\cap \mathcal{R}(f) =\varnothing, \ \forall f\in \boldsymbol F,$ where $S=\{w\in \mathbb{T}:w-\alpha \in \mathcal{NC}\}\cup \{w\in \mathbb{T}:w-\beta \in \mathcal{NC}\}\cup \{w\in \mathbb{T}:w-\gamma \in \mathcal{NC}\}$ and $\mathcal{R}(f)$ denotes the range of $f.$\ Then $f$ is a constant function. Let $\boldsymbol F$ be a family of bicomplex meromorphic functions defined in a domain $\Omega \subset \mathbb{T}.$ Suppose there exist $\alpha, \ \beta, \ \gamma \in \mathbb {T}$ such that\ (a) $\alpha - \beta, \ \beta - \gamma, \ \gamma -\alpha$ are invertible, and\ (b) $S\cap \mathcal{R}(f) =\varnothing, \ \forall f\in \boldsymbol F,$ where $S=\{w\in \mathbb{T}:w-\alpha \in \mathcal{NC}\}\cup \{w\in \mathbb{T}:w-\beta \in \mathcal{NC}\}\cup \{w\in \mathbb{T}:w-\gamma \in \mathcal{NC}\}$ and $\mathcal{R}(f)$ denotes the range of $f.$\ Then $\boldsymbol F$ is normal in $\Omega.$ Following the method of proof of Theorem \[FNTH\] and applying Theorem \[promero\] and the fundamental normality test for meromorphic functions ([@Schiff], Page 74) we can easily conclude that the family $\boldsymbol F$ is normal in $\Omega$. Acknowledgements {#acknowledgements .unnumbered} ---------------- DR is grateful to the Natural Sciences and Engineering Research Council of Canada for financial support. [HD]{} V. Avanissian, Cellule d’harmonicité et prolongement analytique complexe, Travaux en cours, Paris, 1985. P. Baird and J. C. Wood, Harmonic Morphisms and Bicomplex Manifolds, J. Geometry and Physics 61 (2011), no. 1, 46–61. R. Baston, M. Eastwood, Penrose Transform: Its Interaction with representation theory, Clarendon Press, Oxford, 1989. F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Pitman, London, 1982. K. S. Charak and D. Rochon, On Factorization of Bicomplex Meromorphic Functions, Trends in Mathematics, Birkh[ä]{}user Verlag Basel/Switzerland, 2008, 55–68. K. S. Charak, D. Rochon and N. Sharma, Normal Families of Bicomplex Holomorphic Functions, Fractals 17 (2009), no. 2, 267–268. C. T. Chuang and C. C. Yang, Fix-points and Factorization of meromorphic functions, World Scientific, 1990. R. Delanghe, F. Sommen and V. Soucek, Clifford analysis and spinor valued functions, Kluwer Acad. Publ., Dordrecht, 1992. G. B. Price, An introduction to multicomplex spaces and functions, Marcel Dekker Inc., New York, 1991. J. D. Riley, Contributions to the theory of functions of a bicomplex variable, Tôhoku Math. J. 2 (1953), no. 5, 132–165. D. Rochon, A Bicomplex Riemann Zeta Function, Tokyo Journal of Mathematics 27 (2004), no. 2, 357–369. D. Rochon, Dynamique bicomplexe et théorème de Bloch pour fonctions hyperholomorphes, Doctoral Thesis, Université de Montréal, 2001. D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, Anal. Univ. Oradea , fasc. math. 11 (2004), 71–110. W. Rudin, Real and Complex Analysis 3rd ed., New York, McGraw-Hill, 1976. J. Ryan, Complexified Clifford Analysis, Complex Variables 1 (1982), 119–149. B. V. Shabat, Introduction to Complex Analysis part II: Functions of Several Variables, American Mathematical Society, 1992. J. L. Schiff, Normal Families, Springer-Verlag, New York, 1993. G. Sobczyk, The hyperbolic number plane, Coll. Maths. Jour. 26 (1995), no. 4, 268–280. L. Zalcman, Normal Families: New Perspectives, Bulletin (New Series) Amer. Math. Soc. 35 (1998), no. 3, 215–230. [^1]: 2010 Mathematics Subject Classification: 30G, 30G35, 30D30, 32A, 32A30. [^2]: Key words and phrases: Bicomplex numbers, Complex Clifford algebras, Normal families, Bicomplex holomorphic functions, Bicomplex meromorphic functions, Bicomplex Riemann sphere.
--- abstract: 'Calabi-Yau Fermat varieties are obtained from moduli spaces of Lagrangian connect sums of graded Lagrangian vanishing cycles on stability conditions on Fukaya-Seidel categories. These graded Lagrangian vanishing cycles are stable representations of quivers on their [mirror stability conditions]{}.' author: - 'So Okada[^1]' title: Mirror stability conditions and SYZ conjecture for Fermat polynomials --- Introduction ============ For a projective variety $X$, Strominger-Yau-Zaslow conjectured that we can obtain the space $X$ as a moduli space of special Lagrangians with $U(1)$ connections when we have a mirror pair of $X$ and argued locally and physically. The author recommends [@Aur] for a recent illustration on the subject and [@HKKPTVVZ] for a general reference. In this paper, we study the conjecture categorically in the framework of homological mirror symmetry of Kontsevich [@Kon95] and stability conditions of Bridgeland [@Bri07]. The latter notion, which was inspired by Douglas’ so-called $\Pi$-stabilities in superstring theory [@Dou02; @Dou01], categorizes King’s $\theta$-stabilities [@Kin] ([@BriTol Section 5.3],[@Asp Section 7.3.3],[@Ber]). In particular, for stability conditions of quivers without relations, when deformed on the stability manifold if necessarily, stable representations solve self-dual equations twisted by characters of general linear groups [@Kin Section 6]. On derived categories defined in terms of graded Lagrangian vanishing cycles over morsifications of Fermat polynomials $X_{n}:=x_{1}^{n}+\ldots +x_{n}^{n}:\bC^{n}\to \bC$ [@Sei01; @AurKatOrl08], we put pairs of stability conditions which are named [mirrors]{} in Definition \[def:mirror\] and defined on quivers with commuting relations with the following properties (see Theorem \[thm:syz\]). For one in each pair, with framing in Definitions \[def:rest\] and \[def:rest\_rep\] in terms of a finite group of tensor products of Auslander-Reiten transformations, we have a moduli space of stable Lagrangian connect sums of graded Lagrangian vanishing cycles; this moduli space via the Serre-de Rham functor in Definition \[def:twisted\] gives the Calabi-Yau Fermat variety in $\bP^{-1}$ defined by the zero locus of $X_{n}$. For the other one in the pair, graded Lagrangian vanishing cycles are stable. Stable objects above are stable representations of quivers even when we forget relations. In particular, graded Lagrangian vanishing cycles trivially satisfy self-dual equations. In this paper, connections are over transverse intersections of graded Lagrangian vanishing cycles and connect trivial vector bundles on graded Lagrangian vanishing cycles; so, they are maps between vector spaces and consistency with grading make them representations of quivers with relations. Mirror stability conditions {#sec:mir} =========================== Let us recall the notion of stability conditions. For a triangulated category $\cT$, each stability condition is determined by a bounded $t$-structure and a stability function $Z$, which gives central charges of elements of the Grothendieck group of $\cT$, on the heart of the $t$-structure. We go on with explicit examples for our later use. Let $A_{n-1}$ denote the A-type Dynkin quiver of vertices $0,\ldots, n-1$ and arrows $0\to \ldots \to n-1$ and let $A_{n-1}^{\otimes n}$ denote the $n$-fold tensor product of $A_{n-1}$ (see [@Les] for the general definition of tensor products of quivers). Each vertex $a$ of $A_{n-1}^{\otimes n}$ can be labelled by $n$-tuples $a^{1},\ldots, a^{n}$ for $0\leq a^{i} \leq n-1$. We have an arrow $a\to b$ when $b^{i}-a^{i}=0$ for all $i$ but at most one $j$ such that $b^{j}-a^{j}=1$; we let $\lambda(a,b)=j$ if we have such $j$ or $\lambda(a,b)=0$ if $a=b$. We have commuting relations on arrows. For our convenience in this paper, for each vertex $a$ of $A_{n-1}^{\otimes n}$, we call the number $\lambda(a):=\sum a^{i}$ the [index]{} of $a$. For example, the following figure shows commuting arrows on rectangles which connect vertices of indices $0,1,2$ for $A_{4}^{\otimes 5}$. \[fig:partial quintic quiver\] For the triangulated category ${{\mathrm D}^{b}}(\mo A_{n-1}^{\otimes n})$ and the heart $\mo A_{n-1}^{\otimes n}$ of the bounded $t$-structure, we have a stability function which maps simple representations into the upper-half plane of the complex plane. We mean by $Z_{n}$ a stability function on the heart $\mo A_{n-1}^{\otimes n}$ with the following conditions: slopes of simple representations, which are one-dimensional representations over vertices, strictly decreases as their indices increases and central charges of simple representations of the same indices are the same. Choices of such stability functions gives the open submanifold of the stability manifold of ${{\mathrm D}^{b}}(\mo A_{n-1}^{\otimes n})$. The following figure shows one of such stability functions on $\mo A_{4}^{\otimes 5}$. Arrows indicate increases of coordinates of the complex plane. Central charges of simple representations put dots. For example, central charges of simple representations of vertices with the index $1$ put the second left-most dot. \[fig:central\_charge\_p\] For the stability function above, we have the stability condition of ${{\mathrm D}^{b}}(\mo A_{4}^{\otimes 5})$ with the stability function $-\bar{Z_{5}}$ on the same heart $\mo A_{4}^{\otimes 5}$. \[fig:central\_charge\_q\] For our convenience in this paper, we call the stability condition of ${{\mathrm D}^{b}}(\mo A_{n-1}^{\otimes n})$ determined by the stability function $-\bar{Z_{n}}$ on the heart $\mo A_{n-1}^{\otimes n}$ is [mirror]{} to the stability condition determined by the stability function $Z_{n}$ on the same heart $\mo A_{n-1}^{\otimes n}$. Between these mirrors, we have wall-crossing paths on the stability manifold of ${{\mathrm D}^{b}}(\mo A_{n-1}^{\otimes n})$. We put the formal definition as follows. \[def:mirror\] Let $\cA$ be the heart of a bounded $t$-structure of a triangulated category $\cT$ such that $\cA$ is isomorphic to the category of representations of a quiver with or without relations. Let $Z$ be a stability function on the heart such that central charges of simple representations are in the upper-half plane of the complex plane. We call the stability function $-\bar{Z}$ on $\cA$ [mirror]{} to the stability function $Z$, and the stability condition with the stability function $Z$ on $\cA$ [mirror]{} to the other. For a nontrivial directed acyclic graph such as of $A_{n-1}^{\otimes n}$, we have a stability condition with stable objects being simple representations, and we have the mirror stability condition with a representation of a nontrivial support being a stable object. We can state notions in Definition \[def:mirror\] in a more general way; however, without a finiteness property on the heart such as [@Bri07 Proposition 2.3], even if we assume the existence of a stability condition, the existence of the mirror stability condition is obscure. Examples of mirror stability conditions in terms of spherical functors or wall-crossings can be found in [@BayMac; @Bri09; @Bri05; @IshUedUeh; @KajSaiTak; @KonSoi; @Mac; @Ohk; @Oka06a; @Oka06b]. Recap of homological mirror symmetry ==================================== In this paper, derived equivalence $\cong$ means that on both sides of the equivalence, we have compact generating objects with $A_{\infty}$-isomorphic $A_{\infty}$ enhancements; this is an example of so-called derived Morita equivalence. In [@Oka09], we have seen that $X_{n}$ is self-dual up to the equivariance with respect to the group $H_{n}$ which consists of $n$-tuples of $n$-th roots of the unity modulo diagonals and which acts on coordinates. For example, we have ${{\mathrm D}^{b}}_{H_{n}}(\Coh X_{n})\cong \FS(X_{n})$ for the $H_{n}$-equivariant bounded derived category of coherent sheaves of the Calabi-Yau Fermat variety and the Fukaya-Seidel category of $X_{n}$ (see [@FutUed] for a different account). We have a morsification of $X_{n}$ and an $A_{\infty}$ algebra of graded Lagrangian vanishing cycles in terms of Lagrangian intersection theories such that the $A_{\infty}$ algebra is $A_{\infty}$-isomorphic to the Yoneda ($\Ext$) algebra of simple representations of $A_{n-1}^{\otimes n}$. For simple representations $S_{a}$ and $S_{b}$ of vertices $a$ and $b$ of $A_{n-1}^{\otimes n}$ with an arrow $a\to b$, we have an one-dimensional $\Ext^{1}(S_{a},S_{b})$ morphism and these morphisms along arrows anti-commute. For each $0\leq i \leq n-1$, let $O_{n}^{i}$ denote the object $\Omega_{\bP^{n-1}}^{n-1-i}(n-1-i)[i]$ restricted on the Calabi-Yau Fermat variety in $\bP^{n-1}$. Bases of $\Ext^{1}(O_{n}^{i},O_{n}^{i+1})$ can be given by morphisms $d x_{n,i}^{j}$ for $1\leq j\leq n$ such that we have anti-commuting relations $d x_{n,i}^{j} d x_{n,i-1}^{j'}= - d x_{n,i}^{j'} d x_{n,i-1}^{j}$ for $1\leq j, j'\leq n$. Putting (weighted) copies of $d x_{n,i}^{j}$ on arrows $a\to b$ such that $\lambda(a,b)=j$ and $\lambda(a)=i$, it is easy to check that $H_{n}$-equivariant objects of $O_{n}^{i}$ give also the Yoneda algebra. For the dual statement of ${{\mathrm D}^{b}}_{H_{n}}\Coh (X_{n})\cong \FS(X_{n})$, enhancing the argument above, we have defined the $\hat{H}_{n}$ quotient $\FS^{\hat{H}_{n}}(X_{n})$ of $\FS(X_{n})$ as the perfect derived category of the dg $\hat{H}_{n}$-orbit category (see [@Kel05]) of a dg algebra $\cA$ which is $A_{\infty}$ isomorphic to the $A_{\infty}$ algebra of objects in $\hat{H}_{n}$-orbits of graded Lagrangian vanishing cycles so that the dg $\hat{H}_{n}$-orbit category is a dg enhancement of the Yoneda algebra of objects $O_{n}^{i}$. This is a formal application of Pontryagin duality on categories of dg categories, and ${{\mathrm D}^{b}}(\Coh X_{n})\cong \FS^{\hat{H}_{n}}(X_{n})$. As for the existence of such $\cA$, we can take the dg algebra of $\hat{H}_{n}$-graded matrix factorizations of $O_{n}^{i}$ in terms of the dg category of graded matrix factorizations of $X_{n}$ [@Orl]. We have $\FS(X_{n})\cong {{\mathrm D}^{b}}(\mo A_{n-1}^{\otimes n})\cong {{\mathrm D}^{b}}(\mo A_{n-1})^{\otimes n}$ in terms of dg tensor products (see [@Kel06]). We have tensor products of Auslander-Reiten transformations $\tau$ of ${{\mathrm D}^{b}}(\mo A_{n})$ such that $\tau^{n}\cong [2]$. Actions of the finite group consisting of $\tau^{t_1}\otimes \cdots \otimes \tau^{t_{n}}$ such that $\sum t_{i}=0$ coincides with those of $\hat{H}_{n}$ above and the index of each simple representation of $A_{n-1}^{\otimes n}$ stays the same by the actions. Main statement ============== Directly, it is highly nontrivial to discuss stability conditions and moduli spaces on complexes of $O_{n}^{i}$ with nontrivial $A_{\infty}$ structures to take into account (see [@DouGovJay]), even with motivating realizations in terms of coherent sheaves or Lagrangians. By taking equivariance, we have stability conditions on $\mo A_{n-1}^{\otimes n}$, and on ${{\mathrm D}^{b}}_{H_{n}}(\Coh X_{n})$. Still, an issue we face to consider stability conditions on ${{\mathrm D}^{b}}(\Coh X_{n})$ with stability conditions on $\mo A_{n-1}^{\otimes n}$ is that such stability conditions are not invariant under $\hat{H}_{n}$; because, not all objects which consist of the $\hat{H}_{n}$-orbit of a simple representation of $ A_{n-1}^{\otimes n}$ are representations of $A_{n-1}^{\otimes n}$. If they were invariant, then we would have taken advantages of the paper [@Pol] by Polishchuk and the paper [@MacMehSte] by Macrì-Mehrotra-Stellari. We overcome the issue by taking the notion of [framed $\hat{H}_{n}$-invariance]{} on stability conditions in Definition \[def:rest\] and representations and morphisms in Definition \[def:rest\_rep\]; instead of full products of general linear groups over vertices, we take ones which commutes with $\hat{H}_{n}$ actions restricted on simple representations of $A_{n-1}^{\otimes n}$. \[def:rest\] We say that a stability function $Z$ on $\mo A_{n-1}^{\otimes n}$ is [framed $\hat{H}_{n}$-invariant]{}, if central charges of simple representations of $ A_{n-1}^{\otimes n}$ of each index are the same. For example, stability functions $Z_{i}$ and their mirrors in Section \[sec:mir\] are framed $\hat{H}_{n}$-invariant. To define framed $\hat{H}_{n}$-invariant representations, for each representation $E$ of $A_{n-1}^{\otimes n}$, let $E_{a,b}:E_{a}\to E_{b}$ denote commuting linear maps along arrows $a\to b$; in particular, $E_{a,a}$ are identity maps on $E_{a}$. \[def:rest\_rep\] We say that a representation $E$ of $A_{n-1}^{\otimes n}$ is [ framed $\hat{H}_{n}$-invariant]{}, if for vertices $a,b$ of $A_{n-1}^{\otimes n}$ with the same indices, we have [framing isomorphisms]{} $\phi_{a,b}:E_{a}\to E_{b}$ with the following conditions. For vertices $a,b,c, a',c'$ such that $a\to a'$ and $c\to c'$ with $\lambda(a,a')=\lambda(c,c')$ and $\lambda(a)=\lambda(b)=\lambda(c)$, we have $\phi_{a',c'}E_{a,a'}=E_{c,c'}\phi_{b,c}\phi_{a,b}$; i.e., we have the following commuting diagram such that squig arrows indicate isomorphisms and plain arrows indicate maps of a representation. $$\begin{matrix} a & \rightsquigarrow & b & \rightsquigarrow &c\\ \downarrow & & \circlearrowright & &\downarrow \\ a' & & \rightsquigarrow& & c' \end{matrix}.$$ For representations $E$ and $F$ of $A_{n-1}^{\otimes n}$ with framed $\hat{H}_{n}$-invariance, we say that a morphism $f:E\to F$ of $A_{n-1}^{\otimes n}$ is [framed $\hat{H}_{n}$-invariant]{}, if for vertices $b,b'$ such that $\lambda(b)=\lambda(b')$, we have $\phi^{F}_{b,b'}f_{b}\phi^{E}_{b',b}=f_{b'}$. Let $\fmod A_{n-1}^{\otimes n}$ denote the [category of framed $\hat{H}_{n}$-invariant representations]{} which consists of framed $\hat{H}_{n}$-invariant representations and morphisms of $A_{n-1}^{\otimes n}$. In Definition \[def:rest\_rep\], by putting $a=c$ and $a'=c'$, we see that $\phi_{b,c}\phi_{a,b}=\phi_{a,c}$, by putting $a=c$, we have $\phi_{b,a}\phi_{a,b}=\phi_{a,a}$, and by putting $a=b=c$, we have $\phi_{a,a}=1_{E_{a}}$. We have uniqueness of framing isomorphisms in the following sense. \[lem:triv\] Let $E$ be a framed $\hat{H}_{n}$-invariant representation of $A_{n-1}^{\otimes n}$ supported over vertices with indices $0,\ldots, n-1$. For vertices $a_{i}$ of $A_{n-1}^{\otimes n}$ such that $\lambda(a_{i})=i$ for $0\leq i \leq n-1$, we have a framed $\hat{H}_{n}$-invariant isomorphism $t^{E}:E\to E'$ such that $E'$ is a framed $\hat{H}_{n}$-invariant representation of $A_{n-1}^{\otimes n}$ with trivial framing isomorphisms and $t^{E}_{a}=\phi_{a,a_{\lambda(a)}}$ for vertices $a$ of $A_{n-1}^{\otimes n}$. For vertices $b_{i}$ of $A_{n-1}^{\otimes n}$ with $\lambda(b_{i})=i$, we let $E'_{b_{i}}:=E_{a_{i}}$, and for arrows $b_{i}\to b_{i+1}$, we let $E'_{b_{i},b_{i+1}}:=\phi_{b_{i+1},a_{i+1}}E_{b_{i},b_{i+1}}\phi_{a_{i},b_{i}}$. For vertices $b_{i}',b_{i+1}'$ with $\lambda(b_{i},b_{i+1})=\lambda(b'_{i},b'_{i+1})$, we have $E'_{b_{i},b_{i+1}}=E'_{b'_{i},b'_{i+1}}$. For vertices $b'_{i}$ with $\lambda(b_{i-1},b_{i})=\lambda(b'_{i},b_{i+1})$ and $\lambda(b_{i-1},b'_{i})=\lambda(b_{i},b_{i+1})$, we have $E'_{b_{i},b_{i+1}}E'_{b_{i-1},b_{i}} =E'_{b'_{i},b_{i+1}}E'_{b_{i-1},b'_{i}}$. For arrows $b_{i}\to b_{i+1}$, we have $t_{b_{i+1}}E_{b_{i},b_{i+1}}= E'_{b_{i},b_{i+1}} t_{b_{i}}$. For vertices $b_{i}'$ with $\lambda(b_{i})=\lambda(b_{i}')$, we have $\phi^{E'}_{b_{i},b_{i}'}t^{E}_{b_{i}}\phi_{b_{i}',b_{i}}^{E} =t^{E}_{b_{i}'}$. Let us recall the quiver $B_{n-1}$, called the Be[ĭ]{}linson quiver of $\bP^{n-2}$ [@Bei; @Min], which has $n$ vertices $0,\ldots, n-1$ and $n-1$ arrows from $i$ to $i+1$ with commuting relations. For vertices $i,i+1$, we label arrows by $s$ for $1 \leq s \leq n$ so that for maps $E_{i,i+1}^{s}$ on labelled arrows $i\stackrel{s}{\to}i+1$, we have $E_{i,i+1}^{s} E_{i-1,i}^{s'} =E_{i,i+1}^{s'} E_{i-1,i}^{s}$. \[prop:eq\] The full subcategory of $\fmod A_{n-1}^{\otimes n}$ consisting of representations supported over vertices with indices $0, \ldots,n-1$ is equivalent to $\mo B_{n-1}$. For each representation $E$ of $B_{n-1}$, we put the representation $F(E)$ of $A_{n-1}^{\otimes n}$ as follows. For vertices $a,b$ of $A_{n-1}^{\otimes n}$ with arrows $a\to b$, we put linear maps $F(E)_{a,b}:=E_{\lambda(a),\lambda(b)}^{\lambda(a,b)}$ from $F(E)_{a}:=E_{\lambda(a)}$ to $F(E)_{b}:=E_{\lambda(b)}$. For each morphism $f:E\to E'$ of representations $E$ and $E'$ of $B_{n-1}$, we put $F(f):F(E)\to F(E')$ by $F(f)_{a}:=f_{\lambda(a)}$ for each vertex $a$. To obtain an inverse $G$ of $F$, let $a_{i}$ be vertices of $A_{n-1}^{\otimes n}$ such that $\lambda(a_{i})=i$ for $0\leq i \leq n-1$. For $E$ of $\fmod A_{n-1}^{\otimes n}$ supported over vertices with the indices, we put $G(E)_{i}:=E_{a_{i}}$, and for arrows $a\to b$ such that $\lambda(a)=\lambda(a_{i})$, we put the linear map $G(E)_{\lambda(a_{i}),\lambda(a_{i+1})}^{\lambda(a,b)}:= \phi_{b,a_{i+1}}E_{a,b}\phi_{a_{i},a}$, which is independent for choices of such arrows $a\to b$. For $1\leq c,c'\leq n-1$ and $0\leq i-1\leq n-3$, let us note that we have a vertex $b_{i-1},b_{i},b'_{i}, b_{i+1}$ of $A_{n-1}^{\otimes n}$ with $\lambda(b_{i-1})=i-1$, $\lambda(b_{i-1},b_{i})=\lambda(b'_{i},b_{i+1})=c$, and $\lambda(b_{i-1},b'_{i})=\lambda(b_{i},b_{i+1})=c'$; so, we have $G(E)_{i,i+1}^{c'} G(E)_{i-1,i}^{c}= G(E)_{i,i+1}^{c} G(E)_{i-1,i}^{c'}$. For a framed $\hat{H}_{n}$-invariant morphism $f:E\to E'$, we put $G(f):G(E)\to G(E')$ such that $G(f)_{i}= f_{a_{i}}$; we have $G(E')_{i,i+1}^{c}G(f)_{i}=G(f)_{i+1} G(E)_{i,i+1}^{c}$. For the functor $FG$, a framed $\hat{H}_{n}$-invariant morphism $f:E_{1}\to E_{2}$, and $t^{E_{1}},t^{E_{2}}$ in the notation of Lemma \[lem:triv\], we have $t^{E_{1}}_{a}f_{a}=\phi^{E_{1}}_{a,a_{\lambda(a)}}f_{a}=f_{a_{\lambda(a)}} \phi^{E_{2}}_{a, a_{\lambda(a)}} =FG(f)_{a} t^{E_{2}}_{a}$ for each vertex $a$ of $A_{n-1}^{\otimes n}$. For the functor $GF$, we take the identity functor on $\fmod A_{n-1}^{\otimes n}$. Similar statements to the one in Proposition \[prop:eq\] can be obtained by taking other supporting vertices. For $n=3$ and $4$, let us mention that derived categories ${{\mathrm D}^{b}}(\Coh \bP^{n-2})$, which is derived equivalent to ${{\mathrm D}^{b}}(\mo B_{n-1})$, have been described in terms of graded Lagrangian vanishing cycles and Lagrangian intersection theories in [@AurKatOrl08]. With Serre twists in use, let us define the [Serre-de Rham functor]{} $\SdR$ as follows. \[def:twisted\] For each representation $E$ of $\mo B_{n-1}$, we put the chain $\SdR(E)$ of morphisms $\sum_{1 \leq j\leq n}E_{i,i+1}^{j} \otimes d x_{n,i}^{j}: E_{i}\otimes O_{n}^{i} \to E_{i+1}\otimes O_{n}^{i+1}$ for $0\leq i\leq n-2$. For each morphism $f:E\to F$ in $\mo B_{n-1}$, we put the chain map $\SdR(f):\SdR(E)\to \SdR(F)$ such that $f_{i}\otimes \id_{O_{n}^{i}}:E_{i}\otimes O_{n}^{i} \to F_{i}\otimes O_{n}^{i}$ for $0\leq i \leq n-1$. The following is in order. For each representation $E$ of $\mo B_{n-1}$, we have that $\SdR(E)$ is a complex and $\SdR$ is a functor from $\mo B_{n-1}$ to ${{\mathrm D}^{b}}(\Coh X_{n})$. Each $E$ satisfies commuting relations among maps. This translates into zero compositions of consecutive morphisms. Morphisms between objects of $\mo B_{n-1}$ become morphisms of complexes. For a given framed $\hat{H}_{n}$-invariant stability function $Z$ on $\mo A_{n-1}^{\otimes n}$, we define the stability function $Z'$ on $\mo B_{n-1}$ by putting $Z'(i):=Z(a)$ for a vertex $a$ of $A_{n-1}^{\otimes n}$ such that $\lambda(a)=i$. In the following, we write $Z'$ on $\mo B_{n-1}$ also as $Z$ on $\mo B_{n-1}$. \[thm:syz\] For a stability function $Z_{n}$ on the category of framed $\hat{H}_{n}$-invariant representations of $A_{n-1}^{\otimes n}$, we have a moduli space of stable Lagrangian connect sums of graded Lagrangian vanishing cycles for a morsification of $X_{n}$ such that the Serre-de Rham functor localizes the moduli space into the Calabi-Yau Fermat variety in $\bP^{n-1}$. For the mirror of $Z_{n}$ on the category of representations of $A_{n-1}^{n}$, each graded Lagrangian vanishing cycle is stable. For a stability function $Z_{n}$ on $\mo B_{n-1}$ and stable representations $E$ of $B_{n-1}$ with the dimension vector $(1,\ldots, 1)$, commuting relations and the indecomposable property of the representation give nonzero $k^{E}_{i}$ such that $k^{E}_{i} E_{i,i+1}^{s}=E_{i-1,i}^{s}$ for $1\leq s \leq n$ and $0 \leq i \leq n-1$. If objects $O_{n}^{i}$ were unrestricted on the Calabi-Yau Fermat variety, then $\SdR(E)$ would have been Koszul resolutions of skyscraper sheaves of points in $\bP^{n-1}$. For non-isomorphic representations $E$ of $\mo B_{n-1}$, objects $\SdR(E)$ of ${{\mathrm D}^{b}}(\Coh X_{n})$ are non-isomorphic objects supported over distinct points of the Calabi-Yau Fermat variety, unless isomorphic to the zero object outside. For the mirror of $Z_{n}$ on $\mo A_{n-1}^{\otimes n}$, each stable object is isomorphic to a simple object of $\mo A_{n-1}^{\otimes n}$. Stable objects of Theorem \[thm:syz\] in terms of the mirror $Z_{n}$ are simple representations of the quiver $A_{n-1}^{\otimes n}$. We may as well take the mirror of $Z_{n}$ on the category of framed ones of $A_{n-1}^{\otimes n}$; in this case, we obtain polysimple representations of $A_{n-1}^{\otimes n}$ consisting of graded Lagrangian vanishing cycles. For projective spaces, and for Calabi-Yau hypersurfaces of $\bP^{n-1}$ represented as $x_{1}^{n}+\ldots + x_{n}^{n}+\psi \cdot x_{1}\cdots x_{n}:\bC^{n}\to \bC$ for $\psi \in \bC$, we may obtain similar statements to Theorem \[thm:syz\]. Acknowledgements {#acknowledgements .unnumbered} ================ The author thanks Research Institute for Mathematical Sciences of Kyoto University and Institut des Hautes Études Scientifiques for providing him with excellent research environment. He thanks Professors Carqueville, Fukaya, Hori, Kajiura, Keller, Y. Kimura, Kontsevich, Nakajima, Ohashi, Ohkawa, Stoppa, A. Takahashi, Toda, Usnich, Weist, and Yamazaki for their useful discussions or stimulating talks related to this paper. Aspinwall, P. S.: D-branes on Calabi-Yau manifolds. In: Progress in string theory, 1–152, World Sci. Publ., Hackensack, NJ (2005). Auroux, D.: Mirror symmetry and T-duality in the complement of an anticanonical divisor. J. Gökova Geom. Topol. GGT 1 (2007), 51–91. Auroux, D.; Katzarkov, L.; Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math. (2) [167]{}(3), 867–943 (2008). Bayer, A.; Macrì, E.: The space of stability conditions on the local projective plane. arXiv:0912.0043. Be[ĭ]{}linson, A.: Coherent Sheaves on $\bP^n$ and Problems in Linear Algebra, (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69, also English translation: Functional Anal. Appl. 12 (1978), no. 3, 212–214 (1979); ibid. 12 (1978), no. 3, 214–216 (1979). Bergman, A.: Stability conditions and branes at singularities. J. High Energy Phys. 2008, no. 10, 073, 19 pp. Bridgeland, T.: Stability conditions and Kleinian singularities. Int. Math. Res. Not. IMRN 2009, no. 21, 4142–4157. Bridgeland, T.: Stability conditions on triangulated categories. Ann. of Math. (2) [166]{}(2), 317–345 (2007). Bridgeland, T.: $t$-structures on some local Calabi-Yau varieties. J. Algebra 289 (2005), no. 2, 453–483. Bridgeland, T.; Toledano-Laredo, V.: Stability conditions and Stokes factors. arXiv:0801.3974. Douglas, M. R.: Dirichlet branes, homological mirror symmetry, and stability. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing (2002). Douglas, M. R.: D-branes and $\cN=1$ supersymmetry. In: Strings, 2001 (Mumbai), 139–152, Clay Math. Proc., 1, Amer. Math. Soc., Providence, RI (2002). Douglas, M. R.; Govindarajan, S.; Jayaraman, T.; Tomasiello, A.: D-branes on Calabi-Yau manifolds and superpotentials. Comm. Math. Phys. 248 (2004), no. 1, 85–118. Futaki, M.; Ueda, K.: Homological mirror symmetry for Brieskorn-Pham singularities, arXiv:0912.0316. Hori, K.; Katz, S.; Klemm, A.; Pandharipande, R.; Thomas, R.; Vafa, C.; Vakil, R.; Zaslow, E.: Mirror symmetry. With a preface by Vafa, C.. Clay Mathematics Monographs, 1. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2003). Ishii, A.; Ueda, K.; Uehara, H.: Stability conditions on $A_{n}$-singularities. J. Differential Geom. 84 (2010), no. 1, 87–126. Kajiura, H.; Saito, K.; Takahashi, A.: Matrix factorization and representations of quivers. II. Type $ADE$ case. Adv. Math. [211]{}(1), 327–362 (2007). Keller, B.: On differential graded categories, In: International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich (2006). Keller, B.: On triangulated orbit categories. Doc. Math. [10]{}, 551–581 (2005). King, A. D.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel (1995). Kontsevich, M.; Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, arXiv:1006.2706. Leszczyski, Z.: On the representation type of tensor product algebras. Fund. Math. [144]{}, no. 2, 143–161 (1994). Macr[ì]{}, E.: Stability conditions on curves. Math. Res. Lett. 14 (2007), no. 4, 657–672. Macr[ì]{}, E.; Mehrotra, S.; Stellari, P.: Inducing stability conditions. J. Algebraic Geom. 18 (2009), no. 4, 605–649. Minamoto, H.: A noncommutative version of Beilinson’s theorem. J. Algebra 320 (2008), no. 1, 238–252. Ohkawa, R.: Flips of moduli of stable torsion free sheaves with $c_1=1$ on $\bP^2$. arXiv.org:1006.4203. Okada, S.: Homological mirror symmetry for Fermat polynomials, arXiv:0910.2014 (submitted). Okada, S.: On stability manifolds of Calabi-Yau surfaces. Int. Math. Res. Not. 2006, Art. ID 58743, 16 pp. Okada, S.: Stability manifold of $\bP^1$. J. Algebraic Geom. 15 (2006), no. 3, 487–505. Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. arXiv:math/0503632. Polishchuk, A.: Constant families of $t$-structures on derived categories of coherent sheaves. Mosc. Math. J. 7 (2007), no. 1, 109–134, 167. Seidel, P.: More about vanishing cycles and mutation. In: Symplectic geometry and mirror symmetry (Seoul, 2000), 429–465, World Sci. Publ., River Edge, NJ (2001). Strominger, A.; Yau, S.-T.; Zaslow, E.: Mirror symmetry is T-duality. Nuclear Phys. B 479 (1996), no. 1-2, 243–259. [^1]: Supported by JSPS Grant-in-Aid and Global Center of Excellence Program at Kyoto University, Email: okada@kurims.kyoto-u.ac.jp, Address: Research Institute for Mathematical Sciences, Kyoto University, 606-8502, Japan.
--- abstract: \| Fisher’s famous Lady Tasting Tea experiment is often referred to as the first permutation test or as an example of such a test. Permutation tests are special cases of the general group invariance test. Recently it has been emphasized that the set of permutations used within a permutation test should have a group structure, in the algebraic sense. If not, the test can be very anti-conservative. In this paper, however, we note that in the Lady Tasting Tea experiment, the type I error rate is controlled even if the set of permutations used does not correspond to a group. We explain the difference between permutation-based tests that fundamentally rely on a group structure, and permutation-based tests that do not. The latter are tests based on randomization of treatments. When using such tests, it can be useful to consider a randomization scheme that does correspond to a group. In particular, we can use randomization schemes where the number of possible treatment patterns is larger than in standard permutation-based randomization tests. This leads to exact p-values of improved resolution, providing increased power for very small significance levels. We discuss applications in clinical trials and elsewhere.\ \ keywords: Permutation test; Lady Tasting Tea; Group invariance test; Randomization test author: - 'Jesse Hemerik[^1] [^2] and Jelle Goeman[^3]' bibliography: - 'bibliography.bib' title: 'Another look at the Lady Tasting Tea and permutation-based randomization tests' --- Introduction ============ The well-known “Lady Tasting Tea" experiment, decribed in @fisher1935 [Ch. II], is commonly referred to as the first published permutation test [@wald1944statistical; @hoeffding1952large; @anderson2001permutation; @lehmann2005testing; @langsrud2005rotation; @mielke2007permutation; @phipson2010permutation; @winkler2014permutation] or as a representative example of a permutation test [@freedman1983nonstochastic]. Indeed, this test is based on permutations and, like other permutation-based tests, falls under the definition of group invariance tests. This is a general class of tests based on transformations of data, such as permutations or rotations [@langsrud2005rotation]. Details are in Section \[secp\]. In @fisher1935 [Ch. II], the null hypothesis is that a particular lady cannot distinguish between two types of cups of tea with milk: cups in which the tea was added first and cups in which the milk was added first. To test the null hypothesis, which we will denote by $H_0$, the experimenter “mixes eight cups of tea, four in one way and four in the other," and presents them “to the subject for judgment in a random order." The experimental setup is made known to the lady. The lady then tastes from the cups and has to determine which four cups had milk added first. According to most sources, Fisher actually performed the experiment [@box1978ra; @berry2014chronicle]. The test is detailed in Section \[secltt\]. It has been emphasized by @southworth2009properties, among others, that for permutation tests to have proven properties, it is important that the set of permutations used has a group structure, in the algebraic sense, as we discuss in Section \[secp\]. For example, the set of balanced permutations, which is a subset of a permutation group, does not have a group structure, and using it within a permutation test tends to lead to a very anti-conservative test. Balanced permutations (not to be confused with stratified permutations) have been used in several publications [@fan2004normalization; @jones2007genome] and @southworth2009properties warn against their use. Surprisingly, as we will show, the Lady Tasting Tea experiment still controls the type I error rate if the set of permutations used does not have a group structure. How is it possible that this test is a permutation test, but does not require a group structure? As we will show, the reason is that the Lady Tasting Tea experiment can be viewed as a test based on randomization of treatments, i.e., as a randomization test in the sense of for example @kempthorne1969behaviour, @edgington2007randomization and @rosenberger2019randomization. Indeed, the experiment involves an experimenter, who randomizes the true pattern of cups. Unlike archetypical permutation tests, such randomization tests can generally still control the type I error rate, even if the set of permutations used is not a group. For instance, in the Lady Tasting Tea experiment, both the permutations that the experimenter randomly chooses from and the permutations that the lady is told to pick from, need not be groups. Randomization tests which are not based on groups, are uncommon but not new in the literature on randomization tests [@onghena1994randomization; @rosenberger2019randomization]. In this paper, we explain the difference between permutation-based tests that require a group structure and permutation-based tests that do not, the latter being tests based on randomization of treatments. This explicit distinction has not been made before, to our knowledge. The further contributions of this paper are related to this distinction and are as follows. First of all, since permutation-based randomization tests do not require a group structure, it can be useful to consider a randomization scheme that does not correspond to a group. We introduce the idea of using an alternative randomization scheme to increase the number of possible treatment patterns. This increases the resolution of the p-value, thus improving power for very small significance levels $\alpha$. In addition, this paper provides the caveat that the Lady Tasting Tea experiment is rather different from archetypical permutation tests [in the sense of @onghena2018randomization]. Using the Lady Tasting Tea experiment as an example of a permutation test, as is often done, can put readers on the wrong foot, since the reasoning underlying this experiment is not based on a group structure. Referring to the Lady Tasting Tea may have contributed to the confusion that has led researchers to design invalid permutation tests without a group structure [@southworth2009properties]. Instead of referring to @fisher1935 [Ch. II] as an example of a permutation test, it may better to refer to the example in @fisher1936coefficient [pp. 58-59], in which statures of Frenchmen and Englishmen are compared. This example is discussed in Section \[secp\]. This is a typical permutation test, which is not based on randomization of treatments, but on permuting random samples from populations. The argument underlying this test (which is implicit in that article) is based on the group structure of the set of permutations, unlike the argument underlying the Lady Tasting Tea experiment. This paper is built up as follows. In Section \[secp\] we review existing results on permutation and group invariance tests, empasizing the key role of the group structure of the permutations. In Section \[secltt\] we discuss the Lady Tasting Tea experiment, emphasizing why this test does not require a group structure to control the type I error rate. In Section \[secgenct\] we generalize the test of Section \[secltt\], obtaining a general randomization test and mentioning applications. In Section \[highrescc\] we apply the general randomization test in a clinical trial setting, discussing how we can obtain higher-resolution p-values than with a canonical permutation-based test. The performance of our alternative test is illustrated with simulations in Section \[secsim\]. We end with a discussion. Permutation tests and group invariance tests {#secp} ============================================ The terms “permutation test" and “randomization test" have been used somewhat inconsistently in the literature. Sometimes the class of permutation tests is understood to include randomization tests [@edgington2007randomization p.1]. @rosenberger2019randomization write that “Many statisticians use the terms permutation tests and randomization tests interchangeably. The first author has regrettably made this mistake himself." @onghena2018randomization and @kempthorne1969behaviour compare the two terms in detail. A typical example of a permutation test in the sense of @onghena2018randomization is discussed in @fisher1936coefficient [pp. 58-59]. In this thought experiment, measurements of the statures of 100 Englishmen and 100 Frenchmen are considered. These observations are assumed to be randomly sampled from their respective populations. Such a model, where observations are randomly sampled from their populations, is typical for permutation tests in the sense of for example @kempthorne1969behaviour, @onghena2018randomization and @rosenberger2019randomization. Note that in this example, there is no randomization of treatments as in, for example, clinical trials. In the example in @fisher1936coefficient [pp. 58-59], to test whether “the two populations are homogeneous", the difference between the two sample means is computed and this is repeated for each permutation of the 200 observations. The null hypothesis is rejected if the original difference is larger than most of the differences obtained after permutation. We will return to this example below. Permutation tests are special cases of the general group invariance test. The definition of the group invariance test in, for example, @hoeffding1952large, @lehmann2005testing and @hemerik2018exact is rather general, so that randomization tests also fall under it. The principle underlying the group invariance test can also be used to prove properties of various permutation-based multiple testing methods [@westfall1993resampling; @tusher2001significance; @meinshausen2005lower; @hemerik2018false; @hemerik2019permutation]. A general definition of a group invariance test is as follows. Generalizations of this framework, such as two-sided tests, are possible. Let $X$ be data taking values in a sample space $\mathcal{X}$. Consider a set $G$ of permutation maps or other transformations $g:\mathcal{X}\rightarrow \mathcal{X}$. We will assume that $G$ is finite, although generalizations are possible. The set $G$ is assumed to have a group structure with respect to the operation of composition of maps, which means that: $G$ contains the identity map $x\mapsto x$; every element in $G$ has an inverse; and for all $g$, $h\in G$, $g\circ h\in G$ [@hoeffding1952large]. Further, we consider some test statistic $T:\mathcal{X}\rightarrow \mathcal{X}$. Consider a null hypothesis $H_0$ which implies that the joint distribution of all test statistics $T(g(X))$ with $g\in G$ is invariant under all transformations in $G$ of $X$. This holds in particular if the data $X$ are themselves transformation-invariant, i.e., if $$\label{Ginvariance} g(X)\,{\buildrel d \over =}\,X$$ for every $g\in G$. A typical example of such a setting is the thought experiment from @fisher1936coefficient [pp. 58-59], mentioned above. Let $X_1,...,X_{100}$ be the statures of the Englishmen and let $X_{101},...,X_{200}$ be the statures of the Frenchmen. The test statistic considered in @fisher1936coefficient [pp. 58-59] is $$T(X)=\frac{1}{100}\sum_{i=1}^{100} X_i - \frac{1}{100} \sum_{i=101}^{200} X_i. \label{tst}$$ The null hypothesis $H_0$ is that $X_1,...,X_{200}$ are i.i.d.. The group $G$ consists of all permutation maps $g: \mathbb{R}^{200} \rightarrow \mathbb{R}^{200}$. Here, every $g\in G$ is of the form $$(x_1,...,x_{200})\mapsto (x_{\pi_1},..., x_{\pi_{200}} ),$$ where $(\pi_1,...,\pi_{200})$ is a permutation of $(1,...,200)$. Note that $X$ is then $G$-invariant under $H_0$, i.e., holds for every $g\in G$. As another example of group invariance, suppose $X\in \mathbb{R}^{n}$ has independent entries and under $H_0$, the entries are symmetric around 0. Then the distribution of $X$ is invariant under all transformations in $G$ under $H_0$ if we define $G$ to be the group of all sign-flipping maps of the form $$\label{signflip} (x_1,...,x_n)\mapsto (s_1x_1,...,s_nx_n),$$ with $(s_1,...,s_n)\in \{-1,1\}^n$. This test already appears in @fisher1935 [§21], albeit without explicit proof. In both examples above, we can apply the general group invariance test to test $H_0$. This test already appears in the literature [@hoeffding1952large; @lehmann2005testing; @hemerik2018exact], but for completeness we include the result and its proof. Exact testing with randomly sampled permutations will not be discussed here, but is also possible [@hemerik2018exact]. We will write $gX=g(X)$ for short. Let $T^{(1)}(X) \leq ... \leq T^{(\|G\|)}(X)$ be the sorted values $T(gX)$ with $g\in G$. Let $k=\lceil (1-\alpha)\|G\| \rceil$, the smallest integer which is larger than or equal to $(1-\alpha)\|G\|$. \[basic\] Under $H_0$, $\mathbb{P}\big\{T(X)>T^{(k)}(X)\big\}\leq \alpha.$ By the group structure, $Gg=G$ for all $g\in G$. Hence $T^{(k)}(gX)=T^{(k)}(X)$ for all $g\in G$. Let $h$ have the uniform distribution on $G$. Then under $H_0$, the rejection probability is $$\begin{aligned} &\mathbb{P}\big\{T(X)>T^{(k)}(X)\big\}=\\ &\mathbb{P}\big\{T(hX)>T^{(k)}(hX)\big\}=\\ &\mathbb{P}\big\{T(hX)>T^{(k)}(X)\big\}. \end{aligned}$$ The first equality follows from the null hypothesis and the second equality holds since $T^{(k)}(X)=T^{(k)}(hX)$. Since $h$ is uniform on $G$, the above probability equals $$\mathbb{E} \Big [\|G\|^{-1} \cdot \big\|\big \{g\in G: T(gX)> T^{(k)}(X) \big \}\big\| \Big ] \leq \alpha,$$ as was to be shown. Under additional assumptions, the test is exact, i.e., the rejection probability is exactly $\alpha$ under $H_0$. In the above proof we used the group structure, which guarantees the symmetry property $Gg=G$ for all $g\in G$. A different proof, based on conditioning on the pooled sample, is also possible and also requires using this symmetry [first proof of Theorem 1 in @hemerik2018exact]. Write $GX=\{gX:g\in G\}$ and assume for convenience that $gX$ and $g'X$ are distinct with probability 1 if $g,g'\in G$ are distinct. The permutation test is based on the fact that under $H_0$, for every permutation $g\in G$ the probability $\mathbb{P}\{T(gX)>T^{(k)}(X)\}$ is the same. The reason is that under $H_0$, for every $g\in G$, the joint distribution of $(gX,GX)$ is the same. This is because if $g$, $g'\in G$, under $H_0$ we have $$(gX,GX)= (gX,GgX) \,{\buildrel d \over =}\, (X,GX) \,{\buildrel d \over =}\, (g'X,Gg'X) = (g'X,GX).$$ When $Gg=G$ does not hold for all $g\in G$, then the above does not generally hold under $H_0$. The group structure of $G$ implies that $Gg=G$ for all $g\in G$. The reverse implication also holds, under the mild condition that all $g\in G$ are surjective. For example, if $Gg=G$ for all $g\in G$, there are $h,g\in G$ with $hg=g$. It follows that $G$ contains an identity element, and the other group properties also easily follow. We conclude that in the argument underlying the permutation test, the group structure is key. The Lady Tasting Tea and randomization tests ============================================ Here, we first discuss the Lady Tasting Tea experiment, explaining that this test essentially does not rely on a group structure. This experiment is a special case of a general randomization test, which we discuss in Section \[secgenct\]. In Section \[highrescc\] we apply this test to provide higher-resolution p-values in clinical trials. The Lady Tasting Tea experiment {#secltt} ------------------------------- As discussed in the Introduction, in the Lady Tasting Tea experiment, the lady receives eight cups. There are two types of cups and she receives four of each kind. There are $\binom{8}{4}=70$ possible orders, with respect to the two types of cups. Suppose $H_0$ is true. If the lady guesses every pattern with probability $1/70$, then the probability that she chooses the correct order is $1/70$. Even if she has an a priori preference for a certain order, the probability of guessing correct is $1/70$. Indeed, it is assumed that the researcher randomizes the true pattern, i.e., he chooses each pattern with equal probability. Thus, if we reject $H_0$ when the lady guesses the order correctly, then the probability of a type I error is $1/70$. The probability that she labels three of the “tea first" cups correctly is $\binom{4}{3} \binom{4}{1}/70=16/70$ and the probability of two correct picks is $36/70$. Thus, for example, when we reject $H_0$ if at least three picks are correct, the level is $16/70+1/70=17/70$. The test is equivalent to an instance of “Fisher’s exact test" [@yates1934contingency; @fisher1935logic; @berry2014chronicle] with pre-fixed marginal frequencies in the $2\times 2$ table. Mathematically, we can descibe the experiment as follows. Let ${\mathcal{D}}\subset\{0,1\}^8$ be the set of vectors containing four 0’s and four 1’s, so that the cardinality of ${\mathcal{D}}$ is $m:=\|{\mathcal{D}}\|=70$. Let the decision of the lady be encoded as $D'\in \mathcal{D}$ and let $D\in \mathcal{D}$ be the true order, i.e., the random decision by the experimenter. The experimenter’s order $D$ is assumed to be uniformly distributed on ${\mathcal{D}}$. The null hypothesis is $$H_0: \text{ } D' \text{ is independent of } D.$$ Let $\alpha\in(0,1)$ be the desired type I error rate. If $\alpha\in A=\{1/70,17/70,53/70,69/70\}$, then $\alpha$ is called attainable in the Lady Tasting Tea experiment, meaning that we obtain a test of exactly level $\alpha$ [@pesarin2015some]. If $\alpha$ is not attainable, then we obain a test with level strictly less than $\alpha$. Let $T: {\mathcal{D}}\times {\mathcal{D}}\rightarrow \mathbb{R}$ be a test statistic such that high values of $T(D,D')$ indicate that the patterns $D$ and $D'$ are similar, i.e., that there is evidence against $H_0$. Let $$T^{(1)}(D') \leq ... \leq T^{(70)}(D')$$ be the sorted statistics $T(B,D')$ with $B\in {\mathcal{D}}$. Whether the vector of sorted statistics $(T^{(1)},...,T^{(70)})$ actually depends on $D'$ or not, depends on the definition of $T$; in @fisher1935, the test statistic is $$\label{Tfisher} T(D,D')=\sum_{i=1}^8 \{D_i=1\}\cap \{D'_i=1\}$$ and it can be seen the sorted statistics do not depend on $D'$. Let $\lceil (1-\alpha)m \rceil $ be the smallest integer which is at least $(1-\alpha)m$. We have the following result [@fisher1935]. \[ltttest\] The test that rejects $H_0$ if and only if $ T(D,D')> T^{(\lceil (1-\alpha)m \rceil)} $ has size as most $\alpha$. Assume $H_0$ holds. Conditional on $D'$, $D$ is uniformly distributed on ${\mathcal{D}}$ and $T^{(\lceil (1-\alpha)m \rceil) }(D')$ is known. Hence, conditional on $D'$, the rejection probability is $$\mathbb{P}\big( D\in \{ B\in{\mathcal{D}}: T(B,D')> T^{(\lceil (1-\alpha)m \rceil)}(D') \} \big )=$$ $$\frac{1}{m}\|\{ B\ \in{\mathcal{D}}: T(B,D')> T^{(\lceil (1-\alpha)m \rceil)}(D') \} \|\leq \alpha.$$ Thus, marginal over $D'$, the rejection probability is also at most $\alpha$. Observe that when we use the test statistic , then taking $\alpha\in A$ indeed results in an exact test. This follows from the fact that $$T^{(1)}< T^{(2)}=...= T^{(17)}< T^{(18)}=... = T^{(53)} < T^{(54)} =...= T^{(69)} < T^{(70)},$$ by the argument at the beginning of this section. If $\alpha\in(0,1)\setminus A$, the level is strictly smaller than $\alpha$. If the experimenter does not choose randomly from all 70 possible patterns, but uses some smaller set of patterns for him and the lady to choose from, then there may not be any $\alpha\in(0,1)$ for which the test is exact, since the sorted test statistics may depend on $D'$. This is one of the reasons why using the full set of patterns, in combination with a suitable test statistic $T$, is useful. However, to prove Theorem \[ltttest\], we did not need to use the group structure of the permutations. The reason is that in the Lady Tasting Tea experiment, under $H_0$ the randomization $D$ of the researcher is by design independent of the reference set $\{(B,D'): B\in {\mathcal{D}}\}$. Further considerations follow below. A general randomization test {#secgenct} ---------------------------- Theorem \[ltttest\] still applies if the researcher uses a set of permutations that does not correspond to a group. Suppose for example that the researcher picks randomly from some set ${\mathcal{D}}$ of 69 patterns, with or without the lady’s knowledge. Denote the set that the lady chooses from by ${\mathcal{D}}'$. Then $D$ and $D'$ will still be independent and Theorem \[ltttest\] still applies if we let $m=\|{\mathcal{D}}\|=69$ and let $$T^{(1)}(D) \leq ... \leq T^{(69)}(D)$$ be the sorted test statistics $T(B,D')$, $B\in {\mathcal{D}}$. Indeed, conditional on $D'$, $D$ will have a uniform distribution on ${\mathcal{D}}$. In fact, we have the following very general randomization test, of which the Lady Tasting Tea experiment is a special case. We refer to this result as a randomization test since in most applications of the theorem, the variable $D$ will encode experimental randomization of treatments [@kempthorne1969behaviour; @onghena2018randomization]. The idea of the theorem is certainly not new. \[dtest\] Let ${\mathcal{D}}$ and ${\mathcal{D}}'$ be nonempty sets, where ${\mathcal{D}}$ is assumed to be finite. Write $m=\|{\mathcal{D}}\|$. Let $D'$ be a variable taking values in ${\mathcal{D}}'$ and assume $D$ is uniformly distributed on ${\mathcal{D}}$. Let $T: {\mathcal{D}}\times {\mathcal{D}}'\rightarrow \mathbb{R}$ be some test statistic. Consider a null hypothesis $H_0$ which implies that $D'$ is independent of $D$. Let $T^{(1)}(D') \leq ... \leq T^{(m)}(D')$ be the sorted values $T(B,D')$ with $B\in{\mathcal{D}}$. Then the result of Theorem \[ltttest\] still applies. The proof is analogous to that of Theorem \[ltttest\]. Assume $H_0$ holds. Conditional on $D'$, $D$ is uniformly distributed on ${\mathcal{D}}$ and $T^{(\lceil (1-\alpha)m \rceil) }(D')$ is known. Hence, conditional on $D'$, the rejection probability is $$\mathbb{P}\big( D\in \{ B\in {\mathcal{D}}: T(B,D')> T^{(\lceil (1-\alpha)m \rceil)}(D') \} \big ) =$$ $$\frac{1}{m}\|\{ B \in{\mathcal{D}}: T(B,D')> T^{(\lceil (1-\alpha)m \rceil)}(D') \} \| \leq \alpha.$$ Thus, marginally over $D'$, the rejection probability is also at most $\alpha$. We assumed that ${\mathcal{D}}$ is finite, but generalizations to infinite ${\mathcal{D}}$ are possible, as well as generalizations to non-uniform $D$. We can also define a two-sided test. Moreover, under straightforward additional assumptions, we can prove that the test of Theorem \[dtest\] is exact for certain $\alpha$, i.e., that the rejection probability is exactly $\alpha$ under $H_0$. Note that in Theorem \[dtest\], $D'$ might a constant, conditional on $D$. In principle, randomization tests can be used without an assumption that the responses are randomly sampled from populations [@cox2009randomization; @onghena2018randomization; @rosenberger2019randomization]. This is a known property of randomization tests, which we discuss further in the context of clinical trials in Section \[highrescc\]. The general randomization test of Theorem \[dtest\] has many applications. Examples are agricultural experiments and randomized clinical trials. The latter example will be discussed in Section \[highrescc\]. We mention a few other interesting applications here. First of all, Theorem \[dtest\] has implications for the Lady Tasting Tea experiment. In Section \[secltt\], it is assumed that the lady knows beforehand that there are $m$ cups of each type, where $2m$ is the total number of cups she receives. If for some reason she does not know that, then she might label e.g. $m+1$ of the $2m$ items with the same label. Theorem \[dtest\] then says that the type I error probability will nevertheless be at most $\alpha$ under $H_0$. Indeed, in Theorem \[dtest\], ${\mathcal{D}}'$ is allowed to be any set, so in particular it can be larger than ${\mathcal{D}}$. A further application of Theorem \[dtest\] are general sensory tests, of which the Lady Tasting Tea experiment is an example. It is interesting to note that in the literature on sensory tests, Fisher’s experiment has been regarded a “forerunner of modern sensory analysis" [@bi2015revisiting]. For example, @harris1949measurement perform a sensory experiment as follows: “The subject is now presented with eight tumblers, four of which contain a few c.c. of water and four containing a few c.c. of the solution \[...\]. The glasses are arranged at random. The subject is told that four of them contain the substance and four contain water, and he is asked to taste them all and to separate them into the two groups of four." Another application of Theorem \[dtest\] are existing permutation-based randomization tests which are used to evaluate whether some classification algorithm has any predictive ability. Such tests can be used to evaluate algorithms for, for example, text categorization, fraud detection, optical character recognition and medical diagnosis. Tests of this type are discussed in, for instance, @golland2005permutation, @airola2010applying, @ojala2010permutation, @schreiber2013statistical and @rosenblatt2016better. Randomization testing without a group structure: higher-resolution p-values {#highrescc} ----------------------------------------------------------------------------- In randomized clinical trials, often we are interested in comparing two different treatments, for example a drug and a placebo. In such a setting, there is obviously a treatment assignment, which we randomize. In that case, we can use the randomization test of Theorem \[dtest\]. As discussed, we then do not require a group structure to control the type I error rate. We now discuss such a setting in detail. The tests considered here will also be studied with simulations in Section \[secsim\]. Let $n\geq 2$ be an integer, assumed even for convenience, and suppose we have $n$ subjects, $n/2$ of which receive one treatment and $n/2$ of which receive the other treatment. Let $Z=(Z_1,..,Z_n)$ encode the treatments and $Y=(Y_1,...,Y_n)$ the responses. The treatment pattern $Z$ is uniformly sampled from a set $ \mathcal{Z}\subseteq \{0,1\}^n$. In a standard randomized trial, $$\label{refset} \mathcal{Z}= \{z\in \{0,1\}^n: \text{ } z \text{ contains } n/2 \text{ 1's}\}$$ [@lachin1988statistical; @braun2001optimal]. For each $1\leq i \leq n$, the response $Y_i\in \mathbb{R}$ is independent of all other variables except (possibly) $Z_i$. We consider the null hypothesis $H_0$ that $Y$ is independent of $Z$. These assumptions are still rather general. It can be useful to consider a more specific randomization model as in @pitman1937significance [§7], who assumes an additive treatment effect. An important property of randomization models, is that to test whether the treatment has an effect on our particular patients, we do not need to assume that they are random draws from populations. We could consider the patients as fixed and $Y$ as constant, conditional on $Z$ [@pitman1937significance §7]. Indeed, “Any assumption that the units are, say, a random sample from a population of units \[...\] is additional to the specification" of the model [@cox2009randomization]. This property is discussed in detail in @onghena2018randomization and @rosenberger2019randomization. We can invoke Theorem \[dtest\] to obtain a test which controls the type I error rate. Indeed, one can take $D=Z$, $D'=Y$ and ${\mathcal{D}}= \mathcal{Z}$ and note that $D$ is uniformly distributed on ${\mathcal{D}}$. We can also obtain an exact test, i.e., a test that rejects with probability exactly $\alpha$ under $H_0$. Consider the test statistic $T: \mathcal{Z}\times \mathbb{R}^n\rightarrow \mathbb{R}$ that satisfies $$\label{te1} T(Z,Y)=\sum_{\{i: Z_i=1\}}Y_i - \sum_{\{i: Z_i=0\}}Y_i.$$ Recall that $Y$ may be viewed as random or constant, conditional on $Z$. In either case, assume that $Y$ is such that (with probability $1$), for all distinct $z_1, z_2\in \mathcal{Z}$, $T(z_1,Y)\neq T(z_2,Y)$. This is satisfied in particular if $Y_1,...,Y_n$ have continuous distributions. The test is exact if $\alpha\in(0,1)$ is a multiple of $1/\|\mathcal{Z}\|$, where $\|\mathcal{Z}\|$ equals $$N:=\binom{n}{n/2} = \frac{n!}{(n/2)!(n/2)!}.$$ An exact p-value is $$\label{pvrtest} p(Z,Y)=\frac{\|\{z\in \mathcal{Z}: T(z,Y)\geq T(Z,Y) \}\| }{ \|\mathcal{Z}\|},$$ i.e., if $\alpha\in(0,1)$ is a multiple of $1/\|\mathcal{Z}\|$, then $\mathbb{P}(p\leq \alpha)=\alpha$ under $H_0$. A two-sided exact test can be obtained analogously. Since Theorem \[dtest\] applies, the test essentially does not rely on a group structure. Hence, we may consider sampling $Z$ from a set which does not correspond to a group. In a different context, this is also done in @onghena1994randomization, where a set of permutations is used that is strictly smaller than the full set of permutations. This is done to avoid too repetitive treatment patterns such as ABBBBAAA. In our setting, if $n=8$, instead of taking $\mathcal{Z}$ to be the set of all permutations of $(0,0,0,0,1,1,1,1)$ we could sample $Z$ from a subset of these permutations which does correspond to a group, and still obtain an exact test (for certain $\alpha$). As @onghena1994randomization illustrates, this may be useful in some settings. However, in a typical clinical trial there is no evident reason to only use a subset of the permutations, except to limit the number of permutations for computational reasons. A more interesting alternative is to draw $Z$ from a set that is strictly larger than the set in , for example, from the set of all possible labelings, $\{0,1\}^n$. Indeed, if the standard randomization test is used, the smallest possible p-value that can be obtained is $1/N$, due to the discreteness of the p-value. If $n=8$ for example, then $1/N=1/70$. This means that if the significance level is $\alpha=0.01$ for instance, we have a power of 0 to reject $H_0$. Such small $\alpha$ are often used nowadays, for example due to multiple testing. The discreteness of the permutation p-value is a well-known downside of permutation-based tests [@berger2000pros]. If we take $\mathcal{Z}=\{0,1\}^n$, however, then $\|\mathcal{Z}\|=2^8$, so that the smallest possible p-value is $1/2^8=1/256$. If $1/256\leq \alpha<1/70$, this means a uniform improvement in power over the standard randomization test. Under $H_0$, if $\alpha$ is a multiple of $1/2^n$, the test with $\mathcal{Z}=\{0,1\}^n$ rejects with probability exactly $\alpha$. Otherwise the test rejects with probability less than $\alpha$ under $H_0$. For $\mathcal{Z}=\{0,1\}^n$, to our knowledge it is not known what the optimal choice of $T$ is for testing an additive treatment effect. In Section \[secsim\] we will take $$\label{te2} T(Z,Y)=\sum_{\{i: Z_i=1\}}(Y_i-\overline{Y}) - \sum_{\{i: Z_i=0\}}(Y_i-\overline{Y}),$$ where $\overline{Y}=n^{-1}(Y_1+...+Y_n)$. Using this test statistic ensures that under $H_0$, the expected value of $T(Z,Y)$ does not depend on the random labelling $Z$. That it is possible to take $\mathcal{Z}=\{0,1\}^n$ has been noted by several authors [@pocock1979allocation; @kalish1985treatment; @lachin1988properties; @wei1988properties; @suresh2011overview; @rosenberger2019randomization]. They do not recommend this approach, but merely mention it as a possibility, while focusing on more common randomization schemes. Their main argument against taking $\mathcal{Z}=\{0,1\}^n$ seems to be that it is “inefficient" [@pocock1979allocation p.188]. While this is true when $\alpha$ is large enough, the opposite is true when $\alpha$ is rather small. The idea that using $\mathcal{Z}=\{0,1\}^n$ leads to higher-resolution p-values, is not mentioned by these authors. Nowadays, the use of large multiple testing corrections is more common than in the past, so higher-resolution, exact p-values can clearly be of interest. Suppose we use $\mathcal{Z}=\{0,1\}^n$. Then, if we happen to draw $Z=(0,...,0)$ or $Z=(1,...,1)$, the value of the statistic is $0$ and we can have no hope of rejecting $H_0$ (if $\alpha=0.05$). Hence we might exclude $(0,...,0)$ and $(1,...,1)$, and perhaps more elements, from $\mathcal{Z}$. We leave the question of how to choose $\mathcal{Z}$ for future research. In any case, if $\alpha<1/N$, it can be useful to consider a test with $\|\mathcal{Z}\|$ larger than $N$. Note that in practice, we should choose $\mathcal{Z}$ before administering the treatments. Once the treatments have been given, we cannot change our minds about $\mathcal{Z}$. The test based on $\mathcal{Z}=\{0,1\}^n$ is further studied with simulations in Section \[secsim\]. Randomization testing under a random sampling model --------------------------------------------------- For completeness we note the following, but it can be skipped at a first read. Existing permutation tests based on random sampling from populations rely on a group structure. However, in some cases, we can use an alternative approach to avoid the requirement of a group structure also in this setting. The approach is analogous to the test in Section \[highrescc\]. Suppose that we are comparing two populations, for example, a population of cases and a population of controls, or Englishmen and Frenchmen. Let $Z$ be uniformly distributed on $\mathcal{Z}=\{0,1\}^n$ or some subset thereof, as before. Then we could draw from the two populations as indicated by $Z$, i.e., for every $1\leq i \leq n$, the $i$-th individual is drawn from the first population if $Z_i=0$ and from the second population if $Z_i=1$. For $1\leq i \leq n$, let $Y_i$ be the observation for the $i$-th individual, for example his or her stature. We can then perform a test exactly as in Section \[highrescc\], using the test statistic and the p-value . If we take $\mathcal{Z}$ as in , then the test will be equivalent to a standard permutation test. For many other choices of $\mathcal{Z}$, we obtain a novel type of test. If we take for instance $\mathcal{Z}=\{0,1\}^n$, then the number of observations drawn from each population will be random, with only the total number of observations being fixed at $n$. In many situations this would be impractical, for example because there is only a limited, fixed number of cases. We will not pursue such tests further here. Empirical example {#secsim} ================= Here we illustrate the idea in Section \[highrescc\] with a simple simulation study. We considered the two tests in Section \[highrescc\]: a standard randomization test and the alternative test that provides higher-resolution p-values. The data were as in the example in Section \[highrescc\], with $n=8$. Every $Y_i$ was distributed as the absolute value of a $N(0,1)$ variable if $Z_i=0$; if $Z_i=1$ it had the same distribution, but with an increase in mean of $\eta\geq 0$. Under the null hypothesis, $\eta=0$. The first test considered was the standard randomization test. This test uses $N=(n!)/((n/2)!(n/2)!)=70$ permutations. The second test was the one based on all $2^n=256$ relabellings in $\{0,1\}^n$. We used the test statistic . By Theorem \[dtest\], both tests control the type I error rate. Moreover, the first test is exact if $\alpha\in(0,1)$ is a multiple of $1/70$. The second test is exact if $\alpha$ is multiple of $1/256$. In Table \[table:2tests\], for different values of the significance level $\alpha$, the estimated level and power of the two tests are shown. Every estimate in the table is based on $10^4$ repeated simulations. The regular randomization test had no power for $\alpha<1/70$, due to the fact that only 70 relabellings are available with this approach. The test based on all 256 relabellings in $\{0,1\}^n$, however, did have substantial power, as explained in Section \[highrescc\]. In the table, the estimated size for $\alpha=1/256$ is $0.0041$, which is approximately the true size $1/256$. Note that for $\alpha=0.005$, the size and power are the same as for $\alpha=1/256$. The reason is the discreteness of the p-value: $0.005$ lies between $1/256$ and $2/256$. [ l l l l l l l l ]{}\ & &\ & test & 1/256 & .005 & .01 & .02 & .05\ size & test 1 & 0& 0 & 0& .0122 & .0418\ & test 2 & .0041& .0041 & .0070 & .0188 & .0456\ power & test 1 & 0& 0 & 0 & .9011 & .9725\ & test 2 & .5415& .5415 & .7010 & .8463 & .9257\ \[table:2tests\] Discussion ========== In this paper, we have distinguished between two types of permutation-based tests: tests which fundamentally rely on a group structure and tests based on treatment randomization, which do not require a group structure. We have discussed that in settings where treatments are randomly assigned, it can be useful to consider a randomization scheme which does not correspond to a group. In particular, this allows obtaining higher-resolution exact p-values than are possible with standard randomization tests. This paper also provides the caveat that referring to the Lady Tasting Tea experiment as an example of a permutation test can be misleading, since the reasoning underlying this experiment is not based on a group structure. The two types of tests between which we distinguish roughly correspond to respectively “permutation tests" and “randomization tests" in the sense of @onghena2018randomization and @rosenberger2019randomization. As we mentioned, the use of these terms has been rather inconsistent throughout the literature. For example, @edgington2007randomization [p.1] write that “randomization tests are a subclass of statistical tests called permutation tests", while @onghena2018randomization proposes to use the terms for strictly distinct classes of tests. In any case, we propose to use the term “randomization tests" only when there is some form of treatment randomization. This is in line with @kempthorne1969behaviour, @edgington2007randomization, @onghena2018randomization and @rosenberger2019randomization. The purpose of this paper has not been to identify the first permutation test, which would not be straightforward [@berry2014chronicle]. In any case, it is clear that, once the concepts of randomization of treatments and random sampling from populations had been established in the 1920’s [@rubin1990comment; @fisher1925statistical; @neyman1928use], the way was paved for the theoretical development of permutation-based tests. However, until the 1980’s, there was limited interest in permutation-based procedures, due to lack of access to fast computers. Nowadays, the opposite is true [@albajes2019voxel; @hemerik2019permutation; @rao2019permutation]. [^1]: Oslo Centre for Biostatistics and Epidemiology, University of Oslo [^2]: Address for correspondence: Jesse Hemerik, Oslo Centre for Biostatistics and Epidemiology, P.O. Box 1122 Blindern, 0317 Oslo, Norway. e-mail: jesse.hemerik@medisin.uio.no [^3]: Biomedical Data Sciences, Leiden University Medical Center, Einthovenweg 20, 2333 ZC Leiden, The Netherlands
--- abstract: \| Under the key assumption of finite $\rho $-variation, $\rho \in \lbrack 1,2)$, of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), $\rho =1$ resp. $\rho =1/\left( 2H\right) $, we recover and extend the respective results of \[Hu–Nualart; Rough path analysis via fractional calculus; TAMS 361 (2009) 2689-2718\] and \[Deya–Neuenkirch–Tindel; A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion; AIHP (2011)\]. In particular, we establish an a.s. rate $k^{-\left( 1/\rho -1/2-\varepsilon \right) }$, any $\varepsilon >0 $, for Wong-Zakai and Milstein-type approximations with mesh-size $1/k$. When applied to fBM this answers a conjecture in the afore-mentioned references. address: - \| TU Berlin, Fakultät II\ Institut für Mathematik, MA 7-2\ Strasse des 17. Juni 136\ 10623 Berlin\ Germany - \| Weierstrass Institut für Angewandte Analysis und Stochastik\ Mohrenstrasse 39\ 10117 Berlin\ Germany - \| TU Berlin, Fakultät II\ Institut für Mathematik, MA 7-4\ Straße des 17. Juni 136\ 10623 Berlin\ Germany author: - Peter Friz - Sebastian Riedel title: Convergence rates for the full Gaussian rough paths --- [^1] [^2] Introduction ============ Recall that rough path theory [@L98; @LQ02; @FV10] is a general framework that allows to establish existence, uniqueness and stability of differential equations driven by multi-dimensional continuous signals $% x\colon \left[ 0,T\right] \rightarrow \mathbb{R}^{d}$ of low regularity. Formally, a rough differential equation (RDE) is of the form$$dy_{t}=\sum_{i=1}^{d}V_{i}\left( y_{t}\right) \,dx_{t}^{i}\equiv V\left( y_{t}\right) \,dx_{t};\quad y_{0}\in \mathbb{R}^{e} \label{eqn_determ_rde}$$where $\left( V_{i}\right) _{i=1,\ldots ,d}$ is a family of vector fields in $\mathbb{R}^{e}$. When $x$ has finite $p$-variation, $p<2$, such differential equations can be handled by Young integration theory. Of course, this point of view does not allow to handle differential equations driven by Brownian motion, indeed$$\sup_{D\subset \left[ 0,T\right] }\sum_{t_{i}\in D}\left\vert B_{t_{i+1}}-B_{t_{i}}\right\vert ^{2}=+\infty \text{ a.s.,}$$leave alone differential equations driven by stochastic processes with less sample path regularity than Brownian motion (such as fractional Brownian motion (fBM) with Hurst parameter $H<1/2$). Lyons’ key insight was that low regularity of $x$, say $p$-variation or $1/p$-Hölder for some $p\in \lbrack 1,\infty )$, can be compensated by including “enough” higher order information of $x$ such as all increments $$\begin{aligned} \mathbf{x}_{s,t}^{n} &\equiv &\int_{s<t_{1}<\dots <t_{n}<t}dx_{t_{1}}\otimes \ldots \otimes dx_{t_{n}} \label{IIIntro} \\ &\equiv &\sum_{1\leq i_{1},\ldots ,i_{n}\leq d}\left( \int_{s<t_{1}<\dots <t_{n}<t}dx_{t_{1}}^{i_{1}}\ldots dx_{t_{n}}^{i_{n}}\right) \,e_{i_{1}}\otimes \ldots \otimes e_{i_{n}}\in \left( \mathbb{R}^{d}\right) ^{\otimes n}\end{aligned}$$where “enough” means $n\leq \left[ p\right] $ ($\left\{ e_{1},\ldots ,e_{d}\right\} $ denotes just the usual Euclidean basis in $\mathbb{R}^{d}$ here). Subject to some generalized $p$-variation (or $1/p$-Hölder) regularity, the ensemble $\left( \mathbf{x}^{1},\dots ,\mathbf{x}^{\left[ p% \right] }\right) $ then constitutes what is known as a rough path.[^3] In particular, no higher order information is necessary in the Young case; whereas the regime relevant for Brownian motion requires second order - or level $2$ - information (“Lévy’s area”), and so on. Note that the iterated integral on the r.h.s. of (\[IIIntro\]) is not - in general - a well-defined Riemann-Stieltjes integral. Instead one typically proceeds by mollification - given a multi-dimensional sample path $x=X\left( \omega \right) $, consider piecewise linear approximations or convolution with a smooth kernel, compute the iterated integrals and then pass, if possible, to a limit in probability. Following this strategy one can often construct a “canonical” enhancement of some stochastic process to a (random) rough path. Stochastic integration and differential equations are then discussed in a (rough) pathwise fashion; even in the complete absence of a semi-martingale structure. It should be emphasized that rough path theory was - from the very beginning - closely related to higher order Euler schemes. Let $D=\left\{ 0=t_{0}<\ldots <t_{\#D-1}=1\right\} $ be a partition of the unit interval.[^4] Considering the solution $y$ of $\left( \ref% {eqn_determ_rde}\right) $, the step-$N$ Euler approximation $y^{\text{Euler}% ^{N};D}$ is given by$$\begin{aligned} y_{0}^{\text{Euler}^{N};D} &=&y_{0} \\ y_{t_{j+1}}^{\text{Euler}^{N};D} &=&y_{t_{j}}^{\text{Euler}% ^{N};D}+V_{i}\left( y_{t_{j}}^{\text{Euler}^{N};D}\right) \mathbf{x}% _{t_{j},t_{j+1}}^{i}+\mathcal{V}_{i_{1}}V_{i_{2}}\left( y_{t_{j}}^{\text{% Euler}^{N};D}\right) \mathbf{x}_{t_{j},t_{j+1}}^{i_{1},i_{2}} \\ &&+\ldots +\mathcal{V}_{i_{1}}\mathcal{\ldots V}_{i_{N-1}}V_{i_{N}}\left( y_{t_{j}}^{\text{Euler}^{N};D}\right) \mathbf{x}_{t_{j},t_{j+1}}^{i_{1},% \ldots ,i_{N}}\end{aligned}$$at the points $t_{j}\in D$ where we use the Einstein summation convention, $% \mathcal{V}_{i}$ stands for the differential operator $% \sum_{k=1}^{e}V_{i}^{k}\partial _{x_{k}}$ and $\mathbf{x}_{s,t}^{i_{1},% \ldots ,i_{n}}=\int_{s<t_{1}<\dots <t_{n}<t}dx_{t_{1}}^{i_{1}}\ldots dx_{t_{n}}^{i_{n}}$. An extension of the work of A.M. Davie (cf. [@D07], [@FV10]) shows that the step-$N$ Euler scheme[^5] for an RDE driven by a $1/p$-Hölder rough path with step size $1/k$ (i.e. $D=D_{k}=\left\{ \frac{j}{k}:j=0,\ldots ,k\right\} $) and $N\geq \left[ p\right] $ will converge with rate $O\left( \frac{1}{k}\right) ^{\left( N+1\right) /p-1}$. Of course, in a probabilistic context, simulation of the iterated (stochastic) integrals $\mathbf{x}_{t_{j},t_{j+1}}^{n}$ is not an easy matter. A natural simplification of the step-$N$ Euler scheme thus amounts to replace in each step$$\left\{ \mathbf{x}_{t_{j},t_{j+1}}^{n}:n\in \left\{ 1,\dots ,N\right\} \right\} \text{ }\leftrightarrow \text{ }\left\{ \frac{1}{n!}\left( \mathbf{x% }_{t_{j},t_{j+1}}^{1}\right) ^{\otimes n}:n\in \left\{ 1,\dots ,N\right\} \right\}$$which leads to the simplified step-$N$ Euler scheme$$\begin{aligned} y_{0}^{\text{sEuler}^{N};D} &=&y_{0} \\ y_{t_{j+1}}^{\text{sEuler}^{N};D} &=&y_{t_{j}}^{\text{sEuler}% ^{N};D}+V_{i}\left( y_{t_{j}}^{\text{sEuler}^{N};D}\right) \mathbf{x}% _{t_{j},t_{j+1}}^{i}+\frac{1}{2}\mathcal{V}_{i_{1}}V_{i_{2}}\left( y_{t_{j}}^{\text{sEuler}^{N};D}\right) \mathbf{x}_{t_{j},t_{j+1}}^{i_{1}}% \mathbf{x}_{t_{j},t_{j+1}}^{i_{2}} \\ &&+\ldots +\frac{1}{N!}\mathcal{V}_{i_{1}}\mathcal{\ldots V}% _{i_{N-1}}V_{i_{N}}\left( y_{t_{j}}^{\text{sEuler}^{N};D}\right) \mathbf{x}% _{t_{j},t_{j+1}}^{i_{1}}\ldots \mathbf{x}_{t_{j},t_{j+1}}^{i_{N}}.\end{aligned}$$Since $\mathbf{x}_{t_{j},t_{j+1}}^{1}=X_{t_{j},t_{j+1}}\left( \omega \right) =X_{t_{j+1}}\left( \omega \right) -X_{t_{j}}\left( \omega \right) $ this is precisely the effect in replacing the underlying sample path segment of $X$ by its piecewise linear approximation, i.e.$$\left\{ X_{t}\left( \omega \right) :t\in \left[ t_{j},t_{j+1}\right] \right\} \text{ }\leftrightarrow \left\{ X_{t_{j}}\left( \omega \right) +% \frac{t-t_{j}}{t_{j+1}-t_{j}}X_{t_{j},t_{j+1}}\left( \omega \right) :t\in % \left[ t_{j},t_{j+1}\right] \right\} .$$Therefore, as pointed out in [@DT] in the level $N=2$ Hölder rough path context, it is immediate that a Wong-Zakai type result, i.e. a.s. convergence of $y^{\left( k\right) }\rightarrow y$ for $k\rightarrow \infty $ where $y^{\left( k\right) }$ solves$$dy_{t}^{\left( k\right) }=V\left( y_{t}^{\left( k\right) }\right) \,dx_{t}^{\left( k\right) };\quad y_{0}^{\left( k\right) }=y_{0}\in \mathbb{R% }^{e}$$and $x^{\left( k\right) }$ is the piecewise linear approximation of $x$ at the points $\left( t_{j}\right) _{j=0}^{k}=D_{k}$, i.e.$$x_{t}^{\left( k\right) }=x_{t_{j}}+\frac{t-t_{j}}{t_{j+1}-t_{j}}% x_{t_{j},t_{j+1}}\text{\quad if }t\in \left[ t_{j},t_{j+1}\right] \text{, }% t_{j}\in D_{k},$$leads to the convergence of the simplified (and implementable!) step-$N$ Euler scheme. While Wong-Zakai type results in rough path metrics are available for large classes of stochastic processes [@FV10 Chapter 13, 14, 15, 16] our focus here is on Gaussian processes which can be enhanced to rough paths. This problem was first discussed in [@CQ02] where it was shown in particular that piecewise linear approximation to fBM are convergent in $p$-variation rough path metric if and only if $H>1/4$. A practical (and essentially sharp) structural condition for the covariance, namely finite $% \rho $-variation based on rectangular increments for some $\rho <2$ of the underlying Gaussian process was given in [@FV10AIHP] and allowed for a unified and detailed analysis of the resulting class of Gaussian rough paths. This framework has since proven useful in a variety of different applications ranging from non-Markovian Hörmander theory [@CF10] to non-linear PDEs perturbed by space-time white-noise [@Hai10]. Of course, fractional Brownian motion can also be handled in this framework (for $H>1/4$) and we shall make no attempt to survey its numerous applications in engineering, finance and other fields. Before describing our main result, let us recall in more detail some aspects of Gaussian rough path theory (e.g. [@FV10AIHP], [@FV10 Chapter 15], [@FV11]). The basic object is a centred, continuous Gaussian process with sample paths $X\left( \omega \right) =\left( X^{1}\left( \omega \right) ,\ldots ,X^{d}\left( \omega \right) \right) \colon \left[ 0,1\right] \rightarrow $ $\mathbb{R}^{d}$ where $X^{i}$ and $X^{j}$ are independent for $i\neq j$. The law of this process is determined by $R_{X}\colon \left[ 0,1% \right] ^{2}\rightarrow \mathbb{R}^{d\times d}$, the covariance function, given by$$R_{X}\left( s,t\right) =\text{diag}\left( E\left( X_{s}^{1}X_{t}^{1}\right) ,\ldots ,E\left( X_{s}^{d}X_{t}^{d}\right) \right) .$$We need \[def\_2D\_grid\_variation\]Let $f=f\left( s,t\right) $ be a function from $% \left[ 0,1\right] ^{2}$ into a normed space; for $s\leq t,u\leq v$ we define rectangular increments as$$f\left( \begin{array}{c} s,t \\ u,v% \end{array}% \right) =f\left( t,v\right) -f\left( t,u\right) -f\left( s,v\right) +f\left( s,u\right) .$$For $\rho \geq 1$ we then set$$V_{\rho }\left( f,\left[ s,t\right] \times \left[ u,v\right] \right) =\left( \sup_{\substack{ D\subset \left[ s,t\right] \\ \tilde{D}\subset \left[ u,v% \right] }}\sum_{\substack{ t_{i}\in D \\ \tilde{t}_{j}\in \tilde{D}}}% \left\vert f\left( \begin{array}{c} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{array}% \right) \right\vert ^{\rho }\right) ^{1/\rho }$$where the supremum is taken over all partitions $D$ and $\tilde{D}$ of the intervals $\left[ s,t\right] $ resp. $\left[ u,v\right] $. If $V_{\rho }(f,% \left[ 0,1\right] ^{2})<\infty $ we say that $f$ has finite ($2D$) $\rho $-variation. The main result in this context (see e.g. [@FV10 Theorem 15.33], [FV11]{}) now asserts that if there exists $\rho <2$ such that $V_{\rho }\left( R_{X},\left[ 0,1\right] ^{2}\right) <\infty $ then $X$ lifts to an enhanced Gaussian process $\mathbf{X}$ with sample paths in the $p$-variation rough path space $C^{0,p-var}\left( \left[ 0,1\right] ,G^{\left[ p% \right] }\left( \mathbb{R}^{d}\right) \right) $, any $p\in \left( 2\rho ,4\right) $. (This and other notations are introduced in section 2.) This lift is “natural” in the sense that for a large class of smooth approximations $X^{\left( k\right) }$ of $X$ (say piecewise linear, mollifier, Karhunen-Loeve) the corresponding iterated integrals of $% X^{\left( k\right) }$ converge (in probability) to $\mathbf{X}$ with respect to the $p$-variation rough path metric. (We recall from [@FV10] that $% \rho _{p\text{-var}}$, the so-called inhomogeneous $p$-variation metric for $% G^{N}\left( \mathbb{R}^{d}\right) $-valued paths, is called $p$-variation rough path metric when $\left[ p\right] =N$; the Itō-Lyons map enjoys local Lipschitz regularity in this $p$-variation rough path metric.) Moreover, this condition is sharp; indeed fBM falls into this framework with $\rho =1/\left( 2H\right) $ and we known that piecewise-linear approximations to Lévy’s area diverge when $H=1/4$. Our main result (cf. Theorem \[theorem\_main01\]), when applied to (mesh-size $1/k$) piecewise linear approximations $X^{\left( k\right) }$ of $% X$, reads as follows. \[theorem\_main01\_intro\]Let $X=\left( X^{1},\ldots ,X^{d}\right) \colon % \left[ 0,1\right] \rightarrow \mathbb{R}^{d}$ be a centred Gaussian process on a probability space $\left( \Omega ,\mathcal{F},P\right) $ with continuous sample paths where $X^{i}$ and $X^{j}$ are independent for $i\neq j$. Assume that the covariance $R_{X}$ has finite $\rho $-variation for $% \rho \in \lbrack 1,2)$ and $K\geq V_{\rho }\left( R_{X},\left[ 0,1\right] ^{2}\right) $. Then there is an enhanced Gaussian process $\mathbf{X}$ with sample paths a.s. in $C^{0,p-var}\left( \left[ 0,1\right] ,G^{\left[ p\right] }\left( \mathbb{R}^{d}\right) \right) $ for any $p\in \left( 2\rho ,4\right) $ and$\in $$$\left\vert \rho _{p-var}\left( S_{\left[ p\right] }\left( X^{\left( k\right) }\right) ,\mathbf{X}\right) \right\vert _{L^{r}}\rightarrow 0$$for $k\rightarrow \infty $ and every $r\geq 1$ ($\left\vert \cdot \right\vert _{L^{r}}$ denotes just the usual $L^{r}\left( P\right) $-norm for real valued random variables here). Moreover, for any $\gamma >\rho $ such that $\frac{1}{\gamma }+\frac{1}{\rho }>1$ and any $q>2\gamma $ and $% N\in \mathbb{N}$ there is a constant $C=C\left( q,\rho ,\gamma ,K,N\right) $ such that$$\left\vert \rho _{q-var}\left( S_{N}\left( X^{\left( k\right) }\right) ,S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}}\leq Cr^{N/2}\sup_{0\leq t\leq 1}\left\vert X_{t}^{\left( k\right) }-X_{t}\right\vert _{L^{2}}^{1-\frac{\rho }{\gamma }}$$holds for every $k\in \mathbb{N}$. As an immediate consequence we obtain (essentially) sharp a.s. convergence rates for Wong-Zakai approximations and the simplified step-$3$ Euler scheme. Consider a RDE with $C^{\infty }$-bounded vector fields driven by a Gaussian Hölder rough path $\mathbf{X}$. Then mesh-size $1/k$ Wong-Zakai approximations (i.e. solutions of ODEs driven by $X^{\left( k\right) }$) converge uniformly with a.s. rate $k^{-\left( 1/\rho -1/2-\varepsilon \right) }$, any $\varepsilon >0$, to the RDE solution. The same rate is valid for the simplified (and implementable) step-$3$ Euler scheme. See Corollary \[cor\_wong\_zakai\_piecew\_lin\] and Corollary [Cor\_rate\_simple\_euler]{}. Several remarks are in order. - Rough path analysis usually dictates that $N=2$ (resp. $N=3$) levels need to be considered when $\rho \in \lbrack 1,3/2)$ resp. $\rho \in \lbrack 3/2,2)$. Interestingly, the situation for the Wong-Zakai error is quite different here - referring to Theorem \[theorem\_main01\_intro\], when $\rho =1$ we can and will take $\gamma $ arbitrarily large in order to obtain the optimal convergence rate. Since $\rho _{q-\text{var}}$ is a rough path metric only in the case $N=\left[ q\right] \geq \left[ 2\gamma \right] $, we see that we need to consider all levels $N$ which is what Theorem [theorem\_main01\_intro]{} allows us to do. On the other hand, as $\rho $ approaches $2$, there is not so much room left for taking $\gamma >\rho $. Even so, we can always find $\gamma $ with $\left[ \gamma \right] =2$ such that $1/\gamma +1/\rho >1$. Picking $q>2\gamma $ small enough shows that we need $N=\left[ q\right] =4$. - The assumption of $C^{\infty }$-bounded vector fields in the corollary was for simplicity only. In the proof we employ local Lipschitz continuity of the Itō-Lyons map for $q$-variation rough paths (involving $N=\left[ q% \right] $ levels). As is well-known, this requires $\mathrm{Lip}% ^{q+\varepsilon }$-regularity of the vector fields[^6]. Curiously again, we need $C^{\infty }$-bounded vector fields when $\rho =1$ but only $% \mathrm{Lip}^{4+\varepsilon }$ as $\rho $ approaches the critical value $2$. - Brownian motion falls in this framework with $\rho =1$. While the a.s. (Wong-Zakai) rate $k^{-\left( 1/2-\varepsilon \right) }$ is part of the folklore of the subject (e.g. [@GS06]) the $C^{\infty }$-boundedness assumption appears unnecessarily strong. Our explanation here is that our rates are universal (i.e. valid away from one universal null-set, not dependent on starting points, coefficients etc). In particular, the (Wong-Zakai) rates are valid on the level of stochastic flows of diffeomorphisms; we previously discussed these issues in the Brownian context in [@FR11]. - A surprising aspect appears in the proof of theorem [theorem\_main01\_intro]{}. The strategy is to give sharp estimates for the levels $n=1,\ldots ,4$ first, then performing an induction similar to the one used in Lyon’s Extension Theorem ([@L98]) for the higher levels. This is in contrast to the usual considerations of level $1$ to $3$ only (without level $4$!) which is typical for Gaussian rough paths. (Recall that we deal with Gaussian processes which have sample paths of finite $p$-variation, $p\in (2\rho ,4)$, hence $\left[ p\right] \leq 3$ which indicates that we would need to control the first $3$ levels only before using the Extension Theorem.) - Although theorem \[theorem\_main01\_intro\] was stated here for (step-size $1/k$) piecewise linear approximations $\left\{ X^{\left( k\right) }\right\} $, the estimate holds in great generality for (Gaussian) approximations whose covariance satisfies a uniform $\rho $-variation bound. The statements of Theorem \[theorem\_main01\] and Theorem [theorem\_as\_wong\_zakai\_rate]{} reflect this generality. - Wong-Zakai rates for the Brownian rough path (level $2$) were first discussed in [@HN08]. They prove that Wong-Zakai approximations converge (in $\gamma $-Hölder metric) with rate $k^{-\left( 1/2-\gamma -\varepsilon \right) }$ (in fact, a logarithmic sharpening thereof without $% \varepsilon $) provided $\gamma \in \left( 1/3,1/2\right) $. This restriction on $\gamma $ is serious (for they fully rely on “level $2$” rough path theory); in particular, the best “uniform” Wong-Zakai convergence rate implied is $k^{-\left( 1/2-1/3-\varepsilon \right) }=k^{-\left( 1/6-\varepsilon \right) }$ leaving a significant gap to the well-known Brownian a.s. Wong-Zakai rate. - Wong-Zakai (and Milstein) rates for the fractional Brownian rough path (level $2$ only, Hurst parameter $H>1/3$) were first discussed in [@DT]. They prove that Wong-Zakai approximations converge (in $\gamma $-Hölder metric) with rate $k^{-\left( H-\gamma -\varepsilon \right) }$ (again, in fact, a logarithmic sharpening thereof without $\varepsilon $) provided $% \gamma \in \left( 1/3,H\right) $. Again, the restriction on $\gamma $ is serious and the best “uniform” Wong-Zakai convergence rate - and the resulting rate for the Milstein scheme - is $k^{-\left( H-1/3-\varepsilon \right) }$. This should be compared to the rate $k^{-\left( 2H-1/2-\varepsilon \right) }$ obtained from our corollary. In fact, this rate was conjectured in [@DT] and is sharp as may be seen from a precise result concerning Levy’s stochastic area for fBM, see [@NTU10]. The remainder of the article is structured as follows: In Section [section\_notations]{}, we repeat the basic notions of (Gaussian) rough paths theory. Section \[section\_it\_int\_and\_shuffle\] recalls the connection between the shuffle algebra and iterated integrals. In particular, we will use the shuffle structure to see that in order to show the desired estimates, we can concentrate on some iterated integrals which somehow generate all the others. Our main tool for showing $L^{2}$ estimates on the lower levels is multidimensional Young integration which we present in Section \[section\_multidim\_young\]. The main work, namely showing the desired $L^{2}$-estimates for the difference of high-order iterated integrals, is done in Section \[section\_main\_estimates\]. After some preliminary Lemmas in Subsection \[subsection\_special\_cases\], we show the estimates for the lower levels, namely for $n=1,2,3,4$ in Subsection [subsection\_lower\_levels]{} , then give an induction argument in Subsection \[subsection\_higher\_levels\] for the higher levels $n>4$. Section [section\_main\_result]{} contains our main result, namely sharp a.s. convergence rates for a class of Wong-Zakai approximations, including piecewise-linear and mollifier approximations. We further show in Subsection \[subsection\_simple\_euler\] how to use these results in order to obtain sharp convergence rates for the simplified Euler scheme. Notations and basic definitions\[section\_notations\] ===================================================== For $N\in \mathbb{N}$ we define$$T^{N}\left( \mathbb{R}^{d}\right) =\mathbb{R}\oplus \mathbb{R}^{d}\oplus \left( \mathbb{R}^{d}\otimes \mathbb{R}^{d}\right) \oplus \ldots \oplus \left( \mathbb{R}^{d}\right) ^{\otimes N}=\oplus _{n=0}^{N}\left( \mathbb{R}% ^{d}\right) ^{\otimes n}$$and write $\pi _{n}:T^{N}\left( \mathbb{R}^{d}\right) \rightarrow \left( \mathbb{R}^{d}\right) ^{\otimes n}$ for the projection on the $n$-th Tensor level. It is clear that $T^{N}\left( \mathbb{R}^{d}\right) $ is a (finite-dimensional) vector space. For elements $g,h\in T^{N}\left( \mathbb{R% }^{d}\right) $, we define $g\otimes h\in T^{N}\left( \mathbb{R}^{d}\right) $ by$$\pi _{n}\left( g\otimes h\right) =\sum_{i=0}^{n}\pi _{n-i}\left( g\right) \otimes \pi _{i}\left( h\right) .$$One can easily check that $\left( T^{N}\left( \mathbb{R}^{d}\right) ,+,\otimes \right) $ is an associative algebra with unit element $\mathbf{1}% =\exp \left( 0\right) =1+0+0+\ldots +0$ . We call it the truncated tensor algebra of level $N$. A norm is defined by$$\left\vert g\right\vert _{T^{N}\left( \mathbb{R}^{d}\right) }=\max_{n=0,\ldots ,N}\left\vert \pi _{n}\left( g\right) \right\vert$$which turns $T^{N}\left( \mathbb{R}^{d}\right) $ into a Banach space. For $s<t$, we define$$\Delta _{s,t}^{n}=\left\{ \left( u_{1},\ldots ,u_{n}\right) \in \left[ s,t% \right] ^{n}~;~u_{1}<\ldots <u_{n}\right\}$$which is the $n$-simplex on the square $\left[ s,t\right] ^{n}$. We will use $\Delta =\Delta _{0,1}^{2}$ for the $2$-simplex over $\left[ 0,1\right] ^{2}$. A continuous map $\mathbf{x\colon }\Delta \rightarrow T^{N}\left( \mathbb{R% }^{d}\right) $ is called multiplicative functional if for all $s<u<t$ one has $\mathbf{x}_{s,t}=\mathbf{x}_{s,u}\mathbf{\otimes x}_{u,t}.$For a path $x=\left( x^{1},\ldots ,x^{d}\right) \colon \left[ 0,1\right] \rightarrow \mathbb{R}^{d}$ and $s<t$, we will use the notation $% x_{s,t}=x_{t}-x_{s}$. If $x$ has finite variation, we define its $n$-th iterated integral by$$\begin{aligned} \mathbf{x}_{s,t}^{n} &=&\int_{\Delta _{s,t}^{n}}dx\otimes \ldots \otimes dx \\ &=&\sum_{1\leq i_{1},\ldots ,i_{n}\leq d}\int_{\Delta _{s,t}^{n}}dx^{i_{1}}\ldots dx^{i_{n}}e_{i_{1}}\otimes \ldots \otimes e_{i_{n}}\in \left( \mathbb{R}^{d}\right) ^{\otimes n}\end{aligned}$$where $\left\{ e_{1},\ldots ,e_{d}\right\} $ denotes the Euclidean basis in $% \mathbb{R}^{d}$ and $\left( s,t\right) \in \Delta $. The canonical lift $% S_{N}\left( x\right) \colon \Delta \rightarrow T^{N}\left( \mathbb{R}% ^{d}\right) $ is defined by$$\pi _{n}\left( S_{N}\left( x\right) _{s,t}\right) =\left\{ \begin{array}{ccc} \mathbf{x}_{s,t}^{n} & \text{if} & n\in \left\{ 1,\ldots ,N\right\} \\ 1 & \text{if} & n=0.% \end{array}% \right.$$It is well know (as a consequence of Chen’s theorem) that $S_{N}\left( x\right) $ is a multiplicative functional. Actually, one can show that $% S_{N}\left( x\right) $ takes values in the smaller set $G^{N}\left( \mathbb{R% }^{d}\right) \subset T^{N}\left( \mathbb{R}^{d}\right) $ defined by$$G^{N}\left( \mathbb{R}^{d}\right) =\left\{ S_{N}\left( x\right) _{0,1}:x\in C^{1-var}\left( \left[ 0,1\right] ,\mathbb{R}^{d}\right) \right\}$$which is still a group with $\otimes $. If $\mathbf{x},\mathbf{y\colon }% \Delta \rightarrow T^{N}\left( \mathbb{R}^{d}\right) $ are multiplicative functionals and $p\geq 1$ we set $$\rho _{p-var}\left( \mathbf{x},\mathbf{y}\right) :=\max_{n=1,\ldots ,N}\sup_{\left( t_{i}\right) \in \left[ 0,1\right] }\left( \sum_{i}\left\vert \mathbf{x}_{t_{i},t_{i+1}}^{n}-\mathbf{y}% _{t_{i},t_{i+1}}^{n}\right\vert ^{p/n}\right) ^{n/p}.$$This generalizes the $p$-variation distance induced by the usual $p$-variation semi-norm$$\left\vert x\right\vert _{p-var;\left[ s,t\right] }=\left( \sup_{\left( t_{i}\right) \subset \left[ s,t\right] }\sum_{i}\left\vert x_{t_{i+1}}-x_{t_{i}}\right\vert ^{p}\right) ^{1/p}$$for paths $x\colon \left[ 0,1\right] \rightarrow \mathbb{R}^{d}$. The Lie group $G^{N}\left( \mathbb{R}^{d}\right) $ admits a natural norm $\left\Vert \cdot \right\Vert $, called the Carnot-Caratheodory norm (cf. [@FV10 Chapter 7]). If $\mathbf{x\colon }\Delta \rightarrow G^{N}\left( \mathbb{R}^{d}\right) $, we set$$\left\Vert \mathbf{x}\right\Vert _{p-var;\left[ s,t\right] }=\left( \sup_{\left( t_{i}\right) \subset \left[ s,t\right] }\sum_{i}\left\Vert \mathbf{x}_{t_{i},t_{i+1}}\right\Vert ^{p}\right) ^{1/p}.$$ The space $C_{o}^{0,p-var}\left( \left[ 0,1\right] ,G^{N}\left( \mathbb{R}% ^{d}\right) \right) $ is defined as the set of continuous paths $% \mathbf{x\colon }\Delta \rightarrow G^{N}\left( \mathbb{R}^{d}\right) $ for which there exists a sequence of smooth paths $x_{k}\colon \left[ 0,1\right] \rightarrow $ $\mathbb{R}^{d}$ such that $\rho _{p-var}\left( \mathbf{x}% ,S_{N}\left( x_{k}\right) \right) \rightarrow 0~$for $k\rightarrow \infty $. If $N=\left[ p\right] =\max \left\{ n\in \mathbb{N}:n<p\right\} $ we call this the space of (geometric) $p$-rough paths. It is clear by definition that every $p$-rough path is also a multiplicative functional. By Lyon’s First Theorem (or Extension Theorem, see [@L98 Theorem 2.2.1] or [@FV10 Theorem 9.5]) every $p$-rough path $\mathbf{x}$ has a unique lift to a path in $G^{N}\left( \mathbb{R}^{d}\right) $ for $% N\geq \left[ p\right] $. We denote this lift by $S_{N}(\mathbf{x)}$ and call it the Lyons lift. For a $p$-rough path $\mathbf{x}$, we will also use the notation$$\mathbf{x}_{s,t}^{n}=\pi _{n}\left( S_{N}\left( \mathbf{x}\right) _{s,t}\right)$$for $N\geq n$. Note that this is consistent with our former definition in the case where $x$ had finite variation. We will always use small letters for paths $x$ and capital letters for stochastic processes $X$. The same notation introduced here will also be used for stochastic processes. A function $\omega \colon \Delta \rightarrow \mathbb{R}^{+}$ is called a $% \left( 1D\right) $ control if it is continuous and superadditive, i.e. if for all $s<u<t$ one has$$\omega \left( s,u\right) +\omega \left( u,t\right) \leq \omega \left( s,t\right) .$$ If $x\colon \left[ 0,1\right] \rightarrow \mathbb{R}^{d}$ is a continuous path with finite $p$-variation, one can show that$$\left( s,t\right) \mapsto V_{p}\left( x,\left[ s,t\right] \right) ^{p}:=\left\vert x\right\vert _{p-var;\left[ s,t\right] }^{p}$$is continuous and superadditive, hence defines a $1D$-control function. Unfortunately, this is not the case for higher dimensions. Recall Definition \[def\_2D\_grid\_variation\]. If $f\colon \left[ 0,1\right] ^{2}\rightarrow \mathbb{R}$ has finite $p$-variation,$$\left( s,t\right) ,\left( u,v\right) \mapsto V_{p}\left( f,\left[ s,t\right] \times \left[ u,v\right] \right) ^{p}$$in general fails to be superadditive (cf. [@FV11]). Therefore, we will need a second definition. If $A=\left[ s,t\right] \times \left[ u,v\right] $ is a rectangle in $\left[ 0,1\right] ^{2}$, we will use the notation $% f\left( A\right) :=f\left( \begin{array}{c} s,t \\ u,v% \end{array}% \right) $. We call two rectangles essentially disjoint if their intersection is empty or degenerate. A partition $\Pi $ of a rectangle $% R\subset \left[ 0,1\right] ^{2}$ is a finite set of essentially disjoint rectangles whose union is $R$. The family of all such partitions is denoted by $\mathcal{P}\left( R\right) $. \[def\_2D\_controls\]A function $\omega \colon \Delta \times \Delta \rightarrow \mathbb{R}^{+}$ is called a $\left( 2D\right) $ control if it is continuous, zero on degenerate rectangles and super-additive in the sense that for all rectangles $R\subset \left[ 0,1\right] ^{2}$,$$\sum_{i=1}^{n}\omega \left( R_{i}\right) \leq \omega \left( R\right)$$whenever $\{R_{i}:i=1,\ldots ,n\}\in \mathcal{P}\left( R\right) $. $\omega $ is called symmetric if $\omega \left( \left[ s,t\right] \times \left[ u,v\right] \right) =\omega \left( \left[ u,v\right] \times \left[ s,t\right] \right) $ holds for all $s<t$ and $u<v$. If $f\colon \left[ 0,1\right] ^{2}\rightarrow B$ is a continuous function, we say that its $p$-variation is controlled by $\omega $ if $\left\vert f\left( R\right) \right\vert ^{p}\leq \omega \left( R\right) $ holds for all rectangles $R\subset \lbrack 0,1]^{2}$. It is easy to see that if $\omega $ is a $2D$ control, $\left( s,t\right) \mapsto \omega \left( \left[ s,t\right] ^{2}\right) $ defines a $1D$-control. For $f\colon \left[ 0,1\right] ^{2}\rightarrow \mathbb{R}$, $R\subset \lbrack 0,1]^{2}$ a rectangle and $p\geq 1$ we define$$\left\vert f\right\vert _{p-var;R}:=\sup_{\Pi \in \mathcal{P}\left( R\right) }\left( \sum_{A\in \Pi }\left\vert f\left( A\right) \right\vert ^{p}\right) ^{1/p}.$$If $\left\vert f\right\vert _{p-var;\left[ 0,1\right] ^{2}}<\infty $ we say that $f$ has finite controlled $p$-variation. The difference of $2D$ $p$-variation introduced in Definition [def\_2D\_grid\_variation]{} and controlled $p$-variation is that in the former, one only takes the supremum over grid-like partitions whereas in the latter, one takes the supremum over all partitions of the rectangle. By superadditivity, the existence of a control $\omega $ which controls the $p$-variation of $f$ implies that $f$ has finite controlled $p$-variation and $% \left\vert f\right\vert _{p-var;R}\leq \omega \left( R\right) ^{1/p}$. In this case, we can always assume w.l.o.g. that $\omega $ is symmetric, otherwise we just substitute $\omega $ by its symmetrization $\omega _{\text{% sym}}$ given by$$\omega _{\text{sym}}\left( \left[ s,t\right] \times \left[ u,v\right] \right) =\omega \left( \left[ s,t\right] \times \left[ u,v\right] \right) +\omega \left( \left[ u,v\right] \times \left[ s,t\right] \right) .$$The connection between finite variation and finite controlled $p$-variation is summarized in the following theorem. \[theorem\_comp\_contr\_p\_var\]Let $f\colon \left[ 0,1\right] ^{2}\rightarrow \mathbb{R}$ be continuous and $R\subset \left[ 0,1\right] ^{2}$ be a rectangle. 1. We have $$V_{1}\left( f,R\right) =\left\vert f\right\vert _{1-var;R}.$$ 2. For any $p\geq 1$ and $\epsilon >0$ there is a constant $C=C\left( p,\epsilon \right) $ such that$$\frac{1}{C}\left\vert f\right\vert _{\left( p+\epsilon \right) -var;R}\leq V_{p-var}\left( f,R\right) \leq \left\vert f\right\vert _{p-var;R}.$$ 3. If $f$ has finite controlled $p$-variation, then$$R\mapsto \left\vert f\right\vert _{p-var;R}^{p}$$is a $2D$-control. In particular, there exists a $2D$-control $\omega $ such that for all rectangles $R\subset \left[ 0,1\right] ^{2}$ we have $% \left\vert f\left( R\right) \right\vert ^{p}\leq \omega \left( R\right) $, i.e. $\omega $ controls the $p$-variation of $f$. [@FV11 Theorem 1]. In the following, unless mentioned otherwise, $X$ will always be a Gaussian process as in Theorem \[theorem\_main01\_intro\] and $\mathbf{X}$ denotes the natural Gaussian rough path. We will need the following Proposition: \[prop\_moments\_lp\]Let $X$ be as in Theorem \[theorem\_main01\_intro\] and assume that $\omega $ controls the $\rho $-variation of the covariance of $X$, $\rho \in \lbrack 1,2)$. Then for every $n\in \mathbb{N}$ there is a constant $C\left( n\right) =C\left( n,\rho \right) $ such that$$\left\vert \mathbf{X}_{s,t}^{n}\right\vert _{L^{2}}\leq C\left( n\right) \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n}{2\rho }}$$for any $s<t$. For $n=1,2,3$ this is proven in [@FV10 Proposition 15.28]. For $n\geq 4$ and fixed $s<t$, we set $\tilde{X}_{\tau }:=\frac{1}{\omega \left( \left[ s,t% \right] ^{2}\right) ^{\frac{1}{2\rho }}}X_{s+\tau \left( t-s\right) }$. Then $\left\vert R_{\tilde{X}}\right\vert _{\rho -var;\left[ 0,1\right] }^{\rho }\leq 1=:K$ and by the standard (deterministic) estimates for the Lyons lift,$$\frac{\left\vert \mathbf{X}_{s,t}^{n}\right\vert ^{1/n}}{\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1}{2\rho }}}\leq c_{1}\left\Vert S_{n}\left( \mathbf{\tilde{X}}\right) \right\Vert _{p-var;\left[ 0,1\right] }\leq c_{2}\left( n,p\right) \left\Vert \mathbf{\tilde{X}}\right\Vert _{p-var;% \left[ 0,1\right] }$$for any $p\in \left( 2\rho ,4\right) $. Now we take the $L^{2}$-norm on both sides. From [@FV10 Theorem 15.33] we know that $\left\vert \left\Vert \mathbf{\tilde{X}}\right\Vert _{p-var;\left[ 0,1\right] }\right\vert _{L^{2}} $ is bounded by a constant only depending on $p,\rho $ and $K$ which shows the claim. Alternatively (and more in the spirit of the forthcoming arguments), one performs an induction similar (but easier) as in the proof of Proposition \[prop\_key\_estimate\_higher\_levels\]. Iterated integrals and the shuffle algebra[section\_it\_int\_and\_shuffle]{} ============================================================================ Let $x=\left( x^{1},\ldots ,x^{d}\right) \colon \left[ 0,1\right] \rightarrow \mathbb{R}^{d}$ be a path of finite variation. Forming finite linear combinations of iterated integrals of the form$$\int_{\Delta _{0,1}^{n}}dx^{i_{1}}\ldots dx^{i_{n}},\quad i_{1},\ldots ,i_{n}\in \left\{ 1,\ldots ,d\right\} ,n\in \mathbb{N}$$defines a vector space over $\mathbb{R}$. In this section, we will see that this vector space is also an algebra where the product is given simply by taking the usual multiplication. Moreover, we will describe precisely how the product of two iterated integrals looks like. The shuffle algebra ------------------- Let $A$ be a set which we will call from now on the alphabet. In the following, we will only consider the finite alphabet $A=\left\{ a,b,\ldots \right\} =\left\{ a_{1},a_{2},\ldots ,a_{d}\right\} =\left\{ 1,\ldots ,d\right\} $. We denote by $A^{\ast }$ the set of words composed by the letters of $A$, hence $w=a_{i_{1}}a_{i_{2}}\ldots a_{i_{n}},~a_{i_{j}}\in A$. The empty word is denoted by $e$. $A^{+}$ is the set of non-empty words. The length of the word is denoted by $\left\vert w\right\vert $ and $% \left\vert w\right\vert _{a}$ denotes the number of occurrences of the letter $a$. We denote by $\mathbb{R}\left\langle A\right\rangle $ the vector space of noncommutative polynomials on $A$ over $\mathbb{R}$, hence every $% P\in \mathbb{R}\left\langle A\right\rangle $ is a linear combination of words in $A^{\ast }$ with coefficients in $\mathbb{R}$. $(P,w)$ denotes the coefficient in $P$ of the word $w$. Hence every polynomial $P$ can be written as$$P=\sum_{w\in A^{\ast }}(P,w)w$$and the sum is finite since the $(P,w)$ are non-zero only for a finite set of words $w$. We define the degree of $P$ as$$\deg \left( P\right) =\max \left\{ \left\vert w\right\vert ~;~\left( P,w\right) \neq 0\right\} .$$A polynomial is called homogeneous if all monomials have the same degree. We want to define a product on $\mathbb{R}\left\langle A\right\rangle $. Since a polynomial is determined by its coefficients on each word, we can define the product $PQ$ of $P$ and $Q$ by$$(PQ,w)=\sum_{w=uv}(P,u)(Q,v).$$Note that this definition coincides with the usual multiplication in a (noncommutative) polynomial ring. We call this product the concatenation product and the algebra $\mathbb{R}\left\langle A\right\rangle $ endowed with this product the concatenation algebra. There is another product on $\mathbb{R}\left\langle A\right\rangle $ which will be of special interest for us. We need some notation first. Given a word $w=a_{i_{1}}a_{i_{2}}\ldots a_{i_{n}}$ and a subsequence $% U=(j_{1},j_{2},\ldots ,j_{k})$ of $\left( i_{1},\ldots ,i_{n}\right) $, we denote by $w(U)$ the word $a_{j_{1}}a_{j_{2}}\ldots a_{j_{k}}$ and we call $% w(U)$ a subword of $w$. If $w,u,v$ are words and if $w$ has length $% n$, we denote by $\left( \begin{array}{c} w \\ u\quad v% \end{array}% \right) $ the number of subsequences $U$ of $(1,\ldots ,n)$ such that $% w(U)=u $ and $w(U^{c})=v$. The (homogeneous) polynomial$$u\ast v=\sum_{w\in A^{\ast }}\left( \begin{array}{c} w \\ u\quad v% \end{array}% \right) w$$is called the shuffle product of $u$ and $v$. By linearity we extend it to a product on $\mathbb{R}\left\langle A\right\rangle $. In order to proof our main result, we want to use some sort of induction over the length of the words. Therefore, the following definition will be useful. If $U$ is a set of words of the same length, we call a subset $\left\{ w_{1},\ldots ,w_{k}\right\} $ of $U$ a generating set for $U$ if for every word $w\in U$ there is a polynomial $R$ and real numbers $\lambda _{1},\ldots ,\lambda _{k}$ such that$$w=\sum_{j=1}^{k}\lambda _{j}w_{j}+R$$where $R$ is of the form $R=\sum_{u,v\in A^{+}}\mu _{u,v}u\ast v$ for real numbers $\mu _{u,v}$. We say that a word $w$ is* composed by* $a_{1}^{n_{1}},\ldots ,a_{d}^{n_{d}}$ if $w\in \left\{ a_{1},\ldots ,a_{d}\right\} ^{\ast }$ and $% \left\vert w\right\vert _{a_{i}}=n_{i}$ for $i=1,\ldots ,d$, hence every letter appears in the word with the given multiplicity. The aim now is to find a (possibly small) generating set for the set of all words composed by some given letters. The next definition introduces a special class of words which will be important for us. Let $A$ be totally ordered and put on $A^{\ast }$ the alphabetical order. If $w$ is a word such that whenever $w=uv$ for $u,v\in A^{+}$ one has $u<v$, then $w$ is called a Lyndon word. \[prop\_generating\_sets\] 1. For the set $\{$words composed by $a,a,b\}$ a generating set is given by $\{aab\}$. 2. For the set $\{$words composed by $a,a,a,b\}$ a generating set is given by $\{aaab\}$. 3. For the set $\{$words composed by $a,a,b,b\}$ a generating set is given by $\{aabb\}$. 4. For the set $\{$words composed by $a,a,b,c\}$ a generating set is given by $\{aabc,aacb,baac\}$. Consider the alphabet $A=\left\{ a,b,c\right\} $. We choose the order $a<b<c$. A general theorem states that every word $w$ has a unique decreasing factorization into Lyndon words, i.e. $w=l_{1}^{i_{1}}\ldots l_{k}^{i_{k}}$ where $l_{1}>\ldots >l_{k}$ are Lyndon words and $i_{1},\ldots ,i_{k}\geq 1$ (see [@R93 Theorem 5.1 and Corollary 4.7]), and the formula$$\frac{1}{i_{1}!\ldots i_{k}!}l_{1}^{\ast i_{1}}\ast \ldots \ast l_{k}^{\ast i_{k}}=w+\sum_{u<w}\alpha _{u}u$$holds, where $\alpha _{u}$ are some natural integers (see again [@R93 Theorem 6.1]). By repeatedly applying this formula for the words in the sum on the right hand side, it follows that a generating set for each of the sets in $\left( 1\right) $ to $\left( 4\right) $ is given exactly by the Lyndon words composed by these letters. One can easily show that indeed $aab$, $aaab$ and $aabb$ are the only Lyndon words composed by the corresponding letters. The Lyndon words composed by $a,a,b,c$ are $\left\{ aabc,abac,aacb\right\} $ which therefore is a generating set for $\{$words composed by $a,a,b,c\}$. From the shuffle identity$$abac=baac+aabc+aacb-b\ast aac$$it follows that also $\{aabc,aacb,baac\}$ generates this set. The connection to iterated integrals ------------------------------------ Let $x=(x^{1},\ldots ,x^{d})\colon \left[ 0,1\right] \rightarrow \mathbb{R}% ^{d}$ be a path of finite variation and fix $s<t\in \left[ 0,1\right] $. For a word $w=\left( a_{i_{1}}\ldots a_{i_{n}}\right) \in A^{\ast }$, $A=\left\{ 1,\ldots ,d\right\} $ we define$$\mathbf{x}^{w}=\left\{ \begin{array}{ccc} \int_{\Delta _{s,t}^{n}}dx^{i_{1}}\ldots dx^{i_{n}} & \text{if} & w\in A^{+} \\ 1 & \text{if} & w=e% \end{array}% \right. .$$ Let $\left( \mathbb{R}\left\langle A\right\rangle ,+,\ast \right) $ be the shuffle algebra over the alphabet $A$. We define a map $\Phi \colon \mathbb{R% }\left\langle A\right\rangle \rightarrow \mathbb{R}$ by $\Phi \left( w\right) =\mathbf{x}_{s,t}^{w}$ and extend it linearly to polynomials $P\in \mathbb{R}\left\langle A\right\rangle $. The key observation is the following: \[isomorphism\_shufflealgebra\]$\Phi $ is an algebra homomorphism from the shuffle algebra $\left( \mathbb{R}\left\langle A\right\rangle ,+,\ast \right) $ to $\left( \mathbb{R},+,\cdot \right) $. [@R93], Corollary 3.5. The next proposition shows that we can restrict ourselves in showing the desired estimates only for the iterated integrals which generate the others. \[proposition\_key\_shuffle\] Let $\left( X,Y\right) =\left( X^{1},Y^{1},\ldots ,X^{d},Y^{d}\right) $ be a Gaussian process on $\left[ 0,1% \right] $ with paths of finite variation. Let $A=\left\{ 1,\ldots ,d\right\} $ be the alphabet, let $U$ be a set of words of length $n$ and $V=\left\{ w_{1,}\ldots ,w_{k}\right\} $ be a generating set for $U$. Let $\omega $ be a control, $\rho ,\gamma \geq 1$ constants and $s<t\in \left[ 0,1\right] $. Assume that there are constants $C=C\left( \left\vert w\right\vert \right) $ such that $$\left\vert \mathbf{X}_{s,t}^{w}\right\vert _{L^{2}}\leq C\left( \left\vert w\right\vert \right) \omega \left( s,t\right) ^{\frac{\left\vert w\right\vert }{2\rho }}\quad \text{and\quad }\left\vert \mathbf{Y}% _{s,t}^{w}\right\vert _{L^{2}}\leq C\left( \left\vert w\right\vert \right) \omega \left( s,t\right) ^{\frac{\left\vert w\right\vert }{2\rho }}$$holds for every word $w\in A^{\ast }$ with $\left\vert w\right\vert \leq n-1$. Assume also that for some $\epsilon >0$$$\left\vert \mathbf{X}_{s,t}^{w}-\mathbf{Y}_{s,t}^{w}\right\vert _{L^{2}}\leq C\left( \left\vert w\right\vert \right) \epsilon \omega \left( s,t\right) ^{% \frac{1}{2\gamma }}\omega \left( s,t\right) ^{\frac{\left\vert w\right\vert -1}{2\rho }}$$holds for every word $w$ with $\left\vert w\right\vert \leq n-1$ and $w\in V$. Then there is a constant $\tilde{C}$ which depends on the constants $C$, on $n$ and on $d$ such that$$\left\vert \mathbf{X}_{s,t}^{w}-\mathbf{Y}_{s,t}^{w}\right\vert _{L^{2}}\leq \tilde{C}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\omega \left( s,t\right) ^{\frac{n-1}{2\rho }}$$holds for every $w\in U$. We could account for the factor $\omega \left( s,t\right) ^{\frac{1}{2\gamma }}$ in $\epsilon $ here but the present form is how we shall use this proposition later on. Consider a copy $\bar{A}$ of $A$. If $a\in A$, we denote by $\bar{a}$ the corresponding letter in $\bar{A}$. If $w=a_{i_{1}}\ldots a_{i_{n}}\in A^{\ast }$, we define $\bar{w}=\bar{a}_{i_{1}}\ldots \bar{a}_{i_{n}}\in A^{\ast }$ and in the same way we define $\bar{P}\in \mathbb{R}\left\langle \bar{A}\right\rangle $ for $P\in \mathbb{R}\left\langle A\right\rangle $. Now we consider $\mathbb{R}\left\langle A\dot{\cup}\bar{A}\right\rangle $ equipped with the usual shuffle product. Define $\Psi \colon \mathbb{R}% \left\langle A\dot{\cup}\bar{A}\right\rangle \rightarrow \mathbb{R}$ by$$\Psi \left( w\right) =\int_{\Delta _{s,t}^{n}}dZ^{b_{i_{1}}}\ldots dZ^{b_{i_{n}}}$$for a word $w=b_{i_{1}}\ldots b_{i_{n}}$ where$$Z^{b_{j}}=\left\{ \begin{array}{ccc} X^{a_{j}} & \text{for} & b_{j}=a_{j} \\ Y^{\bar{a}_{j}} & \text{for} & b_{j}=\bar{a}_{j}% \end{array}% \right.$$and extend this definition linearly. By Theorem [isomorphism\_shufflealgebra]{}, we know that $\Psi $ is an algebra homomorphism. Take $w\in U$. By assumption, we know that there is a vector $% \lambda =\left( \lambda _{1},\ldots ,\lambda _{k}\right) $ such that$$w-\bar{w}=\sum_{j=1}^{k}\lambda _{j}\left( w_{j}-\bar{w}_{j}\right) +R-\bar{R% }$$where $R$ is of the form $R=\sum_{u,v\in A^{+},\left\vert u\right\vert +\left\vert v\right\vert =n}\mu _{u,v}\,u\ast v$ with real numbers $\mu _{u,v}$. Applying $\Psi $ and taking the $L^{2}$ norm yields$$\begin{aligned} \left\vert \mathbf{X}_{s,t}^{w}-\mathbf{Y}_{s,t}^{w}\right\vert _{L^{2}} &\leq &\sum_{l=1}^{k}\left\vert \lambda _{j}\right\vert \left\vert \mathbf{X}% _{s,t}^{w_{j}}-\mathbf{Y}_{s,t}^{w_{j}}\right\vert _{L^{2}}+\left\vert \Psi \left( R-\bar{R}\right) \right\vert _{L^{2}} \\ &\leq &c_{1}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\omega \left( s,t\right) ^{\frac{n-1}{2\rho }}+\left\vert \Psi \left( R-\bar{R}% \right) \right\vert _{L^{2}}.\end{aligned}$$Now,$$R-\bar{R}=\sum_{u,v}\mu _{u,v}\left( u\ast v-\bar{u}\ast \bar{v}\right) =\sum_{u,v}\mu _{u,v}\left( u-\bar{u}\right) \ast v+\mu _{u,v}\bar{u}\ast \left( v-\bar{v}\right) .$$Applying $\Psi $ and taking the $L^{2}$ norm gives then$$\begin{aligned} \left\vert \Psi \left( R-\bar{R}\right) \right\vert _{L^{2}} &\leq &\sum_{u,v}\left\vert \mu _{u,v}\right\vert \left\vert \left( \mathbf{X}% _{s,t}^{u}-\mathbf{Y}_{s,t}^{u}\right) \mathbf{X}_{s,t}^{v}\right\vert _{L^{2}}+\left\vert \mu _{u,v}\right\vert \left\vert \mathbf{Y}% _{s,t}^{u}\left( \mathbf{X}_{s,t}^{v}-\mathbf{Y}_{s,t}^{v}\right) \right\vert _{L^{2}} \\ &\leq &\sum_{u,v}c_{2}\left( \left\vert \mathbf{X}_{s,t}^{u}-\mathbf{Y}% _{s,t}^{u}\right\vert _{L^{2}}\left\vert \mathbf{X}_{s,t}^{v}\right\vert _{L^{2}}+\left\vert \mathbf{Y}_{s,t}^{u}\right\vert _{L^{2}}\left\vert \mathbf{X}_{s,t}^{v}-\mathbf{Y}_{s,t}^{v}\right\vert _{L^{2}}\right) \\ &\leq &\sum_{u,v}c_{3}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }% }\omega \left( s,t\right) ^{\frac{\left\vert v\right\vert +\left\vert u\right\vert -1}{2\rho }} \\ &\leq &c_{4}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\omega \left( s,t\right) ^{\frac{n-1}{2\rho }}\end{aligned}$$where we used equivalence of $L^{q}$-norms in the Wiener Chaos (cf. [@FV10 Proposition 15.19 and Theorem D.8]). Putting all together shows the assertion. Multidimensional Young-integration and grid-controls[section\_multidim\_young]{} ================================================================================ Let $f\colon \left[ 0,1\right] ^{n}\rightarrow \mathbb{R}$ be a continuous function. If $s_{1}<t_{1},\ldots ,s_{n}<t_{n}$ and $u_{1},\ldots ,u_{n}$ are elements in $\left[ 0,1\right] $, we make the following recursive definition:$$\begin{aligned} f\left( \begin{array}{c} s_{1},t_{1} \\ u_{2} \\ \vdots \\ u_{n}% \end{array}% \right) &:&=f\left( \begin{array}{c} t_{1} \\ u_{2} \\ \vdots \\ u_{n}% \end{array}% \right) -f\left( \begin{array}{c} s_{1} \\ u_{2} \\ \vdots \\ u_{n}% \end{array}% \right) \quad \text{and} \\ f\left( \begin{array}{c} s_{1},t_{1} \\ \vdots \\ s_{k-1},t_{k-1} \\ s_{k},t_{k} \\ u_{k+1} \\ \vdots \\ u_{n}% \end{array}% \right) &:&=f\left( \begin{array}{c} s_{1},t_{1} \\ \vdots \\ s_{k-1},t_{k-1} \\ t_{k} \\ u_{k+1} \\ \vdots \\ u_{n}% \end{array}% \right) -f\left( \begin{array}{c} s_{1},t_{1} \\ \vdots \\ s_{k-1},t_{k-1} \\ s_{k} \\ u_{k+1} \\ \vdots \\ u_{n}% \end{array}% \right) .\end{aligned}$$We will also use the simpler notation$$f\left( R\right) =f\left( \begin{array}{c} s_{1},t_{1} \\ \vdots \\ s_{n},t_{n}% \end{array}% \right)$$for the rectangle $R=\left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \subset \left[ 0,1\right] ^{n}$. Note that for $n=2$ this is consistent with our initial definition of $f\left( \begin{array}{c} s_{1},t_{1} \\ s_{2},t_{2}% \end{array}% \right) $. If $f,g\colon \left[ 0,1\right] ^{n}\rightarrow \mathbb{R}$ are continuous functions, the $n$-dimensional Young-integral is defined by$$\begin{aligned} &&\int_{\left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}% \right] }f\left( x_{1},\ldots ,x_{n}\right) \,dg\left( x_{1},\ldots ,x_{n}\right) \\ &:&=\lim_{\left\vert D_{1}\right\vert ,\ldots ,\left\vert D_{n}\right\vert \rightarrow 0}\sum_{\substack{ \left( t_{i_{1}}^{1}\right) \subset D_{1} \\ % \vdots \\ \left( t_{i_{n}}^{n}\right) \subset D_{n}}}f\left( t_{i_{1}}^{1},\ldots ,t_{i_{n}}^{n}\right) g\left( \begin{array}{c} t_{i_{1}}^{1},t_{i_{1}+1}^{1} \\ \vdots \\ t_{i_{n}}^{n},t_{i_{n}+1}^{n}% \end{array}% \right)\end{aligned}$$if this limit exists. Take $p\geq 1$. The $n$-dimensional $p$-variation of $% f $ is defined by$$V_{p}\left( f,\left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right) =\left( \sup_{\substack{ D_{1}\subset \left[ s_{1},t_{1}\right] \\ \vdots \\ D_{n}\subset \left[ s_{n},t_{n}\right] }}% \sum_{\substack{ \left( t_{i_{1}}^{1}\right) \subset D_{1} \\ \vdots \\ % \left( t_{i_{n}}^{n}\right) \subset D_{n}}}\left\vert f\left( \begin{array}{c} t_{i_{1}}^{1},t_{i_{1}+1}^{1} \\ \vdots \\ t_{i_{n}}^{n},t_{i_{n}+1}^{n}% \end{array}% \right) \right\vert ^{p}\right) ^{1/p}$$and if $V_{p}\left( f,\left[ 0,1\right] ^{n}\right) <\infty $ we say that $f$ has finite ($n$-dimensional) $p$-variation. The fundamental theorem is the following: Assume that $f$ has finite $p$-variation and $g$ finite $q$-variation where $% \frac{1}{p}+\frac{1}{q}>1$. Then the joint Young-integral below exists and there is a constant $C=C\left( p,q\right) $ such that$$\begin{aligned} &&\left\vert \int_{\left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}\right] }f\left( \begin{array}{c} s_{1},u_{1} \\ \vdots \\ s_{n},u_{n}% \end{array}% \right) \,dg\left( u_{1},\ldots ,u_{n}\right) \right\vert \\ &\leq &CV_{p}\left( f,\left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right) V_{q}\left( g,\left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right) .\end{aligned}$$ [@T02], Theorem 1.2 (c). We will mainly consider the case $n=2$, but we will also need $n=3$ and $4$ later on. In particular, the discussion of level $n=4$ will require us to work with $4D$ grid control functions which we now introduce. With no extra complication we make the following general definition. A map $\tilde{\omega}\colon \underbrace{\Delta \times \ldots \times \Delta }% _{\text{n-times}}\rightarrow \mathbb{R}^{+}$ is called a $n$-$D$grid-control if it is continuous and partially super-additive, i.e. for all $\left( s_{1},t_{1}\right) ,\ldots ,\left( s_{n},t_{n}\right) \in \Delta $ and $s_{i}<u_{i}<t_{i}$ we have$$\begin{aligned} &&\tilde{\omega}\left( \left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{i},u_{i}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right) +% \tilde{\omega}\left( \left[ s_{1},t_{1}\right] \times \ldots \times \left[ u_{i},t_{i}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right) \\ &\leq &\tilde{\omega}\left( \left[ s_{1},t_{1}\right] \times \ldots \times % \left[ s_{i},t_{i}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right)\end{aligned}$$for every $i=1,\ldots ,n$. $\tilde{\omega}$ is called symmetric if$$\tilde{\omega}\left( \left[ s_{1},t_{1}\right] \times \ldots \times \left[ s_{n},t_{n}\right] \right) =\tilde{\omega}\left( \left[ s_{\sigma \left( 1\right) },t_{\sigma \left( 1\right) }\right] \times \ldots \times \left[ s_{\sigma \left( n\right) },t_{\sigma \left( n\right) }\right] \right)$$holds for every $\sigma \in S_{n}$. The point of this definition is that $\left\vert f\left( A\right) \right\vert ^{p}\leq \tilde{\omega}\left( A\right) $ for every rectangle $% A\subset \left[ 0,1\right] ^{n}$ implies that $V_{p}\left( f,R\right) ^{p}\leq \tilde{\omega}\left( R\right) $ for every rectangle $R\subset \left[ 0,1\right] ^{n}$. Note that a 2D control in the sense of Definition [def\_2D\_controls]{} is automatically a $2D$ grid-control. The following immediate properties will be used in Section \[section\_n=4\] with $m=n=2$. 1. The restriction of a $\left( m+n\right) $-dimensional grid-control to $% m$ arguments is a $m$-dimensional grid-control. 2. The product of a $m$- and a $n$-dimensional grid-control is a $(m+n)$-dimensional grid-control. Iterated $2D$-integrals ----------------------- In the $1$-dimensional case, the classical Young-theory allows to define iterated integrals of functions with finite $p$-variation where $p<2$. There, the superadditivity of $\left( s,t\right) \mapsto \left\vert \cdot \right\vert _{p-var;\left[ s,t\right] }^{p}$ played an essential role. We will see that Theorem \[theorem\_comp\_contr\_p\_var\] can be used to define and estimate iterated $2D$-integrals. This will play an important role in Section \[section\_main\_estimates\] when we estimate the $L^{2}$-norm of iterated integrals of Gaussian processes. \[lemma\_kernel\_iter\_2D\]Let $f,g\colon \left[ 0,1\right] ^{2}\rightarrow \mathbb{R}$ be continuous where $f$ has finite $p$-variation and $g$ finite controlled $q$-variation with $p^{-1}+q^{-1}>1$. Let $\left( s,t\right) \in \Delta $ and assume that $f\left( s,\cdot \right) =f\left( \cdot ,s\right) =0 $. Define $\Phi \colon \left[ s,t\right] ^{2}\rightarrow \mathbb{R}$ by$$\Phi \left( u,v\right) =\int_{\left[ s,u\right] \times \left[ s,v\right] }f\,dg.$$Then there is a constant $C=C\left( p,q\right) $ such that$$V_{q-var}\left( \Phi ;\left[ s,t\right] ^{2}\right) \leq C\left( p,q\right) V_{p-var}\left( f;\left[ s,t\right] ^{2}\right) \left\vert g\right\vert _{q-var;\left[ s,t\right] ^{2}}.$$ 1. Let $t_{i}<t_{i+1}$ and $\tilde{t}_{j}<\tilde{t}_{j+1}$. Then,$$\Phi \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% =\int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}% _{j+1}\right] }f\,dg.$$Now let $t_{i}<u<t_{i+1}$ and $\tilde{t}_{j}<v<\tilde{t}_{j+1}$. Then one has$$f% \begin{pmatrix} t_{i},u \\ \tilde{t}_{j},v% \end{pmatrix}% =f\left( u,v\right) -f\left( t_{i},v\right) -f\left( u,\tilde{t}_{j}\right) +f\left( t_{i},\tilde{t}_{j}\right) .$$Therefore,$$\begin{aligned} \left\vert \Phi \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% \right\vert &\leq &\left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }f% \begin{pmatrix} t_{i},u \\ \tilde{t}_{j},v% \end{pmatrix}% \,dg\left( u,v\right) \right\vert +\left\vert \int_{\left[ t_{i},t_{i+1}% \right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }f\left( t_{i},v\right) \,dg\left( u,v\right) \right\vert \\ &&+\left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},% \tilde{t}_{j+1}\right] }f\left( u,\tilde{t}_{j}\right) \,dg\left( u,v\right) \right\vert +\left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }f\left( t_{i},\tilde{t}_{j}\right) \,dg\left( u,v\right) \right\vert\end{aligned}$$For the first integral we use Young $2D$-estimates to see that$$\begin{aligned} &&\left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},% \tilde{t}_{j+1}\right] }f% \begin{pmatrix} t_{i},u \\ \tilde{t}_{j},v% \end{pmatrix}% \,dg\left( u,v\right) \right\vert \\ &\leq &c_{1}\left( p,q\right) V_{p}\left( f,\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] \right) V_{q}\left( g,% \left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}% \right] \right) \\ &\leq &c_{1}\left( p,q\right) V_{p}\left( f,\left[ s,t\right] ^{2}\right) \left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }\end{aligned}$$For the second, one has by a Young $1D$-estimate$$\begin{aligned} \left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},% \tilde{t}_{j+1}\right] }f\left( t_{i},v\right) \,dg\left( u,v\right) \right\vert &=&\left\vert \int_{\left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }f\left( t_{i},v\right) \,d\left( g\left( t_{i+1},v\right) -g\left( t_{i},v\right) \right) \right\vert \\ &\leq &c_{2}\sup_{u\in \left[ s,t\right] }\left\vert f\left( u,\cdot \right) \right\vert _{p-var;\left[ s,t\right] }\left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }.\end{aligned}$$Similarly,$$\left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},% \tilde{t}_{j+1}\right] }f\left( u,\tilde{t}_{j}\right) \,dg\left( u,v\right) \right\vert \leq c_{2}\sup_{v\in \left[ s,t\right] }\left\vert f\left( \cdot ,v\right) \right\vert _{p-var;\left[ s,t\right] }\left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}% _{j+1}\right] }.$$Finally,$$\left\vert \int_{\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},% \tilde{t}_{j+1}\right] }f\left( t_{i},\tilde{t}_{j}\right) \,dg\left( u,v\right) \right\vert =\left\vert f\left( t_{i},\tilde{t}_{j}\right) \right\vert \left\vert g% \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% \right\vert \leq \left\vert f\right\vert _{\infty ;\left[ s,t\right] }\left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }.$$Putting all together, we get$$\begin{aligned} &&\left\vert \Phi \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% \right\vert ^{q} \\ &\leq &c_{3}\left( V_{p}\left( f,\left[ s,t\right] ^{2}\right) +\sup_{u\in % \left[ s,t\right] }\left\vert f\left( u,\cdot \right) \right\vert _{p-var;% \left[ s,t\right] }+\sup_{v\in \left[ s,t\right] }\left\vert f\left( \cdot ,v\right) \right\vert _{p-var;\left[ s,t\right] }+\left\vert f\right\vert _{\infty ;\left[ s,t\right] }\right) ^{q} \\ &&\times \left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times % \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }^{q}.\end{aligned}$$Take a partition $D\subset \left[ s,t\right] $ and $u\in \left[ s,t\right] $. Then$$\sum_{t_{i}\in D}\left\vert f\left( u,t_{i+1}\right) -f\left( u,t_{i}\right) \right\vert ^{p}=\sum_{t_{i}\in D}\left\vert f% \begin{pmatrix} s,u \\ t_{i},t_{i+1}% \end{pmatrix}% \right\vert ^{p}\leq V_{p}\left( f,\left[ s,t\right] ^{2}\right) ^{p}$$and hence$$\sup_{u\in \left[ s,t\right] }\left\vert f\left( u,\cdot \right) \right\vert _{p-var;\left[ s,t\right] }\leq V_{p}\left( f,\left[ s,t\right] ^{2}\right) .$$The same way one obtains$$\sup_{v\in \left[ s,t\right] }\left\vert f\left( \cdot ,v\right) \right\vert _{p-var;\left[ s,t\right] }\leq V_{p}\left( f,\left[ s,t\right] ^{2}\right) .$$Finally, for $u,v\in \left[ s,t\right] $,$$\left\vert f\left( u,v\right) \right\vert =\left\vert f% \begin{pmatrix} s,u \\ s,v% \end{pmatrix}% \right\vert \leq V_{p}\left( f,\left[ s,t\right] ^{2}\right)$$and therefore $\left\vert f\right\vert _{\infty ;\left[ s,t\right] }\leq V_{p}\left( f,\left[ s,t\right] ^{2}\right) $. Putting everything together, we end up with$$\left\vert \Phi \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% \right\vert ^{q}\leq c_{4}V_{p}\left( f,\left[ s,t\right] ^{2}\right) ^{q}\left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] }^{q}.$$Hence for every partition $D,\tilde{D}\subset \left[ s,t\right] $ one gets, using superadditivity of $\left\vert g\right\vert _{q-var}^{q}$,$$\begin{aligned} \sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\left\vert \Phi \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% \right\vert ^{q} &\leq &c_{4}V_{p}\left( f,\left[ s,t\right] ^{2}\right) ^{q}\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\left\vert g\right\vert _{q-var;\left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}% _{j+1}\right] }^{q} \\ &\leq &c_{4}V_{p}\left( f,\left[ s,t\right] ^{2}\right) ^{q}\left\vert g\right\vert _{q-var;\left[ s,t\right] ^{2}}^{q}.\end{aligned}$$Passing to the supremum over all partitions shows the assertion. This lemma allows us to define iterated $2D$-integrals. Let $f,g_{1},\ldots ,g_{n}\colon \left[ 0,1\right] ^{2}\rightarrow \mathbb{R}$. An iterated $2D$-integral is given by $\int_{\Delta _{s,t}^{1}\times \Delta _{s^{\prime },t^{\prime }}^{1}}f\,dg_{1}=\int_{\left[ s,t\right] \times \left[ s^{\prime },t^{\prime }\right] }f\left( u,v\right) \,dg_{1}\left( u,v\right) $ for $% n=1 $ and recursively defined by $$\int_{\Delta _{s,t}^{n}\times \Delta _{s^{\prime },t^{\prime }}^{n}}f\,dg_{1}\ldots dg_{n}:=\int_{\left[ s,t\right] \times \left[ s^{\prime },t^{\prime }\right] }\left( \int_{\Delta _{s,u}^{n-1}\times \Delta _{s^{\prime },v}^{n-1}}f\,dg_{1}\ldots dg_{n-1}\right) \,dg_{n}\left( u,v\right)$$for $n\geq 2$. \[prop\_rho\_var\_iter\_2D\]Let $f,g_{1},g_{2},\ldots \colon \left[ 0,1% \right] ^{2}\rightarrow \mathbb{R}$ and $p,q_{1},q_{2},\ldots $ be real numbers such that $p^{-1}+q_{1}^{-1}>1$ and $q_{i}^{-1}+q_{i+1}^{-1}>1$ for every $i\geq 1$. Assume that $f$ has finite $p$-variation and $g_{i}$ has finite $q_{i}$-variation for $i=1,2,\ldots $ and that for $\left( s,t\right) \in \Delta $ we have $f\left( s,\cdot \right) =f\left( \cdot ,s\right) =0$. Then for every $n\in \mathbb{N}$ there is a constant $C=C\left( p,q_{1},\ldots ,q_{n}\right) $ such that$$\left\vert \int_{\Delta _{s,t}^{n}\times \Delta _{s,t}^{n}}f\,dg_{1}\ldots dg_{n}\right\vert \leq CV_{p}\left( f,\left[ s,t\right] ^{2}\right) V_{q_{1}}\left( g_{1},\left[ s,t\right] ^{2}\right) \ldots V_{q_{n}}\left( g_{n},\left[ s,t\right] ^{2}\right) .$$ Define $\Phi ^{\left( n\right) }\left( u,v\right) =\int_{\Delta _{s,u}^{n}\times \Delta _{s,v}^{n}}f\,dg_{1}\ldots dg_{n}$. We will show a stronger result; namely that for every $n\in \mathbb{N}$ and $q_{n}^{\prime }>q_{n}$ there is a constant $C=C\left( p,q_{1},\ldots ,q_{n},q_{n}^{\prime }\right) $ such that$$V_{q_{n}^{\prime }}\left( \Phi ^{\left( n\right) },\left[ s,t\right] ^{2}\right) \leq CV_{p}\left( f,\left[ s,t\right] ^{2}\right) V_{q_{1}}\left( g_{1},\left[ s,t\right] ^{2}\right) \ldots V_{q_{n}}\left( g_{n},\left[ s,t\right] ^{2}\right) .$$To do so, let $\tilde{q}_{1},\tilde{q}_{2},\ldots $be a sequence of real numbers such that $\tilde{q}_{j}>q_{j}$ and $\frac{1}{\tilde{q}_{j-1}}+\frac{% 1}{\tilde{q}_{j}}>1$ for every $j=1,2,\ldots $ where we set $\tilde{q}_{0}=p$. We make an induction over $n$. For $n=1$, we have $\tilde{q}_{1}>q_{1}$ and $\frac{1}{p}+\frac{1}{\tilde{q}_{1}}>1$, hence from Theorem [theorem\_comp\_contr\_p\_var]{} we know that $g_{1}$ has finite controlled $% \tilde{q}_{1}$-variation and Lemma \[lemma\_kernel\_iter\_2D\] gives us$$V_{\tilde{q}_{1}}\left( \Phi ^{\left( 1\right) };\left[ s,t\right] ^{2}\right) \leq c_{1}V_{p}\left( f;\left[ s,t\right] ^{2}\right) \left\vert g_{1}\right\vert _{\tilde{q}_{1};\left[ s,t\right] ^{2}}\leq c_{2}V_{p}\left( f;\left[ s,t\right] ^{2}\right) V_{q_{1}}\left( g_{1};\left[ s,t\right] ^{2}\right) .$$W.l.o.g, we may assume that $q_{1}^{\prime }>\tilde{q}_{1}>q_{1}$, otherwise we choose $\tilde{q}_{1}$ smaller in the beginning. From $V_{q_{1}^{\prime }}\left( \Phi ^{\left( 1\right) };\left[ s,t\right] ^{2}\right) \leq V_{% \tilde{q}_{1}}\left( \Phi ^{\left( 1\right) };\left[ s,t\right] ^{2}\right) $ the assertion follows for $n=1$. Now take $n\in \mathbb{N}$. Note that$$\Phi ^{\left( n\right) }\left( u,v\right) =\int_{\left[ s,u\right] \times % \left[ s,v\right] }\Phi ^{\left( n-1\right) }\,dg_{n}$$and clearly $\Phi ^{\left( n-1\right) }\left( s,\cdot \right) =\Phi ^{\left( n-1\right) }\left( \cdot ,s\right) =0$. We can use Lemma [lemma\_kernel\_iter\_2D]{} again to see that$$\begin{aligned} V_{\tilde{q}_{n}}\left( \Phi ^{\left( n\right) },\left[ s,t\right] ^{2}\right) &\leq &c_{3}V_{\tilde{q}_{n-1}}\left( \Phi ^{\left( n-1\right) };% \left[ s,t\right] ^{2}\right) \left\vert g_{n}\right\vert _{\tilde{q}% _{n}-var;\left[ s,t\right] ^{2}} \\ &\leq &c_{4}V_{\tilde{q}_{n-1}}\left( \Phi ^{\left( n-1\right) };\left[ s,t% \right] ^{2}\right) V_{q_{n}}\left( g_{n};\left[ s,t\right] ^{2}\right) .\end{aligned}$$Using our induction hypothesis shows the result for $\tilde{q}_{n}$. By choosing $\tilde{q}_{n}$ smaller in the beginning if necessary, we may assume that $q_{n}^{\prime }>\tilde{q}_{n}$ and the assertion follows. The main estimates\[section\_main\_estimates\] ============================================== In the following section, $\left( X,Y\right) =\left( X^{1},Y^{1},\ldots ,X^{d},Y^{d}\right) $ will always denote a centred continuous Gaussian process where $\left( X^{i},Y^{i}\right) $ and $\left( X^{j},Y^{j}\right) $ are independent for $i\neq j$. We will also assume that the $\rho $-variation of $R_{\left( X,Y\right) }$ is finite for a $\rho <2$ and controlled by a symmetric $2D$-control $\omega $ (this in particular implies that the $\rho $-variation of $R_{X},R_{Y}$ and $R_{X-Y}$ is controlled by $% \omega $, see [@FV10 Section 15.3.2]). Let $\gamma >\rho $ such that $% \frac{1}{\rho }+\frac{1}{\gamma }>1$. The aim of this section is to show that for every $n\in \mathbb{N}$ there are constants $C\left( n\right) $ such that$\footnote{% We prefer to write it in this notation instead of writing $\omega \left( % \left[ s,t\right] ^{2}\right) ^{\frac{1}{2\gamma }+\frac{n-1}{2\rho }}$ to emphasize the different roles of the two terms. The first term will play no particular role and just comes from interpolation whereas the second one will be crucial when doing the induction step from lower to higher levels in Proposition \ref{prop_key_estimate_higher_levels}.}$$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}\left( \left( \mathbb{R}^{d}\right) ^{\otimes n}\right) }\leq C\left( n\right) \epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1}{% 2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{2\rho }% }\quad \text{for every }s<t \label{eqn_key_estimate}$$where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$(see Definition \[def\_V\_infty\] below for the exact definition of $V_{\infty }$). Equivalently, we might show $\left( \ref% {eqn_key_estimate}\right) $ coordinate-wise, i.e. proving that the same estimate holds for $\left\vert \mathbf{X}^{w}-\mathbf{Y}^{w}\right\vert _{L^{2}\left( \mathbb{R}\right) }$ for every word $w$ formed by the alphabet $A=\left\{ 1,\ldots ,d\right\} $. In some special cases, i.e. if a word $w$ has a very simple structure, we can do this directly using multidimensional Young integration. This is done in Subsection \[subsection\_special\_cases\]. Subsection \[subsection\_lower\_levels\] shows $\left( \ref% {eqn_key_estimate}\right) $ for $n=1,2,3,4$ coordinate-wise, using the shuffle algebra structure for iterated integrals and multidimensional Young integration. In Subsection \[subsection\_higher\_levels\], we show $\left( % \ref{eqn_key_estimate}\right) $ coordinate-free for all $n>4$, using an induction argument very similar to the one Lyon’s used for proving the Extension Theorem (cf. [@L98]). We start with giving a $2$-dimensional analogue for the one-dimensional interpolation inequality. \[def\_V\_infty\]If $f\colon \left[ 0,1\right] ^{2}\rightarrow B$ is a continuous function in a Banach space and $\left( s,t\right) \times \left( u,v\right) \in \Delta \times \Delta $ we set$$V_{\infty }\left( f,\left[ s,t\right] \times \left[ u,v\right] \right) =\sup_{A\subset \left[ s,t\right] \times \left[ u,v\right] }\left\vert f\left( A\right) \right\vert .$$ \[lemma\_2D\_interpolation\]For $\gamma >\rho \geq 1$ we have the interpolation inequality$$V_{\gamma -var}\left( f,\left[ s,t\right] \times \left[ u,v\right] \right) \leq V_{\infty }\left( f,\left[ s,t\right] \times \left[ u,v\right] \right) ^{1-\rho /\gamma }V_{\rho -var}\left( f,\left[ s,t\right] \times \left[ u,v% \right] \right) ^{\rho /\gamma }$$for all $\left( s,t\right) ,\left( u,v\right) \in \Delta $. Exactly as $1D$-interpolation, see [@FV10 Proposition 5.5]. Some special cases\[subsection\_special\_cases\] ------------------------------------------------ If $Z\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ is a process with smooth sample paths, we will use the notation$$\mathbf{Z}_{s,t}^{\left( n\right) }=\int_{\Delta _{s,t}^{n}}dZ\ldots dZ$$for $s<t$. \[lemma\_rho\_var\_same\_letters\] Let $X\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ be a centred Gaussian process with continuous paths of finite variation and assume that the $\rho $-variation of the covariance $% R_{X}$ is controlled by a $2D$-control $\omega $. For fixed $s<t$, define$$f\left( u,v\right) =E\left( \mathbf{X}_{s,u}^{\left( n\right) }\mathbf{X}% _{s,v}^{\left( n\right) }\right) .$$Then there is a constant $C=C\left( \rho ,n\right) $ such that$$V_{\rho }\left( f,\left[ s,t\right] ^{2}\right) \leq C\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n}{\rho }}.$$ Let $t_{i}<t_{i+1}$, $\tilde{t}_{j}<\tilde{t}_{j+1}$. Then$$f% \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% =E\left( \left( \mathbf{X}_{s,t_{i+1}}^{\left( n\right) }-\mathbf{X}% _{s,t_{i}}^{\left( n\right) }\right) \left( \mathbf{X}_{s,\tilde{t}% _{j+1}}^{\left( n\right) }-\mathbf{X}_{s,\tilde{t}_{j}}^{\left( n\right) }\right) \right) .$$We know that $\mathbf{X}^{\left( n\right) }=\frac{\left( X\right) ^{n}}{n!}$. From the identity$$b^{n}-a^{n}=\left( b-a\right) \left( a^{n-1}+a^{n-2}b+\ldots +\ldots ab^{n-2}+b^{n-1}\right)$$we deduce that$$f% \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% =\frac{1}{\left( n!\right) ^{2}}\sum_{k,l=0}^{n-1}E\left( X_{t_{i},t_{i+1}}X_{\tilde{t}_{j},\tilde{t}_{j+1}}\left( X_{s,t_{i+1}}\right) ^{n-1-k}\left( X_{s,t_{i}}\right) ^{k}\left( X_{s,% \tilde{t}_{j+1}}\right) ^{n-1-l}\left( X_{s,\tilde{t}_{j}}\right) ^{l}\right) .$$We want to apply Wick’s formula now (cf. [@J97 Theorem 1.28]). If $Z,% \tilde{Z}\in \left\{ X_{s,t_{i+1}},X_{s,t_{i}},X_{s,\tilde{t}_{j+1}},X_{s,% \tilde{t}_{j}}\right\} $ we know that$$\begin{aligned} \left\vert E\left( X_{t_{i},t_{i+1}}Z\right) \right\vert ^{\rho } &\leq &\omega \left( \left[ t_{i},t_{i+1}\right] \times \left[ s,t\right] \right) \\ \left\vert E\left( X_{t_{i},t_{i+1}}X_{\tilde{t}_{j},\tilde{t}_{j+1}}\right) \right\vert ^{\rho } &\leq &\omega \left( \left[ t_{i},t_{i+1}\right] \times % \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] \right) \\ \left\vert E\left( Z\tilde{Z}\right) \right\vert ^{\rho } &\leq &\omega \left( \left[ s,t\right] ^{2}\right)\end{aligned}$$and the same holds for $X_{\tilde{t}_{j},\tilde{t}_{j+1}}$. Now take two partitions $D,\tilde{D}\in \left[ 0,1\right] $. Then, by Wick’s formula and the estimates above,$$\begin{aligned} \sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\left\vert f% \begin{pmatrix} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{pmatrix}% \right\vert ^{\rho } &\leq &c_{1}\left( \rho ,n\right) \omega \left( \left[ s,t\right] ^{2}\right) ^{n-2}\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}% }\omega \left( \left[ t_{i},t_{i+1}\right] \times \left[ s,t\right] \right) \omega \left( \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] \times \left[ s,t% \right] \right) \\ &&+c_{2}\left( \rho ,n\right) \omega \left( \left[ s,t\right] ^{2}\right) ^{n-1}\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\omega \left( \left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] \right) \\ &\leq &c_{3}\omega \left( \left[ s,t\right] ^{2}\right) ^{n}.\end{aligned}$$ \[lemma\_diff\_allthesame\]Let $\left( X,Y\right) $ be a centred Gaussian process in $\mathbb{R}^{2}$ with continuous paths of finite variation. Assume that the $\rho $-variation of $R_{\left( X,Y\right) }$ is controlled by a $2D$-control $\omega $ for $\rho <2$ and take $\gamma >\rho $. Then for every $n\in \mathbb{N}$ there is a constant $C=C\left( n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{\left( n\right) }-\mathbf{Y}_{s,t}^{\left( n\right) }\right\vert _{L^{2}}\leq C\left( n\right) \epsilon \omega \left( % \left[ s,t\right] ^{2}\right) ^{\frac{1}{2\gamma }}\omega \left( \left[ s,t% \right] ^{2}\right) ^{\frac{n-1}{2\rho }}$$for any $s<t$ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t% \right] ^{2}\right) ^{1-\rho /\gamma }$. By induction. For $n=1$ we simply have from Lemma [lemma\_2D\_interpolation]{}$$\begin{aligned} \left\vert X_{s,t}-Y_{s,t}\right\vert _{L^{2}}^{2} &=&E\left[ \left( X_{s,t}-Y_{s,t}\right) \left( X_{s,t}-Y_{s,t}\right) \right] \leq V_{\gamma -var}\left( R_{X-Y},\left[ s,t\right] ^{2}\right) \\ &\leq &\epsilon ^{2}V_{\rho -var}\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{\rho /\gamma }\leq \epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1}{\gamma }}\end{aligned}$$For $n\in \mathbb{N}$ we use the identity$$\mathbf{X}_{s,t}^{\left( n\right) }-\mathbf{Y}_{s,t}^{\left( n\right) }=% \frac{1}{n}\left( X_{s,t}\mathbf{X}_{s,t}^{\left( n-1\right) }-Y_{s,t}% \mathbf{Y}_{s,t}^{\left( n-1\right) }\right)$$and hence$$\begin{aligned} \left\vert \mathbf{X}_{s,t}^{\left( n\right) }-\mathbf{Y}_{s,t}^{\left( n\right) }\right\vert _{L^{2}} &\leq &c_{1}\left( \left\vert X_{s,t}-Y_{s,t}\right\vert _{L^{2}}\left\vert \mathbf{X}_{s,t}^{\left( n-1\right) }\right\vert _{L^{2}}+\left\vert \mathbf{X}_{s,t}^{\left( n-1\right) }-\mathbf{Y}_{s,t}^{\left( n-1\right) }\right\vert _{L^{2}}\left\vert Y_{s,t}\right\vert _{L^{2}}\right) \\ &\leq &c_{2}\epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1}{% 2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{2\rho }}.\end{aligned}$$ Assume that $\left( Z^{1},Z^{2}\right) $ is a centred, continuous Gaussian process in $\mathbb{R}^{2}$ with smooth sample paths and that both components are independent. Then (at least formally, cf. [@FV10AIHP]),$$\begin{aligned} \left\vert \int_{0}^{1}Z_{0,u}^{1}\,dZ_{u}^{2}\right\vert _{L^{2}}^{2} &=&E \left[ \left( \int_{0}^{1}Z_{0,u}^{1}\,dZ_{u}^{2}\right) ^{2}\right] =E\left[ \int_{\left[ 0,1\right] ^{2}}Z_{0,u}^{1}Z_{0,v}^{1}\,dZ^{2}\,dZ_{v}^{2}% \right] \label{eqn_L2_norm_into_integral1} \\ &=&\int_{\left[ 0,1\right] ^{2}}E\left[ Z_{0,u}^{1}Z_{0,v}^{1}\right] \,dE% \left[ Z_{u}^{2}Z_{v}^{2}\right] =\int_{\left[ 0,1\right] ^{2}}R_{Z^{1}}% \begin{pmatrix} 0 & \cdot \\ 0 & \cdot% \end{pmatrix}% \,dR_{Z^{2}} \label{eqn_L2_norm_into_integral2}\end{aligned}$$where the integrals in the second row are $2D$ Young-integrals (to make this rigorous, one uses that the integrals are a.s. limits of Riemann sums and that a.s. convergence implies convergence in $L^{1}$ in the (inhomogeneous) Wiener chaos). These kinds of computations together with our estimates for $% 2D$ Young-integrals will be heavily used from now on. \[lemma\_diff\_alldifferent\]Let $\left( X,Y\right) =\left( X^{1},Y^{1},\ldots ,X^{d},Y^{d}\right) $ be a centred Gaussian process with continuous paths of finite variation where $\left( X^{i},Y^{i}\right) \ $and $\left( X^{j},Y^{j}\right) $ are independent for $i\neq j$. Assume that the $% \rho $-variation of $R_{\left( X,Y\right) }$ is controlled by a $2D$-control $\omega $ for $\rho <2$. Let $w$ be a word of the form $w=$ $i_{1}\cdots i_{n}$ where $i_{1},\ldots ,i_{n}\in \left\{ 1,\ldots ,d\right\} $ are all distinct. Take $\gamma >\rho $ such that $\frac{1}{\rho }+\frac{1}{\gamma }% >1 $. Then there is a constant $C=C\left( \rho ,\gamma ,n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{w}-\mathbf{Y}_{s,t}^{w}\right\vert _{L^{2}}\leq C\left( n\right) \epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{% \frac{1}{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{% 2\rho }}$$for any $s<t$ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t% \right] ^{2}\right) ^{1-\rho /\gamma }$. By the triangle inequality,$$\begin{aligned} \left\vert \mathbf{X}_{s,t}^{w}-\mathbf{Y}_{s,t}^{w}\right\vert _{L^{2}} &=&\left\vert \int_{\Delta _{s,t}^{n}}\,dX^{i_{1}}\ldots dX^{i_{n}}-\int_{\Delta _{s,t}^{n}}\,dY^{i_{1}}\ldots dY^{i_{n}}\right\vert _{L^{2}} \\ &\leq &\sum_{k=1}^{n}\left\vert \int_{\Delta _{s,t}^{n}}\,dY^{i_{1}}\ldots dY^{i_{k-1}}\,d\left( X^{i_{k}}-Y^{i_{k}}\right) \,dX^{i_{k+1}}\ldots dX^{i_{n}}\right\vert _{L^{2}}.\end{aligned}$$From independence, Proposition \[prop\_rho\_var\_iter\_2D\] and Lemma [lemma\_2D\_interpolation]{} $$\begin{aligned} &&\left\vert \int_{\Delta _{s,t}^{n}}\,dY^{i_{1}}\ldots dY^{i_{k-1}}\,d\left( X^{i_{k}}-Y^{i_{k}}\right) \,dX^{i_{k+1}}\ldots dX^{i_{n}}\right\vert _{L^{2}}^{2} \\ &=&\int_{\Delta _{s,t}^{n}\times \Delta _{s,t}^{n}}\,dR_{Y^{i_{1}}}\ldots dR_{Y^{i_{k-1}}}\,dR_{X^{i_{k}}-Y^{i_{k}}}\,dR_{X^{i_{k+1}}}\ldots dR_{X^{i_{n}}} \\ &\leq &c_{1}V_{\rho }\left( R_{Y^{i_{1}}},\left[ s,t\right] ^{2}\right) \ldots V_{\rho }\left( R_{Y^{i_{k-1}}},\left[ s,t\right] ^{2}\right) V_{\gamma }\left( R_{X^{i_{k}}-Y^{i_{k}}},\left[ s,t\right] ^{2}\right) \\ &&\times V_{\rho }\left( R_{X^{i_{k+1}}},\left[ s,t\right] ^{2}\right) \ldots V_{\rho }\left( R_{X^{i_{n}}},\left[ s,t\right] ^{2}\right) \\ &\leq &c_{1}V_{\gamma }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{\rho }}\leq c_{1}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1}{\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{\rho }}.\end{aligned}$$The first inequality above is an immediate generalization of the calculations made in $\left( \ref{eqn_L2_norm_into_integral1}\right) $ and $% \left( \ref{eqn_L2_norm_into_integral2}\right) $. Note that the respective random terms are not only pairwise but mutually independent here since we are dealing with a Gaussian process $\left( X,Y\right) $. Interchanging the limits is allowed since convergence in probability implies convergence in $% L^{p}$, any $p>0$, in the Wiener chaos. Lower levels\[subsection\_lower\_levels\] ----------------------------------------- ### $n=1,2$ \[prop\_main\_estimates\_n1\_n2\]Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma \[lemma\_diff\_alldifferent\]. Then there are constants $C\left( 1\right) ,C\left( 2\right) $ which depend on $\rho $ and $% \gamma $ such that$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}}\leq C\left( n\right) \epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{% \frac{1}{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{% 2\rho }}$$holds for $n=1,2$ and every $\left( s,t\right) \in \Delta $ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. The coordinate-wise estimates are just special cases of Lemma [lemma\_diff\_allthesame]{} and Lemma \[lemma\_diff\_alldifferent\]. ### $n=3$ \[prop\_main\_estimates\_n3\]Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma \[lemma\_diff\_alldifferent\]. Then there is a constant $C\left( 3\right) $ which depends on $\rho $ and $\gamma $ such that$$\left\vert \mathbf{X}_{s,t}^{3}-\mathbf{Y}_{s,t}^{3}\right\vert _{L^{2}}\leq C\left( 3\right) \epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{% \frac{1}{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{2}{% 2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. We have to show the estimate for $\mathbf{X}^{i,j,k}-\mathbf{Y}^{i,j,k}$ where $i,j,k\in \left\{ 1,\ldots ,d\right\} $. From Proposition [proposition\_key\_shuffle]{} and \[prop\_generating\_sets\] it follows that it is enough to show the estimate for $\mathbf{X}^{w}-\mathbf{Y}^{w}$ where$$w\in \left\{ iii,ijk,iij:i,j,k\in \left\{ 1,\ldots ,d\right\} ~\text{distinct% }\right\} \text{.}$$The cases $w=iii$ and $w=ijk$ are special cases of Lemma [lemma\_diff\_allthesame]{} and Lemma \[lemma\_diff\_alldifferent\]. The rest of this section is devoted to show the estimate for $w=iij$. \[lemma\_half\_estimates\] Let $\left( X,Y\right) \colon \left[ 0,1\right] \rightarrow \mathbb{R}^{2}$ be a centred Gaussian process and consider$$f\left( u,v\right) =E\left( \left( X_{u}-Y_{u}\right) X_{v}\right) .$$Assume that the $\rho $-variation of $R_{\left( X,Y\right) }$ is controlled by a $2D$-control $\omega $ where $\rho \geq 1$. Let $s<t$ and consider a rectangle $\left[ \sigma ,\tau \right] \times \left[ \sigma ^{\prime },\tau ^{\prime }\right] \subset \left[ s,t\right] ^{2}$. Let $\gamma >\rho $. Then$$V_{\gamma -var}\left( f,\left[ \sigma ,\tau \right] \times \left[ \sigma ^{\prime },\tau ^{\prime }\right] \right) \leq \epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{1/2\left( 1/\rho -1/\gamma \right) }\omega \left( % \left[ \sigma ,\tau \right] \times \left[ \sigma ^{\prime },\tau ^{\prime }% \right] \right) ^{1/\gamma }$$where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. Let $u<v$ and $u^{\prime }<v^{\prime }\in \left[ s,t\right] $. Then$$\begin{aligned} \left\vert E\left( \left( X_{u,v}-Y_{u,v}\right) X_{u^{\prime },v^{\prime }}\right) \right\vert &\leq &\left\vert X_{u,v}-Y_{u,v}\right\vert _{L^{2}}\left\vert X_{u^{\prime },v^{\prime }}\right\vert _{L^{2}} \\ &\leq &V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1/2}V_{\rho -var}\left( R_{\left( X,Y\right) },\left[ s,t\right] ^{2}\right) ^{1/2}\end{aligned}$$ and hence$$\sup_{u<v,u^{\prime }<v^{\prime }}\left\vert E\left( \left( X_{u,v}-Y_{u,v}\right) X_{u^{\prime },v^{\prime }}\right) \right\vert \leq V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1/2}\omega \left( % \left[ s,t\right] ^{2}\right) ^{\frac{1}{2\rho }}.$$Now take a partition $D$ of $\left[ \sigma ,\tau \right] $ and a partition $% \tilde{D}$ of $\left[ \sigma ^{\prime },\tau ^{\prime }\right] $. Then$$\begin{aligned} &&\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\left\vert E\left( \left( X_{t_{i},t_{i+1}}-Y_{t_{i},t_{i+1}}\right) X_{\tilde{t}_{j},\tilde{t}% _{j+1}}\right) \right\vert ^{\gamma } \\ &\leq &\sup_{u<v,u^{\prime }<v^{\prime }}\left\vert E\left( \left( X_{u,v}-Y_{u,v}\right) X_{u^{\prime },v^{\prime }}\right) \right\vert ^{\gamma -\rho }\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\left\vert E\left( \left( X_{t_{i},t_{i+1}}-Y_{t_{i},t_{i+1}}\right) X_{\tilde{t}_{j},% \tilde{t}_{j+1}}\right) \right\vert ^{\rho } \\ &\leq &V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1/2\left( \gamma -\rho \right) }\omega \left( \left[ s,t\right] ^{2}\right) ^{1/2\left( \gamma /\rho -1\right) }\omega \left( \left[ \sigma ,\tau \right] \times \left[ \sigma ^{\prime },\tau ^{\prime }\right] \right)\end{aligned}$$and taking the supremum over all partitions shows the result. \[lemma\_rhovar\_diff\_n2\]Let $\left( X,Y\right) \colon \left[ 0,1\right] \rightarrow \mathbb{R}^{2}$ be a centred Gaussian process with continuous paths of finite variation. Assume that the $\rho $-variation of $R_{\left( X,Y\right) }$ is controlled by a $2D$-control $\omega $ where $\rho \geq 1$. Consider the function$$g\left( u,v\right) =E\left[ \left( \mathbf{X}_{s,u}^{\left( 2\right) }-% \mathbf{Y}_{s,u}^{\left( 2\right) }\right) \left( \mathbf{X}_{s,v}^{\left( 2\right) }-\mathbf{Y}_{s,v}^{\left( 2\right) }\right) \right] .$$Then for every $\gamma >\rho $ there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$V_{\gamma -var}\left( g,\left[ s,t\right] ^{2}\right) \leq C\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma +1/\rho }$$holds for every $\left( s,t\right) \in \Delta $ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. Let $u<v$ and $u^{\prime }<v^{\prime }$. Then$$\begin{aligned} g% \begin{pmatrix} u,v \\ u^{\prime },v^{\prime }% \end{pmatrix} &=&E\left[ \left( \left( \mathbf{X}_{s,v}^{\left( 2\right) }-\mathbf{X}% _{s,u}^{\left( 2\right) }\right) -\left( \mathbf{Y}_{s,v}^{\left( 2\right) }-% \mathbf{Y}_{s,u}^{\left( 2\right) }\right) \right) \left( \left( \mathbf{X}% _{s,v^{\prime }}^{\left( 2\right) }-\mathbf{X}_{s,u^{\prime }}^{\left( 2\right) }\right) -\left( \mathbf{Y}_{s,v^{\prime }}^{\left( 2\right) }-% \mathbf{Y}_{s,u^{\prime }}^{\left( 2\right) }\right) \right) \right] \\ &=&\frac{1}{2^{2}}E\left[ \left( \left( X_{s,v}^{2}-X_{s,u}^{2}\right) -\left( Y_{s,v}^{2}-Y_{s,u}^{2}\right) \right) \left( \left( X_{s,v^{\prime }}^{2}-X_{s,u^{\prime }}^{2}\right) -\left( Y_{s,v^{\prime }}^{2}-Y_{s,u^{\prime }}^{2}\right) \right) \right] .\end{aligned}$$Now,$$\begin{aligned} \left( X_{s,v}^{2}-X_{s,u}^{2}\right) -\left( Y_{s,v}^{2}-Y_{s,u}^{2}\right) &=&X_{u,v}\left( X_{s,u}+X_{s,v}\right) -Y_{u,v}\left( Y_{s,u}+Y_{s,v}\right) \\ &=&X_{u,v}\left( X_{s,u}-Y_{s,u}\right) +\left( X_{u,v}-Y_{u,v}\right) Y_{s,u} \\ &&+X_{u,v}\left( X_{s,v}-Y_{s,v}\right) +\left( X_{u,v}-Y_{u,v}\right) Y_{s,v}.\end{aligned}$$The same way one gets$$\begin{aligned} \left( X_{s,v^{\prime }}^{2}-X_{s,u^{\prime }}^{2}\right) -\left( Y_{s,v^{\prime }}^{2}-Y_{s,u^{\prime }}^{2}\right) &=&X_{u^{\prime },v^{\prime }}\left( X_{s,u^{\prime }}-Y_{s,u^{\prime }}\right) +\left( X_{u^{\prime },v^{\prime }}-Y_{u^{\prime },v^{\prime }}\right) Y_{s,u^{\prime }} \\ &&+X_{u^{\prime },v^{\prime }}\left( X_{s,v^{\prime }}-Y_{s,v^{\prime }}\right) +\left( X_{u^{\prime },v^{\prime }}-Y_{u^{\prime },v^{\prime }}\right) Y_{s,v^{\prime }}.\end{aligned}$$Now we expand the product of both sums and take expectation. For the first term we obtain, using the Wick formula and Lemma \[lemma\_half\_estimates\],$$\begin{aligned} &&\left\vert E\left( X_{u,v}\left( X_{s,u}-Y_{s,u}\right) X_{u^{\prime },v^{\prime }}\left( X_{s,u^{\prime }}-Y_{s,u^{\prime }}\right) \right) \right\vert \\ &\leq &\left\vert E\left( X_{u,v}X_{u^{\prime },v^{\prime }}\right) E\left[ \left( X_{s,u}-Y_{s,u}\right) \left( X_{s,u^{\prime }}-Y_{s,u^{\prime }}\right) \right] \right\vert \\ &&+\left\vert E\left[ X_{u,v}\left( X_{s,u^{\prime }}-Y_{s,u^{\prime }}\right) \right] E\left[ X_{u^{\prime },v^{\prime }}\left( X_{s,u}-Y_{s,u}\right) \right] \right\vert \\ &&+\left\vert E\left[ X_{u^{\prime },v^{\prime }}\left( X_{s,u^{\prime }}-Y_{s,u^{\prime }}\right) \right] E\left[ X_{u,v}\left( X_{s,u}-Y_{s,u}\right) \right] \right\vert \\ &\leq &V_{\rho -var}\left( R_{\left( X,Y\right) },\left[ u,v\right] \times % \left[ u^{\prime },v^{\prime }\right] \right) V_{\gamma -var}\left( R_{X-Y},% \left[ s,t\right] ^{2}\right) \\ &&+2V_{\gamma -var}\left( R_{\left( X,X-Y\right) },\left[ u,v\right] \times % \left[ s,t\right] \right) V_{\gamma -var}\left( R_{\left( X,X-Y\right) },% \left[ u^{\prime },v^{\prime }\right] \times \left[ s,t\right] \right) \\ &\leq &\epsilon ^{2}\omega \left( \left[ u,v\right] \times \left[ u^{\prime },v^{\prime }\right] \right) ^{1/\rho }\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma } \\ &&+2\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\rho -1/\gamma }\omega \left( \left[ u,v\right] \times \left[ s,t\right] \right) ^{1/\gamma }\omega \left( \left[ u^{\prime },v^{\prime }\right] \times \left[ s,t\right] \right) ^{1/\gamma }.\end{aligned}$$Now take two partitions $D,\tilde{D}$ of $\left[ s,t\right] $. With our calculations above,$$\begin{aligned} &&\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\left\vert E\left( X_{t_{i},t_{i+1}}\left( X_{s,t_{i}}-Y_{s,t_{i}}\right) X_{\tilde{t}_{j},% \tilde{t}_{j+1}}\left( X_{s,\tilde{t}_{j}}-Y_{s,\tilde{t}_{j}}\right) \right) \right\vert ^{\gamma } \\ &\leq &c_{1}\epsilon ^{2\gamma }\omega \left( \left[ s,t\right] ^{2}\right) \sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\omega \left( \left[ t_{i},t_{i+1}\right] \times \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] \right) ^{\gamma /\rho } \\ &&+c_{2}\epsilon ^{2\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{\gamma /\rho -1}\sum_{t_{i}\in D,\tilde{t}_{j}\in \tilde{D}}\omega \left( % \left[ t_{i},t_{i+1}\right] \times \left[ s,t\right] \right) \omega \left( % \left[ \tilde{t}_{j},\tilde{t}_{j+1}\right] \times \left[ s,t\right] \right) \\ &\leq &c_{3}\epsilon ^{2\gamma }\left( \omega \left( \left[ s,t\right] ^{2}\right) \omega \left( \left[ s,t\right] ^{2}\right) ^{\gamma /\rho }+\omega \left( \left[ s,t\right] ^{2}\right) ^{\gamma /\rho -1}\omega \left( \left[ s,t\right] ^{2}\right) ^{2}\right) .\end{aligned}$$The other terms are treated exactly the same way. Taking the supremum over all partitions shows the result. The next corollary completes the proof of Proposition [prop\_main\_estimates\_n3]{}. Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma [lemma\_diff\_alldifferent]{}. Then there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$\left\vert \mathbf{X}_{s,t}^{i,i,j}-\mathbf{Y}_{s,t}^{i,i,j}\right\vert _{L^{2}}\leq C\epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1% }{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{2}{2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ and $i\neq j$ where $% \epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. From the triangle inequality,$$\left\vert \mathbf{X}_{s,t}^{i,i,j}-\mathbf{Y}_{s,t}^{i,i,j}\right\vert _{L^{2}}\leq \left\vert \int_{\left[ s,t\right] }\left( \mathbf{X}% _{s,u}^{i,i}-\mathbf{Y}_{s,u}^{i,i}\right) \,dY_{u}^{j}\right\vert _{L^{2}}+\left\vert \int_{\left[ s,t\right] }\mathbf{Y}_{s,u}^{i,i}\,d\left( X^{j}-Y^{j}\right) _{u}\right\vert _{L^{2}}.$$For the first integral, we use independence to move the expectation inside the integral as seen in the proof of Lemma \[lemma\_diff\_alldifferent\], then we use $2D$ Young integration and Lemma \[lemma\_rhovar\_diff\_n2\] to obtain the desired estimate. The second integral is estimated in the same way using Lemma \[lemma\_rho\_var\_same\_letters\]. ### $n=4\label{section_n=4}$ \[prop\_main\_estimates\_n4\]Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma \[lemma\_diff\_alldifferent\]. Then there is a constant $C\left( 4\right) $ which depends on $\rho $ and $\gamma $ such that$$\left\vert \mathbf{X}_{s,t}^{4}-\mathbf{Y}_{s,t}^{4}\right\vert _{L^{2}}\leq C\left( 4\right) \epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{% \frac{1}{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{3}{% 2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. From Proposition \[proposition\_key\_shuffle\] and \[prop\_generating\_sets\] one sees that it is enough to show the estimate for $\mathbf{X}^{w}-\mathbf{Y% }^{w}$ where$$w\in \left\{ iiii,ijkl,iijj,iiij,iijk,jiik:i,j,k,l\in \left\{ 1,\ldots ,d\right\} ~\text{distinct}\right\} .$$The cases $w=iiii$ and $w=ijkl$ are special cases of Lemma [lemma\_diff\_allthesame]{} and Lemma \[lemma\_diff\_alldifferent\]. Hence it remains to show the estimate for$$w\in \left\{ iijj,iiij,iijk,jiik:i,j,k\in \left\{ 1,\ldots ,d\right\} ~\text{% pairwise distinct}\right\} \text{.}$$This is the content of the remaining section. Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma [lemma\_diff\_alldifferent]{}. Then there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$\left\vert \mathbf{X}_{s,t}^{i,i,j,k}-\mathbf{Y}_{s,t}^{i,i,j,k}\right\vert _{L^{2}}\leq C\epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1% }{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{3}{2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ where $i,j,k$ are distinct and $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. From the triangle inequality,$$\begin{aligned} &&\left\vert \mathbf{X}_{s,t}^{i,i,j,k}-\mathbf{Y}_{s,t}^{i,i,j,k}\right% \vert _{L^{2}} \\ &=&\left\vert \int_{\left\{ s<u<v<t\right\} }\mathbf{X}_{s,u}^{i,i}% \,dX_{u}^{j}\,dX_{v}^{k}-\int_{\left\{ s<u<v<t\right\} }\mathbf{Y}% _{s,u}^{i,i}\,dY_{u}^{j}\,dY_{v}^{k}\right\vert _{L^{2}} \\ &\leq &\left\vert \int_{\left\{ s<u<v<t\right\} }\left( \mathbf{X}% _{s,u}^{i,i}-\mathbf{Y}_{s,u}^{i,i}\right) \,dX_{u}^{j}\,dX_{v}^{k}\right\vert _{L^{2}}+\left\vert \int_{\left\{ s<u<v<t\right\} }\mathbf{Y}_{s,u}^{i,i}\,d\left( X^{j}-Y^{j}\right) _{u}\,dX_{v}^{k}\right\vert _{L^{2}} \\ &&+\left\vert \int_{\left\{ s<u<v<t\right\} }\mathbf{Y}_{s,u}^{i,i}% \,dY_{u}^{j}\,d\left( X^{k}-Y^{k}\right) _{v}\right\vert _{L^{2}}.\end{aligned}$$For the first integral, we use Proposition \[prop\_rho\_var\_iter\_2D\] and Lemma \[lemma\_rhovar\_diff\_n2\] to obtain$$\begin{aligned} \left\vert \int_{\left\{ s<u<v<t\right\} }\left( \mathbf{X}_{s,u}^{i,i}-% \mathbf{Y}_{s,u}^{i,i}\right) \,dX_{u}^{j}\,dX_{v}^{k}\right\vert _{L^{2}}^{2} &=&\int_{\Delta _{s,t}^{2}\times \Delta _{s,t}^{2}}E\left[ \left( \mathbf{X}_{s,\cdot }^{i,i}-\mathbf{Y}_{s,\cdot }^{i,i}\right) \left( \mathbf{X}_{s,\cdot }^{i,i}-\mathbf{Y}_{s,\cdot }^{i,i}\right) \right] \,dR_{X^{j}}\,dR_{X^{k}} \\ &\leq &c_{1}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma +1/\rho }\omega \left( \left[ s,t\right] ^{2}\right) ^{2/\rho }.\end{aligned}$$For the other two integrals we also use Proposition [prop\_rho\_var\_iter\_2D]{} together with Lemma \[lemma\_rho\_var\_same\_letters\] to obtain the same estimate. \[lemma\_rhovar\_diff\_n3\]Let $\left( X,Y\right) \colon \left[ 0,1\right] \rightarrow \mathbb{R}^{2}$ be a centred Gaussian process with continuous paths of finite variation. Assume that the $\rho $-variation of $R_{\left( X,Y\right) }$ is controlled by a $2D$-control $\omega $ where $\rho \geq 1$. Consider the function$$g\left( u,v\right) =E\left[ \left( \mathbf{X}_{s,u}^{\left( 3\right) }-% \mathbf{Y}_{s,u}^{\left( 3\right) }\right) \left( \mathbf{X}_{s,v}^{\left( 3\right) }-\mathbf{Y}_{s,v}^{\left( 3\right) }\right) \right] .$$Then for every $\gamma >\rho $ there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$V_{\gamma -var}\left( g,\left[ s,t\right] ^{2}\right) \leq C\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma +2/\rho }$$holds for every $\left( s,t\right) \in \Delta $ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{\left( 1-\rho /\gamma \right) }$. Similar to the one of Lemma \[lemma\_rhovar\_diff\_n2\] applying again Wick’s formula. Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma [lemma\_diff\_alldifferent]{}. Then there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$\left\vert \mathbf{X}_{s,t}^{i,i,i,j}-\mathbf{Y}_{s,t}^{i,i,i,j}\right\vert _{L^{2}}\leq C\epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1% }{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{3}{2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ and $i\neq j$ where $% \epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{\left( 1-\rho /\gamma \right) }$. The triangle inequality gives$$\begin{aligned} \left\vert \mathbf{X}_{s,t}^{i,i,i,j}-\mathbf{Y}_{s,t}^{i,i,i,j}\right\vert _{L^{2}} &=&\left\vert \int_{\left[ s,t\right] }\mathbf{X}% _{s,u}^{i,i,i}\,dX_{u}^{j}-\int_{\left[ s,t\right] }\mathbf{Y}% _{s,u}^{i,i,i}\,dY_{u}^{j}\right\vert \\ &\leq &\left\vert \int_{\left[ s,t\right] }\left( \mathbf{X}_{s,u}^{i,i,i}-% \mathbf{Y}_{s,u}^{i,i,i}\right) \,dX_{u}^{j}\right\vert _{L^{2}}+\left\vert \int_{\left[ s,t\right] }\mathbf{Y}_{s,u}^{i,i,i}\,d\left( X^{j}-Y^{j}\right) _{u}\right\vert _{L^{2}}.\end{aligned}$$For the first integral, we move the expectation inside the integral, use $2D$ Young integration and Lemma \[lemma\_rhovar\_diff\_n3\] to conclude the estimate. The second integral is estimated the same way applying Lemma [lemma\_rho\_var\_same\_letters]{}. It remains to show the estimates for $\mathbf{X}^{w}-\mathbf{Y}^{w}$ where $% w\in \left\{ iijj,jiik\right\} $. We need to be a bit careful here for the following reason: It is clear that $\mathbf{X}_{0,1}^{i,i,j}=\int_{\left[ 0,1% \right] }\mathbf{X}_{u}^{i,i}\,dX_{u}^{j}$. One might expect that also $% \mathbf{X}_{0,1}^{j,i,i}=\int_{\left[ 0,1\right] }X_{u}^{j}\,d\mathbf{X}% _{u}^{i,i}$ holds, but this is not true in general. Indeed, just take $% f\left( u\right) =g\left( u\right) =u$. Then$$\int_{0}^{1}f\left( u\right) \,d\left( \int_{0}^{u}g\left( v\right) \,dg\left( v\right) \right) =\frac{1}{2}\int_{0}^{1}u\,d\left( u^{2}\right) =\int_{0}^{1}u^{2}\,du=\frac{1}{3}$$but$$\int_{\Delta _{0,1}^{2}}f\left( u\right) \,dg\left( u\right) \,dg\left( v\right) =\int_{\Delta _{0,1}^{3}}du_{1}\,du_{2}\,du_{3}=\frac{1}{6}.$$One the other hand, if $g$ is smooth, we can use Fubini to see that$$\begin{aligned} \int_{\Delta _{0,1}^{2}}f\left( u\right) \,dg\left( u\right) \,dg\left( v\right) &=&\int_{\left[ 0,1\right] ^{2}}f\left( u\right) g^{\prime }\left( u\right) g^{\prime }\left( v\right) 1_{\left\{ u<v\right\} }\,du\,dv \\ &=&\frac{1}{2}\int_{\left[ 0,1\right] ^{2}}f\left( u\right) g^{\prime }\left( u\right) g^{\prime }\left( v\right) 1_{\left\{ u<v\right\} }\,du\,dv \\ &&+\frac{1}{2}\int_{\left[ 0,1\right] ^{2}}f\left( v\right) g^{\prime }\left( v\right) g^{\prime }\left( u\right) 1_{\left\{ v<u\right\} }\,du\,dv \\ &=&\frac{1}{2}\int_{\left[ 0,1\right] ^{2}}\left( f\left( u\right) 1_{\left\{ u<v\right\} }+f\left( v\right) 1_{\left\{ v<u\right\} }\right) g^{\prime }\left( u\right) g^{\prime }\left( v\right) \,du\,dv \\ &=&\frac{1}{2}\int_{\left[ 0,1\right] ^{2}}f\left( u\wedge v\right) g^{\prime }\left( u\right) g^{\prime }\left( v\right) \,du\,dv \\ &=&\frac{1}{2}\int_{\left[ 0,1\right] ^{2}}f\left( u\wedge v\right) \,d\left( g\left( u\right) g\left( v\right) \right)\end{aligned}$$where the last integral is a $2D$ Young integral. Hence we have seen that an iterated $1D$-integral can be transformed into a usual $2D$-integral. We will use this trick for the remaining estimates. \[lemma\_rho\_var\_invar\_symm\]Let $f\colon \left[ 0,1\right] ^{2}\rightarrow \mathbb{R}$ be a continuous function. Set$$\bar{f}\left( u_{1},u_{2},v_{1},v_{2}\right) =f\left( u_{1}\wedge u_{2},v_{1}\wedge v_{2}\right) .$$ 1. Let $u_{1}<\tilde{u}_{1},u_{2}<\tilde{u}_{2},v_{1}<\tilde{v}_{1},v_{2}<% \tilde{v}_{2}$ be all in $\left[ 0,1\right] $. Then$$\bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ v_{1},\tilde{v}_{1} \\ v_{2},\tilde{v}_{2}% \end{array}% \right) =f% \begin{pmatrix} u,\tilde{u} \\ v,\tilde{v}% \end{pmatrix}%$$where we set$$\begin{aligned} \left[ u,\tilde{u}\right] &=&\left\{ \begin{array}{ccc} \left[ u_{1},\tilde{u}_{1}\right] \cap \left[ u_{2},\tilde{u}_{2}\right] & \text{if} & \left[ u_{1},\tilde{u}_{1}\right] \cap \left[ u_{2},\tilde{u}_{2}% \right] \neq \emptyset \\ \left[ 0,0\right] & \text{if} & \left[ u_{1},\tilde{u}_{1}\right] \cap \left[ u_{2},\tilde{u}_{2}\right] =\emptyset% \end{array}% \right. . \\ \left[ v,\tilde{v}\right] &=&\left\{ \begin{array}{ccc} \left[ v_{1},\tilde{v}_{1}\right] \cap \left[ v_{2},\tilde{v}_{2}\right] & \text{if} & \left[ v_{1},\tilde{v}_{1}\right] \cap \left[ v_{2},\tilde{v}_{2}% \right] \neq \emptyset \\ \left[ 0,0\right] & \text{if} & \left[ v_{1},\tilde{v}_{1}\right] \cap \left[ v_{2},\tilde{v}_{2}\right] =\emptyset% \end{array}% \right.\end{aligned}$$ 2. For $s<t$, $\sigma <t$ and $p\geq 1$ we have$$V_{p}\left( f,\left[ s,t\right] \times \left[ \sigma ,\tau \right] \right) =V_{p}\left( \bar{f},\left[ s,t\right] ^{2}\times \left[ \sigma ,\tau \right] ^{2}\right) .$$ <!-- --> 1. By definition of the higher dimensional increments,$$\begin{aligned} \bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ v_{1} \\ v_{2}% \end{array}% \right) &=&\bar{f}\left( \begin{array}{c} \tilde{u}_{1} \\ \tilde{u}_{2} \\ v_{1} \\ v_{2}% \end{array}% \right) -\bar{f}\left( \begin{array}{c} \tilde{u}_{1} \\ u_{2} \\ v_{1} \\ v_{2}% \end{array}% \right) -\bar{f}\left( \begin{array}{c} u_{1} \\ \tilde{u}_{2} \\ v_{1} \\ v_{2}% \end{array}% \right) +\bar{f}\left( \begin{array}{c} u_{1} \\ u_{2} \\ v_{1} \\ v_{2}% \end{array}% \right) \\ &=&f\left( \tilde{u}_{1}\wedge \tilde{u}_{2},v_{1}\wedge v_{2}\right) -f\left( \tilde{u}_{1}\wedge u_{2},v_{1}\wedge v_{2}\right) \\ &&-f\left( u_{1}\wedge \tilde{u}_{2},v_{1}\wedge v_{2}\right) +f\left( u_{1}\wedge u_{2},v_{1}\wedge v_{2}\right) .\end{aligned}$$By a case distinction, one sees that this is equal to $f\left( \tilde{u}% ,v_{1}\wedge v_{2}\right) -f\left( u,v_{1}\wedge v_{2}\right) $. One goes on with$$\begin{aligned} \bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ v_{1},\tilde{v}_{1} \\ v_{2},\tilde{v}_{2}% \end{array}% \right) &=&\bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ \tilde{v}_{1} \\ \tilde{v}_{2}% \end{array}% \right) -\bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ \tilde{v}_{1} \\ v_{2}% \end{array}% \right) -\bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ v_{1} \\ \tilde{v}_{2}% \end{array}% \right) +\bar{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ v_{1} \\ v_{2}% \end{array}% \right) \\ &=&h\left( \tilde{v}_{1}\wedge \tilde{v}_{2}\right) -h\left( \tilde{v}% _{1}\wedge v_{2}\right) -h\left( v_{1}\wedge \tilde{v}_{2}\right) +h\left( v_{1}\wedge v_{2}\right) \\ &=&h\left( \tilde{v}\right) -h\left( v\right)\end{aligned}$$where $h\left( \cdot \right) =f\left( \tilde{u},\cdot \right) -f\left( u,\cdot \right) .$Hence$$h\left( \tilde{v}\right) -h\left( v\right) =f\left( \tilde{u},\tilde{v}% \right) -f\left( u,\tilde{v}\right) -f\left( \tilde{u},v\right) +f\left( u,v\right) =f\left( \begin{array}{c} u,\tilde{u} \\ v,\tilde{v}% \end{array}% \right) .$$ 2. Let $D$ be a partition of $\left[ s,t\right] $ and $\tilde{D}$ a partition of $\left[ \sigma ,\tau \right] $. Then by 1,$$\sum_{t_{i}\in D,\tilde{t}\in \tilde{D}}\left\vert f\left( \begin{array}{c} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{array}% \right) \right\vert ^{p}=\sum_{t_{i}\in D,\tilde{t}\in \tilde{D}}\left\vert \bar{f}\left( \begin{array}{c} t_{i},t_{i+1} \\ t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{array}% \right) \right\vert ^{p}\leq \left( V_{p}\left( \bar{f},\left[ s,t\right] ^{2}\times \left[ \sigma ,\tau \right] ^{2}\right) \right) ^{p},$$hence $V_{p}\left( f,\left[ s,t\right] \times \left[ \sigma ,\tau \right] \right) \leq V_{p}\left( \bar{f},\left[ s,t\right] ^{2}\times \left[ \sigma ,\tau \right] ^{2}\right) $. Now let $D_{1},D_{2}$ be partitions of $\left[ s,t\right] $ and $\tilde{D}_{1},\tilde{D}_{2}$ be partitions of $\left[ \sigma ,\tau \right] $. Set $D=D_{1}\cup D_{2}$, $\tilde{D}=\tilde{D}% _{1}\cup \tilde{D}_{2}$. Then $D$ is a partition of $\left[ s,t\right] $ and $\tilde{D}$ a partition of $\left[ \sigma ,\tau \right] $ (see Figure 1 below). By (1),$$\sum_{\substack{ t_{i_{1}}^{1}\in D_{1},t_{i_{2}}^{2}\in D_{2} \\ \tilde{t}% _{j_{1}}^{1}\in \tilde{D}_{1},\tilde{t}_{j_{2}}^{2}\in \tilde{D}_{2}}}% \left\vert f\left( \begin{array}{c} t_{i_{1}}^{1},t_{i_{1}+1}^{1} \\ t_{i_{2}}^{2},t_{i_{2}+1}^{2} \\ \tilde{t}_{j_{1}}^{1},\tilde{t}_{j_{1}+1}^{1} \\ \tilde{t}_{j_{2}}^{2},\tilde{t}_{j_{2}+1}^{2}% \end{array}% \right) \right\vert ^{p}=\sum_{t_{i}\in D,\tilde{t}\in \tilde{D}}\left\vert f\left( \begin{array}{c} t_{i},t_{i+1} \\ \tilde{t}_{j},\tilde{t}_{j+1}% \end{array}% \right) \right\vert ^{p}\leq \left( V_{p}\left( f,\left[ s,t\right] \times % \left[ \sigma ,\tau \right] \right) \right) ^{p}$$and we also get $V_{p}\left( \bar{f},\left[ s,t\right] ^{2}\times \left[ \sigma ,\tau \right] ^{2}\right) \leq V_{p}\left( f,\left[ s,t\right] \times % \left[ \sigma ,\tau \right] \right) $. \[lemma\_rho\_var\_4D\_estimates\]Let $\left( X,Y\right) \colon \left[ 0,1% \right] \rightarrow \mathbb{R}^{2}$ be a centred Gaussian process with continuous paths of finite variation and assume that $\omega $ is a symmetric control which controls the $\rho $-variation of $R_{\left( X,Y\right) }$ where $\rho \geq 1$. Take $\left( s,t\right) \in \Delta $, $% \gamma >\rho $ and set $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t% \right] ^{2}\right) ^{1-\rho /\gamma }$. 1. Set $f\left( u_{1},u_{2},v_{1},v_{2}\right) =E\left[ X_{u_{1}}X_{u_{2}}X_{v_{1}}X_{v_{2}}\right] $. Then there is a constant $% C_{1}=C_{1}\left( \rho \right) $ and a symmetric $4D$ grid-control $\tilde{% \omega}_{1}$ which controls the $\rho $-variation of $f$ and$$V_{\rho }\left( f,\left[ s,t\right] ^{4}\right) \leq \tilde{\omega}% _{1}\left( \left[ s,t\right] ^{4}\right) ^{1/\rho }=C_{1}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{2}{\rho }}.$$ 2. Set $\tilde{f}\left( u_{1},u_{2},v_{1},v_{2}\right) =E\left[ \mathbf{X}% _{s,u_{1}\wedge u_{2}}^{\left( 2\right) }\mathbf{X}_{s,v_{1}\wedge v_{2}}^{\left( 2\right) }\right] $. Then there is a constant $% C_{2}=C_{2}\left( \rho \right) $ such that$$V_{\rho }\left( \tilde{f},\left[ s,t\right] ^{4}\right) \leq C_{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{2}{\rho }}.$$ 3. Set$$g\left( u_{1},u_{2},v_{1},v_{2}\right) =E\left[ \left( X_{u_{1}}X_{u_{2}}-Y_{u_{1}}Y_{u_{2}}\right) \left( X_{v_{1}}X_{v_{2}}-Y_{v_{1}}Y_{v_{2}}\right) \right] .$$Then there is a constant $C_{3}=C_{3}\left( \rho ,\gamma \right) $ and a symmetric $4D$ grid-control $\tilde{\omega}_{2}$ which controls the $\gamma $-variation of $g$ and$$V_{\gamma }\left( g,\left[ s,t\right] ^{4}\right) \leq \tilde{\omega}% _{2}\left( \left[ s,t\right] ^{4}\right) ^{1/\gamma }=C_{3}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma +1/\rho }.$$ 4. Set$$\tilde{g}\left( u_{1},u_{2},v_{1},v_{2}\right) =E\left[ \left( \mathbf{X}% ^{\left( 2\right) }-\mathbf{Y}^{\left( 2\right) }\right) _{s,u_{1}\wedge u_{2}}\left( \mathbf{X}^{\left( 2\right) }-\mathbf{Y}^{\left( 2\right) }\right) _{s,v_{1}\wedge v_{2}}\right] .$$Then there is a constant $C_{4}=C_{4}\left( \rho ,\gamma \right) $ such that$$V_{\gamma }\left( \tilde{g},\left[ s,t\right] ^{4}\right) \leq C_{4}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma +1/\rho }.$$ <!-- --> 1. Let $u_{1}<\tilde{u}_{1}$, $u_{2}<\tilde{u}_{2}$, $v_{1}<\tilde{v}_{1}$, $v_{2}<\tilde{v}_{2}$. By the Wick-formula,$$\begin{aligned} &&\left\vert E\left[ X_{u_{1},\tilde{u}_{1}}X_{u_{2},\tilde{u}_{2}}X_{v_{1},% \tilde{v}_{1}}X_{v_{2},\tilde{v}_{2}}\right] \right\vert ^{\rho } \\ &\leq &3^{\rho -1}\left\vert E\left[ X_{u_{1},\tilde{u}_{1}}X_{u_{2},\tilde{u% }_{2}}\right] E\left[ X_{v_{1},\tilde{v}_{1}}X_{v_{2},\tilde{v}_{2}}\right] \right\vert ^{\rho }+3^{\rho -1}\left\vert E\left[ X_{u_{1},\tilde{u}% _{1}}X_{v_{1},\tilde{v}_{1}}\right] E\left[ X_{u_{2},\tilde{u}_{2}}X_{v_{2},% \tilde{v}_{2}}\right] \right\vert ^{\rho } \\ &&+3^{\rho -1}\left\vert E\left[ X_{u_{1},\tilde{u}_{1}}X_{v_{2},\tilde{v}% _{2}}\right] E\left[ X_{u_{2},\tilde{u}_{2}}X_{v_{1},\tilde{v}_{1}}\right] \right\vert ^{\rho } \\ &\leq &3^{\rho -1}\omega \left( \left[ u_{1},\tilde{u}_{1}\right] \times % \left[ u_{2},\tilde{u}_{2}\right] \right) \omega \left( \left[ v_{1},\tilde{v% }_{1}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \\ &&+3^{\rho -1}\omega \left( \left[ u_{1},\tilde{u}_{1}\right] \times \left[ v_{1},\tilde{v}_{1}\right] \right) \omega \left( \left[ u_{2},\tilde{u}_{2}% \right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \\ &&+3^{\rho -1}\omega \left( \left[ u_{1},\tilde{u}_{1}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \omega \left( \left[ u_{2},\tilde{u}_{2}% \right] \times \left[ v_{1},\tilde{v}_{1}\right] \right) \\ &=&:\tilde{\omega}_{1}\left( \left[ u_{1},\tilde{u}_{1}\right] \times \left[ u_{2},\tilde{u}_{2}\right] \times \left[ v_{1},\tilde{v}_{1}\right] \times % \left[ v_{2},\tilde{v}_{2}\right] \right) .\end{aligned}$$It is easy to see that $\tilde{\omega}_{1}$ is a symmetric grid-control and that it fulfils the stated property. 2. A direct consequence of Lemma \[lemma\_rho\_var\_same\_letters\] and Lemma \[lemma\_rho\_var\_invar\_symm\]. 3. We have$$X_{u_{1}}X_{u_{2}}-Y_{u_{1}}Y_{u_{2}}=\left( X_{u_{1}}-Y_{u_{1}}\right) X_{u_{2}}+Y_{u_{1}}\left( X_{u_{2}}-Y_{u_{2}}\right) .$$Hence for $u_{1}<\tilde{u}_{1},u_{2}<\tilde{u}_{2},v_{1}<\tilde{v}_{1},v_{2}<% \tilde{v}_{2}$,$$\begin{aligned} \tilde{f}\left( \begin{array}{c} u_{1},\tilde{u}_{1} \\ u_{2},\tilde{u}_{2} \\ v_{1},\tilde{v}_{1} \\ v_{2},\tilde{v}_{2}% \end{array}% \right) &=&E\left[ \left( X-Y\right) _{u_{1},\tilde{u}_{1}}X_{u_{2},\tilde{u}% _{2}}\left( X-Y\right) _{v_{1},\tilde{v}_{1}}X_{v_{2},\tilde{v}_{2}}\right] \\ &&+E\left[ Y_{u_{1},\tilde{u}_{1}}\left( X-Y\right) _{u_{2},\tilde{u}% _{2}}\left( X-Y\right) _{v_{1},\tilde{v}_{1}}X_{v_{2},\tilde{v}_{2}}\right] \\ &&+E\left[ \left( X-Y\right) _{u_{1},\tilde{u}_{1}}X_{u_{2},\tilde{u}% _{2}}Y_{v_{1},\tilde{v}_{1}}\left( X-Y\right) _{v_{2},\tilde{v}_{2}}\right] \\ &&+E\left[ Y_{u_{1},\tilde{u}_{1}}\left( X-Y\right) _{u_{2},\tilde{u}% _{2}}Y_{v_{1},\tilde{v}_{1}}\left( X-Y\right) _{v_{2},\tilde{v}_{2}}\right]\end{aligned}$$For the first term we have, using Lemma \[lemma\_half\_estimates\], $$\begin{aligned} &&\left\vert E\left[ \left( X-Y\right) _{u_{1},\tilde{u}_{1}}X_{u_{2},\tilde{% u}_{2}}\left( X-Y\right) _{v_{1},\tilde{v}_{1}}X_{v_{2},\tilde{v}_{2}}\right] \right\vert ^{\gamma } \\ &\leq &3^{\gamma -1}\left\vert E\left[ \left( X-Y\right) _{u_{1},\tilde{u}% _{1}}X_{u_{2},\tilde{u}_{2}}\right] \right\vert ^{\gamma }\left\vert E\left[ \left( X-Y\right) _{v_{1},\tilde{v}_{1}}X_{v_{2},\tilde{v}_{2}}\right] \right\vert ^{\gamma } \\ &&+3^{\gamma -1}\left\vert E\left[ \left( X-Y\right) _{u_{1},\tilde{u}% _{1}}\left( X-Y\right) _{v_{1},\tilde{v}_{1}}\right] \right\vert ^{\gamma }\left\vert E\left[ X_{u_{2},\tilde{u}_{2}}X_{v_{2},\tilde{v}_{2}}\right] \right\vert ^{\gamma } \\ &&+3^{\gamma -1}\left\vert E\left[ \left( X-Y\right) _{u_{1},\tilde{u}% _{1}}X_{v_{2},\tilde{v}_{2}}\right] \right\vert ^{\gamma }\left\vert E\left[ X_{u_{2},\tilde{u}_{2}}\left( X-Y\right) _{v_{1},\tilde{v}_{1}}\right] \right\vert ^{\gamma } \\ &\leq &3^{\gamma -1}\epsilon ^{2\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{\gamma }{\rho }-1}\omega \left( \left[ u_{1},\tilde{u}% _{1}\right] \times \left[ u_{2},\tilde{u}_{2}\right] \right) \omega \left( % \left[ v_{1},\tilde{v}_{1}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \\ &&+3^{\gamma -1}\epsilon ^{2\gamma }\omega \left( \left[ u_{1},\tilde{u}_{1}% \right] \times \left[ v_{1},\tilde{v}_{1}\right] \right) \omega \left( \left[ u_{2},\tilde{u}_{2}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) ^{\frac{\gamma }{\rho }} \\ &&+3^{\gamma -1}\epsilon ^{2\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{\gamma }{\rho }-1}\omega \left( \left[ u_{1},\tilde{u}% _{1}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \omega \left( % \left[ u_{2},\tilde{u}_{2}\right] \times \left[ v_{1},\tilde{v}_{1}\right] \right) \\ &\leq &3^{\gamma -1}\epsilon ^{2\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{\gamma }{\rho }-1}\left( \omega \left( \left[ u_{1},% \tilde{u}_{1}\right] \times \left[ u_{2},\tilde{u}_{2}\right] \right) \omega \left( \left[ v_{1},\tilde{v}_{1}\right] \times \left[ v_{2},\tilde{v}_{2}% \right] \right) \right. \\ &&+\omega \left( \left[ u_{1},\tilde{u}_{1}\right] \times \left[ v_{1},% \tilde{v}_{1}\right] \right) \omega \left( \left[ u_{2},\tilde{u}_{2}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \\ &&\left. +\omega \left( \left[ u_{1},\tilde{u}_{1}\right] \times \left[ v_{2},\tilde{v}_{2}\right] \right) \omega \left( \left[ u_{2},\tilde{u}_{2}% \right] \times \left[ v_{1},\tilde{v}_{1}\right] \right) \right) \\ &=&:\tilde{\omega}\left( \left[ u_{1},\tilde{u}_{1}\right] \times \left[ u_{2},\tilde{u}_{2}\right] \times \left[ v_{1},\tilde{v}_{1}\right] \times % \left[ v_{2},\tilde{v}_{2}\right] \right) .\end{aligned}$$$\tilde{\omega}$ is a symmetric grid-control and fulfils the stated property. The other terms are treated in the same way. 4. Follows from Lemma \[lemma\_rhovar\_diff\_n2\] and Lemma [lemma\_rho\_var\_invar\_symm]{}. Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma [lemma\_diff\_alldifferent]{}. Then there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$\left\vert \mathbf{X}_{s,t}^{i,i,j,j}-\mathbf{Y}_{s,t}^{i,i,j,j}\right\vert _{L^{2}}\leq C\epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1% }{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{3}{2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ and $i\neq j$ where $% \epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. As seen before, we can use Fubini to obtain$$\mathbf{X}_{s,t}^{i,i,j,j}=\int_{\Delta _{s,t}^{2}}\mathbf{X}% _{s,u_{1}}^{i,i}\,dX_{u_{1}}^{j}\,dX_{u_{2}}^{j}=\frac{1}{2}\int_{\left[ s,t% \right] ^{2}}\mathbf{X}_{s,u_{1}\wedge u_{2}}^{i,i}\,d\left( X_{u_{1}}^{j}X_{u_{2}}^{j}\right)$$and hence$$\begin{aligned} \left\vert \mathbf{X}_{s,t}^{i,i,j,j}-\mathbf{Y}_{s,t}^{i,i,j,j}\right\vert _{L^{2}} &\leq &\frac{1}{2}\left\vert \int_{\left[ s,t\right] ^{2}}\left( \mathbf{X}_{s,u_{1}\wedge u_{2}}^{i,i}-\mathbf{Y}_{s,u_{1}\wedge u_{2}}^{i,i}\right) \,d\left( X_{u_{1}}^{j}X_{u_{2}}^{j}\right) \right\vert _{L^{2}} \\ &&+\frac{1}{2}\left\vert \int_{\left[ s,t\right] ^{2}}\mathbf{Y}% _{s,u_{1}\wedge u_{2}}^{i,i}\,d\left( X_{u_{1}}^{j}X_{u_{2}}^{j}-Y_{u_{1}}^{j}Y_{u_{2}}^{j}\right) \right\vert _{L^{2}}.\end{aligned}$$We use a Young $4D$-estimate and the estimates of Lemma [lemma\_rho\_var\_4D\_estimates]{} to see that$$\begin{aligned} &&\left\vert \int_{\left[ s,t\right] ^{2}}\left( \mathbf{X}_{s,u_{1}\wedge u_{2}}^{i,i}-\mathbf{Y}_{s,u_{1}\wedge u_{2}}^{i,i}\right) \,d\left( X_{u_{1}}^{j}X_{u_{2}}^{j}\right) \right\vert _{L^{2}}^{2} \\ &=&\int_{\left[ s,t\right] ^{4}}E\left[ \left( \mathbf{X}_{s,u_{1}\wedge u_{2}}^{i,i}-\mathbf{Y}_{s,u_{1}\wedge u_{2}}^{i,i}\right) \left( \mathbf{X}% _{s,v_{1}\wedge v_{2}}^{i,i}-\mathbf{Y}_{s,v_{1}\wedge v_{2}}^{i,i}\right) % \right] \,dE\left[ X_{u_{1}}^{j}X_{u_{2}}^{j}X_{v_{1}}^{j}X_{v_{2}}^{j}% \right] \\ &\leq &c_{1}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{3/\rho }.\end{aligned}$$The second term is estimated in the same way using again Lemma [lemma\_rho\_var\_4D\_estimates]{}. \[lemma\_ext\_young\_rho\_var\]Let $f\colon \lbrack 0,1]^{2}\rightarrow \mathbb{R}$ and $g\colon \lbrack 0,1]^{2}\times \lbrack 0,1]^{2}\rightarrow \mathbb{R}$ be continuous where $g$ is symmetric in the first and the last two variables. Let $\left( s,t\right) \in \Delta $ and assume that $f\left( s,\cdot \right) =f(\cdot ,s)=0$. Assume also that $f$ has finite $p$-variation and that the $q$-variation of $g$ is controlled by a symmetric $% 4D $ grid-control $\tilde{\omega}$ where $\frac{1}{p}+\frac{1}{q}>1$. Define$$\Psi \left( u,v\right) =\int_{[s,u]^{2}\times \lbrack s,v]^{2}}f(u_{1}\wedge u_{2},v_{1}\wedge v_{2})\,dg\left( u_{1},u_{2};v_{1},v_{2}\right)$$Then there is a constant $C=C\left( p,q\right) $ such that$$V_{q}\left( \Psi ;\left[ s,t\right] ^{2}\right) \leq CV_{p}\left( f;\left[ s,t\right] ^{2}\right) \tilde{\omega}\left( \left[ s,t\right] ^{4}\right) ^{1/q}.$$ Set$$\tilde{f}\left( u_{1},u_{2},v_{1},v_{2}\right) =f(u_{1}\wedge u_{2},v_{1}\wedge v_{2}).$$Let $u<v$ and $u^{\prime }<v^{\prime }$. Note that$$\begin{aligned} &&1_{[s,v]^{2}\times \lbrack s,v^{\prime }]^{2}}-1_{[s,u]^{2}\times \lbrack s,v^{\prime }]^{2}}-1_{[s,v]^{2}\times \lbrack s,u^{\prime }]^{2}}+1_{[s,u]^{2}\times \lbrack s,u^{\prime }]^{2}} \\ &=&1_{\left( [s,v]^{2}\setminus \lbrack s,u]^{2}\right) \times \lbrack s,v^{\prime }]^{2}}-1_{\left( [s,v]^{2}\setminus \lbrack s,u]^{2}\right) \times \lbrack s,u^{\prime }]^{2}} \\ &=&1_{\left( [s,v]^{2}\setminus \lbrack s,u]^{2}\right) \times \left( \lbrack s,v^{\prime }]^{2}\setminus \lbrack s,u^{\prime }]^{2}\right) }\end{aligned}$$If we take out the square $\left[ s,u\right] ^{2}$ of the larger square $% \left[ s,v\right] ^{2}$, what is left is the union of three essentially disjoint squares. More precisely,$$\overline{\left[ s,v\right] ^{2}\setminus \left[ s,u\right] ^{2}}=\left[ u,v% \right] ^{2}\cup \left( \left[ s,u\right] \times \left[ u,v\right] \right) \cup \left( \left[ u,v\right] \times \left[ s,u\right] \right) .$$The same holds for $u^{\prime }$ and $v^{\prime }$. Hence,$$\begin{aligned} &&\left( \overline{[s,v]^{2}\setminus \lbrack s,u]^{2}}\right) \times \left( \overline{[s,v^{\prime }]^{2}\setminus \lbrack s,u^{\prime }]^{2}}\right) \\ &=&\left( [u,v]^{2}\cup \left( \lbrack s,u]\times \lbrack u,v]\right) \cup \left( \lbrack u,v]\times \lbrack s,u]\right) \right) \\ &&\times \left( \lbrack u^{\prime },v^{\prime }]^{2}\cup \left( \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]\right) \cup \left( \lbrack u^{\prime },v^{\prime }]\times \lbrack s,u^{\prime }]\right) \right) \\ &=&\left( [u,v]^{2}\times \lbrack u^{\prime },v^{\prime }]^{2}\right) \cup \left( \lbrack u,v]^{2}\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]\right) \cup \left( \lbrack u,v]^{2}\times \lbrack u^{\prime },v^{\prime }]\times \lbrack s,u^{\prime }]\right) \\ &&\cup \left( \lbrack s,u]\times \lbrack u,v]\times \lbrack u^{\prime },v^{\prime }]^{2}\right) \cup \left( \lbrack s,u]\times \lbrack u,v]\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]\right) \\ &&\cup \left( \lbrack s,u]\times \lbrack u,v]\times \lbrack u^{\prime },v^{\prime }]\times \lbrack s,u^{\prime }]\right) \\ &&\cup \left( \lbrack u,v]\times \lbrack s,u]\times \lbrack u^{\prime },v^{\prime }]^{2}\right) \cup \left( \lbrack u,v]\times \lbrack s,u]\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]\right) \\ &&\cup \left( \lbrack u,v]\times \lbrack s,u]\times \lbrack u^{\prime },v^{\prime }]\times \lbrack s,u^{\prime }]\right)\end{aligned}$$and all these are unions of essentially disjoint sets. Using continuity and the symmetry of $\tilde{f}$ and $g$ we have then$$\begin{aligned} \Psi \begin{pmatrix} u,v \\ u^{\prime },v^{\prime }% \end{pmatrix} &=&\int_{\left( [s,v]^{2}\setminus \lbrack s,u]^{2}\right) \times \left( \lbrack s,v^{\prime }]^{2}\setminus \lbrack s,u^{\prime }]^{2}\right) }% \tilde{f}\,dg \\ &=&\int_{[u,v]^{2}\times \lbrack u^{\prime },v^{\prime }]^{2}}\tilde{f}% \,dg+2\int_{[u,v]^{2}\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}\tilde{f}\,dg \\ &&+2\int_{[s,u]\times \lbrack u,v]\times \lbrack u^{\prime },v^{\prime }]^{2}}\tilde{f}\,dg+4\int_{[s,u]\times \lbrack u,v]\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}\tilde{f}\,dg.\end{aligned}$$For the first integral we use Young $4D$-estimates. Since $\tilde{f}\left( s,\cdot ,\cdot ,\cdot \right) =\ldots =\tilde{f}\left( \cdot ,\cdot ,\cdot ,s\right) =0$, we can proceed as in the proof of Lemma [lemma\_kernel\_iter\_2D]{} and use Lemma \[lemma\_rho\_var\_invar\_symm\] to see that$$\begin{aligned} \left\vert \int_{\lbrack u,v]^{2}\times \lbrack u^{\prime },v^{\prime }]^{2}}% \tilde{f}\,dg\right\vert &\leq &c_{1}V_{p}\left( f,\left[ s,t\right] ^{2}\right) V_{q}\left( g,[u,v]^{2}\times \lbrack u^{\prime },v^{\prime }]^{2}\right) \\ &\leq &c_{1}V_{p}\left( f,\left[ s,t\right] ^{2}\right) \tilde{\omega}\left( [u,v]^{2}\times \lbrack u^{\prime },v^{\prime }]^{2}\right) ^{1/q}\end{aligned}$$For the second integral, we have$$\begin{aligned} &&\int_{[u,v]^{2}\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}\tilde{f}\,dg \\ &=&\int_{[u,v]^{2}\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}f(u_{1}\wedge u_{2},v_{1}\wedge v_{2})\,dg\left( u_{1},u_{2};v_{1},v_{2}\right) \\ &=&\int_{[u,v]^{2}\times \lbrack s,u^{\prime }]}f(u_{1}\wedge u_{2},v_{1})\,d \left[ g\left( u_{1},u_{2};v_{1},v^{\prime }\right) -g\left( u_{1},u_{2};v_{1},u^{\prime }\right) \right]\end{aligned}$$We now use a Young $3D$-estimate to see that$$\begin{aligned} \left\vert \int_{\lbrack u,v]^{2}\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}\tilde{f}\,dg\right\vert &\leq &c_{2}V_{p}\left( f\left( \cdot \wedge \cdot ,\cdot \right) ,\left[ s,t\right] ^{3}\right) \\ &&\times V_{q}\left( g\left( \cdot ,\cdot ;\cdot ,v^{\prime }\right) -g\left( \cdot ,\cdot ;\cdot ,u^{\prime }\right) ,\left[ u,v\right] ^{2}\times \left[ s,u^{\prime }\right] \right)\end{aligned}$$As in Lemma \[lemma\_rho\_var\_invar\_symm\], one can show that $V_{p}\left( f\left( \cdot \wedge \cdot ,\cdot \right) ,\left[ s,t\right] ^{3}\right) =V_{p}\left( f,\left[ s,t\right] ^{2}\right) $. For $g$, we have$$\begin{aligned} V_{q}\left( g\left( \cdot ,\cdot ;\cdot ,v^{\prime }\right) -g\left( \cdot ,\cdot ;\cdot ,u^{\prime }\right) ,\left[ u,v\right] ^{2}\times \left[ s,u^{\prime }\right] \right) &\leq &V_{q}\left( g,\left[ u,v\right] ^{2}\times \left[ s,u^{\prime }\right] \times \left[ u^{\prime },v^{\prime }% \right] \right) \\ &\leq &\tilde{\omega}\left( \left[ u,v\right] ^{2}\times \left[ s,t\right] \times \left[ u^{\prime },v^{\prime }\right] \right) ^{1/q}.\end{aligned}$$Hence$$\left\vert \int_{\lbrack u,v]^{2}\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}\tilde{f}\,dg\right\vert \leq c_{2}V_{p}\left( f,% \left[ s,t\right] ^{2}\right) \tilde{\omega}\left( \left[ u,v\right] ^{2}\times \left[ s,t\right] \times \left[ u^{\prime },v^{\prime }\right] \right) ^{1/q}.$$Similarly, using Young $3D$ and $2D$ estimates, we get$$\left\vert \int_{\lbrack s,u]\times \lbrack u,v]\times \lbrack u^{\prime },v^{\prime }]^{2}}\tilde{f}\,dg\right\vert \leq c_{3}V_{p}\left( f,\left[ s,t\right] ^{2}\right) \tilde{\omega}\left( \left[ s,t\right] \times \left[ u,v\right] \times \left[ u^{\prime },v^{\prime }\right] ^{2}\right) ^{1/q}$$and$$\left\vert \int_{\lbrack s,u]\times \lbrack u,v]\times \lbrack s,u^{\prime }]\times \lbrack u^{\prime },v^{\prime }]}\tilde{f}\,dg\right\vert \leq c_{4}V_{p}\left( f,\left[ s,t\right] ^{2}\right) \tilde{\omega}\left( \left[ s,t\right] \times \left[ u,v\right] \times \lbrack s,t]\times \lbrack u^{\prime },v^{\prime }]\right) ^{1/q}.$$Putting all together, using the symmetry of $\tilde{\omega}$ we have shown that$$\left\vert \Psi \begin{pmatrix} u,v \\ u^{\prime },v^{\prime }% \end{pmatrix}% \right\vert ^{q}\leq c_{5}V_{p}\left( f,\left[ s,t\right] ^{2}\right) ^{q}% \tilde{\omega}\left( \left[ u,v\right] \times \lbrack u^{\prime },v^{\prime }]\times \left[ s,t\right] ^{2}\right) .$$Since $\tilde{\omega}_{2}\left( \left[ u,v\right] \times \lbrack u^{\prime },v^{\prime }]\right) :=\tilde{\omega}\left( \left[ u,v\right] \times \lbrack u^{\prime },v^{\prime }]\times \left[ s,t\right] ^{2}\right) $ is a $% 2D$ grid-control this shows the claim. We are now able to prove the remaining estimate. Let $\left( X,Y\right) $, $\omega $, $\rho $ and $\gamma $ as in Lemma [lemma\_diff\_alldifferent]{}. Then there is a constant $C=C\left( \rho ,\gamma \right) $ such that$$\left\vert \mathbf{X}_{s,t}^{j,i,i,k}-\mathbf{Y}_{s,t}^{j,i,i,k}\right\vert _{L^{2}}\leq C\epsilon \omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{1% }{2\gamma }}\omega \left( \left[ s,t\right] ^{2}\right) ^{\frac{3}{2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ and $i,j,k$ pairwise distinct where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) ^{1-\rho /\gamma }$. From$$\int_{\Delta _{s,w}^{2}}X_{s,u_{1}}^{j}\,dX_{u_{1}}^{i}\,dX_{u_{2}}^{i}=% \frac{1}{2}\int_{\left[ s,w\right] ^{2}}X_{s,u_{1}\wedge u_{2}}^{j}\,d\left( X_{u_{1}}^{i}X_{u_{2}}^{i}\right)$$we see that$$\mathbf{X}_{s,t}^{j,i,i,k}=\frac{1}{2}\int_{s}^{t}\left( \int_{\left[ s,w% \right] ^{2}}X_{s,u_{1}\wedge u_{2}}^{j}\,d\left( X_{u_{1}}^{i}X_{u_{2}}^{i}\right) \right) \,dX_{w}^{k}.$$Hence$$\begin{aligned} &&\left\vert \mathbf{X}_{s,t}^{j,i,i,k}-\mathbf{Y}_{s,t}^{j,i,i,k}\right% \vert _{L^{2}} \\ &\leq &\frac{1}{2}\left\vert \int_{s}^{t}\Psi _{1}\left( w\right) \,dX_{w}^{k}\right\vert _{L^{2}}+\frac{1}{2}\left\vert \int_{s}^{t}\Psi _{2}\left( w\right) \,dX_{w}^{k}\right\vert _{L^{2}}+\frac{1}{2}\left\vert \int_{s}^{t}\Psi _{3}\left( w\right) \,d\left( X^{k}-Y^{k}\right) _{w}\right\vert _{L^{2}}\end{aligned}$$where$$\begin{aligned} \Psi _{1}\left( w\right) &=&\int_{\left[ s,w\right] ^{2}}\left( X_{s,u_{1}\wedge u_{2}}^{j}-Y_{s,u_{1}\wedge u_{2}}^{j}\right) \,d\left( X_{u_{1}}^{i}X_{u_{2}}^{i}\right) \\ \Psi _{2}\left( w\right) &=&\int_{\left[ s,w\right] ^{2}}Y_{s,u_{1}\wedge u_{2}}^{j}\,d\left( X_{u_{1}}^{i}X_{u_{2}}^{i}-Y_{u_{1}}^{i}Y_{u_{2}}^{i}\right) \\ \Psi _{3}\left( w\right) &=&\int_{\left[ s,w\right] ^{2}}Y_{s,u_{1}\wedge u_{2}}^{j}\,d\left( Y_{u_{1}}^{i}Y_{u_{2}}^{i}\right) .\end{aligned}$$We start with the first integral. From independence and Young $2D$-estimates,$$\begin{aligned} \left\vert \int_{s}^{t}\Psi _{1}\left( w\right) \,dX_{w}^{k}\right\vert _{L^{2}}^{2} &=&\int_{\left[ s,t\right] ^{2}}E\left[ \Psi _{1}\left( w_{1}\right) \Psi _{1}\left( w_{2}\right) \right] \,dE\left[ X_{w_{1}}^{k}X_{w_{2}}^{k}\right] \\ &\leq &c_{1}V_{\rho }\left( E\left[ \Psi _{1}\left( \cdot \right) \Psi _{1}\left( \cdot \right) \right] ,\left[ s,t\right] ^{2}\right) V_{\rho }\left( R_{X^{k}}\left[ s,t\right] ^{2}\right) .\end{aligned}$$Now,$$\begin{aligned} &&E\left[ \Psi _{1}\left( w_{1}\right) \Psi _{1}\left( w_{2}\right) \right] \\ &=&\int_{\left[ s,w_{1}\right] ^{2}\times \left[ s,w_{2}\right] ^{2}}E\left[ \left( X_{s,u_{1}\wedge u_{2}}^{j}-Y_{s,u_{1}\wedge u_{2}}^{j}\right) \,\left( X_{s,v_{1}\wedge v_{2}}^{j}-Y_{s,v_{1}\wedge v_{2}}^{j}\right) % \right] dE\left[ X_{u_{1}}^{i}X_{u_{2}}^{i}X_{v_{1}}^{i}X_{v_{2}}^{i}\right] .\end{aligned}$$In Lemma \[lemma\_rho\_var\_4D\_estimates\] we have seen that the $\rho $-variation of $E\left[ X_{\cdot }^{i}X_{\cdot }^{i}X_{\cdot }^{i}X_{\cdot }^{i}\right] $ is controlled by a symmetric grid-control $\tilde{\omega}_{1}$. Hence we can apply Lemma \[lemma\_ext\_young\_rho\_var\] to conclude that$$\begin{aligned} V_{\rho }\left( E\left[ \Psi _{1}\left( \cdot \right) \Psi _{1}\left( \cdot \right) \right] ,\left[ s,t\right] ^{2}\right) &\leq &c_{2}V_{\gamma }\left( R_{X-Y};\left[ s,t\right] ^{2}\right) \tilde{\omega}_{1}\left( \left[ s,t% \right] ^{4}\right) ^{1/\rho } \\ &\leq &c_{3}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{2/\rho }.\end{aligned}$$Clearly, $V_{\rho }\left( R_{X^{k}}\left[ s,t\right] ^{2}\right) \leq \omega \left( \left[ s,t\right] ^{2}\right) ^{1/\rho }$ and therefore$$\left\vert \int_{s}^{t}\Psi _{1}\left( w\right) \,dX_{w}^{k}\right\vert _{L^{2}}^{2}\leq c_{4}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{3/\rho }.$$Now we come to the second integral. From independence,$$\begin{aligned} \left\vert \int_{s}^{t}\Psi _{2}\left( w\right) \,dX_{w}^{k}\right\vert _{L^{2}}^{2} &=&\int_{\left[ s,t\right] ^{2}}E\left[ \Psi _{2}\left( w_{1}\right) \Psi _{2}\left( w_{2}\right) \right] \,dE\left[ X_{w_{1}}^{k}X_{w_{2}}^{k}\right] . \\ &\leq &c_{5}V_{\gamma }\left( E\left[ \Psi _{2}\left( \cdot \right) \Psi _{2}\left( \cdot \right) \right] ,\left[ s,t\right] ^{2}\right) V_{\rho }\left( R_{X^{k}}\left[ s,t\right] ^{2}\right) .\end{aligned}$$Now$$\begin{aligned} &&E\left[ \Psi _{2}\left( w_{1}\right) \Psi _{2}\left( w_{2}\right) \right] \\ &=&\int_{\left[ s,w_{1}\right] ^{2}\times \left[ s,w_{2}\right] ^{2}}E\left[ Y_{s,u_{1}\wedge u_{2}}^{j}Y_{s,v_{1}\wedge v_{2}}^{j}\right] \,dE\left[ \left( X_{u_{1}}^{i}X_{u_{2}}^{i}-Y_{u_{1}}^{i}Y_{u_{2}}^{i}\right) \left( X_{v_{1}}^{i}X_{v_{2}}^{i}-Y_{v_{1}}^{i}Y_{v_{2}}^{i}\right) \right] \\ &=&:\int_{\left[ s,w_{1}\right] ^{2}\times \left[ s,w_{2}\right] ^{2}}E\left[ Y_{s,u_{1}\wedge u_{2}}^{j}Y_{s,v_{1}\wedge v_{2}}^{j}\right] \,dg\left( u_{1},u_{2},v_{1},v_{2}\right) .\end{aligned}$$In Lemma \[lemma\_rho\_var\_4D\_estimates\] we have seen that the $4D$ $\gamma $-variation of $g$ is controlled by a symmetric $4D$ grid-control $\tilde{% \omega}_{2}$ where$$\tilde{\omega}_{2}\left( \left[ s,t\right] ^{4}\right) ^{1/\gamma }=c_{6}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\rho +1/\gamma }.$$Hence$$V_{\gamma }\left( E\left[ \Psi _{2}\left( \cdot \right) \Psi _{2}\left( \cdot \right) \right] ,\left[ s,t\right] ^{2}\right) \leq c_{7}V_{\rho }\left( R_{Y^{j}};\left[ s,t\right] ^{2}\right) \tilde{\omega}_{2}\left( % \left[ s,t\right] ^{4}\right) ^{1/\gamma }\leq c_{8}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{2/\rho +1/\gamma }.$$This gives us$$\left\vert \int_{s}^{t}\Psi _{2}\left( w\right) \,dX_{w}^{k}\right\vert _{L^{2}}^{2}\leq c_{9}\epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{3/\rho }.$$For the third integral we see again that$$\begin{aligned} \left\vert \int_{s}^{t}\Psi _{3}\left( w\right) \,d\left( X^{k}-Y^{k}\right) _{w}\right\vert _{L^{2}}^{2} &=&\int_{\left[ s,t\right] ^{2}}E\left[ \Psi _{3}\left( w_{1}\right) \Psi _{3}\left( w_{2}\right) \right] \,dE\left[ \left( X^{k}-Y^{k}\right) _{w_{1}}\left( X^{k}-Y^{k}\right) _{w_{2}}\right] \\ &\leq &c_{10}V_{\rho }\left( E\left[ \Psi _{3}\left( \cdot \right) \Psi _{3}\left( \cdot \right) \right] ,\left[ s,t\right] ^{2}\right) V_{\gamma }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) .\end{aligned}$$From$$E\left[ \Psi _{3}\left( w_{1}\right) \Psi _{3}\left( w_{2}\right) \right] =\int_{\left[ s,w_{1}\right] ^{2}\times \left[ s,w_{2}\right] ^{2}}E\left[ Y_{s,u_{1}\wedge u_{2}}^{j}Y_{s,v_{1}\wedge v_{2}}^{j}\right] \,dE\left[ Y_{u_{1}}^{i}Y_{u_{2}}^{i}Y_{v_{1}}^{i}Y_{v_{2}}^{i}\right]$$we see that we can apply Lemma \[lemma\_ext\_young\_rho\_var\] to obtain$$V_{\rho }\left( E\left[ \Psi _{3}\left( \cdot \right) \Psi _{3}\left( \cdot \right) \right] ,\left[ s,t\right] ^{2}\right) \leq c_{11}V_{\rho }\left( R_{Y^{j}};\left[ s,t\right] ^{2}\right) \omega \left( \left[ s,t\right] ^{2}\right) ^{2/\rho }\leq c_{11}\omega \left( \left[ s,t\right] ^{2}\right) ^{3/\rho }.$$Clearly, $V_{\gamma }\left( R_{X-Y},\left[ s,t\right] ^{2}\right) \leq \epsilon ^{2}\omega \left( \left[ s,t\right] ^{2}\right) ^{1/\gamma }$ and hence$$\left\vert \int_{s}^{t}\Psi _{3}\left( w\right) \,d\left( X^{k}-Y^{k}\right) _{w}\right\vert _{L^{2}}^{2}\leq c_{12}\epsilon ^{2}\omega \left( \left[ s,t% \right] ^{2}\right) ^{1/\gamma }\omega \left( \left[ s,t\right] ^{2}\right) ^{3/\rho }$$which gives the claim. Even though Proposition \[prop\_main\_estimates\_n1\_n2\], [prop\_main\_estimates\_n3]{} and \[prop\_main\_estimates\_n4\] are only formulated for Gaussian processes with sample paths of finite variation, the estimate $\left( \ref{eqn_key_estimate}\right) $ is valid also for general Gaussian rough paths for $n=1,2,3,4$. Indeed, this follows from the fact that Gaussian rough paths are just defined as $L^{2}$ limits of smooth paths, cf. [@FV10AIHP]. Higher levels\[subsection\_higher\_levels\] ------------------------------------------- Once we have shown our desired estimates for the first four levels, we can use induction to obtain also the higher levels. This is done in the next proposition. \[prop\_key\_estimate\_higher\_levels\]Let $X$ and $Y$ be Gaussian processes as in Theorem \[theorem\_main01\_intro\]. Let $\rho $, $\gamma $ be fixed and $\omega $ be a control. Assume that there are constants $\tilde{C}=% \tilde{C}\left( n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{n}\right\vert _{L^{2}},\left\vert \mathbf{Y}% _{s,t}^{n}\right\vert _{L^{2}}\leq \tilde{C}\left( n\right) \frac{\omega \left( s,t\right) ^{\frac{n}{2\rho }}}{\beta \left( \frac{n}{2\rho }\right) !% }$$holds for $n=1,\ldots ,\left[ 2\rho \right] $ and constants $C=C\left( n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}}\leq C\left( n\right) \epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}% \frac{\omega \left( s,t\right) ^{\frac{n-1}{2\rho }}}{\beta \left( \frac{n-1% }{2\rho }\right) !}$$holds for $n=1,\ldots ,\left[ 2\rho \right] +1$ and every $\left( s,t\right) \in \Delta $. Here, $\epsilon >0$ and $\beta $ is a positive constant such that$$\beta \geq 4\rho \left( 1+2^{\left( \left[ 2\rho \right] +1\right) /2\rho }\left( \zeta \left( \frac{\left[ 2\rho \right] +1}{2\rho }\right) -1\right) \right)$$ where$\ \zeta $ is just the usual Riemann zeta function. Then for every $% n\in \mathbb{N}$ there is a constant $C=C\left( n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}}\leq C\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n-1}{2\rho }}}{\beta \left( \frac{n-1}{2\rho }\right) !}$$holds for every $\left( s,t\right) \in \Delta $. From Proposition \[prop\_moments\_lp\] we know that for every $n\in \mathbb{N% }$ there are constants $\tilde{C}\left( n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{n}\right\vert _{L^{2}},\left\vert \mathbf{Y}% _{s,t}^{n}\right\vert _{L^{2}}\leq \tilde{C}\frac{\omega \left( s,t\right) ^{% \frac{n}{2\rho }}}{\beta \left( \frac{n}{2\rho }\right) !}$$holds for all $s<t$. We will proof the assertion by induction over $n$. The induction basis is fulfiled by assumption. Suppose that the statement is true for $k=1,\ldots ,n$ where $n\geq \left[ 2\rho \right] +1$. We will show the statement for $n+1$. Let $D=\left\{ s=t_{0}<t_{1}<\ldots <t_{j}=t\right\} $ be any partition of $\left[ s,t\right] $. Set$$\begin{aligned} \mathbf{\bar{X}}_{s,t} &:&=\left( 1,\mathbf{X}_{s,t}^{1},\ldots ,\mathbf{X}% _{s,t}^{n},0\right) \in T^{n+1}\left( \mathbb{R}^{d}\right) , \\ \mathbf{\bar{X}}_{s,t}^{D} &:&=\mathbf{\bar{X}}_{s,t_{1}}\otimes \ldots \otimes \mathbf{\bar{X}}_{t_{j-1},t}\end{aligned}$$and the same for $\mathbf{Y}$. We know that $\lim_{\left\vert D\right\vert \rightarrow 0}\mathbf{\bar{X}}_{s,t}^{D}=S_{n+1}\left( \mathbf{X}\right) _{s,t}$ a.s. and the same holds for $\mathbf{Y}$ (indeed, this is just the definition of the Lyons lift, cf. [@L98 Theorem 2.2.1]). By multiplicativity, $\pi _{k}\left( \mathbf{\bar{X}}_{s,t}^{D}\right) =\mathbf{% X}_{s,t}^{k}$ for $k\leq n$. We will show that for any dissection $D$ we have$$\left\vert \pi _{n+1}\left( \mathbf{\bar{X}}_{s,t}^{D}-\mathbf{\bar{Y}}% _{s,t}^{D}\right) \right\vert _{L^{2}}\leq C\left( n+1\right) \epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n}{2\rho }}}{\beta \left( \frac{n}{2\rho }\right) !}.$$We use the notation $\left( \mathbf{X}^{D}\right) ^{k}:=\pi _{k}\left( \mathbf{\bar{X}}^{D}\right) $. Assume that $j\geq 2$. Let $D^{\prime }$ be the partition of $\left[ s,t\right] $ obtained by removing a point $t_{i}$ of the dissection $D$ for which$$\omega \left( t_{i-1},t_{i+1}\right) \leq \left\{ \begin{array}{ccc} \frac{2\omega \left( s,t\right) }{j-1} & \text{for} & j\geq 3 \\ \omega \left( s,t\right) & \text{for} & j=2% \end{array}% \right.$$holds (Lemma 2.2.1 in [@L98] shows that there is indeed such a point). By the triangle inequality,$$\left\vert \left( \mathbf{X}^{D}-\mathbf{Y}^{D}\right) ^{n+1}\right\vert _{L^{2}}\leq \left\vert \left( \mathbf{X}^{D}-\mathbf{X}^{D^{\prime }}\right) ^{n+1}-\left( \mathbf{Y}^{D}-\mathbf{Y}^{D^{\prime }}\right) ^{n+1}\right\vert _{L^{2}}+\left\vert \left( \mathbf{X}^{D^{\prime }}-% \mathbf{Y}^{D^{\prime }}\right) ^{n+1}\right\vert _{L^{2}}.$$We estimate the first term on the right hand side. As seen in the proof of [@L98 Theorem 2.2.1], $\left( \mathbf{X}_{s,t}^{D}-\mathbf{X}% _{s,t}^{D^{\prime }}\right) ^{n+1}=\sum_{l=1}^{n}\mathbf{X}% _{t_{i-1},t_{i}}^{l}\mathbf{X}_{t_{i},t_{i+1}}^{n+1-l}$. Set $\mathbf{R}^{l}=% \mathbf{Y}^{l}\mathbf{-X}^{l}$. Then$$\begin{aligned} &&\left( \mathbf{X}_{s,t}^{D}-\mathbf{X}_{s,t}^{D^{\prime }}\right) ^{n+1}-\left( \mathbf{Y}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D^{\prime }}\right) ^{n+1} \\ &=&\sum_{l=1}^{n}\mathbf{X}_{t_{i-1},t_{i}}^{l}\mathbf{X}% _{t_{i},t_{i+1}}^{n+1-l}-\left( \mathbf{X}_{t_{i-1},t_{i}}^{l}+\mathbf{R}% _{t_{i-1},t_{i}}^{l}\right) \left( \mathbf{X}_{t_{i},t_{i+1}}^{n+1-l}+% \mathbf{R}_{t_{i},t_{i+1}}^{n+1-l}\right) \\ &=&\sum_{l=1}^{n}-\mathbf{X}_{t_{i-1},t_{i}}^{l}\mathbf{R}% _{t_{i},t_{i+1}}^{n+1-l}-\mathbf{R}_{t_{i-1},t_{i}}^{l}\mathbf{Y}% _{t_{i},t_{i+1}}^{n+1-l}.\end{aligned}$$By the triangle inequality, equivalence of $L^{q}$-norms in the Wiener Chaos, our moment estimate for $\mathbf{X}^{k}$ and $\mathbf{Y}^{k}$ and the induction hypothesis,$$\begin{aligned} &&\left\vert \left( \mathbf{X}_{s,t}^{D}-\mathbf{X}_{s,t}^{D^{\prime }}\right) ^{n+1}-\left( \mathbf{Y}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D^{\prime }}\right) ^{n+1}\right\vert _{L^{2}} \\ &\leq &c_{1}\left( n+1\right) \sum_{l=1}^{n}\left\vert \mathbf{X}% _{t_{i-1},t_{i}}^{l}\right\vert _{L^{2}}\left\vert \mathbf{R}% _{t_{i},t_{i+1}}^{n+1-l}\right\vert _{L^{2}}+\left\vert \mathbf{R}% _{t_{i-1},t_{i}}^{l}\right\vert _{L^{2}}\left\vert \mathbf{Y}% _{t_{i},t_{i+1}}^{n+1-l}\right\vert _{L^{2}} \\ &\leq &c_{2}\left( n+1\right) \sum_{l=1}^{n}\epsilon \omega \left( t_{i},t_{i+1}\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( t_{i-1},t_{i}\right) ^{\frac{l}{2\rho }}}{\beta \left( \frac{l}{2\rho }% \right) !}\frac{\omega \left( t_{i},t_{i+1}\right) ^{\frac{n-l}{2\rho }}}{% \beta \left( \frac{n-l}{2\rho }\right) !} \\ &&+\epsilon \omega \left( t_{i-1},t_{i}\right) ^{\frac{1}{2\gamma }}\frac{% \omega \left( t_{i-1},t_{i}\right) ^{\frac{l-1}{2\rho }}}{\beta \left( \frac{% l-1}{2\rho }\right) !}\frac{\omega \left( t_{i},t_{i+1}\right) ^{\frac{n+1-l% }{2\rho }}}{\beta \left( \frac{n+1-l}{2\rho }\right) !} \\ &\leq &2c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }% }\sum_{l=0}^{n}\frac{\omega \left( t_{i-1},t_{i}\right) ^{\frac{l}{2\rho }}}{% \beta \left( \frac{l}{2\rho }\right) !}\frac{\omega \left( t_{i},t_{i+1}\right) ^{\frac{n-l}{2\rho }}}{\beta \left( \frac{n-l}{2\rho }% \right) !} \\ &=&\frac{4\rho }{\beta ^{2}}c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1% }{2\gamma }}\frac{1}{2\rho }\sum_{l=0}^{n}\frac{\omega \left( t_{i-1},t_{i}\right) ^{\frac{l}{2\rho }}}{\left( \frac{l}{2\rho }\right) !}% \frac{\omega \left( t_{i},t_{i+1}\right) ^{\frac{n-l}{2\rho }}}{\left( \frac{% n-l}{2\rho }\right) !} \\ &\leq &4\rho c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}% \frac{\omega \left( t_{i-1},t_{i+1}\right) ^{\frac{n}{2\rho }}}{\beta ^{2}\left( \frac{n}{2\rho }\right) !}\end{aligned}$$where we used the neo-classical inequality (cf. [@HH10]) and superadditivity of the control function. Hence for $j\geq 3$,$$\begin{aligned} \left\vert \left( \mathbf{X}_{s,t}^{D}-\mathbf{X}_{s,t}^{D^{\prime }}\right) ^{n+1}-\left( \mathbf{Y}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D^{\prime }}\right) ^{n+1}\right\vert _{L^{2}} &\leq &4\rho c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( t_{i-1},t_{i+1}\right) ^{% \frac{n}{2\rho }}}{\beta ^{2}\left( \frac{n}{2\rho }\right) !} \\ &\leq &\left( \frac{2}{j-1}\right) ^{\frac{n}{2\rho }}4\rho c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n}{2\rho }}}{\beta ^{2}\left( \frac{n}{2\rho }\right) !}.\end{aligned}$$For $j=2$ we get$$\left\vert \left( \mathbf{X}_{s,t}^{D}-\mathbf{X}_{s,t}^{D^{\prime }}\right) ^{n+1}-\left( \mathbf{Y}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D^{\prime }}\right) ^{n+1}\right\vert _{L^{2}}\leq 4\rho c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n}{2\rho }}}{% \beta ^{2}\left( \frac{n}{2\rho }\right) !}$$but then $D^{\prime }=\left\{ s,t\right\} $ and therefore $\left\vert \left( \mathbf{X}_{s,t}^{D^{\prime }}-\mathbf{Y}_{s,t}^{D^{\prime }}\right) ^{n+1}\right\vert _{L^{2}}=0$. Hence by successively dropping points we see that$$\left\vert \left( \mathbf{X}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D}\right) ^{n+1}\right\vert _{L^{2}}\leq \left( 1+\sum_{j=3}^{\infty }\left( \frac{2}{% j-1}\right) ^{\frac{n}{2\rho }}\right) 4\rho c_{2}\epsilon \omega \left( s,t\right) ^{\frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n}{% 2\rho }}}{\beta ^{2}\left( \frac{n}{2\rho }\right) !}$$holds for all partitions $D$. Since $n\geq \left[ 2\rho \right] +1$,$$\sum_{j=3}^{\infty }\left( \frac{2}{j-1}\right) ^{\frac{n}{2\rho }}\leq \sum_{j=3}^{\infty }\left( \frac{2}{j-1}\right) ^{\frac{\left[ 2\rho \right] +1}{2\rho }}\leq 2^{\frac{\left[ 2\rho \right] +1}{2\rho }}\left( \zeta \left( \frac{\left[ 2\rho \right] +1}{2\rho }\right) -1\right)$$and thus$$\left\vert \left( \mathbf{X}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D}\right) ^{n+1}\right\vert _{L^{2}}\leq \frac{4\rho \left( 1+2^{\frac{\left[ 2\rho % \right] +1}{2\rho }}\left( \zeta \left( \frac{\left[ 2\rho \right] +1}{2\rho }\right) -1\right) \right) }{\beta }c_{2}\epsilon \omega \left( s,t\right) ^{% \frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n}{2\rho }}}{\beta \left( \frac{n}{2\rho }\right) !}.$$By the choice of $\beta $, we get the uniform bound$$\left\vert \left( \mathbf{X}_{s,t}^{D}-\mathbf{Y}_{s,t}^{D}\right) ^{n+1}\right\vert _{L^{2}}\leq c_{2}\epsilon \omega \left( s,t\right) ^{% \frac{1}{2\gamma }}\frac{\omega \left( s,t\right) ^{\frac{n}{2\rho }}}{\beta \left( \frac{n}{2\rho }\right) !}$$which holds for all partitions $D$. Noting that a.s. convergence implies convergence in $L^{2}$ in the Wiener chaos, we obtain our claim by sending $% \|D\|\rightarrow 0$. \[cor\_diff\_estimates\]Let $\left( X,Y\right) $, $\omega $, $\rho $ and $% \gamma $ as in Lemma \[lemma\_diff\_alldifferent\]. Then for all $n\in \mathbb{N}$ there are constants $C=C\left( \rho ,\gamma ,n\right) $ such that$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}}\leq C\epsilon \omega \left( \lbrack s,t]^{2}\right) ^{\frac{1}{2\gamma }}\omega \left( \lbrack s,t]^{2}\right) ^{\frac{n-1}{2\rho }}$$holds for every $\left( s,t\right) \in \Delta $ where $\epsilon ^{2}=V_{\infty }\left( R_{X-Y},\left[ 0,1\right] ^{2}\right) ^{1-\rho /\gamma }$. For $n=1,2,3,4$ this is the content of Proposition [prop\_main\_estimates\_n1\_n2]{}, \[prop\_main\_estimates\_n3\] and [prop\_main\_estimates\_n4]{}. By making the constants larger if necessary, we also get$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}}\leq c\left( n\right) \epsilon \omega \left( \lbrack s,t]^{2}\right) ^{\frac{1}{% 2\gamma }}\frac{\omega \left( \lbrack s,t]^{2}\right) ^{\frac{n-1}{2\rho }}}{% \beta \left( \frac{n-1}{2\rho }\right) !}$$with $\beta $ chosen as in Proposition \[prop\_key\_estimate\_higher\_levels\]. We have already seen that$$\left\vert \mathbf{X}_{s,t}^{n}\right\vert _{L^{2}},\left\vert \mathbf{Y}% _{s,t}^{n}\right\vert _{L^{2}}\leq \tilde{c}\left( n\right) \frac{\omega \left( \lbrack s,t]^{2}\right) ^{\frac{n}{2\rho }}}{\beta \left( \frac{n}{% 2\rho }\right) !}$$holds for constants $\tilde{c}\left( n\right) $ where $n=1,2,3$. Since $\rho <2$, we have $[2\rho ]+1\leq 4$. From Proposition [prop\_key\_estimate\_higher\_levels]{} we can conclude that$$\left\vert \mathbf{X}_{s,t}^{n}-\mathbf{Y}_{s,t}^{n}\right\vert _{L^{2}}\leq c\left( n\right) \epsilon \omega \left( \lbrack s,t]^{2}\right) ^{\frac{1}{% 2\gamma }}\frac{\omega \left( \lbrack s,t]^{2}\right) ^{\frac{n-1}{2\rho }}}{% \beta \left( \frac{n-1}{2\rho }\right) !}$$holds for every $n\in \mathbb{N}$ and constants $c\left( n\right) $. Setting $C\left( n\right) =\frac{c\left( n\right) }{\beta \left( \frac{n-1}{2\rho }% \right) !}$ gives our claim. Main result\[section\_main\_result\] ==================================== Assume that $X$ is a Gaussian process as in Theorem [theorem\_main01\_intro]{} with paths of finite $p$-variation. Consider a sequence $\left( \Lambda _{k}\right) _{k\in \mathbb{N}}$ of continuous operators$$\Lambda _{k}\colon C^{p-var}\left( \left[ 0,1\right] ,\mathbb{R}\right) \rightarrow C^{1-var}\left( \left[ 0,1\right] ,\mathbb{R}\right) .$$If $x=\left( x^{1},\ldots ,x^{d}\right) \in C^{p-var}\left( \left[ 0,1\right] ,\mathbb{R}^{d}\right) $, we will write $\Lambda _{k}\left( x\right) =\left( \Lambda _{k}\left( x^{1}\right) ,\ldots ,\Lambda _{k}\left( x^{d}\right) \right) $. Assume that $\Lambda _{k}$ fulfils the following conditions: 1. $\Lambda _{k}\left( x\right) \rightarrow x$ in the $\left\vert \cdot \right\vert _{\infty }$-norm if $k\rightarrow \infty $ for every $x\in C^{p-var}\left( \left[ 0,1\right] ,\mathbb{R}^{d}\right) .$ 2. If $R_{X}$ has finite controlled $\rho $-variation, then, for some $% C=C\left( \rho \right) $,$$\sup_{k,l\in \mathbb{N}}\left\vert R_{\left( \Lambda _{k}\left( X\right) ,\Lambda _{l}\left( X\right) \right) }\right\vert _{\rho -var;\left[ 0,1% \right] ^{2}}\leq C\left\vert R_{X}\right\vert _{\rho -var;\left[ 0,1\right] ^{2}}.$$ Our main result is the following: \[theorem\_main01\]Let $X$ be a Gaussian process as in Theorem [theorem\_main01\_intro]{} for $\rho <2$ and $K\geq V_{\rho }\left( R_{X},\left[ 0,1\right] ^{2}\right) $. Then there is an enhanced Gaussian process $% \mathbf{X}$ with sample paths in $C^{0,p-var}\left( \left[ 0,1\right] ,G^{% \left[ p\right] }\left( \mathbb{R}^{d}\right) \right) $ w.r.t. $\left( \Lambda _{k}\right) _{k\in \mathbb{N}}$ where $p\in \left( 2\rho ,4\right) $, i.e.$$\left\vert \rho _{p-var}\left( S_{\left[ p\right] }\left( \Lambda _{k}\left( X\right) \right) ,\mathbf{X}\right) \right\vert _{L^{r}}\rightarrow 0$$for $k\rightarrow \infty $ and every $r\geq 1$. Moreover, choose $\gamma $ such that $\gamma >\rho $ and $\frac{1}{\gamma }+\frac{1}{\rho }>1$. Then for $q>2\gamma $ and every $N\in \mathbb{N}$ there is a constant $C=C\left( q,\rho ,\gamma ,K,N\right) $ such that$$\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) ,S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}}\leq Cr^{N/2}\sup_{0\leq t\leq 1}\left\vert \Lambda _{k}\left( X\right) _{t}-X_{t}\right\vert _{L^{2}\left( \mathbb{R}^{d}\right) }^{1-\frac{\rho }{% \gamma }}$$holds for every $k\in \mathbb{N}$. The first statement is a fundamental result about Gaussian rough paths, see [@FV10 Theorem 15.33]. For the second, take $\delta >0$ and set$$\gamma ^{\prime }=\left( 1+\delta \right) \gamma \quad \text{and\quad }\rho ^{\prime }=\left( 1+\delta \right) \rho .$$By choosing $\delta $ smaller if necessary we can assume that $\frac{1}{\rho ^{\prime }}+\frac{1}{\gamma ^{\prime }}>1$ and $q>2\gamma ^{\prime }$. Set$$\omega _{k,l}\left( A\right) =\left\vert R_{\left( \Lambda _{k}\left( X\right) ,\Lambda _{l}\left( X\right) \right) }\right\vert _{\rho ^{\prime }-var;A}^{\rho ^{\prime }}$$for a rectangle $A\subset \left[ 0,1\right] ^{2}$ and $$\epsilon _{k,l}=V_{\infty }\left( R_{\left( \Lambda _{k}\left( X\right) -\Lambda _{l}\left( X\right) \right) },\left[ 0,1\right] ^{2}\right) ^{\frac{% 1}{2}-\frac{\rho ^{\prime }}{2\gamma ^{\prime }}}=V_{\infty }\left( R_{\left( \Lambda _{k}\left( X\right) -\Lambda _{l}\left( X\right) \right) },% \left[ 0,1\right] ^{2}\right) ^{\frac{1}{2}-\frac{\rho }{2\gamma }}.$$From Theorem \[theorem\_comp\_contr\_p\_var\] we know that $\omega _{k,l}$ is a $2D$ control function which controls the $\rho ^{\prime }$-variation of $% R_{\left( \Lambda _{k}\left( X\right) ,\Lambda _{l}\left( X\right) \right) }$. From Corollary \[cor\_diff\_estimates\] we can conclude that there is a constant $c_{1}$ such that$$\left\vert \pi _{n}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) _{s,t}-S_{N}\left( \Lambda _{l}\left( X\right) \right) _{s,t}\right) \right\vert _{L^{2}}\leq c_{1}\epsilon _{k,l}\omega _{k,l}\left( \left[ s,t% \right] ^{2}\right) ^{\frac{1}{2\gamma ^{\prime }}}\omega _{k,l}\left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{2\rho ^{\prime }}}$$holds for every $n=1,\ldots ,N$, $\left( s,t\right) \in \Delta $ and $k,l\in \mathbb{N}$. Now,$$\begin{aligned} \omega _{k,l}\left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{2\rho ^{\prime }}} &=&\left( \frac{\omega _{k,l}\left( \left[ s,t\right] ^{2}\right) }{\omega _{k,l}\left( \left[ 0,1\right] ^{2}\right) }\right) ^{% \frac{n-1}{2\rho ^{\prime }}}\omega _{k,l}\left( \left[ 0,1\right] ^{2}\right) ^{\frac{n-1}{2\rho ^{\prime }}} \\ &\leq &\omega _{k,l}\left( \left[ s,t\right] ^{2}\right) ^{\frac{n-1}{% 2\gamma ^{\prime }}}\omega _{k,l}\left( \left[ 0,1\right] ^{2}\right) ^{% \frac{n-1}{2\rho ^{\prime }}-\frac{n-1}{2\gamma ^{\prime }}}.\end{aligned}$$From Theorem \[theorem\_comp\_contr\_p\_var\] and our assumptions on the $% \Lambda _{k}$ we know that $$\omega _{k,l}\left( \left[ 0,1\right] ^{2}\right) ^{1/\rho ^{\prime }}\leq c_{2}\left\vert R_{X}\right\vert _{\rho ^{\prime }-var;\left[ 0,1\right] ^{2}}\leq c_{3}V_{\rho }\left( R_{X},\left[ 0,1\right] ^{2}\right) \leq c_{4}\left( \rho ,\rho ^{\prime },K\right) .$$holds uniformly over all $k,l$. Hence$$\left\vert \pi _{n}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) _{s,t}-S_{N}\left( \Lambda _{l}\left( X\right) \right) _{s,t}\right) \right\vert _{L^{2}}\leq c_{5}\epsilon _{k,l}\omega _{k,l}\left( \left[ s,t% \right] ^{2}\right) ^{\frac{n}{2\gamma ^{\prime }}}.$$Proposition \[prop\_moments\_lp\] shows with the same argument that$$\left\vert \pi _{n}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) _{s,t}\right) \right\vert _{L^{2}}\leq c_{6}\omega _{k,l}\left( \left[ s,t% \right] ^{2}\right) ^{\frac{n}{2\rho ^{\prime }}}\leq c_{7}\omega _{k,l}\left( \left[ s,t\right] ^{2}\right) ^{\frac{n}{2\gamma ^{\prime }}}$$for every $k\in \mathbb{N}$ and the same holds for $S_{N}\left( \Lambda _{l}\left( X\right) \right) _{s,t}$. From [@FV10 Proposition 15.24] we can conclude that there is a constant $c_{8}$ such that$$\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) ,S_{N}\left( \Lambda _{l}\left( X\right) \right) \right) \right\vert _{L^{r}}\leq c_{8}r^{N/2}\epsilon _{k,l}$$holds for all $k,l\in \mathbb{N}$. In particular, we have shown that $\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) \right) _{k\in \mathbb{N}}$ is a Cauchy sequence in $L^{r\text{ }}$and it is clear that the limit is given by the Lyons lift $S_{N}\left( \mathbf{X}\right) $ of the enhanced Gaussian process $\mathbf{X}$. Now fix $k\in \mathbb{N}$. For every $l\in \mathbb{N}$,$$\begin{aligned} \left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) ,S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}} &\leq &\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) ,S_{N}\left( \Lambda _{l}\left( X\right) \right) \right) \right\vert _{L^{r}} \\ &&+\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{l}\left( X\right) \right) ,S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}} \\ &\leq &c_{8}r^{N/2}\epsilon _{k,l}+\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{l}\left( X\right) \right) ,S_{N}\left( \mathbf{X}% \right) \right) \right\vert _{L^{r}}.\end{aligned}$$It is easy to see that$$\epsilon _{k,l}\rightarrow V_{\infty }\left( R_{\left( \Lambda _{k}\left( X\right) -X\right) },\left[ 0,1\right] ^{2}\right) ^{\frac{1}{2}-\frac{\rho }{2\gamma }}\quad \text{for }l\rightarrow \infty$$and since$$\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{l}\left( X\right) \right) ,S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}}\rightarrow 0\quad \text{for }l\rightarrow \infty$$we can conclude that$$\left\vert \rho _{q-var}\left( S_{N}\left( \Lambda _{k}\left( X\right) \right) ,S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}}\leq c_{8}r^{N/2}V_{\infty }\left( R_{\left( \Lambda _{k}\left( X\right) -X\right) },\left[ 0,1\right] ^{2}\right) ^{\frac{1}{2}-\frac{\rho }{2\gamma }}$$holds for every $k\in \mathbb{N}$. Finally, we have for $\left[ \sigma ,\tau % \right] \times \left[ \sigma ^{\prime },\tau ^{\prime }\right] \subset \left[ 0,1\right] ^{2}$$$\left\vert R_{\left( \Lambda _{k}\left( X\right) -X\right) }\left( \begin{array}{c} \sigma ,\tau \\ \sigma ^{\prime },\tau ^{\prime }% \end{array}% \right) \right\vert _{\mathbb{R}^{d\times d}}\leq 4\sup_{0\leq s<t\leq 1}\left\vert R_{\left( \Lambda _{k}\left( X\right) -X\right) }\left( s,t\right) \right\vert _{\mathbb{R}^{d\times d}}$$and hence$$V_{\infty }\left( R_{\left( \Lambda _{k}\left( X\right) -X\right) },\left[ 0,1\right] ^{2}\right) \leq 4\sup_{0\leq s<t\leq 1}\left\vert R_{\left( \Lambda _{k}\left( X\right) -X\right) }\left( s,t\right) \right\vert _{% \mathbb{R}^{d\times d}}.$$Furthermore, for any $s<t$,$$\left\vert R_{\left( \Lambda _{k}\left( X\right) -X\right) }\left( s,t\right) \right\vert _{\mathbb{R}^{d\times d}}\leq \left\vert \Lambda _{k}\left( X\right) _{s}-X_{s}\right\vert _{L^{2}\left( \mathbb{R}% ^{d}\right) }\left\vert \Lambda _{k}\left( X\right) _{t}-X_{t}\right\vert _{L^{2}\left( \mathbb{R}^{d}\right) }\leq \sup_{0\leq t\leq 1}\left\vert \Lambda _{k}\left( X\right) _{t}-X_{t}\right\vert _{L^{2}\left( \mathbb{R}% ^{d}\right) }^{2}$$and therefore$$V_{\infty }\left( R_{\left( \Lambda _{k}\left( X\right) -X\right) },\left[ 0,1\right] ^{2}\right) ^{\frac{1}{2}-\frac{\rho }{2\gamma }}\leq c_{9}\sup_{0\leq t\leq 1}\left\vert \Lambda _{k}\left( X\right) _{t}-X_{t}\right\vert _{L^{2}\left( \mathbb{R}^{d}\right) }^{1-\frac{\rho }{% \gamma }}$$which shows the result. The next Theorem gives pathwise convergence rates for the Wong-Zakai error for suitable approximations of the driving signal. \[theorem\_as\_wong\_zakai\_rate\]Let $X$ be as in Theorem [theorem\_main01\_intro]{} for $\rho <2$, $K\geq V_{\rho }\left( R_{X},\left[ 0,1% \right] ^{2}\right) $ and $X^{\left( k\right) }=\Lambda _{k}\left( X\right) $. Consider the SDEs$$\begin{aligned} dY_{t} &=&V(Y_{t})\,d\mathbf{X}_{t},\quad Y_{0}\in \mathbb{R}^{n} \label{eqn_RDE_GP_general} \\ dY_{t}^{\left( k\right) } &=&V(Y_{t}^{\left( k\right) })\,dX_{t}^{\left( k\right) },\quad Y_{0}^{\left( k\right) }=Y_{0}\in \mathbb{R}^{n} \label{eqn_RS_general}\end{aligned}$$where $\left\vert V\right\vert _{Lip^{\theta }}\leq \nu <\infty $ for a $% \theta >2\rho $. Assume that there is a constant $C_{1}$ and a sequence $% \left( \epsilon _{k}\right) _{k\in \mathbb{N}}\subset \dbigcup\limits_{r\geq 1}l^{r}$ such that$$\sup_{0\leq t\leq 1}\left\vert X_{t}^{\left( k\right) }-X_{t}\right\vert _{L^{2}}^{2}\leq C_{1}\epsilon _{k}^{1/\rho }~\text{for all }k\in \mathbb{N}.$$Choose $\eta ,q$ such that$$0\leq \eta <\min \left\{ \frac{1}{\rho }-\frac{1}{2},\frac{1}{2\rho }-\frac{1% }{\theta }\right\} \quad \text{and\quad }q\in \left( \frac{2\rho }{1-2\rho \eta },\theta \right) .$$Then both SDEs $\left( \ref{eqn_RDE_GP_general}\right) $ and $\left( \ref% {eqn_RS_general}\right) $ have unique solutions $Y$ and $Y^{\left( k\right) } $ and there is a finite random variable $C$ and a null set $M$ such that$$\left\vert Y^{\left( k\right) }\left( \omega \right) -Y\left( \omega \right) \right\vert _{\infty ;\left[ 0,1\right] }\leq \left\vert Y^{\left( k\right) }\left( \omega \right) -Y\left( \omega \right) \right\vert _{q-var;\left[ 0,1% \right] }\leq C\left( \omega \right) \epsilon _{k}^{\eta } \label{eqn_pathwise_wong_zakai}$$holds for all $k\in \mathbb{N}$ and $\omega \in \Omega \setminus M$. The random variable $C$ depends on $\rho ,q,\eta ,\nu ,\theta ,K,C_{1}$, the sequence $\left( \epsilon _{k}\right) _{k\in \mathbb{N}}$ and the driving process $X$ but not on the equation itself. The same holds for the set $M$. Note that this means that we have universal rates, i.e. the set $M$ and the random variable $C$ are valid for all starting points (and also vector fields subject to a uniform $Lip^{\theta }$-bound). In particular, our convergence rates apply to solutions viewed as $C^{l}$-diffeomorphisms where $l=\left[ \theta -q\right] $, cf. [@FV10 Theorem 11.12] and [FR11]{}. Note that $\gamma >\rho $ and $\frac{1}{\rho }+\frac{1}{\gamma }>1$ is equivalent to $0<\frac{1}{2\rho }-\frac{1}{2\gamma }<\frac{1}{\rho }-\frac{1% }{2}$. Hence there is a $\gamma _{0}>\rho $ such that $\eta =\frac{1}{2\rho }% -\frac{1}{2\gamma _{0}}$ and $\frac{1}{\rho }+\frac{1}{\gamma _{0}}>1$. Furthermore, $2\gamma _{0}=\frac{2\rho }{1-2\rho \eta }<q$. Choose $\gamma _{1}>\gamma _{0}$ such that still $2\gamma _{1}<q$ and $\eta <\frac{1}{2\rho }-\frac{1}{2\gamma _{1}}<$ $\frac{1}{\rho }-\frac{1}{2}$ , hence $\frac{1}{% \rho }+\frac{1}{\gamma _{1}}>1$ hold. Set $\alpha :=$ $\frac{1}{2\rho }-% \frac{1}{2\gamma _{1}}-\eta >0$. From Theorem \[theorem\_main01\] we know that for every $r\geq 1$ and $N\in \mathbb{N}$ there is a constant $c_{1}$ such that $$\left\vert \rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}\right) \right) \right\vert _{L^{r}}\leq c_{1}r^{N/2}\sup_{0\leq t\leq 1}\left\vert X_{t}^{\left( k\right) }-X_{t}\right\vert _{L^{2}}^{1-% \frac{\rho }{\gamma }}\leq c_{2}r^{N/2}\epsilon _{k}^{\frac{1}{2\rho }-\frac{% 1}{2\gamma }}$$holds for every $k\in \mathbb{N}$. Hence$$\left\vert \frac{\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}\right) \right) }{\epsilon _{k}^{\eta }}\right\vert _{L^{r}}\leq c_{2}r^{N/2}\epsilon _{k}^{\alpha }$$for every $k\in \mathbb{N}$. From the Markov inequality, for any $\delta >0$, $$\sum_{k=1}^{\infty }P\left[ \frac{\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}\right) \right) }{\epsilon _{k}^{\eta }}% \geq \delta \right] \leq \frac{1}{\delta ^{r}}\sum_{k=1}^{\infty }\left\vert \frac{\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}% \right) \right) }{\epsilon _{k}^{\eta }}\right\vert _{L^{r}}^{r}\leq c_{3}\sum_{k=1}^{\infty }\epsilon _{k}^{\alpha r}$$ By assumption, we can choose $r$ large enough such that the series converges. With Borel-Cantelli we can conclude that$$\frac{\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}% \right) \right) }{\epsilon _{k}^{\eta }}\rightarrow 0$$outside a null set $M$ for $k\rightarrow \infty $. We set$$C_{2}:=\sup_{k\in \mathbb{N}}\frac{\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}\right) \right) }{\epsilon _{k}^{\eta }}% <\infty \quad \text{a.s.}$$Since $C_{2}$ is the supremum of $\mathcal{F}$-measurable random variables it is itself $\mathcal{F}$-measurable. Now set $N=\left[ q\right] $ which turns $\rho _{q-var}$ into a rough path metric. Note that since $\theta >2\rho $, $\left( \ref{eqn_RDE_GP_general}\right) $ and $\left( \ref% {eqn_RS_general}\right) $ have indeed unique solutions $Y$ and $Y^{\left( k\right) }$. We substitute the driver $\mathbf{X}$ by $S_{N}(\mathbf{X})$ resp. $X^{\left( k\right) }$ by $S_{N}(X^{\left( k\right) })$ in the above equations, now considered as RDEs in the $q$-rough paths space. Since $% \theta >q$, both (RDE-) equations have again unique solutions and it is clear that they coincide with $Y$ and $Y^{\left( k\right) }$. From$$\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),\mathbf{1}\right) \leq \rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}\right) \right) +\rho _{q-var}\left( S_{N}\left( \mathbf{X}\right) ,\mathbf{1}% \right) \leq C_{1}+\rho _{q-var}\left( S_{N}\left( \mathbf{X}\right) ,% \mathbf{1}\right)$$we see that for every $\omega \in \Omega \setminus M$ the $S_{N}(X^{\left( k\right) }\left( \omega \right) )$ are uniformly bounded for all $k$ in the topology given by the metric $\rho _{q-var}$. Thus we can apply local Lipschitz-continuity of the Itō-Lyons map (see [@FV10 Theorem 10.26]) to see that there is a random variable $C_{3}$ such that $$\left\vert Y^{\left( k\right) }-Y\right\vert _{q-var;\left[ 0,1\right] }\leq C_{3}\rho _{q-var}\left( S_{N}(X^{\left( k\right) }),S_{N}\left( \mathbf{X}% \right) \right) \leq C_{3}\cdot C_{2}\epsilon _{k}^{\eta }$$holds for every $k\in \mathbb{N}$ outside $M$. Finally,$$\left\vert Y_{t}^{\left( k\right) }-Y_{t}\right\vert =\left\vert Y_{0,t}^{\left( k\right) }-Y_{0,t}\right\vert \leq \left\vert Y^{\left( k\right) }-Y\right\vert _{q-var;\left[ 0,t\right] }\leq \left\vert Y^{\left( k\right) }-Y\right\vert _{q-var;\left[ 0,1\right] }$$is true for all $t\in \left[ 0,1\right] $ and the claim follows. Mollifier approximations ------------------------ Let $\phi $ be a mollifier function with support $\left[ -1,1\right] $, i.e. $\phi \in C_{0}^{\infty }\left( \left[ -1,1\right] \right) $ is positive and $\left\vert \phi \right\vert _{L^{1}}=1$. If $x\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ is a continuous path, we denote by $\bar{x}\colon \mathbb{R}\rightarrow \mathbb{R}$ its continuous extension to the whole real line, i.e. $$\bar{x}_{u}=\left\{ \begin{array}{ccc} x_{0} & \text{for} & x\in (-\infty ,0] \\ x_{u} & \text{for} & x\in \left[ 0,1\right] \\ x_{1} & \text{for} & x\in \lbrack 1,\infty )% \end{array}% \right.$$For $\epsilon >0$ set $$\begin{aligned} \phi _{\epsilon }\left( u\right) &:&=\frac{1}{\epsilon }\phi \left( u/\epsilon \right) \quad \text{and} \\ x_{t}^{\epsilon } &:&=\int_{\mathbb{R}}\phi _{\epsilon }\left( t-u\right) \bar{x}_{u}\,du.\end{aligned}$$Let $\left( \epsilon _{k}\right) _{k\in \mathbb{N}}$ be a sequence of real numbers such that $\epsilon _{k}\rightarrow 0$ for $k\rightarrow \infty $. Define$$\Lambda _{k}\left( x\right) :=x^{\epsilon _{k}}.$$In [@FV10], Chapter 15.2.3 it is shown that the sequence $\left( \Lambda _{k}\right) _{k\in \mathbb{N}}$ fulfils the conditions of Theorem [theorem\_main01]{}. Let $X$ be as in Theorem \[theorem\_main01\_intro\] and assume that there is a constant $C$ such that $V_{\rho }\left( R_{X};\left[ s,t\right] ^{2}\right) \leq C\left\vert t-s\right\vert ^{1/\rho }$ holds for all $s<t$. Choose $\left( \epsilon _{k}\right) _{k\in \mathbb{N}}\in $ $% \dbigcup\limits_{r\geq 1}l^{r}$ and set $X^{\left( k\right) }=X^{\epsilon _{k}}$. Then the solutions $Y^{\left( k\right) }$ of the SDE $\left( \ref% {eqn_RS_general}\right) $ converge pathwise to the solution $Y$ of $\left( % \ref{eqn_RDE_GP_general}\right) $ in the sense of $\left( \ref% {eqn_pathwise_wong_zakai}\right) $ with rate $O\left( \epsilon _{k}^{\eta }\right) $ where $\eta $ is chosen as in Theorem [theorem\_as\_wong\_zakai\_rate]{}. It suffices to note that for every $\epsilon >0$, $Z\in \left\{ X^{1},\ldots ,X^{d}\right\} $ and $t\in \left[ 0,1\right] $ we have$$\begin{aligned} E\left[ \left\vert Z_{t}^{\epsilon }-Z_{t}\right\vert ^{2}\right] &=&E\left[ \left( \int_{\mathbb{R}}\phi _{\epsilon }\left( t-u\right) \left( \bar{Z}% _{u}-Z_{t}\right) \,du\right) ^{2}\right] \\ &=&E\left[ \left( \int_{\left[ t-\epsilon ,t+\epsilon \right] }\phi _{\epsilon }\left( t-u\right) \left( \bar{Z}_{u}-Z_{t}\right) \,du\right) ^{2}\right] \\ &=&E\left[ \int_{\left[ t-\epsilon ,t+\epsilon \right] ^{2}}\phi _{\epsilon }\left( t-u\right) \phi _{\epsilon }\left( t-v\right) \left( \bar{Z}% _{u}-Z_{t}\right) \left( \bar{Z}_{v}-Z_{t}\right) \,du\,dv\right] \\ &=&\int_{\left[ t-\epsilon ,t+\epsilon \right] ^{2}}\phi _{\epsilon }\left( t-u\right) \phi _{\epsilon }\left( t-v\right) E\left[ \left( \bar{Z}% _{u}-Z_{t}\right) \left( \bar{Z}_{v}-Z_{t}\right) \right] \,du\,dv \\ &\leq &\sup_{\substack{ t\in \left[ 0,1\right] \\ \left\vert h_{1}\right\vert ,\left\vert h_{2}\right\vert \leq \epsilon }}\left\vert E% \left[ \left( \bar{Z}_{t+h_{1}}-Z_{t}\right) \left( \bar{Z}% _{t+h_{2}}-Z_{t}\right) \right] \right\vert \\ &\leq &\sup_{\substack{ t\in \left[ 0,1\right] \\ \left\vert h\right\vert \leq \epsilon }}E\left[ \left( \bar{Z}_{t+h}-Z_{t}\right) ^{2}\right] \leq c_{1}\epsilon ^{1/\rho }\end{aligned}$$from which follows that $\sup_{0\leq t\leq 1}\left\vert X_{t}^{\epsilon _{k}}-X_{t}\right\vert _{L^{2}}^{2}\leq c_{1}\epsilon _{k}^{1/\rho }$. We conclude with Theorem \[theorem\_as\_wong\_zakai\_rate\]. Piecewise linear approximations ------------------------------- If $D=\{0=t_{0}<t_{1}<\ldots <t_{\#D-1}=1\}$ is a partition of $[0,1]$ and $% x\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ a continuous path, we denote by $x^{D}$ the piecewise linear approximation of $x$ at the points of $D$, i.e. $x^{D}$ coincides with $x$ at the points $t_{i}$ and if $t_{i}\leq t<t_{i+1}$ we have$$\frac{x_{t_{i+1}}^{D}-x_{t}^{D}}{t_{i+1}-t}=\frac{x_{t_{i+1}}-x_{t_{i}}}{% t_{i+1}-t_{i}}.$$Let $\left( D_{k}\right) _{k\in \mathbb{N}}$ be a sequence of partitions of $% \left[ 0,1\right] $ such that $\left\vert D_{k}\right\vert :=\max_{t_{i}\in D_{k}}\left\{ \left\vert t_{i+1}-t_{i}\right\vert \right\} \rightarrow 0$ for $k\rightarrow \infty $. If $x\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ is continuous, we define$$\Lambda _{k}\left( x\right) :=x^{D_{k}}.$$In [@FV10 Chapter 15.2.3] it is shown that $\left( \Lambda _{k}\right) _{k\in \mathbb{N}}$ fulfils the conditions of Theorem \[theorem\_main01\]. If $R_{X}$ is the covariance of a Gaussian process, we set$$\left\vert D\right\vert _{R_{X},\rho }=\left( \max_{t_{i}\in D}V_{\rho }\left( R_{X};\left[ t_{i},t_{i+1}\right] ^{2}\right) \right) ^{\rho }.$$ \[cor\_wong\_zakai\_piecew\_lin\]Let $X$ be as in Theorem [theorem\_main01\_intro]{}. Choose a sequence of partitions $\left( D_{k}\right) _{k\in \mathbb{N}}$ of the interval $\left[ 0,1\right] $ such that $\left( \left\vert D_{k}\right\vert _{R_{X},\rho }\right) _{k\in \mathbb{N}}\in $ $% \dbigcup\limits_{r\geq 1}l^{r}$ and set $X^{\left( k\right) }=X^{D_{k}}$. Then the solutions $Y^{\left( k\right) }$ of the SDE $\left( \ref% {eqn_RS_general}\right) $ converge pathwise to the solution $Y$ of $\left( % \ref{eqn_RDE_GP_general}\right) $ in the sense of $\left( \ref% {eqn_pathwise_wong_zakai}\right) $ with rate $O\left( \epsilon _{k}^{\eta }\right) $ where $\left( \epsilon _{k}\right) _{k\in \mathbb{N}}=\left( \left\vert D_{k}\right\vert _{R_{X},\rho }\right) _{k\in \mathbb{N}}$ and $% \eta $ is chosen as in Theorem \[theorem\_as\_wong\_zakai\_rate\]. Let $D$ be any partition of $\left[ 0,1\right] $ and $t\in \left[ t_{i},t_{i+1}\right] $ where $t_{i},t_{i+1}\in D$. Take $Z\in \left\{ X^{1},\ldots ,X^{d}\right\} $. Then$$Z_{t}^{D}-Z_{t}=Z_{t_{i},t_{i+1}}\frac{t-t_{i}}{t_{i+1}-t_{i}}-Z_{t_{i},t}.$$Therefore$$\left\vert Z_{t}^{D}-Z_{t}\right\vert _{L^{2}}\leq \left\vert Z_{t_{i},t_{i+1}}\right\vert _{L^{2}}+\left\vert Z_{t_{i},t}\right\vert _{L^{2}}\leq 2V_{\rho }\left( R_{X};\left[ t_{i},t_{i+1}\right] ^{2}\right) ^{1/2}\leq 2\left\vert D\right\vert _{R_{X},\rho }^{\frac{1}{2\rho }}.$$We conclude with Theorem \[theorem\_as\_wong\_zakai\_rate\]. Let $X=B^{H}$ be the fractional Brownian motion with Hurst parameter $H\in (1/4,1/2]$. Set $\rho =\frac{1}{2H}<2$. Then one can show that $R_{X}$ has finite $\rho $-variation and $V_{\rho }\left( R_{X};\left[ s,t\right] ^{2}\right) \leq c\left( H\right) \left\vert t-s\right\vert ^{1/\rho }$ for all $\left( s,t\right) \in \Delta $ (see [@FV11], Example 1). Assume that the vector fields in $\left( \ref{eqn_RDE_GP_general}\right) $ are sufficiently smooth by which we mean that $1/\rho -1/2\leq 1/\left( 2\rho \right) -1/\theta $, i.e. $$\theta \geq \frac{2\rho }{\rho -1}=\frac{1}{1/2-H}.$$Let $\left( D_{k}\right) _{k\in \mathbb{N}}$ be the sequence of uniform partitions. By Corollary \[cor\_wong\_zakai\_piecew\_lin\], for every $\eta <2H-1/2$ there is a random variable $C$ such that$$\left\vert Y^{\left( k\right) }-Y\right\vert _{\infty }\leq C\left( \frac{1}{% k}\right) ^{\eta }\quad \text{a.s.}$$hence we have a Wong-Zakai convergence rate arbitrary close to $2H-1/2$. In particular, for the Brownian motion, we obtain a rate close to $1/2$, see also [@GS06] and [@FR11]. For $H$ $\rightarrow 1/4$, the convergence rate tends to $0$ which reflects the fact that the Lévy area indeed diverges for $H=1/4$, see [@CQ02]. The simplified step-$N$ Euler scheme[subsection\_simple\_euler]{} ----------------------------------------------------------------- Consider again the SDE$$dY_{t}=V(Y_{t})\,dX_{t},\quad Y_{0}\in \mathbb{R}^{n}$$interpreted as a pathwise RDE driven by the lift $\mathbf{X}$ of a Gaussian process $X$ which fulfils the conditions of Theorem [theorem\_main01\_intro]{}. Let $D$ be a partition of $\left[ 0,1\right] $. We recall the simplified step-$N$ Euler scheme from the introduction:$$\begin{aligned} Y_{0}^{\text{sEuler}^{N};D} &=&Y_{0} \\ Y_{t_{j+1}}^{\text{sEuler}^{N};D} &=&Y_{t_{j}}^{\text{sEuler}% ^{N};D}+V_{i}\left( Y_{t_{j}}^{\text{sEuler}^{N};D}\right) X_{t_{j},t_{j+1}}^{i}+\frac{1}{2}\mathcal{V}_{i_{1}}V_{i_{2}}\left( Y_{t_{j}}^{\text{sEuler}^{N};D}\right) X_{t_{j},t_{j+1}}^{i_{1}}X_{t_{j},t_{j+1}}^{i_{2}} \\ &&+\ldots +\frac{1}{N!}\mathcal{V}_{i_{1}}\mathcal{\ldots V}% _{i_{N-1}}V_{i_{N}}\left( Y_{t_{j}}^{\text{sEuler}^{N};D}\right) X_{t_{j},t_{j+1}}^{i_{1}}\ldots X_{t_{j},t_{j+1}}^{i_{N}}\end{aligned}$$where $t_{j}\in D$. In this section, we will investigate the convergence rate of this scheme. For simplicity, we will assume that$$V_{\rho }\left( R_{X};\left[ s,t\right] ^{2}\right) =O\left( \left\vert t-s\right\vert ^{1/\rho }\right)$$which can always be achieved at the price of a deterministic time-change based on $$\left[ 0,1\right] \ni t\mapsto \frac{V_{\rho }\left( R_{X};\left[ 0,t\right] ^{2}\right) ^{\rho }}{V_{\rho }\left( R_{X};\left[ 0,1\right] ^{2}\right) ^{\rho }}\in \left[ 0,1\right] .$$Set $D_{k}=\left\{ \frac{i}{k}:i=0,\ldots ,k\right\} $. \[Cor\_rate\_simple\_euler\]Let $p>2\rho $ and assume that $\left\vert V\right\vert _{Lip^{\theta }}<\infty $ for $\theta >p$. Choose $\eta $ and $% N $ such that$$\eta <\min \left\{ \frac{1}{\rho }-\frac{1}{2},\frac{1}{2\rho }-\frac{1}{% \theta }\right\} \quad \text{and\quad }N\leq \left[ \theta \right] .$$Then there are random variables $C_{1}$ and $C_{2}$ such that$$\max_{t_{j}\in D_{k}}\left\vert Y_{t_{j}}-Y_{t_{j}}^{\text{sEuler}% ^{N};D_{k}}\right\vert \leq C_{1}\left( \frac{1}{k}\right) ^{\eta }+C_{2}\left( \frac{1}{k}\right) ^{\frac{N+1}{p}-1}\quad \text{a.s. for all }% k\in \mathbb{N}\text{.}$$ Recall the step-$N$ Euler scheme from the introduction (or cf. [@FV10 Chapter 10]). Set $X^{\left( k\right) }=X^{D_{k}}$ and let $Y^{\left( k\right) }$ be the solution of the SDE $\left( \ref{eqn_RS_general}\right) $. Then $% Y_{t_{j}}^{\text{sEuler}^{N};D_{k}}=\left( Y^{\left( k\right) }\right) _{t_{j}}^{\text{Euler}^{N};D_{k}}$ for every $t_{j}\in D_{k}$ and therefore, using the triangle inequality,$$\max_{t_{j}\in D_{k}}\left\vert Y_{t_{j}}-Y_{t_{j}}^{\text{sEuler}% ^{N};D_{k}}\right\vert \leq \sup_{t\in \left[ 0,1\right] }\left\vert Y_{t}-Y_{t}^{\left( k\right) }\right\vert +\max_{t_{j}\in D_{k}}\left\vert Y_{t_{j}}^{\left( k\right) }-\left( Y^{\left( k\right) }\right) _{t_{j}}^{% \text{Euler}^{N};D_{k}}\right\vert .$$By the choice of $D_{k}$ we have $\left\vert D_{k}\right\vert _{R_{X},\rho }=O\left( k^{-1}\right) $. Applying Corollary \[cor\_wong\_zakai\_piecew\_lin\] we obtain for the first term $\left\vert Y-Y^{\left( k\right) }\right\vert _{\infty }=O\left( k^{-\eta }\right) $. Refering to [FV10]{} we see that the second term is of order $O\left( k^{-\left( \frac{N+1% }{p}-1\right) }\right) $. Assume that the vector fields are sufficiently smooth, i.e. $\theta \geq \frac{2\rho }{\rho -1}$. Then we obtain an error of $O\left( k^{-(2/p-1/2)}\right) +O\left( k^{-\left( \frac{N+1}{p}-1\right) }\right) $, any $p>2\rho $. That means that in the case $\rho =1$, the step-$2$ scheme (i.e. the simplified Milstein scheme) gives an optimal convergence rate of (almost) $1/2$. For $\rho \in (1,2)$, the step-$3$ scheme gives an optimal rate of (almost) $1/\rho -1/2$. In particular, we see that using higher order schemes does not improve the convergence rate since in that case, the Wong-Zakai error persists. In the fractional Brownian motion case, the simplified Milstein scheme gives an optimal convergence rate of (almost) $% 1/2 $ for the Brownian motion and for $H\in (1/4,1/2)$ the step-$3$ scheme gives an optimal rate of (almost) $2H-1/2$. This answers a conjecture stated in [@DT]. [99]{} Ben Arous, G.: Flots et series de Taylor stochastiques, Probab. Theory Related Fields 81, 29-77, 1989. Cass, T.; Friz, P.. Densities for Rough Differential Equations under Hoermander’s Condition; Annals of Mathematics, 2010 (Volume 171, no. 3), 2115–2141. Coutin, L., Qian, Z.: Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122(1):108-140, 2002. Davie, A.M.: Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express. AMRX, (2): Art. ID abm009, 40, 2007. Deya, A., Neuenkirch, A., Tindel, S.: A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion, Annales de l’Institut Henri Poincaré, to appear. Friz, P., Oberhauser, H.: Rough paths limits of the Wong-Zakai type with a modified drift term, J. Funct. Anal., 256(10):3236-3256, 2009. Friz, P., Riedel, S.: Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows, Bulletin des Sciences Mathématiques (proceeding in memory of P. Malliavin), DOI 10.1016/j.bulsci.2011.07., 2011. Friz, P., Victoir, N.: Differential Equations Driven by Gaussian Signals, Annales de l’Institut Henri Poincare (B) Probability and Statistics, May 2010, Vol. 46, No. 2, 369–413. Friz, P., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths, Cambridge University Press, 2010. Friz, P., Victoir, N.: A note on higher dimensional $p$-variation, Electronic Journal of Probability, to appear. Gyöngy, I., Shmatkov, A.: Rate of Convergence of Wong-Zakai Approximations for Stochastic Partial Differential Equations, Appl. Math. Optim. 54:315-341, 2006. Hairer, M.: Rough stochastic PDE; arXiv:1008.1708v1; to appear in Comm. Pure Applied Math. Hara, K., Hino, M.: Fractional order Taylor’s series and the neo-classical inequality, Bull. Lond. Math. Soc. 42, 467-477, 2010. Hu, Y., Nualart, D.: Rough Path Analysis via Fractional Calculus, Trans. Amer. Math. Soc. 361(5):2689-2718, 2009. Inahama, Y.: A moment estimate of the derivative process in rough path theory, arXiv:1007.4651v1. Janson, S.: Gaussian Hilbert spaces, Cambridge University Press, 1997. Lyons, T.: Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14, no. 2, 215–310, 1998. Lyons, T., Qian, Z.: Flow of diffeomorphisms induced by a geometric multiplicative functional, Probab. Theory Related Fields 112, no. 1, 91-119, 1998. Lyons, T., Qian, Z.: System Control and Rough Paths, Oxford University Press, 2002. Neuenkirch, A., Tindel, S., Unterberger, J.: Discretizing the fractional Lévy area, Stochastic Process. Appl., Vol. 120, Issue 2, 223-254, 2010. Reutenauer, C.: Free Lie Algebras, Clarendon Press, New York, 1993. Towghi, N.: Multidimensional extension of L.C. Young’s inequality, JIPAM J. Inequal. Pure Appl. Math., 3(2): Article 22, 13 pp. (electronic), 2002. [^1]: P.K. Friz has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr. 258237. [^2]: S. Riedel is supported by an IRTG (Berlin-Zurich) PhD-scholarship and a scholarship from the Berlin Mathematical School (BMS) [^3]: A basic theorem of rough path theory asserts that further iterated integrals up to any level $N\geq \left[ p\right] $, i.e.$$S_{N}\left( \mathbf{x}\right) :=(\mathbf{x}^{n}:n\in \left\{ 1,\dots ,N\right\} )$$are then deterministically determined and the map $\mathbf{x}\mapsto S_{N}\left( \mathbf{x}\right) $, known as Lyons lift, is continuous in rough path metrics. [^4]: A general time horizon $\left[ 0,T\right] $ is handled by trivial reparametrization of time. [^5]: ... which one would call Milstein scheme when $N=2$ ... [^6]: ...in the sense of E. Stein; cf. [@LQ02; @FV10] for instance.
--- author: - 'Alessio Franci$^$, Guillaume Drion$^$, Rodolphe Sepulchre' title: Robust and tunable bursting requires slow positive feedback --- Running title {#running-title .unnumbered} ------------- Bursting requires slow positive feedback Authors and affiliations {#authors-and-affiliations .unnumbered} ------------------------ Alessio Franci is with Department of Mathematics, Universidad Nacional Autónoma de México, Mexico. Guillaume Drion is with the Institut Montefiore, Université de Liege, Belgium. Rodolphe Sepulchre is with the Department of Engineering, University of Cambridge, United Kingdom. Equal contribution {#equal-contribution .unnumbered} ------------------ $^$ These authors contributed equally to this work. Corresponding author {#corresponding-author .unnumbered} -------------------- Alessio Franci, afranci@ciencias.unam.mx, tel. +5215562054874 Funding {#funding .unnumbered} ------- The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645 and from DGAPA-UNAM under the grant PAPIIT RA105816. [We highlight that the robustness and tunability of a bursting model critically relies on currents that provide slow positive feedback to the membrane potential. Such currents have the ability of making the total conductance of the circuit [negative]{} in a time scale that is termed [slow]{} because intermediate between the [fast]{} time scale of the spike upstroke and the [ultraslow]{} time scale of even slower adaptation currents. We discuss how such currents can be assessed either in voltage-clamp experiments or in computational models. We show that, while frequent in the literature, mathematical and computational models of bursting that lack the slow negative conductance are fragile and rigid. Our results suggest that modeling the slow negative conductance of cellular models is important when studying the neuromodulation of rhythmic circuits at any broader scale. ]{} New and noteworthy {#new-and-noteworthy .unnumbered} ------------------ Nervous system functions rely on the modulation of neuronal activity between different rhythmic patterns. The mechanisms of this modulation are still poorly understood. Using computational modeling, we show the critical role of currents that provide slow negative conductance, distinct from the fast negative conductance necessary for spike generation. The significance of the slow negative conductance for neuromodulation is often overlooked, leading to computational models that are rigid and fragile. Introduction {#introduction .unnumbered} ============ While the function of neuronal bursting is still debated and probably diverse, the continuous modulation between distinct firing patterns is an important signaling component of many nervous functions. Those include muscle contraction orchestrated by central pattern generators [@Marder2012], control of sleep, wakefulness and attention in thalamocortical circuits [@McCormick1997; @Sherman2001; @Bezdudnaya2006], and sensing [@Krahe2004]. Voltage recordings in those references suggest robust and continuous modulations between spiking and bursting. All transitions share a sharp separation between the low frequency of spikes in tonic firing and the high frequency of spikes during bursts. They are observed across a broad range of neuronal and bursting types. The mechanisms of this regulation are still poorly understood. At the physiological level, they seem to involve a variety of ionic currents and neuromodulators, see e.g. the review [@Marder2007]. At the modeling level, most textbooks on computational and mathematical neuroscience include a chapter on bursting [@Izhikevich2007 Chapter 9],[@Ermentrout2010a Chapter 5], but the mathematical theory of bursting is based on a classification of different types of bursting according to different bifurcation mechanisms. Each bursting type is associated to a different bifurcation mechanism with little importance given to the transitions between bursting types and, most importantly, to the connection between mathematical transitions and physiological modulation. To the best of our knowledge, a mathematical theory of bursting [modulation]{} and how it relates to physiological mechanisms has not been addressed until the recent paper [@Franci2014]. In this paper we use a state-of-the-art conductance-based model, widely used in computational neuromodulation studies, to highlight a modeling feature of bursting that is critical to robustness and modulation. Specifically, mimicking the classical voltage-clamp experiment, we study the conductance of the neuron by analysing the total current response to a voltage step perturbation around the threshold potential. We aim to show that, irrespective of the modeling details, the robustness and modulation properties of the model primarily rely on the ability to modulate the slow temporal component of the conductance from positive (in spiking mode) to negative (in bursting mode). This modulation only occurs [transiently]{} in a time scale that is [slow]{} compared to the [fast]{} time scale of the spike upstroke. The voltage-clamp signature of this slow conductance makes it model-independent and easy to assess experimentally. The presence of a [slow]{} negative conductance in a circuit, distinct from its [fast]{} negative conductance, is easily overlooked because of its [transient]{} nature. We highlight the [transient]{} nature of this property with a computational experiment that only changes the kinetics of calcium channel activation from slow to fast, without affecting the balance of currents at steady-state. We show that all the modulation properties of the bursting model are lost when the calcium activation is fast, just because the slow negative conductance is no longer distinct from the fast negative conductance. A model can exhibit a slow negative conductance only if it includes an inward current with slow activation or an outward current with slow inactivation. Such currents have been named slow regenerative in [@Franci2013]. By definition, a model that does not include slow regenerative currents cannot exhibit a slow negative conductance in any voltage range. We show that such models abound in the literature of bursting. This is because the slow negative conductance, while essential to robust modulation, is not necessary to bursting [per se]{}. But we illustrate on a number of published models that bursting models that lack a source of slow negative conductance are both fragile and rigid: they are very sensitive to small parameter variations, and those parameter variations disrupt the bursting pattern altogether rather than modulating the shape of the bursting pattern. In sharp contrast, published models that include a slow negative conductance are robust and tunable: small parameter variations do not disrupt the bursting pattern and specific parameter variations modulate the bursting shape between different bursting types. The total conductance of a neuronal circuit is modulated in a given time scale by a balance between currents of negative and positive conductance, or equivalently, by a balance between currents providing positive and negative feedback to the membrane potential. Our results suggest that modulating the sign of the slow conductance of a model is necessary to the regulation of bursting, meaning that slow regenerative channels are a natural target for neuromodulators involved in bursting modulation, in line with a number of experimental studies [@Marder2007]. We also provide an analysis of our results in terms of phase portraits and bifurcation theory, the classical language of bursting theory. Phase portraits of regenerative and restorative models are indeed fundamentally different [@Drion2012; @Franci2012]. We show that only in the presence of a slow negative conductance a same phase portrait is both robust to parameter variations [and]{} compatible with various bursting types that have traditionally been associated to distinct models. This comparison suggests the relevance of analyzing bursting as circuits regulated by a balance of positive and negative feedbacks in distinct time scales as a complement to the traditional classification based on bifurcation theory. While the analysis in this paper is performed at the single cell level, there is growing evidence, see e.g. the recent paper [@dethier2015] that slow positive feedback at the cellular level critically impacts the robustness and tunability of rhythmic circuits as well. This suggests that accounting for the modeling feature highlighted in this paper is relevant for neuromodulation studies at every scale and therefore a feature that merits attention both from experimentalists and modelers. Methods {#methods .unnumbered} ======= All simulations and analyses were performed using the Julia programming language. The Julia code is available as Extended Data and can be downloaded at https://github.com/elsesma/eNeuro2017-Code. Figure \[fig:1\]A is generated using the STG model described in [@Goldman2001]. Briefly, the model is composed of a leak current $I_{leak}$, a transient sodium current $I_{Na}$, a T-type calcium current $I_{Ca,T}$, a S-type calcium current $I_{Ca,S}$, a delayed rectifier potassium current $I_{K,DR}$, a transient potassium current $I_{A}$, a calcium activated potassium current $I_{K,Ca}$. Parameters used in the simulations are as follows. (a): $C=1\,\mu F\cdot cm^{-2}$, $V_{Na}=50\,mV$, $V_K=-80\,mV$, $V_{Ca}=80\,mV$, $V_{leak}=-50\,mV$, $\bar g_{leak}=0.1\,mS\,cm^{-2}$, $\bar g_{Na}=700\,mS\,cm^{-2}$, $\bar g_{Ca,T}=6\,mS\,cm^{-2}$, $\bar g_{Ca,S}=9\,mS\,cm^{-2}$, $\bar g_{A}=30\,mS\,cm^{-2}$, $\bar g_{K,DR}=80\,mS\,cm^{-2}$, $\bar g_{K,Ca}=25\,mS\,cm^{-2}$. (b): same parameters as (a) except $\bar g_{Ca,T}=1\,mS\,cm^{-2}$, $\bar g_{Ca,S}=1.5\,mS\,cm^{-2}$, $\bar g_{A}=240\,mS\,cm^{-2}$. (c): same parameters as (a) except $\bar g_{Ca,T}=3\,mS\,cm^{-2}$, $\bar g_{Ca,S}=4.5\,mS\,cm^{-2}$, $\bar g_{A}=26\,mS\,cm^{-2}$. (d): same parameters as (a) except $\bar g_{Ca,T}=7\,mS\,cm^{-2}$, $\bar g_{Ca,S}=10.5\,mS\,cm^{-2}$, $\bar g_{A}=225\,mS\,cm^{-2}$. Burstiness is defined as $\frac{\text{spikes per burst}\times\text{intraburst frequency}}{\text{bursting period}}$. Voltage steps in the voltage clamp experiments are from $-40mV$ to $-39mV$. Figure \[fig:1\]B is generated with the same model and parameters as Figure 1A except that $\tau_{m_{Ca,T}}$ and $\tau_{m_{Ca,S}}$ are scaled by $0.5$ in the center parameter chart and $m_{Ca,T}=m_{{Ca,T}_\infty}(V),m_{Ca,T}=m_{{Ca,T}_\infty}(V)$ (instantaneous calcium activation) in the right parameter chart. Voltage clamp steps are from $-39mV$ to $-40mV$. Nominal models in Figure \[FIG:2\] are given as follows. The STG model is the same as Figure \[fig:1\]A with maximal conductance parameters: $\bar g_{leak}=0.1\,mS\,cm^{-2}$, $\bar g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=6.5.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=9.75\,mS\,cm^{-2}$, $\bar g_{A}=100\,mS\,cm^{-2}$, $\bar g_{K,DR}=80\,mS\,cm^{-2}$, $\bar g_{K,Ca}=40\,mS\,cm^{-2}$. The Plant R15 model and parameters are the same as given in [@Rinzel1987a]. The pancreatic beta [cell]{} model and parameters are the same as described in [@Chay1983]. The thalamocortical (TC) model and parameters are the same as given in [@Wang1994]. The CA1 model and parameters are the same as given in [@Golomb2006]. The modified CA1+ model is obtained from the nominal model by: the persistent sodium current activation is made dynamic with time constant equal to $6$ times the original delayed rectifier activation time constant; the original delayed rectifier activation time constant is scaled by $4$; the cell capacitance is scaled by $0.4$. Figure \[FIG:3\]A is generated using the same STG model as Figure \[fig:1\]A. Parameters used in the simulations are as in Figure \[fig:1\]A except the following. Left trace: $\bar g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=1.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=4.\,mS\,cm^{-2}$, $\bar g_{A}=10\,mS\,cm^{-2}$, $\bar g_{K,DR}=40\,mS\,cm^{-2}$, $\bar g_{K,Ca}=8\,mS\,cm^{-2}$. Center trace: $g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=1.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=7.\,mS\,cm^{-2}$, $\bar g_{A}=8\,mS\,cm^{-2}$, $\bar g_{K,DR}=40\,mS\,cm^{-2}$, $\bar g_{K,Ca}=13\,mS\,cm^{-2}$. Right trace: $g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=10.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=8.\,mS\,cm^{-2}$, $\bar g_{A}=10\,mS\,cm^{-2}$, $\bar g_{K,DR}=120\,mS\,cm^{-2}$, $\bar g_{K,Ca}=40\,mS\,cm^{-2}$.\ Figure \[FIG:3\]B is generated using the same STG model as Figure \[fig:1\]A. Parameters used in the simulations are as in Figure \[fig:1\]A except the following. Parabolic case: $\bar g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=1.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=32.\,mS\,cm^{-2}$, $\bar g_{A}=40\,mS\,cm^{-2}$, $\bar g_{K,DR}=150\,mS\,cm^{-2}$, $\bar g_{K,Ca}=200\,mS\,cm^{-2}$. Square-wave case: $g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=10.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=8.\,mS\,cm^{-2}$, $\bar g_{A}=10\,mS\,cm^{-2}$, $\bar g_{K,DR}=120\,mS\,cm^{-2}$, $\bar g_{K,Ca}=50\,mS\,cm^{-2}$. Tapered case: $g_{Na}=1200\,mS\,cm^{-2}$, $\bar g_{Ca,T}=1.\,mS\,cm^{-2}$, $\bar g_{Ca,S}=40.\,mS\,cm^{-2}$, $\bar g_{A}=40\,mS\,cm^{-2}$, $\bar g_{K,DR}=200\,mS\,cm^{-2}$, $\bar g_{K,Ca}=200\,mS\,cm^{-2}$. Figure \[FIG:5\]A,B are generated using the same STG model as Figure \[fig:1\]A. The bifurcation diagrams are computed by setting the calcium-activated potassium channel activation variable as the bifurcation parameter, and all other ultraslow variables at constant values ($h_{Ca,T} = h_{Ca,S} = 0.15$, $h_A = 0.05$). Bifurcation diagrams of Figure \[FIG:5\]A are computed using the original STG model in spiking and bursting modes. Bifurcation diagrams of Figure \[FIG:5\]B are computed using the STG model in bursting mode for different values of the calcium channel activation time constant. Figure \[FIG:5\]C is computed by simulating the STG model (left) and CA1 model (right) in bursting mode configuration for different values of the membrane capacitance ($C_m = 1\mu F/cm^2$ corresponds to the original value in both cases). Figure \[FIG:5\]D is computed by simulating the CA1 mode in bursting mode configuration and after changes in various model parameters as indicated in the figure. The bifurcation diagrams are computed for the original model and after an increase (top right) or a decrease (bottom right) in membrane capacitance. Figure \[FIG:6\]A top is generated using the STG model described in [@Liu1998]. Briefly, the model is composed of a leak current $I_{leak}$, a transient sodium current $I_{Na}$, a T-type calcium current $I_{Ca,T}$, a S-type calcium current $I_{Ca,S}$, a delayed rectifier potassium current $I_{K,DR}$, a transient potassium current $I_{A}$, a calcium activated potassium current $I_{K,Ca}$, and hyperpolarization-activated cyclic nucleotide-€“gated $I_{H}$ current. Parameters used in the simulations are as follow. Tonic firing: $C=1\,\mu F\cdot cm^{-2}$, $V_{Na}=50\,mV$, $V_K=-80\,mV$, $V_{Ca}=80\,mV$, $V_{leak}=-50\,mV$, $\bar g_{leak}=0.01\,mS\,cm^{-2}$, $\bar g_{Na}=800\,mS\,cm^{-2}$, $\bar g_{Ca,T}=1\,mS\,cm^{-2}$, $\bar g_{Ca,S}=1\,mS\,cm^{-2}$, $\bar g_{A}=50\,mS\,cm^{-2}$, $\bar g_{K,DR}=90\,mS\,cm^{-2}$, $\bar g_{K,Ca}=60\,mS\,cm^{-2}$, $\bar g_{H}=0.1\,mS\,cm^{-2}$. Bursting: same parameters as tonic except $\bar g_{Ca,T}=4\,mS\,cm^{-2}$, $\bar g_{Ca,S}=8\,mS\,cm^{-2}$. Voltage steps in the voltage clamp experiments are from $-44mV$ to $-42mV$. Figure \[FIG:6\]A bottom is generated using the same STG model as Figure 1A. Parameters used in the simulations are as in \[fig:1\]A except $\bar g_{Na}=800\,mS\,cm^{-2}$, $\bar g_{Ca,T}=10\,mS\,cm^{-2}$, $\bar g_{Ca,S}=8\,mS\,cm^{-2}$, $\bar g_{A}=10\,mS\,cm^{-2}$, $\bar g_{K,DR}=120\,mS\,cm^{-2}$, $\bar g_{K,Ca}=50\,mS\,cm^{-2}$ (tonic mode) or $\bar g_{Na}=800\,mS\,cm^{-2}$, $\bar g_{Ca,T}=1\,mS\,cm^{-2}$, $\bar g_{Ca,S}=1\,mS\,cm^{-2}$, $\bar g_{A}=10\,mS\,cm^{-2}$, $\bar g_{K,DR}=120\,mS\,cm^{-2}$, $\bar g_{K,Ca}=50\,mS\,cm^{-2}$ (bursting mode). Voltage steps in the voltage clamp experiments are from $-44mV$ to $-42mV$. Figure \[FIG:6\]B bottom is generated using the Plant R15 aplysia model as described in [@Rinzel1987a]. Briefly, the model is composed of a leak current $I_{leak}$, a transient sodium current $I_{Na}$, a persistent calcium current $I_{Ca}$, a delayed rectifier potassium current $I_{K,DR}$, a calcium activated potassium current $I_{K,Ca}$. Parameters used in the simulation are as follows. $C=0.8\,\mu F\cdot cm^{-2}$, $V_{Na}=30\,mV$, $V_K=-75\,mV$, $V_{Ca}=140\,mV$, $V_{leak}=-40\,mV$, $\bar g_{leak}=0.003\,mS\,cm^{-2}$, $\bar g_{Na}=4\,mS\,cm^{-2}$, $\bar g_{K,DR}=4\,mS\,cm^{-2}$, $\bar g_{Ca}=0.006\,mS\,cm^{-2}$, $\bar g_{K,Ca}=0.04\,mS\,cm^{-2}$. Voltage steps in the voltage clamp experiments are from $-80mV$ to $-40mV$. Figure \[FIG:6\]B top is generated using the same model and parameters as Figure \[FIG:6\]B bottom, except that the calcium current activation is 100 times faster. Results {#results .unnumbered} ======= A transient signature of robust and tunable bursting {#sec:2 .unnumbered} ---------------------------------------------------- Figure \[fig:1\]A uses the computational model of [@Goldman2001] to illustrate a classical physiological transition from tonic firing to bursting. The model includes seven voltage- and time-dependent conductances as well as a leak conductance (see Methods). The modulation from tonic firing to bursting is obtained by varying the balance of calcium and A-type potassium voltage-gated currents. This modulation defines modulatory paths in the parameter space of the two maximal conductances $\bar{g}_{Ca}$ and $\bar{g}_A$ (Path a-b in Figure \[fig:1\]A ). The same plane contains degeneracy paths where modulation of the maximal conductances results in almost no change in the resulting neuronal activity (Path c-d in Figure \[fig:1\]A ). The coexistence of degeneracy and modulatory paths has been shown to be critical for robust neuromodulation [@Marder2014]. A computational model that reproduces such features does not only exhibit a bursting trace for a well chosen set of conductance parameters. In addition, the bursting rhythm is robust and tunable in the parameter space of maximal conductances. ![A. The slow transient in a voltage clamp experiment near threshold is a reliable signature to discriminate between bursting and tonic firing. Left panel: parameter chart of burstiness as a function of the parameters $g_A$ and $g_Ca$ in the STG model [@Goldman2001]. Burstiness was computed as described in the Method Section. Right panel: voltage clamp experiment close to threshold potential in tonic mode (a) and bursting mode (b). [Voltage steps are from $-40mV$ to $-39mV$.]{} The slow transient of the [current response to a voltage step]{} is increasing in spiking mode (a signature of slow positive conductance) and decreasing in bursting mode (a signature of slow negative conductance). The signature is modulated along a modulation path (a-b) and conserved along a degeneracy path (c-d, not shown) in the parameter space of maximal conductances. B. Effects of decreasing calcium current activation time constant on the parameter chart and the voltage clamp experiment in A. The decreasing phase of the slow transient vanishes as calcium activation kinetics, the only source of slow negative conductance in the model, varies from slow to fast. In the parameter charts, reduction of the calcium activation time constants shrinks the parameter region where the model can be modulated. In the limit of instantaneous activation, the model has lost its modulation properties and in particular the transition from tonic firing to bursting. []{data-label="fig:1"}](Fig1){width="90.00000%"} Figure \[fig:1\]B highlights that this tunability property is completely lost by changing a single parameter in the model, namely, the time constant of activation of the calcium channels. Before interpreting this result, we stress that the modeling difference between Figure \[fig:1\]A and Figure \[fig:1\]B is purely [dynamical]{} in nature: it does not affect the [static]{} behavior of the model, that is, the model equations at equilibrium. This means in particular an identical balance of ionic currents at equilibrium and an identical I-V curve. To unfold the [transient]{} mechanism responsible for the structural change between Figure \[fig:1\]A and Figure \[fig:1\]B, we mimick the classical voltage-clamp experiment of electrophysiology: we clamp the voltage at a constant value close to threshold potential ($V_{th}\sim-40mV$ in this model) and apply a small voltage step perturbation $\Delta V_m$ at time $t=0$. The current step response $\Delta I_m(t)$ provides us with the temporal evolution of the local [conductance]{} $\Delta I_m(t)/ \Delta V_m $ of the model around the threshold potential. This total conductance is the aggregate conductance resulting from all the ionic current variations at a given time and around a given voltage. In Figure \[fig:1\]A, we see that the [transient]{} behavior of the local conductance is markedly different in the spiking configuration (a) and in the bursting configuration (b). In the spiking configuration, the current step response carries the usual signature of an excitable circuit: an initial phase characterised by a fast inverse response, followed by a slow monotone convergence to equilibrium. In the bursting configuration, the current response exhibits an additional slow inverse response, distinct from the initial fast inverse response. If we decompose the current response into fast, slow, and ultraslow transient phases, it is the slow transient that discriminates bursting from spiking. In Figure \[fig:1\]B, the distinct slow transient signature of bursting is progressively lost as the time constant of calcium activation is decreased. This is because the two distinct inverse responses progressively merge. In the limiting case of an instantaneous calcium activation, they simply add up in the fast time scale. This phenomenon is easy to explain in the computational model used in Figure \[fig:1\]: the first ([fast]{}) inverse response of the current results from the fast activation of sodium channels whereas the second ([slow]{}) inverse response results from the slow activation of calcium channels. The two successive inverse responses are distinct in the voltage clamp experiment of Figure \[fig:1\]A because the time scales of the calcium channel activations are significantly slower than the time scale of the sodium channel activation [@Kostyuk1977], [@Hille2001 p.127]. In contrast, the two successive inverse responses merge in Figure \[fig:1\]B because the time scale of calcium channel activation merges with the time scale of sodium channel activation. A robust and tunable bursting model must include a source of slow negative conductance {#sec:3 .unnumbered} -------------------------------------------------------------------------------------- In the seminal work of Hodgkin and Huxley, the voltage clamp experiment described in the previous section was applied to the squid giant axon and served as a foundation to model the voltage dependence and the dynamics of ionic conductances. The [early inverse]{} current response was attributed to a [fast negative]{} conductance modeled by the [fast]{} activation of an inward (sodium) current. Likewise, the [late monotone]{} convergence was attributed to a [slow positive]{} conductance modeled both by the [slow]{} inactivation of sodium channels and the [slow]{} activation of an outward (potassium) current. We proceed in the same manner to explain the slow transient signature of a bursting neuron: it requires a [voltage-gated current]{} providing a [slow negative conductance]{}, distinct from the [fast]{} negative conductance. A negative conductance is provided by the activation of an inward current or the inactivation of an outward current. It is [slow]{} if the corresponding channel kinetics is distinctively slower than the fast activation of sodium channels [and distinctively faster than adaptation current kinetics. Typical slow conductance time constants are in the range $5-20\,{\rm ms}$]{}. Physiological contributors of such currents include the whole family of calcium currents with slow activation as well as resurgent sodium channels [@Swensen2003]. They also include any outward current that inactivates slowly, [such as some potassium channels]{} [@Storm1990]. Such channels have been named [slow regenerative]{} in the paper [@Franci2013]. Our computational experiment suggests that a robust and tunable bursting neuronal model must include a source of slow negative conductance. For a conductance-based model, this means that the gated ionic currents must include at least one type of slow regenerative channel. In the simulated STG model, only calcium currents contribute to the slow negative conductance. They do so because their activation is slow. The modulation path in Figure \[fig:1\]A amplifies the slow negative conductance of the total current from (a) to (b) by modulating the balance between slow regenerative (calcium) channels and slow restorative (potassium) channels. In Figure \[fig:1\]B, this modulation property is lost because the calcium channels become fast regenerative. Modulation of the total conductance in the slow time scale from positive to negative is no longer possible because the model has lost its only source of slow regenerative channels. In a conductance-based model, modulation of the total slow negative conductance is possible only in the presence of slow regenerative channels. The voltage-clamp experiment in the previous section is a general method to assess the negative slow conductance of a circuit, irrespective of the modeling details of the model. The reader is referred to the recent paper [@Drion2015] for a method that quantitatively assesses the slow negative conductance (or any other conductance) of an arbitrary one-port circuit at a given voltage, either computationally or experimentally. Robust versus fragile bursting {#robust-versus-fragile-bursting .unnumbered} ------------------------------ The absence of slow negative conductance has a dramatic consequence on the robustness of the bursting model to parameter perturbations. Figure \[FIG:2\] illustrates the striking contrast between the fragility of models that lack slow negative conductance and the robustness of models that include slow negative conductance. The chosen perturbation is a uniform scaling of all maximal conductance parameters, which is mathematically equivalent to a scaling of the membrane capacitance. ![Bursting models that lack currents providing slow negative conductance are fragile: tiny parameter variations disrupt the nominal rhythm. Green models (STG and R15) do include slow regenerative channels providing slow negative conductance. Red models (p$\beta$C, TC, and CA1) lack slow negative conductance. [Top panels]{}: only the bursting traces of green models are robust to a uniform scale of the maximal conductance vector ($\bar{\mathbf{g}}\mapsto\,0.8\bar{\mathbf{g}}$ or $\bar{\mathbf{g}}\mapsto\,1.2\bar{\mathbf{g}}$). The red model CA1 is turned into the robust green model CA1+ by making the calcium activation slow. See Methods for details. [Bottom panels]{}: random uniform scaling of the vector of maximal conductances induce large variability in the rhythm properties only in models lacking slow negative conductance. The scatter plots are obtained by scaling the maximal condutance vector $\bar{\mathbf{g}}$ by a uniformly distributed random number in the range $[0.8,1.2]$. Variability plots are absolute for the mean spike height (left) and logarithmic for the burst period (center) and number of spikes per burst (right): $\frac{\text{(spike height)}}{\text{(spike height)}_{\rm nominal}}$, $\log\left(\frac{\text{(burst period)}}{\text{(burst period)}_{\rm nominal}}\right)$, $\log\left(\frac{\text{(spikes-per-burst)}}{\text{(spikes-per-burst)}_{\rm nominal}}\right)$.[]{data-label="FIG:2"}](Fig2){width="0.9\columnwidth"} The STG [@Goldman2001] and the R15 [@Rinzel1987a] models are two classical bursting models of the literature that are robust to the perturbation. Both include slow regenerative channels by modeling calcium channels that activate [slowly]{}. The three models p$\beta$C [@Chay1983], TC [@Wang1994], and CA1 [@Golomb2006] are three published models that are fragile to the perturbation. Small deviations from the nominal parameter set produce large variations in different properties of the rhythm. The three models lack any source of slow negative conductance. They all include calcium channels or other regenerative channels but assume an instantaneous activation, making them [fast]{} regenerative instead of [slow]{} regenerative. The model CA1+ is a modification of the published CA1 model. In the modified model, the activation of persistent sodium channels was modified from fast to slow, making the persistent sodium channels slow regenerative instead of fast regenerative (see Methods for details). This only modification was sufficient to recover the robustness of models that have a slow negative conductance. Tunable versus rigid bursting {#tunable-versus-rigid-bursting .unnumbered} ----------------------------- Bursting models that include a slow negative conductance are not only robust, but they are robustly tunable. This means that the shape of the bursting trace can be tightly controlled by modulating maximal conductance values (i.e. channel densities). Figure \[FIG:3\] illustrates those modulation properties with the same published STG model as in the previous sections. Figure \[FIG:3\]A illustrates the modulation of bursting quality: the intraburst frequency and plateau properties are continuously modulated while maintaining other features of the burst unaffected, such as the interburst frequency or the mean voltage during resting and spiking phases. Figure \[FIG:3\]B illustrates the modulation of the bursting shape, while maintaining the mean intraburst and interburst frequencies. The continuous modulation recovers three distinct types of bursting usually referred to as “square-wave", “parabolic", and “tapered". ![image](Fig3){width="90.00000%"} The modulation properties illustrated in Figure \[FIG:3\] do not result from a systematic exploration of the parameter space, a task already formidable for the chosen STG model [@Prinz2003]. Instead, they only rely on modulating the ratio of the maximal values of the total slow negative conductance and of the total ultra-slow positive conductance following the methodology of dynamic input conductances [@Drion2015]. Shaping the dynamic conductances relatively to each other is easy and intuitive because the tuning parameters are few and directly map to the bursting behavior. This qualitative tuning is then easily translated into physiologically plausible modulations. For instance, modulation of the slow negative conductance relative to the ultraslow positive conductance in Figure \[FIG:3\] was achieved by modifying only the five following maximal conductances: $\bar{g}_{Ca,T}, \bar{g}_{Ca,S}, \bar{g}_{A}, \bar{g}_{Kd}, \bar{g}_{KCa}$. In each case, the modulation is robust, that is, not sensitive to small parameter variations in the large-dimensional space of the conductance-based model parameters. The robust modulation illustrated in Figure \[FIG:3\] is in sharp contrast with the rigidity of bursting models that lack a source of slow negative conductance. The nominal bursting trace of the three models p$\beta$C, TC, and CA1 in Figure \[FIG:2\] is rigid because the relationship between intraburst and interburst frequencies as well as the relationship between the mean voltage of the resting and spiking modes is extremely constrained. The resulting burst is not only fragile; it is also rigid, making it difficult to modulate the burstiness or the bursting type as in Figure \[FIG:3\]. The geometric analysis in the next section provides additional insight to those limitations. Connection with phase portrait and bifurcation analysis {#connection-with-phase-portrait-and-bifurcation-analysis .unnumbered} ------------------------------------------------------- The critical role of the slow negative conductance for modulation and robustness will now be examined in the light of geometric analysis. We rely on the common simplification that a three time-scale bursting attractor can be analyzed via the bifurcation diagram of a two time-scale phase portrait. The variables of the phase portrait are the fast voltage and a slow variable aggregating all the slow variables. The bifurcation parameter is a representative ultraslow variable. Bursting is modeled as ultraslow adaptation of the bifurcation parameter across a parameter range where a stable fixed point (the [resting]{} state) and a stable limit cycle (the [spiking]{} state) coexist in the phase portraits. The role of the slow negative conductance has been previously analyzed in fast-slow phase portraits (see [@Drion2012],[@Franci2012],[@Franci2013]) and in mathematical three-time scale models of bursting [@Franci2014]. The results of this qualitative analysis are summarized in Figure \[FIG:4\]. The reduced phase portraits are called regenerative when they include a slow negative conductance and restorative otherwise. Figure \[FIG:4\]A shows that the two types of phase portrait are qualitatively different. In restorative phase portraits, the $V$-nullcline has the classical $N$-shape of spiking neuronal models (red). In regenerative phase portraits, this shape is mirrored (green). The reader is referred to [@Drion2012],[@Franci2012],[@Franci2013] for a detailed analysis of why the mirrored $V$-nullcline requires a slow negative conductance. Both restorative and regenerative phase portraits exhibit bistability between a fixed point and a limit cycle and both phase portraits rely on the same bifurcations: they are of the same saddle-node/saddle homoclinic type according to the classification of [@Izhikevich2007 Figure 9.24]. However, the difference in their $V$-nullclines strongly affects the robustness and the tunability of the bistable attractor. In the regenerative phase portrait (green), the stable manifold of the saddle point (dark green) is a separatrix that sharply divides the phase portrait into two distinct regions, each of which corresponds to the basin of attraction of one of the two attractors: the stable fixed point and the stable limit cycle. The two stable attractors can be shaped independently from each other by deforming the nullclines away from the separatrix and modulating the ratio of the fast and slow time scales. Robust bistability is maintained across a broad range of variations (Figure \[FIG:4\]A, top right). In sharp contrast, the bistability of a restorative phase portrait (green) requires both a specific intersection of the nullclines and a specific ratio between the fast and slow time scales. The bistability is fragile to any perturbation of this specific tuning (Figure \[FIG:4\]A, bottom right). As a result, the bifurcation diagram associated to bursting only exists in a narrow parameter range. ![The phase portrait geometry of regenerative and restorative bursting. A. While sharing the same saddle node (SN) and saddle-homoclinic (SH) bifurcations, the phase portraits of rest-spike bistable models are radically affected by the slow negative conductance. Regenerative phase portraits (green) are robust to variations of the fast and slow time scales. Restorative phase portraits (red) are rest-spike bistable only for a well-chosen ratio of time scales. Stable fixed points are marked as full dots, unstable fixed points as empty dots, saddle points as crosses. Limit cycles and typical trajectories are sketched as blue oriented lines. The stable manifold of saddle points is sketched as a green oriented line. Bifurcation diagrams are the same as Figure \[FIG:5\].A bottom (regenerative case) and Figure \[FIG:5\].D left (restorative case). The mirrored N-shaped nullcline of the regenerative case ensures robustness with respect to variations in the time scale separation between the membrane potential and the recovery variable. The N-shaped nullcline of the restorative case makes the phase portrait fragile to the ratio of fast and slow time scales. Small deviations from the nominal ratio destroy the rest-spike bistability, either via a saddle-node homocinic bifurcation (cf. [@Izhikevich2007 Figure 6.44]) or via Hopf bifurcation around the up equilibrium. B. Regenerative phase portraits can be continuously deformed to match different types of bursters. Only the square-wave burster is compatible with restorative phase portraits.[]{data-label="FIG:4"}](Fig4){width="90.00000%"} Figure \[FIG:4\]B illustrates how the geometry of the bistable phase portrait impacts not only the robustness but also the tunability of the bursting attractor. The resting and spiking attractors can be shaped independently in a regenerative phase portrait because deforming the $V$-nullcline near the fixed point does not affect the limit cycle and vice versa. As a result, the values of the membrane potential at rest and in spiking mode can be tuned independently, leading to the generation of both non plateau bursting and plateau (or square-wave) bursting depending on the maximal conductance parameter set (Figure \[FIG:4\]B, left). This flexibility does not exist in restorative phase portraits. In particular, the resting state is always more hyperpolarized than the spiking state, forcing a bursting trace of the square-wave type (Figure \[FIG:4\]B, right). The qualitative analysis above is based on a low-dimensional mathematical model but the conclusions persist in higher-dimensional computational models. Figure \[FIG:5\] illustrates the various bifurcation diagrams associated to the numerical observations in the previous sections. Fig \[FIG:5\]A illustrates how the modulation from tonic spiking to bursting in Figure \[fig:1\]A affects the corresponding bifurcation diagrams. For the tonic spiking mode (point (a) in Figure \[fig:1\]A), the bifurcation diagram is the bifurcation diagram of a spiking model: the equilibrium curve is monotone, and there is no bistable range. For the bursting mode (point (b) in Figure \[fig:1\]A), the diagram instead exhibits the bistable range of a robust saddle-homoclinic burster consistent with the regenerative phase portraits of Figure \[FIG:4\]A. As illustrated in Figure \[FIG:5\]B, the bistable range of the burster is progressively lost when the slow conductance becomes fast. Here the bistability is lost not because of a deformation of the equilibrium curve (consistent with the fact that the [static]{} properties of the model are unchanged) but because the two bifurcations that determine the parameter range of bistability (saddle-node and saddle-homoclinic) progressively merge to a saddle-node homoclinic (SNH) bifurcation. Near the SNH bifurcation, the fragility of the bistable range is consistent with the fragility of restorative phase portraits of Figure \[FIG:4\]A. ![Bifurcation analysis of bursting with and without slow negative conductance. A. Membrane potential-ultraslow adaptation variable ($m_{K,Ca}$) bifurcation diagram and associated trajectories in tonic (top) and burst (bottom) modes in the STG model of Figure \[fig:1\]. Other ultra-slow variables of the model ($h_{Ca,T}$, $h_{Ca,S}$ and $h_A$) are fixed at physiologically plausible constant values (see Methods). B. Deformations of the bursting traces (top) and bifurcation diagram (center) of the STG model with parameter as in A bottom under reduction of the calcium activation time constants. The parameter range of bistability gradually shrinks as the time constant of calcium activation decreases to zero (bottom). C. Effects of varying the cell membrane capacitance in models with (left) and without (right) slow negative conductance. In the STG model, bursting persists for arbitrary small values of the capacitance. In the CA1 model, bursting only persists in a tiny window around the nominal value of the published model. D. The fragility of bursting in CA1 model is illustrated with respect to different parameters. The different perturbations cause similar alterations of the bifurcation diagram and of the corresponding rhythm: reduction or elimination of spiking in one direction, reduction or elimination of the slow bursting oscillation in the other direction. Bifurcation diagrams correspond to: nominal case (left), increased membrane capacitance (top right), decreased membrane capacitance (bottom right).[]{data-label="FIG:5"}](Fig5){width="90.00000%"} The impact of the bistable parameter range on the fragility of bursting is further illustrated in Figure \[FIG:5\]C, where we reexamine the robustness of bursting models to a perturbation of the membrane capacitance (see Figure \[FIG:2\]). This perturbation affects the time-scale separation between the fast and slow variables of reduced phase portraits. Bursting in the STG model is robust to this perturbation as far as the capacitance is low enough to allow for spike generation, consistently with the regenerative phase portraits of Figure \[FIG:4\]A. The time-scale separation between fast and slow variables can be increased at will without destroying the bistable bifurcation parameter range that is essential to bursting (Figure \[FIG:5\]C, left). In sharp contrast, bursting in the CA1 model is fragile to the same perturbation, consistently with the restorative phase portraits of Figure \[FIG:4\]A. The bistable parameter range quickly disappears, perturbing the SN-SH bifurcation diagram of a bursting attractor into one of two possible scenarios: either the SNIC bifurcation diagram of a slow spiker (smaller values of membrane capacitance) or the SN-SN bifurcation diagram of a slow rhythm that switches between a low and high resting states (larger values of membrane capacitance) (Figure \[FIG:5\]C, right). In the case where the capacitance is low enough to allow for spike generation, the robust firing pattern of the CA1 model is a slow spiking pattern, not a bursting one. Finally, Figure \[FIG:5\]D illustrates that this fragility is generic and not specific to a particular parameter. The robustness of the nominal CA1 bursting model was tested against five different parameter perturbations. In each case, the perturbation is well in the range of physiological variability and it produces the same alteration of the bifurcation diagram (the effect of a $5$ mV change in the potassium reversal potential is particularly striking). The bursting attractor is [fragile]{}, in the sense that small parameter variations disrupt the rhythm of the nominal model. It is also [rigid]{}, in the sense that different parameter variations always disrupt the rhythm in the same way (i.e. deform the bifurcation diagram in the same way), disrupting either the fast or the slow oscillation of the bursting rhythm. Discussion {#discussion .unnumbered} ========== A transient signature characterizes the transition from spiking to bursting {#a-transient-signature-characterizes-the-transition-from-spiking-to-bursting .unnumbered} --------------------------------------------------------------------------- The computational experiments in this paper suggest the critical role of the [slow]{} conductance in a robust and tunable bursting model. The classical voltage clamp experiment of electrophysiology near threshold provides a clear signature that the sign of this conductance is modulated during a continuous modulation from spiking to bursting. We now discuss several reasons why the role of the slow negative conductance is often overlooked both in experimental and modeling studies. The specific signature of a slow negative conductance has been reported experimentally, at least in two published papers: [@Rodriguez2013 Figures 3 and 5] use this signature to assess the modulatory effects of the peptide CabPK in regulating a cellular and circuit bursting rhythm and [@Swensen2005 Figure 11] uses the same signature to assess the cooperation of calcium currents and persistent sodium currents in Purkinje cell burst excitability. One important reason why this signature is not more prevalent in experimental studies is that most voltage-clamp experiments nowadays study the current response to a quasi-static voltage [ramp]{} rather than a voltage [step]{}. There are several reasons to prefer a ramp to a step in an experimental setup. For instance, a single ramp input experiment might be sufficient to extract the entire [static]{} I-V curve. Unfortunately, a ramp input will mask the transient signature reported in this paper. The voltage-clamp experiments in [@Rodriguez2013; @Swensen2005] are rare instances of [step]{} input experiments. More generally, it is the [transient]{} nature of the slow conductance that makes it difficult to assess from the [stationary]{} indicators traditionally associated to experimental investigations of bursting. Most importantly, such indicators include the monotonicity of the I-V curve [@Wilson1974], [@Chen2000], [@Butera2005], [@Lewis1984] or the analysis of slow oscillatory potentials (SOPs) [@Amini1999], [@Canavier2007], [@Zhang2013], [@Wang1992], [@Wang1993a], [@Skinner1994]. Figure \[FIG:6\]A contrasts the dynamical voltage clamp signature with the monotonicity properties of the static I-V curve. It provides four model conditions for which the electrophysiological distinction between tonic firing and bursting is unambiguously predicted by the slow transient voltage clamp signature but not by the $I-V$ curve. This is because the I-V curve only depends on the stationary value of a voltage-clamp experiment, but not on the [transient]{} current response. By definition, the slope of the I-V curve at a given voltage is the local conductance of the model at steady-state: for a step of small amplitude, this corresponds to the asymptotic value for large $t$ of the ratio $\Delta I_m(t) / \Delta V_m$. A negative conductance in the I-V curve is thus synonym of an inverse [steady-state]{} response in the voltage clamp experiment. The slow transient of the voltage clamp is a [transient]{} feature that cannot be inferred from the steady-state response of the voltage clamp. ![A. The transient signature of slow negative conductance cannot be recovered from the static I-V curve. A. A region of negative conductance in the I-V curve is neither necessary nor sufficient for bursting. In the STG model described in [@Liu1998] model (top) the I-V curve possesses a region of negative conductance both in tonic and burst modes. In the STG model of [@Goldman2001], the I-V curve is monotone both in tonic and burst modes. See Methods for details about the simulations. B. Slow oscillatory potentials in the absence of sodium do not necessarily discriminate between fast and slow negative conductances. The Plant R15 aplysia model described in [@Rinzel1987a] exhibits slow oscillatory potential under blockade of sodium channels both in tonic (top) and burst (bottom) modes. See Methods for details about the simulations.[]{data-label="FIG:6"}](Fig6){width="75.00000%"} Bursting and its modulation are also often studied experimentally through slow oscillatory potentials (SOPs) observed in the absence of spikes (by blocking sodium channels). But a slow oscillation is not a reliable signature of bursting per se either. Fig \[FIG:6\]B illustrates that a slow oscillation does not necessarily discriminate between tonic firing and bursting because either a fast or a slow current can be responsible for the regenerative upstroke of the slow oscillation. Fast or instantaneous activation of a calcium channel as in Figure \[fig:1\]B will generate neither the slow transient voltage clamp signature nor bursting. But it provides a steady-state inward current that can be sufficient to destabilize the resting potential and generate a slow oscillatory potential in the absence of sodium channels. Once again, the dynamic role of a given current cannot be inferred from its static properties. The negative slow conductance is also often overlooked in modeling and computational studies. Its role can only be captured in models that respect the time scale separation between fast and slow regenerative channels. This time scale separation is well acknowledged in the ion channel literature. For instance, activation and inactivation of calcium channels is often described as similar to activation and inactivation of sodium channels, but up to fifty times [slower]{} for some of them [@Kostyuk1977], [@Hille2001 p.127]. But it is often neglected in mathematical and computational modeling. For instance, Figure 5.6 in the textbook [@Izhikevich2007] refers to both sodium and calcium activation as [fast]{}. The section on calcium channels in the recent textbook [@Ermentrout2010a] also suggests that calcium and sodium channels have similar dynamics. In computational modeling, it is widespread practice to set both the calcium and sodium activation to steady-state when reducing the complexity of a model. See Table \[TAB1\] for a list of important papers that make that assumption. All those references suggest that the role of slow negative conductances in robustness and neuromodulation is underappreciated. [Reference]{} [Slow regenerative gating variable set to steady state]{} ------------------- --------------------------------------------------------------- [@Terman2002] Activation of T-type and high-threshold $Ca^{2+}$ channels [@Rubin2004] Activation of T-type $Ca^{2+}$ channels [@Butera1999] Activation of persistent $Na^+$ channels [@Butera1999a] Activation of persistent $Na^+$ channels [@Pospischil2008] Activation of T-type $Ca^{2+}$ channels [@Rush1994] Activation of T-type $Ca^{2+}$ channels [@Smith2000] Activation of T-type $Ca^{2+}$ channels [@Kubota2011] Activation of T-type and high-threshold $Ca^{2+}$ channels [@Golomb1997] Activation of persistent $Na^+$ channels [@Wang1994] Activation of T-type $Ca^{2+}$ channels [@Golomb2006] Activation of persistent $Na^+$ channels : List of [published models lacking a slow negative conductance because the activation of slow regenerative channels is set to steady-sate.]{}[]{data-label="TAB1"} Classification versus modulation of bursting models {#classification-versus-modulation-of-bursting-models .unnumbered} --------------------------------------------------- Mathematical models of bursting often omit the slow negative conductance because they only include the minimal number of currents that is necessary to bursting. The rationale is simple: a spiking model only requires two distinct currents to model the fast negative and the slow positive conductances. In such a model, the modulation from rest to spike is achieved by modulating the constant applied current. A third current is then enough to model the additional ultraslow positive conductance that converts the spiker into a burster. In this approach, bursting is seen as the result of ultraslow adaptation between resting and spiking. This minimal motif only requires three distinct ionic currents and is at the core of textbook expositions of bursting such as Chapter 9 in [@Keener2009], Chapter 9 in [@Izhikevich2007], and Chapter 5 in [@Ermentrout2010a]. It was originally proposed in the work of Chay and Keyzer [@Chay1983] on secretory (pancreatic) cells. None of those models include slow regenerative channels, meaning that none of those models includes a source of slow negative conductance. The fact that a minimal model of bursting does not require a slow negative conductance probably reinforces the common practice of considering instantaneous calcium activations in computational models. If minimal models of the literature show that a slow negative conductance is not necessary to bursting, our results suggest that bursting models that lack a slow negative conductance are necessarily fragile and rigid. Robustness and tunability are not addressed in mathematical textbooks, which focus on [classification]{} rather than [modulation]{}. Starting with the seminal work of Rinzel [@Rinzel1985; @Rinzel1987], the mathematical theory of bursting has relied on a classification based on the possible bifurcations that can govern the transition between rest and spike. The recent work of Izhikevich [@Izhikevich2007 page 376] provides up to sixteen different such mechanisms. Our results show that robustness and tunability are properties that are distinct from a mathematical classification based on bifurcations. For instance, Figure \[FIG:4\] illustrates that a saddle-node / saddle-homoclinic burster can be fragile or robust. And it also illustrates that a burster can be continuously modulated between different shapes without affecting the two bifurcations that determine its mathematical class. The feedback motif of robust and tunable bursting {#the-feedback-motif-of-robust-and-tunable-bursting .unnumbered} ------------------------------------------------- Our analysis of bursting modulation in terms of [conductances]{} in different time scales and different voltage ranges has a more general interpretation in terms of distinct [feedback]{} loops. When a current source has a positive conductance, it provides negative feedback to the circuit because it counteracts variations of the membrane potential. When it has a negative conductance, it provides positive feedback to the circuit because it amplifies variations of the membrane potential. With this terminology, the main message of this paper is that a minimal motif of bursting is a three feedback motif whereas a four feedback motif is required for robust modulation. The fundamental role of the slow negative conductance is interpreted as the fundamental role of a slow positive feedback. It is common to associate excitability to a two feedback motif : a fast positive feedback for the spike upstroke and a slow negative feedback for the refractory period. The minimal motif of bursting only adds a third ultraslow negative feedback for the ultraslow adaptation between rest and spike. Instead, our results highlight that bursting is modulated by a balance of negative and positive feedbacks in the slow time scale. The interpretation of the results in terms of feedback loops links our results to similar findings in other areas of systems biology, see e.g. [@Ferrell2008]. Positive feedback loops are the essence of switches and thresholds. Our emphasis on distinct sources of fast and slow positive feedbacks has therefore the interpretation of the necessity of two rather than one thresholds for the robustness and modulation of bursting. Each threshold accounts for two discrete states of the circuit. An excitable circuit relies on one threshold, which separates two discrete states: rest and spike. Our results suggest that a tunable bursting circuit relies on both a fast and a slow thresholds. The two thresholds determine four discrete states: rest, tonic spiking, slow spiking, and bursting. In the absence of a slow negative conductance, the circuit has only one source of positive feedback, leading to a single threshold that makes the distinction between spiking and bursting fragile and rigid. The distinction between one and two thresholds also has a clear interpretation on the phase portraits of Figure \[FIG:4\] . The thresholds correspond to points of ultrasensitivity where small perturbations of the initial condition leads to large differences in the resulting trajectory. Regenerative phase portraits have distinct [fast]{} and [slow]{} thresholds. In contrast, restorative phase portraits have only one threshold, that requires a specific ratio between the fast and slow time scales. We stress that the distinction between four distinct feedback loops of a bursting motif do not necessarily match the physiological distinction between four distinct ionic currents. For instance, sodium channels usually provide a source of fast positive feedback through their activation and a source of slow negative feedback through their inactivation. More generally, a same current can contribute to several of the four feedback loops [@Franci2013; @Drion2015]. But a particular modulation scenario will usually have a clear interpretation in terms of the four feedback loops. Central to this paper, the modulation from spiking to bursting will inevitably involve a balance between the slow positive feedback provided by slow regenerative channels and the slow negative feedback provided by slow restorative channels. The paper [@Drion2015] introduces the concept of [dynamic]{} input conductances to map the modulation of feedback loops to the modulation of conductance parameters in an arbitrary conductance-based model. What is central to the message of the present paper is that the slow positive feedback is key to a feedback motif that robustly accounts for modulation from spiking to bursting. The essential role of the slow negative conductance is consistent with singularity theory {#the-essential-role-of-the-slow-negative-conductance-is-consistent-with-singularity-theory .unnumbered} ----------------------------------------------------------------------------------------- The recent paper [@Franci2014] uses singularity theory [@Golubitsky1985] to propose a mathematical analysis of modulation in bursting models. It shows that conductance-based models that have tunable bursting capabilities have a normal form organised by a codimension three winged-cusp singularity. All the attractors that can be generated in the vicinity of this singularity can be described in a four dimensional parameter space : three unfolding parameters and the bifurcation parameter. Those abstract parameters aggregate all possible behaviors of the original model, regardless of the number of physiological parameters. They define the parameters that suffice to shape the attractors of the model. It is a remarkable mathematical prediction from singularity theory that the four shaping parameters match the gains of the four feedback loops, or equivalently, the four distinct conductances discussed in the present paper. In particular, the bifurcation parameter of the model [@Franci2014] precisely has the interpretation of the balance between positive and negative feedback in the slow time scale. It is this parameter that governs the transition from spiking to bursting in the normal form. Singularity theory therefore identifies this one parameter as the fundamental tuning parameter of a tunable burster. This mathematical prediction was verified in six different published models of the literature [@Franci2013]. A key message of the present paper is that this parameter cannot be tuned in a model that lacks currents with slow negative conductance. Modelling bursting without a slow negative conductance necessarily leads to models with less unfolding parameters than the codimension of its organizing center, a typical [ground for caution]{} in singularity theory [@Golubitsky1985 Section IV.1]. Amini, B., Clark, J., and Canavier, C. (1999). Calcium dynamics underlying pacemaker-like and burst firing oscillations in midbrain dopaminergic neurons: a computational study. , 82(5):2249–2261. Bezdudnaya, T., Cano, M., Bereshpolova, Y., Stoelzel, C. R., Alonso, J.-M., and Swadlow, H. A. (2006). Thalamic burst mode and inattention in the awake lgnd. , 49(3):421–432. Butera, R., Rubin, J., Terman, D., and Smith, J. (2005). Oscillatory bursting mechanisms in respiratory pacemaker neurons and networks. In [Bursting: The Genesis of Rhythm in the Nervous System]{}, pages 303–347. World Scientific Press, London. Butera, R. J., Rinzel, J., and Smith, J. C. (1999a). Models of respiratory rhythm generation in the pre-b[ö]{}tzinger complex. i. bursting pacemaker neurons. , 82(1):382–397. Butera, R. J., Rinzel, J., and Smith, J. C. (1999b). Models of respiratory rhythm generation in the pre-b[ö]{}tzinger complex. ii. populations of coupled pacemaker neurons. , 82(1):398–415. Canavier, C., Oprisan, S., Callaway, J., Ji, H., and Shepard, P. (2007). Computational model predicts a role for erg current in repolarizing plateau potentials in dopamine neurons: implications for modulation of neuronal activity. , 98(5):3006–3022. Chay, T. R. and Keizer, J. (1983). Minimal model for membrane oscillations in the pancreatic beta-cell. , 42(2):181. Chen, Y. and Tsai, M. (2000). Action potential bursts in central snail neurons elicited by d-amphetamine: roles of ionic currents. , 96(1):237 – 248. Dethier, J., Drion, G., Franci, A., and Sepulchre, R. (2015). A positive feedback at the cellular level promotes robustness and modulation at the circuit level. , 114:2472–2484. Drion, G., Franci, A., Dethier, J., and Sepulchre, R. (2015). Dynamic input conductances shape neuronal spiking. , 2(1). Drion, G., Franci, A., Seutin, V., and Sepulchre, R. (2012). A novel phase portrait for neuronal excitability. , 7(8):e41806. Ermentrout, G. B. and Terman, D. H. (2010). , volume 35. Springer Science & Business Media. Franci, A., Drion, G., and Sepulchre, R. (2012). An organizing center in a planar model of neuronal excitability. , 11(4):1698–1722. Franci, A., Drion, G., and Sepulchre, R. (2014). Modeling the modulation of neuronal bursting: a singularity theory approach. , 13(2):798–829. Franci, A., Drion, G., Seutin, V., and Sepulchre, R. (2013). A balance equation determines a switch in neuronal excitability. , 9(5):e1003040. Goldman, M. S., Golowasch, J., Marder, E., and Abbott, L. F. (2001). Global structure, robustness, and modulation of neuronal models. , 21(14):5229–5238. Golomb, D. and Amitai, Y. (1997). Propagating neuronal discharges in neocortical slices: computational and experimental study. , 78(3):1199–1211. Golomb, D., Yue, C., and Yaari, Y. (2006). Contribution of persistent na+ current and m-type k+ current to somatic bursting in ca1 pyramidal cells: combined experimental and modeling study. , 96(4):1912–1926. Golubitsky, M. and Schaeffer, D. G. (1985). Singularities and groups in bifurcation theory. , 51. Hille, B. (2001). . Sinauer Associates Inc.,U.S. Izhikevich, E. M. (2007). . MIT press. Keener, J. and Sneyd, J. (2009). . Springer Verlag. Kostyuk, P., Krishtal, O., and Shakhovalov, Y. (1977). Separation of sodium and calcium currents in the somatic membrane of mollusc neurones. with an appendix by yu a. shakhovalov. , 270(3):545. Krahe, R. and Gabbiani, F. (2004). Burst firing in sensory systems. , 5(1):13–23. Kubota, S. and Rubin, J. E. (2011). Nmda-induced burst firing in a model subthalamic nucleus neuron. , 106(2):527–537. Lewis, D., Evans, G., and Wilson, W. (1984). Dopamine reduces slow outward current and calcium influx in burst-firing neuron r15 of aplysia. , 4(12):3014–3020. Liu, Z., Golowasch, J., Marder, E., and Abbott, L. (1998). A model neuron with activity-dependent conductances regulated by multiple calcium sensors. , 18(7):2309–2320. Marder, E. (2012). Neuromodulation of neuronal circuits: back to the future. , 76(1):1–11. Marder, E. and Bucher, D. (2007). Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs. , 69:291–316. Marder, E., O’Leary, T., and Shruti, S. (2014). Neuromodulation of circuits with variable parameters: Single neurons and small circuits reveal principles of state-dependent and robust neuromodulation. , 37:329–347. McCormick, D. and Bal, T. (1997). Sleep and arousal: thalamocortical mechanisms. , 20(1):185–215. Pospischil, M., Toledo-Rodriguez, M., Monier, C., Piwkowska, Z., Bal, T., Fr[é]{}gnac, Y., Markram, H., and Destexhe, A. (2008). Minimal hodgkin–huxley type models for different classes of cortical and thalamic neurons. , 99(4):427–441. Prinz, A., Billimoria, C., and Marder, E. (2003). Alternative to hand-tuning conductance-based models: construction and analysis of databases of model neurons. , 90(6):3998–4015. Rinzel, J. (1985). Bursting oscillations in an excitable membrane model. In [Ordinary and partial differential equations*]{}, pages 304–316. Springer. Rinzel, J. (1987). A formal classification of bursting mechanisms in excitable systems. , 71:267–281. Rinzel, J. and Lee, Y. S. (1987). Dissection of a model for neuronal parabolic bursting. , 25(6):653–675. Rodriguez, J., Blitz, D., and Nusbaum, M. (2013). Convergent rhythm generation from divergent cellular mechanisms. , 33(46):18047–18064. Rubin, J. E. and Terman, D. (2004). High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. , 16(3):211–35. Rush, M. E. and Rinzel, J. (1994). Analysis of bursting in a thalamic neuron model. , 71(4):281–291. Sherman, S. M. (2001). Tonic and burst firing: dual modes of thalamocortical relay. , 24(2):122–126. Skinner, F., Kopell, N., and Marder, E. (1994). Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks. , 1(1-2):69–87. Smith, G. D., Cox, C. L., Sherman, S. M., and Rinzel, J. (2000). Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-fire-or-burst model. , 83(1):588–610. Storm, J. (1990). Potassium currents in hippocampal pyramidal cells. , 83:161–187. Swensen, A. and Bean, B. (2003). Ionic mechanisms of burst firing in dissociated purkinje neurons. , 23(29):9650–9663. Swensen, A. M. and Bean, B. P. (2005). Robustness of burst firing in dissociated purkinje neurons with acute or long-term reductions in sodium conductance. , 25(14):3509–3520. Terman, D., Rubin, J. E., Yew, A. C., and Wilson, C. J. (2002). Activity patterns in a model for the subthalamopallidal network of the basal ganglia. , 22(7):2963–76. Tsai, T. Y.-C., Choi, Y. S., Ma, W., Pomerening, J. R., Tang, C., and Ferrell, J. E. (2008). Robust, tunable biological oscillations from interlinked positive and negative feedback loops. , 321(5885):126–129. Wang, X.-J. (1994). Multiple dynamical modes of thalamic relay neurons: rhythmic bursting and intermittent phase-locking. , 59(1):21–31. Wang, X.-J. and Rinzel, J. (1992). Alternating and synchronous rhythms in reciprocally inhibitory model neurons. , 4(1):84–97. Wang, X.-J. and Rinzel, J. (1993). Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons. , 53(4):899–904. Wilson, W. A. and Wachtel, H. (1974). Negative resistance characteristic essential for the maintenance of slow oscillations in bursting neurons. , 186(4167):932–934. Zhang, C. and Lewis, T. (2013). Phase response properties of half-center oscillators. , 35(1):55–74.
--- author: - \| K. Kutak\ Deutsches Elektronen Synchrotron, D-22603 Hamburg, Germany\ E-mail: title: 'Studies of forward jets and production of W, Z bosons at LHC energies' --- Introduction ============ Experiments at the Large Hadron Collider (LHC) will allow to test standard model at very high energies. Here we are interested in Quantum Chromodynamics (QCD) processes like forward jet production and interplay of QCD with Electro-Weak interactions which lead to $W$, $Z$ bosons production in hadronic collision. The first of listed processes is of interest since it will allow for better understanding of partonic structure of the proton at extreme energies where new phenomena like saturation [@sat; @sat2; @dent] may occur. The other process is an important candle of Electro-Weak theory which has to be tested at various energy scales in order to have full understanding of it. This process will also serve as one to calibrate calorimeters and to measure luminosities.\ The large center of mass energy at the LHC will require a need to apply QCD resummation approaches capable to account for multiple scales in the problem. Namely, one has to account for logarithms of type $\alpha_s\ln(\mu/\Lambda_{QCD})$ where $\mu$ might be $p_T$ of jet or mass of $W/Z$ and logarithms coming from the fact that at least parton densities in one of the incoming proton will be probed at very small longitudinal momentum fraction $x$ giving rise to logarithms of type $\alpha_s^n\ln^m1/x$. The theoretical framework to resume consistently both kinds of logarithmic corrections in QCD calculations is based on high-energy factorization at fixed transverse momentum. [@hef]. This formulation depends on unintegrated distributions for parton splitting, obeying appropriate evolution equations, and short-distance, process-dependent matrix elements. The unintegrated-level evolution is given by evolution equations in rapidity, or angle, parameters. Different forms of the evolution, valid in different kinematic regions, are available, see [@jcc-lc08; @fhfeb07], and references therein, for recent work in this area and reviews. In this article we present recently obtained results for hard matrix elements needed in application of factorisation formulas for both of introduced above processes. In Sec. 2 we introduce the basic structure of jet production in the LHC forward region. In Sec. 2.1 we consider associated parton showering effects. In Sec. 2.3 we consider effects from the short-distance matrix elements that control the resummation of logarithmically enhanced corrections in $\sqrt{s} / E_T$, where $E_T$ is the hard jet transverse energy. In Sec. 3 we briefly discuss central production of $W$, $Z$ bosons focusing on observables which are sensitive to different showering methods and different assumptions on initial state of colliding partons. We give concluding remarks in Sec. 4. Forward jets ============ The hadroproduction of a forward jet associated with hard final state $X$ is pictured in Fig. \[fig:forwp\]. The kinematics of the process is characterized by the large ratio of sub-energies $s_2 / s_1 \gg 1 $ and highly asymmetric longitudinal momenta in the partonic initial state, $q_A \cdot p_B \gg q_B \cdot p_A$. At the LHC the use of forward calorimeters allows one to measure events where jet transverse momenta $p_\perp > 20$ GeV are produced several units of rapidity apart, $\Delta y \greatersim 4 \div 6$ [@cmsfwd; @aslano; @heralhc]. Working at polar angles that are small but sufficiently far from the beam axis not to be affected by beam remnants, one measures azimuthal plane correlations between high-$p_\perp$ events (Fig. \[fig:azimcorr\]) widely separated in rapidity [@heralhc; @preprint]. The presence of multiple large-momentum scales implies that, as recognized in [@muenav; @vddtang; @stirl94], reliable theoretical predictions for forward jets can only be obtained after summing logarithmic QCD corrections at high energy to all orders in $\alpha_s$[^1]. This motivates efforts [@webetal99; @orrsti; @stirvdd; @andsab] to construct new, improved algorithms for Monte Carlo event generators capable of describing jet production beyond the central rapidity region. In the LHC forward kinematics, realistic phenomenology of hadronic jet final states requires taking account of both logarithms of the large rapidity interval (of high-energy type) and logarithms of the hard transverse momentum (of collinear type). The theoretical framework to resume consistently both kinds of logarithmic corrections in QCD calculations is based on high-energy factorization at fixed transverse momentum [@hef]. Ref. [@preprint] investigates forward jets in this framework. It presents the short-distance matrix elements needed to evaluate the factorization formula, including all partonic channels, in a fully exclusive form. On one hand, once convoluted with the BFKL off-shell gluon Green’s function according to the method of [@hef], these matrix elements control the summation of high-energy logarithmic corrections to the jet cross sections. They contain contributions both to the next-to-leading-order BFKL kernel [@fadlip98] and to the jet impact factors [@mc98; @schw0703]. On the other hand, they can be used in a shower Monte Carlo generator implementing parton-branching kernels at unintegrated level (see e.g. [@jadach09; @hj_ang] for recent works) to generate fully exclusive events. The high-energy factorized form [@hef; @preprint; @mc98] of the forward-jet cross section is represented in Fig. \[fig:sec2\]a. Initial-state parton configurations contributing to forward production are asymmetric, with the parton in the top subgraph being probed near the mass shell and large $ x $, while the parton in the bottom subgraph is off-shell and small-$x$. The jet cross section differential in the final-state transverse momentum $Q_\perp$ and azimuthal angle $\varphi$ is given schematically by [@hef; @preprint; @mc98] $$\label{forwsigma} {{d \sigma } \over { d Q_\perp^2 d \varphi}} = \sum_a \int \ \phi_{a/A} \ \otimes \ {{d {\widehat \sigma} } \over { d Q_\perp^2 d \varphi }} \ \otimes \ \phi_{g^/B} \;\; ,$$ where $\otimes$ specifies a convolution in both longitudinal and transverse momenta, $ {\widehat \sigma} $ is the hard scattering cross section, calculable from a suitable off-shell continuation of perturbative matrix elements, $ \phi_{a/A} $ is the distribution of parton $a$ in hadron $A$ obtained from near-collinear shower evolution, and $ \phi_{g^/B} $ is the gluon unintegrated distribution in hadron $B$ obtained from non-collinear, transverse momentum dependent shower evolution. In the next section we comment on the initial-state shower evolution. In Sec. 2.2 we turn to hard-scattering contributions. Parton shower evolution ----------------------- Parton distributions can be obtained by parton-shower Monte Carlo methods via branching algorithms based on collinear evolution of the jets developing from the hard event [@mc_lectures]. The branching probability can be given in terms of two basic quantities (Fig. \[fig:pshower\]), the splitting functions at the vertices of the parton cascade and the form factors to go from one vertex to the other. An important ingredient of this approach is the inclusion of soft-gluon coherence effects [@mc_lectures; @dokrev; @mc89] through angular ordering of the emissions in the shower. Corrections to collinear-ordered showers, however, arise in high-energy processes with multiple hard scales [@heralhc; @mw92; @bo04], as is the case with the production of jets at forward rapidities in Fig. \[fig:forwp\]. In particular, new color-coherence effects set in in this regime due to emissions from internal lines in the branching decay chain [@heralhc; @mc98; @anderss96] that involve space-like partons carrying small longitudinal momentum fractions. The picture of the coherent branching is modified in this case because the emission currents become dependent on the total transverse momentum transmitted down the initial-state parton decay chain [@hef; @mc98; @mw92; @bo04; @jung04]. Correspondingly, one needs to work at the level of unintegrated splitting functions and partonic distributions [@jcc-lc08; @hj_rec] in order to take into account color coherence not only for large $x $ but also for small $x$ in the angular region (Fig. \[fig:coh\]) $$\label{cohregion} \alpha / x > \alpha_1 > \alpha \hspace{0.3 cm} ,$$ where the angles $\alpha$ for the partons radiated from the initial-state shower are taken with respect to the initial beam jet direction, and increase with increasing off-shellness. The case of LHC forward jet production is a multiple-scale problem where coherence effects of the kind above enter, in the factorization formula (\[forwsigma\]), both the short-distance factor $ {\widehat \sigma} $ and the long-distance factor $\phi$. Contributions from the coherence region (\[cohregion\]) are potentially enhanced by terms $ \alpha_s^n \ln^{m} \sqrt{s} / p_\perp $ where $\sqrt{s}$ is the total center-of-mass energy and $p_\perp$ is the jet transverse momentum[^2]. These contributions represent corrections to the angular ordering implemented in collinear showers and are not included at present in standard Monte Carlo generators [@mc_lectures]. Work to develop methods for unintegrated shower evolution, capable of including such corrections, is underway by several authors. The proposal [@jadach09] incorporates NLO corrections to flavor non-singlet QCD evolution in an unintegrated-level Monte Carlo. The approach is based on the generalized ladder expansion of [@CFP], which is extended to the high-energy region in [@ch94]. This approach could in principle be applied generally, including flavor singlet evolution, and used to treat also forward hard processes. The factorizing hard cross sections ----------------------------------- Logarithmic corrections for large rapidity $y \sim \ln s / p_\perp^2$ are resummed to all orders in $\alpha_s$ via Eq. (\[forwsigma\]), by convoluting (Fig. \[fig:sec2\]) unintegrated distribution functions with well-prescribed short-distance matrix elements, obtained from the high-energy limit of higher-order scattering amplitudes [@preprint; @mc98]. With reference to Fig. \[fig:sec2\]b, in the forward production region we have $(p_4+ p_6)^2 \gg (p_3 +p_4)^2 $ and longitudinal momentum ordering, so that $$\label{fwdkin} p_5 \simeq (1 - \xi_1 ) p_1 \;\;\; , \;\;\;\;\; p_6 \simeq (1 - \xi_2 ) p_2 - k_\perp \;\;\; , \;\;\;\;\; \xi_1 \gg \xi_2 \;\; .$$ Here $\xi_1$ and $\xi_2$ are longitudinal momentum fractions, and $ k_\perp $ is the di-jet transverse momentum in the laboratory frame. It is convenient to define the rapidity-weighted average $Q_\perp = (1-\nu) p_{\perp 4} - \nu p_{\perp 3}$, with $\nu = (p_2 \cdot p_4) / p_2 \cdot (p_1 -p_5) $. In Fig. \[fig:sec2\]b Eq. (\[forwsigma\]) factorizes the high-energy $q g$ amplitude in front of the (unintegrated) distribution from the splitting in the bottom subgraph. The factorization in terms of this parton splitting distribution is valid at large $y$ not only in the collinear region but also in the large-angle emission region [@hef]. As a result the rapidity resummation is carried out consistently with perturbative high-$Q_\perp$ corrections [@hef; @mc98] at any fixed order in $\alpha_s$. The explicit expressions for the relevant high-energy amplitudes are given in [@preprint]. Figs. \[fig:forwplot\] and \[fig:forwplot1\] illustrate features of the factorizing matrix elements, partially integrated over final states. We plot distributions differential in $Q_\perp$ and azimuthal angle $\varphi$ ($\cos \varphi = Q_\perp \cdot k_\perp / \| Q_\perp \| \| k_\perp \| $) for the case of the $q g$ channel. Fig. \[fig:forwplot\] shows the dependence on $k_\perp$, which measures the distribution of the third jet recoiling against the leading di-jet system. Fig. \[fig:forwplot1\] shows the energy dependence. The region $ k_\perp / Q_\perp \to 0$ in Fig. \[fig:forwplot\] corresponds to the leading-order process with two back-to-back jets. The resummation of the higher-order logarithmic corrections for large $y \sim \ln s / p_\perp^2$ is precisely determined [@hef; @mc98] by integrating the u-pdfs over the $ k_\perp$-distribution in Fig. \[fig:forwplot\]. So the results in Fig. \[fig:forwplot\] illustrate quantitatively the significance of contributions with $k_\perp \simeq Q_\perp$ in the large-$y$ region. The role of coherence from multi-gluon emission is to set the dynamical cut-off at values of $ k_\perp $ of order $ Q_\perp $. Non-negligible effects arise at high energy from the finite-$k_\perp $ tail. These effects are not included in collinear-branching generators (and only partially in fixed-order perturbative calculations), and become more and more important as the jets are observed at large rapidity separations. The dependence on the azimuthal angle in Figs. \[fig:forwplot\] and \[fig:forwplot1\] is also relevant, as forward jet measurements will rely on azimuthal plane correlations between jets far apart in rapidity (Fig.\[fig:azimcorr\]). Results for all other partonic channels are given in [@preprint]. After including parton showering [@prepar], quark- and gluon-initiated contributions are of comparable size in the LHC forward kinematics: realistic phenomenology requires including all channels. Note also that since the forward kinematics selects asymmetric parton momentum fractions, effects due to the $ x \to 1$ endpoint behavior [@fhfeb07] at the fully unintegrated level may become relevant as well. W, Z boson production ===================== Central production of $W/Z$ bosons at LHC energies will be dominated by gluonic component of partonic system. The longitudinal components of incoming gluons four momenta will be small and of the same order what will result in the central production. As in the forward jet case the framework to consistently account for this kinematical set up is provided by $k_\perp$ factorisation approach where $\sigma(g^g^\to q(W/Z)\overline{q})$ is an off shell continuation of hard matrix element for $\sigma(gg\to q(W/Z)\overline{q})$ [@deak; @zotov] which allows to include effects coming from finite transversal momenta of gluons. The interesting observable to calculate for this production process is the angular distance between highest $p_\perp$ jet and $p_\perp$ of $Z$ or $W$, which allows for estimation of uncertainties of theoretical predictions. On Fig. \[fig:ptz\] (left) we show comparison of calculation based on $k_\perp$ factorisation approach [@deak; @zotov] and collinear approach at LO and NLO. We see that the distributions are considerably different for small angles in case of $k_\perp$ factorisation approach and collinear LO. While NLO collinear is quite similar to the one obtained in $k_\perp$ factorisation approach. The reason for that comes from the momentum conservation in LO collinear approach which forbids events in region from $0$ to $\pi/2$. This not the case in $k_\perp$ factorisation approach and NLO collinear calculation where the additional transversal momentum flow allows momenta of $Z$, $b$ and $\overline b$ to be unbalanced. The $k_\perp$ factorisation formula while applied to this process takes form: $$\label{bosons} \sigma=\phi_{a^/A} \ \otimes \widehat \sigma(g^g^\to q(W/Z)\,\overline{q}) \phi_{b^/B} \;\; ,$$ The other important distribution is the differential cross section in the transversal momentum of $Z$ $b\overline b$ system \[fig:ptz\] (right). This distributions are rather different at small $p_{Zb\overline b\perp}$ region. The difference is due to the fact that in the $k_\perp$ factorisation approach subleading terms of all orders from point of view of collinear approach are taken into account. We also see that at higher values of $p_{Zb\overline b\perp}$ both approaches agree. [![Distributions of the distance in azimuthal angle of Z and highest p$_\perp$ quark or antiquark. Calculation with massive $b$-quarks (left). Comparison of cross sections in transverse momentum of the produced $Z$ gauge boson. Calculation with massless $b$-quarks (right)[]{data-label="fig:ptz"}](mi1.eps "fig:"){width="0.49\columnwidth"}]{} [![Distributions of the distance in azimuthal angle of Z and highest p$_\perp$ quark or antiquark. Calculation with massive $b$-quarks (left). Comparison of cross sections in transverse momentum of the produced $Z$ gauge boson. Calculation with massless $b$-quarks (right)[]{data-label="fig:ptz"}](mi2.eps "fig:"){width="0.49\columnwidth"}]{} \[fig:ptz\] Conclusions =========== Forward + central detectors at the LHC allow jet correlations to be measured across rapidity intervals of several units, $\Delta y \greatersim 4 \div 6$. Such multi-jet states can be relevant to new particle discovery processes as well as new aspects of standard model physics. Existing sets of forward-jet data in ep collisions, much more limited than the potential LHC yield, indicate that neither conventional parton-showering Monte Carlos nor next-to-leading-order QCD calculations are capable of describing forward jet phenomenology. Improved methods to evaluate QCD predictions are needed to treat the multi-scale region implied by the forward kinematics. In this article we have discussed ongoing progress, examining in particular factorization properties of multi-parton matrix elements in the forward region, and prospects to include parton-showering effects with gluon coherence not only in the collinear region but also in the large-angle emission region. We also have discussed predictions for $W$ and $Z$ central production based on different showering schemes. I thank the conference organizers and the conference staff for the nice atmosphere at the meeting. The results presented in Sec. 2 of this article have been obtained in collaboration with M. Deák, F. Hautmann and H. Jung while the results presented in Sec.3 were obtained by M. Deák and F. Schwennsen [99]{} L. V. Gribov, E. M. Levin and M. G. Ryskin, [Phys. Rep.]{} [100]{} (1983) 1; I. I. Balitsky, Nucl. Phys. [B463]{} (1996) 99; Y. V. Kovchegov, [Phys.Rev]{} [D60]{} (1999) 034008; J. Kwiecinski, K.Kutak [Eur.Phys.J.]{} C29:521,2003; J. Bartels, K. Kutak [Eur.Phys.J.]{} C53:533-548,2008; J. L. Albacete, N. Armesto, J. G. Milhano, C. A. Salgado, U. A. Wiedemann [Phys.Rev.]{} D71 (2005) 014003; C. Marquet, G. Soyez [Nucl.Phys]{} A760 (2005) 208-222; K. Kutak, [Phys.Lett.]{} B675:332-335,2009 E. Iancu, M.S. Kugeratski and D.N. Triantafyllopoulos, Nucl. Phys. A[808]{} (2008) 95; Y. Hatta, E. Iancu and A.H. Mueller, JHEP [0801]{} (2008) 026; E. Iancu, C. Marquet and G. Soyez, Nucl. Phys. A [780]{} (2006) 52; C. Marquet and R. B. Peschanski, Phys. Lett. B [587]{} (2004) 201. D. d’Enterria, Eur. Phys. J. A [31]{} (2007) 816; S. Cerci and D. d’Enterria, arXiv:0812.2665. S. Catani, M. Ciafaloni and F. Hautmann, Phys. Lett. B[307]{} (1993) 147; Nucl. Phys. B[366]{} (1991) 135; Phys. Lett. B[242]{} (1990) 97. J.C. Collins, hep-ph/0106126, in Proc. Workshop DIS01; arXiv:0808.2665 \[hep-ph\], in Proc. F. Hautmann, Phys. Lett. B[655]{} (2007) 26; arXiv:0708.1319; J.C. Collins and F. Hautmann, JHEP [0103]{} (2001) 016; Phys. Lett. B [472]{} (2000) 129. CMS Coll., CERN-LHCC-2006-001 (2006); CMS PAS FWD-08-001 (2008). X. Aslanoglou et al., CERN-CMS-NOTE-2008-022 (2008); Eur. Phys. J. C [52]{} (2008) 495. H. Jung et al., Proc. Workshop “[HERA and the LHC]{}”, arXiv:0903.3861 \[hep-ph\]. M. Deák, F. Hautmann, H. Jung and K. Kutak, JHEP09(2009) 121. A.H. Mueller and H. Navelet, Nucl. Phys. B[282]{} (1987) 727. V. Del Duca, M.E. Peskin and W.K. Tang, Phys. Lett. B[306]{} (1993) 151. W.J. Stirling, Nucl. Phys. B[423]{} (1994) 56. C. Ewerz, L.H. Orr, W.J. Stirling and B.R. Webber, J. Phys. G[26]{} (2000) 696; J. Forshaw, A. Sabio Vera and B.R. Webber, J. Phys. G[25]{} (1999) 1511. L.H. Orr and W.J. Stirling, Phys. Lett. B[436]{} (1998) 372. J.R. Andersen, V. Del Duca, S. Frixione, F. Maltoni, C.R. Schmidt and W.J. Stirling, hep-ph/0109019; J.R. Andersen, V. Del Duca, S. Frixione, C.R. Schmidt and W.J. Stirling, JHEP [0102]{} (2001) 007. J.R. Andersen, arXiv:0906.1965 \[hep-ph\], J.R. Andersen and A. Sabio Vera, Phys. Lett. B[567]{} (2003) 116. V.S. Fadin and L.N. Lipatov, Phys. Lett. B[429]{} (1998) 127; G. Camici and M. Ciafaloni, Phys. Lett. B[430]{} (1998) 349. M. Ciafaloni, Phys. Lett. B[429]{} (1998) 363. F. Schwennsen, hep-ph/0703198; J. Bartels, A. Sabio Vera and F. Schwennsen, arXiv:0709.3249, JHEP [0611]{} (2006) 051; J. Bartels, D. Colferai and G.P. Vacca, Eur. Phys. J. C[24]{} (2002) 83. S. Jadach and M. Skrzypek, arXiv:0905.1399 \[hep-ph\]. F. Hautmann and H. Jung, JHEP [0810]{} (2008) 113; arXiv:0804.1746 \[hep-ph\]. A.H. Mueller, Nucl. Phys. B Proc. Suppl. [18C]{} (1990) 125. W.K. Tang, Phys. Lett. B[278]{} (1992) 363; J. Bartels, A. De Roeck and M. Loewe, Z. Phys. [C54]{} (1992) 635; J. Kwiecinski, A.D. Martin and P.J. Sutton, Phys. Rev. D[46]{} (1992) 921; S. Catani, M. Ciafaloni and F. Hautmann, Nucl. Phys. B Proc. Suppl. [29A]{} (1992) 182. A. Aktas et al., Eur. Phys. J. C[46]{} (2006)27; S. Chekanov et al., Phys. Lett. B[632]{} (2006)13. B.R. Webber, hep-ph/9510283, in Proc. Workshop DIS95. B.R. Webber, [CERN Academic Training Lectures]{} (2008). Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller and S.I. Troian, [Perturbative QCD]{}, Ed. Frontieres, Gif-sur-Yvette (1991). M. Ciafaloni, in [Perturbative Quantum Chromodynamics]{}, ed. A.H. Mueller (World Scientific, Singapore, 1989) G. Marchesini and B.R. Webber, Nucl. Phys. [B386]{} (1992) 215. B. Andersson et al., Eur. Phys. J. C [25]{} (2002) 77. B. Andersson, G. Gustafson and J. Samuelsson, Nucl. Phys. B[467]{} (1996) 443. H. Jung, Mod. Phys. Lett. A [19]{} (2004) 1. F. Hautmann and H. Jung, Nucl. Phys. Proc. Suppl. [184]{} (2008) 64 \[arXiv:0712.0568 \[hep-ph\]\]; arXiv:0808.0873 \[hep-ph\]; F. Hautmann, Acta Phys. Polon. B [40]{} (2009) 2139. G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B[175]{} (1980) 27. S. Catani and F. Hautmann, Nucl. Phys. B[427]{} (1994) 475; Phys. Lett. B[315]{} (1993) 157. M. Ciafaloni and D. Colferai, JHEP [0509]{} (2005) 069; M. Ciafaloni, D. Colferai, G.P. Salam and A.M. Stasto, Phys. Lett. B[635]{} (2006) 320; G. Altarelli, R.D. Ball and S. Forte, arXiv:0802.0968 \[hep-ph\]. M. Deák, F. Hautmann, H. Jung and K. Kutak, in preparation. M. Deák, F. Schwennsen, JHEP [0809*]{} (2008) 035; S.P. Baranov, A.V. Lipatov, N.P. Zotov Phys. Rev. D78 (2008) 014025. [^1]: Analogous observation applies to forward jets associated to deeply inelastic scattering [@mueproc90c; @forwdis92]. Indeed, measurements of forward jet cross sections at Hera [@heraforw] have illustrated that either fixed-order next-to-leading calculations or standard shower Monte Carlos [@heraforw; @web95; @webetal99], e.g. or , are not able to describe forward jet $ep$ data. [^2]: Terms with $m > n$ are known to drop out from inclusive processes due to strong cancellations associated with coherence, so that, for instance, the anomalous dimensions $\gamma^{ i j}$ for space-like evolution receive at most single-logarithmic corrections at high energy [@fadlip98; @ch94]. This need not be the case for exclusive jet distributions, where such cancellations are not present and one may expect larger enhancements.
High energy electron scattering from nuclei can provide important information on the wave function of nucleons in the nucleus. In particular, with simple assumptions about the reaction mechanism, scaling functions can be deduced that, if shown to scale (i.e. are independent of length scale or momentum transfer), can provide information about the momentum and energy distribution of nucleons in a nucleus. Several theoretical studies [@annrev; @frank; @cdasim1; @ben1] have indicated that such measurements may provide direct access to short-range nucleon-nucleon correlations. The concept of y-scaling in electron-nucleus scattering was first introduced by West [@west] and Kawazoe et al. [@kawa]. They showed that in the impulse approximation, if quasielastic scattering from a nucleon in the nucleus was the dominant reaction mechanism, a scaling function $F(y)$ could be extracted from the measured cross section which was related to the momentum distribution of the nucleons in the nucleus. In the simplest approximation the corresponding scaling variable $y$ is the minimum momentum of the struck nucleon along the direction of the virtual photon. In general the scaling function depends on both $y$ and momentum transfer - $F(y,Q^2)$ - but at sufficiently high $Q^2$ ($-Q^2$ is the square of the four-momentum transfer) the dependence on $Q^2$ should vanish yielding scaling. However the simple impulse approximation picture breaks down when the final-state interactions (FSI) of the struck nucleon with the rest of the nucleus are included. Previous calculations [@fsi1; @fsi2; @ben2; @fsi3; @fsi4; @fsi5; @fsi6; @fsi8] suggest that the contributions from final state interactions should vanish at sufficiently high $Q^2$. A previous SLAC measurement [@ne3] suggested an approach to the scaling limit for heavy nuclei but only for low values of $\|y\| < 0.3$ GeV/c at momentum transfers up to 3 (GeV/c)$^2$. The data presented here represent a significant increase in the Q$^2$ range compared to previous measurements while also extending the coverage in $y$. The present data were obtained in Hall C at the Thomas Jefferson National Accelerator Facility (TJNAF), using 4.045 GeV electron beams with intensities from 10 - 80 $\mu$A. The absolute beam energy was calibrated to 0.08% using 0.8 GeV elastic scattering from carbon and BeO targets and 4.0 GeV elastic scattering from hydrogen. The beam current was monitored with three calibrated resonant cavities. The beam energy resolution was better than 0.05% as defined by the accelerator acceptance. Solid targets of C (2.1 and 5.9% of a radiation length), Fe (1.5 and 5.8% of a radiation length) and Au (5.8% of a radiation length) with natural isotopic abundance were used. Data were also taken with liquid targets of hydrogen and deuterium (nominally 4 and 15 cm in length). Scattering from hydrogen allows a cross check of the absolute normalization of the cross section; results from the deuterium target will be presented elsewhere. Less than 1% density variations were observed for the liquid targets due to beam heating for incident beam currents up to 55 $\mu$A (maximum current used for the liquid targets) when the 200 $\mu$m $\times$ 200 $\mu$m beam was rastered by a pair of electro-magnets to the typical spot-size of $\pm$ 1.2 mm. The scattered electrons were detected with the High Momentum Spectrometer (HMS) at angles of $15^\circ, 23^\circ, 30^\circ, 37^\circ, 45^\circ$ and $55^\circ$ and the Short Orbit Spectrometer (SOS) at an angle of $74^\circ$. Both spectrometers took data simultaneously with nearly identical detector systems configured for electron detection. Each detector system included two planes of plastic scintillator for triggering, two six-element drift chambers for tracking information as well as a gas Čerenkov detector and Pb glass calorimeter for particle identification. The measured tracks were required to reconstruct to the target location. For the HMS, additional cuts were applied to eliminate events produced on the pole pieces of the spectrometer magnets. Cuts were also applied to select electrons and reject $\pi^-$ using the signals from the Cerenkov detector and Calorimeter. The combined efficiency of all the cuts was $> 98\%$. The binned events were corrected for spectrometer acceptance using an acceptance function generated by a Monte Carlo calculation [@jra] that included all apertures within the spectrometer. This calculation accurately reproduced the distributions and cross section from hydrogen elastic scattering. Estimated systematic uncertainties due to the acceptance are $< 2.5\%$. Tracking efficiencies were typically 94% - 97%. Background from mis-identified $\pi^-$ was negligible for the HMS and $< 3\%$ in the worst case for the SOS. High energy photons produced principally from $\pi^0$ decay can result in secondary electrons following pair production by the photons in the target material. This background, estimated by measuring positron yields with the spectrometer magnetic fields reversed, was negligible for spectrometer angles $< 55^\circ$, but was 3 - 10% at $55^\circ$ and 20 - 100% at $74^\circ$. The larger values for the contribution of this background are for the 6% radiation length targets and result in an estimated systematic error of 5 - 10%. However, because the large backgrounds are only present in kinematic regions where the cross section is very small, the statistical uncertainties dominate the total uncertainty. Because of the large acceptance of the spectrometers ($>6$ msr) and the rapid variation of the cross section with $\theta$, there can be a significant variation of the cross section over the acceptance. In order to extract cross sections vs. energy transfer $\nu$ at fixed scattering angle a bin centering correction must be applied. This is accomplished with a model of the cross section [@jra] that is constrained to reproduce the angle and energy transfer dependence of the measurements. The cross section model was also used to apply radiative corrections using the iterative technique of Refs. [@stein] and [@dhp]. Variations in the form of the model were used to estimate systematic uncertainties in these corrections. The total estimated systematic uncertainties in the bin-centering and radiative corrections were 1-2% and 2.5% respectively. Lastly a Coulomb correction was applied for the change in the incident and scattered energy due to the Coulomb acceleration from the nuclear charge. This correction was significant ($\sim 10\%$ for Fe and $\sim 20\%$ for Au) for the largest scattering angles of the present experiment. Fig. \[sigma\] shows the measured cross sections vs. energy loss $\nu$ for Fe, where for each angle the $Q^2$ value at Bjorken $x = Q^2/2M\nu = 1$ is given (this value corresponds to elastic scattering from a free nucleon). Because of the significant smearing due to the Fermi motion and the large contribution from other inelastic processes (e.g. $\pi$ production, resonance production and deep inelastic scattering) at these relatively high $Q^2$, there is little evidence of a quasielastic peak. In fact the sharp bend in the spectrum at $\theta = 15^\circ$ is the only distinctive feature resulting from quasielastic scattering. At larger angles the additional inelastic processes cause even this feature to disappear. It should be noted however that quasielastic scattering is still expected to contribute significantly to the cross section for $\nu < Q^2/2M$ ($x > 1$). The minimum measured cross sections were limited by count rate and represent a factor of $ > 100$ improvement in sensitivity compared to the previous experiment [@ne3]. This improvement is largely due to the higher beam currents and larger acceptance spectrometers available at TJNAF. 8.5 cm 6 cm The scaling function is defined as the ratio of the measured cross section to the off-shell electron-nucleon cross section multiplied by a kinematic factor: $$F(y) = {d^2\sigma \over d\Omega d\nu}[Z \sigma_p + N \sigma_n]^{-1}{q \over (M^2 + (y + q)^2)^\frac{1}{2}}$$ Where Z and N are the number of protons and neutrons in the target nucleus, the off-shell cross sections $\sigma_p$ and $\sigma_n$ are taken from $\sigma_{CC1}$ from Ref. [@deforest] using the elastic form factors from Ref. [@gari], $q$ is the three-momentum transfer and M is the mass of the proton. The $y$ variable is defined through the equation [@pacesalme]: $$\nu + M_A = (M^2 + q^2 + y^2 + 2yq)^{\frac{1}{2}} + (M_{A-1}^2 + y^2)^\frac{1}{2}$$ where $M_A$ is the mass of the target nucleus and $M_{A-1}$ is the ground state mass of the $A-1$ nucleus. The scaling function for Fe is shown in Fig. \[yfe\] for all measured angles. While the cross section as a function of $Q^2$ and $\nu$ varies over many orders of magnitude (see Fig. 1), the scaling function for values of $y < -0.1$ GeV/c shows a clear approach to a universal curve where the data can be represented by a function that depends only on $y$. The breakdown of scaling for values of $y$ near zero is due to the dominance of other inelastic processes beyond quasielastic scattering. 8.5 cm 6 cm The approach to scaling is also shown in Figs. \[yscale1\] and \[yscale2\], where the $Q^2$ dependence of $F(y)$ at several fixed values of $y$ is presented. For $y = -0.2$ to $-0.5$ GeV/c there is a clear approach to scaling as $Q^2$ is increased. This is the first evidence for $y$-scaling in heavy nuclei for $y < -0.3$ GeV/c. There are, in addition, significant scaling violations observed at both low and high $Q^2$. The increase in $F(y)$ with $Q^2$ for $y = 0, -0.1$ GeV/c (Fig. \[yscale1\]) is clearly due to the inelastic processes mentioned above. A similar effect was observed [@ne18] previously, but only for $y \sim 0$. Calculations that include both quasielastic and other inelastic processes [@ben2; @fsi8] indicate that at $y = 0$ these other process dominate the reaction for $Q^2 > 2$ (GeV/c)$^2$. 8.5 cm 8.5 cm At large negative $y$ (Fig. \[yscale2\]) there is a decrease in $F(y)$ with increasing $Q^2$ as the scaling is approached. This behavior contradicts the approach to scaling expected within the impulse approximation (where the scaling limit is approached from below because of incomplete kinematic coverage at low $Q^2$), and suggests the influence of final state interactions. A recent calculation [@simpriv] indicates that the component of the FSI resulting from the scattered nucleon interacting with the mean-field of the nucleus should be a strongly decreasing function of $Q^2$ and become negligible for $Q^2 > 3$ (GeV/c)$^2$. An additional component in the calculation, due to interaction with a correlated nucleon, has a much weaker $Q^2$ dependence and may persist to the $Q^2$ range of the present experiment. The present data suggest a scaling that is consistent with an approach to the impulse approximation scaling limit, but cannot exclude contributions from FSI that are $Q^2$-independent. Comparison of the scaling functions for C, Fe and Au show very similar distributions. This can be seen in Fig. \[adep\], where all targets are plotted vs. $Q^2$ for a fixed value of $y = - 0.3$ GeV/c. The small A-dependence seen in these data is suggestive of a universal response for all medium-mass nuclei as might be expected in a kinematic region dominated by short-range correlations. 7.5 cm In summary, we have measured the inclusive cross section at $x > 1$ for electrons scattering from C, Fe and Au targets to $Q^2 \simeq 7$ (GeV/c)$^2$, a significant increase compared to the previous experiment. When analyzed in terms of the $y$-scaling function the data show an approach to scaling for $Q^2 > 3$ (GeV/c)$^2$. At these values of $Q^2$ a scaling limit can be expected within a simple impulse approximation. In addition a scaling behavior is observed for the first time at very large negative $y$ ($y = -0.5$ GeV/c). This is a regime where the nucleon momentum distribution is expected to be dominated by short-range nucleon-nucleon correlations. It is interesting to note that contributions from short-range final-state interactions may also result in a scaling-like behavior due to the small $Q^2$-dependence of these effects, and that these contributions are also dominated by short-range nucleon-nucleon correlations. We gratefully acknowledge the staff and management of TJNAF for their efforts in delivering the electron beam. We also acknowledge helpful discussions with O. Benhar and C. Ciofi degli Atti. This research was supported by the National Science Foundation, the Department of Energy and the Swiss National Science Foundation. $^$ Present Address: Physics Division, Argonne National Laboratory, Argonne, IL 60439. $^\#$ Present Address: Thomas Jefferson National Accelerator Facility, Newport News, VA 23606. $^+$ Present Address: College of William and Mary, Williamsburg, VA 23187. $^\&$ Present Address: Mississippi State University, Mississippi State, MS 39762. D. Day, et al. Ann. Rev. Nucl. Part. Sci. [40]{}, 357 (1990). L. L. Frankfurt et al., Phys. Rev. [C48]{}, 2451 (1993). C. C. degli Atti and S. Simula, Phys. Lett. [B325]{}, 276 (1994). O. Benhar et al., Phys. Lett. [B343]{}, 47 (1995). G. B. West, Phys. Rep. [18]{}, 263 (1975). Y. Kawazoe, G. Takeda and H. Matsuzaki, Prog. Theo. Phys. [54]{}, 1394 (1975). B. L. Ioffe, V. Khoze, and L. N. Lipatov, [Hard Processes]{} (Horth-Holland, Amsterdam, 1984). E. Pace, G. Salmé, and G. B. West, Phys. Lett. [B273]{}, 205 (1991). O. Benhar, et al., Phys. Rev. [C44]{}, 2328 (1991). O. Greenberg, Phys. Rev. [D47]{}, 331 (1993). B. L. Ioffe, ZhEFT Lett. [58]{}, 930 (1993). E. Pace, G. Salmé, and A. S. Rinat, Nucl. Phys. [A572]{}, 1 (1993). S. A. Gurvitz and A. S. Rinat, Phys. Rev. [C47]{}, 2901 (1993). A. S. Rinat and M. F. Taragin, Nucl. Phys. [A620]{}, 417 (1997). Erratum: Nucl. Phys. [A624]{}, 773 (1997). D. B. Day et al., Phys. Rev. Lett. [59]{}, 427 (1987). J. Arrington Ph.D. thesis, California Institute of Technology: www.krl.caltech.edu/$\sim$johna/thesis, (1998). S. Stein et al., Phys. Rev. [D12]{}, 1884 (1975). D. H. Potterveld, Ph.D. thesis, California Institute of Technology, 1989. T. DeForest, Nucl. Phys. [A392]{}, 232 (1983). M. Gari and W. Krümpelmann, Phys. Lett. [B141]{}, 295 (1984). E. Pace and G. Salmé, Phys. Lett. [B110]{}, 411 (1982). J. Arrington et al., Phys. Rev. [C53*]{}, 2248 (1996). S. Simula, private communication (1998).
--- abstract: 'We apply the measurement reduction technique to optimally reconstruct an object image from multiplexed ghost images (GI) while taking into account both GI correlations and object image sparsity. We show that one can reconstruct an image in that way even if the object is illuminated by a small photon number. We consider frequency GI multiplexing using coupled parametric processes. We revealed that the imaging condition depends on the type of parametric process, namely, whether down- or up-conversion is used. Influence of information about sparsity in discrete cosine transform and Haar transform bases on reconstruction quality is studied. In addition, we compared ordinary and ghost images when the detectors are additionally illuminated by noise photons in a computer experiment, which showed increased noise immunity of GI, especially with processing via the proposed technique.' author: - 'D.A.Balakin[^1]' - 'A.V.Belinsky' - 'A.S.Chirkin' bibliography: - 'gi2018bibliography.bib' title: Object reconstruction from multiplexed quantum ghost images using reduction technique --- Introduction {#introduction .unnumbered} ============ By now, to enhance human visual capability a vast high-tech base including highly sensitive, high-precision and high-speed cameras have been developed. Nevertheless, there still are objects whose direct optical observation is difficult. They are primarily halftone biological objects that are especially sensitive to illumination and thus, have to be investigated very delicately. Ghost imaging (GI) are one way of solving this problem, as it allows to obtain object images without direct observation of its spatial structure. For GI, correlated light beams are necessary. GI enables extraction of object information from spatial correlations between beams, one of which (in the object arm) interacts with the object, while the other one (in the reference arm) does not. In the object arm, a bucket detector is used, which provides only information about the total intensity of the transmitted radiation. The other beam does not interact with the object, but is detected by a CCD matrix, which permits measuring the spatial correlation function of intensity between two arms. The information about transparency or reflectivity distribution of the research object is extracted from photocount correlations in the object and reference arms [@belinsky_klyshko_1994], see also [@gatti_et_al_2004; @gatti_et_al_2007; @chan_et_al_2009; @erkman_shapiro_2010; @shapiro_boyd_2012]. In this paper, we study application of multicomponent entangled quantum light states that let us produce several GI simultaneously (to multiplex GI) [@chirkin_jetp_letters_15; @mgi_icono_lat_2016; @mgi_correlations_jrlr_2017; @ghost_images_jetp] by using radiation with different frequencies in reference arms. Mutual correlations of multiplexed images are used as additional information to improve image processing in the presence of fluctuations. There are various ways of producing multi-frequency entangled light beams. The required states can be obtained, e.g., in consecutive coupled parametric interactions in nonlinear crystals located either outside [@rodionov_chirkin_2004; @ferraro_et_al_2004] or inside [@olsen_drummond_2005] an optical resonator, in nonlinear waveguide structures [@solntsev_et_al_2012; @kruse_et_al_2013] where modes are coupled through evanescent modes, in a spatially modulated pump beam [@daems_et_al_2010]. The considered GI multiplexing employs four-frequency entangled quantum states formed through parametric decay of pump photons into two photons with different frequencies that are mixed in the same crystal with pump photons, which produces photons with sum frequencies [@chirkin_shutov_2007; @chirkin_shutov_2009]. Quantum theory of this process has been systematically developed in recent years [@saygin_chirkin_2010; @saygin_chirkin_2011; @saygin_et_al_2012; @tlyachev_et_al_2013]. Note that in [@duan_et_al_2013; @zhang_et_al_2015; @chan_et_al_2009] GI were multiplexed via multi-frequency noncoherent radiation sources to simultaneously produce several GI that are superimposed afterwards. Recently, polarization multiplexing has been used in several works on GI (see [@polar_mult_gi] and references there), in particular, to improve the reconstructed image quality. The ghost image processing methods considered in the literature usually rely on regularization. The regularizing functional is a characteristic of image sparsity in a given basis [@imaging_small_n_photons; @compressive_ghost_imaging; @gong_han_2012; @hi-res_gi_sparsity], and the minimized functional itself is the least squares one [@compressive_ghost_imaging; @gong_han_2012; @hi-res_gi_sparsity] or likelihood function [@imaging_small_n_photons]. Alternatively, a sparsity characteristic (e.g. the $L^1$ norm in a given basis) is minimized [@katz_et_al] under the constraint that measuring the image reconstructed in that way would give the results actually obtained. Since such functional is not connected the error of the interpretation result, the obtained estimate is, generally speaking, not the optimal one. Unlike [@imaging_small_n_photons], measurement reduction method, including its proposed version, does not require only Poisson photocount distribution, and unlike [@imaging_small_n_photons; @compressive_ghost_imaging; @katz_et_al; @gong_han_2012; @hi-res_gi_sparsity], image sparsity in any basis is not required. Note the main differences between this article and publications [@mgi_icono_lat_2016; @mgi_correlations_jrlr_2017; @ghost_images_jetp], in which multiplexed GI processing using measurement reduction technique was employed as well. Firstly, in these works the situation was considered when the only information about transparency distribution available to the researcher was that its values belong to a unit interval. In this article, it is considered that the researcher also has information about transparency distribution sparsity in a given basis and wants to take advantage of it to improve estimation quality. Secondly, as this information enables reconstruction of acceptable quality even with a small number of photons illuminating the object, multiplexed ghost imaging with a small number of photons ($\sim 1 \div 10$ photons per pixel) and processing of acquired images is modeled (see Sec. \[sec:computer-modelling\]). Thirdly, the presented version of measurement reduction technique differs from the one used in [@mgi_icono_lat_2016; @mgi_correlations_jrlr_2017; @ghost_images_jetp] in that projection (to take into account the information about the object) minimizes Mahalanobis distance instead of Euclidean distance, see Eqn. below. Finally, fourthly, in the studied multiplexed ghost imaging setup the object arm is lensless. This leads to imaging conditions depending on the type of parametric coupling of photon frequencies in the object arm and the reference arms. The article structure is as follows. In section \[sec:imaging\], we discuss GI multiplexing setup with lensless object arm and with lenses in reference arms. In section \[sec:image-processing\] the measurement reduction method is outlined. The information about the object that is available to the researcher and that is employed in reduction is summarized in subsection \[sec:image-information\]. In subsection \[sec:reduction-algorithm\], the algorithm of GI processing using reduction method that takes this information into account is described. Computer modeling results are given in section \[sec:computer-modelling\]. Main results of the article are summarized in the conclusion. Frequency multiplexing of quantum ghost images {#sec:imaging} ============================================== ![Multiplexed ghost imaging setup. NC is the nonlinear convertor; BS is the beam splitter; $\omega_{p}$ is pump frequency; $\omega_{1}, \ldots, \omega_{4}$ are frequencies of produced entangled photons; $O$ is the object; BD is the bucket detector in the object arm; $L_j$ are lenses with focal lengths $f_j$; CCD$_j$ are CCD in reference arms; C$_j$ are intensity correlators, $j = 2, 3, 4$[]{data-label="fig:ghost-imaging"}](ghost_imaging){width="\linewidth"} GI multiplexing setup is shown in fig. \[fig:ghost-imaging\]. The illumination is provided by coupled parametric processes that produce four-frequency entangled light fields. Pump radiation incident into the nonlinear convertor (nonlinear photon crystal) has frequency $\omega_{p}$. In the crystal, pump photons decay into two photons with related frequencies $\omega_1$ and $\omega_2$: $\omega_p = \omega_1 + \omega_2$. Four-frequency fields appear as a result of further conversion of a part of photons with frequencies $\omega_{1}$ and $\omega_{2}$ to photons with frequencies $\omega_{3}$ and $\omega_{4}$ in frequency-mixing processes: $$\label{eqn:mixing} \begin{aligned} \omega_{p} + \omega_{1} &= \omega_{3},\\ \omega_{p} + \omega_{2} &= \omega_{4}. \end{aligned}$$ Efficient energy exchange between interacting light waves in these processes can be achieved in aperiodically nonlinear photon crystals, e.g. in $\mathrm{Li Nb O}_3$, in the in quasi-phase matched regime, in which phase matching $\Delta k_j$ between interacting waves are compensated by the vectors of the inverse nonlinear lattice [@chirkin_shutov_2007; @chirkin_shutov_2009]. Note that the considered process was recently realized via a setup with two nonlinear photon crystals in the work [@suchowski_et_al_2016], where the spectrum of a photon pair at frequency above pump frequency was studied. Let ghost images be obtained by means of the optical system setup shown in fig. \[fig:ghost-imaging\], where a detector integrating radiation over the entire aperture is used in the object arm. It is assumed that the length of nonlinear photon crystal is chosen so that the transversal wave number amplification band of the nonlinear convertor substantially exceeds the width of wave spectrum of the object image. Details of four-frequency entangled state formation were considered in [@chirkin_jetp_letters_15; @mgi_icono_lat_2016; @mgi_correlations_jrlr_2017; @ghost_images_jetp]. In the setup of fig. \[fig:ghost-imaging\] the object is illuminated by radiation with frequency $\omega_1$, which is detected by the bucket detector (BD) over the entire beam aperture, and therefore lacks spatial resolution. Radiation with other frequencies $\omega_{2}, \omega_{3}, \omega_{4}$ after their spatial separation enters reference arms with lenses in them. Focal lengths $f_j$ of lenses and their positions between the beam splitter (BS) and CCD cameras are chosen according to imaging conditions. These conditions depend on the type of relation to the frequency of object illumination. For frequencies $\omega_{2}$, $\omega_{4}$ the imaging condition has the form (cf. the analogous condition in [@belinsky_klyshko_1994] for equal frequencies) $$\label{eqn:image24} \frac{1}{f_j} = \frac{1}{l_{j2}} + \frac{1}{l_{j1} + (\lambda_{1}/\lambda_{j})l_{11}}, j = 2, 4,$$ while for frequency $\omega_3$ it reads (cf. the analogous condition in [@vyunishev_et_al_2015] for equal frequencies) $$\label{eqn:image3} \frac{1}{f_3}= \frac{1}{l_{32}}+\frac{1}{l_{31}-(\lambda_{1}/\lambda_{3})l_{11}}.$$ In expressions , the length $l_{11}$ is the distance from BS to the object, $l_{j1}$ is the distance from BS to the lens, $l_{j2}$ is the distance from the lens to the detector CCD$_j$, $\lambda_j$ is the length of the wave of corresponding frequency. The derivation of Eqns. , is described below. They are a generalization of known ones to the case of different frequencies of radiation illuminating the object and radiation in reference arms. The theory of formation of entangled quantum four-beam states in processes was developed in [@saygin_chirkin_2010; @saygin_chirkin_2011; @saygin_et_al_2012]. Fourier components of Bose operators of field at nonlinear crystal output are represented in matrix form: $$\label{eqn:8} \hat{\mathbf{a}}({\mathbf{q}}, l) = Q({\mathbf{q}}, l){\hat{\mathbf{v}}}({\mathbf{q}}),$$ Here $\hat{\mathbf{a}}$ and $\hat{\mathbf{v}}$ are columns of Bose operators at crystal output and input, respectively. They are of the form $\hat{\mathbf{a}} \operatorname{\overset{\text{def}}{=}}(\hat{a}_{1}, \hat{a}_{2}^{\dagger}, \hat{a}_{3}, \hat{a}_{4}^{\dagger})^{T}$, where $T$ denotes transposition, $\hat{a}_1 = \hat{a}_1({\mathbf{q}}, l)$, $\hat{a}_{2}^{\dagger}= \hat{a}_{2}^{\dagger}(-{\mathbf{q}}, l)$, $\hat{a}_3 = \hat{a}_3({\mathbf{q}}, l)$, $\hat{a}_{4}^{\dagger} = \hat{a}_{4}^{\dagger}(-{\mathbf{q}}, l)$, $l$ is the length of nonlinear crystal. Operators in the column $\hat{\mathbf{v}} \operatorname{\overset{\text{def}}{=}}(\hat{v}_{1}, \hat{v}_{2}^{\dagger}, \hat{v}_{3}, \hat{v}_{4}^{\dagger})^{T}$ refer to vacuum field state. $Q$ is a $4 \times 4$ matrix whose elements $Q_{mn}$ describe field conversion from frequency $\omega_n$ to frequency $\omega_m$. The form of the matrix $Q$ and its properties in the quasioptical approximation are given in [@saygin_chirkin_2010]. The elements of $Q$ depend on crystal length, pump intensity and transversal wave number ${\mathbf{q}}$. The opertor $\hat{a}_{j}({\mathbf{q}}, z)$ is the annihilation operator of plane mode photons with frequency $\omega_j$ and transversal wave vector ${\mathbf{q}}$: $$\label{eqn:7} \hat{a}_{j}({\mathbf{q}}, z) = \frac{1}{2\pi} \iint\limits_{-\infty}^{+\infty} \hat{A}_j ({\boldsymbol{\rho}}, z) \exp(-i {\mathbf{q}}{\boldsymbol{\rho}}) d{\boldsymbol{\rho}},$$ where $\hat{A}_j ({\boldsymbol{\rho}}, z)$ is the slowly varying amplitude operator of the positive-frequency field $$\label{eqn:7a} \hat{E}^\dagger_{j}({\mathbf{r}}, z, t) = \hat{A}_j ({\mathbf{r}}, z, t) \exp(-i(\omega_{j} t - k_{j}z)),$$ $k_{j}$ is wave number, $z$ is the direction of propagation of interacting waves in the nonlinear crystal. After BS, their amplitude operators are defined by the following relations $$\label{eqn:11} \hat{B}_{j}({\mathbf{r}}_{j}) = \int H_{j}({\mathbf{r}}_j, {\boldsymbol{\rho}}_{j}){\hat{A}}_{j}({\boldsymbol{\rho}}_{j}, l) d{\boldsymbol{\rho}}_{j},$$ in the detector plane, integration being over the light beam aperture. $H_{j}({\mathbf{r}}_j, {\boldsymbol{\rho}})$ is the medium response function for radiation propagation from the crystal to the detector in $j$-th arm. We assume for simplicity that beam splitting takes place directly at nonlinear crystal output. In other words, BS is considered to be thin. For the object arm $$\label{eqn:resp1} H_{1}({\mathbf{r}}_1, {\boldsymbol{\rho}}_{1}) = \int\limits_{-\infty}^{+\infty} H_{1}({\mathbf{r}}_1 - {\boldsymbol{\rho}}_{1}'; l_{12}) T({\boldsymbol{\rho}}_{1}') H_{j}( {\boldsymbol{\rho}}_{1}' - {\boldsymbol{\rho}}_{1}; l_{11}) d{\boldsymbol{\rho}}_{1}'.$$ Here $T({\boldsymbol{\rho}}_{1}')$ is the object transmission coefficient, $H_{1}({\mathbf{r}}- {\boldsymbol{\rho}}; l_{1j})$ is the Green’s function $$\label{eqn:resp2} \int H_{1}({\mathbf{r}}- {\boldsymbol{\rho}}; l_{1j}) = -i\frac{k_1}{2\pi l_{1j}} \exp\left(i\frac{k_{1}({\mathbf{r}}- {\boldsymbol{\rho}})^2}{2 l_{1j}}\right).$$ As noted above, $l_{11}$ is the distance between BS and the object and $l_{12}$ is the distance between the object and the bucket detector. Response functions of reference arms containing thin lenses with focal length $f_j$ can be represented as (see [@intro_stat_radiophys_optics; @goodman_fourier_optics]) $$\label{eqn:resp3} H_{j}({\mathbf{r}}_j, {\boldsymbol{\rho}}_{j}) = -i \frac{k_j}{2\pi L_{j}} \exp\left(i\frac{k_{j}}{2 L_{j}} \left[({\mathbf{r}}_j - {\boldsymbol{\rho}}_j)^2 - (l_{j1} {\mathbf{r}}_j^2 + l_{j2} {\boldsymbol{\rho}}_j^2)/ f_j\right] \right),$$ where $$L_j = l_{j1} + l_{j2} - l_{j1} l_{j2}/f_j.$$ Intensity operators of the obtained beams are $\hat{I}_j({\mathbf{r}}_j) = \hat{B}^\dagger_j({\mathbf{r}}_j) \hat{B}_j({\mathbf{r}}_j)$. Mutual intensity correlation functions of the object arm and reference arms, taking into account Gaussian field statistics, are determined by the following formulas: for radiation with frequency $\omega_3$ the correlation function is $$G_{13}({\mathbf{r}}_1, {\mathbf{r}}_3) = \langle \hat{I}_1({\mathbf{r}}_1) \hat{I}_3({\mathbf{r}}_3) \rangle - \langle \hat{I}_1({\mathbf{r}}_1) \rangle \langle \hat{I}_3({\mathbf{r}}_3) \rangle = \| \langle \hat{B}_1 ({\mathbf{r}}_1) \hat{B}_3^\dagger ({\mathbf{r}}_3) \rangle \|^2,$$ while for radiation with frequencies $\omega_2$ or $\omega_4$ the correlation function is $$G_{1j}({\mathbf{r}}_1, {\mathbf{r}}_j) = \langle \hat{I}_1({\mathbf{r}}_1) \hat{I}_j({\mathbf{r}}_j) \rangle - \langle \hat{I}_1({\mathbf{r}}_1) \rangle \langle \hat{I}_j({\mathbf{r}}_j) \rangle = \| \langle \hat{B}_1 ({\mathbf{r}}_1) \hat{B}_j ({\mathbf{r}}_j) \rangle \|^2, j = 2, 4.$$ The difference in the definitions of the correlation functions under consideration is due to the type of parametric conversion and the initial vacuum fluctuations. As a consequence, only the vacuum operators in antinormal ordering contribute to correlations. Under imaging conditions , the expressions can be transformed to $$\label{eqn:corr-func-imag-cond-3} G_{13}({\mathbf{r}}_1, {\mathbf{r}}_3) = \| \Gamma_3 \|^2 \left\| \frac{l_{31} (\lambda_1/\lambda_3) l_{11}}{\lambda_1 l_{12} l_{32}} T\left( - \alpha_3 {\mathbf{r}}_3 \right) \right\|^2, \alpha_3 \operatorname{\overset{\text{def}}{=}}\frac{l_{31} - (\lambda_1/\lambda_3) l_{11}}{l_{32}},$$ $$\label{eqn:corr-func-imag-cond-24} G_{1j}({\mathbf{r}}_1, {\mathbf{r}}_j) = \| \Gamma_j \|^2 \left\| \frac{l_{j1} (\lambda_1/\lambda_j) l_{11}}{\lambda_1 l_{12} l_{j2}} T\left( - \alpha_j {\mathbf{r}}_j \right) \right\|^2, \alpha_j \operatorname{\overset{\text{def}}{=}}\frac{l_{j1} + (\lambda_1/\lambda_j) l_{11}}{l_{j2}}.$$ These formulas are derived under the assumption that at the nonlinear crystal output, mutual correlation functions of the radiation $$\Gamma_{1j}({\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_j) = \langle \hat{A}_1({\boldsymbol{\rho}}_1, l) \hat{A}_j({\boldsymbol{\rho}}_j, l) \rangle = (2 \pi)^{-1} \int Q_{11j}({\mathbf{q}}) \exp( i {\mathbf{q}}({\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_j)) d{\mathbf{q}}, j = 2, 4,$$ $$\Gamma_{13}({\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_3) = \langle \hat{A}_1({\boldsymbol{\rho}}_1, l) \hat{A}_j^\dagger({\boldsymbol{\rho}}_3, l) \rangle = (2 \pi)^{-1} \int Q_{113}({\mathbf{q}}) \exp( -i {\mathbf{q}}({\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_3)) d{\mathbf{q}},$$ where $Q_{11n}({\mathbf{q}}) \operatorname{\overset{\text{def}}{=}}Q_{11}({\mathbf{q}}) Q_{n1}^{}({\mathbf{q}}) + Q_{13}({\mathbf{q}})Q_{n3}^{}({\mathbf{q}})$, $n = 1, 2, 3$, are substituted by $\delta$-functions $$\Gamma_{1n}({\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_n) = \Gamma_n \delta({\boldsymbol{\rho}}_1 - {\boldsymbol{\rho}}_n), \Gamma_n = \int \Gamma_{1n}({\boldsymbol{\rho}}) d{\boldsymbol{\rho}}.$$ These substitutions are valid if the radiation correlation radius is much smaller than a characteristic spatial scale of object image change. The expressions , coincide, up to a factor before the image transmission coefficient, with the expression obtained for another experimental setup [@mgi_correlations_jrlr_2017; @ghost_images_jetp]. In [@mgi_correlations_jrlr_2017; @ghost_images_jetp] ghost image correlations determined by fourth-order intensity correlations (eight-order field ones) were studied as well. Obviously, in the setup under consideration they will be the same as in [@mgi_correlations_jrlr_2017; @ghost_images_jetp]. After integration over the area $s$ of the beam in the object arm (over $d{\mathbf{r}}_1$) intensity correlation functions of the second order, in accordance with , , becomes $$\label{eqn:corr-func-2-order} G_{j}({\mathbf{r}}_{j}) \sim s \left\| T( - \alpha_j {\mathbf{r}}_{j}) \right\|^{2},$$ while the GI correlation function determined by eighth-order field correlation function becomes $$\label{eqn:corr-func-4-order} K_{ij}^{\textnormal{GI}}({\mathbf{r}}_{i}, {\mathbf{r}}_{j}) \sim s^2 \|T(-\alpha_i {\mathbf{r}}_i)\|^2 \|T(-\alpha_j {\mathbf{r}}_j)\|^2.$$ Naturally, the coefficients $\alpha_2, \alpha_3, \alpha_4$ can be made equal by choice of setup parameters. In addition, in the following formulas and the factors dependent on the measurement unit choice will be omitted for brevity. As mentioned above, the correlation functions derived above provide the information about the measuring (image acquisition) process that is used in the measurement reduction technique along with the information about the object. The following section focuses on the measurement reduction technique itself and the information about the object. Not all of the information mentioned above is equal in importance: only the correlation function and finiteness of $L^2$ norm of the correlation function for all $i, j = 2, 3, 4$ are strictly necessary for image reconstruction. Nevertheless, additional information about both the measuring process (in our case, the form of correlation function ) and the object (sparsity of its transparency distribution) can vastly improve reconstruction quality, as it will be shown below. Processing of acquired images {#sec:image-processing} ============================= The output of $i$-th correlator, denoted as $\xi^{(i)}({\mathbf{r}})$, can be considered as the impact of a measuring transducer (MT) on the input signal $f({\mathbf{r}}) \sim \|T(-{\mathbf{r}})\|^2$. Here and below, unlike the previous section, $f$ denotes the vector describing the object transparency distribution instead of focal length. We assume for simplicity that $\alpha_2 = \alpha_3 = \alpha_4 = 1$. We will consider piecewise constant images, i.e. transparency is constant within each pixel. Areas of constant transparency and constant brightness corresponding to pixels are considered to be ordered in an arbitrary but fixed way. Due to that it is sufficient for us to consider a finite number of values of ${\mathbf{r}}$. Thus, $f$ as the vector of transparencies is an element of finite-dimensional Euclidean space $\mathcal{F}$. An image processing algorithm ought to provide the most accurate estimate of the feature of the original image $f$ that is of interest to the researcher based on obtained data $\xi$, which consists of acquired ghost images $\xi^{(i)}({\mathbf{r}})$, $i = 2, 3, 4$. Measurement reduction method allows to obtain such an estimate. Let us formulate the measurement model as $$\label{eqn:measurement-model} \xi = \mathbf{A} f + \nu,$$ where $f$ is an priori unknown vector that describes the transparency distribution of the object, $\nu$ is measurement error with zero expectation, $\operatorname{\mathbb{E}}\nu = 0$, which means absence of systematic measurement error, and covariance matrix $\mathbf{\Sigma}_{\nu} = \operatorname{\mathbb{E}}\nu \nu^$. The matrix $\mathbf{A}$ describes ghost imaging and GI acquisition: the matrix element $\mathbf{A}_{ij}$ is equal to the mean output of $i$-th detector for unit transparency of $j$-th element of the object and zero transparency of other object elements (i.e. whose indices differ from $j$). The dimension of vector $f$ is the number of pixels in the object image, while the dimension of $\xi$ is the number of pixels in all CCD. The condition of systematic measurement error absense $\operatorname{\mathbb{E}}\nu = 0$ means, in particular, that the expectation of the component of measurement results caused by detector dark noises is subtracted from the measurement results, similar to [@gi_compressed_sensing_substr_const]. Matrices $\mathbf{A}$ and $\mathbf{\Sigma}_{\nu}$ are related to the correlation functions considered above. The measuring setup employs correlators that measure correlations between the object arm and other arms. Therefore, the matrix $\mathbf{A}$, which models the impact of MT on the image, is a block matrix and consists of three blocks describing correlator outputs, i.e. correlations between the object arm and reference arms: $$\label{eqn:a-op-form} \mathbf{A} = \begin{pmatrix} \mathbf{B}_{2} \mathbf{C}_{2}\\ \mathbf{B}_{3} \mathbf{C}_{3}\\ \mathbf{B}_{4} \mathbf{C}_{4} \end{pmatrix}.$$ Under the conditions used to derive the intensity correlation functions and the matrices $\mathbf{C}_2$–$\mathbf{C}_4$ are identity ones multiplied by pixel size and the factor before $\|T({\mathbf{r}}_{i})\|^{2}$ in expression for the correlation function $G_{j}$. The matrices $\mathbf{B}_2$–$\mathbf{B}_4$ model the detectors. Specifically, the matrix element $(\mathbf{B}_i)_{pk}$ is equal to the output of the detector in $i$-th arm at $p$-th position for unit brightness of $k$-th pixel and zero brightness of other pixels. Noise covariance matrix has block form as well: $$\label{eqn:sigma-op-form} \mathbf{\Sigma}_\nu = \begin{pmatrix} \mathbf{B}_2\mathbf{\Sigma}_{22}(f)\mathbf{B}_2^ & \mathbf{B}_2\mathbf{\Sigma}_{23}(f)\mathbf{B}_3^* & \mathbf{B}_2\mathbf{\Sigma}_{24}(f)\mathbf{B}_4^\\ \mathbf{B}_3\mathbf{\Sigma}_{32}(f)\mathbf{B}_2^ & \mathbf{B}_3\mathbf{\Sigma}_{33}(f)\mathbf{B}_3^* & \mathbf{B}_3\mathbf{\Sigma}_{34}(f)\mathbf{B}_4^\\ \mathbf{B}_4\mathbf{\Sigma}_{42}(f)\mathbf{B}_2^ & \mathbf{B}_4\mathbf{\Sigma}_{43}(f)\mathbf{B}_3^* & \mathbf{B}_4\mathbf{\Sigma}_{44}(f)\mathbf{B}_4^* \end{pmatrix} + \mathbf{\Sigma}_{\nu'}.$$ Here the element with indices $k$, $k'$ of the block $\mathbf{\Sigma}_{ij}$ is equal to the integral of $K_{ij}^{\textnormal{GI}}$ over the values of ${\mathbf{r}}_{i}$ belonging to $k$-th pixel and over the values of ${\mathbf{r}}_{j}$ belonging to $k'$-th pixel, for the same pixel ordering as in the matrix $\mathbf{A}$. Hence, the dependence of on $f$ is caused by the dependence on $\|T(\cdot)\|^2$ of the correlation function $K_{ij}^{\textnormal{GI}}$ . The term $\mathbf{\Sigma}_{\nu'}$ is the covariance matrix of the noise component $\nu'$ that is unrelated to ghost imaging, e.g. thermal noise in circuits and digitization error. Most of the noise arising before the correlators is suppressed by them if noise in object and reference arms is independent, but this does not apply to noise arising after the correlators. Besides, due to finite coincidence circuit match time some of noise photons contribute to the noise as well, see discussion of fig. \[fig:gi-vs-ordinary-10\] below. It should be noted that the algorithm proposed below can be applied for an image multiplexing method that differs from the one considered in section \[sec:imaging\] if the measurement model has the form . Specifically the expectation of measurement result has to be the product of a matrix $\mathbf{A}$ and the transparency distribution vector of the measured object, and the error has to be able to be considered additive. For that, the fourth-order intensity correlation function (an analog of ) has to linearly depend on the transparency distribution, and the eighth-order intensity correlation function (an analog of ) has to “sufficiently weakly” depend on the transparency distribution so that an unknown covariance matrix could be estimated using measurement results. If, in addition to that, photon detections in reference arms are conditionally independent under fixed output of the bucket detector in the object arm (output of a detector in a reference arm does not affect output of detectors in other reference arms), then what was said about the form of matrices $\mathbf{A}$ and $\mathbf{\Sigma}_{\nu}$ remains valid. The estimation problem consists of reconstruction of the most accurate estimate of the signal $\mathbf{U} f$ from the measurement result $\xi$, where the matrix $\mathbf{U}$ describes a measuring device that is ideal (for the researcher). We consider the case when the researcher is interested in reconstruction of the object image itself, and imaging does not distort the object, therefore, $\mathbf{U} = I$. Since measurement results linearly depend on $f$, to solve the estimation problem we can use the model $[\mathbf{A}, \mathbf{\Sigma}_\nu, \mathbf{U}]$ described in [@pytyev_ivs], see also [@pytyev_chulichkov_1998; @pytyev_et_al_2004; @pytyev_2010; @reduction_vmu]. If the estimation process is described by a linear operator $R$ ($R \xi$ is the result of processing the measurement $\xi$), the corresponding mean squared error (MSE) in the worst case of $f$, $h(R, \mathbf{U}) = \sup\limits_{f \in \mathcal{F}} \operatorname{\mathbb{E}}\lVert R \xi - \mathbf{U} f \rVert^2$, as shown in [@pytyev_ivs], is minimal for $R$ that is equal to the linear unbiased reduction operator $$\label{eqn:reduction-operator} R_* \operatorname{\overset{\text{def}}{=}}\mathbf{U} (\mathbf{A}^* \mathbf{\Sigma}_{\nu}^{-1} \mathbf{A})^- \mathbf{A}^* \mathbf{\Sigma}_{\nu}^{-1},$$ where ${}^-$ denotes pseudoinverse. $h(R_, \mathbf{U}) = \operatorname{tr}\mathbf{U} (\mathbf{A}^ \mathbf{\Sigma}_{\nu}^{-1} \mathbf{A})^{-1} \mathbf{U}^$, and the covariance matrix of the linear reduction estimate $R_ \xi$ is $$\label{eqn:reduction-cov-op} \mathbf{\Sigma}_{R_* \xi} = \mathbf{U} (\mathbf{A}^* \mathbf{\Sigma}_{\nu}^{-1} \mathbf{A})^{-1} \mathbf{U}^.$$ Estimation is possible (MSE is finite) if the condition $\mathbf{U} (I - \mathbf{A}^- \mathbf{A}) = 0$ holds, where, as noted above, $\mathbf{A}$ characterizes the real* measuring device, while $\mathbf{U}$ characterizes an ideal one with the point spread function required by the researcher, and, therefore, any desired resolution, if this condition if fulfilled. Note that, unlike fluorescence-based superresolution techniques, see e.g. [@solomon_et_al_2018], the proposed technique does not require attaching fluorescent molecules to the object. However, as a rule, the better the desired resolution of the ideal measuring device compared to the resolution of the real one, the larger MSE of the obtained estimate. By choosing $\mathbf{U}$ one can select an acceptable (to him) compromise between obtained resolution and noise magnitude. In the case under consideration, as seen from , diagonal elements of each block (which, up to a nonzero factor, are equal to the factor before $\|T(-{\mathbf{r}}_j)\|^2$ in the expression for correlation function $G_{j}$) are nonzero. Therefore, each block $\mathbf{A}_j$ is non-degenerate, so for non-degenerate $\mathbf{B}_j$ the reduction error takes only finite values. For a different multiplexing method and thus, different form of the matrix $\mathbf{A}$ this is generally not so. The measurement reduction technique for the case when it is known that the value $u$ of the feature of interest is an arbitrary element not of the entire $\mathcal{U}$ but of its convex closed subset $\mathcal{U}_{{\textnormal{pr}}}$ was considered in [@reduction_vmu; @lomo_readings]. The estimate refinement which takes advantage of this information is determined by solving the equation $$\label{eqn:reduction-estimate-in-set} \hat{u} = \Pi_{\mathbf{\Sigma}_{R_* \xi}}\left(\tilde{R}_{\mathbf{\Sigma}_{R_* \xi}} \left( \xi^T, \hat{u}^T \right)^T\right)$$ for $\hat{u}$, where $\tilde{R}_{\mathbf{\Sigma}_{R_* \xi}}$ is the measurement reduction operator for a MT $\left(\mathbf{A}^T, \mathbf{U}^T\right)^T$ and noise with covariance matrix $\begin{pmatrix}\mathbf{\Sigma}_{\nu} & 0\\ 0 & \mathbf{\Sigma}_{R_* \xi}\end{pmatrix}$, and the operator $$\label{eqn:mahalanobis-projection} \Pi_{\mathbf{\Sigma}_{R_* \xi}}(u) \operatorname{\overset{\text{def}}{=}}\operatorname{argmin}\limits_{v \in \mathcal{U}_{{\textnormal{pr}}}} (v - u, {\mathbf{\Sigma}_{R_ \xi}}^{-1} (v - u))$$ describes projection onto $\mathcal{U}_{{\textnormal{pr}}}$ by minimizing Mahalanobis distance $\lVert \mathbf{\Sigma}_{R_* \xi}^{-1/2} \cdot \rVert$ that is related to covariance matrix $\mathbf{\Sigma}_{R_* \xi}$ of error of the linear reduction estimate $R_* \xi$. Note that the version of reduction technique proposed in [@ghost_images_jetp] and in [@reduction_vmu] for similar information used minimization of the “ordinary” Euclidean distance instead of Mahalanobis distance. In [@lomo_readings], the advantages of minimizing Mahalanobis distance instead of Euclidean distance during projection are shown. In that case the covariance matrix of linear reduction estimate error is an upper bound on the covariance matrix of the obtained estimate. Representation of the object information that is available to the researcher {#sec:image-information} ---------------------------------------------------------------------------- It is obvious that a priori $\|T(-{\mathbf{r}}_j)\|^2 \in [0, 1]$, hence $f \in [0, 1]^{\dim \mathcal{F}}$, $\mathbf{U} f \in [0, 1]^{\dim \mathcal{F}}$. It is assumed that the transparency distribution of the object is not “entirely” arbitrary: transparencies of neighboring pixels usually do not differ much, so the image is sparse (many of its components are zero) in a given (a priori known) basis, similarly to compressed sensing ghost imaging [@imaging_small_n_photons; @hi-res_gi_sparsity; @gi_compressed_sensing_substr_const]. The researcher also knows the matrix $\mathbf{A}$ that describes image acquisition conditions and, up to the vector $f$, the matrix $\mathbf{\Sigma}_{\nu}$ that describes measurement errors. Note that the worst case of $f$ is realized if all pixels are equally transparent. For a different multiplexing method one considers the worst case in the sense of reduction MSE of the object in step \[itm:first-estimate\] of the algorithm below. Reduction algorithm {#sec:reduction-algorithm} ------------------- The proposed algorithm of multiplexed GI processing using measurement reduction technique that is based on the indicated prior information has the following form. 1. \[itm:first-estimate\] Calculation of linear unbiased reduction estimate $R_* \xi$ based on the acquired GI, assuming for calculation of covariance matrix that all pixels have the same brightness. 2. Refinement of the estimate $R_* \xi$ using the information $\mathcal{U}_{{\textnormal{pr}}} = [0, 1]^{\dim \mathcal{F}}$ by the method by fixed-point iteration, i.e. by consecutive application of the mapping with $\Pi_{\mathbf{\Sigma}_{R_* \xi}} (R_* \xi)$ as the initial approximation. We denote the obtained estimate by $\hat{u}$. 3. Application of the sparsity-inducing transformation $T$ to $\hat{u}$. “Sparsity-inducing” means that the transformation is chosen by the researcher so that, in his opinion, the transform of the true transparency distribution of the object is sparse. 4. \[itm:thresholding\] Calculation of maximal (in the worst case of $f$) variances $\sigma_{T \hat{u}}^2 = (\sigma_{(T \hat{u})_1}^2, \dots, \sigma_{(T \hat{u})_{\dim \mathcal{F}}}^2)$ of the components of $T \hat{u}$, i.e. the diagonal matrix elements of $T \mathbf{\Sigma}_{R_* \xi} T^$, and calculation of $T \hat{u}_{\textnormal{thr}}$: $(T \hat{u}_{\textnormal{thr}})_i \operatorname{\overset{\text{def}}{=}}0$ if $\|(T \hat{u})_i\| < \lambda \sigma_{(T \hat{u})_i}$, otherwise $(T \hat{u}_{\textnormal{thr}})_i \operatorname{\overset{\text{def}}{=}}(T \hat{u})_i$. 5. Inverse transformation $T^{-1}$ of $T \hat{u}_{\textnormal{thr}}$ (if $T$ is a unitary transformation, then $T^{-1} = T^$), i.e. calculation of $\hat{u}_{\textnormal{thr}} \operatorname{\overset{\text{def}}{=}}T^{-1} T \hat{u}_{\textnormal{thr}}$. 6. Calculation of the projection $\Pi_{\mathbf{\Sigma}_{R_* \xi}} (\hat{u}_{\textnormal{thr}})$ that is considered to be the result of processing obtained ghost images. The value of $\lambda \geq 0$ is a parameter of the algorithm. It reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$. Step \[itm:thresholding\] can be considered as testing a statistical hypothesis, according to which $(T \mathbf{U} f)_i = 0$ (for the alternative hypothesis that $(T \mathbf{U} f)_i \neq 0$) for all $i$. In this paper to do that we employ in step \[itm:thresholding\] a simple criterion based on Chebyshev’s inequality: if $(T \mathbf{U} f)_i = 0$, then $\Pr\left(\|(T \hat{u})_i\| \geq \lambda \sigma_{(T \hat{u})_i}\right) \leq \lambda^{-2}$. Due to that one can suppose that image distortion is insignificant for, at least, $\lambda \leq 1$, as such distortion would be indistinguishable from the noise. Step \[itm:thresholding\] can be also interpreted as replacement of the original matrix $\mathbf{U}$ with one whose kernel contains the estimate components after the specified transform that are affected by the noise the most. In [@ghost_images_jetp] the matrix $\mathbf{U}$ was chosen to suppress the noise more, even at the cost of potential image distortion (e.g. worse resolution), by discarding the most noisy components of the image. Unlike [@mgi_icono_lat_2016; @mgi_correlations_jrlr_2017; @ghost_images_jetp], here we consider components of the image in a basis specified by the researcher instead of the eigenbasis [@pytyev_ivs ch. 8] of the measurement interpretation model, i.e. a basis determined by error properties. In this article the basis is defined by the transformation whose result for the true transparency distribution is sparse, but the discarded components are determined, as in [@ghost_images_jetp], by the measurement error. Thus, to improve estimation quality not only information about the noise is used, but also information about the object, namely, the properties of the transparency distribution (in the opinion of the researcher) and its features of interest. Computer modeling results {#sec:computer-modelling} ========================= The results of processing of obtained GI as described above are shown in fig. \[fig:two-slits\] and \[fig:phys-msu\]. The detectors in reference arms are identical ones that are three times as large as an element of the object image. Therefore, image processing via measurement reduction increases resolution in addition to noise suppression. Modeling was carried out for the same parameters of the optical setup as in [@ghost_images_jetp]: beam wave numbers $k_1 = 6 \cdot 10^4~\textnormal{cm}^{-1}$, $k_3 = 1.7 \cdot 10^5~\textnormal{cm}^{-1}$, crystal parameter $\beta = 10~\textnormal{cm}^{-1}$, crystal parameter $\xi = \gamma / \beta = 0.4$. ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}](two_slits-src_img) [2]{} ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}](two_slits-gi2 "fig:") ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}](two_slits-gi3 "fig:") ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}](two_slits-gi4 "fig:") ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}](two_slits-red) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}]({{two_slits-red-dct-1.25}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}]({{two_slits-red-dct-2.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}]({{two_slits-red-haar-1.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}]({{two_slits-red-haar-2.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $1$ photon per pixel on average, () are its acquired GI, and (–) are image reduction results: () is the result of reduction without sparsity information, (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases. The parameter $\lambda \geq 0$ of the image processing algorithm reflects a compromise between noise suppression (the larger the value of $\lambda$, the greater the noise suppression) and distortion of images whose components are close to $0$[]{data-label="fig:two-slits"}]({{two_slits-red-haar-3.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}](phys-msu-src_img) [2]{} ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}](phys-msu-gi2 "fig:") ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}](phys-msu-gi3 "fig:") ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}](phys-msu-gi4 "fig:") ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}]({{phys-msu-red-dct-1.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}]({{phys-msu-red-dct-1.5}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}]({{phys-msu-red-dct-2.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}]({{phys-msu-red-haar-1.0}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}]({{phys-msu-red-haar-1.5}}) ![GI processing by the developed algorithm: () is the object, $64$x$64$ pixels, that is illuminated by $10$ photons per pixel on average, () are its acquired GI, and (–) are results of reduction using information about sparsity in (–) discrete cosine transform (DCT), (–) Haar transform bases[]{data-label="fig:phys-msu"}]({{phys-msu-red-haar-2.0}}) One can see that additional information about sparsity allows to suppress noise more but its impact on obtained resolution is weak. As expected, for $0 \leq \lambda \leq 1$ the distortion is undistinguishable from the noise. Further increase of $\lambda$ leads to better noise suppression (cf., e.g., fig. \[fig:two-slits-red\] and \[fig:two-slits-red-dct-1.25\], \[fig:phys-msu-red-dct-1.0\] and \[fig:phys-msu-red-dct-1.5\]), but also leads to more severe distortions caused by discarding “significant” image components as well (cf., e.g., fig. \[fig:two-slits-red-dct-2.0\] and \[fig:phys-msu-red-dct-2.0\]). For large $\lambda$, their influence outweighs the improvement of image quality due to noise suppression, as small-scale image details are suppressed as well. Therefore, the optimal value of $\lambda$ depends on one’s intentions: one should choose the maximal value of $\lambda$ that preserves the details of interest. To do that, one can model acquisition of a test image that contains the required details and choose the largest value of $\lambda$ that preserves them, or specify the value of $\lambda$ after comparing reduction results for different $\lambda$. In the case of an object with sharp transparency changes (fig. \[fig:two-slits\]) the additional information allowed to suppress false signal where the object is opaque, but only for Haar transform (discrete cosine transform (DCT) causes increased false signal in that region). The transform whose result for the transparency distribution of the object is sparse that is usually employed in ghost image processing by the means of compressed sensing is DCT [@gi_compressed_sensing_substr_const; @imaging_small_n_photons; @hi-res_gi_sparsity]. In [@sparsity_property_influence], several transforms (identity transform, discrete wavelet transform and DCT) were reviewed and the advantages of DCT were shown. However, it seems that Haar transform may be preferable in the case of a transparency distribution that contains areas of weakly changing transparency with sharp borders if these areas are large compared to the resolution of the ideal measuring transducer and the location of the borders is important to the researcher. This assumption is verified by fig. \[fig:two-slits-red-haar-1.0\]–, where one can see that Haar transform in this case, as opposed to fig. \[fig:phys-msu\], allows larger $\lambda$ values without causing significant distortions, cf., e.g., fig. \[fig:two-slits-red-dct-2.0\] and , where the usage of DCT causes blurring of transversal slit borders for the same value of $\lambda$. ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}](phys-msu-extra-o-10) ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}](phys-msu-extra-o-10-red) ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}](phys-msu-extra-o-10-dct) [2]{} ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.1-gi2}} "fig:") ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.1-gi3}} "fig:") ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.1-gi4}} "fig:") ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.1-red-dct}}) [2]{} ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.5-gi2}} "fig:") ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.5-gi3}} "fig:") ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.5-gi4}} "fig:") ![Ordinary and ghost image processing by the developed algorithm. () is the ordinary image of the object from fig. \[fig:phys-msu-src\] obtained by illuminating it by $10$ photons per pixel on average and $10$ noise photons, and () and () are the results of its reduction () without sparsity information and () with information about sparsity in DCT base. (, ) are GI impacted by $1$ and $5$ noise photons, respectively. (, ) are the results of their reduction with information about sparsity in DCT base[]{data-label="fig:gi-vs-ordinary-10"}]({{phys-msu-extra-10-0.5-red-dct}}) In fig. \[fig:gi-vs-ordinary-10\] GI are compared with ordinary images if noise photons, which do not carry information about the object, but increase noise, are present. Due to employing correlations to acquire GI, noise photons usually do not affect detected images, as this requires simultaneous detection of a noise photon by one detector and another photon by a different detector. Nevertheless, due to finite coincidence windows and finite widths of the light filters before the detectors the noise photons do increase the measurement errors. One can see that due to suppression of most noise photons the quality of the reconstructed image is better than the quality of the image reconstructed using the ordinary image for the same number of noise photons. Moreover, when taking advantage of sparsity information ghost imaging allows to exploit larger $\lambda$ values and thus, to suppress the noise more (cf., e.g., fig. \[fig:phys-msu-extra-o-10-dct\] and \[fig:phys-msu-extra-red-dct-10-0.5\]). In this case multiplexing provides the means for further noise suppression if noise photons in different arms are detected independently. Therefore, formalization the researcher’s information about sparsity of the object transparency distribution by Haar transform is preferable if it has areas of weakly changing transparency with sharp borders that are large compared to the resolution of the ideal measuring transducer and the location of the borders is important to the researcher. DCT is preferable if the transparency distribution has small transparency changes that have to be present in the estimate, e.g. biological objects without high-contrast borders. The values of $\lambda \sim 1 \div 1.5$ are optimal if small-scale details are present and are of interest. Otherwise, larger $\lambda$ values are advisable. Conclusion {#conclusion .unnumbered} ========== The actual problem of increasing noise immunity is exacerbated by photon transmission and detecton in photocounting mode due to higher information content of each photon or its absense. Multiplexing of ghost images allows to reduce the noise level, since it increases the amount of transmitted information, enabling improvement the quality of processing of acquired data. In this case the additional information available to the researcher about the measurement process and about the object allows further noise suppression under the same detection conditions. Alternatively, one can make the detection conditions worse (e.g. to reduce the number of photons) while preserving the same estimation quality. The additional information about the measuring process in this work is the correlation functions of multiplexed ghost images. The additional information about the object is the information that the object transparency distribution is not arbitrary, namely, transparencies of neighboring pixels, as a rule, differ only slightly. This information is formalized as sparsity of the result of a given transform (e.g. DCT) of the transparency distribution, similar to compressed sensing. In compressed sensing, as a rule, the measurement error is modeled as an arbitrary vector with bounded norm. Instead, in the proposed method it is modeled as a random vector, and selection of the estimate components which are considered to be zero is based on the statistical properties of the estimate components, namely, their variances. The use of covariances of the estimate components in addition to their variances is a subject of further research. We consider that computer modeling based on the developed algorithm showed high efficiency of the developed reduction technique of ghost image processing in the sense of improvement of both their quality and their noise immunity. It is of interest to apply this technique in the field of quantum image processing for parametric amplification of images and frequency conversion. The authors are grateful for help to T.Yu.Lisovskaya. This work was supported by RFBR grant 18-01-00598-A. This is a pre-print of an article published in Quantum Information Processing. The final authenticated version is available online at: <https://doi.org/10.1007/s11128-019-2193-x> [^1]: Corresponding author e-mail: balakin\_d\_a@physics.msu.ru
--- abstract: 'We present the first comparison of profiles of H et He resonance lines observed by SUMER with theoretical profiles computed with our non-LTE radiative transfer code. We use the [[H]{} <span style="font-variant:small-caps;">[i]{}</span>]{} Lyman $\beta$, [[H]{} <span style="font-variant:small-caps;">[i]{}</span>]{} Lyman $\epsilon$, and [[He]{} <span style="font-variant:small-caps;">[i]{}</span>]{} $\lambda$584 Å lines. Our code allows us to obtain the plasma parameters in prominences in conjunction with a multi-line, multi-element set of observations. The plasma temperature in the prominence core is $\sim 8600$ K and the pressure is 0.03 dyn cm$^{-2}$. The Ly$\beta$ line is formed in a higher temperature region (more than 11000 K).' address: - 'Institute of Mathematical and Physical Sciences, University of Wales Aberystwyth, UK' - 'Institut d’Astrophysique Spatiale, CNRS–Université Paris Sud, Orsay, FR' author: - 'Labrosse, N.' - 'Vial, J.-C.' - 'Gouttebroze, P.$^2$' title: Plasma diagnostic of a solar prominence from hydrogen and helium resonance lines --- Introduction ============ The H and He resonance lines have a strong diagnostic potential. However, observations are difficult to interpret as the prominence plasma is optically thick at these wavelengths and out of LTE. It is thus necessary to use a non-LTE radiative transfer code to analyse the data. Observations and modelling ========================== Observations were made during the 13th MEDOC campaign (IAS) on 2004, June 15. The SUMER slit centre was located at $X=870{\mbox{$^{\prime\prime}$}}$, $Y=465{\mbox{$^{\prime\prime}$}}$. On Fig. \[images\] we note the det. A anomaly around pixel 30: a dark area surrounded by bright pixels. We select one large area between pixels 70 and 100 to represent the prominence plasma. The Ly$\beta$ line exhibits a central reversal on several pixels. ![Intensity in the Ly$\beta$, Ly$\epsilon$, and [[He]{} <span style="font-variant:small-caps;">[i]{}</span>]{} 584 lines as a function of wavelength and position along the slit.[]{data-label="images"}](plotimage-colall.eps){width="7cm"} The numerical code solves the equations of radiative transfer and statistical equilibrium for H, [[He]{} <span style="font-variant:small-caps;">[i]{}</span>]{}, and [[He]{} <span style="font-variant:small-caps;">[ii]{}</span>]{}. Details are given in Labrosse & Gouttebroze (2001, 2004). The prominence is represented by a 1D vertical plane-parallel slab. We consider 2 types of prominence atmospheres: constant temperature and pressure within the prominence, or with a prominence-to-corona transition region (PCTR). In the latter case the temperature and pressure profiles are of the same type as proposed by Anzer & Heinzel (1999). Results ======= The top plots on Fig. \[fit6\] show each line profile observed by SUMER and the theoretical profile which best fits the observed profile. We obtain a different model for each line (see Tab. \[results\]). Each one of the three selected models is isothermal and isobaric. The PCTR models considered here cannot reproduce the observed profiles for this prominence. We also searched for a unique model that simultaneously reproduces the three line profiles. The result is shown on the three bottom panels of Fig. \[fit6\]. The selected model is the same as the one which best fits the observed [[He]{} <span style="font-variant:small-caps;">[i]{}</span>]{} 584 line alone (our model \# 3), but it fails to reproduce the Ly$\beta$ line. There is a long-standing problem regarding the Ly-$\beta$ line (see, e.g., Patsourakos & Vial 2002). Some possible explanations are: 1) We may need a different type of temperature profile for this particular prominence. 2) The fine structure of the prominence should be incorporated in the modelling. 3) The incident radiation illuminating the structure is strongly affected by the presence of several active regions on the disk seen on MDI images of the same day. ![Comparisons between observed (green) and theoretical (red) profiles. Intensities are in W m$^{-2}$ sr$^{-1}$ Å$^{-1}$.[]{data-label="fit6"}](fit6-70-100.eps){width="9cm"} \[results\] [Model]{} [Line]{} [$T$]{} [$p$]{} [$M$]{} [$D$]{} [$V_t$]{} [$H$]{} [$\tau_{912}$]{} [$\tau_{504}$]{} ----------- --------------- --------- --------- ---------------- --------- ----------- --------- ------------------ ------------------ 1 Ly-$\beta$ 11155 0.075 3.48 10$^{-6}$ 584 19 3627 2 1 2 Ly-$\epsilon$ 6677 0.414 1.12 10$^{-5}$ 127 13 73198 26 3 3 584 8602 0.029 4.50 10$^{-7}$ 128 18 94333 0.5 0.1 : Parameters for selected models. Parameters: temperature $T$ (K), pressure $p$ (dyncm$^{-2}$), total column mass $M$ (gcm$^{-2}$), slab thickness $D$ (km), microturbulent velocity $V_t$ (kms$^{-1}$), height above the limb $H$ (km). Conclusion ========== The plasma temperature in the core of the observed prominence is found to be $\sim 8600$ K and the pressure is 0.03 dyn cm$^{-2}$. The Ly$\beta$ line is formed in a higher temperature region (more than 11000 K). The [[He]{} <span style="font-variant:small-caps;">[i]{}</span>]{} line at 584 Å has been succesfully used for the diagnostic of the prominence. The combination of the helium resonance lines with the Lyman lines of hydrogen provides new constraints for prominence models. Anzer, U. & Heinzel, P, 1999, A&A, 349, 974 Labrosse, N. & Gouttebroze, P., 2001, A&A, 380, 323 Labrosse, N. & Gouttebroze, P., 2004, ApJ, 617, 614 Patsourakos, S. & Vial, J.-C., 2002, Sol. Phys., 208, 253
--- abstract: 'To advance fundamental understanding, and ultimate application, of transition-metal dichalcogenide (TMD) monolayers, it is essential to develop capabilities for the synthesis of high-quality single-layer samples. Molecular beam epitaxy (MBE), a leading technique for the fabrication of the highest-quality epitaxial films of conventional semiconductors has, however, typically yielded only small grain sizes and sub-optimal morphologies when applied to the van der Waals growth of monolayer TMDs. Here, we present a systematic study on the influence of adatom mobility, growth rate, and metal:chalcogen flux on the growth of NbSe$_2$, VSe$_2$ and TiSe$_2$ using MBE. Through this, we identify the key drivers and influence of the adatom kinetics that control the epitaxial growth of TMDs, realising four distinct morphologies of the as-grown compounds. We use this to determine optimised growth conditions for the fabrication of high-quality monolayers, ultimately realising the largest grain sizes of monolayer TMDs that have been achieved to date via MBE growth.' author: - Akhil Rajan - Kaycee Underwood - Federico Mazzola - 'Philip D.C. King' title: Morphology Control of Epitaxial Monolayer Transition Metal Dichalcogenides --- Introduction ============ Transition metal dichalcogenides (TMDs), composed of a transition-metal (M) layer sandwiched between two chalcogen (X) layers, represent a particularly diverse materials family. In bulk, such covalently-bonded MX$_2$ monolayers are stacked with weak van der Waal’s bonding between neighbouring layers. Depending on the filling of the transition-metal $d$-orbitals, a large variety of electronic properties are found, including semiconductors, metals, charge-density wave systems, superconductors, and topologically non-trivial materials.[@Chhowalla2013; @wang2012electronics; @xu2014spin; @bahramy2018ubiquitous] Excitingly, their properties can be significantly modified by changing material’s thickness down to the monolayer limit. Famous examples include a thickness-tuned cross-over from an indirect to a direct band gap in MoS$_2$ [@Mak2010; @splendiani2010emerging], the realisation of extremely high exciton binding energies as a result of reduced dielectric screening in the monolayer limit of various MX$_2$ semiconductors [@raja2017coulomb], and the emergence of a novel Ising superconductivity in single-layer NbSe$_2$, arising due to the combination of broken inversion symmetry and strong spin-orbit coupling.[@xi2016ising; @bawden2016spin] The group IV and V TMDs, which are the focus of the current work, are perhaps most famous for the charge density wave (CDW) phases which they host. The group-V systems are $d^1$ metals, with large Fermi surfaces which undergo charge-ordering instabilities upon cooling.[@wilson1974charge] NbSe$_2$ and TaSe$_2$ additionally exhibit a superconducting instability at low temperature [@Yokoya2518; @kershaw1967preparation; @revolinsky1965superconductivity], while VSe$_2$ does not. The group IV system TiSe$_2$ also hosts a CDW-like phase [@di1976electronic], despite it being a very narrow-gap semiconductor [@WatsonPRL], and there has been substantial discussion over whether this compound may be considered as a rare realisation of an excitonic insulator.[@wilson1977concerning; @cercellier2007evidence; @kogar2017signatures] There has been substantial debate over how such interacting electronic states and phases evolve when thinned to a single monolayer. The charge-ordering temperatures have been reported to increase, decrease, or even vary non-monotonically with reducing sample thickness.[@doi:10.1063/1.4893027; @Xi2015; @Yu2015; @Yoshidae1500606; @Chen2015; @Ugeda2015; @doi:10.1021/acs.nanolett.8b01649] A robust ferromagnetic phase was reported to occur in the monolayer limit of VSe$_2$ [@Bonilla2018; @xu2013ultrathin], although several recent studies question this conclusion.[@doi:10.1021/acs.nanolett.8b01649; @chen2018unique; @fumega2019absence] To enable reaching a coherent understanding of the evolution of such quantum many-body states in monolayer TMDs, it is essential to develop improved methodologies for their materials growth. To this end, here, we report the fabrication of epitaxial monolayers of NbSe$_2$, VSe$_2$, and TiSe$_2$ using molecular-beam epitaxy (MBE). We investigate the effect of growth temperature, growth rate, and metal:chalcogen flux ratios on the uniformity and morphology of the monolayer films grown. We identify a key role of the transition-metal adatom mobility in dictating the growth dynamics, and through this develop strategies for the optimal growth of large area, high-quality epitaxial monolayers of transition-metal dichalcogenides. Results and discussion ====================== ![image](Fig1) Our approach is summarised in Figure \[Fig1\]. We employ highly-oriented pyrolytic graphite (HOPG) as a substrate throughout, which was cleaved immediately prior to loading into the growth system for each growth (see experimental section). This gives rise to a somewhat spotty (1$\times$1) RHEED pattern (Figure \[Fig1\](c)). Atomic-force microscopy (AFM) measurements (Figure \[Fig1\](e)) indicate a smooth surface, with occasional cleavage steps. Epitaxial TMD monolayers were grown on the cleaved substrate surface by co-evaporation of the transition-metal and chalcogen. The sticking coefficient of Se at the growth temperatures used (300-900 $^\circ$C) is very low compared to the metal species due to the huge differences in vapour pressures and the chemical environment. This necessitates a very high Se to metal flux ratio, which is also crucial in preventing the formation of 3D metal clusters via metal-metal bonding. A recent kinetic Monte Carlo simulation of the growth of WSe$_2$ has estimated the mean dwelling time of a Se adatom on the surface before desorption to be over four orders of magnitude less than that of the metal adatoms.[@Nie_2016] Similarly, the mean diffusion distance of Se as compared to the metal adatoms was two orders of magnitude shorter. During the growths performed for this work, we have therefore maintained a metal:Se flux ratio of at least 1:60, although for most parts of the study, a ratio as high as 1:500 was used. As the growth progresses, the RHEED pattern of the HOPG substrate begins to slowly fade, whilst a new pattern starts to appear which we attribute to the TMD epilayer. Towards the end of the growth, the new features become strong and streaky (Figure \[Fig1\](d)) confirming a flat morphology of the monolayer surface. From the spacing of the TMD RHEED streaks, we can extract a lattice constant for the TiSe$_2$ monolayer shown in Figure \[Fig1\](d) of $3.52\pm0.05$ [Å]{}. This is in excellent agreement with the in-plane bulk lattice constant of TiSe$_2$, despite the nearly 30 $\%$ lattice mismatch with the HOPG substrate. Equivalent results were obtained for VSe$_2$ and NbSe$_2$, confirming that the TMD monolayers are grown without strain and misfit dislocations, facilitated by a relaxed substrate-epilayer interaction at the interface via van-der-Waals epitaxy. Large-area AFM imaging (Figure \[Fig1\](f)) indicates that growth yields a number of islands distributed across the sample surface. Around defects on the substrate (such as grain boundaries between neighbouring lateral domains with random in-plane rotational alignment, which are known to form in HOPG), there are a large number of nucleation sites and inhomogeneous growth is observed. Away from such substrate grain boundaries, there are a lower density of larger islands. In the following, we focus on the growth dynamics which dictate the morphology, size, and structure of these isolated islands, and elucidate the key parameters that can be tuned in order to optimise these. ![image](Fig2) Figure \[Fig2\] shows AFM images of monolayer NbSe$_2$, VSe$_2$, and TiSe$_2$ grown on HOPG at growth temperatures of between 300$^\circ$C and 600$^\circ$C (throughout, we report the growth temperature as the temperature measured by a thermocouple positioned behind the sample plate). With an increasing growth temperature, the sticking coefficient decreases which results in a lower growth rate as evident from the smaller coverage of the epilayer (particularly clear for VSe$_2$). More importantly, changing both growth temperature and the transition metal atom leads to pronounced changes in the morphology of the as-grown monolayer islands. For NbSe$_2$, growth at the lowest temperature studied here leads to randomly-branched growth with very small feature sizes. We refer to this morphology as dendritic. With increasing growth temperature, a somewhat more symmetrical, but still branched, morphology of the growing islands is observed, while at the highest growth temperature ($T_g$) of $600^\circ$C, small triangular islands are formed, with a side length of ca. 50 nm. We note that at around $600^\circ$C growth temperature, NbSe$_2$ was previously reported to undergo a phase transition from the 1H (at low growth temperature) to the 1T (at higher growth temperature) polymorph, as judged from changes in the electronic structure measured using angle-resolved photoemission.[@MottNbSe2] Our own photoemission measurements (Supplementary Fig. S1) indicate that for growths at 500$^\circ$C and below, our samples are purely in the 1H phase, whilst at a growth temperature of 600$^\circ$C, we still have predominantly 1H phase, but with a partial admixture of regions of 1T phase. No clear morphological differences are evident in different regions of the AFM scans shown in the bottom right panel of Figure \[Fig2\], suggesting that the polytype does not have a major impact on the island morphology here, although this remains an interesting topic for future detailed exploration. VSe$_2$ and TiSe$_2$ are both expected to be stable in the 1T polymorph for all growth temperatures studied here. For both of these compounds, randomly branched growth is not observed at the lowest temperatures studied, in contrast to NbSe$_2$. Rather, at a growth temperature of 300$^\circ$C, a symmetrically branched growth mode is obtained. As is particularly clear for VSe$_2$, the growing islands have tree-like morphologies, with additional branching evident on the side of a growing spur, reminiscent of self-similarity. We thus attribute the symmetrically-branched structures as arising from a fractal growth mode. With increasing temperature, a trend towards a triangular growth mode is again observed. For a given growth temperature, the largest island sizes are observed for TiSe$_2$, with the smallest for NbSe$_2$. [@Note1] The transition from branched to a triangular growth mode also occurs at lower growth temperatures for TiSe$_2$ vs. NbSe$_2$. The formation and evolution of these structures can be understood on the basis of varying adatom surface diffusion lengths. At a given growth temperature, the transition-metal mobility is the lowest for Nb atoms, while the Ti atoms are the most mobile. Moreover, higher growth temperatures lead to an increase in thermally promoted adatom surface diffusion, yielding longer surface diffusion lengths of both the metal and chalcogen species. The dendritic and fractal growth modes observed here can thus be understood due to the kinetic limitations of the adatoms at very low temperatures within a simple model of diffusion-limited aggregation. [@PhysRevLett.47.1400] An adatom randomly diffuses on the substrate surface until it comes in contact with an already formed cluster or a nucleation site and sticks at the first point of contact. Once condensed at the edge of an island, edge diffusion is restricted or negligible at lower temperatures and this results in the formation of dendrites.[@PhysRevLett.67.3279] The dendritic growth observed for NbSe$_2$ at $T_g=300^\circ$C can thus be attributed to the extremely low mobility of Nb adatoms at lower temperatures. With increasing growth temperature, thermal excitation of the adatoms enables a moderate edge mobility. Randomly attached adatoms become more mobile and diffuse preferentially towards higher-symmetry bonding sites, enabling the steady coalescence of nucleating islands into morphologically more compact fractals. The mobility and directionality at this stage is still limited, however, and so the transition between the two growth morphologies is subtle. A key diagnostic is the increased symmetry of the fractal mode as compared to the dendritic one, similar to the morphological changes observed in the initial stages of growth of elemental metals on surfaces.[@Brune1994] The transition is evident here with increasing growth temperature above 300$^\circ$C for NbSe$_2$. Dendrites are not, however, formed during VSe$_2$ or TiSe$_2$ growths even at $T_g=300^\circ$C, due to the relatively higher surface diffusion lengths of V and Ti adatoms as compared to Nb ones at that temperature. The growth of compact triangular domains at higher temperatures differs from diffusion-limited aggregation. For the more compact growth, adatoms diffuse to an existing cluster, and then relax to a lower energy site through edge diffusion. As the rate of relaxation increases with respect to the rate of adatom diffusion to the cluster, a stoichiometric transition occurs from fractal growth to the more thermodynamically favourable triangular island growth mode. As the growth progresses, various islands begin to develop from different nucleation sites and they compete for the available adatoms. This naturally explains the steady transition from fractal to triangular domain growth mode evident for all three materials at intermediate/high growth temperatures discussed above. It is also evident, however, that there is a large difference in island size and density between the different compounds. This can again be understood from the varying adatom mobility of the different transition metals at a given growth temperature: In the absence of nucleation sites, an impinging atom diffuses until (within the surface dwell time) it comes in contact with another diffusing adatom which results in the formation of a seed. The mobility and stability of these seeds depend on the growth temperature. As the growth progresses, the number of seeds increases linearly until the the density is comparable to normal adatoms. At this point, island growth competes with any seed formation. At higher growth temperatures, the surface diffusion lengths of adatoms become larger than the mean island separation distances, which results in the adatoms diffusing into existing islands.[@BRUNE1998125] The significantly higher nucleation density present in NbSe$_2$ as compared to both VSe$_2$ and TiSe$_2$ can thus also be attributed to a lower thermally activated diffusion hopping rate of Nb vs. V or Ti adatoms at comparable temperatures. ![image](Fig3) The above results demonstrate the major impact that variations in the adatom surface diffusion length, governed by changing growth temperature and transition-metal atom, have on the morphology of TMD monolayers grown by MBE. Nonetheless, other parameters can also influence the fractal to triangular domain transitions observed above. In the following, we focus on TiSe$_2$ and NbSe$_2$ as these show the extremes of behaviour of transition-metal surface diffusion. Figure \[Fig3\] shows the morphology of TiSe$_2$ and NbSe$_2$ monolayers grown at temperatures between 300$^\circ$C and 500$^\circ$C under two different Se fluxes, corresponding to a Se beam equivalent pressure (BEP) of $\sim2\times10^{-8}$ mbar and $\sim2\times10^{-7}$ mbar, respectively. The AFM scans from the growth in the more Se-rich conditions is reproduced from Figure \[Fig2\] to aid comparison. ![image](Fig4) While qualitatively the same transitions from dendritic to fractal to triangular growth modes are still evident, this evolution is slowed down when growing using the lower Se flux. A noticeable change for the high Se, high temperature growth is the increased domain sizes as compared to growth in a lower Se flux. We stress that even at the lower Se BEP of $\sim2\times10^{-8}$ mbar, this still corresponds to a very Se-rich growth condition with a high metal:Se ratio of $\sim$ 1:60. Nontheless, given the extremely volatile nature of Se, we find that further increase above this value leads to a significant increase in the effective sticking co-efficient, particularly at the highest growth temperatures. Considering the constant metal fluxes used here, it is evident that the excess Se impinging on the surface takes part in bonding with the metal adatoms and by means of edge diffusion forms the energetically favourable triangular domains. The increased surface diffusion lengths at higher temperature enables the formations of larger islands. It is also evident that in the absence of any excess Se, the extra metal adatoms otherwise available for bonding do not form any metallic clusters, possibly due to a combination of lower sticking coefficients and higher adatom mobilities. We also note that even at the lowest temperature we have used, the Se sticking coefficient has a huge dependence on the metal fluxes and hence Se atoms do not take part in the growth in the absence of the metal adatoms. Given the pronounced influence of adatom mobility on the morphology of the synthesised epilayers outlined above, it is of interest to also investigate the influence of growth rate. In the following, we thus fix the growth temperature to 500$^\circ$C and the Se BEP to $\sim\!2\times10^{-7}$ mbar, and vary the impinging transition-metal flux. Figure \[Fig4\](a) shows $1\times1$ $\mu$m$^2$ AFM images of TiSe$_2$ samples for which the Ti effusion cell temperature was varied from 1330$^\circ$C to 1350$^\circ$C. We note that since the transition metal flux is very low (BEP $\sim\!6\times10^{-10}$ mbar), the change in Ti flux is modest for this change in cell temperature, and the error in measuring the BEP is high at these low values, we report simply the cell temperature used here. To compensate the changes in surface coverage due to a varying metal flux, growth times were adjusted accordingly (from 55 minutes for the sample grown with 1350$^\circ$C cell temperature to 90 minutes for the sample growth with 1330$^\circ$C cell temperature) to maintain approximately equivalent coverage for the different growth rates used. Two key features are evident in the case of TiSe$_2$ (Figure \[Fig4\](a)). First, the triangular domains seen on all three AFM images consist of a larger monolayer island with side length varying from $\sim\!0.9 - 1.2$ $\mu$m and a smaller bilayer island on top. The monolayer island is not a perfect triangle, but shows some deformations along its edges. This is most pronounced for the fastest growth rate (highest Ti cell temperature), but similar deformations are evident even for the samples grown more slowly. The growth temperature here ($T_g=500^\circ$C) is at around the temperature at which the transition from fractal to triangular growth was found for TiSe$_2$ in Figure \[Fig2\](a). Our growth-rate dependent studies here suggest that, at around this transition temperature, the growth of a monolayer is initiated by the formation of three fractal islands separated by 120$^\circ$ rotation. These fractal islands are originated from the same nucleation site and as the growth progresses, slowly evolve and merge to form a large triangular island. As seen from Figure \[Fig4\](a), this transformation is highly growth rate dependent: at faster growth rates, there is not enough time for the adatoms to participate in edge diffusion, and the domains remain more fractal, while at slower growth rates, the adatoms have a longer time for edge diffusion, thus facilitating the formation of larger triangular grains. Slow growth rates are thus clearly preferable to generating large triangular islands. These are not isolated monolayers, however, but have a small bilayer region forming at the middle of the island. The bilayer exhibits good epitaxial registry with the underlying monolayer. Interestingly, unlike the monolayer, we find that the bilayer region forms as a near-perfect triangle immediately from the initial stages of growth. This is likely due to the differences in growth kinetics when a layer is grown on a graphite substrate vs. a monolayer substrate of the same kind as the growing epilayer, which thus acts a favourable substrate. A reduction in size of the bilayer is evident with increasing growth rate. We speculate that this reflects the fact that, for the faster growth rates, the three 120$^\circ$ rotated fractal legs of the underlying monolayer have lateral dimensions less than the surface diffusion lengths of adatoms. Thus, adatoms which absorb on the monolayer surface can diffuse to the edge of the monolayer, where they can then participate in edge diffusion of the monolayer itself. Less adatoms thus contribute to forming a bilayer region. In contrast, for the slower growth rates, the monolayer becomes more triangular and its centre becomes further away from any nucleation edges, which in turn ultimately favours nucleation of a second layer atop the monolayer. This suggests that the formation of bilayer patches can be reduced by again increasing the surface diffusion length of the adsorbed adatoms, such that they reach the edge of the growing monolayer island within their surface diffusion time, and thus participate in edge diffusion, resulting in the formation of larger monolayers without bilayer growth. Consistent with this, we note that for TiSe$_2$ growth at a temperature of 600$^\circ$C (Figure \[Fig2\]), for which the adatom mobility is consequently increased as compared to the growths shown in Figure \[Fig4\], no bilayer formation is observed. In fact, these monolayer islands, with edge length of ca. 600 nm, are – to our knowledge – the largest pure monolayers (i.e., without partial bi- or multi-layer coverage) of any TMDs achieved to date via MBE growth. Figure \[Fig4\](b) shows equivalent growth-rate dependent measurements for NbSe$_2$. Here, a larger change in flux from 0.5 to 2.5 nA (as measured by a flux monitor integrated into the electron-beam evaporator used for the evaporation of Nb) was used, with the corresponding growth times changed from 540 to 35 minutes, respectively, in order to maintain approximately equivalent surface coverage. As evident in Figure \[Fig4\](b), such changes in the NbSe$_2$ growth rate have a significant influence on both the onset of nucleation and the sizes of islands. At faster growth rates, there is an increased number of nucleation sites and resulting islands. However, when the growth is slowed down, the nucleation site density decreases as the adatoms have more time to migrate over longer distances, increasing their probability of a subsequent encounter with an existing island. This enhancement in the surface migration length also gives rise to larger monolayer islands. As for TiSe$_2$, there is evidence of some fractal to triangular domain transformation occurring at the lowest growth rates. Nonetheless, there are no clear triangular domains formed for NbSe$_2$ here, as the 500$^\circ$C growth temperature used is still below the temperature where this transition occurs (Figure \[Fig2\]), due to the significantly lower adatom surface diffusion lengths of Nb atoms as compared to Ti. It is clear, however, from the measurements shown in Figure \[Fig4\] that the transition from a fractal to triangular growth mode is not simply a function of the surface adatom mobilities, but can also be strongly modified by the growth rate used, as well as the ratio of metal:chalcogen flux, as shown in Figure \[Fig3\]. ![image](Fig5){width="\textwidth"} From the above, it is thus clear that enhancing adatom mobility and utilising slow growth rates are key to obtaining more compact and thermodynamically-favourable morphological configurations of the epitaxial TMD islands, and for realising true monolayer growth without additional bilayer patches. To explore this further, and to investigate whether other close-packed configurations may be obtained, we have synthesised TiSe$_2$ monolayers using even higher growth temperatures. Figure \[Fig5\] shows AFM images of the resulting TiSe$_2$ monolayers grown at 600, 750, 800, 850 and 900 $^\circ$C. We find that the triangular domains discussed above slowly transform into hexagons, via a gradual truncation of the tips of the original triangular domain with increasing growth temperature. The process starts when increasingly energetic atoms attached to the three corners of a triangle undergo edge diffusion at elevated temperatures. In CVD synthesis of WSe$_2$, a transition from islands of triangular morphology to hexagonal islands was previously observed to be associated with a cross-over from monolayer to multi-layer structures.[@doi:10.1021/acsnano.5b01301] A transition to hexagonal monolayer patches was also reported during the CVD growth of MoS$_2$, where the change in morphology was attributed to the changes in the Mo:S ratio of the precursors.[@doi:10.1021/cm5025662] In contrast, the triangular to hexagonal transition observed in our work can be attributed simply to the increasing adatom mobilities with temperature, and thus reflects the intrinsic stability of the hexagonal morphology of the as-grown layer given high adatom diffusion lengths. Conclusions =========== ![image](Fig6) We summarise our key findings in the schematic phase diagram shown in Figure \[Fig6\]. For low adatom mobilities, an undesirable dendritic growth mode is found. Within the parameter range investigated here, this was only observed for NbSe$_2$, pointing to the additional challenges for TMD growth associated with the low diffusion lengths of the heavier transition-metals, which have lower vapour pressures as compared to the lighter transition-metals. Nonetheless, morphological control for such systems is still possible. With reducing growth rate or increasing surface diffusion lengths promoted via increased growth temperature, or higher intrinsic adatom mobility of different transition metals, the dendritic growth mode transforms into a fractal mode, with tree-like branching morphologies. Within the fractal growth region, there is a clear dependence of the size of the monolayer islands on both growth rate and adatom mobility; smaller fractals are obtained when materials with smaller adatom mobilities are grown at lower temperatures under faster growth rates. The fractals get larger with an increasing adatom mobility and with reducing growth rate. Upon further increasing the adatom diffusion lengths and lowering the growth rate, a more thermodynamically favourable compact triangular domain growth regime can be achieved. The growth conditions in this region further promotes the transformation of neighbouring fractal domains into single triangular islands. Finally a regime where the growth of the most stable and thermodynamically favourable hexagonal domains is be obtained can be found for the highest adatom mobilities. A clear and steady transition region is observed between the triangular and hexagonal growth regimes. Ultimately, our study therefore indicates that, to achieve large monolayer triangular or hexagonal domains, growth should proceed at high substrate temperature to promote surface adatom mobility, and at low growth rate to increase time available for surface diffusion. The high growth temperatures in turn necessitates the use of very high Se overpressures, to compensate surface desorption due to the extremely high vapour pressure of this element. The required growth conditions will vary for a given transition-metal atom used: for NbSe$_2$, growth at an extremely slow rate of ca. 0.05 ML/h was required to obtain domain sizes of ca. 150 nm for a $\sim\!0.5$ ML coverage, while for TiSe$_2$, island sizes of over 1 $\mu$m$^2$ could be achieved for a similar surface coverage at a much faster growth rate of ca. 0.5 ML/h. For TiSe$_2$, via use of the optimised growth conditions as determined here, we were able to achieve the largest monolayer islands of a TMD grown by molecular-beam epitaxy to date. While we studied three specific TMDs here, our conclusions should be generally applicable to the growth of other TMDs using this method. Our study thus paves the way to the synthesis of improved-quality epilayers in challenging systems such as the $4d$ and $5d$ systems, which are of interest, for example, for their optoelectronic properties, strong spin-orbit interactions, and possibilities to stabilise exoitc quantum states.[@Chhowalla2013; @xu2014spin; @law20171t] Moreover, by further extending the parameter range studied here, our results suggest the route to even larger island sizes of the lighter $3d$ systems, which may consequently be able to approach the grain sizes achieved in other monolayer preparation methods such as mechanical exfoliation. Experimental Section ==================== Materials were grown on HOPG substrates using a DCA R450 MBE system. The growth chamber has a base pressure of $\sim$ 1 $\times$ 10$^{-10}$ mbar and a background pressure of $\sim$ 3 $\times$ 10$^{-9}$ mbar during growth. HOPG substrates were chosen for the growth due to their similar crystal symmetry to the TMD epilayer, weak van-der-Waal’s interactions between the substrate surface and epilayer and the thermal stability of HOPG at the highest growth temperatures used for this work. Fresh HOPG surfaces were exfoliated in atmosphere before rapidly transferring into a vacuum load lock. Substrates are first degassed at $\sim$ 200 $^\circ$C in the load lock overnight before transferring to the growth chamber. The quality of the substrate surface was monitored using $in-situ$ reflection high energy electron diffraction (RHEED). Prior to growth, the substrate is further annealed at 600 - 950 $^\circ$C for $\sim$ 20 minutes before cooling to growth temperature, which varied from 300 to 900 $^\circ$C. For transition metal sources, high temperature effusion cells containing 4N pure V, 3N5 pure Ti and an electron-beam evaporator containing 3N5 pure Nb were used. A valved cracker cell was used to generate 5N pure Se flux. The cracker zone of the Se source was maintained at an elevated temperature of 500 $^\circ$C during growth, to generate cracked Se monomers or dimers and to prevent condensation near the valve. During a typical growth, V and Ti fluxes were maintained at $\sim$ 6 $\times$ 10$^{-10}$ mbar beam-equivalent pressure (BEP), which was measured by positioning a retractable beam flux monitoring ion gauge in front of the substrate, just before growth. The Nb flux was measured using a flux monitor built into the e-beam assembly. A Nb flux of 1.5 nA is used for typical growths, unless otherwise specified. For this study we used a varying Se BEP from $\sim$ 1 $\times$ 10$^{-8}$ to $\sim$ 3 $\times$ 10$^{-7}$ mbar. During growth, the sample surfaces were monitored using the RHEED operated at 15 keV. Surface morphology analysis was performed after removing the as-grown sample from vacuum, in atmospheric conditions. A typical sample is exposed to air for $\sim$ 2-4 hours before being scanned. A Bruker Multimode atomic force microscope (AFM) was used to examine the morphology of the epilayers. Samples were scanned in tapping mode using a Si tip. The step height obtained using AFM for the monolayers studied here was in the range of $\sim$ 7.5 $\AA$, which is in very good agreement with the height value of typical transition metal dichalcogenides, where a monolayer is composed of three layers of atoms. Acknowledgments {#acknowledgments .unnumbered} =============== We thank David Jones, Chao Dun Tan, and Georg Haehner for access to the AFM system (funded via an EPSRC equipment grant: EP/L017008/1) used in this work, and for providing experimental support. We gratefully acknowledge support from The Leverhulme Trust (Grant no. RL-2016-006), The Royal Society, and the European Research Council (Grant No. ERC-714193-QUESTDO). K.U. acknowledges EPSRC for PhD studentship support through grant no. EP/L015110/1. [10]{} \[1\][`#1`]{} M. Chhowalla, H. S. Shin, G. Eda, L.-J. Li, K. P. Lo, H. Zhang. Nat. Chem. 2013, 5, 4 263. Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, M. S. Strano. Nat. Nanotechnol. 2012, 7, 11 699. X. Xu, W. Yao, D. Xiao, T. F. Heinz. Nat. Phys. 2014, 10, 5 343. M. S. Bahramy, O. J. Clark, B.-J. Yang, J. Feng, L. Bawden, J. M. Riley, I. Markovic, F. Mazzola, V. Sunko, D. Biswas, S. P. Cooil, M. Jorge, J. W. Wells, M. Leandersson, T. Balasubramanian, J. Fujii, I. Vobornik, J. E. Rault, T. K. Kim, M. Hoesch, K. Okawa, M. Asakawa, T. Sasagawa, T. Eknapakul, W. Meevasana, P. D. C. King. Nat. Mater. 2017, 17 21. K. F. Mak, C. Lee, J. Hone, J. Shan, T. F. Heinz. Phys. Rev. Lett. 2010, 105, 13 136805. A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, F. Wang. Nano Lett. 2010, 10, 4 1271. A. Raja, A. Chaves, J. Yu, G. Arefe, H. M. Hill, A. F. Rigosi, T. C. Berkelbach, P. Nagler, C. Sch[ü]{}ller, T. Korn, C. Nuckolls, J. Hone, L. E. Brus, T. F. Heinz, D. R. Reichman, A. Chernikov. Nat. Commun. 2017, 8 15251. X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forr[ó]{}, J. Shan, K. F. Mak. Nat. Phys. 2016, 12, 2 139. L. Bawden, S. P. Cooil, F. Mazzola, J. M. Riley, L. J. Collins-McIntyre, V. Sunko, K. W. B. Hunvik, M. Leandersson, C. M. Polley, T. Balasubramanian, T. K. Kim, M. Hoesch, J. W. Wells, G. Balakrishnan, M. S. Bahramy, P. D. C. King. Nat. Commun. 2016, 7 11711. J. Wilson, F. Di Salvo, S. Mahajan. Phys. Rev. Lett. 1974, 32, 16 882. T. Yokoya, T. Kiss, A. Chainani, S. Shin, M. Nohara, H. Takagi. Science 2001, 294, 5551 2518. R. Kershaw, M. Vlasse, A. Wold. Inorg. Chem. 1967, 6, 8 1599. E. Revolinsky, G. Spiering, D. Beerntsen. J. Phys. Chem. Solids 1965, 26, 6 1029. F. J. Di Salvo, D. Moncton, J. Waszczak. Phys. Rev. B 1976, 14, 10 4321. M. D. Watson, O. J. Clark, F. Mazzola, I. Markovi ć, V. Sunko, T. K. Kim, K. Rossnagel, P. D. C. King. Phys. Rev. Lett. 2019, 122 076404. J. A. Wilson. Solid State Commun. 1977, 22, 9 551. H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. Despont, M. G. Garnier, H. Beck, P. Aebi, L. Patthey, H. Berger, L. Forró. Phys. Rev. Lett. 2007, 99 146403. A. Kogar, M. S. Rak, S. Vig, A. A. Husain, F. Flicker, Y. I. Joe, L. Venema, G. J. MacDougall, T. C. Chiang, E. Fradkin, J. van Wezel, P. Abbamonte. Science 2017, 358, 6368 1314. J. Yang, W. Wang, Y. Liu, H. Du, W. Ning, G. Zheng, C. Jin, Y. Han, N. Wang, Z. Yang, M. Tian, Y. Zhang. Appl. Phys. Lett. 2014, 105, 6 063109. X. Xi, L. Zhao, Z. Wang, H. Berger, L. Forr[ó]{}, J. Shan, K. F. Mak. Nat. Nanotechnol. 2015, 10 765. Y. Yu, F. Yang, X. F. Lu, Y. J. Yan, Y.-H. Cho, L. Ma, X. Niu, S. Kim, Y.-W. Son, D. Feng, S. Li, S.-W. Cheong, X. H. Chen, Y. Zhang. Nat. Nanotechnol. 2015, 10 270. M. Yoshida, R. Suzuki, Y. Zhang, M. Nakano, Y. Iwasa. Sci. Adv. 2015, 1, 9. P. Chen, Y.-H. Chan, X.-Y. Fang, Y. Zhang, M. Y. Chou, S.-K. Mo, Z. Hussain, A.-V. Fedorov, T.-C. Chiang. Nat. Commun. 2015, 6 8943. M. M. Ugeda, A. J. Bradley, Y. Zhang, S. Onishi, Y. Chen, W. Ruan, C. Ojeda-Aristizabal, H. Ryu, M. T. Edmonds, H.-Z. Tsai, A. Riss, S.-K. Mo, D. Lee, A. Zettl, Z. Hussain, Z.-X. Shen, M. F. Crommie. Nat. Phys. 2015, 12 92. J. Feng, D. Biswas, A. Rajan, M. D. Watson, F. Mazzola, O. J. Clark, K. Underwood, I. Marković, M. McLaren, A. Hunter, D. M. Burn, L. B. Duffy, S. Barua, G. Balakrishnan, F. Bertran, P. Le Fèvre, T. K. Kim, G. van der Laan, T. Hesjedal, P. Wahl, P. D. C. King. Nano Lett. 2018, 18, 7 4493. M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil, R. Das, T. Eggers, H. R. Gutierrez, M.-H. Phan, M. Batzill. Nat. Nanotechnol. 2018, 13, 4 289. K. Xu, P. Chen, X. Li, C. Wu, Y. Guo, J. Zhao, X. Wu, Y. Xie. Angew. Chem. International Edition 2013, 52, 40 10477. P. Chen, W. W. Pai, Y.-H. Chan, V. Madhavan, M.-Y. Chou, S.-K. Mo, A.-V. Fedorov, T.-C. Chiang. Phys. Rev. Lett. 2018, 121, 19 196402. A. O. Fumega, M. Gobbi, P. Dreher, W. Wan, C. Gonz[á]{}lez-Orellana, M. Pe[ñ]{}a-D[í]{}az, C. Rogero, J. Herrero-Mart[í]{}n, P. Gargiani, M. Ilin, M. M. Ugeda, V. Pardo, S. Blanco-Canosa. arXiv preprint arXiv:1907.02034 2019. Y. Nie, C. Liang, K. Zhang, R. Zhao, S. M. Eichfeld, P.-R. Cha, L. Colombo, J. A. Robinson, R. M. Wallace, K. Cho. 2D Mater. 2016, 3, 2 025029. Y. Nakata, K. Sugawara, R. Shimizu, Y. Okada, P. Han, T. Hitosugi, K. Ueno, T. Sato, T. Takahashi. Npg Asia Mater. 2016, 8 e321. The low coverage for VSe$_2$ at the highest growth temperatures here precludes us making a definitive statement at the highest growth temperatures, but generally VSe$_2$ lies between the case for NbSe$_2$ and TiSe$_2$. T. A. Witten, L. M. Sander. Phys. Rev. Lett. 1981, 47 1400. R. Q. Hwang, J. Schröder, C. Günther, R. J. Behm. Phys. Rev. Lett. 1991, 67 3279. H. Brune, C. Romainczyk, H. R[ö]{}der, K. Kern. Nature 1994, 369, 6480 469. H. Brune. Surf. Sci. Rep. 1998, 31, 4 125 . B. Liu, M. Fathi, L. Chen, A. Abbas, Y. Ma, C. Zhou. ACS Nano 2015, 9, 6 6119. S. Wang, Y. Rong, Y. Fan, M. Pacios, H. Bhaskaran, K. He, J. H. Warner. Chem. Mater. 2014, 26, 22 6371. K. Law, P. A. Lee. Proc. Natl. Acad. Sci. 2017, 114, 27 6996.
--- abstract: 'We propose a model based on coupled multiplicative stochastic processes to understand the dynamics of competing species in an ecosystem. This process can be conveniently described by a Fokker-Planck equation. We provide an analytical expression for the marginalized stationary distribution. Our solution is found in excellent agreement with numerical simulations and compares rather well with observational data from tropical forests.' author: - Simone Pigolotti - Alessandro Flammini - Amos Maritan title: \| A Stochastic Model for the Species Abundance Problem\ in an Ecological Community --- Introduction ============ One of the the most widespread quantities employed in Ecology to describe the biodiversity in a given ecosystem is the distribution of species abundance. In operational terms it can be defined as the histogram of the number of species (in a well defined temporal and geographical context) consisting of a generic number of individuals, or, from a more theoretical perspective, as the probability that a generic species is composed by a certain number of individuals. Data collected in different locations suggests that the relative species abundance distributions show a certain degree of similarity [@macarthur]. To elucidate the causes that determine the shapes of these distributions and therefore their similarity is a problem of the uttermost importance and not only of theoretical nature: to understand the motives that influences the relative rarity or commonness of different species can be of great help in determining policies for the conservation of the endangered ones. The first studies on this subject can be dated back to the ’40 and are due to Fisher [@fisher] and Preston [@preston]. Their works were focused on finding distributions that could fit well particular data set in an empirical way. In particular, Preston [@preston] argued that the probability of finding species with a certain number of of individuals $x$ should be lognormal distributed, while Fisher [@fisher] proposed a function of the form $e^{-ax}/x$, with $a<<1$, the so-called Fisher log series. Later, MacArthur [@macarthur] firstly pointed out that similar distributions are found in very different ecosystems, suggesting that the shape of such distributions is to a large extent determined by very basic, general and ecosystem-independent mechanisms. This in turn hinted to the possibility to predict the shape of such distributions with simple and general models, without taking into account too many specific details of the ecosystem under consideration. Several models have been proposed that spoused this view [@hubbel; @sole; @amos]. Many of them restrict to modeling a single ecological community, a collection of similar species that feed on the same pool of resources in a local area. This definition implies that species belonging to the same community interact mainly in a competitive way: in particular, there are no prey-predator relationships among them. The particular case of single ecological community can be framed in the wider context of a neutrality hypothesis. The concept of neutrality was firstly introduced in the framework of a biomolecular evolution theory by Kimura [@kimura], and then extended to other fields of biology. In the words of Hubbell [@hubbel], an ecological theory can be considered neutral when “[...treats organisms in the community as essentially identical in their per capita probabilities of giving birth, dying, migrating and speciating. This neutrality is defined at the individual level, not the species level ...]{}” The question whether there exist ecological communities satisfying this assumption is still rather controversial [@Tilman], therefore it is crucial to understand what are the consequences of this zero order hypothesis [@Harte]. In the context of a neutral hypothesis it is reasonable to describe the number of offspring to which any given individual gives place to as a stochastic variable. As a consequence, the number of individuals in a species at a given time can be regarded as a multiplicative random process. Here we present a model aimed at reproducing the features of species abundance distributions under a minimal set of assumptions: neutrality and the possibility to describe the birth process as a multiplicative random processes. The model translates in a Fokker-Plank equation for the species abundance distribution and is amenable to an analytical treatment. The solutions found are compared with the experimental data we avail of. The shape of these solutions depend on one parameter, and give in the two limiting cases both a lognormal-like curve and the Fisher log series. The paper is organized as follows. In the second section we will present the model and comment the assumptions made. In the third one we will take the continuum time limit of our model, and will provide an analytical solution for the marginalized stationary probability distribution function. In the last two sections we compare our results with the experimental data and comment them. The Model ========= Let us consider an ecological community consisting of a fixed number, $s$, of species. According to MacArthur and Wilson theory of island biogeography [@mcarthur], the number of species in a community approaches a dynamical equilibrium between immigration, speciation and extinction. We assume that we can neglect the fluctuations around this equilibrium value: in our model, when a species go extinct, it is immediately replaced by another one. We also assume that the net effect of the competitive interaction between species in the community is just to keep also the total number of individuals in the community fixed: the resources available are enough to support just $N$ individuals across all the species. This last assumption implies that the populations of the species undergo a zero sum dynamics. This hypothesis is well confirmed by experimental data [@preston; @mcarthur]; at the end of section III we will show that relaxing these constraints does lead to similar conclusions in the large $N$ limit. We introduce the $s$ variables $x_i^t$, representing the population of the $i$-th specie at (discrete) time $t$, with the condition: $$\sum_{i=1}^{s}x_i^t =N \qquad \forall t$$ Let $P(\lambda)$ be the probability that an individual in the community has $\lambda$ offspring during one time step. Here neutrality plays a key role: our assumption implies that $P(\lambda)$ is the same for all individuals. The population of the $i$-th species evolves according to the following equation: $$\label{pois} x_i^{t+1}=N \frac{\sum_{k=1}^{[ x_i^t]}\lambda_{k,i}^t+b}{\sum_{j=1}^s \left( \sum_{k=1}^{[ x_i^t]} \lambda_{k,i}^t+b\right) }$$ where $[\ ]$ means the integer part. We are assuming that the existence of species with a non integer number of individuals is not too drastic. This might lead to round-off problems only for rare species. At each time step (generation) we just sum the number of offspring of every individual belonging to that species, and then add a small quantity $b$. This quantity becomes relevant only for small $x_i$, and this describes the behavior of species near their extinction threshold. We are assuming that the net effect of extinctions, immigration and speciation can be modeled in a simple way with this term, whose effect is to force the $x_i$’s to be greater than zero. Indeed, for $b = 0$, our system admits an absorbing state with only one $x_i$ equal to $N$ and the others equal to $0$, the so-called monodominance [@hubbel]. Notice that species are only coupled through the denominator, that simply preserves the normalization condition. The number of individual of each species will be typically large, so we apply the central limit theorem to the sum of random variables in this equation, obtaining the following model: $$x_i^{t+1}=N \frac{\bar{\lambda}x_i^t+\sigma\sqrt{x_i^t}\xi_i^t+b}{\sum_{j=1}^s \left(\bar{\lambda}x_j^t+\sigma\sqrt{x_j^t}\xi_j^t+b \right)}$$ where $\bar{\lambda}$ and $\sigma$ are the mean value and the r.m.s.d. of the distribution $P(\lambda)$, and the $\xi$’s are uncorrelated gaussian variables with zero mean and unit variance. It is worth noting the relation between our model and the multiplicative process introduced by Kesten in [@kesten]. Kesten studied random multiplicative processes of the form $X_{t+1}=\lambda_t X_t +b_t$, where $X_t$ is the variable and both $\lambda$ and $b$ are random variables. He found that, depending on the mean value of $\lambda$ and on the boundary conditions, one retrieves a lognormal or a power-law regime. Models for ecology and economics based on this kind of processes were proposed by Sornette [@sorn] and Solomon [@solomon]. In our model the number of individuals of different species can be thought as following coupled Kesten-like processes. The coupling is a consequence of the constrain that keeps fixed to $N$ the number of individuals in the community and that is enforced in equation (1) by the factor $N$ and by the denominator. The Continuum Limit =================== In order to obtain some analytical result, we do the continuous time limit of this model, by introducing the time interval $dt$ in the following way: $$\begin{aligned} \lambda & \rightarrow &1 + \lambda dt \nonumber \\ b &\rightarrow & b\ dt \\ \nonumber \sigma &\rightarrow & \sigma\ dt \nonumber\end{aligned}$$ By means of this substitution, our model becomes: $$x_i^{t+dt}=\frac{x_i^t+dt (\bar{\lambda}x_i^t+\sigma\sqrt{x_i^t}\xi_i^t+b)}{1+\frac{dt}{N}\sum_{j=1}^s(\bar{\lambda}x_j^t+\sigma\sqrt{x_j^t}\xi_j^t+b)}$$ Expanding the denominator and using the fact that $\sum_j x_j=N$, we get the Langevin equation: $$\dot{x_i}= f_i(x)+\sum_{j=1}^s B_{ij}(x)\xi_j$$ where: $$\begin{aligned} f_i(x_i) = b(1-\frac{s}{N}x_i) \nonumber \\ B_{ij}(\underline{x}) = (\delta_{ij}-\frac{x_i}{N})\sqrt{x_j} \end{aligned}$$ The Fokker-Planck equation [@gardiner] associated to this Langevin equation is : $$\label{fp} \dot{P}(\underline{x},t)=-\sum_{i=1}^s \partial_i \left[-f_i P(\underline{x},t) + D \sum_j \partial_j (g_{ji}(\underline{x})P(\underline{x},t))\right]$$ with $D=\frac{\sigma^2}{2}$ and: $$g_{ij}(\underline{x})=g_{ji}(\underline{x})=\sum_k B_{ik}B_{jk}=(\delta_{ij}-\frac{x_j}{N})x_i$$ We search for a solution of this equation satisfying detailed balance (i.e. $P^{st}f_i=D\sum_j\partial_j(g_{ij}P^{st})$). Defining the marginalized probability distribution function: $$p(x)=\int_0^\infty \prod_{j \neq i} dx_j P^{st}(\underline{x})$$ we can easily obtain an equation for $p(x)$. $$b\left(1-\frac{sx}{N}\right) p(x) = D \frac{d}{dx}\left[ \left(x-\frac{x^2}{N}\right) p(x)\right]$$ This equation can be easily solved, giving: $$p(x) \propto x^{\beta-1}\left(1-\frac{x}{N}\right)^{\beta(s-\frac{1}{N})-1} \qquad \beta=\frac{b}{D}$$ Notice that this distribution correctly shows the monodominance behavior $\delta(0)$ or $\delta(N)$ in the limit $\beta \rightarrow 0$. Finally, if we fix $\mu=\frac{\beta s}{N}$, in the limit for $N \rightarrow \infty$ we obtain: $$\label{soluzione} p(x)=\frac{\mu^\beta}{\Gamma(\beta)x^{1-\beta}}e^{-\mu x}$$ In fig. 1 we plot simulation of the stationary p.d.f. for various value of the parameter $\beta$, and check the validity of (\[soluzione\]). ![simulation of marginalized stationary p.d.f. for various values of the diffusion coefficient $D$, compared with theoretical curves. For all curves $b=1$, $\bar{\lambda}=1$ $s=100$,$N=10^9$ Curves are binned linearly with binning size $\delta x =10^4$. Notice that as $D$ increases the curve approaches the Fisher log series. ](curves2.eps){width="8.5cm"} Instead of having a system of stochastic differential equation, it is possible to take into account the interaction of a species with the ecosystem in an averaged way. Let us consider the Langevin equation: $$\dot{x}(t) = b+ \bar{\lambda}x - \gamma x + D \sqrt{x} \xi$$ where the parameter $\gamma$ takes into account the effect of competition. In order to have normalizable solutions, we have to require that $\gamma>\bar{\lambda}$. When this condition holds, it is straightforward to show that the stationary p.d.f. satisfying detailed balance is the same as (\[soluzione\]), with $\mu=-(\bar{\lambda}-\gamma)/D$. Notice that in this case, the detailed balance solution is exact; it is also remarkable that the stationary distribution (\[soluzione\]) can be achieved without fixing neither the number of species, nor the number of individuals. Comparison with experimental data ================================= Among the most reliable data on single-trophic species distribution of species abundance are tropical forest census [@condit]. In order to make a coarse graining, a Preston plot is used: data are collected via a logarithmic binning in base 2, and species at the edge between two consecutive binning are equally divided between them. Since we have a continuous probability density, we compared the histogram with the integral over the bins of the distribution with the experimental data, and made a least-square fit of the parameters $\beta$ and $\mu$, plus the normalization. We found a good agreement of our predicted curve with the histogram; in FIG.2 it is shown the comparison between our solution and the lognormal. Notice that the two distributions have the same number of fitted parameter. It would be interesting to compare our distribution with data collected form other kind of ecosystems, and to try to clarify the dependence of our free parameter $\beta$ from ecological quantities like the immigration pressure, the speciation rate and the extinction threshold. Discussion and Perspectives =========================== The model we introduce admits a family of stationary p.d.f. depending on the parameter $\beta$. This parameter fully determines the shape of the distribution: for $\beta << 1$ one recovers the Fisher log series, while for $\beta$ large, one obtains a lognormal-like distribution. As we already pointed out, both these distributions are well known in the population biology literature as possible candidate to be the ‘right’ distributions found in nature. There is some analogy between our model and the Kesten process. Indeed, also the Kesten process admits two different regimes, one lognormal and one with a power law tail. The main differences is that in our case the multiplicative random process is applied to the square root of the variables, rather than to the variable itself. As a consequence, in the Kesten case, the exponent of the power law tail of the stationary distribution is always greater than one, while the small $\beta$ regime of our system is characterized by a power law tail over many decades, with an exponent that is always less than $1$: the cutoff due to the conserved number of individuals ensures the normalization of these long-tailed distributions. It is remarkable that our distribution is the same found in studies made by Kerner in the ’50 [@kerner] on the invariant measure in a system of Lotka-Volterra equations with purely asymmetric couplings. In that works the interactions are only of predator-prey type, and the system is deterministic, while we are considering a stochastic system with purely competitive coupling. The discover of the same distribution in such different models suggests that it might exist some deeper and more general mechanism determining the statistical behavior of ecosystems, regardless of the type of interactions among species. This work is a byproduct of many discussions with J.Banavar and I.Volkov. [99]{} R. MacArthur, PNAS, USA, [43]{}:293 (1957) R. A. Fisher, A. S. Corbet, and C. B. Williams, Journal of Animal Ecology [12]{} 42-58 (1943) F. W. Preston, Ecology [29]{}, 254-283 (1948) S. P. Hubbell, [The Unified Neutral Theory of Biodiversity and Biogeography]{}. Princeton, NJ 2001. Princeton University Press. A. McKane, ,D. Alonso & R. V. Solé, Phys.Rev.E [62]{}, 8466-8484 (2000) I. Volkov, [et al.]{}, Nature [424]{}, 1035-1037. (2003) M. Kimura, [The Neutral Theory of Molecular Evolution]{}. Cambridge Univ. Press, Cambridge UK 1983. D. Tilman, [Plant Strategies and the Dynamics and Structure of Plant Communities]{}. Princeton Univ. Press, Princeton, NJ, (1988). J. Harte, Nature [424]{} 1006-1007.,(2003) R. MacArthur, & E. O. Wilson, [The theory of island biogeography]{}. Princeton, NJ 1967: Princeton University Press. H. Kesten, Acta Math. [131]{}, 207 (1973). D. Sornette, Physica A [250]{}, 295-314.,(1998) S. Solomon, cond-mat/9901250 (1999) C.W. Gardiner, [Handbook of stochastic methods]{}. Springer-Verlag, Berlin 1985. R. Condit, S.P. Hubbell and R.B. Foster, Journal of Tropical Ecology 12: 231-256 (1996). E.H. Kerner, Bull.Math.Biophys. [19]{} 121-46 (1957); Bull.Math.Biophys [21]{} 217-55 (1959).
--- abstract: 'We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.' address: \| Department of Mathematics\ Harvard University\ 1 Oxford St.\ Cambridge, MA 02138 author: - Thomas Koberda title: 'Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification' --- Introduction, motivation and statement of results ================================================= The mapping class group of a surface $\Sigma$ is defined to be the group of self-homeomorphisms up to isotopy, and is usually denoted $\operatorname{Mod}(\Sigma)$. We will often consider $\Sigma$ with a marked point which must be preserved by the isotopies, in which case the associated mapping class group is denoted $\operatorname{Mod}^1(\Sigma)$. A major unsolved problem in the theory of mapping class groups is whether or not they are linear (cf. [@F]). It is known that if $\Sigma$ is closed of genus $2$ then $\operatorname{Mod}(\Sigma)$ is linear by [@BB2]. Some care must be taken when $\Sigma$ has punctures or boundary components in order to give a proper definition of the mapping class group. When $\Sigma$ is an open disk with $n$ marked points (sometimes thought of as punctures), $\operatorname{Mod}(\Sigma)$ is usually defined to be the group of compactly supported homeomorphisms up to compactly supported isotopy and is often denoted by $B_n$, the braid group on $n$ strands. Braid groups are known to be linear by [@B]. There is an analogous problem for $\operatorname{Aut}(F_n)$, the automorphism group of a free group of finite rank. $\operatorname{Aut}(F_n)$ contains a natural copy of $B_n$, but it is known that $\operatorname{Aut}(F_n)$ is not linear by [@FP]. It is therefore natural to look at representations of the mapping class group, especially ones which arise in topological and geometric contexts, and analyze their faithfulness. One obvious candidate is the representation of $\operatorname{Mod}(\Sigma)$ as a group of automorphisms of $H_1(\Sigma,{\mathbb{Z}})$. We obtain a representation $$\operatorname{Mod}(\Sigma)\to Sp_{2g}({\mathbb{Z}})$$ when $\Sigma$ is closed of genus $g$, and otherwise $$\operatorname{Mod}(\Sigma)\to GL_n({\mathbb{Z}})$$ where $n=\operatorname{rk}H_1(\Sigma,{\mathbb{Z}})$. This representation is given by taking a mapping class $\psi$, lifting it to a homeomorphism of $\Sigma$, and then looking at its action on the homology of $\Sigma$. Therefore, one would be justified in writing the representation as $$\operatorname{Mod}(\Sigma)\to \operatorname{Aut}(H_1(\Sigma,{\mathbb{Z}})).$$ The homology representation of the mapping class group carries a great deal of information about mapping classes. For instance, no finite order automorphism is contained in the kernel of the homology representation. This is a consequence of the standard fact from the theory of mapping class groups (see [@FM], for instance) that a finite order automorphism must fix a complex structure on $\Sigma$, and it is standard from algebraic geometry that holomorphic automorphisms of an algebraic curve act nontrivially on the homology of the curve. Unfortunately, the homology representation also forgets a lot of information. It is well-known that the homology representation of the mapping class group has a very large kernel, called the [Torelli group]{} and is often denoted $\mathcal{I}(\Sigma)$. The Torelli group is very complicated – to study it, one often considers a filtration on $\mathcal{I}(\Sigma)$ called the [Johnson filtration]{} (see [@J] and the references therein). The Johnson filtration is defined in terms of the lower central series $\{\gamma_i(\pi_1(\Sigma))\}$. We shall assume that the Johnson filtration lies inside of $\operatorname{Mod}^1(\Sigma)$ in order to avoid technical difficulties involved in giving proper definitions. Recall that $\gamma_0(\pi_1(\Sigma))=\pi_1(\Sigma)$ and $\gamma_i(\pi_1(\Sigma))=[\pi_1(\Sigma),\gamma_{i-1}(\pi_1(\Sigma))]$. The $k^{th}$ term $J_k$ of the Johnson filtration is the kernel of the natural map $$\operatorname{Mod}^1(\Sigma)\to\operatorname{Aut}(\pi_1(\Sigma)/\gamma_k(\pi_1(\Sigma)).$$ Note that $J_1=\mathcal{I}(\Sigma)$. Johnson established that when $\Sigma$ is not a thrice- or four-times-punctured sphere or a once-punctured torus, $J_k$ is nontrivial for all $k$ and does not consist entirely of inner automorphisms. The fact that $$\bigcap_k J_k=\{1\}$$ is a consequence of the fact that $$\bigcap_k\gamma_k(\pi_1(\Sigma))=\{1\},$$ a fact which is originally due to Magnus (see [@MKS] for a classical treatment). The goal of this paper is to show that if one is willing to consider the actions of mapping classes on the homology of finite covers of $\Sigma$, one can recover much of the information that the homological representation forgets. This is where the “asymptotic" part of the title comes from: we see more and more nontrivial mapping classes acting nontrivially on the homology of covers as we look at higher and higher covers of $\Sigma$. Asymptotic properties, especially asymptotic linearity, have been studied by many authors: see [@A], for example. Before we state the main results, we fix some notation. Let $\Sigma=\Sigma_{g,n}$ be a connected hyperbolic type surface with genus $g$ and $n$ punctures. By hyperbolic type, we mean that $\Sigma$ admits a finite volume complete hyperbolic metric, though we will not use any aspects of hyperbolic geometry in the main part of the paper. All surfaces we consider will be orientable but we will not appeal to any fixed orientation. Fix a basepoint $\in\Sigma^0$, the interior of $\Sigma$. Throughout we will denote the fundamental group of $\Sigma$ by $\pi_1(\Sigma,)$, though we will suppress the basepoint in the notation. Recall that the marked mapping class group of $\Sigma$ is defined as $$\operatorname{Mod}^1(\Sigma)=\pi_0(\operatorname{Homeo}^+(\Sigma),),$$ the group of orientation preserving homeomorphisms of $\Sigma$ up to isotopy, along with a distinguished marked fixed during the isotopy and by each homeomorphism. We do not necessarily puncture the surface at the marked point. We identify homotopy classes of curves based at $$ with elements of $\pi_1(\Sigma,)$ and the free homotopy classes of essential closed curves in $\Sigma$ with conjugacy classes in $\pi_1(\Sigma)$. This allows us to identify $\operatorname{Mod}^1(\Sigma)$ with a subgroup of $\operatorname{Aut}(\pi_1(\Sigma))$ and $\operatorname{Mod}(\Sigma)$ with a subgroup of $\operatorname{Out}(\pi_1(\Sigma))$. We perform this identification once and for all, so that the lifts of mapping classes to covers are unambiguous as far as their actions on homology are concerned. A note on terminology: if $\Sigma'\to\Sigma$ is a regular cover and $\pi_1(\Sigma')<\pi_1(\Sigma)$ is a characteristic subgroup, we say that the cover is a [characteristic cover]{}. We call the quotient of a group by a characteristic subgroup a [characteristic quotient]{}. Our first main result is: \[t:nontrivial\] Let $\psi\in\operatorname{Mod}^1(\Sigma)$ and let ${\Gamma}$ be a finite characteristic quotient of $\pi_1(\Sigma)$ and let $\Sigma_{{\Gamma}}$ denote the associated covering space of $\Sigma$. If $1\neq\psi\in\operatorname{Aut}({\Gamma})$ then $\psi$ acts nontrivially on $H_1(\Sigma_{{\Gamma}},{\mathbb{Z}})$. The fundamental group of a surface admits many characteristic quotients. We can take the deck group to be solvable, nilpotent, or even a $p$-group. When we decorate a cover with an adjective such as solvable, nilpotent, etc. we mean that the deck group has this property. The fundamental observation is that if we take a sequence of quotients which exhaust $\pi_1(\Sigma)$, then each automorphism of $\pi_1(\Sigma)$ acts nontrivially on one of the quotients. We immediately obtain: Let $\{\Sigma_i\}$ be a sequence of exhausting, finite characteristic covers of $\Sigma$ and $\psi\in\operatorname{Mod}^1(\Sigma)$. Then $\psi$ induces a nontrivial automorphism of $H_1(\Sigma_i,{\mathbb{Z}})$ for some $i$. In particular, we may assume $\psi$ acts nontrivially on the homology of a solvable, nilpotent or even $p$-cover for any prime $p$. Let $\Sigma'\to\Sigma$ be a characteristic cover with deck group ${\Gamma}$. It is not immediately clear that the action of $\psi$ on $H_1(\Sigma',{\mathbb{Z}})$ will not coincide with the action of some element of ${\Gamma}$. To this end, we have the following extension of Theorem \[t:nontrivial\]: \[t:inner\] Let $\psi\in \operatorname{Mod}^1(\Sigma)$. Suppose that for every finite characteristic cover $\Sigma'\to\Sigma$ the action of $\psi$ on $H_1(\Sigma',{\mathbb{Z}})$ coincides with that of an element of the deck group. Then $\psi$ induces an inner automorphism of $\pi_1(\Sigma)$. Theorem \[t:nontrivial\], remains true for arbitrary automorphisms of $\pi_1(\Sigma)$ regardless of whether or not they are actually induced by homeomorphisms of $\Sigma$. We will generally not distinguish notationally between a homeomorphism and its isotopy class. We will consistently appeal to the fact that the map $\operatorname{Mod}^1(\Sigma)\to\operatorname{Aut}(\pi_1(\Sigma))$ is injective. We remark briefly that if we forget $\in\Sigma$ during isotopies, we get the standard mapping class group $\operatorname{Mod}(\Sigma)$. Since changing the basepoint for an automorphism $\psi$ of $\pi_1(\Sigma)$ is tantamount to replacing $\psi$ by a conjugate, we obtain an injective map $\operatorname{Mod}(\Sigma)\to\operatorname{Out}(\pi_1(\Sigma))$. It is well-known that the map $\operatorname{Mod}^1(\Sigma)\to\operatorname{Mod}(\Sigma)$ does not split (cf. [@Bi]). Theorem \[t:nontrivial\] can be viewed as positive evidence towards a question which McMullen asked the author. McMullen has since answered the question in the strongest negative sense possible in [@Mc]. To state the question properly, we develop some notation. Let $\psi\in\operatorname{Mod}(\Sigma)$ be a pseudo-Anosov mapping class and let $K_{\psi}=K$ be its [geometric dilatation]{}. A reader unfamiliar with pseudo-Anosov homeomorphisms should consult the definitions given below. The dilatation of the pseudo-Anosov map $\psi$ is the exponential of the entropy for the least-entropy representative of $\psi$ in its isotopy class. Consider the collection of $\psi$-invariant finite covers of $\Sigma$, $\{\Sigma'\}\to\Sigma$. We fix a lift of $\psi$ to each $\Sigma'$, and we let $K_H(\Sigma')$ be the [homological dilatation]{} of $\psi$, namely its spectral radius as an automorphism of $H_1(\Sigma',{\mathbb{R}})$. Since $K$ and $K_H$ are can be defined in terms of word growth in groups, it follows that $K$ is constant over this family and both $K_H$ and $K$ are independent of the choice of lift (cf. [@FLP]). McMullen’s question can be stated as: \[q:homdil\] Is $$\sup_{\Sigma'\to\Sigma}K_H(\Sigma')=K?$$ The original motivation for this question was the work of Friedl and Vidussi in [@FV], which shows that the fibering of $3$-manifolds is detected by twisted Alexander polynomials (see [@FK] for more details about the twisted Alexander polynomial). Since twisted Alexander polynomials encode a large amount of data about finite covers, it seems that finite covers of a surface should provide a wealth of information about mapping classes. He was also motivated by the work of Kazhdan in [@Ka], also [@Rh], where it is shown that a hyperbolic metric on a Riemann surface can be recovered from the Jacobians of its finite covers. McMullen proves: Suppose that $\psi$ is a pseudo-Anosov homeomorphism of a surface with dilatation $K$. Then either $K$ is the spectral radius of the action of $\psi$ on a finite cover of $\Sigma$, or there is an $0\leq{\alpha}<1$ such that $$\sup_{\Sigma'\to\Sigma}K_H(\Sigma')={\alpha}K.$$ Even though Question \[q:homdil\] is completely answered, one can still attempt to understand the action of $\operatorname{Mod}^1(\Sigma)$ on the homology of finite covers. We will see later that at least $$\sup_{\Sigma'\to\Sigma}K_H(\Sigma')\leq K,$$ so that the action of a mapping class on the virtual homology of $\Sigma$ is a reasonable lower bound for the value of its dilatation. This fact can also be seen in [@Roy2], but we will develop a theory which is more general and is divorced from the fact that $\psi$ is a homeomorphism of an actual surface. We will prove: \[t:bound\] Let $M$ be a compact manifold equipped with a metric which is compatible with the manifold topology, and let $\psi$ be a $K$–Lipschitz homeomorphism of $M$. Let $K_{H,i}$ denote the homological dilatation of $\psi$ on the $i^{th}$ homology with real coefficients. Then $K_{H,i}\leq K^i$. The standard homology representation of the mapping class group carries a great deal of information regarding the action of homeomorphisms on simple closed curves on $\Sigma$. Recall the [Nielsen-Thurston classification]{} of mapping classes (cf. [@CB], [@FLP], [@FM] for instance). It says that each mapping class $\psi\in\operatorname{Mod}(\Sigma)$ can be classified by its action on the set of simple closed curves on a surface: a mapping class $\psi$ is called [finite order]{} if has finite order in $\operatorname{Mod}(\Sigma)$. It is a nontrivial fact that then it has a representative which is a finite order homeomorphism of $\Sigma$. A mapping class $\psi$ is called [reducible]{} if there is a finite collection $\mathcal{C}$ of disjoint, distinct, nonperipheral isotopy classes of simple closed curves which is preserved by $\psi$. An Euler characteristic argument shows that the order of $\mathcal{C}$ can be bounded in terms of the topology of $\Sigma$. In particular, some power of $\psi$ fixes the isotopy class of a nonperipheral simple closed curve. A mapping class $\psi$ is called [pseudo-Anosov]{} if it has infinite order and is not reducible. One of the main results of this paper is that the homology of all finite covers of $\Sigma$ is sufficiently rich to reveal the Nielsen-Thurston of each mapping class. Before stating the theorems precisely, we will give some more background and setup. The homology representation of $\operatorname{Mod}(\Sigma)$ can also determine whether certain mapping classes are pseudo-Anosov. Recall that there is a well-known [Casson-Bleiler criterion]{} which can certify that certain mapping classes are pseudo-Anosov (see [@CB], also [@Ma]). To use the criterion, one take the image of $\psi$ under the homological representation computes its characteristic polynomial $p_{\psi}(t)$. The criterion asserts: Suppose that $p_{\psi}(t)$ is irreducible over ${\mathbb{Q}}$, none of its zeros are roots of unity, and that that $p_{\psi}(t)$ is not a polynomial in $t^n$ for any $n>1$. Then $\psi$ is a pseudo-Anosov mapping class. We will prove in a precise sense that no naïve generalization of the Casson-Bleiler criterion could possibly hold. The Casson-Bleiler criterion will not detect all pseudo-Anosov homeomorphisms, as it is known that that $\mathcal{I}(\Sigma)$ contains pseudo-Anosov mapping classes. The fact that $\mathcal{I}(\Sigma)$ contains pseudo-Anosov classes is a reflection of a more general fact due to Ivanov (see [@I]): \[p:ivanov\] Let $\Sigma$ be a hyperbolic type surface which is not the thrice-punctured sphere. Suppose $H<\operatorname{Mod}(\Sigma)$ is non-central and normal (the former condition is unnecessary when the genus of $\Sigma$ is greater than $2$). Then every coset of $H$ in $\operatorname{Mod}(\Sigma)$ contains a pseudo-Anosov mapping class. In particular, it follows not only that $\mathcal{I}(\Sigma)$ and each term of the Johnson filtration contains pseudo-Anosov elements, but that one can fix any symplectic matrix and find pseudo-Anosov mapping classes which induce that automorphism on the homology of $\Sigma$. The discussion above shows that the usual homological representation of the mapping class group forgets a lot of data about a mapping class. The point of this paper is that a large part of the lost data can be recovered if we allow ourselves to consider a virtual homological representation. Before we state the remaining main results, we say a few more general things. Let $G$ be a group $\psi\in\operatorname{Aut}(G)$ and let $G'<G$ be $\psi$-invariant. Then $\psi$ restricts to an automorphism of $G'$. When $G'$ is characteristic (in which we call the cover given by the quotient $G\to G/G'$ a characteristic cover), we get a restriction map $\operatorname{Aut}(G)\to\operatorname{Aut}(G')$, and it is well-known that this map is in general neither injective nor surjective. When $G'$ has finite index in $\pi_1(\Sigma)$, the restriction map is injective, at least when restricted to $\operatorname{Mod}^1(\Sigma)$. Indeed, clearly the map is a homeomorphism, and by [@I] it is enough to show that no pseudo-Anosov homeomorphism is contained in the kernel. A pseudo-Anosov homeomorphism stabilizes a measured foliation which lifts to every finite cover (see [@FLP]) and is therefore characterized by its local geometry. This implies that the lift of $\psi$ is pseudo-Anosov and hence nontrivial. Theorem \[t:nontrivial\] and the homological Nielsen-Thurston classification can be conveniently stated in terms of a certain representation. Let $F$ be a field which we assume to have characteristic zero in order to avoid technical difficulties. Let $\Sigma'$ be a finite characteristic cover of $\Sigma$. We suppose that $\Sigma'$ is an element in a class $\mathcal{K}$ of finite characteristic covers of $\Sigma$. We may define the [pro-$\mathcal{K}$ $F$-homology]{} of $\Sigma$ by taking $$H(\Sigma)=\varprojlim H_1(\Sigma',F),$$ as $\Sigma'$ ranges over all elements of $\mathcal{K}$. If $P$ denotes the set of punctures of $\Sigma'$, we can define the relative pro-$\mathcal{K}$ homology of $\Sigma$ by $$H_R(\Sigma)=\varprojlim H_1(\Sigma',P,F).$$ The field of coefficients and $\mathcal{K}$ will generally be apparent from context and we will suppress them from the notation. We will not exploit any abstract properties of $H(\Sigma)$ other than the fact that it is a vector space of infinite dimension when the characteristic of $F$ is zero. In this context, Theorem \[t:nontrivial\] can be restated as saying that if $\mathcal{K}$ exhausts $G$ then $H(\Sigma)$ is a faithful representation of $\operatorname{Aut}(G)$. Note that $H(\Sigma)$ comes equipped with a natural action of ${\widehat{G}}$, the pro-$\mathcal{K}$ completion of $\pi_1(\Sigma)$. It is clear from the definitions that this action is compatible with the action of $\operatorname{Aut}(\pi_1(\Sigma))$. Indeed, if $\Sigma'\to\Sigma$ is a finite regular cover with deck group ${\Gamma}$ then ${\Gamma}$ acts on itself and on $\pi_1(\Sigma')$ by conjugation. The action descends to the abelianization of $\pi_1(\Sigma')$. If $$\Sigma''\to\Sigma'\to\Sigma$$ is a characteristic tower of covers with the total deck group given by ${\Gamma}$, then the conjugation action of ${\Gamma}$ on $\pi_1(\Sigma')$ factors through the quotient map from ${\Gamma}$ to the deck group of $\Sigma'/\Sigma$. Thus, the action of the pro-$\mathcal{K}$ completion of $G$ acts on all $\mathcal{K}$ covers in a way which is compatible with the covering maps. If $\psi$ is an automorphism of $G$, then $\psi$ acts on each $\mathcal{K}$-cover since these covers are all characteristic. The action of ${\widehat{G}}$ is given by conjugation within $\pi_1(\Sigma)$, so the action of $\psi$ is compatible with the action of ${\widehat{G}}$. By reversing the algebraic limit in the definition of $H(\Sigma)$, we may define the [pro-$\mathcal{K}$ $F$-cohomology]{} of $\Sigma$. In general it will be clear if we mean pro-$\mathcal{K}$ homology or cohomology, so we will not distinguish between these notationally. The other main result of this paper is that the representation $H(\Sigma)$ can detect the Nielsen-Thurston classification of mapping classes. Since the Nielsen-Thurston classification is characterized by actions of homeomorphisms on isotopy classes of curves, one way to proceed is to homologically encode closed curves on $\Sigma$ inside of $H(\Sigma)$. We will do this by defining certain vectors of [finite type]{}, whose $\pi_1(\Sigma)$-orbits are in bijective correspondence with free homotopy classes in $\pi_1(\Sigma)$. With this terminology, we can state the next theorem: \[t:nt\] Let $\mathcal{K}$ be any exhausting class of finite solvable characteristic covers of $\Sigma$ and let $H(\Sigma)$ be the pro-$\mathcal{K}$ rational cohomology of $\Sigma$. Let $\psi\in\operatorname{Mod}(\Sigma)$, and choose a lift of $\psi$ to $\operatorname{Mod}^1(\Sigma)$. The action of $\psi$ on $H(\Sigma)$ determines the Nielsen-Thurston classification of $\psi$ as follows: 1. $\psi$ has finite order if and only if some power of $\psi$ preserves the $\pi_1(\Sigma)$-orbits of all finite type vectors in $H(\Sigma)$. 2. $\psi$ is reducible if and only if some power of $\psi$ fixes the $\pi_1(\Sigma)$-orbit of some non-peripheral finite type vector in $H(\Sigma)$. 3. $\psi$ is pseudo-Anosov if and only if the previous two conditions fail. The powers in items $(1)$ and $(2)$ are the order of $\psi$ and a power of $\psi$ which fixes the isotopy class of an essential simple closed curve, respectively. Unfortunately, the finite-type vectors do not give much insight into the action of a mapping class on the homology or cohomology of the cover, since the finite type vectors do not give rise to a subrepresentation of $H(\Sigma)$, but we shall show in a precise sense that they are the only way to encode homotopy classes of loops in $\Sigma$ as vectors in $H(\Sigma)$. There is another homological version of the Nielsen-Thurston classification. To state it, we need some setup which applies to general faithful representations of $\operatorname{Mod}(\Sigma)$: Let ${\Gamma}$ be a characteristic quotient of $\pi_1(\Sigma)$. If $\rho$ is a finite-dimensional representation of ${\Gamma}$ in characteristic zero, or more generally a direct limit of finite-dimensional representations in characteristic zero, then at any finite stage we may decompose the representation as a direct sum of irreducible modules. If $\rho$ is an irreducible representation of ${\Gamma}$ and ${\alpha}\in\operatorname{Aut}({\Gamma})$, we get a new representation of ${\Gamma}$ via $\rho\circ{\alpha}$. To each representation $\rho$ we can associate its character $\chi$. $\operatorname{Aut}({\Gamma})$ acts on the characters of ${\Gamma}$ by precomposition. We say two characters $\chi$ and $\chi'$ are [equivalent]{} on a cyclic subgroup $\langle g\rangle<{\Gamma}$ if $\chi$ and $\chi'$ are equal as functions on $\langle g\rangle$, and inequivalent otherwise. \[t:nt2\] Let $H(\Sigma)$ be the pro-$\mathcal{K}$ complex cohomology, where $\mathcal{K}$ is the class of all characteristic finite covers of $\Sigma$. The action of $\psi\in\operatorname{Mod}(\Sigma)$ on the finite representations of ${\widehat{G}}$ in $H(\Sigma)$ determines the Nielsen-Thurston classification of $\psi$ as follows: 1. $\psi$ is finite order if and only if some power of $\psi$ induces the trivial automorphism of the representations of the deck group for every $\mathcal{K}$–cover of $\Sigma$. 2. $\psi$ is reducible if and only if there is a power of $\psi$ and a $1\neq g\in \pi_1(\Sigma)$ such that on every finite $\mathcal{K}$–cover of $\Sigma$ and every representation of the deck group, $\chi$ and $\chi\circ\psi$ are equivalent on the image of $\langle g\rangle$. 3. $\psi$ is pseudo-Anosov if and only if the previous two conditions fail. Note that even though the conditions use the action of $\operatorname{Mod}^1(\Sigma)$ on certain quotients of $\pi_1(\Sigma)$, the characterizations of the mapping classes does actually exploit the action of automorphisms on the homology of finite covers. This is because every representation of the deck group of a finite cover of $\Sigma$ occurs as a summand of the homology of the cover. The homological Nielsen-Thurston classification uses the fact that certain mapping classes permute the representations of the deck groups in a certain way. We will formulate and prove an analogous result for automorphisms of free groups in Section \[s:free\]. The terminology of the Nielsen-Thurston classification is reminiscent of representation-theoretic terminology, and it is interesting to see what the relationship between topology and representation theory is. What Theorems \[t:nt\] and \[t:nt2\] make precise is that a mapping class is reducible if and only if the associated action of $\psi$ on $H(\Sigma)$ has (more or less) a trivial finite-type subrepresentation or if there exists fixed subrepresentation of every finite characteristic quotient of $\pi_1(\Sigma)$. It is natural to ask whether the following strengthened version of Theorem \[t:nt\] is true: $\psi$ is pseudo-Anosov if and only if there is a finite cover of $\Sigma$ for which $\psi$ together with the deck transformation group act irreducibly on the rational homology. This is not true in a very strong sense: \[t:reducible\] Let $\psi\in\operatorname{Mod}^1(\Sigma)$ and $\Sigma'\to\Sigma$ a finite regular cover with deck group ${\Gamma}$ which is $\psi$-invariant. Suppose that $\psi$ acts irreducibly on $H_1(\Sigma',F)$, where $F$ is a field of characteristic zero. Then ${\Gamma}$ is trivial. Furthermore, there exists a power $n>0$ such that if we replace $\psi$ by $\psi^n$, then there is at least one irreducible $(\psi^n,{\Gamma})$-module for each irreducible ${\Gamma}$-module over $F$. In particular, every pseudo-Anosov homeomorphism acts reducibly on every nontrivial finite cover of $\Sigma$, and any particular infinite order mapping class can be replaced by a power which acts with a prescribed number of invariant subspaces on some finite cover. We remark that Theorem \[t:reducible\] could easily be deduced from the material in [@KS]. Theorem \[t:reducible\] should be thought of as a dramatic failure of a straightforward generalization of the Casson-Bleiler criterion. Acknowledgements ================ The idea for this paper came out of numerous discussions with Curt McMullen, and the author thanks him for his patience and ceaseless help. The author thanks Joan Birman for her patience and help with various incarnations of this paper. The author also thanks Nir Avni, Benson Farb, Stefan Friedl, Eriko Hironaka, Dan Margalit, and Andrew Putman for various helpful conversations. The author especially thanks Aaron Silberstein for extremely inspiring and helpful conversations on this topic. The author thanks Alexandra Pettet and Juan Souto for inspiring the material contained in the Section \[s:free\]. Finally, the author thanks the anonymous referees for the time they spent thoroughly understanding this work, and for their numerous helpful, insightful and thorough comments and suggestions. The author is partially supported by a NSF Graduate Research Fellowship. Generalities on residual finiteness and automorphisms ===================================================== In this section, let $G$ be a finitely generated residually finite group. The following is well-known and originally due to Baumslag (cf. [@LySch]): $\operatorname{Aut}(G)$ is residually finite. If $H<G$ is a finite index subgroup, then $G$ admits a finite index characteristic subgroup contained in $H$. This claim uses the fact that $G$ is finitely generated. If ${\alpha}\in\operatorname{Aut}(G)$ is nontrivial, then there is a $g\in G$ such that $g\neq{\alpha}(g)$. Since $G$ is residually finite, there is a finite index subgroup $H$ of $G$ which does not contain $g$ and ${\alpha}(g)g^{-1}$. There is a finite index characteristic subgroup $H'<H<G$, and ${\alpha}$ descends to an automorphism of $G/H'$. Since $g$ and ${\alpha}(g)g^{-1}$ are both nonidentity elements of $G/H'$, we have that ${\alpha}(g)\neq g$ in $G/H'$. It follows that ${\alpha}$ is nontrivial in a finite quotient of $\operatorname{Aut}(G)$. It is not true that if $G$ is residually finite then $\operatorname{Out}(G)$ is necessarily residually finite. By the work of Wise in [@W], any finitely generated group is a subgroup of $\operatorname{Out}(G)$ for some group $G$. We will need the following observation to prove the homological characterizations of the Nielsen-Thurston classification (see [@G] for a similar argument): Let $G$ be a finitely generated residually finite group. The following three statements are equivalent: 1. $\operatorname{Out}(G)$ is residually finite. 2. The closure of the image of $G$ in $\widehat{\operatorname{Aut}(G)}$ is separated from all non-inner automorphisms. 3. Any non-inner automorphism of $G$ descends to a non-inner automorphism of some finite quotient of $G$. Since $G$ is residually finite, we have that $\operatorname{Aut}(G)$ injects into its profinite completion $\widehat{\operatorname{Aut}(G)}$. Strictly speaking, this is true if $G$ has no center, which is certainly the case when $G$ is a free or surface group. Otherwise, we need to quotient out by the center of $G$ (since the center of $G$ is viewed as a group of trivial inner automorphisms). To say that ${\alpha}$ coincides with an inner automorphism on every characteristic quotient of $G$ is simply to say that ${\alpha}$ is an accumulation point of the image of $G$ under the composition $$G\to\operatorname{Aut}(G)\to\widehat{\operatorname{Aut}(G)}.$$ Thus, ${\alpha}$ cannot be separated from all inner automorphisms on any finite quotient of $G$ if and only if the closure of image of $G$ in $\widehat{\operatorname{Aut}(G)}$ contains ${\alpha}$. Taking the topological quotient of $\widehat{\operatorname{Aut}(G)}$ by the closure of the image of $G$ gives a Hausdorff space $X$. It is evident that $X$ is the profinite completion of $\operatorname{Out}(G)$ with respect to a cofinal sequence of finite index subgroups. $\operatorname{Out}(G)$ is residually finite if and only if $\operatorname{Out}(G)$ injects into its profinite completion, and this happens only if the profinite topology on $\operatorname{Out}(G)$ is Hausdorff. Note that cosets of finite index subgroups of $\operatorname{Aut}(G)$ form a basis for the profinite topology on $\operatorname{Aut}(G)$. Therefore ${\alpha}$ will be separated from the identity in the profinite completion of $\operatorname{Out}(G)$ if and only if some coset of a finite index subgroup of $\operatorname{Aut}(G)$ separates ${\alpha}$ from the image of $G$. The claim follows. It is well-known that both $\operatorname{Out}(F_n)$ and $\operatorname{Mod}(\Sigma)$ are residually finite (see [@G], also [@FM]). The fact that surface and free groups are residually finite can be found in [@LySch], for instance. Let $\psi$ be an automorphism of a surface group or a free group $G$. Then there is a finite characteristic quotient ${\Gamma}$ of $G$ such that $\psi\in\operatorname{Aut}({\Gamma})$ does not coincide with any inner automorphism of ${\Gamma}$. Let $G$ be a group which is residually $\mathcal{K}$. We say that $G$ is [conjugacy separable]{} with respect to the class of $\mathcal{K}$–groups if for every pair $g,h\in G$ of representatives from different conjugacy classes, there is a $\mathcal{K}$–quotient $K_{g,h}$ of $G$ such that the images of $g$ and $h$ are not conjugate. We will need the following well-known result: Let $G$ be a free or a surface group. Then $G$ is conjugacy separable with respect to the class of finite $p$-groups. A characteristic zero asymptotically faithful representation of $\operatorname{Mod}^1(\Sigma)$ and highly reducible actions on virtual homology =============================================================================================================================================== In this section, we shall prove Theorem \[t:nontrivial\]. There are several proofs, and we will restrict ourselves to the simplest. \[l:cw\] Let $G$ be a free group of finite rank or a finitely generated surface group, and let ${\Gamma}$ be a finite quotient of $G$ with kernel $K$. Then ${\Gamma}$ acts faithfully on $H_1(K,{\mathbb{Z}})$. Let $X,Y$ be a $K(G,1)$ and a $K(K,1)$ respectively. If $\gamma\in{\Gamma}$ is nontrivial, then $\gamma$ acts fixed-point freely on $Y$, so the Lefschetz number of $\gamma$ must be zero. Let $n=\|{\Gamma}\|$. Homology in degrees zero and two is easy to describe, and $\gamma$ acts trivially on them. In this way, we obtain the character $\chi$ of the representation of ${\Gamma}$ on $H_1(Y,{\mathbb{Z}})$: $\chi(1)=2n(g-1)+2$ or $n(d-1)+1$, depending on whether $X$ is homotopy equivalent to a surface of genus $g$ or a wedge of $d$ circles, and $\chi({\Gamma})=2$ or $1$ respectively when ${\Gamma}\neq 1$. Since characters uniquely determine the isomorphism class of a representation, it follows that the representation of ${\Gamma}$ consists of $(2g-2)$ copies of the regular representation and two copies of the trivial representation (respectively $(d-1)$ and one). The proof of Lemma \[l:cw\] is stronger than what we need for Theorem \[t:nontrivial\], but we will use this extra information later. Lemma \[l:cw\] also follows from the fact that ${\Gamma}$ is a finite group of automorphisms and hence ${\Gamma}$ acts nontrivially on the homology of the corresponding cover (cf. [@FM], for instance). The previous lemma can be seen as a generalization of the following result which was the original motivation for the proof of Theorem \[t:nontrivial\]: \[l:nontrivial\] Let $\gamma\in G$. Then there is a finite $p$-group quotient of $G$ with kernel $K$ such that $0\neq [\gamma]\in H_1(K,{\mathbb{Z}})$. Suppose $\gamma$ is homologically trivial. Then there is a finite $p$-group quotient $P$ of $G$ such that $1\neq \gamma\in Z(P)$. Consider $\Sigma_{P/Z(P)}$, the cover of the base surface corresponding to $P/Z(P)$. Since $\pi_1(\Sigma_{P/Z(P)})$ admits $Z(P)$ as a quotient, it follows that $\gamma$ is nontrivial in $\pi_1(\Sigma_{P/Z(P)})^{ab}$, whence the claim. If $c\subset\Sigma$ is a simple closed curve, one can explicitly produce covers where the homology class of $c$ is nontrivial. If $c$ is nonseparating, then its homology class is already nontrivial. Therefore we may assume that $c$ separates $\Sigma$, and $c$ is determined up to a homeomorphism of $\Sigma$ by the splitting on $H_1(\Sigma,{\mathbb{Z}})$ it determines. Suppose that $\Sigma$ is closed. Clearly we may arrange $\Sigma$ so that its “holes" are linearly ordered, and $c$ lies somewhere between the first and the last hole. Take two simple closed curves which travel “through the hole", one for each of the fist and last hole. There is a finite cover $\Sigma'\to\Sigma$ given by counting the sum of the algebraic intersection numbers with each of these curves modulo $2$. It is easy to verify that $c$ has two distinct lifts to $\Sigma'$, and that each of them is nonseparating. It is trivial to generalize this construction so that the covering has degree $n$ for any positive integer $n$. A similar construction holds when $\Sigma$ is not closed and has a finite set $P$ of punctures. The modification we need to perform is to do intersection theory in $H_1(\Sigma,P,{\mathbb{Z}})$ and use modular algebraic intersection numbers with cycles in $H_1(\Sigma,P,{\mathbb{Z}})$ to find the desired covers. Lemma \[l:cw\] shows that if $K<\pi_1(\Sigma)$ is a finite index normal subgroup and $\rho$ is an irreducible representation of ${\Gamma}=\pi_1(\Sigma)/K$ over a field $F$ of characteristic zero of dimension $n_{\rho}$, then $\rho$ occurs in the representation of ${\Gamma}$ on $H_1(K,F)$ with multiplicity $(2g-2)n_{\rho}$ when $G$ is not free and $\rho$ is nontrivial, with multiplicity $(d-1)n_{\rho}$ when $G$ is free and $\rho$ is nontrivial, and with multiplicity $\operatorname{rk}G$ when $\rho$ is trivial. We can use this observation to prove that in general, the action of a mapping class on the homology of a finite cover is highly reducible: Notice that there is a finite index subgroup of $H_1(\Sigma,{\mathbb{Z}})$, a basis for which is given by curves which lift to $\Sigma'$. We can assume that these curves are Poincaré dual to a basis for the pullback of $H^1(\Sigma,{\mathbb{Z}})$. Tensoring with $F$, we get a subspace of $H_1(\Sigma',F)$ which is isomorphic to the pullback of $H^1(\Sigma,F)$ by duality. This is evidently a proper subspace and is invariant under $\psi$ and ${\Gamma}$. The second of these claims follows since $H_1(\Sigma',F)$ decomposes as a direct sum of irreducible modules corresponding to irreducible characters $\chi$ of $\rho$: $$H_1(\Sigma,F)\cong\bigoplus_{\chi}V_{\chi}.$$ The identified subspace which is dual to the pullback of the cohomology of the base corresponds to the trivial representation. The action of $\psi$ twists the representations of ${\Gamma}$, so that $\rho_{\psi}(\gamma):=\rho(\psi(\gamma))$ may not be equal to $\rho$. If $\rho$ is trivial, then the two are obviously equal. It follows that the isotypic component of trivial character is $\psi$-invariant, and this is the dual of the pullback of the cohomology of the base. Since $\psi$ is an automorphism of the finite group ${\Gamma}$, some power of $\psi$ acts trivially on ${\Gamma}$ and hence preserves the isotypic components corresponding to all irreducible characters of ${\Gamma}$. The second claim follows. The method of the previous proof is soft and applies to groups in general. It seems that whenever $\psi$ is an automorphism of a group with at least some finite index $\psi$-invariant subgroups, then the action of $\psi$ on the homology of these subgroups will be highly reducible. Let ${\Gamma}$ be as in the statement of the theorem and suppose the contrary, so that $\psi(d)=d$ for all $d\in H_1(\Sigma_{{\Gamma}},{\mathbb{Z}})$. Choose $d$ which rationally generates a regular representation, which we can assume to be an integral class by replacing it with a multiple if necessary. If $\gamma\in{\Gamma}$, we have $$\gamma\cdot d=\psi(\gamma\cdot d)=\psi(\gamma)\cdot d,$$ which is a contradiction since $d$ generates a regular representation. An immediate corollary of the proof of Theorem \[t:nontrivial\] is the following general statement: Let $G$ be a residually finite group and suppose that each $g\in G$ acts nontrivially by conjugation on the homology of some finite index subgroup $G'_g$ of $G$. Then each $\psi\in\operatorname{Aut}(G)$ acts nontrivially on the homology of some finite index subgroup $G'_{\psi}$ of $G$. After some conversations with the author about early drafts of this paper, Stefan Friedl independently found a proof of Theorem \[t:nontrivial\] which is essentially the same. The various corollaries to Theorem \[t:nontrivial\] follow from the fact that $G$ is residually solvable, nilpotent, $p$, etc. For certain mapping classes one can explicitly exhibit finite covers of $\Sigma$ to which these mapping classes lift and act nontrivially on the integral homology of the cover. We can do this explicitly for any Dehn twist, and the idea is identical to the lifting of separating curves to separating ones: Let $p$ be a prime and $c\subset\Sigma$ a simple closed curve on $\Sigma$. Let $T_c$ denote a Dehn twist about $c$. Then there is a degree $p$ cover $\Sigma_p$ of $\Sigma$ to which $T_c$ lifts and acts nontrivially on $H_1(\Sigma_p,{\mathbb{Z}})$ when $p\neq 2$, and a degree $4$ cover with this property otherwise. We may evidently assume that $c$ is separating. Construct a cover $\Sigma_p$ to which $c$ lifts to $p$ nonseparating curves. It is clear that when $p>2$ that the simultaneous lifted Dehn twist about these curves acts nontrivially on the homology of $\Sigma_p$. This can be easily seen by constructing a nonseparating curve which intersects only $2$ of the lifts of $c$. When $p=2$, $\Sigma_p$ will not do the trick since then $T_c$ lifts to a twist about a bounding pair, which is still an element of the Torelli group. By taking a $4$-fold cover, we find $T_c$ lifts out of the Torelli group. It is conceivable that the methods of the previous proposition could be used to give another more topological proof of Theorem \[t:nontrivial\], but there seem to be many technical difficulties arising due to twists about single curves lifting to twists about multicurves. We can easily see the following, however: Let $\psi\in\operatorname{Mod}(\Sigma)$ be a Dehn twist about a simple closed curve, or more generally a product of Dehn twists about a multicurve in $\Sigma$. Then $\psi$ acts with infinite order on the homology of a finite cover of $\Sigma$. We may assume that the cover is a $p$-cover. In particular, $\psi$ does not lift to an inner automorphism on a finite cover. Recall that if $1\neq\psi\in\operatorname{Mod}(\Sigma)$ (or $\in\operatorname{Out}(F_n)$), then we can lift $\psi$ to an automorphism of $\pi_1(\Sigma)$ (or $F_n)$ and find a finite quotient ${\Gamma}$ of $\pi_1(\Sigma)$ ($F_n$) on which $\psi$ acts by a non-inner automorphism. We thus obtain the extension of Theorem \[t:nontrivial\] which we mentioned in the introduction: Each homeomorphism of $\Sigma$ which is not isotopic to the identity acts nontrivially on the homology of some finite cover $\Sigma'$ of $\Sigma$. Furthermore, we may assume that the action does not coincide with the action of any element of the deck group of the covering. There is a finite characteristic quotient ${\Gamma}$ of $\pi_1(\Sigma)$ which has the property that $\psi(g)$ is not conjugate to $g$ for any $g\in{\Gamma}$. It follows that there is a $d\in H_1(\Sigma,{\mathbb{Z}})$ such that $\psi(g\cdot d)\neq (h^{-1}gh)\cdot d$ for any $h\in{\Gamma}$. The exact same argument holds for non-inner automorphisms of free groups. Encoding closed curves on $\Sigma$ in $H(\Sigma)$ {#s:encode} ================================================= In this section we describe two natural ways of taking an essential closed curve $\gamma\subset\Sigma$ and producing a unique piece of data in $H(\Sigma)$. In this sense, we shall encode the set of curves in $H(\Sigma)$. The first method produces so-called [finite type vectors]{}. We will associate to each element $\gamma$ of $\pi_1(\Sigma)$ a unique vector $v_{\gamma}\in H(\Sigma)$. Since elements of $\pi_1(\Sigma)$ are in bijective correspondence with based homotopy classes of loops in $\Sigma$, conjugacy classes of elements of $\pi_1(\Sigma)$ will correspond to free homotopy classes of loops in $\Sigma$. The set of finite type vectors in $H(\Sigma)$ will have a natural action of $\pi_1(\Sigma)$, and as such that $\pi_1(\Sigma)$–orbits of finite type vectors will be in bijective correspondence with free homotopy classes of essential closed curves in $\Sigma$. This method associates to each based curve a vector, but unfortunately the span of the finite type vectors is not a subrepresentation of $H(\Sigma)$ in any natural way. The second method is more representation-theoretic. To each finite characteristic cover $\Sigma'\to\Sigma$ with deck group ${\Gamma}$ we will consider the ${\mathbb{C}}{\Gamma}$–module $H(\Sigma',{\mathbb{C}})$. Each irreducible representation of ${\Gamma}$ occurs as a direct summand of $H(\Sigma',{\mathbb{C}})$. Let $\gamma\subset\Sigma$ be an essential closed curve. On $\Sigma'$, there are two possibilities. Either $\gamma$ lifts to a union of closed curves or it does not. The first case occurs exactly when we represent $\gamma$ as a based homotopy class of loops and have that $\gamma$ is contained in the kernel of the map $\pi_1(\Sigma)\to{\Gamma}$. When $\gamma$ lifts to a curve, then each lift of $\gamma$ represents a homology class. Now suppose that $\gamma$ does not lift. Representing $\gamma$ by a based loop, we get an action of $\gamma$ on each irreducible representation of ${\Gamma}$ in $H_1(\Sigma',{\mathbb{C}})$ or $H^1(\Sigma',{\mathbb{C}})$. Taking the trace of the action, we get a complex number which does not depend on the choice of lift. Thus to a based homotopy class of loops $\gamma$ we associate the values of $\chi(\gamma)$, where $\chi$ ranges over all irreducible characters of characteristic quotients of $\pi_1(\Sigma)$. We will now be more explicit about these constructions. Let $1\neq \gamma\in \pi_1(\Sigma)$ and suppose that ${\Gamma}$ is an abelian quotient of $\pi_1(\Sigma)$. Assume furthermore that $\gamma$ is non-peripheral. We can write $\gamma=[\gamma]c$, where $[\gamma]$ is the homology class of $\gamma$ and $c\in [\pi_1(\Sigma),\pi_1(\Sigma)]$. We identify $[\gamma]$ with a cohomology class which we also call $[\gamma]$ via intersection number (taken relative to the punctures if $\Sigma$ is not closed). Now consider $H^1(\Sigma',{\mathbb{Q}})$. We see that algebraic intersection number with $c$ identifies $c$ with a cohomology class $[c]$ in $H^1(\Sigma',{\mathbb{Q}})$, which projects trivially onto the pullback of the cohomology of $\Sigma$. The finite type vector in $H^1(\Sigma',{\mathbb{Q}})$ should be the vector $$[\gamma]+[c],$$ where we have written $$H^1(\Sigma',{\mathbb{Q}})=H^1(\Sigma,{\mathbb{Q}})\oplus V$$ for some $V$. Without making some choices, this expression is not unique. We shall soon make precise the choices that need to be made to make the expression canonical. We are now in a position to define the map $\iota$ for the profinite solvable rational cohomology of $\Sigma$ when $\Sigma$ is closed. When $\Sigma$ is not closed, we need to consider cohomology relative to the punctures. We will associate to $g\in\pi_1(\Sigma)$ a vector $v_g\in H(\Sigma)$. The strategy is to take an element $g\in\pi_1(\Sigma)$ and to look at its “components" in each $S_i=\pi_1(\Sigma)/D_i(\pi_1(\Sigma))$, the universal $i$-step solvable quotients of $\pi_1(\Sigma)$. Here $D_i(\pi_1(\Sigma))$ is the $i^{th}$ term of the derived series of $\pi_1(\Sigma)$: $$D_0(\pi_1(\Sigma))=\pi_1(\Sigma),$$ $$D_n(\pi_1(\Sigma))=[D_{n-1}(\pi_1(\Sigma)),D_{n-1}(\pi_1(\Sigma))].$$ Comparing the components of $g$ in $S_i$ and $S_{i+1}$ gives us an element in $$D_i(\pi_1(\Sigma))/D_{i+1}(\pi_1(\Sigma)),$$ and therefore a homology class of $D_i(\pi_1(\Sigma))$. First, choose a set of coset representatives for every finite solvable quotient of $\pi_1(\Sigma)$. It is possible to arrange the choice compatibly, so that if $T_{{\Gamma}}$ is a transversal for ${\Gamma}$ and ${\Gamma}'$ is a quotient of ${\Gamma}$, then $T_{{\Gamma}'}$ is identified with a subset of $T_{{\Gamma}}$. We thus obtain a set of coset representatives for $D_i(\pi_1(\Sigma))$ for all $i$ as well, since $D_i(\pi_1(\Sigma))$ is the intersection of all finite index subgroups of $\pi_1(\Sigma)$ which contain $D_i(\pi_1(\Sigma))$. An example of such a choice of coset representatives is to take a basis for $D_i(\pi_1(\Sigma))/D_{i+1}(\pi_1(\Sigma))$ for all $i$, which we then pull back to $\pi_1(\Sigma)$ in some way. For each finite quotient $F$ of $D_i(\pi_1(\Sigma))/D_{i+1}(\pi_1(\Sigma))$, simply choose elements of $D_i(\pi_1(\Sigma))/D_{i+1}(\pi_1(\Sigma))$ which map onto each $f\in F$. Let ${\Gamma}$ be a finite solvable quotient of $\pi_1(\Sigma)$. We filter ${\Gamma}$ by its derived series to get $${\Gamma}={\Gamma}_0>{\Gamma}_1>\cdots>{\Gamma}_n=\{1\}.$$ Each quotient $Q_i={\Gamma}/{\Gamma}_i$ of ${\Gamma}$ gives us a cover $\Sigma_{Q_i}$ of $\Sigma$, and we obtain a tower of covers $$\Sigma_{{\Gamma}}\to \Sigma_{Q_{n-1}}\to\cdots\to\Sigma_{Q_1}\to\Sigma,$$ where each of the successive deck groups is abelian. The rational cohomology of $\Sigma_{Q_{i+1}}$ consists of the pullback of $H^1(\Sigma_{Q_i},{\mathbb{Q}})$, together with some new cohomology equipped with a nontrivial action of the abelian group ${\Gamma}_i/{\Gamma}_{i+1}$. Let $g\in\pi_1(\Sigma)$. The homology class of $g$ is $[\gamma]$, which we represent by a pre-chosen coset representative $\gamma$ of $\pi_1(\Sigma)$. The choice of $\gamma$ is canonical after a choice of coset representatives. Then $c_1=\gamma^{-1}g\in D_1(\pi_1(\Sigma))$, the first term of the derived series of $\pi_1(\Sigma)$. Suppose $c_1$ is nontrivial in ${\Gamma}$. Then there is a smallest $i$ such that $c_1$ becomes a homology class in $H_1(\Sigma_{Q_i},{\mathbb{Q}})$, and we record the Poincaré dual of this homology class as the entry of $\iota(g)$ corresponding to the cover $\Sigma_{Q_i}\to\Sigma$. In general, suppose that in $\pi_1(\Sigma)/D_{n+1}(\Sigma)$ we have a representation of the image $g=\gamma\cdot c_1\cdots c_n$, where each $c_i\in D_i(\pi_1(\Sigma))$. Again, the $c_i$ are canonical after a choice of coset representatives. If ${\Gamma}$ is any finite $n$-step solvable quotient of $\pi_1(\Sigma)$, we define the entries of $\iota(g)$ for all the intermediate covers coming from quotients of ${\Gamma}$ as follows. We take such a quotient $Q$, look at the longest terminal segment $c_i\cdots c_n$ which is trivial in $Q$, and record the homology class of $c_i\cdots c_n$ in $H_1(\Sigma,{\mathbb{Q}})$. Dualizing, we get a cohomology class, and thus a definition of $\iota$. When $\Sigma$ has a finite set $P$ of punctures, we need to look at homology and cohomology relative to the punctures to be able to get a duality isomorphism. This is because Poincaré duality gives an isomorphism $H_1(\Sigma,{\mathbb{Z}})\to H^1(\Sigma,P,{\mathbb{Z}})$. To summarize, $\iota$ is a map $\pi_1(\Sigma)\to H(\Sigma)$. $H(\Sigma)$ can be thought of as infinite tuples whose entries are classes in $H^1(\Sigma',{\mathbb{Q}})$ for some characteristic solvable cover $\Sigma'\to \Sigma$ with deck group ${\Gamma}$. We assume that $\iota$ has been defined for every characteristic cover lying between $\Sigma'$ and $\Sigma$. Let $g\in\pi_1(\Sigma)$. Since ${\Gamma}$ is solvable, we have specified a unique coset representative for $g\in{\Gamma}$, so that $g=t\cdot c$, where $c\in\pi_1(\Sigma')$. The entry of $\iota(g)$ corresponding to $\Sigma'$ is the cohomology class given by algebraic intersection number with $c$. Finally, we call a finite-type vector $v_g$ [peripheral]{} if $g$ is in the free homotopy class of a small loop about a puncture of $\Sigma$. The vector $v_g$ is non-peripheral if it is not peripheral. The map $\iota$ is well-defined. This is an immediate consequence of the fact that we chose a compatible set of coset representatives for all finite solvable quotients of $\pi_1(\Sigma)$. The definition of $\iota$ shows that the image of $\pi_1(\Sigma)$ under $\iota$ is invariant under the action of ${\widehat{G}}$, the profinite completion of $\pi_1(\Sigma)$. \[l:injective\] The map $\iota:\pi_1(\Sigma)\to H(\Sigma)$ is injective. Since $H(\Sigma)$ is a vector space, $\iota$ cannot a homomorphism if it is to be injective, so it is not sufficient to show that no $\gamma\in \pi_1(\Sigma)$ is trivial under $\iota$. It is clear, however that if $1\neq\gamma\in \pi_1(\Sigma)$, then $\iota(\gamma)$ is nontrivial in light of Lemma \[l:nontrivial\]. Let $\gamma_1,\gamma_2$ be distinct homotopy classes of curves in $\Sigma$. It follows that for some $i$, the expansions $([\gamma_1],c_{1,1},c_{2,1},\ldots,c_{i,1})$ and $([\gamma_2],c_{1,2},c_{2,2},\ldots,c_{i,2})$ for $\gamma_1$ and $\gamma_2$ in $\pi_1(\Sigma)/D_{i+1}(\pi_1(\Sigma))$ must differ because $\pi_1(\Sigma)$ is residually solvable. Since $\pi_1(\Sigma)$ is residually $p$-by-torsion-free abelian, there is a finite $p$-cover of $\Sigma$ such that the cohomology classes dual to $c_{i,1}$ and $c_{i,2}$ are different, so that $\iota(\gamma_1)\neq\iota(\gamma_2)$. We call the image of $\pi_1(\Sigma)$ the finite type vectors in $H(\Sigma)$. As we remarked in the introduction, free homotopy classes in $\Sigma$ are in bijective correspondence with conjugacy classes in $\pi_1(\Sigma)$, and hence with $\pi_1(\Sigma)$–orbits of finite type vectors in $H(\Sigma)$. For the sake of clarity we remark that $\pi_1(\Sigma)$ acts by conjugation on the cohomology of each finite cover of $\Sigma$, and we represent finite type vectors as sums of vectors coming from various representations of the deck group of each cover. Since these representations are clearly $\pi_1(\Sigma)$–invariant, we see that the conjugation action of $\pi_1(\Sigma)$ on itself and $H(\Sigma)$ commutes with the operation of taking finite type vectors. To make $\psi\in\operatorname{Mod}(\Sigma)$ act on the finite type vectors, we start with $\gamma\subset\Sigma$ an essential closed curve and its image $\psi(\gamma)$. To both of these curves we have associated $\pi_1(\Sigma)$–orbits of finite type vectors $v_{\gamma}$ and $v_{\psi(\gamma)}$. The action of $\psi$ on finite type vectors should take the orbit of $v_{\gamma}$ to the orbit of $v_{\psi(\gamma)}$. Let us now make a few remarks about the second method. Since $\pi_1(\Sigma)$ is residually $\mathcal{K}$, where $\mathcal{K}$ is the class of finite, solvable, nilpotent, $p$-groups, etc., we have that each $\gamma\in\pi_1(\Sigma)$ is nontrivial in a $\mathcal{K}$–quotient of $\pi_1(\Sigma)$. In particular for each $\gamma\in\pi_1(\Sigma)$, there is a $\mathcal{K}$–quotient ${\Gamma}$ of $\pi_1(\Sigma)$ and an irreducible character $\chi$ of ${\Gamma}$ such that $\chi(\gamma)\neq 0$. Furthermore, the irreducible characters of $\gamma$ over ${\mathbb{C}}$ span the vector space of class functions on $\gamma$, so that if $\gamma$ and $\gamma'$ are not conjugate in ${\Gamma}$ then there is an irreducible character which separates them. The Nielsen-Thurston classification of mapping classes and homology {#s:nt} =================================================================== We are now in a position to give the proofs of the two Nielsen-Thurston classifications. Let $\psi\in\operatorname{Mod}(\Sigma)$ have finite order. Lifting $\psi$ to $\operatorname{Aut}(\pi_1(\Sigma))$, some power of $\psi$ is an inner automorphism of $\pi_1(\Sigma)$. Since we have a bijection between free homotopy classes of curves and $\pi_1(\Sigma)$-orbits of finite type vectors, the characterization of finite order mapping classes is immediate. This can also be seen from the fact that surface groups are conjugacy separable with respect to the class of finite solvable quotients, so that if $g_1,g_2\in\pi_1(\Sigma)$ are not conjugate then they will be non-conjugate in a finite solvable quotient of $\pi_1(\Sigma)$ (in fact the strongest possible form of this statement is true: they will be non-conjugate in a finite $p$-group quotient of $\pi_1(\Sigma)$). Reducible mapping classes clearly preserve the $\pi_1(\Sigma)$–orbit of a finite type nonperipheral vector. Suppose conversely that the conjugacy class of a finite type vector is preserved by $\psi$, but that $\psi$ is pseudo-Anosov. Then $\psi$ preserves the conjugacy class of a based homotopy class of curves in $\Sigma$. Lift $\psi$ to $\operatorname{Aut}(\pi_1(\Sigma))$. On the level of elements of $\pi_1(\Sigma)$, if $1\neq g\in \pi_1(\Sigma)$, we have $\ell(\psi^n_(g))\sim K^n$, where $K$ is the pseudo-Anosov dilatation of $\psi$ and $\ell$ denotes the word length in $\pi_1(\Sigma)$ (see [@FLP]). The idea behind those asymptotics is the fact that there is a natural metric $\ell_{\mu}$ on $\Sigma$ coming from the invariant foliations of $\psi$ if $\psi$ is pseudo-Anosov. It turns out that this metric is equivalent to the hyperbolic metric $\ell_h$, or precisely that for any nontrivial class of curves $\gamma$, there exist positive constants $c$ and $C$ such that $$c\leq \frac{\ell_h(\gamma)}{\ell_{\mu}(\gamma)}\leq C.$$ This means that the length of the shortest word representing $\psi^n(g)$ within its conjugacy class grows exponentially. It follows in our case that $\psi$ cannot be pseudo-Anosov. We are now ready for the second version of the Nielsen-Thurston classification. Let $\psi$ be a reducible mapping class which fixes the conjugacy class of an element $c$ in $\pi_1(\Sigma)$. Then lifting $\psi$ to an automorphism of $\pi_1(\Sigma)$, we may assume that $\psi$ fixes $c$. Note that if $\psi$ is inner then $\psi$ preserves all of the representations of the deck group on any cover. It follows that the action of $\operatorname{Mod}^1(\Sigma)$ on the representations of the deck group of each cover descends to an action of $\operatorname{Mod}(\Sigma)$. If the conjugacy class of $c\in \pi_1(\Sigma)$ is invariant under $\psi$, then we see from the proof of Theorem \[t:nt\] that $\psi$ is reducible. If $\psi(c)$ is conjugate to $c$, then for each character of the deck group ${\Gamma}$, we have that $\chi(\psi(c))=\chi(c)$. Conversely, if $\psi$ does not preserve the conjugacy class of $c$ then this becomes visible on some cover of $\Sigma$ since $\pi_1(\Sigma)$ is conjugacy-separable. In particular, there will be a finite characteristic quotient ${\Gamma}$ of $\pi_1(\Sigma)$ such that the images of $c$ and $\psi(c)$ are not conjugate. Since the irreducible characters over ${\mathbb{C}}$ form a basis for the class functions of ${\Gamma}$ and since every irreducible representation of ${\Gamma}$ occurs as a summand of $H_1(\Sigma_{{\Gamma}},{\mathbb{C}})$, we see that there is a character of $\chi$ such that $\chi(c)$ and $\chi(\psi(c))$ do not coincide. The characterization of finite-order mapping classes is clear. Free groups and detecting the classification of free group automorphisms {#s:free} ======================================================================== Analogously to the Nielsen-Thurston classification, there is a classification of free group automorphisms that can be described using the geometry of Outer space. An (outer) automorphism $\phi$ of the free group on $n$ generators $F_n$ is called [finite order]{} if it has finite order in $\operatorname{Aut}(F_n)$ ($\operatorname{Out}(F_n)$). Recall that $\operatorname{Out}(F_n)$ and $\operatorname{Aut}(F_n)$ act on Outer and Auter space respectively, which are defined as simplicial complexes that parametrize isometry classes of graphs and isometry classes of graphs with basepoint respectively. An automorphism or outer automorphism $\phi$ has finite order if and only if it fixes a point in Auter or Outer space, respectively. The analogue of a reducible mapping class is a [reducible]{} automorphism, which is defined as one which fixes a subgraph of some representative graph ${\Gamma}$ satisfying $\pi_1({\Gamma})=F_n$, with the requirement that the subgraph not be a forest. An automorphism $\phi$ is called [irreducible]{} if it is not reducible. For an accessible introduction to the classification, see [@Be]. By analogy to the construction of $H(\Sigma)$, we may take an exhausting inverse system $\mathcal{K}$ of finite $p$-power index subgroups of $F_n$, abelianize the kernels simultaneously, tensor with ${\mathbb{C}}$ and take the inverse limit. Let us call the resulting vector space $H(F_n)$. We have: The action of $Aut(F_n)$ on $H(F_n)$ is faithful. The main result of this section is the following: \[t:main\] The representation $H(F_n)$ with complex coefficients detects irreducible automorphisms and finite order automorphisms. We will need to appeal to the following characterization of reducible automorphisms which can be found in [@BH]: \[l:bh\] Let $\phi\in Out(F_n)$. Then $\phi$ is reducible if and only if there are free factors $F_{n_i}$, $1\leq i\leq k$, $n_1<n$, such that $F_{n_1}\cdots F_{n_k}$ is a free factor of $F_n$ and $\phi$ cyclically permutes the conjugacy classes of the $F_{n_i}$’s. Let $\psi$ be a reducible automorphism of $F$. By Lemma \[l:bh\], we may assume that there is a free factor decomposition $AB$ of $F$ such that the conjugacy class of $A$ is preserved by $\psi$. If $a\in A$ is any particular element, we may lift $\psi$ to an automorphism of $F$ which sends $a$ to $A$. If $\psi$ is irreducible, then for any candidate free decomposition of $F=AB$, we can find an $a\in A$ such that $\psi(a)$ is not conjugate to an element of $A$. The next lemma shows the finite index subgroups of $F_n$ detect the failure of a free splitting of $F_n$ to be preserved by an automorphism. Write $F_n=AB$. Suppose $x\in F_n$ is not conjugate to any element of $A$. Then for every $a\in A$ there is a finite quotient of $F_n$ such that image of $x$ is not conjugate to $a$. We may assume that this quotient is a $p$-group for any prime $p$. This follows from the conjugacy separability of the free group (see [@LySch]). The free group is conjugacy separable with respect to the class of finite $p$-groups, whence the second claim. Let $\psi$ be irreducible and $AB$ any candidate splitting. Lift $\psi$ to $\operatorname{Aut}(F_n)$. Let $F'<F_n$ be any characteristic subgroup with deck group ${\Gamma}$. As before (cf. Lemma \[l:cw\]), we have that ${\Gamma}$ acts on $H_1(F',{\mathbb{Q}})$, and that each irreducible representation of ${\Gamma}$ occurs as a direct summand. Fix $a\in A$ such that $\psi(a)$ is not conjugate to any element of $A$. For each $a'\in A$, there is a characteristic $p$-group quotient ${\Gamma}$ of $F_n$ and an irreducible character $\chi$ of ${\Gamma}$ such that $\chi(\psi(a))$ and $\chi(a')$ do not coincide (we are appealing strongly to the fact that we are doing representation theory over ${\mathbb{C}}$). It follows that the representations of the deck groups of all finite $p$-group quotients of $F_n$ on the complex homology of finite index covers witnesses the fact that $\psi$ does not preserve $AB$. If $\psi$ has finite order, then replacing $\psi$ by a power and lifting to $\operatorname{Aut}(F_n)$ shows that for each finite quotient ${\Gamma}$ of $F_n$, each $x\in {\Gamma}$, and each irreducible character $\chi$ of ${\Gamma}$, $\chi(\psi(x))=\chi(x)$. It follows that $\psi$ must be inner. Using representations to approximate $K_{\psi}$ =============================================== Though we have obtained a representation-theoretic characterization of the Nielsen-Thurston classification and a faithful representation of the mapping class group, the objects in this paper are difficult to work with. Understanding the finite nilpotent covers of a thrice-punctured sphere is no easier than understanding all two-generated finite nilpotent groups. Therefore, even the simplest examples present a lot of difficulty in the setup we consider here. We can say a few things, though. For instance, consider the braid groups $B_n$, identified with the the mapping class groups of $n$-times punctured disks. There is a natural homological representation to consider here: the Burau representation. Indeed, we have a homomorphism from $F_n\to{\mathbb{Z}}$ that takes a word in a fixed standard generating set for $F_n$ to its exponent sum. This homomorphism gives rise to a covering space $X$ called the [Burau cover]{}, and by taking values in ${\mathbb{Z}}/m{\mathbb{Z}}$ we get the [Burau cover modulo $m$]{}. The modulo $m$ Burau covers together form the finite Burau covers. These obviously correspond to covers where small loops about the punctures are all unwound the same number (namely $m$) of times. The braid group acts on $F_n$ preserving this homomorphism. We thus get a representation of the braid group on the covering corresponding to the kernel of this homomorphism, which is the classical Burau representation, $V_n$. It is well known that $V_3$ is faithful and that $V_n$ is not faithful for $n\geq 5$ (see [@B2] and the references therein). The moment $V_n$ is not faithful, there are pseudo-Anosov mapping classes whose nontriviality is not detected by the Burau representation (cf. [@I]), much less their dilatations. Conversely, a representation that contains no pseudo-Anosov classes in its kernel is faithful. Since the kernel and image of the homomorphism are $B_n$ invariant, we see that if $X$ is the cover of $\Sigma$ corresponding to the kernel of the homomorphism, $B_n$ acts on $H_1(X,{\mathbb{Q}})$ by ${\mathbb{Z}}[t^{\pm1}]$–linear maps. We can set the parameter to be a root of unity and thus obtain a representation of the braid group on the homology of the multiply punctured disk with twisted coefficients. Such homology groups arise naturally from covering spaces. Letting $t$ be a primitive $n^{th}$ root of unity gives rise to the twisted homology given by the Burau cover modulo $n$. It is easy to see the following proposition, whose phrasing was first suggested to the author by McMullen: \[t:bsup\] Let $\psi\in B_n$ be a pseudo-Anosov braid and $S^1$ denote the unit complex numbers. Then $\sup_{t\in S^1}\rho(V_n(\psi))\leq K$, and the supremum represents finite cyclic covers of $\Sigma=\Sigma_{0,n,1}$ that have equal branching over each of the punctures. The first claim in the proposition follows from general principles about Lipschitz maps acting on metric manifolds which we stated above as Theorem \[t:bound\] together with the observation that there is a metric for which a pseudo-Anosov map with dilatation $K$ is $K$–Lipschitz. Though multiply punctured surfaces are not compact, there are compact manifolds with boundary which are deformation retracts of multiply punctured surfaces, and thus Theorem \[t:bound\] holds for surfaces with punctures by applying an appropriate limiting process. Precisely, we can remove ${\epsilon}/2$–neighborhoods of the punctures to get a compact manifold with boundary, and we can find a new homeomorphism on the truncated surface which coincides with a $K$–Lipschitz homeomorphism $f$ outside of an ${\epsilon}$-neighborhood of the punctures. We then let ${\epsilon}\to 0$. It will be clear from the argument below that even considering surfaces with punctures will not introduce significant technical obstacles. The proof proceeds by producing a bound for simplicial chains, and estimating the growth of the action of the homeomorphism $f^n$ on the vector space of real simplicial chains. We can then estimate the spectral radius $\rho$ of the action of $f$ on the chains and then deduce an estimate of the spectral radius of the induced action on the real homology. Choose a simplicial decomposition of $X$ with a very fine simplices, so that any point in $X$ is $\ll1/K$ from the barycenter of a simplex. Then, if $\gamma$ is a loop in $X$, we will be able to homotope it to a path lying in the $1$-skeleton of $X$ without increasing the length by more than some constant factor $C$ that will work for all of $X$. Choose the subdivision also so that all the $1$–simplices have approximately the same size. This can be done as follows: consider the length spectrum of all $1$–simplices for some decomposition. This is a finite set since $X$ is compact. Consider any two positive real numbers $s,t$. For any ${\epsilon}>0$, there exist integers $m,n$ such that $\|s/m-t/n\|<{\epsilon}$. By an easy induction, for any ${\epsilon}$ and any finite collection of positive real numbers, we can find a sequence of integers such that the quotients differ pairwise by no more than ${\epsilon}$. Therefore, given any ${\epsilon}$, we can subdivide the $1$-skeleton of our simplicial complex so that the lengths of any two $1$-simplices differ by no more than ${\epsilon}$. Now let $z$ be a $1$–cycle. Writing $z$ as a weighted union of $1$-simplices, we may talk about the length $\ell$ of $z$. Let $f_$ denote the linear map on real simplicial chains induced by $f$, and consider $f^n_(z)$ and $f^n(z)$. Since $f$ is $K$–Lipschitz, the length of $f^n(z)$ is no more than $K^n\cdot\ell$. On the other hand we can homotope $f^n(z)$ to the $1$–skeleton, thus increasing its length to no more than $C\cdot K^n\cdot\ell$. Choosing ${\epsilon}$ small enough, we see that if $z$ required $m$ $1$–simplices to be expressed as a $1$-chain, $f^n_(z)$ requires no more than $C/(1-{\epsilon})\cdot K^n\cdot m$ simplices. Let $G$ be a finitely generated group, and $g:G\to H$ a surjective homomorphism. There is a well-defined length function on $G$ given by the graph metric on a fixed Cayley graph for $G$, and it induces a length function on $H$. Furthermore, it is clear that the induced length function is bounded by the length function on $\ell_G$, i.e. $\ell_G(\gamma)\geq\ell_H(g(\gamma))$ for all $\gamma\in G$. Since homology is a quotient of a subgroup of the $1$–chains, we obtain $\rho_1(f_)\leq K$. The proof in general is analogous. The metric gives us a way to measure the volume of simplices: the volume of a very small $m$-cube is given by the $m^{th}$ power of the side lengths. Therefore, $f$ will scale the volume element by no more than $K^m$. As before, we can cut up $m$-simplices so that their volumes are all similar. So, if $z$ is an $m$-cycle, we can homotope $f^n(z)$ to sit in the $m$-skeleton of $X$, increasing its volume by no more than a factor of some constant $C$. The constant $C$ can be estimated as follows: if an $m$-chain $c$ intersects the interior of an $m'$-simplex $S$ with $m'>m$, then we perform a homotopy $rel\,\partial S$ to push $c$ to the $m'-1$-skeleton $S\cap X_{m'-1}$ and proceed inductively. Subdividing the interior of $S$ if necessary, we can assume the homotopy does not change the volume of $c\cap S$ by much. The subdivision of the $m$-skeleton into simplices of similar size can be done afterwards without altering the validity of the proof. This proves the claim. The second claim in proposition \[t:bsup\], though intuitively clear, needs some argument, and a complete proof can be found in [@BB]. Note that there is a two-sheeted covering of the thrice punctured disk that gives a thrice punctured torus with one boundary component. Furthermore, any simple pole of any quadratic differential is resolved, so that we get a quadratic differential on a once-punctured torus. To see that poles of quadratic differentials can generally be resolved to be regular points on a finite cover we have the following result: \[l:monogon\] Suppose that $\psi$ stabilizes a Teichmüller geodesic determined by a quadratic differential $q=q(z)\, dz^2$ which has only simple poles and even order zeros. Then there is a finite unbranched cover $\Sigma'$ of $\Sigma$ such that the lift of $q$ has only even order zeros. In particular, there exists a finite unbranched cover $\Sigma'\to\Sigma$ such that for any lift of $\psi$ to $\Sigma'$, the homological and geometric dilatations of $\psi$ coincide. Let $P$ be the set of punctures of $\Sigma$. At each point in $P$ we may assume the stable foliation $\mathcal{F}$ determined by $q$ has either an even-pronged singularity or a one-pronged singularity. In the latter case, we may simply fill in the missing point since the quadratic differential associated to $\mathcal{F}$ has a zero at that point (so that the puncture is a removable singularity). Therefore we may assume that all the points in $P$ are poles of the associated quadratic differential. By passing to a characteristic cover of $\Sigma$, we may assume $\|P\|$ is even. Such a cover might be given by taking the cover associated to $H_1(\Sigma,{\mathbb{Z}}/2{\mathbb{Z}})$ if $\|P\|=1$. If $\|P\|>1$, label the punctures $p_1,\ldots,p_n$. Take a small loop about each puncture and record its homology class. Send the homology classes of the loops about $p_1,\ldots,p_{n-1}$ to $1\in{\mathbb{Z}}/2{\mathbb{Z}}$. Since the sum of the the homology classes of these loops must be zero, if $n$ is even we may send the homology class of the loop about $p_n$ to $1\in{\mathbb{Z}}/2{\mathbb{Z}}$. Otherwise, we are forced to send it to $0\in{\mathbb{Z}}/2{\mathbb{Z}}$. Since the first cover is characteristic and since punctures of $\Sigma$ are $\psi$-invariant, $\psi$ lifts to the covers determined by these homomorphisms. After taking this cover, it is obvious that all but possibly one puncture in $P$ will lift to a regular point of the foliation. Since the cover has even degree, the one puncture over which the cover did not “branch" lifts to an even number of punctures. Repeating the second cover construction, we construct a further unbranched cover (which can clearly be refined to a $\psi$-invariant or even characteristic cover, since these punctures are poles of the quadratic differential associated to $\mathcal{F}$) where a small loop about each puncture is unwound to half a loop about one of the punctures in the cover. Once we have unwound each puncture as above, each pole of the quadratic differential becomes a regular point and hence can be filled in as a removable singularity. It follows that after passing to another finite cover if necessary, $q$ becomes the global square of a holomorphic $1$-form $\omega$, whence the claim. We may view $B_3/Z(B_3)$ (where the center is generated by a twist about the boundary of the disk) as a subgroup of the mapping class group of the once-punctured torus. The homological representation theory of this group is well-understood, especially in connection to the Nielsen-Thurston classification. The situation is already much more complicated for the four-times punctured disk. Let $\{\sigma_1,\ldots,\sigma_{n-1}\}$ denote the standard generators of the braid group $B_n$. Then $\beta=\sigma_1\sigma_2\sigma_3^{-1}\in B_4$ is pseudo-Anosov (see [@HK] for a large class of examples in this same flavor.) According to Hironaka and Kin the dilatation of $\beta$ is the largest root of the polynomial $$1-t-2t^2-2t^3-t^4+t^5,$$ which is approximately $2.29663$. Applying the machinery of proposition \[t:bsup\], we get that the Burau matrix $V_4(\beta)$ is $$M= \begin{pmatrix} -t&-t^2&-t^2\\1&0&0\\0&1&1-1/t \end{pmatrix}.$$ The characteristic polynomial of $M$ is $$t+u-tu+t^2u-u^2+u^2/t+tu^2+u^3.$$ The supremum of the spectral radii of $M=M(t)$ as $t$ varies over $S^1$ is approximately $2.17401$. By choosing a small mesh size, the estimate will be close to the actual value. It is possible to show that the inequality between the supremum of the spectral radii of $M(t)$ and $K$ is strict in this case by showing that $2.17401$ is already very close to the actual Burau supremum. It is a theorem of Band and Boyland in [@BB] that the spectral radius of a Burau matrix specialized at a root of unity is either equal to the dilatation when $t=-1$ or is strictly smaller than the dilatation. Our computation shows that the difference between the spectral radii and the dilatation can be bounded away from zero. The question of whether $K$ is achieved if the covers are allowed to vary over all covers of $\Sigma$ is a consequence of [@Mc]. The strict inequality of Proposition \[t:bsup\] does not change if we pass to the Lawrence-Krammer representation, which is a well-known faithful representation of the braid group. For more detail, see for instance [@B]. Recall that the configuration space of pairs of points in a space $X$ is the set $$(X\times X\setminus\Delta)/({\mathbb{Z}}/2{\mathbb{Z}})),$$ where the group action permutes the coordinates. In the case of a disk punctured at $p_1,\ldots,p_n$, the configuration space $C$ is a $4$-manifold and inherits a natural action of the braid group. The representation itself is the action of $B_n$ on the second (usual) homology of a certain ${\mathbb{Z}}^2$-cover of $C$, viewed as a ${\mathbb{Z}}[t^{\pm 1},q^{\pm 1}]$–module. We now give a verbatim account of the setup in which Bigelow works in [@B]. If ${\alpha}$ is a path in $C$, we may view ${\alpha}$ as $\{{\alpha}_1,{\alpha}_2\}$, where we mean unordered pairs of points. Let $$a=\frac{1}{2\pi i}\sum_{j=1}^n\left(\int_{{\alpha}_1}\frac{dz}{z-p_j}+\int_{{\alpha}_2}\frac{dz}{z-p_j}\right)$$ and $$b=\frac{1}{\pi i}\int_{{\alpha}_1-{\alpha}_2}\frac{dz}{z}.$$ These quantities are $B_n$-invariant, so that $B_n$ acts on $H_2(Z,{\mathbb{Z}})$ as a ${\mathbb{Z}}[t^{\pm 1},q^{\pm 1}]$–module, where $Z$ is the covering space corresponding to the map ${\alpha}\mapsto q^at^b$. Bigelow provides explicit matrices for the corresponding representation, which allows for relatively simple computation of the supremum of the homological dilatations over finite intermediate covers between $C$ and $Z$ (the proof of this is again analogous to Proposition \[t:bsup\]). For the Hironaka-Kin example, we obtain a $6\times 6$ matrix which we do not reproduce here. Once we obtain the supremum, we must take the square root, since the action is on second homology. Theorem \[t:bound\] applies for the Lawrence-Krammer representation, since the action of a homeomorphism on the covering space is inherited from the action on the coordinates. If the action comes from a pseudo-Anosov homeomorphism with dilatation $K$, those actions can be taken to be $K$–Lipschitz. Theorem \[t:bound\] guarantees that this estimate will not exceed the dilatation. We obtain the supremum $2.17433<2.29663$. Questions and examples {#s:examples} ====================== A basic question which the preceding discussion leaves open is the following: Let $\psi$ be an infinite order mapping class. Does $\psi$ act with infinite order on the (co)homology of some finite cover? If $\psi$ has positive entropy, can $\psi$ be made to act with spectral radius greater than one? We can formulate analogous questions to this one and Question \[q:homdil\] for outer automorphisms of free groups. The first part of this question is really only open for an automorphisms of surfaces which are generalized pseudo-Anosov, in the following sense. Given $\psi\in\operatorname{Mod}(\Sigma)$, find a canonical reduction system $\mathcal{C}$ for $\psi$. Then on every component of $\Sigma\setminus\mathcal{C}$, $\psi$ either acts trivially (up to isotopy) or as a pseudo-Anosov homeomorphism, possibly after passing to a power of $\psi$. If there is a $\mathcal{C}'\subset \mathcal{C}$ such that $\psi$ does a combination of Dehn twists about $\mathcal{C}'$, then we have already shown that $\psi$ will act with infinite order on the homology of a finite cover. By a Dehn twist about $c\in\mathcal{C}'$, we mean that in $\Sigma\setminus\mathcal{C}$, $c$ forms two boundary components in $\Sigma\setminus\mathcal{C}$, and we require that the corresponding components of $\Sigma\setminus\mathcal{C}$ have trivial restriction of $\psi$. Here we are using that $\mathcal{C}$ is indeed a canonical reduction system. Note that $\psi$ has to have infinite order in $\operatorname{Mod}(\Sigma)$, as no inner automorphism can act with infinite order on the homology of a finite cover. If the answer to this question or the analogous one for free groups is true, then $\psi$ will act with infinite order on $V_{\chi}^{2g-2}\subset H^1(\Sigma_{{\Gamma}},{\mathbb{Q}})$ (resp. on the homology of some finite index subgroup of $F_n$ with quotient ${\Gamma}$), where $V_{\chi}$ is an irreducible representation of ${\Gamma}$ and $V_{\chi}^{2g-2}$ is its isotypic component. A rather dramatic example of this fact is the following observation: Recall that $\mathcal{I}(\Sigma)$ admits the Johnson filtration. It can be easily shown that any element of $J_1$ acts trivially on $\gamma_k(G)/\gamma_{k+1}(G)$ for all $k$ (cf. [@BL]). Let $\psi\in J_k\setminus J_{k+1}$. It follows that there is an element $g\in G$ and $z\in\gamma_k(G)/\gamma_{k+1}(G)$ such that $\psi(g)=gz$ in $G/\gamma_{k+1}(G)$. For all $n\neq 0$, we have $\psi(g^n)=g^nz^n$ in that quotient. Now let $\Sigma'$ be a finite cover of $\Sigma$ where $z$ lifts to a nontrivial homology class and let $n$ be such that $g^n$ lifts to a homology class. Consider $d=g^n$ as a homology class in $\Sigma'$, and suppose that $\psi(g)=g\cdot z$. It follows that $\psi(d)$ is equal to $d+ z'$, where $z'$ is the sum of the conjugates of $z$ by all powers of $g$, where $g$ is viewed as a deck transformation of the covering. The representation of the subgroup $\langle g\rangle$ where the homology class of $z$ is located can be assumed to be a trivial representation if and only if there is a finite quotient ${\Gamma}'$ of $G$ such that both $g$ and $z$ are in the kernel of the quotient map, and if both $g$ and $z$ are nontrivial in the abelianization of the kernel. If the conjugation action is nontrivial, then the sum of the conjugates of $z$ is zero. Indeed, tensoring with ${\mathbb{C}}$ shows that $g^k$ acts by $\zeta_n^k$, where $\zeta_n$ is an $n^{th}$ root of unity, whence the claim. Finally, one would like to produce a more useful homological version of the Nielsen-Thurston classification. Recall that there is the so-called Casson-Bleiler criterion for a map to be pseudo-Anosov (see [@CB]). We have seen that a naïve generalization of the Casson-Bleiler criterion cannot hold by Theorem \[t:reducible\]. Is there a characterization of pseudo-Anosov maps which is homologically visible on some finite cover? Some $p$-power cover? Some remarks on the uniqueness of finite type vectors in $H(\Sigma)$ ==================================================================== Though it is disappointing that finite type vectors in $H(\Sigma)$ do not form a subrepresentation, we will now show that they are about as good a construction as one can expect. If we want to encode homotopy classes of curves as vectors in $H(\Sigma)$, there are certain natural conditions we should put on the image of each homotopy class. Imposing just a few relatively mild assumptions greatly restricts the possible ways to encode $\pi_1(\Sigma)$. Let $\iota:\pi_1(\Sigma)\to H(\Sigma)$ be an injective map. Suppose that $\iota$ satisfies the following two conditions: 1. Let $\Sigma'\to \Sigma$ be one of the finite covers occurring in the construction of $H(\Sigma)$, and let ${\Gamma}$ be the deck group of the cover. If $g\in\pi_1(\Sigma)$ is in the kernel of $\pi_1(\Sigma)\to{\Gamma}$, the entry of $\iota(g)$ corresponding to $\Sigma'$ is the image of $g$ in $H_1(\Sigma',{\mathbb{Z}})$. 2. Let $\Sigma'\to\Sigma$ be a finite solvable cover. If $g_1,g_2\in \pi_1(\Sigma')$ are homologous, then $\iota(g_1)-\iota(g_2)=\iota(g_2^{-1}g_1)$ in the entry corresponding to $\Sigma'$. Then $\iota$ coincides with the $\iota$ defined in Section \[s:encode\], which is well-defined up to some choices of coset representatives. Let $g\in \pi_1(\Sigma)$. On $\Sigma$, we must first take the homology class of $g$. By identifying coset representatives for each term of the derived series of $\pi_1(\Sigma)$, we found a canonical element $\gamma$ of $\pi_1(\Sigma)$ which is homologous to $g$. By condition (2), we found the smallest solvable cover of $\Sigma$ where $\gamma^{-1}g$ becomes a homology class and repeated the construction. The representation $H(\Sigma)$ has some rather pathological properties which make the characterization of mapping classes by their action on $H(\Sigma)$ rather subtle. For instance: Suppose that $\psi\in\operatorname{Mod}(\Sigma)$ fixes a positive dimensional subspace of $H^1(\Sigma,{\mathbb{Q}})$. After lifting $\psi$ to $\operatorname{Mod}^1(\Sigma)$ arbitrarily, $\psi$ fixes a vector in $H(\Sigma)$. This is obvious, since we can pullback the cohomology of $\Sigma$ to each cover $\Sigma'$ and extend a vector in $H^1(\Sigma,{\mathbb{Q}})$ to $H^1(\Sigma',{\mathbb{Q}})$ by zero. Analogously, we can replace $H^1(\Sigma',{\mathbb{Q}})$ by $H_1(\Sigma',{\mathbb{Q}})$, except we then need to rescale by the degree of the cover $\Sigma'\to\Sigma$. Any more interesting data to be found in the virtual homological representation of $\operatorname{Mod}^1(\Sigma)$ has to be found in the action of an automorphism on the isotypic components of the representations of a finite characteristic quotient of $\pi_1(\Sigma)$. Describing these actions seems rather difficult, certainly no easier than understanding the action of an automorphism on a characteristic quotient of $\pi_1(\Sigma)$. Certain invariants, such as the dilatation of a pseudo-Anosov homeomorphism, may be hidden somewhere in these representations, but it is not at all obvious where. If $\psi$ is not virtually a homological pseudo-Anosov mapping class for instance (any Torelli mapping class of a closed surface will do), then the homological dilatation of $\psi$ will be strictly smaller than the geometric dilatation of $\psi$ (cf. [@BB], [@KS], [@LT]). [99]{} Jørgen Ellegaard Andersen. Asymptotic faithfulness of the quantum ${\rm SU}(n)$ representations of the mapping class groups. [Ann. of Math]{}. 2, 163, no. 1, 347–368, 2006. Gavin Band and Philip Boyland. The Burau estimate for the entropy of a braid. Available at arXiv:math/0612716, 2006. Hyman Bass and Alexander Lubotzky. Linear-central filtrations on groups. [The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions]{}. Contemp. Math. 169, pp. 45–98, 1994. Mladen Bestvina. A Bers-like proof of the existence of trans tracks for free group automorphisms. arXiv:1001.0325v1 \[math.GR\], 2010. Mladen Bestvina and Michael Handel. Train tracks and automorphisms of free groups. [Annals of Mathematics]{}, 135 (1992), pp. 1-51. Stephen J. Bigelow. Braid groups are linear. [J. Amer. Math. Soc.]{}, 14(2):471-486 (electronic), 2001. Stephen J. Bigelow. The Burau representation is not faithful for $n=5$. [Geom. Topol.]{} 3, 397–404, 1999. Stephen J. Bigelow and Ryan D. Budney. The mapping class group of a genus two surface is linear. [Algebr. Geom. Topol.]{} 1, 699-708, 2001. Joan Birman. [Braids, links and mapping class groups]{}. Ann. of Math. Studies, no. 82, Princeton University Press, 1975. Andrew J. Casson and Steven A. Bleiler. [Automorphisms of surfaces after Nielsen and Thurston]{}. London Mathematical Society Student Texts, 9, 1988. Claude Chevalley and André Weil. Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers. [Abh. Math. Sem. Univ. Hamburg]{} 10 (1934), pp. 358–361. A. Fathi and F. Laudenbach and V. Poénaru. [Travaux de Thurston sur les Surfaces]{}. Soc. Math. de France, Paris, Astérisque 66-67, 1979. Benson Farb, editor. [Problems on mapping class groups and related topics]{}. [Proc. Symp. Pure and Applied math.]{}, 74, 2006. Benson Farb and Dan Margalit. [A Primer on the mapping class group]{}. Online book, 2009. Edward Formanek and Claudio Procesi. The automorphism group of a free group is not linear. [J. Algebra]{} 149, no. 2, 494–499, 1992. Stefan Friedl and Taehee Kim. Thurston norm, fibered manifolds and twisted Alexander polynomials. To be published in [Topology]{}, arXiv:math, GT/0505682.s, 2006. Stefan Friedl and Stefano Vidussi. Twisted Alexander polynomials detect fibered $3$-manifolds. arXiv:math, GT/0805.1234v1, 2008. E.K. Grossman. On the residual finiteness of certain mapping class groups. [J. London Math. Soc.]{}, 9, 160–164, 1974. Eriko Hironaka and Eiko Kin. A family of pseudo-Anosov braids with small dilatation. [Algebraic Geometry and Topology]{}, 6:699-738 (electronic), 2006. Nikolai Ivanov. Subgroups of Teichmüller modular groups. [Translations of Mathematical Monographs]{}, 115, 1992. Dennis Johnson. A survey of the Torelli group. [Low-dimensional topology, San Francisco, Calif.]{} 165–179, 1981. Thomas Koberda and Aaron Michael Silberstein. Representations of Galois groups on the homology of surfaces. Submitted. Available at arXiv:0905.3002. D.A. Kazhdan. On arithmetic varieties. In [Lie groups and their representations]{} (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971), 151–217. Halsted, 1975. Erwan Lanneau and Jean-Luc Thiffeault. On the minimum dilatation of pseudo-Anosov homeomorphisms on surfaces of small genus. Available at arXiv:0905.1302, 2009. To appear in [Annales de l’Institut Fourier]{}. Roger C. Lyndon and Paul E. Schupp. [Combinatorial group theory]{}. Springer, New York, 1977. Justin Malestein. Pseudo-Anosov homeomorphisms and the lower central series of a surface group. [Algebr. Geom. Topol.]{} 7, 1921–1948, 2007. Wilhelm Magnus, Abraham Karrass and Donald Solitar. [Combinatorial group theory]{}. Dover Publications, Inc., Mineola, NY, 1975. Howard Masur. Closed trajectories for quadratic differentials with an application to billiards. [Duke Mathematical Journal]{}, 53, no. 2, 307–314, 1986. Curtis T. McMullen. Entropy on Riemann surfaces and the Jacobians of finite covers. Preprint, 2010. R.C. Penner and J.L. Harer. [Combinatorics of train tracks.]{} Annals of Mathematics Studies, no. 125, [Princeton University Press]{}, 1992. J.A. Rhodes. Sequences of metrics on compact Riemann surfaces. [Duke Math. J.]{} 72, 725–738, 1993. H.L. Royden. Automorphisms and isometries of Teichmüller space. [Advances in the Theory of Riemann Surfaces (proc. Conf., Stony Brook, NY, 1969)]{}, 369–383, Ann. of Math. Studies, no. 66, Princeton University Press, 1971. H.L. Royden. Invariant metrics on Teichmüller space. [Contributions to analysis ( a collection of papers dedicated to Lipman Bers)]{}, 393–399, 1974. Peter Scott. Subgroups of surface groups are almost geometric. [J. London Math. Soc.]{}, 2, 17, 555–565, 1978. Daniel T. Wise. A residually finite version of Rips’ construction. [Bull. London Math. Soc.]{} 35, 1, 23–29, 2003.
--- abstract: 'We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to $1$. We prove these results by using the idea of an annular itinerary. In the case of a general [[transcendental entire function]{}]{} we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.' address: \| Department of Mathematics and Statistics\ The Open University\ Walton Hall\ Milton Keynes MK7 6AA\ UK author: - 'D. J. Sixsmith' bibliography: - '../../Research.bib' title: Dimensions of slowly escaping sets and annular itineraries for exponential functions --- Introduction ============ This paper is principally concerned with [[transcendental entire function]{}]{}s in the exponential family, defined by $${E_\lambda}(z) = \lambda e^z, {\quad\text{for }}\lambda \in \mathbb{C}\backslash\{0\}.$$ For a general [[transcendental entire function]{}]{} $f$, the Fatou set $F(f)$ is defined as the set $z~\in~\mathbb{C}$ such that $\{f^n\}_{n\isnatural}$ is a normal family in a neighbourhood of $z$. The Julia set $J(f)$ is the complement in $\mathbb{C}$ of $F(f)$. An introduction to the properties of these sets was given in [@MR1216719]. The escaping set, which was first studied for a general [[transcendental entire function]{}]{} in [@MR1102727], is defined by $$I(f) = \{z : f^n(z)\rightarrow\infty\text{ as }n\rightarrow\infty\}.$$ Many authors have studied the Hausdorff dimension of $I({E_\lambda})$, or subsets of this set. We refer to [@falconer] for a definition of Hausdorff dimension, which we denote here by $\dim_H$. A key result is that of McMullen [@MR871679], who showed that $\dim_H J({E_\lambda}) = 2$. It is well-known that it follows from his construction that $\dim_H I({E_\lambda}) = 2$. When $\lambda \in (0, e^{-1})$ it is known – see [@MR758892; @MR873428] – that $J({E_\lambda})$ consists of an uncountable set of unbounded curves known as a Cantor bouquet, and that each curve, except possibly for its finite endpoint, lies in $I({E_\lambda})$. In two celebrated papers [@MR1680622; @MR1696203] Karpi[ń]{}ska proved the paradoxical fact that the set consisting of these curves excluding their finite endpoints has Hausdorff dimension $1$, whereas the set of finite endpoints has Hausdorff dimension $2$. A somewhat related result is that of Karpi[ń]{}ska and Urba[ń]{}ski [@MR2197375] who defined subsets of $I({E_\lambda})$ of Hausdorff dimension $d$, for each $d \in (1,2)$. In fact, all these papers considered a subset $A({E_\lambda})$ of $I({E_\lambda})$ known as the fast escaping set. The fast escaping set was introduced in [@MR1684251], and can be defined [@Rippon01102012] for a general [[transcendental entire function]{}]{} $f$ by $$\label{Adef} A(f) = \{z : \text{there exists } \ell \isnatural \text{ such that } \|f^{n+\ell}(z)\| \geq M^n(R,f), \text{ for } n \isnatural\}.$$ Here the maximum modulus function is defined by $M(r,f) = \max_{\|z\|=r} \|f(z)\|,$ for $r \geq 0.$ We write $M^n(r,f)$ to denote repeated iteration of $M(r,f)$ with respect to the variable $r$. In (\[Adef\]), $R > 0$ is such that $M^n(R,f)\rightarrow\infty$ as $n\rightarrow\infty$. It is well-known that the sets constructed in [@MR1680622; @MR1696203] and [@MR871679] lie in $A({E_\lambda})$. We show in Section \[SKU\] that this is also the case for the sets defined in [@MR2197375]. It seems that little is known about the dimension of subsets of $I({E_\lambda})\backslash A({E_\lambda})$. Indeed, very little is known about the dimension of $J(f) \cap(I(f)\backslash A(f))$ for any [[transcendental entire function]{}]{} $f$, with three notable exceptions. Bishop [@Bish2] constructed a [[transcendental entire function]{}]{} $f_1$ such that $\dim_H I(f_1)\backslash A(f_1) = 0$. At the other extreme, Eremenko and Lyubich [@MR918638 Example 4] constructed a [[transcendental entire function]{}]{} $f_2$ such that $J(f_2) \cap(I(f_2)\backslash A(f_2))$ has positive area. The remaining exception concerns the Eremenko-Lyubich class, $\mathcal{B}$, which is defined as the class of [[transcendental entire function]{}]{}s for which the set of singular values is bounded. Clearly ${E_\lambda}\in\mathcal{B}$, for $\lambda\ne 0$. If $f \in\mathcal{B}$, then $I(f) \subset J(f)$ [@MR1196102 Theorem 1] and so $$\label{setthing} J(f) \cap(I(f)\backslash A(f)) = I(f)\backslash A(f), {\quad\text{for }}f \in\mathcal{B}.$$ Bergweiler and Peter [@MR3054344] studied the dimension of subsets of $I(f)$ consisting of points for which there is a completely general upper bound on the rate of escape. The following is part of [@MR3054344 Theorem 1]. \[TBP\] Suppose that $f \in \mathcal{B}$ and that $(p_n)_{n\geq 0}$ is a sequence of real numbers tending to infinity. Define $$\operatorname{Esc}(f,(p_n)) = \{ z \in I(f) : \text{ there exists } N \geq 0 \text{ such that } \|f^n(z)\| \leq p_n, \text{ for } n \geq N\}.$$ Then $\dim_H \operatorname{Esc}(f,(p_n)) \geq 1.$ Suppose that $f\in\mathcal{B}$. In contrast to Bishop’s result, it follows from Theorem \[TBP\] and (\[setthing\]) that $\dim_H J(f) \cap (I(f)\backslash A(f)) \geq 1$. Rempe and Stallard [@MR2587450] showed that there exists a [[transcendental entire function]{}]{} $f_3 \in \mathcal{B}$ such that $\dim_H I(f_3) = 1$. It follows from Theorem \[TBP\] and (\[setthing\]) that $\dim_H J(f_3) \cap (I(f_3)\backslash A(f_3)) = 1$.\ In this paper we show that various subsets of $J({E_\lambda}) \cap (I({E_\lambda})\backslash A({E_\lambda}))$ have Hausdorff dimension exactly equal to $1$. We do not use Theorem \[TBP\], since all our results give an exact value for the Hausdorff dimension of sets defined by a two-sided inequality on the rate of escape. However, it seems plausible that there is some relationship between these results. For a general [[transcendental entire function]{}]{} $f$ we define the uniformly slowly escaping set by $$\label{LUdef} {L_U}(f) = \{z : \exists N\isnatural, \ R > 1, \ 0 < C_1 < C_2 \text{ s.t. } C_1R^{n} \leq \|f^{n}(z)\| \leq C_2R^{n}, \text{ for } n\geq N \}.$$ Roughly speaking, this set consists of those points for which the rate of escape is eventually uniformly slow. Our first result concerns the Hausdorff dimension of ${L_U}({E_\lambda})$. \[Tmain\] Suppose that $\lambda\ne 0$. Then $\dim_H {L_U}({E_\lambda}) = 1$. In Section \[Sese\] we give, for a general [[transcendental entire function]{}]{}, a necessary and sufficient condition for the uniformly slowly escaping set to be non-empty. We also prove that when the uniformly slowly escaping set is not empty, it has a number of familiar properties which show that, in general, this is a dynamically interesting set. For a general [[transcendental entire function]{}]{} $f$, ${L_U}(f)$ is a subset of the slow escaping set, introduced by Rippon and Stallard [@MR2792984], and defined by $$\label{Ldef} L(f) = \{z \in I(f) : \text{ there exists } R > 1 \text{ s.t. } \|f^{n}(z)\| \leq R^{n}, \text{ for } n\isnatural \}.$$ It was shown in [@MR2792984] that $L(f)\ne\emptyset$, that $J(f)$ is dense in $L(f)$ and also that $J(f)=\partial L(f)$. It follows from Theorem \[TBP\] that $\dim_H L({E_\lambda}) \geq 1$. Nothing more seems to be known about the actual dimension of $L({E_\lambda})$. As a step in that direction, we consider, for a general [[transcendental entire function]{}]{} $f$, a set which is a relatively large subset of $L(f)$ and which contains ${L_U}(f)$. First, for $p\isnatural$, let $\log^{+p}$ denote $p$ iterations of the $\log^+$ function, which is defined by $$\log^+ (x) = \begin{cases} \log x, &\text{if } x \geq 1, \\ 0, &\text{ otherwise}. \end{cases}$$ For a general [[transcendental entire function]{}]{}, $f$, we define$$\label{Lflatdef} {L_A}(f) = \{z : \exists R > 1, \ N, p\isnatural \text{ s.t. } n^{\log^{+p}(n)} \leq \|f^{n}(z)\| \leq R^n, \text{ for } n\geq N \}.$$ Note that the orbits of points in ${L_A}(f)$ are constrained to lie within certain annuli. The fact that $${L_U}(f) \subset {L_A}(f)\subset L(f) \subset I(f)\backslash A(f)$$ follows from (\[LUdef\]), (\[Ldef\]), (\[Lflatdef\]) and well-known properties of the maximum modulus function. Our result concerning the dimension of ${L_A}({E_\lambda})$ is as follows. \[Tthird\] Suppose that $\lambda\ne 0$. Then $\dim_H {L_A}({E_\lambda}) = 1$. Theorem \[Tthird\] is a consequence of the size of the annuli in the definition of ${L_A}({E_\lambda})$. In particular, Theorem \[Tthird\] shows that if $\dim_H L({E_\lambda}) > 1$, then the vast majority of points in $L({E_\lambda})$ must have an extremely slowly escaping subsequence. The techniques that we use to prove Theorem \[Tthird\] also allow us to construct subsets of $I({E_\lambda})\backslash(L({E_\lambda})\cup A({E_\lambda}))$ which have Hausdorff dimension equal to $1$. For example, for a [[transcendental entire function]{}]{} $f$, Rippon and Stallard [@MR2792984] defined the moderately slow escaping set by $$M(f) = \{z \in I(f) : \text{there exists } C > 0 \text{ such that } \|f^{n}(z)\| \leq \exp(e^{Cn}), \text{ for } n\isnatural \}.$$ In a similar way to (\[Lflatdef\]), we define a subset of $M(f)\backslash L(f)$ by $$\label{Mflatdef}{M_A}(f) = \{z : \exists N, p\isnatural \text{ s.t. } e^{n\log^{+p}(n)} \leq \|f^{n}(z)\| \leq \exp(e^{pn}), \text{ for } n\geq N \}.$$ Our result concerning the dimension of ${M_A}({E_\lambda})$ is as follows. \[Tfourth\] Suppose that $\lambda\ne 0$. Then $\dim_H {M_A}({E_\lambda}) = 1$. We prove our results using the idea of an . Before defining this concept, we briefly discuss a different type of itinerary which has frequently been used to study the dynamics of functions in the exponential family. Since $\|{E_\lambda}(z)\| = \|\lambda\| e^{{\operatorname{Re}}(z)}$, it follows that the orbit of a point in $I({E_\lambda})$ must eventually remain in the right half-plane $\mathbb{H} = \{z : {\operatorname{Re}}(z) > 0\}$. Many authors – see, for example, [@MR1721835; @MR758892] and [@MR2197375] – have considered itineraries of points in $I({E_\lambda})$ defined in the following way. First we partition $\mathbb{H}$ into half-open strips $$\label{Vdef} V_n = \{ z \in \mathbb{H} : (2n-1)\pi \leq {\operatorname{Im}}(z) < (2n+1)\pi, \text{ for } n\in\mathbb{Z}\}.$$ Suppose that $\underline{s} = s_0 s_1 s_2 \ldots$ is a sequence of integers. We say that a point $z$ has itinerary $\underline{s} = \underline{s}(z) = s_0 s_1 s_2 \ldots$ if ${E_\lambda}^n(z) \in V_{s_n}$, for $n\geq 0$. For some types of itinerary it can be shown that the set of points with such an itinerary is – in some sense – large. For example, it follows from McMullen’s proof [@MR871679] (and see also [@MR1680622]) that the set $$\{ z \in I({E_\lambda}) : 2\pi\|s_n(z)\| \geq \|{E_\lambda}^n(z)\|/2, \text{ for } n\geq 0\}$$ has Hausdorff dimension $2$. The concept of an annular itinerary was introduced by Rippon and Stallard [@2013arXiv1301.1328R]. Suppose that $f$ is a general [[transcendental entire function]{}]{}, and let $(R_n)_{n\geq 0}$ be a strictly increasing sequence of positive real numbers such that $R_n\rightarrow\infty$ as $n\rightarrow\infty$. The strips $V_n$ in (\[Vdef\]) are replaced by half-open annuli $$A_n = \{ z : R_{n-1} \leq \|z\| < R_n\}, {\quad\text{for }}n\isnatural,$$ and $A_0$ is defined as $\{ z : \|z\| < R_0\}$. Suppose that $\underline{t} = t_0 t_1 t_2 \ldots$ is a sequence of non-negative integers. If $f^n(z) \in A_{t_n}$, for $n\geq 0$, then we say that the point $z$ has annular itinerary $\underline{t} = \underline{t}(z) = t_0 t_1 t_2 \ldots$ with respect to the partition $(A_n)_{n\geq 0}$. Rippon and Stallard [@2013arXiv1301.1328R] let $R_0>0$ be sufficiently large that $M^n(R_0,f)\rightarrow\infty$ as $n\rightarrow\infty$, and then set $R_n = M^n(R_0,f)$, for $n\isnatural$. They showed that, with this choice of partition $(A_n)_{n\geq 0}$, there is a very broad class of annular itineraries such that the set of points with such an itinerary contains a point in $J(f)$. For more information regarding the properties of these annular itineraries, we refer to [@2013arXiv1301.1328R]. Annular itineraries are a natural choice when studying points which escape to infinity with different rates. The used in our paper are defined using annuli of constant modulus, which seems a natural choice when considering points in the slow escaping set. First we choose a value of $R > 1$, and then set $R_n = R^{n+1}$, for $n\geq 0$. This construction of the partition $(A_n)_{n\geq 0}$ should be considered to be in place throughout the remainder of this paper. Note that this construction depends on $R$. Here, and elsewhere, we suppress some dependencies for simplicity of notation, and retain only dependencies which need to remain explicit. We use the following notation $$I_{R}(\underline{t}) = \{ z : z \text{ has {\sai} } \underline{t} \text{ with respect to the partition } (A_n)_{n\geq 0}\}.$$ We are interested in a particular type of . We say that an $\underline{t} = t_0 t_1 t_2 \ldots$ is non-zero if $t_n \ne 0$, for $n\geq 0$, escaping if $t_n\rightarrow\infty$ as $n\rightarrow\infty$, admissible if $e^{t_n} > t_{n+1}$, for $n\geq 0$, and slowly-growing if $$\label{G3} \lim_{n\rightarrow\infty} \frac{t_n}{\sum_{k=1}^{n-1} t_k} = 0.$$ Our main result regarding is as follows. \[Tfirst\] Suppose that $\lambda\ne 0$, $R > 1$ and $\underline{t}$ is an escaping . Then $\dim_H I_{R}(\underline{t}) \leq 1.$ Moreover, there exists $R_0 = R_0(\lambda) > 1$ such that if, in addition, $R \geq R_0$ and $\underline{t}$ is non-zero, admissible and slowly-growing, then $\dim_H I_{R}(\underline{t})~=~1.$ It seems surprising that, for a large class of , the sets of points with the same all have the same Hausdorff dimension. We note that there are annular itineraries of arbitrarily slow growth which satisfy the conditions of Theorem \[Tfirst\]. In other words, if $(p_n)_{n\geq 0}$ is a sequence of positive integers such that $p_n\rightarrow\infty$ as $n\rightarrow\infty$, then there exists an annular itinerary $\underline{t} = t_0 t_1 t_2 \ldots$ and $R>1$ such that $\dim_H I_R(\underline{t}) = 1$ and $t_n \leq p_n$, for $n\geq 0$. We comment briefly on the final two conditions in the second part of Theorem \[Tfirst\]. The condition that the annular itinerary be admissible is required to ensure that $I_{R}(\underline{t})$ is not empty. It is unclear if the condition (\[G3\]) is essential, though it is required for our method of proof. It is a straightforward calculation to show that a sequence $(t_n)_{n\isnatural}$ which satisfies this condition also satisfies $$\label{growthcond} \log t_n = o(n) \text{ as } n\rightarrow\infty.$$ However, the condition (\[growthcond\]) is weaker than the condition (\[G3\]). For example, consider the sequence defined by $$t_n = 2^{m^2}, {\quad\text{for }}(m-1)^3 \leq n < m^3, \ m\isnatural.$$ It can be shown that this sequence satisfies (\[growthcond\]) but not (\[G3\]). The techniques of this paper do not allow us to replace (\[G3\]) with the apparently simpler condition (\[growthcond\]). Finally, we note that the dimension of subsets of $J({E_\lambda})$ which lie outside of $I({E_\lambda})$ was studied in [@MR1680622 Theorem 2] and [@MR1992945]. In addition, Pawelec and Zdunik [@2014arXiv1405.7784P] recently showed that, for certain values of $\lambda$, there exist indecomposable continua in $J({E_\lambda})$ which are of Hausdorff dimension $1$. These continua intersect with the fast escaping set. We refer to [@2014arXiv1405.7784P] for further details.\ The structure of this paper is as follows. First, in Section \[Shann\], we give some preliminary lemmas. In Section \[Slower\] we prove a theorem which gives a lower bound on the Hausdorff dimension of $I_{R}(\underline{t})$ for a certain type of annular itinerary. In Section \[Supper\] we prove a theorem which gives an upper bound on the Hausdorff dimension of certain sets. All our dimension results are consequences of these two theorems. In Section \[Ssgi\] we prove Theorem \[Tmain\], Theorem \[Tthird\], Theorem \[Tfourth\] and Theorem \[Tfirst\]. In Section \[Sese\], we state and prove two results about the uniformly slowly escaping set. Finally, in Section \[SKU\], we discuss, briefly, the result of Karpi[ń]{}ska and Urba[ń]{}ski mentioned earlier. Preliminary lemmas {#Shann} ================== We start this section with two lemmas concerning functions in the exponential family. We define closed annuli and half-annuli, for $0 < r_1 < r_2$, by $$\label{Hdef} A(r_1, r_2) = \{z : r_1 \leq \|z\| \leq r_2\}\text{ and }H(r_1, r_2) = \{z \in A(r_1,r_2) : \operatorname{Re}(z) \geq 0 \}.$$ For $r>0$ and $a\in\mathbb{C}$, we write $B(a, r)$ for the open disc $\{ z : \|z - a\| < r\}$. The first lemma provides an estimate on the density of preimages of one half-annulus in another; see Figure \[f1\]. Here, for measurable sets $U$ and $V$, we define $$\operatorname{dens}(U, V) = \frac {\operatorname{area} (U \cap V)}{\operatorname{area}(V)},$$ where area$(U)$ denotes the Lebesgue measure of $U$. ![The set ${E_\lambda}^{-1}(S_2) \cap S_1$. One preimage component of $S_2$ is shown with a slightly darker background. The two rectangles constructed in the proof of Lemma \[estlem\] are shown with a dashed boundary. Note that $R_3$ is not necessarily larger than $R_2$.[]{data-label="f1"}](pic1.pdf){width="12cm" height="9cm"} \[estlem\] Suppose that $0 < R_1 < R_2$ and $0 < R_3 < R_4$ are such that $$\label{est0} R_2 > \max\left\{2 R_1, \ R_1 + 16\pi, \ 3 \log \frac{R_4}{\|\lambda\|}\right\},$$ and $$\label{est4} R_3 > \|\lambda\|.$$ Let $S_1 = {H(R_1, R_2)}$, $S_2 = {H(R_3, R_4)}$, and let $D$ be the union of all the components of ${E_\lambda}^{-1}(S_2)$ which are contained in $S_1$. Then $$\label{denseq} \operatorname{dens}(D, S_1) \geq \frac{1} {2\pi R_2}\log \frac{R_4}{R_3}.$$ Each component of ${E_\lambda}^{-1}(S_2)$ is a rectangle of the form, for $n \in\mathbb{Z}$, $$\label{rectdef} \left\{ z : \log\frac{R_3}{\|\lambda\|} \leq \operatorname{Re}(z) \leq \log\frac{R_4}{\|\lambda\|}, \ \left(2n-\frac{1}{2}\right)\pi \leq \operatorname{Im}(z) + \arg(\lambda) \leq \left(2n+\frac{1}{2}\right)\pi \right\}.$$ Suppose that the inequalities (\[est0\]) and (\[est4\]) both hold. Consider two large rectangles, each with sides parallel to the coordinate axes. One rectangle has a vertex at the point in the upper half-plane where the vertical line $\{ z : \operatorname{Re}(z) = \log\frac{R_3}{\|\lambda\|}\}$ meets the circle $B(0, R_1)$; note that if $R_1 \leq \log\frac{R_3}{\|\lambda\|}$ we put this vertex at $R_1$. The diagonally opposite vertex of this rectangle is at the point in the upper half-plane where the vertical line $\{ z : \operatorname{Re}(z) = \log\frac{R_4}{\|\lambda\|}\}$ meets the circle $B(0, R_2)$. The second rectangle is the complex conjugate of the first one. Let $h$ be the height of each rectangle. It follows by an application of Pythagoras’s theorem to this rectangle, and by (\[est0\]), that $$\begin{aligned} h &= \left(R_2^2 - \left(\log \frac{R_4}{\|\lambda\|}\right)^2\right)^{\frac{1}{2}} - \left(\max\left\{0, \ R_1^2 - \left(\log \frac{R_3}{\|\lambda\| }\right)^2\right\}\right)^{\frac{1}{2}} \\ &\geq \frac{7}{8}R_2 - R_1 \geq \frac{3}{4}(R_2 - R_1).\end{aligned}$$ It follows by (\[est0\]) and (\[rectdef\]) that each rectangle contains at least $\frac{1}{4\pi}(R_2 - R_1)$ components of ${E_\lambda}^{-1}(S_2)$. Hence $S_1$ contains at least $\frac{1}{2\pi}(R_2 - R_1)$ components of ${E_\lambda}^{-1}(S_2)$, each of which is a closed rectangle of height $\pi$ and width at least $\log \frac{R_4}{R_3}$. Equation (\[denseq\]) follows from this, and the fact that area$(S_1) = \frac{1}{2}\pi(R_2^2 - R_1^2)$. For a domain $V$ and a [[transcendental entire function]{}]{} $f$, univalent in $V$, we define the distortion of $f$ in $V$ by $$D_V(f) = \frac{\sup_{z\in V}\|f'(z)\|}{\inf_{z\in V}\|f'(z)\|}.$$ For functions in the exponential family, the following facts are immediate. Suppose that $F$ is a set such that $\inf\{\|z\| : z \in F \} = r_1 > 0$ and $\sup\{\|z\| : z \in F \} = r_2$, and that $V$ is a component of ${E_\lambda}^{-1}(F)$ such that ${E_\lambda}$ is univalent in $V$. Then $$\label{estlem2A} \|{E_\lambda}'(z)\| \geq r_1, {\quad\text{for }}z \in V,$$ and $$\label{estlem2B} D_V({E_\lambda}) = \frac{r_2}{r_1}.$$ We also use two well-known properties of Hausdorff dimension. For the first see, for example, [@falconer]. \[Lcountstab\] Suppose that $(F)_{i\in I}$ is a collection of subsets of $\mathbb{C}$, and that $I$ is a finite or countable set. Then $$\dim_H \bigcup_{i\in I} F_i = \sup_{i\in I} \ \{\dim_H F_i\}.$$ The second property is used frequently but we are not aware of a reference. \[Lsing\] Suppose that $f$ is a non-constant [[transcendental entire function]{}]{} and that $U \subset \mathbb{C}$. Then $$\dim_H f(U) = \dim_H f^{-1} (U) = \dim_H U.$$ A lower bound on the Hausdorff dimension {#Slower} ======================================== In this section we prove the following theorem which gives a lower bound on the Hausdorff dimension of $I_{R}(\underline{t})$ for a certain type of annular itinerary. \[lbtheo\] Suppose that $\lambda\ne 0$. Then there exists $R_0 = R_0(\lambda) > 1$ such that, if $R \geq R_0$ and $\underline{t}$ is an escaping, non-zero, admissible and slowly-growing , then $\dim_H I_{R}(\underline{t}) \geq 1.$ To prove Theorem \[lbtheo\], we use a well-known construction and result of McMullen. Let $(\mathcal{E}_n)_{n\geq 0}$ be a sequence of finite collections of pairwise disjoint compact subsets of $\mathbb{C}$ such that the following both hold: (i) If $F \in \mathcal{E}_{n+1}$, then there exists a unique $G \in \mathcal{E}_{n}$ such that $F \subset G$; (ii) If $G \in \mathcal{E}_{n}$, then there exists at least one $F \in \mathcal{E}_{n+1}$ such that $G \supset F$. We write $$\label{Edef} D_n = \bigcup_{F\subset \mathcal{E}_n} F, \text{ for } n\geq 0, \quad \text{and} \quad D = \bigcap_{n\geq 0} D_n.$$ McMullen’s result is the following [@MR871679 Proposition 2.2]. Here, for a set $U$, $\operatorname{diam} U$ denotes the Euclidean diameter of $U$. \[mcmlemma\] Suppose that there exists a sequence of finite collections of pairwise disjoint compact sets, $(\mathcal{E}_n)_{n\geq 0}$, which satisfies conditions (i) and (ii) above, and let $D$ and $(D_n)_{n\geq 0}$ be as defined in (\[Edef\]). Suppose also that $(\Delta_n)_{n\geq 0}$ and $(d_n)_{n\geq 0}$ are sequences of positive real numbers, with $d_n \rightarrow 0$ as $n\rightarrow\infty$, such that, for each $n\geq 0$ and for each $F \in \mathcal{E}_n$, we have $$\operatorname{dens}(D_{n+1}, F) \geq \Delta_n\quad\text{and}\quad\operatorname{diam} F \leq d_n.$$ Then $$\label{mcmulleneq} \dim_H D \geq 2 - \limsup_{n\rightarrow\infty} \frac{\sum_{m=0}^{n} \| \log \Delta_m\|}{\| \log d_n\|}.$$ In [@MR871679] the upper bound of summation in (\[mcmulleneq\]) was given as $n+1$. However, the stronger result (\[mcmulleneq\]) – which is required in the proof of Theorem \[lbtheo\] – follows from McMullen’s proof, and has been given in, for example, [@MR2609307 Lemma 4.4] and [@MR2458811 Lemma 4.3]. We also use the following. This is a version of [@areapaper Lemma 5.2], which itself is a detailed version of [@MR871679 Proposition 3.1]. \[distortion.lemma\] Suppose that $f$ is a [[transcendental entire function]{}]{}, and there exists a set $U\subset\mathbb{C}$ and constants $\alpha>1$ and $M > 0$ such that $$\label{unifexp} \|f'(z)\| > \alpha \quad\text{ and }\quad \left\|\frac{f''(z)}{f'(z)}\right\| < M, {\quad\text{for }}z \in U.$$ Suppose also that there exists $s \in (0, (4M)^{-1})$ such that if $B \subset U$ is a disc of diameter $s$, then $f$ is conformal in a neighbourhood of $B$. Suppose finally that $(B_m)_{m\in\{1, 2, \ldots, n\}}$ is a sequence of sets contained in $U$, each of diameter less than $s$, and such that $$B_{m+1} \subset f(B_m), {\quad\text{for }}m\in\{1, 2, \ldots, n-1\}.$$ For $m\in\{1, 2, \ldots, n\}$, let $\phi_m$ be the inverse branch of $f$ which maps $f(B_m)$ to $B_m$, and set $V = \phi_1 \circ \phi_2 \circ \cdots \circ \phi_{n}(f(B_n))$. Then there exists $\tau=\tau(M, s, \alpha)>1$ such that $$D_V(f^n)\leq \tau.$$ Note that in [@areapaper Lemma 5.2] the sets $B_n$ are squares of side $s$. The proof of the above result follows in exactly the same way, and is omitted.\ We deduce the following. \[c1\] There exist absolute constants $s_0 > 0$ and $\tau_0 > 1$ such that the following holds. Suppose that $\lambda\ne 0$, $n\isnatural$ and $V$ is a set such that $${E_\lambda}^m(V) \subset \left\{ z : \operatorname{Re}(z) > \log\frac{2}{\|\lambda\|} \right\} \text{ and } \operatorname{diam} {E_\lambda}^m(V) < s_0, {\quad\text{for }}0 \leq m < n.$$ Then $D_V({E_\lambda}^n)\leq \tau_0$. This result follows from Lemma \[distortion.lemma\] with $f = {E_\lambda}$ , $U = \left\{ z : \operatorname{Re}(z) > \log\frac{2}{\|\lambda\|} \right\}$, $\alpha=M=2$, and $B_{m} = {E_\lambda}^{m-1}(V)$, for $1 \leq m \leq n$. We now give the proof of Theorem \[lbtheo\]. Roughly speaking, our method of proof is as follows. First we set a value of $R_0$ sufficiently large to enable us to use Lemma \[estlem\]. We then define a set which is contained in $I_{R}(\underline{t})$ and apply McMullen’s result to obtain a lower bound on the Hausdorff dimension of this set. Let $s_0$ be the constant in Corollary \[c1\]. We choose $$\label{R0choice} R_0>\max\left\{e, \|\lambda\|, \frac{2}{s_0} \right\},$$ sufficiently large that $$\label{Rineq} R(R-1) > 16 \pi + 2\quad\text{ and }\quad R > 3 \log \frac{R^2}{\|\lambda\|}+1, {\quad\text{for }}R \geq R_0.$$ Suppose that $R \geq R_0$, and that $\underline{t}$ is an escaping, non-zero, admissible and slowly-growing . To use Lemma \[mcmlemma\] we need to work with compact and disjoint sets. In order to do this, and recalling the definition (\[Hdef\]), we define disjoint closed half-annuli $$H_n = H(R^n+1, R^{n+1}-1), {\quad\text{for }}n\isnatural.$$ Since $\underline{t}$ is admissible and non-zero, we deduce by (\[R0choice\]) and (\[Rineq\]) that, for $n\geq 0$, we have $$\label{sineq} R^{t_n + 1} > R e^{t_n} > R t_{n+1} > 3 t_{n+1} \log R + 3 \log \frac{R}{\|\lambda\|} + 1 > 3 \log \left(\frac{R^{t_{n+1}+1}-1}{\|\lambda\|}\right) + 1.$$ Since $\underline{t}$ is non-zero, we deduce from (\[R0choice\]), (\[Rineq\]) and (\[sineq\]) that the hypotheses of Lemma \[estlem\] are satisfied with $S_1 = H_{t_n}$ and $S_2 = H_{t_{n+1}}$, for $n\geq 0$. In order to use Lemma \[mcmlemma\], we define a sequence of finite collections of pairwise disjoint compact sets as follows. First set $$\mathcal{E}_0 = \{ H_{t_0} \},$$ and, for $n\geq 0$, $$\mathcal{E}_{n+1} = \{ F : F \subset G, \text{ for some } G \in \mathcal{E}_n, \text{ and } {E_\lambda}^{n+1}(F) = H_{t_{n+1}}\}.$$ Let $D_n$, for $n\geq 0$, and $D$ be the sets defined in (\[Edef\]). It follows from (\[Edef\]) that $D \subset I_{R}(\underline{t}).$ It is sufficient, therefore, to show that $\dim_H D \geq 1$. It follows from Lemma \[estlem\] that the conditions (i) and (ii) stated prior to Lemma \[mcmlemma\] are both satisfied. It remains to estimate the diameters and the densities stated in Lemma \[mcmlemma\]. Note that to apply equation (\[mcmulleneq\]) we may omit the definition of a finite number of these estimates. Since $\underline{t}$ is escaping, we can let $N_0\geq 2$ be sufficiently large that $t_{n-1} \geq 2$, for $n\geq N_0$. Suppose that $n\geq N_0$ and that $F \in \mathcal{E}_n$. Note that ${E_\lambda}^n(F) = H_{t_n}$. We first find an upper bound on the diameter of $F$. Since $$\text{diam } {E_\lambda}^n(F) = \text{diam } H_{t_n} < 2R^{t_n+1},$$ we have, by (\[estlem2A\]), that $$\label{diambound} \operatorname{diam } F \leq 2R^{t_n+1} \frac{1}{R^{t_1}} \frac{1}{R^{t_2}} \cdots \frac{1}{R^{t_n}} = 2R^{1 - \sum_{m=1}^{n-1} t_m}.$$ We set $d_n = 2R^{1 - \sum_{m=1}^{n-1} t_m}$. Since $\underline{t}$ is escaping, we deduce that $$\label{diams} d_n \rightarrow 0 \quad\text{and}\quad \|\log d_n\| = \log R \ \sum_{m=1}^{n-1} t_m\left(1 + o(1)\right) \text{ as } n\rightarrow\infty.$$ We next show that the distortion of ${E_\lambda}^n$ on $F$ is bounded independently of $n$ and $F$. Once again by (\[estlem2A\]), and by (\[R0choice\]), we have $$\operatorname{diam } {E_\lambda}^{m}(F) \leq 2R^{t_n+1} \frac{1}{R^{t_{n-1}}} \frac{1}{R^{t_n}}\leq \frac{2}{R} < s_0, {\quad\text{for }}0 \leq m < n-1.$$ Suppose that $0 \leq m < n-1$ and that $z \in {E_\lambda}^m(F)$. Since $\underline{t}$ is non-zero, we have $\|{E_\lambda}(z)\| \geq R$. We deduce by (\[R0choice\]) that $$\operatorname{Re}(z) \geq \log \frac{R}{\|\lambda\|} > \log \frac{2}{\|\lambda\|}.$$ Hence $${E_\lambda}^m(F) \subset \left\{ z : \operatorname{Re}(z) > \log\frac{2}{\|\lambda\|} \right\}, {\quad\text{for }}0 \leq m < n-1.$$ We deduce by Corollary \[c1\] that $D_F({E_\lambda}^{n-1})\leq \tau_0$. Moreover, it follows from (\[estlem2B\]) that $D_{{E_\lambda}^{n-1}(F)}({E_\lambda})\leq R$. Thus $$\label{disteq} D_F({E_\lambda}^{n})\leq D_F({E_\lambda}^{n-1}) \ D_{{E_\lambda}^{n-1}(F)}({E_\lambda}) \leq \tau_0 R.$$ We use (\[disteq\]) to find a lower bound on dens$(D_{n+1}, F)$. Note that ${E_\lambda}^n(D_{n+1})$ consists of those components of ${E_\lambda}^{-1}(H_{t_{n+1}})$ which are contained in $H_{t_n}$. Hence, by (\[denseq\]), (\[R0choice\]) and (\[disteq\]), we have $$\begin{aligned} \operatorname{dens}(D_{n+1}, F) &\geq \frac{1}{(\tau_0 R)^2} \operatorname{dens}({E_\lambda}^n(D_{n+1}), {E_\lambda}^n(F)) \\ &\geq \frac{1}{(\tau_0 R)^2} \frac{1}{2\pi\left(R^{t_n+1}-1\right)} \log \frac{R^{t_{n+1}+1}-1}{R^{t_{n+1}}+1} \\ &\geq \frac{\log R} {4\tau_0^2\pi R^{t_n+3}}. $$ We set $\Delta_n = \frac{\log R} {4\tau_0^2\pi R^{t_n+3}}$. Note that $$\label{dists} \|\log \Delta_n\| = t_n\log R\left(1+o(1)\right) \text{ as } n\rightarrow\infty.$$ Since $\underline{t}$ is escaping and slowly-growing, it follows, by (\[G3\]), (\[diams\]) and (\[dists\]), that $$\begin{aligned} \limsup_{n\rightarrow\infty} \frac{\sum_{m=0}^{n} \|\log \Delta_m\|}{\|\log d_n\|} &= \limsup_{n\rightarrow\infty} \frac{\log R \ \sum_{m=1}^{n} t_m(1+o(1))}{\log R \ \sum_{m=1}^{n-1} t_m(1+o(1))} \\ &= \limsup_{n\rightarrow\infty} \left(\frac{t_n(1+o(1))}{\sum_{m=1}^{n-1} t_m(1+o(1))} + \frac{\sum_{m=1}^{n-1} t_m(1+o(1))}{\sum_{m=1}^{n-1} t_m(1+o(1))}\right) \\ &= 1.\end{aligned}$$ We deduce by Lemma \[mcmlemma\] that $\dim_H D \geq 1$, as required. An upper bound on Hausdorff dimension {#Supper} ===================================== In this section we prove a theorem which gives an upper bound on the Hausdorff dimension of certain sets, and so is, in a sense, complementary to Theorem \[lbtheo\]. First we define the sets. Suppose that, for each $p\isnatural$, $(g_{p,n})_{n\isnatural}$ and $(h_{p,n})_{n\isnatural}$ are sequences of positive real numbers such that $$\label{bigenough} h_{p,n} \geq g_{p,n}, {\quad\text{for }}n\isnatural.$$ For $\lambda\ne 0$ define the set $T_{g,h}$ by $$\label{Tdef} T_{g,h} = \{z : \text{there exist } N, p\isnatural \text{ s.t. } {E_\lambda}^{n}(z) \in A(g_{p,n}, h_{p,n}), \text{ for } n \geq N \}.$$ Our theorem is as follows. \[ubtheo\] Suppose that $\lambda\ne 0$, and that for each $p\isnatural$, $(g_{p,n})_{n\isnatural}$ and $(h_{p,n})_{n\isnatural}$ are sequences of positive real numbers such that (\[bigenough\]) is satisfied, $$\label{ghbasic} g_{p,n} \rightarrow\infty \text{ as } n \rightarrow\infty,$$ and $$\label{gheq} \frac{\log g_{p,n}}{\log^+ \log h_{p,n+1}} \rightarrow\infty \text{ as } n \rightarrow\infty.$$ Then $\dim_H T_{g, h} \leq 1.$ Choose $$\label{betadef} \beta > \max\left\{\sqrt{1+\pi^2}, \ \frac{1}{s_0}\right\}$$ and set $$c_0 = \|\lambda\|e^{1+\beta}\max\left\{\frac{2}{\|\lambda\|},\ \exp(\beta^2)\right\}$$ and $$\label{c1def} c_1 = \log\frac{c_0}{\|\lambda\|} = 1+\beta+ \max\left\{\log\frac{2}{\|\lambda\|},\ \beta^2\right\},$$ where $s_0$ is the constant in Corollary \[c1\]. By (\[ghbasic\]), for each $p\isnatural$ we can choose $\ell_p \isnatural$ such that $$\label{geq} g_{p,n} > c_0, {\quad\text{for }}n\geq \ell_p.$$ Define sets $$\label{Sdef} S_{\nu, p} = \{ z : {E_\lambda}^n(z) \in A(g_{p,\nu+n}, h_{p,\nu+n}), \text{ for } n\geq 0\}, {\quad\text{for }}\nu, p \isnatural.$$ Fix a value of $p\isnatural$, and suppose that $\nu\geq \ell_p$. We claim that $$\label{Sclaim} \dim_H S_{\nu,p} \leq 1.$$ Before proving (\[Sclaim\]) we show that the result of the lemma can be deduced from this equation. First we claim that $$\label{Tclaim} T_{g, h} \subset \bigcup_{p\isnatural} \bigcup_{\nu\geq \ell_p} {E_\lambda}^{-\nu}(S_{\nu,p}).$$ For, suppose that $z \in T_{g,h}$, in which case there exist $N, p\isnatural$ such that $${E_\lambda}^{n}(z)~\in~A(g_{p,n}, h_{p,n}), {\quad\text{for }}n \geq N.$$ It follows that there exists $\nu \geq \max\{N, \ell_p\}$ such that ${E_\lambda}^{n}({E_\lambda}^\nu(z)) \in A(g_{p,n+\nu}, h_{p,n+\nu})$, for $n \geq 0$, and so ${E_\lambda}^\nu(z) \in S_{\nu,p}$. This establishes equation (\[Tclaim\]). Theorem \[ubtheo\] follows from (\[Sclaim\]) and (\[Tclaim\]), by Lemma \[Lcountstab\] and Lemma \[Lsing\].\ If remains to show that (\[Sclaim\]) holds for $\nu \geq \ell_p$. We establish this result using a sequence of covers of $S_{\nu, p}$ and basic properties of Hausdorff dimension. We suppress the variable $p$ for simplicity. First we note some properties of $S_{\nu}$ and then use these properties to define a sequence of covers of this set. It follows from (\[geq\]), and the fact that $\nu \geq \ell$, that $$\|{E_\lambda}({E_\lambda}^n(z))\| = \|\lambda\|e^{\operatorname{Re}({E_\lambda}^n(z))} > c_0, {\quad\text{for }}z \in S_{\nu}, \ n\geq 0.$$ We deduce by (\[c1def\]) that $$\label{nicepoints} \operatorname{Re}({E_\lambda}^n(z)) > \log\frac{c_0}{\|\lambda\|} = c_1, {\quad\text{for }}z \in S_{\nu}, \ n\geq 0.$$ Define compact sets $$\label{Wmndef} W_{n,m} = A(e^{m-1}g_{n}, e^{m}g_{n}) \cap \{ z : \operatorname{Re}(z)\geq c_1\}, {\quad\text{for }}n, m \isnatural.$$ We observe that each component of ${E_\lambda}^{-1}(W_{n,m})$, for $m, n \isnatural$, is contained in a distinct rectangle of the form, $$\label{rectangles} \left\{ z : -1 \leq \operatorname{Re}(z) - m - \log g_n + \log \|\lambda\| \leq 0, \ -\pi/2 \leq \operatorname{Im}(z) + \arg(\lambda) + 2k\pi \leq \pi/2 \right\},$$ for some $k \in\mathbb{Z}$. Hence, by (\[betadef\]), if $F$ is a component of ${E_\lambda}^{-1}(W_{n,m})$, then $$\label{thediam} \operatorname{diam } F \leq \beta, {\quad\text{for }}m, n \isnatural.$$ For simplicity of notation, define sets of integers $$\label{cardalphan} \alpha_n = \left\{ 1, 2, \ldots, \left[\log \frac{h_{n}}{g_{n}}\right]+1\right\}, {\quad\text{for }}n\isnatural,$$ where $[x]$ denotes the integer part of $x$. Note that $$\label{eqann} A(g_{n}, h_{n}) \cap \{ z : \operatorname{Re}(z)\geq c_1\} \subset \bigcup_{m\in\alpha_n} W_{n,m}, {\quad\text{for }}n\isnatural.$$ Now set $$\mathcal{E}_0 = \{ W_{\nu,m} : m \in \alpha_{\nu}\},$$ and, for $n\geq 0$, $$\mathcal{E}_{n+1} = \{F : {E_\lambda}^{n+1}(F) = W_{\nu+n+1,m} \text{ for some } m \in \alpha_{\nu+n+1} \text{ and } F \ \cap \ G \ne \emptyset \text{ for some } G \in \mathcal{E}_n\}.$$ It follows by (\[nicepoints\]) and (\[eqann\]) that $$S_{\nu} \subset \bigcup_{F \in \mathcal{E}_n} F, {\quad\text{for }}\text{each } n\geq 0.$$ This completes the definition of the sequence of covers of $S_{\nu}$.\ Next we study the properties of the sets in these covers, particularly the diameters of these sets. Suppose that $q\isnatural$. We claim that the following hold for $n \geq q$ and $F \in \mathcal{E}_n$; $$\label{nicediams} \operatorname{diam } {E_\lambda}^{n-q}(F) \leq \beta^{3-2q},$$ $$\label{nicerealparts0} {E_\lambda}^{n-q}(F) \subset \left\{ z : \operatorname{Re}(z) \geq c_1 - \sum_{k=1}^q \beta^{3-2k}\right\},$$ and $$\label{nicerealparts} {E_\lambda}^{n-q}(F) \subset \left\{ z : \operatorname{Re}(z) > \max\left\{\log\frac{2}{\|\lambda\|},\beta^2\right\}\right\}.$$ First we note that (\[nicerealparts\]) follows from (\[betadef\]), (\[c1def\]) and (\[nicerealparts0\]). We prove (\[nicediams\]) and (\[nicerealparts0\]) by induction on $q$. We consider first the case that $q=1$. Suppose that $n\isnatural$ and that $F \in \mathcal{E}_n$. By (\[thediam\]), ${E_\lambda}^{n-1}(F)$ has diameter at most $\beta$. Moreover, since $${E_\lambda}^{n-1}(F) \cap W_{\nu+n-1,m'} \ne \emptyset, {\quad\text{for }}\text{some } m' \in \alpha_{\nu+n-1},$$ we deduce by (\[Wmndef\]) that ${E_\lambda}^{n-1}(F) \subset \{ z : \operatorname{Re}(z) \geq c_1 - \beta\}$. This establishes (\[nicediams\]) and (\[nicerealparts0\]) in the case that $q=1$. Now, suppose that (\[nicediams\]) and (\[nicerealparts0\]) have been been established for $1 \leq q\leq s$, for some $s\isnatural$. Suppose that $n\geq s+1$, that $F \in \mathcal{E}_n$ and that $G \in \mathcal{E}_{n-1}$ is such that $F \cap G \ne \emptyset$. First, we deduce by (\[estlem2A\]), and by (\[nicediams\]) and (\[nicerealparts\]) with $s$ in place of $q$, that $$\label{nextdiam} \operatorname{diam } {E_\lambda}^{n-(s+1)}(F) \leq \frac{\operatorname{diam } {E_\lambda}^{n-s}(F)}{\inf \{\|z\| : z \in {E_\lambda}^{n-s}(F)\}}\leq \frac{\beta^{3-2s}}{\beta^2}=\beta^{3-2(s+1)}.$$ Second, applying (\[nicerealparts0\]) with $G$ in place of $F$, $n-1$ in place of $n$, and $s$ in place of $q$, we deduce that $${E_\lambda}^{(n-1)-s}(G) \subset \left\{ z : \operatorname{Re}(z) \geq c_1 - \sum_{k=1}^s \beta^{3-2k}\right\}.$$ Since ${E_\lambda}^{n-(s+1)}(F) \cap {E_\lambda}^{(n-1)-s}(G) \ne \emptyset$, it follows by (\[nextdiam\]) that $${E_\lambda}^{n-(s+1)}(F) \subset \left\{ z : \operatorname{Re}(z) \geq c_1 - \sum_{k=1}^s \beta^{3-2k} - \beta^{3-2(s+1)} \right\}.$$ By induction, this completes the proof of (\[nicediams\]) and (\[nicerealparts0\]). We observe that it follows from (\[nicediams\]) that the diameters of the sets in $\mathcal{E}_n$ tend uniformly to zero as $n\rightarrow\infty$.\ Suppose next that $\epsilon > 0$. By the definition of Hausdorff dimension, together with the observation above regarding the diameters of the sets in $\mathcal{E}_n$, our proof of (\[Sclaim\]) is complete if we can show that $$\sum_{F\in\mathcal{E}_{n+1}} (\text{diam }F)^{1+\epsilon} \leq \sum_{F\in\mathcal{E}_{n}} (\text{diam }F)^{1+\epsilon}, {\quad\text{for }}\text{large }n,$$ or indeed if we can show that, for all sufficiently large $n$, we have for each $G \in \mathcal{E}_n$ that $$\label{outcome} \sum_{\substack{F\in\mathcal{E}_{n+1}, \\ F \cap G \ne \emptyset}} \left(\frac{\text{diam } F}{\text{diam } G}\right)^{1+\epsilon} \leq 1.$$ Suppose that $n\isnatural$ and that $G \in \mathcal{E}_n$, in which case ${E_\lambda}^{n}(G) = W_{\nu+n,m}$, for some $m\in\alpha_{\nu+n}$. Suppose also that $F~\in~\mathcal{E}_{n+1}$ intersects with $G$, in which case ${E_\lambda}^{n}(F)$ intersects with $W_{\nu+n,m}$ and is a preimage component of $W_{\nu+n+1,m'}$, for some $m'\in\alpha_{\nu+n+1}$. We note the following two simple estimates. First, it follows from (\[rectangles\]) that for each $m'\in\alpha_{\nu+n+1}$, there are at most $O(e^{m}g_{\nu+n})$ preimage components of $W_{\nu+n+1,m'}$ which intersect with $W_{\nu+n,m}$. It follows from this estimate and from (\[cardalphan\]), that the total number of elements of $\mathcal{E}_{n+1}$ which intersect with $G$ is at most $$\label{numpreims} O\left(e^{m}g_{\nu+n}\log\frac{h_{\nu+n+1}}{g_{\nu+n+1}}\right) \text{ as } n\rightarrow\infty.$$ Second, it is immediate that $$\label{diam2} (\operatorname{diam } {E_\lambda}^{n}(G))^{-1} = O((e^{m}g_{\nu+n})^{-1}) \text{ as } n\rightarrow\infty.$$ Next we estimate the distortion of ${E_\lambda}^{n}$ in $F \cup G$. Suppose that $n\geq 2$. We deduce by (\[betadef\]) and (\[nicediams\]) that $$\operatorname{diam} {E_\lambda}^m(G) < s_0, {\quad\text{for }}0 \leq m < n-1.$$ It follows by (\[nicerealparts\]) and Corollary \[c1\] that $D_G({E_\lambda}^{n-1})\leq \tau_0$. Moreover, it follows from (\[estlem2B\]) that $D_{{E_\lambda}^{n-1}(G)}({E_\lambda})\leq e$. We deduce that $D_G({E_\lambda}^{n}) = O(1) \text{ as } n\rightarrow\infty$. By a similar argument $D_F({E_\lambda}^{n}) = O(1) \text{ as } n\rightarrow\infty$. Hence, since $F \cap G \ne \emptyset$, we have $$\label{disteq3} D_{F \cup G} ({E_\lambda}^{n}) = O(1) \text{ as } n\rightarrow\infty.$$ Combining these estimates, we deduce from (\[thediam\]), (\[numpreims\]), (\[diam2\]) and (\[disteq3\]) that $$\begin{aligned} \sum_{\substack{F\in\mathcal{E}_{n+1}, \\ F \cap G \ne \emptyset}} \left(\frac{\text{diam } F}{\text{diam } G}\right)^{1+\epsilon} &\leq O(1) \sum_{\substack{F\in\mathcal{E}_{n+1}, \\ F \cap G \ne \emptyset}}\left(\frac{\text{diam } {E_\lambda}^{n}(F)}{\text{diam } {E_\lambda}^{n}(G)}\right)^{1+\epsilon} \\ &= O\left(e^{m}g_{\nu+n}\log\frac{h_{\nu+n+1}}{g_{\nu+n+1}} . (e^{m}g_{\nu+n})^{-(1+\epsilon)}\right) \\ &\leq O\left(g_{\nu+n}^{-\epsilon} \log h_{\nu+n+1}\right).\end{aligned}$$ We deduce by (\[gheq\]) that (\[outcome\]) holds, as required. Dimension results {#Ssgi} ================= In this section we prove Theorem \[Tmain\], Theorem \[Tthird\], Theorem \[Tfourth\] and Theorem \[Tfirst\]. In each case we use Theorem \[lbtheo\] to show that the Hausdorff dimension is bounded below by $1$, and we use Theorem \[ubtheo\] to show that the Hausdorff dimension is bounded above by $1$. It is somewhat simpler to prove Theorem \[Tthird\] before Theorem \[Tmain\]. Choose $R \geq R_0$, where $R_0$ is as in the statement of Theorem \[lbtheo\], and let $\underline{t}$ be the sequence $$\label{tdef} \underline{t} = 1 \ 1 \ 1 \ 2 \ 3 \ \ldots \ (n-1) \ \ldots.$$ We deduce by Theorem \[lbtheo\] that $\dim_H I_{R}(\underline{t}) \geq 1$. Suppose that $z \in I_{R}(\underline{t})$, in which case $R^{n-1} \leq \|{E_\lambda}^n(z)\| \leq R^{n}$, for $n\geq 2$. It follows that $z \in{L_A}({E_\lambda})$. Hence $I_{R}(\underline{t}) \subset {L_A}({E_\lambda})$, and so $\dim_H {L_A}({E_\lambda}) \geq 1$.\ Next, for each $p\isnatural$, let sequences $(g_{p,n})_{n\isnatural}$ and $(h_{p,n})_{n\isnatural}$ be defined by $$\label{ghdef}g_{p,n}=n^{\log^{+p}(n) } \text{ and } h_{p,n} = e^{pn}, {\quad\text{for }}n\isnatural.$$ We deduce by Theorem \[ubtheo\] that $\dim_H T_{g, h}\leq 1$. Suppose that $z \in {L_A}({E_\lambda})$, in which case, by (\[Lflatdef\]), there exist $p, N\isnatural$ and $R>1$ such that $$n^{\log^{+p}(n)} \leq \|{E_\lambda}^{n}(z)\| \leq R^{n}, {\quad\text{for }}n\geq N.$$ If $p$ is sufficiently large, then $R^n \leq e^{pn}$, for $n\isnatural$. We deduce that $z \in T_{g, h}$, where $g, h$ are as defined in (\[ghdef\]). It follows that ${L_A}({E_\lambda}) \subset T_{g, h}$, and so $\dim_H {L_A}({E_\lambda}) \leq 1$. This completes the proof of Theorem \[Tthird\]. Choose $R \geq R_0$, where $R_0$ is as in the statement of Theorem \[lbtheo\], and let $\underline{t}$ be as defined in (\[tdef\]). Recall that $\dim_H I_{R}(\underline{t}) \geq 1$. We deduce Theorem \[Tmain\] from Theorem \[Tthird\] and the fact that $I_{R}(\underline{t}) \subset {L_U}({E_\lambda}) \subset {L_A}({E_\lambda}).$ Choose $R \geq R_0$, where $R_0$ is as in the statement of Theorem \[lbtheo\], and let $\underline{t'}$ be the sequence $$\underline{t'} = 1 \ 4 \ 9 \ \ldots \ (n+1)^2 \ \ldots.$$ We deduce by Theorem \[lbtheo\] that $\dim_H I_{R}(\underline{t'}) \geq 1$. Suppose that $z \in I_{R}(\underline{t'})$, in which case $R^{(n+1)^2} \leq \|{E_\lambda}^n(z)\| \leq R^{(n+1)^2+1}$, for $n\geq 0$. It follows that $z \in{M_A}({E_\lambda})$. Hence $I_{R}(\underline{t'}) \subset {M_A}({E_\lambda})$, and so $\dim_H {M_A}({E_\lambda}) \geq 1$.\ Next, for each $p\isnatural$, let sequences $(g'_{p,n})_{n\isnatural}$ and $(h'_{p,n})_{n\isnatural}$ be defined by $$\label{ghdashdef} g'_{p,n}=e^{n\log^{+p}(n)} \text{ and } h'_{p,n} = \exp(e^{pn}), {\quad\text{for }}n\isnatural.$$ We deduce by Theorem \[ubtheo\] that $\dim_H T_{g', h'}\leq 1$. Suppose that $z \in {M_A}({E_\lambda})$, in which case, by (\[Mflatdef\]), there exist $p, N\isnatural$ and $R>1$ such that $$e^{n\log^{+p}(n)} \leq \|{E_\lambda}^{n}(z)\| \leq \exp(e^{pn}), {\quad\text{for }}n\geq N.$$ We deduce that $z \in T_{g', h'}$, where $g', h'$ are as defined in (\[ghdashdef\]). It follows that ${M_A}({E_\lambda}) \subset T_{g', h'}$, and so $\dim_H {M_A}({E_\lambda}) \leq 1$. This completes the proof of Theorem \[Tfourth\]. Finally we prove Theorem \[Tfirst\]. Suppose that $f$ is a [[transcendental entire function]{}]{}, $R > 1$ and $\underline{t} = t_0 t_1 t_2 \ldots$ is an escaping . For each $p\isnatural$ we define sequences $(g_{p,n})_{n\isnatural}$ and $(h_{p,n})_{n\isnatural}$ by $g_{p,n} = R^{t_n}, \text{ and } h_{p,n} = R^{t_n+1}.$ We deduce by Theorem \[ubtheo\] that $\dim_H T_{g, h} \leq 1$. The first part of the theorem follows since $I_{R}(\underline{t}) \subset T_{g, h}$. The second part of Theorem \[Tfirst\] is an immediate consequence of this, together with Theorem \[lbtheo\]. The uniformly slowly escaping set {#Sese} ================================= In this section, for a general [[transcendental entire function]{}]{} $f$, we give two results on the set ${L_U}(f)$. First, we give a necessary and sufficient condition for ${L_U}(f)$ to be non-empty. Here $m(r, f) = \min_{\|z\| = r} \|f(z)\|$ denotes the minimum modulus of $f$, for $r > 0$. \[Texists\] Suppose that $f$ is a [[transcendental entire function]{}]{}. Then $${L_U}(f)\cap J(f)\ne\emptyset$$ if and only if there exist positive constants $c$ and $r_0$, and $d > 1$ such that $$\label{RSeq} \text{for all } r \geq r_0 \text{ there exists } \rho \in (r, dr) \text{ such that } m(\rho, f) \leq c.$$ Moreover, if $L_U(f) \ne \emptyset$, then ${L_U}(f)\cap J(f)\ne\emptyset$. The condition (\[RSeq\]) was used in [@MR2792984], also in relation to points tending to infinity at a specified rate. As observed in [@MR2792984], this condition holds whenever $f$ is bounded on a path to infinity. It is well-known that this is the case for functions in the class $\mathcal{B}$ in particular. Second, we show that when ${L_U}(f)$ is not empty, it has a number of familiar properties which show that, in general, this is a dynamically interesting set. We say that a set $S$ is completely invariant if $z \in S$ implies that $f(z) \in S$ and also that $f^{-1}(\{z\}) \subset S$. \[Tprops\] Suppose that $f$ is a [[transcendental entire function]{}]{}, and that ${L_U}(f)~\ne~\emptyset$. Then the following all hold. (i) ${L_U}(f)$ is completely invariant; (ii) If $U$ is a Fatou component of $f$ and $U \cap {L_U}(f) \ne \emptyset$, then $U \subset {L_U}(f)$; (iii) ${L_U}(f)$ is dense in $J(f)$ and $J(f) = \partial$${L_U}(f)$. An example of a [[transcendental entire function]{}]{} with a Fatou component contained in ${L_U}(f)$ is the function $f(z) = 2 - \log 2 + 2z - e^z,$ given by Bergweiler [@MR1317494]. It is straightforward to deduce from the arguments in [@MR1317494] that $f$ has a Baker domain in ${L_U}(f)$ and also wandering Fatou components in ${L_U}(f)$. We refer to [@MR1216719] for definitions. The proof of Theorem \[Texists\] requires the following [@MR2792984 Theorem 2]. \[LRS\] Suppose that $f$ is a [[transcendental entire function]{}]{}. Then $f$ has the property that, for all positive sequences $(a_n)_{n\isnatural}$ such that $a_n\rightarrow\infty$ as $n\rightarrow\infty$ and $a_{n+1} = O(M(a_n, f))$ as $n\rightarrow\infty$, there exist $\zeta\in J(f)$ and $C > 1$ such that $$a_n \leq \|f^n(\zeta)\| \leq Ca_n, {\quad\text{for }}n\isnatural,$$ if and only if there exist positive constants $c$ and $r_0$, and $d > 1$ such that (\[RSeq\]) holds. Suppose that $f$ is a [[transcendental entire function]{}]{}. If there exist positive constants $c$ and $r_0$, and $d > 1$ such that (\[RSeq\]) holds, then it follows immediately from Lemma \[LRS\] that ${L_U}(f) \cap J(f) \ne \emptyset$. The other direction proceeds very similarly to the proof of the corresponding direction in [@MR2792984 Theorem 2]. We give some details for completeness. Suppose that there do not exist positive constants $c$ and $r_0$, and $d > 1$ such that (\[RSeq\]) holds. Then there exists a sequence of annuli $A(r_n,R_n)$, where $0 < r_n < R_n$, such that $r_n\rightarrow\infty$ as $n\rightarrow\infty$, $R_n/r_n\rightarrow\infty$ as $n\rightarrow\infty$ and $$m(r, f) > 1, {\quad\text{for }}r_n < r < R_n, \ n \isnatural.$$ As shown in the proof of [@MR2792984 Theorem 2], it follows that there exists $\delta \in (0,1)$ and $N\isnatural$ such that $$\label{meq} m(r, f)>M(r, f)^\delta, {\quad\text{for }}2r_n < r < \frac{1}{2}R_n, \ n \geq N.$$ We shall deduce that ${L_U}(f) = \emptyset$. Suppose, by way of contradiction, that there exists $z\in{L_U}(f)$. It follows from (\[LUdef\]) that there exists $C>0$ and $N\isnatural$ such that $$\label{zinLU} f^n(z)\ne 0 \text{ and } \left\|\frac{f^{n+1}(z)}{f^n(z)}\right\| \leq C, {\quad\text{for }}n\geq N.$$ We deduce that, for infinitely many values of $n\isnatural$, there exists $p(n)\isnatural$ such that $f^n(z) \in A(2r_{p(n)}, \frac{1}{2}R_{p(n)})$. It follows by (\[meq\]) that $$\left\|\frac{f^{n+1}(z)}{f^n(z)}\right\| > \frac{M(\|f^n(z)\|,f)^\delta}{\|f^n(z)\|}, {\quad\text{for }}\text{infinitely many values of } n\isnatural.$$ Since, as is well-known, $$\frac{M(r, f)^\delta}{r}\rightarrow\infty \text{ as } r\rightarrow\infty,$$ this is a contradiction to (\[zinLU\]). We deduce that ${L_U}(f) \cap J(f) = \emptyset$ if and only if (\[RSeq\]) holds. Finally, if ${L_U}(f) \ne \emptyset$, then (\[RSeq\]) holds, and so ${L_U}(f) \cap J(f) \ne \emptyset$. This completes the proof of Theorem \[Texists\]. In order to prove Theorem \[Tprops\], we require the following well-known distortion lemma; see, for example, [@MR1216719 Lemma 7]. \[dlemm\] Suppose that $f$ is a [[transcendental entire function]{}]{}, and that $U \subset I(f)$ is a simply connected Fatou component of $f$. Suppose that $K$ is a compact subset of $U$. Then there exist $C > 1$ and $N_0\isnatural$ such that $$\frac{1}{C}\|f^n(z)\| \leq\|f^n(w)\| \leq C\|f^n(z)\|, {\quad\text{for }}w, z \in K, n \geq N_0.$$ We also use the following, which is a special case of [@MR2792984 Lemma 10]. We say that a set $S$ is backwards invariant if $z \in S$ implies that $f^{-1}(\{z\}) \subset S$. \[Jlemm\] Suppose that $f$ is a [[transcendental entire function]{}]{}, and that $E~\subset~\mathbb{C}$ contains at least three points. Suppose also that $E$ is backwards invariant under $f$, that $\operatorname{int} E\cap J(f) = \emptyset$, and that every component of $F(f)$ that meets $E$ is contained in $E$. Then $\partial E = J(f)$. Suppose that $f$ is a [[transcendental entire function]{}]{}, and that ${L_U}(f)\ne\emptyset$, in which case, by Theorem \[Texists\], ${L_U}(f)\cap J(f)\ne\emptyset$. First we observe that part (i) of the theorem follows immediately from the definition of ${L_U}(f)$.\ For part (ii) of the theorem, suppose that $U$ is a Fatou component of $f$, such that $U \cap {L_U}(f) \ne \emptyset$. It follows by normality that $U \subset I(f)$. Suppose that $U$ is multiply connected in which case [@Rippon01102012 Theorem 1.2] we have that $U\subset A(f)$. However, it is known [@MR1684251] that if $z \in A(f)$ then $$\frac{\log \log \|f^n(z)\|}{n}\rightarrow\infty \text{ as } n\rightarrow\infty,$$ in which case $z \notin {L_U}(f)$. We deduce that $U$ is simply connected. Suppose that $z \in U \cap {L_U}(f)$. Then there exist $N\isnatural, \ R > 1$ and $0 < C_1 < C_2$ such that $C_1R^{n} \leq \|f^{n}(z)\| \leq C_2R^{n}, \text{ for } n\geq N.$ Suppose that $K$ is a compact subset of $U$ containing $z$. Then, by Lemma \[dlemm\] there exist $C > 1$ and $N_0\isnatural$ such that $$\frac{C_1}{C} R^{n} \leq \|f^{n}(w)\| \leq C_2CR^{n}, \text{ for } w \in K, \ n\geq \max\{N, N_0\}.$$ Hence $K \subset {L_U}(f)$, and so $U \subset {L_U}(f)$. This completes the proof of part (ii).\ For part (iii) of the theorem, we note that ${L_U}(f)\cap J(f)$ is an infinite set, since for each $z \in {L_U}(f)\cap J(f)$ at least one of the points $z, f(z)$ or $f^2(z)$ must have an infinite backwards orbit. It is known [@MR1216719 Theorem 4] that the set of repelling periodic points of $f$ is dense in $J(f)$. Since, by definition, ${L_U}(f)$ contains no periodic points, $\operatorname{int} {L_U}(f) \subset F(f)$. The result follows by part (i) and part (ii), and by Lemma \[Jlemm\] applied with $E={L_U}(f) \cap J(f)$ and then with $E={L_U}(f)$. This completes the proof of Theorem \[Tprops\]. Results of Karpi[ń]{}ska and Urba[ń]{}ski {#SKU} ========================================= As mentioned in the introduction, Karpi[ń]{}ska and Urba[ń]{}ski [@MR2197375] studied the size of various subsets of $I({E_\lambda})$. For integers $k\geq 0$ and $l\geq k$, and $\epsilon > 0$, they defined sets $$D_\epsilon^{k,l} = \left\{ z \in I({E_\lambda}): {\operatorname{Re}}({E_\lambda}^n(z)) > q, \text{ for } n \geq k \text{, and } \|\operatorname{Im}({E_\lambda}^n(z))\| \leq \frac{\|{E_\lambda}^n(z)\|}{(\log \|{E_\lambda}^n(z)\|)^\epsilon}, \text{ for } n\geq l\right\},$$ where $q$ is fixed and large. Their main result is the following. For every $\epsilon > 0$ and all integers $0 \leq k \leq l,$ $$\dim_H D_\epsilon^{k,l} = 1 + \frac{1}{1 + \epsilon}.$$ In this section we show that the sets $D_\epsilon^{k,l}$ lie in the fast escaping set of ${E_\lambda}$. First, we define the domain $V = \{ z : \operatorname{Re}(z) > \frac{1}{2}\|z\|\}$. Note that $$\label{ifinV} \|{E_\lambda}(z)\| = \|\lambda\|e^{\operatorname{Re}(z)} > \|\lambda\|e^{\frac{1}{2}\|z\|} = M\left(\frac{1}{2}\|z\|, {E_\lambda}\right), {\quad\text{for }}z \in V.$$ Suppose that $\epsilon > 0$ and $0 \leq k \leq l$, and that $z \in D_\epsilon^{k,l}$. It follows from the definition of $D_\epsilon^{k,l}$ that there exists $N\isnatural$ such that ${E_\lambda}^{n+N}(z) \in V$, for $n\geq 0$. Set $R = \frac{1}{2}\|{E_\lambda}^N(z)\|$, and define $$\mu(r) = \frac{1}{2}M(r, {E_\lambda}), {\quad\text{for }}r \geq 0.$$ We may assume that $R$ is sufficiently large that $\mu(r) > r$, for $r \geq R$. It follows from (\[ifinV\]) that $$\|{E_\lambda}^{n}({E_\lambda}^N(z))\| > \mu^n(R), {\quad\text{for }}n\geq 0,$$ and so, by [@Rippon01102012 Theorem 2.9], that $z \in A({E_\lambda})$, as required.\ Acknowledgment: The author is grateful to Phil Rippon and Gwyneth Stallard for all their help with this paper. The author is also grateful to Chris Bishop for useful discussions.
--- abstract: \| Bosbach states represent a way of probabilisticly evaluating the formulas from various (commutative or non-commutative) many-valued logics. They are defined on the algebras corresponding to these logics with values in $[0,1]$. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of $[0,1]$, in this paper we introduce Bosbach states defined on residuated lattices with values in residuated lattices. We are led to two types of generalized Bosbach states, with distinct behaviours. The properties of generalized Bosbach states, proven in the paper, may serve as an algebraic foundation for developping some probabilistic many-valued logics. [Keywords:]{} Bosbach states, residuated lattices, MV-algebras, $s$-Cauchy completion, metric completion. [MSC 2010:]{} Primary 06F35. Secondary 06D35. author: - \| George GEORGESCU and Claudia MUREŞAN\ University of Bucharest\ Faculty of Mathematics and Computer Science\ Academiei 14, RO 010014, Bucharest, Romania\ Emails: georgescu@funinf.cs.unibuc.ro, c.muresan@yahoo.com title: Generalized Bosbach States --- Introduction ============ Classical probability theory is based on the hypothesis that the sets of events associated with random experiments have a structure of a Boolean algebra. This fact derives from the thesis that the random experiment follows the rules of classical logic. An important part of probability theory can be developped by considering probabilities on arbitrary Boolean algebras ([@cuon], [@frem]) . It can happen for random experiments to follow the rules of another logical system. Then the sets of events will have the structure of the Lindenbaum-Tarski algebra associated to that logical system. In the case of infinite-valued Łukasiewicz logic, the sets of events will have a structure of MV-algebra ([@cdom]). The study of probabilities defined on MV-algebras (which are called [MV-states]{}) has been started in [@mun] and then continued by numerous authors (see, for instance, [@br1], [@br3], [@ioanal2]). Together with these, there have been studied different types of states defined on pseudo-MV-algebras ([@dvus]), BL-algebras ([@br2]), pseudo-BL-algebras ([@gg]), Rl-monoids ([@dvura], [@dvura2]), residuated lattices ([@lcciu], [@lcciu1]), pseudo-BCK-algebras ([@kuhr]) etc.. Bosbach states, introduced in [@gg], have as domain a pseudo-BL-algebra $A$ and as codomain the real interval $[0,1]$. The axioms of the Bosbach states are expressed in terms of the two implications of $A$ and of the addition in $\R $. But states can be thought of in another way. By identifying an event with the sentence that describes that event, states will become functions defined on the set of the sentences of the logical system and having as target set the real interval $[0,1]$. This way states can be regarded as a type of semantics. This point of view suggests us to consider $[0,1]$ as a standard algebra of a logical system and to report the definition of states to this algebra. The present work starts from the observation that Bosbach states can be defined using the canonical structure of standard MV-algebra of $[0,1]$. By replacing the MV-algebra $[0,1]$ with an arbitrary residuated lattice $L$, we aim to find a concept of a state (called [generalized Bosbach state]{}) defined on an arbitrary residuated lattice $A$ and with $L$ as target set. To this end, we will express the definition of the Bosbach state in several equivalent forms. By comparing these equivalent forms we will obtain two notions of generalized Bosbach states: of type I and of type II. We will notice that type I states are not order-preserving. By considering order preservation as an essential property for any notion of state, we will be studying especially order-preserving type I states. We will study in parallel order-preserving type I states and type II states. By analyzing the way in which some properties of Bosbach states can be extended to type I and type II states, we will notice a strong asymmetry between them. This paper is organized as follows. In Section \[preliminaries\] we present some basic definitions and results from the theory of residuated lattices. Section \[bosbach\] contains the definition of generalized Bosbach states (of type I and of type II), preceeded by a detailed discussion on its motivation. We give several examples and we prove some arithmetic properties of generalized Bosbach states, as well as some characterizations of them. Section \[filtcan\] deals with the properties of the canonical filter associated with a generalized Bosbach state and of the corresponding quotient residuated lattice. These are related to the notion of state-morphism, which generalizes the one from [@dvus], [@dvura], [@gg] to the more general context of this paper. In Section \[riecan\] we introduce generalized Riečan states. These extend the concept of Riečan state from [@br2], [@gg], [@dvura], [@lcciu]. We analyze the link between generalized Riečan states and generalized Bosbach states of type I and II. In Section \[convergente\] we are treating the continuity of generalized Bosbach states. In [@ggap] the authors introduced the similarity convergence in the context of residuated lattices. Based on this similarity convergence, we are defining three types of continuity for generalized Bosbach states and we establish links between them. To each order-preserving type I state we can associate canonically a similarity relation, which allows us to accomplish, in the general case of the present paper, a construction that generalizes the metric completion of an MV-algebra. The last section of this paper contains a sketch of some connections between generalized Bosbach states and some many-valued logical systems. Preliminaries ============= In this section we recall some notions and arithmetic properties of several varieties of residuated lattices and some from the theory of filters and congruences of residuated lattices. We refer the reader to [@beloh], [@cdom], [@haj], [@ior], [@kow]. A [residuated lattice]{} is an algebraic structure of the form $(A,\vee ,\wedge ,\odot ,\rightarrow ,0,1)$, in which: $(A,\vee ,\wedge ,0,1)$ is a bounded lattice, $(A,\odot ,1)$ is a commutative monoid and, for all $a,b,c\in A$, $a\leq b\rightarrow c$ iff $a\odot b\leq c$ ([the law of residuation]{}). For any residuated lattice $A$ and any $a,b\in A$, we denote $\neg \, a=a\rightarrow 0$ ([the negation]{}) and $a\leftrightarrow b=(a\rightarrow b)\wedge (b\rightarrow a)$ ([the biresiduum]{} or [the equivalence]{}). We will also denote $d_A(a,b)=a\leftrightarrow b$. The next two lemmas collect several arithmetic properties of residuated lattices. [[@kow], [@haj], [@pic], [@tur]]{} For any residuated lattice $A$ and any $a,b,c,d\in A$, we have: 1. \[l2.2(1)\] $a\rightarrow 1=1$; 2. \[l2.2(2)\] $1\rightarrow a=a$; 3. \[l2.2(3)\] $a\leq b$ iff $a\rightarrow b=1$; 4. \[l2.2(4)\] if $a\leq b$ then $b\rightarrow c\leq a\rightarrow c$ and $c\rightarrow a\leq c\rightarrow b$; 5. \[l2.2(5)\] $a\odot b\leq a\wedge b\leq d_A(a,b)\leq a\rightarrow b$; 6. \[l2.2nenum\] $b\leq a\rightarrow b$; 7. \[l2.2(6)\] $a\odot (a\rightarrow b)\leq b$; 8. \[l2.2nenum2\] if $a\leq c$ and $b\leq d$, then $a\odot b\leq c\odot d$; 9. \[l2.2(7)\] $a\rightarrow (b\rightarrow c)=(a\odot b)\rightarrow c=b\rightarrow (a\rightarrow c)$; 10. \[l2.2(8)\] $(a\vee b)\odot c=(a\odot c)\vee (b\odot c)$; 11. \[l2.2(9)\] $(a\vee b)\rightarrow c=(a\rightarrow c)\wedge (b\rightarrow c)$ and $c\rightarrow (a\wedge b)=(c\rightarrow a)\wedge (c\rightarrow b)$; moreover, for any nonempty set $I$ and any family $(a_i)_{i\in I}\subseteq A$ such that $\bigvee _{i\in I}a_i$ exists, $(\bigvee _{i\in I}a_i)\rightarrow c=\bigwedge _{i\in I}(a_i\rightarrow c)$. \[l2.2\] [[@beloh], [@tur]]{} In a residuated lattice, the biresiduum has the following properties, for all $a,b,c,x,y\in A$: 1. \[lA(1)\] $d_A(a,b)=1$ iff $a=b$; 2. \[lA(2)\] $d_A(a,b)=d_A(b,a)$; 3. \[lA(3)\] $d_A(a,b)\odot d_A(b,c)\leq d_A(a,c)$; 4. \[lA(4)\] $d_A(a,b)\leq d_A(\neg \, a,\neg \, b)$; 5. \[lA(5)\] $d_A(a,b)\odot d_A(x,y)\leq d_A(a\circ x,b\circ y)$ for each $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$. \[lA\] [[@kow], [@haj], [@pic], [@tur]]{} For any residuated lattice $A$ and any $a,b\in A$, we have: 1. \[0neg1\] $\neg \, 0=1$ and $\neg \, 1=0$; 2. \[l2.3(1)\] $a\leq \neg \, b$ iff $a\odot b=0$; 3. \[l2.3(2)\] $a\leq \neg \, \neg \, a$ and $\neg \, \neg \, \neg \, a=a$; 4. \[l2.3(3)\] if $a\leq b$ then $\neg \, b\leq \neg \, a$; 5. \[l2.3(4)\] $a\rightarrow b\leq \neg \, b\rightarrow \neg \, a$; 6. \[l2.3(5)\] $\neg \, (a\odot b)=a\rightarrow \neg \, b=b\rightarrow \neg \, a$. \[l2.3\] Important classes of residuated lattices can be introduced starting from the notion of t-norm. A [t-norm]{} is a binary operation $\odot $ on $[0,1]$ with the properties of being associative, commutative, order-preserving and with $1$ as identity. If a t-norm $\odot $ is left-continuous, then we can consider the operation residuum $\rightarrow $ on $[0,1]$, defined by $a\rightarrow b=\max \{c\in [0,1]\|c\odot a\leq b\}$. Then $([0,1],\max ,\min ,\odot ,\rightarrow ,0,1)$ is a residuated lattice. A residuated lattice $A$ is called an [MTL-algebra]{} iff, for all $a,b\in A$, $(a\rightarrow b)\vee (b\rightarrow a)=1$. If $\odot $ is a left-continuous t-norm, then $([0,1],\max ,\min ,\odot ,\rightarrow ,0,1)$ is an MTL-algebra. [[@ego]]{} If $A$ is an MTL-algebra and $a,b\in A$, then $a\vee b=((a\rightarrow b)\rightarrow b)\wedge ((b\rightarrow a)\rightarrow a)$. \[lMTL\] A [BL-algebra]{} is an MTL-algebra $A$ with the property that, for all $a,b\in A$, $a\wedge b=a\odot (a\rightarrow b)$. If $\odot $ is a continuous t-norm, then $([0,1],\max ,\min ,\odot ,\rightarrow ,0,1)$ is a BL-algebra. We list below the three fundamental continuous t-norms and their residua: - [the Łukasiewicz t-norm]{}: $a\odot _{L}b=\max \{0,a+b-1\}$, $a\rightarrow _{L}b=\min \{1,1-a+b\}$; - [the $G\ddot{o}del$ t-norm]{}: $a\odot _{G}b=\min \{a,b\}$, $a\rightarrow _{G}b=\begin{cases}1, & {\rm if}\ a\leq b,\\ b, & {\rm otherwise;}\end{cases}$ - [the product]{} or [Gaines t-norm]{}: $a\odot _{P}b=a\cdot b$, $a\rightarrow _{P}b=\begin{cases}1, & {\rm if}\ a\leq b,\\ b/a, & {\rm otherwise.}\end{cases}$ An [MV-algebra]{} is an algebra $(A,\oplus,\neg \, ,0)$ with one binary operation $\oplus$, one unary operation $\neg \, $ and one constant 0 such that: $(A,\oplus,0)$ is a commutative monoid and, for all $a,b\in A$, $\neg \, \neg \, a=a$, $a\oplus \neg \, 0=\neg \, 0$, $\neg \, (\neg \, a\oplus b)\oplus b=\neg \, (\neg \, b\oplus a)\oplus a$. If $A$ is an MV-algebra, then the binary operations $\odot$, ${\wedge}$, ${\vee}$, $\rightarrow $ and the constant 1 are defined by the following relations: for all $a,b\in A$, $a\odot b=\neg \, (\neg \, a\oplus \neg \, b)$, $a{\wedge}b=(a\oplus \neg \, b)\odot b$ , $a{\vee}b=(a\odot \neg \, b)\oplus b$, $a\rightarrow b=\neg\, a\oplus b$, $1=\neg \, 0$. According to [@pic Theorem 3.2, page 99], MV-algebras are exactly the involutive BL-algebras, that is: an MV-algebra is a BL-algebra $A$ with the property that, for all $a\in A$, $\neg \, \neg \, a=a$. $([0,1],\max ,\min ,\odot _{L},\rightarrow _{L},0,1)$ is an MV-algebra, called [the standard MV-algebra]{}. [[@ioanal]]{} Let $A$ be an MV-algebra and $a,b,c\in A$. Then: 1. \[mvsum\] $a\oplus \neg \, a=1$; 2. \[mvdemorgan1\] $\neg \, (a\odot b)=\neg \, a\oplus \neg \, b$; 3. \[mvdemorgan2\] $\neg \, (a\oplus b)=\neg \, a\odot \neg \, b$; 4. \[mvdistrib\] $a\oplus (b\wedge c)=(a\oplus b)\wedge (a\oplus c)$; 5. \[mvvee\] $c\rightarrow (a\vee b)=(c\rightarrow a)\vee (c\rightarrow b)$; moreover, for any nonempty set $I$ and any family $(a_i)_{i\in I}\subseteq A$ such that $\bigvee _{i\in I}a_i$ exists, $c\rightarrow (\bigvee _{i\in I}a_i)=\bigvee _{i\in I}(c\rightarrow a_i)$; 6. \[mvwedge\] for any nonempty set $I$ and any family $(a_i)_{i\in I}\subseteq A$ such that $\bigwedge _{i\in I}a_i$ exists, $(\bigwedge _{i\in I}a_i)\rightarrow c=\bigvee _{i\in I}(a_i\rightarrow c)$. \[mvlema\] A [Heyting algebra]{} is a residuated lattice $A$ such that, for all $a,b\in A$, $a\odot b=a\wedge b$. In a Heyting algebra $A$ we have: $a\wedge (a\rightarrow b)=a\wedge b$ for all $a,b\in A$ (for a proof see, for instance, [@pic Proposition 1.20, page 17]). A [$G\ddot{o}del$ algebra]{} is a BL-algebra $A$ such that, for all $a,b\in A$, $a\odot b=a\wedge b$, that is: both a Heyting algebra and a BL-algebra. $([0,1],\max ,\min ,\odot _{G},\rightarrow _{G},0,1)$ is a ${\rm G\ddot{o}del}$ algebra. A [product]{} or [PL-algebra]{} is a BL-algebra $A$ that satisfies the following two conditions: - for all $a\in A$, $a\wedge \neg \, a=0$; - for all $a,b,c\in A$, $(\neg \, \neg \, c\odot ((a\odot c)\rightarrow (b\odot c)))\rightarrow (a\rightarrow b)=1$. $([0,1],\max ,\min ,\odot _{P},\rightarrow _{P},0,1)$ is a product algebra. A residuated lattice $A$ is said to be [involutive]{} iff $\neg \, \neg \, a=a$ for all $a\in A$. A residuated lattice $A$ is said to be [divisible]{} iff $a\wedge b=a\odot (a\rightarrow b)$ for all $a,b\in A$. A divisible and involutive residuated lattice is an MV-algebra. [[@cdom], [@haj], [@ior]]{} A residuated lattice $A$ is an MV-algebra iff, for all $a,b\in A$, $(a\rightarrow b)\rightarrow b=(b\rightarrow a)\rightarrow a$. In this case, for all $a,b\in A$, $a\vee b=(a\rightarrow b)\rightarrow b=(b\rightarrow a)\rightarrow a$. \[l2.4\] Throughout the remaining part of this section, let $A$ be a residuated lattice. A [filter]{} of $A$ is a nonempty subset $F$ of $A$ such that, for all $a,b\in A$: - $a,b\in F$ implies $a\odot b\in F$; - $a\in F$ and $a\leq b$ imply $b\in F$. A filter $F$ of $A$ is said to be [proper]{} iff $F\neq A$, which is equivalent to the fact that $0\notin F$. A proper filter $P$ of $A$ is called a [prime filter]{} iff, for all $a,b\in A$, if $a\vee b\in P$, then $a\in P$ or $b\in P$. A maximal element of the set of all proper filters of $A$ is called a [maximal filter]{}. If $F$ is a filter of $A$, then the congruence $\equiv (\mod F)$ associated to $F$ is defined by: for all $a,b\in A$, $a\equiv b(\mod F)$ iff $d_A(a,b)\in F$. It is obvious that $a\equiv b(\mod F)$ iff $a\rightarrow b\in F$ and $b\rightarrow a\in F$. We recall that residuated lattices form an equational class, which ensures us that the quotient set $A/_{\equiv (\mod F)}$ is a residuated lattice, which we denote by $A/F$. For all $a\in A$, we will denote by $a/F$ the congruence class of $A$ with respect to $\equiv (\mod F)$. It is easily seen that: $a/F=1/F$ iff $a\in F$ ([@haj]). A subset $F$ of $A$ is a filter iff $1\in F$ and, for all $a,b\in A$, $a\in F$ and $a\rightarrow b\in F$ imply $b\in F$. [[@kow], [@pic]]{} A proper filter $F$ of $A$ is maximal iff, for all $a\in A\setminus F$, there exists a nonzero natural number $n$ such that $\neg \, (a^{n})\in F$. \[l2.5\] $A$ is said to be [simple]{} iff it has exactly two filters. [[@kow], [@pic]]{} $A$ is simple iff, for all $a\in A\setminus \{1\}$, there exists a nonzero natural number $n$ such that $a^{n}=0$. \[l2.6\] If $s:A\rightarrow L$ is a function, then by the [kernel of $s$]{} we will understand the set $\{a\in A\|s(a)=1\}$, which we will denote ${\rm Ker}(s)$. Notice that, if $s$ is a residuated lattice morphism, then: $s$ is injective iff ${\rm Ker}(s)=\{1\}$. Generalized Bosbach States {#bosbach} ========================== In this section we will present two generalizations for the Bosbach states defined on residuated lattices. We will start from the observation that in the definition of Bosbach states we report essentially to the MV-algebra structure of $[0,1]$. By writing the axioms of Bosbach states in different equivalent ways, there will result two distinct ways of generalizing Bosbach states when we replace the standard MV-algebra $[0,1]$ with an arbitrary residuated lattice. Throughout this section, let $A$ be a residuated lattice. [[@gg], [@lcciu]]{} Let $s:A\rightarrow [0,1]$ be a function such that $s(0)=0$ and $s(1)=1$. Then the following are equivalent: 1. \[p3.1(1)\] for all $a,b\in A$, $1+s(a\wedge b)=s(a\vee b)+s(d_A(a,b))$; 2. \[p3.1(2)\] for all $a,b\in A$, $1+s(a\wedge b)=s(a)+s(a\rightarrow b)$; 3. \[p3.1(3)\] for all $a,b\in A$, $s(a)+s(a\rightarrow b)=s(b)+s(b\rightarrow a)$. \[p3.1\] The proposition above has been proven in [@fla] for Bosbach states defined on pseudo-BL-algebras. Then it was extended to more general cases ([@lcciu1], [@dvura], [@kuhr]). A [Bosbach state]{} on $A$ is a function $s:A\rightarrow [0,1]$ such that $s(0)=0$, $s(1)=1$ and $s$ verifies the equivalent conditions from Proposition \[p3.1\]. \[d3.2\] [[@gg], [@lcciu]]{} Let $s:A\rightarrow [0,1]$ be a Bosbach state. Then, for all $a,b\in A$, we have: 1. \[l3.2(1)\] $s(\neg \, a)=1-s(a)$; 2. \[l3.2(2)\] $s$ is order-preserving: $a\leq b$ implies $s(a)\leq s(b)$; 3. \[l3.2(3)\] $s(a)+s(b)=s(a\vee b)+s(a\wedge b)$. \[l3.2\] Let $s:A\rightarrow [0,1]$ be a Bosbach state. Then, from the fact that $s$ is order-preserving and from Lemma \[l2.2\], (\[l2.2(5)\]) and (\[l2.2nenum\]), we deduce that, for all $a,b\in A$: 1. \[rnenum(1)\] $1-s(a\vee b)+s(a\wedge b)=s(a\vee b)\rightarrow _{L}s(a\wedge b)$ (because $s(a\wedge b)\leq s(a\vee b)$); 2. \[rnenum(2)\] $1-s(d_A(a,b))+s(a\wedge b)=s(d_A(a,b))\rightarrow _{L}s(a\wedge b)$ (because $s(a\wedge b)\leq s(d_A(a,b))$); 3. \[rnenum(3)\] $1-s(a)+s(a\wedge b)=s(a)\rightarrow _{L}s(a\wedge b)$ (because $s(a\wedge b)\leq s(a)$); 4. \[rnenum(4)\] $1-s(a\rightarrow b)+s(a\wedge b)=s(a\rightarrow b)\rightarrow _{L}s(a\wedge b)$ (because $s(a\wedge b)\leq s(a\rightarrow b)$); 5. \[rnenum(5)\] $1-s(a\rightarrow b)+s(b)=s(a\rightarrow b)\rightarrow _{L}s(b)$ (because $s(b)\leq s(a\rightarrow b)$). It follows: - condition (\[p3.1(1)\]) of Proposition \[p3.1\] is equivalent to each of the following two equalities: $(1^{\prime })$ for all $a,b\in A$, $s(d_A(a,b))=s(a\vee b)\rightarrow _{L}s(a\wedge b)$; $(1^{\prime \prime })$ for all $a,b\in A$, $s(a\vee b)=s(d_A(a,b))\rightarrow _{L}s(a\wedge b)$; - condition (\[p3.1(2)\]) of Proposition \[p3.1\] is equivalent to each of the following two equalities: $(2^{\prime })$ for all $a,b\in A$, $s(a\rightarrow b)=s(a)\rightarrow _{L}s(a\wedge b)$; $(2^{\prime \prime })$ for all $a,b\in A$, $s(a)=s(a\rightarrow b)\rightarrow _{L}s(a\wedge b)$; - condition (\[p3.1(3)\]) of Proposition \[p3.1\] is equivalent to the following equality: $(3^{\prime })$ for all $a,b\in A$, $s(a\rightarrow b)\rightarrow _{L}s(b)=s(b\rightarrow a)\rightarrow _{L}s(a)$. Each of the equalities $(1^{\prime })$, $(1^{\prime \prime })$, $(2^{\prime })$, $(2^{\prime \prime })$ and $(3^{\prime })$ can suggest a way to extend the definition of the Bosbach state when the standard MV-algebra $[0,1]$ is replaced by an arbitrary residuated lattice. First, we shall compare these conditions in the general case when the codomain of $s$ is an arbitrary residuated lattice. In the following, let $(L,\vee ,\wedge ,\odot ,\rightarrow ,0,1)$ be a residuated lattice and $s:A\rightarrow L$ be an arbitrary function. If $s(0)=0$ and $s(1)=1$, then the following are equivalent: 1. \[p3.3(1)\] for all $a,b\in A$, $s(d_A(a,b))=s(a\vee b)\rightarrow s(a\wedge b)$; 2. \[p3.3(2)\] for all $a,b\in A$ with $b\leq a$, $s(a\rightarrow b)=s(a)\rightarrow s(b)$; 3. \[p3.3(3)\] for all $a,b\in A$, $s(a\rightarrow b)=s(a)\rightarrow s(a\wedge b)$; 4. \[p3.3(4)\] for all $a,b\in A$, $s(a\rightarrow b)=s(a\vee b)\rightarrow s(b)$. \[p3.3\] Let $a,b\in A$. (\[p3.3(1)\])$\Rightarrow $(\[p3.3(2)\]): Assume $b\leq a$. Then, by Lemma \[l2.2\], (\[l2.2(3)\]), we have that $d_A(a,b)=a\rightarrow b$, so $s(a\rightarrow b)=s(d_A(a,b))=s(a\vee b)\rightarrow s(a\wedge b)=s(a)\rightarrow s(b)$. (\[p3.3(2)\])$\Rightarrow $(\[p3.3(1)\]): By Lemma \[l2.2\], (\[l2.2(9)\]) and (\[l2.2(3)\]), $(a\vee b)\rightarrow (a\wedge b)=(a\rightarrow (a\wedge b))\wedge (b\rightarrow (a\wedge b))=(a\rightarrow a)\wedge (a\rightarrow b)\wedge (b\rightarrow a)\wedge (b\rightarrow b)=(a\rightarrow b)\wedge (b\rightarrow a)=d_A(a,b)$, and $a\wedge b\leq a\vee b$, so $s(d_A(a,b))=s((a\vee b)\rightarrow (a\wedge b))=s(a\vee b)\rightarrow s(a\wedge b)$. (\[p3.3(2)\])$\Rightarrow $(\[p3.3(3)\]): By Lemma \[l2.2\], (\[l2.2(9)\]) and (\[l2.2(3)\]), $a\rightarrow (a\wedge b)=a\rightarrow b$, and $a\wedge b\leq a$, so $s(a\rightarrow b)=s(a\rightarrow (a\wedge b))=s(a)\rightarrow s(a\wedge b)$. (\[p3.3(3)\])$\Rightarrow $(\[p3.3(2)\]): Trivial. (\[p3.3(2)\])$\Leftrightarrow $(\[p3.3(4)\]): Analogous to (\[p3.3(2)\])$\Leftrightarrow $(\[p3.3(3)\]). If $s(0)=0$ and $s(1)=1$, then the following are equivalent: 1. \[p3.4(1)\] for all $a,b\in A$, $s(a\vee b)=s(d_A(a,b))\rightarrow s(a\wedge b)$; 2. \[p3.4(2)\] for all $a,b\in A$, $s(a)=s(a\rightarrow b)\rightarrow s(a\wedge b)$; 3. \[p3.4(3)\] for all $a,b\in A$ with $b\leq a$, $s(a)=s(a\rightarrow b)\rightarrow s(b)$; 4. \[p3.4(4)\] for all $a,b\in A$, $s(a\vee b)=s(a\rightarrow b)\rightarrow s(b)$; 5. \[p3.4(5)\] for all $a,b\in A$, $s(a\rightarrow b)\rightarrow s(b)=s(b\rightarrow a)\rightarrow s(a)$. \[p3.4\] (\[p3.4(1)\])$\Rightarrow $(\[p3.4(3)\]): If $b\leq a$, then $a\vee b=a$, $d_A(a,b)=a\rightarrow b$ (by Lemma \[l2.2\], (\[l2.2(3)\])) and $a\wedge b=b$. (\[p3.4(3)\])$\Rightarrow $(\[p3.4(1)\]): As in the proof of Proposition \[p3.3\], $d_A(a,b)=(a\vee b)\rightarrow (a\wedge b)$, and, since $a\wedge b\leq a\vee b$, we have $s(a\vee b)=s((a\vee b)\rightarrow (a\wedge b))\rightarrow s(a\wedge b)=s(d_A(a,b))\rightarrow s(a\wedge b)$. (\[p3.4(2)\])$\Rightarrow $(\[p3.4(3)\]): Trivial. (\[p3.4(3)\])$\Rightarrow $(\[p3.4(2)\]): Since $a\wedge b\leq a$ and, as in the proof of Proposition \[p3.3\], $a\rightarrow (a\wedge b)=a\rightarrow b$, we have: $s(a)=s(a\rightarrow (a\wedge b))\rightarrow s(a\wedge b)=s(a\rightarrow b)\rightarrow s(a\wedge b)$. (\[p3.4(3)\])$\Leftrightarrow $(\[p3.4(4)\]): Analogous to the proof of (\[p3.4(3)\])$\Leftrightarrow $(\[p3.4(2)\]). (\[p3.4(4)\])$\Rightarrow $(\[p3.4(5)\]): $s(a\rightarrow b)\rightarrow s(b)=s(a\vee b)=s(b\vee a)=s(b\rightarrow a)\rightarrow s(a)$. (\[p3.4(5)\])$\Rightarrow $(\[p3.4(2)\]): If $b\leq a$, then, by Lemma \[l2.2\], (\[l2.2(2)\]) and (\[l2.2(3)\]), $s(a)=1\rightarrow s(a)=s(1)\rightarrow s(a)=s(b\rightarrow a)\rightarrow s(a)=s(a\rightarrow b)\rightarrow s(b)$. Propositions \[p3.3\] and \[p3.4\] suggest the following generalizations of Bosbach states: $s$ is called a [generalized Bosbach state of type I]{} (or, in brief, a [state of type I]{} or [a type I state]{}) iff it verifies the equivalent conditions from Proposition \[p3.3\]. $s$ is called a [generalized Bosbach state of type II]{} (or, in brief, a [state of type II]{} or [a type II state]{}) iff it verifies the equivalent conditions from Proposition \[p3.4\]. $s$ is called a [generalized Bosbach state of type III]{} (or, in brief, a [state of type III]{} or [a type III state]{}) iff it is both a generalized Bosbach state of type I and a generalized Bosbach state of type II. \[defsbg\] Any residuated lattice morphism $s:A\rightarrow L$ is an order-preserving type I state. The identity morphism $id_A:A\rightarrow A$ is a type II state iff $A$ is an MV-algebra. Indeed, any residuated lattice morphism verifies condition (\[p3.3(2)\]) from Proposition \[p3.3\]. For the remark concerning the identity morphism see Corollary \[c3.13\]. \[sbgex1\] In [@ciudvuhyc Definition 3.1], the notion of state-operator on a BL-algebra is introduced. Condition (\[p3.3(3)\]) from Proposition \[p3.3\] is exactly axiom (2) from this definition, thus any state-operator is a type I state. Moreover, according to [@ciudvuhyc Lemma 3.5, (c)], any state-operator is an order-preserving type I state. Let $A$ be a Heyting algebra and $a\in A$. We denote by $s_a:A\rightarrow A$ the function defined by: for all $x\in A$, $s_a(x)=a\rightarrow x$. For all $x,y\in A$, $s_a(x)\rightarrow s_a(x\wedge y)=(a\rightarrow x)\rightarrow (a\rightarrow (x\wedge y))=(a\wedge (a\rightarrow x))\rightarrow (x\wedge y)=(a\wedge x)\rightarrow (x\wedge y)=((a\wedge x)\rightarrow x)\wedge ((a\wedge x)\rightarrow y)=1\wedge ((a\wedge x)\rightarrow y)=(a\wedge x)\rightarrow y=a\rightarrow (x\rightarrow y)=s_a(x\rightarrow y)$, by Lemma \[l2.2\], (\[l2.2(7)\]), a property of Heyting algebras from Section \[preliminaries\] and Lemma \[l2.2\], (\[l2.2(9)\]) and (\[l2.2(3)\]). Thus $s_a$ is an order-preserving type I state, by Lemma \[l2.2\], (\[l2.2(4)\]) and Proposition \[p3.3\], (\[p3.3(3)\]). \[sbgex2\] Let $(A,\leq ,0,1)$ be a bounded chain. By denoting, for all $x,y\in A$, $x\wedge y=\inf \{x,y\}$, $x\vee y=\sup \{x,y\}$ and $x\rightarrow y=\begin{cases}1, & x\leq y,\\ y, & x>y,\end{cases}$ $(A,\vee ,\wedge ,0,1)$ becomes a Heyting algebra. The verification is immediate; this is an example of Heyting algebra from [@bal]. Let $a\in A\setminus \{0\}$, $[0,a)=\{x\in A\|x<a\}$ and $f:[0,a)\rightarrow A$ a strictly order-preserving function with $f(0)=0$. We consider the function $f_a:A\rightarrow A$, defined by: for all $x\in A$, $f_a(x)=\begin{cases}f(x), & x<a,\\ 1, & x\geq a.\end{cases}$ Then $f_a$ is an order preserving type I state. Indeed, $f_a$ is obviously order-preserving and let $x,y\in A$ with $y\leq x$. We have to prove that $f_a(x\rightarrow y)=f_a(x)\rightarrow f_a(y)$, which is clear for $x=y$, as Lemma \[l2.2\], (\[l2.2(3)\]) shows. So let $y<x$ now. Since $x\rightarrow y=y$, we have to prove that $f_a(y)=f_a(x)\rightarrow f_a(y)$. We have three cases: - $y<x<a$. Then $f_a(x)\rightarrow f_a(y)=f(x)\rightarrow f(y)=f(y)=f_a(y)$. - $y<a\leq x$. Then $f_a(x)\rightarrow f_a(y)=1\rightarrow f_a(y)=f(y)=f_a(y)$, by Lemma \[l2.2\], (\[l2.2(2)\]). - $a\leq y<x$. Then $f_a(x)\rightarrow f_a(y)=1\rightarrow 1=1=f_a(y)$, by Lemma \[l2.2\], (\[l2.2(2)\]). So $f_a$ is a type I state, by Proposition \[p3.3\], (\[p3.3(2)\]). Now assume that the chain $A$ is a complete lattice and let $s:A\rightarrow A$ be an arbitrary strictly order-preserving type I state. We denote $a=\inf \{x\in A\|s(x)=1\}$ and let $0\leq y<x<a$. Then $s(y)=s(x)\rightarrow s(y)$ and $s(y)<1$, so, by the law of residuation and Lemma \[l2.2\], (\[l2.2nenum2\]), it follows that $s(y)<s(x)$. Therefore, $f=s\mid _{[0,a)}:[0,a)\rightarrow A$ is strictly order-preserving and, obviously, $s=f_a$. \[sbgex3\] Let $(A,\leq ,0,1)$, with the Heyting algebra structure from Example \[sbgex3\]. Let $s:A\rightarrow A$ be a type II state. Then, for all $x,y\in A$ with $y<x$, $s(x)=s(x\rightarrow y)\rightarrow s(y)=s(y)\rightarrow s(y)=1$, by Proposition \[p3.4\], (\[p3.4(3)\]) and Lemma \[l2.2\], (\[l2.2(3)\]). So $s(x)=\begin{cases}0, & x=0,\\ 1, & x>0.\end{cases}$ \[sbgex4\] If $s$ is a generalized Bosbach state of type I, then, for all $a,b\in A$: 1. \[p3.6(1)\] $s(\neg \, a)=\neg \, s(a)$; 2. \[p3.6(2)\] $s(a\vee b)\rightarrow s(a)=s(b)\rightarrow s(a\wedge b)$; 3. \[p3.6(3)\] $s((a\rightarrow b)\rightarrow b)=s(a\rightarrow b)\rightarrow s(b)$; 4. \[p3.6(4)\] $s((a\rightarrow b)\rightarrow b)=(s(a\vee b)\rightarrow s(b))\rightarrow s(b)$; 5. \[p3.6(5)\] $s(a\vee b)\rightarrow (s(a)\wedge s(b))=(s(a)\vee s(b))\rightarrow s(a\wedge b)$; 6. \[p3.6(6)\] $s(a)\odot s(a\rightarrow (a\odot b))\leq s(a\odot b)$. \[p3.6\] (\[p3.6(1)\]): $s(\neg \, a)=s(a\rightarrow 0)=s(a)\rightarrow s(0)=s(a)\rightarrow 0=\neg \, s(a)$ (see Proposition \[p3.3\], (\[p3.3(2)\])). (\[p3.6(2)\]): By Proposition \[p3.3\], (\[p3.3(3)\]) and (\[p3.3(4)\]). (\[p3.6(3)\]): By Proposition \[p3.3\], (\[p3.3(2)\]), and Lemma \[l2.2\], (\[l2.2nenum\]). (\[p3.6(4)\]): By (\[p3.6(3)\]) and Proposition \[p3.3\], (\[p3.3(4)\]), $s((a\rightarrow b)\rightarrow b)=s(a\rightarrow b)\rightarrow s(b)=(s(a\vee b)\rightarrow s(b))\rightarrow s(b)$. (\[p3.6(5)\]): By Lemma \[l2.2\], (\[l2.2(9)\]) and Proposition \[p3.6\], (\[p3.6(2)\]), $s(a\vee b)\rightarrow (s(a)\wedge s(b))=(s(a\vee b)\rightarrow s(a))\wedge (s(a\vee b)\rightarrow s(b))=(s(b)\rightarrow s(a\wedge b))\wedge (s(a)\rightarrow s(a\wedge b))=(s(a)\vee s(b))\rightarrow s(a\wedge b)$. (\[p3.6(6)\]): By Lemma \[l2.2\], (\[l2.2(5)\]), Proposition \[p3.3\], (\[p3.3(2)\]) and Lemma \[l2.2\], (\[l2.2(6)\]), $a\odot b\leq a$, so $s(a\rightarrow (a\odot b))=s(a)\rightarrow s(a\odot b)$, hence $s(a)\odot s(a\rightarrow (a\odot b))=s(a)\odot (s(a)\rightarrow s(a\odot b))\leq s(a\odot b)$. In the case when $L$ is the standard MV-algebra $[0,1]$, order-preserving type I states $s:A\rightarrow [0,1]$ coincide with Bosbach states on $A$, as the identity (\[p3.3(3)\]) from Proposition \[p3.3\] is equivalent to the identity $(2^{\prime })$, and type II states $s:A\rightarrow [0,1]$ coincide with Bosbach states on $A$, as the identity (\[p3.4(5)\]) from Proposition \[p3.4\] is equivalent to the identity $(3^{\prime })$. \[r3.7\] Let $B$ be a residuated lattice, $s:B\rightarrow L$ be a function and $f:A\rightarrow L$ be a residuated lattice morphism. Then, by Proposition \[p3.3\], (\[p3.3(3)\]), if $s$ is a type I state, then $s\circ f:A\rightarrow L$ is a type I state, and if, moreover, $s$ is order-preserving and $f$ is order-preserving, then $s\circ f$ is order-preserving. By Proposition \[p3.4\], (\[p3.4(2)\]), if $s$ is a type II state, then $s\circ f$ is a type II state. Thus, if $s$ is a type III state, then $s\circ f$ is a type III state. \[rmorf\] Let $s:A\rightarrow L$ be an order-preserving type I state. Then, for all $a,b,x,y\in A$: 1. \[p3.8(1)\] $s(a)\odot s(b)\leq s(a\odot b)$; 2. \[p3.8(2)\] $s(a)\ominus s(b)\leq s(a\ominus b)$; 3. \[p3.8(3)\] $s(a\rightarrow b)\leq s(a)\rightarrow s(b)$; 4. \[p3.8(4)\] $s(a\rightarrow b)\odot s(b\rightarrow a)\leq d_L(s(a),s(b))$; 5. \[p3.8(5)\] $s(d_A(a,b))\leq d_L(s(a),s(b))$; 6. \[p3.8(6)\] $s(d_A(a,x))\odot s(d_A(b,y))\leq d_L(s(d_A(a,b)),s(d_A(x,y)))$. \[p3.8\] (\[p3.8(1)\]) By the law of residuation, the fact that $s$ is order-preserving, Lemma \[l2.2\], (\[l2.2(5)\]), Proposition \[p3.3\], (\[p3.3(2)\]) and again the law of residuation, $b\leq a\rightarrow (a\odot b)$, hence $s(b)\leq s(a\rightarrow (a\odot b))=s(a)\rightarrow s(a\odot b)$, therefore $s(a)\odot s(b)\leq s(a\odot b)$. (\[p3.8(2)\]) By Proposition \[p3.6\], (\[p3.6(1)\]) and (\[p3.8(1)\]) from the current proposition, $s(a)\ominus s(b)=s(a)\odot \neg \, s(b)=s(a)\odot s(\neg \, b)\leq s(a\odot \neg \, b)=s(a\ominus b)$. (\[p3.8(3)\]) By Proposition \[p3.3\], (\[p3.3(3)\]) and Lemma \[l2.2\], (\[l2.2(4)\]), $s(a\rightarrow b)\leq s(a)\rightarrow s(a\wedge b)\leq s(a)\rightarrow s(b)$. (\[p3.8(4)\]) By (\[p3.8(3)\]) from the current proposition and (\[l2.2nenum2\]) and (\[l2.2(5)\]) from Lemma \[l2.2\], $s(a\rightarrow b)\odot s(b\rightarrow a)\leq (s(a)\rightarrow s(b))\odot (s(b)\rightarrow s(a))\leq d_L(s(a),s(b))$. (\[p3.8(5)\]) By the fact that $s$ is order-preserving and (\[p3.8(3)\]), $s(d_A(a,b))=s((a\rightarrow b)\wedge (b\rightarrow a))\leq s(a\rightarrow b)\wedge s(b\rightarrow a)\leq (s(a)\rightarrow s(b))\wedge (s(b)\rightarrow s(a))=d_L(s(a),s(b))$. (\[p3.8(6)\]) By (\[p3.8(1)\]) and (\[p3.8(5)\]) from the current proposition, along with Lemma \[lA\], (\[lA(5)\]), $s(d_A(a,x))\odot s(d_A(b,y))\leq s(d_A(a,x)\odot d_A(b,y))\leq s(d_A(d_A(a,b),d_A(x,y)))\leq d_L(s(d_A(a,b)),s(d_A(x,y)))$. Let $s:A\rightarrow L$ be a type II state. Then, for all $a,b\in A$: 1. \[p3.9(1)\] $b\leq a$ implies $s(b)\leq s(a)$ (that is $s$ is order-preserving); 2. \[p3.9(2)\] $s(a)=\neg \, s(\neg \, a)$; 3. \[p3.9(3)\] $s(\neg \, \neg \, a)=s(a)=\neg \, \neg \, s(a)$; 4. \[p3.9(4)\] $s(a\rightarrow b)=s((a\rightarrow b)\rightarrow b)\rightarrow s(b)$; 5. \[p3.9(6)\] $s(\neg \, a)=\neg \, s(a)$; 6. \[p3.9(5)\] $s(a\odot b)=\neg \, s(a\rightarrow \neg \, b)$. \[p3.9\] (\[p3.9(1)\]) By Lemma \[l2.2\], (\[l2.2nenum\]) and Proposition \[p3.4\], (\[p3.4(3)\]), $s(b)\leq s(a\rightarrow b)\rightarrow s(b)=s(a)$. (\[p3.9(2)\]) By Proposition \[p3.4\], (\[p3.4(3)\]), $s(a)=s(a\rightarrow 0)\rightarrow 0=s(\neg \, a)\rightarrow 0=\neg \, s(\neg \, a)$. (\[p3.9(3)\]) By (\[p3.9(2)\]) and Lemma \[l2.3\], (\[l2.3(2)\]), $s(\neg \, \neg \, a)=\neg \, s(\neg \, \neg \, \neg \, a)=\neg \, s(\neg \, a)=s(a)$, and also $\neg \, \neg \, s(a)=\neg \, \neg \, \neg \, s(\neg \, a)=\neg \, s(\neg \, a)=s(a)$. (\[p3.9(4)\]) By Lemma \[l2.2\], (\[l2.2nenum\]) and Proposition \[p3.4\], (\[p3.4(3)\]). (\[p3.9(6)\]) By (\[p3.9(4)\]) and (\[p3.9(3)\]), $s(\neg \, a)=s(a\rightarrow 0)=s((a\rightarrow 0)\rightarrow 0)\rightarrow s(0)=s(\neg \, \neg \, a)\rightarrow 0=\neg \, s(\neg \, \neg \, a)=\neg \, s(a)$. (\[p3.9(5)\]) By (\[p3.9(2)\]) and Lemma \[l2.3\], (\[l2.3(5)\]), $s(a\odot b)=\neg \, s(\neg \, (a\odot b))=\neg \, s(a\rightarrow \neg \, b)$. Let $A$ be a totally ordered product algebra and $s:A\rightarrow A$ a type II state. Since, for all $a\in A$, $a\wedge \neg \, a\in \{a,\neg \, a\}$, it follows that, for all $a\in A\setminus \{0\}$, $\neg \, a=0$. By Lemma \[l2.3\], (\[0neg1\]), for all $a\in A$, $s(\neg \, a)=\begin{cases}s(1)=1, & {\rm if}\ a=0,\\ s(0)=0, & {\rm if}\ a\neq 0.\end{cases}$ By Proposition \[p3.9\], (\[p3.9(6)\]) and Lemma \[l2.3\], (\[0neg1\]), it follows that there exists a unique type II state $s:A\rightarrow A$, namely, for all $a\in A$, $s(a)=\neg \, s(\neg \, a)=\begin{cases}0, & {\rm if}\ a=0,\\ 1, & {\rm if}\ a\neq 0,\end{cases}$ as this is indeed a type II state, by Proposition \[p3.4\], (\[p3.4(3)\]) and Lemma \[l2.2\], (\[l2.2(2)\]) and (\[l2.2(3)\]). \[r3.6\] In general, if $s:A\rightarrow L$ is a state of type I, then it is not necessarily order-preserving (that is: $a,b\in A$ and $a\leq b$ do not necessarily imply $s(a)\leq s(b)$) and, even if it is order-preserving, it is not necessarily a state of type II. Indeed, let us consider the following example of residuated lattice from [[@kow], [@gal], [@ior]]{}: $A=\{0,a,b,c,d,1\}$, with the following partial order relation and operations: (60,100)(0,0) (37,11) (35,0)[$0$]{} (37,11)[(3,4)[12]{}]{} (49,27) (53,24)[$d$]{} (49,27)[(0,1)[20]{}]{} (49,47) (53,44)[$c$]{} (49,47)[(-3,4)[12]{}]{} (37,63) (41,63)[$a$]{} (37,11)[(-1,1)[26]{}]{} (11,37) (3,34)[$b$]{} (11,37)[(1,1)[26]{}]{} (37,63)[(0,1)[20]{}]{} (37,83) (35,85)[$1$]{} [cc]{} $\rightarrow $ $0$ $a$ $b$ $c$ $d$ $1$ ---------------- ----- ----- ----- ----- ----- ----- $0$ $1$ $1$ $1$ $1$ $1$ $1$ $a$ $0$ $1$ $b$ $c$ $c$ $1$ $b$ $c$ $1$ $1$ $c$ $c$ $1$ $c$ $b$ $1$ $b$ $1$ $a$ $1$ $d$ $b$ $1$ $b$ $1$ $1$ $1$ $1$ $0$ $a$ $b$ $c$ $d$ $1$ & $\odot $ $0$ $a$ $b$ $c$ $d$ $1$ ---------- ----- ----- ----- ----- ----- ----- $0$ $0$ $0$ $0$ $0$ $0$ $0$ $a$ $0$ $a$ $b$ $d$ $d$ $a$ $b$ $0$ $b$ $b$ $0$ $0$ $b$ $c$ $0$ $d$ $0$ $d$ $d$ $c$ $d$ $0$ $d$ $0$ $d$ $d$ $d$ $1$ $0$ $a$ $b$ $c$ $d$ $1$ Let us determine the generalized Bosbach states $s_i:A\rightarrow A$. The type I states from $A$ to $A$ are: $x$ $0$ $a$ $b$ $c$ $d$ $1$ ---------- ----- ----- ----- ----- ----- ----- $s_1(x)$ $0$ $a$ $0$ $1$ $a$ $1$ $s_2(x)$ $0$ $a$ $b$ $c$ $d$ $1$ $s_3(x)$ $0$ $1$ $0$ $1$ $1$ $1$ $s_4(x)$ $0$ $1$ $b$ $c$ $c$ $1$ $s_5(x)$ $0$ $1$ $c$ $b$ $b$ $1$ $s_6(x)$ $0$ $1$ $1$ $0$ $0$ $1$ Out of these, the only order-preserving ones are $s_2$, $s_3$, $s_4$, $s_5$ and $s_6$. Indeed, $s_1$ is not order-preserving, as $c\leq a$ and $s_1(c)=1>s_1(a)=a$. The type II states from $A$ to $A$ are $s_3$, $s_4$, $s_5$ and $s_6$. Let $A$ and $L$ be divisible residuated lattices and $s:A\rightarrow L$ an order-preserving type I state. Then, for all $a,b\in A$: 1. \[p3.10(1)\] $s(a\odot b)=s(a)\odot s(a\rightarrow (a\odot b))$; 2. \[p3.10(2)\] $s(a\wedge b)=s(a)\odot s(a\rightarrow b)$. \[p3.10\] (\[p3.10(1)\]) By Lemma \[l2.2\], (\[l2.2(5)\]) and Proposition \[p3.3\], (\[p3.3(2)\]), $s(a)\odot s(a\rightarrow (a\odot b))=s(a)\odot (s(a)\rightarrow s(a\odot b))=s(a)\wedge s(a\odot b)=s(a\odot b)$. (\[p3.10(2)\]) $a\rightarrow (a\odot (a\rightarrow b))=a\rightarrow (a\wedge b)=(a\rightarrow a)\wedge (a\rightarrow b)=a\rightarrow b$, by (\[l2.2(9)\]) and (\[l2.2(3)\]) from Lemma \[l2.2\]. Let $A$ be a residuated lattice, $L$ an MV-algebra and $s:A\rightarrow L$ an order-preserving type I state. Then, for all $a,b\in A$, $s(a\vee b)=s((a\rightarrow b)\rightarrow b)=s((b\rightarrow a)\rightarrow a)$. \[l3.11\] By Proposition \[p3.6\], (\[p3.6(4)\]), Lemma \[l2.4\] and Lemma \[l2.2\], (\[l2.2(3)\]) and (\[l2.2(2)\]), $s((a\rightarrow b)\rightarrow b)=(s(a\vee b)\rightarrow s(b))\rightarrow s(b)=(s(b)\rightarrow s(a\vee b))\rightarrow s(a\vee b)=1\rightarrow s(a\vee b)=s(a\vee b)=s(b\vee a)=s((b\rightarrow a)\rightarrow a)$. Let $A$ be a residuated lattice, $L$ an MV-algebra and $s:A\rightarrow L$ a function. Then the following are equivalent: 1. \[p3.12(1)\] $s$ is an order-preserving type I state; 2. \[p3.12(2)\] $s$ is a type II state. \[p3.12\] (\[p3.12(1)\])$\Rightarrow $(\[p3.12(2)\]): Let $s$ be an order-preserving type I state. By Proposition \[p3.6\], (\[p3.6(3)\]) and Lemma \[l3.11\], for all $a,b\in A$, $s(a\rightarrow b)\rightarrow s(b)=s((a\rightarrow b)\rightarrow b)=s((b\rightarrow a)\rightarrow a)=s(b\rightarrow a)\rightarrow s(a)$, hence, by Proposition \[p3.4\], (\[p3.4(5)\]), $s$ is a type II state. (\[p3.12(2)\])$\Rightarrow $(\[p3.12(1)\]): Let $s$ be a type II state and $a,b\in A$. Then, by Proposition \[p3.4\], (\[p3.4(2)\]), $s(a\rightarrow b)\rightarrow s(a\wedge b)=s(a)$, therefore, by Lemma \[l2.4\], Lemma \[l2.2\], (\[l2.2(5)\]), Proposition \[p3.9\], (\[p3.9(1)\]) and Lemma \[l2.2\], (\[l2.2(3)\]) and (\[l2.2(2)\]), $s(a)\rightarrow s(a\wedge b)=(s(a\rightarrow b)\rightarrow s(a\wedge b))\rightarrow s(a\wedge b)=(s(a\wedge b)\rightarrow s(a\rightarrow b))\rightarrow s(a\rightarrow b)=1\rightarrow s(a\rightarrow b)=s(a\rightarrow b)$. Hence, by Proposition \[p3.3\], (\[p3.3(3)\]), $s$ is a type I state. By Proposition \[p3.9\], \[p3.9(1)\], $s$ is also order-preserving. Let $A$ be a residuated lattice. Then the following are equivalent: 1. \[c3.13(1)\] $A$ is an MV-algebra; 2. \[c3.13(2)\] any order-preserving type I state $s:A\rightarrow A$ is a type II state. \[c3.13\] (\[c3.13(1)\])$\Rightarrow $(\[c3.13(2)\]): By Proposition \[p3.12\]. (\[c3.13(2)\])$\Rightarrow $(\[c3.13(1)\]): The identity $id_A:A\rightarrow A$ obviously is an order-preserving type I state. Hence it is also a type II state, and this condition on the identity is sufficient for this implication to take place. By Proposition \[p3.4\], (\[p3.4(5)\]), for all $a,b\in A$, we have: $(a\rightarrow b)\rightarrow b=(b\rightarrow a)\rightarrow a$. By Lemma \[l2.4\], it follows that $A$ is an MV-algebra. Let $s:A\rightarrow L$ be a type III state. Then, for all $a,b\in A$, $s((a\rightarrow b)\rightarrow b)=s((b\rightarrow a)\rightarrow a)$. \[p3.14\] By Proposition \[p3.6\], (\[p3.6(3)\]) and Proposition \[p3.4\], (\[p3.4(5)\]), $s((a\rightarrow b)\rightarrow b)= s(a\rightarrow b)\rightarrow s(b)=s(b\rightarrow a)\rightarrow s(a)=s((b\rightarrow a)\rightarrow a)$. In Proposition \[p3.14\], do we have $s((a\rightarrow b)\rightarrow b)=s((b\rightarrow a)\rightarrow a)=s(a\vee b)$? Let $A$ be an MV-algebra, $L$ a residuated lattice and $s:A\rightarrow L$ a function such that $s(0)=0$ and $s(1)=1$. Then the following are equivalent: 1. \[pnoua(1)\] $s$ is an order-preserving type I state; 2. \[pnoua(2)\] for all $a,b\in A$, we have: \(a) $s(\neg \, a)=\neg \, s(a)$; \(b) $s(a\rightarrow b)\rightarrow (s(a)\rightarrow s(b))=1$; \(c) $s(a\oplus b)=(s(a)\rightarrow s(a\odot b))\rightarrow s(b)$. \[pnoua\] (\[pnoua(1)\])$\Rightarrow $(\[pnoua(2)\]): Let $s:A\rightarrow L$ be an order-preserving type I state. (a) is Proposition \[p3.6\], (\[p3.6(1)\]) and (b) results from Proposition \[p3.8\], (\[p3.8(3)\]) and Lemma \[l2.2\], (\[l2.2(3)\]). Let us prove (c) now. By Lemma \[mvlema\], (\[mvdemorgan2\]), (\[mvdemorgan1\]), (\[mvdistrib\]) and (\[mvsum\]), for all $a,b\in A$, we have: $(a\rightarrow (a\odot b))\rightarrow b=\neg \, (\neg \, a\oplus (a\odot b))\oplus b=(\neg \, \neg \, a\odot \neg \, (a\odot b))\oplus b=(\neg \, \neg \, a\odot (\neg \, a\oplus \neg \, b))\oplus b=(a\odot (\neg \, a\oplus \neg \, b))\oplus b= (a\wedge \neg \, b)\oplus b=(a\oplus b)\wedge (\neg \, b\oplus b)=a\oplus b$. But, by Lemma \[l2.2\], (\[l2.2(5)\]) and the law of residuation, $a\odot b\leq a$ and $b\leq a\rightarrow (a\odot b)$, hence, by Proposition \[p3.3\], (\[p3.3(2)\]), $s(a\oplus b)=s((a\rightarrow (a\odot b))\rightarrow b)=s(a\rightarrow (a\odot b))\rightarrow s(b)=(s(a)\rightarrow s(a\odot b))\rightarrow s(b)$. (\[pnoua(2)\])$\Rightarrow $(\[pnoua(1)\]): Assume that $s$ satisfies (a), (b) and (c). (b) immediately implies that $s$ is order-preserving, as shown by Lemma \[l2.2\], (\[l2.2(3)\]). Now let $a,b\in A$ with $b\leq a$, thus $\neg \, a\odot b=0$, by Lemma \[l2.3\], (\[l2.3(2)\]) and (\[l2.3(1)\]). By applying (a) and (c) we obtain: $s(a\rightarrow b)=s(\neg \, a\oplus b)=(s(\neg \, a)\rightarrow s(\neg \, a\odot b))\rightarrow s(b)=(s(\neg \, a)\rightarrow s(0))\rightarrow s(b)=(s(\neg \, a)\rightarrow 0)\rightarrow s(b)=\neg \, s(\neg \, a)\rightarrow s(b)=s(\neg \, \neg \, a)\rightarrow s(b)=s(a)\rightarrow s(b)$. Therefore $s$ is an order-preserving type I state. Conditions (a)-(c) from the previous proposition represent the algebraic form of the axioms $(FP_1)-(FP_3)$ from [@fla1 page 327] in the context of probabilistic many-valued logic FP(Ł$_k$,Ł), where Ł$_k$ is the $k$-valued Łukasiewicz logic and Łis the infinite-valued Łukasiewicz logic. \[r3.26\] Notice that in the example from Remark \[r3.6\] all type II states from $A$ to $A$ are type I states. This is the case for all the numerous examples of finite residuated lattices we considered, whose generalized Bosbach states we determined by means of a small computer program, including the cases where the domain was different from the codomain. In addition to that, it can be easily proven that, for any pair of residuated lattices $A$ and $L$ which are each determined by one of the three fundamental continuous t-norms, all type II states from $A$ to $L$ are type I states. However, we have been unable to prove this in the general case and therefore we mention it as an open problem. Prove that, if $s:A\rightarrow L$ is a type II state, then $s$ is a type I state. Obviously, the definition of type I and type II states can be extended to non-commutative residuated lattices, pseudo-BCK-algebras, pseudo-hoops and so on. It remains to be investigated, for each o these cases, to what extent an interesting theory of generalized Bosbach states can be developped. Properties of Generalized Bosbach States {#filtcan} ======================================== In this section we study properties of the quotient residuated lattice $A/{\rm Ker}(s)$, where ${\rm Ker}(s)$ is the canonical filter associated with a (type I or type II) generalized Bosbach state $s:A\rightarrow L$. We introduce the notion of state-morphism in our context, then the state-morphisms are characterized in terms of ${\rm Ker}(s)$ and $A/{\rm Ker}(s)$. Let $A$ and $L$ be two nontrivial residuated lattices. Let $s:A\rightarrow L$ be an order-preserving type I state or a type II state. Then ${\rm Ker}(s)$ is a proper filter of $A$. \[l4.1\] Obviously, $1\in {\rm Ker}(s)$ and $0\notin {\rm Ker}(s)$. Now let $a,b\in A$ such that $a,a\rightarrow b\in {\rm Ker}(s)$, that is $s(a)=s(a\rightarrow b)=1$. We must prove that $b\in {\rm Ker}(s)$, that is $s(b)=1$. If $s$ is an order-preserving type I state, then, by Proposition \[p3.3\], (\[p3.3(3)\]) and Lemma \[l2.2\], (\[l2.2(2)\]), $1=s(a\rightarrow b)=s(a)\rightarrow s(a\wedge b)=1\rightarrow s(a\wedge b)=s(a\wedge b)\leq s(b)$, thus $s(b)=1$. If $s$ is a type II state, then, by Lemma \[l2.2\], (\[l2.2(2)\]), Proposition \[p3.4\], (\[p3.4(5)\]) and Lemma \[l2.2\], (\[l2.2(3)\]), $s(b)=1\rightarrow s(b)=s(a\rightarrow b)\rightarrow s(b)=s(b\rightarrow a)\rightarrow s(a)=s(b\rightarrow a)\rightarrow 1=1$. Let $s:A\rightarrow L$ be an order-preserving type I state or a type II state and $a,b\in A$. If $a/{\rm Ker}(s)=b/{\rm Ker}(s)$, then $s(a)=s(b)=s(a\vee b)=s(a\wedge b)$. \[l4.3\] Assume $a/{\rm Ker}(s)=b/{\rm Ker}(s)$, that is $d_A(a,b)\in {\rm Ker}(s)$, which means that $s(d_A(a,b))=1$. If $s$ is an order-preserving type I state, then, by Proposition \[p3.3\], (\[p3.3(1)\]) and Lemma \[l2.2\], (\[l2.2(3)\]), $1=s(d_A(a,b))=s(a\vee b)\rightarrow s(a\wedge b)$, so $s(a\vee b)\leq s(a\wedge b)$. But $s(a\wedge b)\leq s(a),s(b)\leq s(a\vee b)$, as $s$ is order-preserving. Therefore $s(a)=s(b)=s(a\vee b)=s(a\wedge b)$. If $s$ is a type II state, then, by Proposition \[p3.4\], (\[p3.4(1)\]) and Lemma \[l2.2\], (\[l2.2(2)\]), $s(a\vee b)=s(d_A(a,b))\rightarrow s(a\wedge b)=1\rightarrow s(a\wedge b)=s(a\wedge b)$. But, by Proposition \[p3.9\], (\[p3.9(1)\]), $s(a\wedge b)\leq s(a),s(b)\leq s(a\vee b)$. Therefore $s(a)=s(b)=s(a\vee b)=s(a\wedge b)$. Let $s:A\rightarrow L$ be an order-preserving type I state and $a,b\in A$. Then the following are equivalent: 1. \[p4.4(1)\] $a/{\rm Ker}(s)=b/{\rm Ker}(s)$; 2. \[p4.4(2)\] $s(a\vee b)=s(a\wedge b)$; 3. \[p4.4(3)\] $s(a)=s(b)=s(a\vee b)$; 4. \[p4.4(4)\] $s(a)=s(b)=s(a\wedge b)$. \[p4.4\] The implications (\[p4.4(1)\])$\Rightarrow $(\[p4.4(2)\]),(\[p4.4(3)\]),(\[p4.4(4)\]) result from Lemma \[l4.3\], and the implications, (\[p4.4(2)\])$\Rightarrow $(\[p4.4(3)\]),(\[p4.4(4)\]) result from the fact that $s$ is order-preserving. (\[p4.4(3)\])$\Rightarrow $(\[p4.4(4)\]): By Lemma \[l2.2\], (\[l2.2(3)\]), and Proposition \[p3.6\], (\[p3.6(2)\]), $1=s(a\vee b)\rightarrow s(a)=s(b)\rightarrow s(a\wedge b)$, hence $s(b)\leq s(a\wedge b)$. But $s(a\wedge b)\leq s(b)$, as $s$ is order-preserving. So that $s(b)=s(a\wedge b)$. (\[p4.4(3)\])$\Rightarrow $(\[p4.4(4)\]): By Proposition \[p3.6\], (\[p3.6(2)\]) and Lemma \[l2.2\], (\[l2.2(3)\]), $s(a\vee b)\rightarrow s(a)=s(b)\rightarrow s(a\wedge b)=1$, thus $s(a\vee b)\leq s(a)$. But $s$ is order-preserving and so $s(a)\leq s(a\vee b)$. Hence $s(a)=s(a\vee b)$. (\[p4.4(3)\])$\Rightarrow $(\[p4.4(1)\]): By Proposition \[p3.3\], (\[p3.3(4)\]) and Lemma \[l2.2\], (\[l2.2(3)\]), $s(a\rightarrow b)=s(a\vee b)\rightarrow s(b)=1$, so $a\rightarrow b\in {\rm Ker}(s)$. Analogously, $b\rightarrow a\in {\rm Ker}(s)$. Thus $a/{\rm Ker}(s)=b/{\rm Ker}(s)$. Let $s:A\rightarrow L$ be an order-preserving type I state, respectively a type II state. We consider the quotient residuated lattice $A/{\rm Ker}(s)$. By Lemma \[l4.3\], we can define a function $\overline{s}:A/{\rm Ker}(s)\rightarrow L$, for all $a\in A$, $\overline{s}(a/{\rm Ker}(s))=s(a)$. It easily follows that $\overline{s}$ is an order-preserving type I state, respectively a type II state. Assume that the residuated lattice $L$ is involutive and $s:A\rightarrow L$ is an order-preserving type I state. Then $A/{\rm Ker}(s)$ is involutive. \[p4.5\] Let $a\in A$. By Proposition \[p3.6\], (\[p3.6(1)\]), $s(\neg \, \neg \, a)=\neg \, \neg \, s(a)=s(a)$. By Lemma \[l2.3\], (\[l2.3(2)\]), $a\vee \neg \, \neg \, a=\neg \, \neg \, a$, so $s(a\vee \neg \, \neg \, a)=s(\neg \, \neg \, a)$. It follows that $s(a\vee \neg \, \neg \, a)=s(\neg \, \neg \, a)=s(a)$, so, by Proposition \[p4.4\], $\neg \, \neg \, a/{\rm Ker}(s)=a/{\rm Ker}(s)$, thus $A/{\rm Ker}(s)$ is involutive. Assume that $A$ is divisible, $L$ is involutive and $s:A\rightarrow L$ is an order-preserving type I state. Then $A/{\rm Ker}(s)$ is an MV-algebra. \[c4.6\] It is easily seen that $A/{\rm Ker}(s)$ is divisible, and, by Proposition \[p4.5\], it is also involutive, hence it is an MV-algebra. Let $A$ be an MTL-algebra, $L$ an MV-algebra and $s:A\rightarrow L$ an order-preserving type I state. Then $A/{\rm Ker}(s)$ is an MV-algebra. \[p4.7\] Let $a,b\in A$. By Lemma \[l3.11\], $s(a\vee b)=s((a\rightarrow b)\rightarrow b)=s((b\rightarrow a)\rightarrow a)$. Let $x=(a\rightarrow b)\rightarrow b$ and $y=(b\rightarrow a)\rightarrow a$. By Lemma \[lMTL\], $a\vee b=x\wedge y$. It follows, by Lemma \[l3.11\], that $s(x)=s(y)=s(x\wedge y)$. By Proposition \[p4.4\] and Lemma \[l2.4\], $x/{\rm Ker}(s)=y/{\rm Ker}(s)$, therefore $A/{\rm Ker}(s)$ is an MV-algebra. If $s:A\rightarrow L$ is a type III state, then $A/{\rm Ker}(s)$ is involutive. \[p4.8\] Let $a\in A$. By Proposition \[p3.9\], (\[p3.9(3)\]), $s(a)=s(\neg \, \neg \, a)$ and, by Lemma \[l2.3\], (\[l2.3(2)\]), $a\vee \neg \, \neg \, a=\neg \, \neg \, a$, hence $s(a\vee \neg \, \neg \, a)=s(\neg \, \neg \, a)=s(a)$. By Proposition \[p4.4\], $\neg \, \neg \, a/{\rm Ker}(s)=a/{\rm Ker}(s)$, thus $A/{\rm Ker}(s)$ is involutive. Let $s:A\rightarrow L$ be an arbitrary function. Let us consider the properties: $(\alpha )$ for all $a,b\in A$, $s(a\vee b)=s(a)\vee s(b)$; $(\beta )$ for all $a,b\in A$, $s(a\wedge b)=s(a)\wedge s(b)$; $(\gamma )$ for all $a,b\in A$, $s(a\rightarrow b)=s(a)\rightarrow s(b)$; $(\delta )$ for all $a,b\in A$, $s(a\odot b)=s(a)\odot s(b)$. Assume that $s:A\rightarrow L$ is an order-preserving type I state. Then each of the conditions $(\alpha )$ and $(\beta )$ implies $(\gamma )$. \[l4.9\] $(\alpha )\Rightarrow (\gamma )$ By Proposition \[p3.3\], (\[p3.3(4)\]) and Lemma \[l2.2\], (\[l2.2(9)\]) and (\[l2.2(3)\]), $s(a\rightarrow b)=s(a\vee b)\rightarrow s(b)=(s(a)\vee s(b))\rightarrow s(b)=(s(a)\rightarrow s(b))\wedge (s(b)\rightarrow s(b))=s(a)\rightarrow s(b)$. $(\beta )\Rightarrow (\gamma )$ By Proposition \[p3.3\], (\[p3.3(3)\]) and Lemma \[l2.2\], (\[l2.2(9)\]) and (\[l2.2(3)\]), $s(a\rightarrow b)=s(a)\rightarrow s(a\wedge b)=s(a)\rightarrow (s(a)\wedge s(b))=(s(a)\rightarrow s(b))\wedge (s(b)\rightarrow s(b))=s(a)\rightarrow s(b)$. Let $L$ be an involutive residuated lattice and $s:A\rightarrow L$ an order-preserving type I state. Then $(\beta )$ implies $(\alpha )$. \[l4.10\] Let $a,b\in A$. By Proposition \[p3.6\], (\[p3.6(1)\]) and Lemma \[l2.2\], (\[l2.2(9)\]), $\neg \, s(a\vee b)=s(\neg \, (a\vee b))=s(\neg \, a\wedge \neg \, b)=s(\neg \, a)\wedge s(\neg \, b)=\neg \, s(a)\wedge \neg \, s(b)=\neg \, (s(a)\vee s(b))$, hence $\neg \, \neg \, s(a\vee b)=\neg \, \neg \, (s(a)\vee s(b))$, so that $s(a\vee b)=s(a)\vee s(b)$, since $L$ is involutive. Let $L$ be an MV-algebra and $s:A\rightarrow L$ an order-preserving type I state. Then conditions $(\alpha )$ and $(\gamma )$ are equivalent. \[p4.11\] $(\alpha )\Rightarrow (\gamma )$ By Lemma \[l4.9\]. $(\gamma )\Rightarrow (\alpha )$ Let $a,b\in A$. By Lemma \[l3.11\] and Lemma \[l2.4\], $s(a\vee b)=s((a\rightarrow b)\rightarrow b)=(s(a)\rightarrow s(b))\rightarrow s(b)=s(a)\vee s(b)$. Let $s:A\rightarrow L$ be an order-preserving type I state. Then: 1. \[l4.12(1)\] if $(\gamma )$ then, for all $a,b\in A$, $\neg \, s(a\odot b)=\neg \, (s(a)\odot s(b))$; 2. \[l4.12(2)\] if $L$ is involutive, then $(\gamma )$ implies $(\delta )$. \[l4.12\] (\[l4.12(1)\]) Let $a,b\in A$. By Lemma \[l2.3\], (\[l2.3(5)\]), $\neg \, (a\odot b)=a\rightarrow \neg \, b$. Thus, by Proposition \[p3.6\], (\[p3.6(1)\]), $\neg \, s(a\odot b)=s(\neg \, (a\odot b))=s(a\rightarrow \neg \, b)=s(a)\rightarrow s(\neg \, b)=s(a)\rightarrow \neg \, s(b)=\neg \, (s(a)\odot s(b))$. (\[l4.12(2)\]) By (\[l4.12(1)\]). Let $A$ be a divisible residuated lattice, $L$ be an MV-algebra and $s:A\rightarrow L$ an order-preserving type I state. Then $(\alpha )$, $(\beta )$ and $(\gamma )$ are equivalent. \[c4.13\] By Proposition \[p4.11\], $(\alpha )\Leftrightarrow (\gamma )$. By Lemma \[l4.9\], $(\beta )\rightarrow (\gamma )$. It remains to show: $(\gamma )\Rightarrow (\beta )$ Let $a,b\in A$. By Lemma \[l4.12\], $s(a\wedge b)=s(a\odot (a\rightarrow b))=s(a)\odot s(a\rightarrow b)=s(a)\odot (s(a)\rightarrow s(b))=s(a)\wedge s(b)$. A function $s:A\rightarrow L$ is called a [state-morphism]{} iff it fulfills $(\alpha )$, $(\beta )$, $(\gamma )$, $s(0)=0$ and $s(1)=1$. \[d4.12\] Any state-morphism is an order-preserving type I state. \[rsm\] By Proposition \[p3.3\], (\[p3.3(2)\]), any state-morphism is a type I state. By $(\alpha )$ and $(\beta )$, it is also a lattice morphism, thus an order-preserving function. If $L$ is the standard MV-algebra $[0,1]$, then Definition \[d4.12\] coincides with the concept of state-morphism from [@dindvu], [@dvura] etc.. Let $s:A\rightarrow L$ be an order-preserving type I state. If $A/{\rm Ker}(s)$ is totally ordered, then $s$ is a state-morphism. \[p4.13\] Let $a,b\in A$. Then $a/{\rm Ker}(s)\leq b/{\rm Ker}(s)$ or $b/{\rm Ker}(s)\leq a/{\rm Ker}(s)$. Assume, for example, that $a/{\rm Ker}(s)\leq b/{\rm Ker}(s)$, thus $(a\rightarrow b)/{\rm Ker}(s)=1/{\rm Ker}(s)$, that is $a\rightarrow b\in {\rm Ker}(s)$, that is $s(a\rightarrow b)=1$, by Lemma \[l2.2\], (\[l2.2(3)\]). By Remark \[rsm\], Proposition \[p3.3\], (\[p3.3(3)\]) and (\[p3.3(4)\]) and Lemma \[l2.2\], (\[l2.2(3)\]), $1=s(a\rightarrow b)=s(a)\rightarrow s(a\wedge b)=s(a\vee b)\rightarrow s(b)$, thus $s(a)\leq s(a\wedge b)$ and $s(a\vee b)\leq s(b)$. Since $s$ is order-preserving, it follows that $s(a)=s(a\wedge b)\leq s(a\vee b)=s(b)$, thus $s(a\vee b)=s(a)\vee s(b)$ and $s(a\wedge b)=s(a)\wedge s(b)$. By Lemma \[l4.9\], we also have $s(a\rightarrow b)=s(a)\rightarrow s(b)$, therefore $s$ is a state-morphism. Let $s:A\rightarrow L$ be an order-preserving type I state. If $A/{\rm Ker}(s)$ is an MV-algebra and ${\rm Ker}(s)$ is a maximal filter of $A$, then $s$ is a state-morphism. \[c4.14\] If ${\rm Ker}(s)$ is a maximal filter of $A$, then $A/{\rm Ker}(s)$ is a simple MV-algebra, thus totally ordered (see [@cito2]). By Proposition \[p4.13\], it follows that $s$ is a state-morphism. Assume that $L$ is totally ordered and $s:A\rightarrow L$ is a state-morphism. Then $A/{\rm Ker}(s)$ is totally ordered. \[p4.15\] Let $a,b\in A$. Then $s(a)\leq s(b)$ or $s(b)\leq s(a)$, so that, by Lemma \[l2.2\], (\[l2.2(3)\]), $s(a\rightarrow b)=s(a)\rightarrow s(b)=1$ or $s(b\rightarrow a)=s(b)\rightarrow s(a)=1$, thus $a/{\rm Ker}(s)\leq b/{\rm Ker}(s)$ or $b/{\rm Ker}(s)\leq a/{\rm Ker}(s)$. Let $L$ be totally ordered and $s:A\rightarrow L$ be an order-preserving type I state. Then: $s$ is a state-morphism iff $A/{\rm Ker}(s)$ is totally ordered. \[c4.16\] By Propositions \[p4.13\] and \[p4.15\]. If $L$ is totally ordered and $s:A\rightarrow L$ is a state-morphism then ${\rm Ker}(s)$ is a prime filter of $A$. By Proposition \[p4.15\], $A/{\rm Ker}(s)$ is totally ordered, thus, by [@pic Proposition 1.41, (iii)], ${\rm Ker}(s)$ is a prime filter. If $A$ is an MTL-algebra, $L$ is totally ordered and $s:A\rightarrow L$ an order-preserving type I state, then: $s$ is a state-morphism iff ${\rm Ker}(s)$ is a prime filter. Apply Corollary \[c4.16\] and [@beloh Lemma 2.61]. Let $L$ be a simple residuated lattice and $s:A\rightarrow L$ a state-morphism. Then ${\rm Ker}(s)$ is a maximal filter of $A$. \[p4.17\] Let $a\in A\setminus {\rm Ker}(s)$, thus $s(a)\neq 1$. By Lemmas \[l4.1\] and \[l2.5\], it is sufficient to prove that there exists an $n\in \N ^{}$ such that $\neg \, (a^{n})\in {\rm Ker}(s)$. By Lemma \[l2.6\], there exists an $n\in \N ^{}$ such that $(s(a))^n=0$. By Lemma \[l4.12\], (\[l4.12(1)\]), $s(\neg \, (a^n))=s(a^n\rightarrow 0)=s(a^n)\rightarrow s(0)=s(a^n)\rightarrow 0=\neg \, s(a^n)=\neg \, s(a)^n=1$, so $\neg \, (a^{n})\in {\rm Ker}(s)$. It is known that the standard MV-algebra $[0,1]$ is simple ([@ciudvuhyc]). Then from Proposition \[p4.17\] we get the following known result ([@dvura2], [@lcciu1]): a Bosbach state $s:A\rightarrow [0,1]$ is a state-morphism iff ${\rm Ker}(s)$ is a maximal filter in $A$. \[r4.18\] Glivenko Property and Riečan states {#riecan} =================================== In this section we study the relation between generalized Bosbach states on a residuated lattice $A$ with Glivenko property and generalized Bosbach states on the involutive residuated lattice ${\rm Reg}(A)$ of the regular elements of $A$. We define the notion of generalized Riečan state and we relate type I states and generalized Riečan states. In the following, let $A$ be a residuated lattice and ${\rm Reg}(A)=\{\neg \, a\|a\in A\}=\{a\in A\|a=\neg \, \neg \, a\}$ the set of the [regular elements]{} of $A$. $A$ is said to be [involutive]{} iff $A={\rm Reg}(A)$. For all $a,b\in A$, we denote $a\vee ^{}b=\neg \, \neg \, (a\vee b)$, $a\wedge ^{}b=\neg \, \neg \, (a\wedge b)$, $a\odot ^{}b=\neg \, \neg \, (a\odot b)$. We say that $A$ has [Glivenko property]{} iff, for all $a,b\in A$, $\neg \, \neg \, (a\rightarrow b)=a\rightarrow \neg \, \neg \, b$. [[@cito2 Theorem 2.1, page 163]]{} The following are equivalent: \(i) $A$ has Glivenko property; \(ii) $({\rm Reg}(A),\vee ^{},\wedge ^{},\odot ^{},\rightarrow ,0,1)$ is an involutive residuated lattice and $\neg \, \neg \, :A\rightarrow {\rm Reg}(A)\ :\ a\rightarrow \neg \, \neg \, a$ is a surjective morphism of residuated lattices. \[‘glivnegneg‘\] Heyting algebras and BL-algebras have Glivenko property. Until mentioned otherwise, let $A$ be a residuated lattice with Glivenko property. We define $\varphi =\neg \, \neg \, $ to be the surjective morphism from the proposition above. Let $L$ be a residuated lattice. If $s:A\rightarrow L$ is a type I state (respectively a type II state), then, obviously, $s\mid _{{\rm Reg}(A)}:{\rm Reg}(A)\rightarrow L$ is a type I state (respectively a type II state). Let $s:{\rm Reg}(A)\rightarrow L$ be an arbitrary function. We define $\tilde{s}:A\rightarrow L$ by $\tilde{s}(a)=s(\varphi (a))$ for all $a\in A$. Assume that $A$ has Glivenko property and $L$ is involutive. If $s:{\rm Reg}(A)\rightarrow L$ is a type I state (respectively a type II state), then $\tilde{s}:A\rightarrow L$ is a type I state (respectively a type II state). \[p5.1\] Assume that $s$ is a type I state. Then, for all $a,b\in A$, $\tilde{s}(a\rightarrow b)=s(\varphi (a\rightarrow b))=s(\varphi (a))\rightarrow s(\varphi (b))=s(\varphi (a))\rightarrow s(\varphi (a)\wedge \varphi (b))=s(\varphi (a))\rightarrow s(\varphi (a\wedge b))=\tilde{s}(a)\rightarrow \tilde{s}(a\wedge b)$. So $\tilde{s}$ is a type I state. Assume that $s$ is a type II state. Then, for all $a,b\in A$, $\tilde{s}(a\rightarrow b)\rightarrow \tilde{s}(b)=s(\varphi (a\rightarrow b))\rightarrow s(\varphi (b))=s(\varphi (a)\rightarrow \varphi (b))\rightarrow s(\varphi (b))=s(\varphi (b)\rightarrow \varphi (a))\rightarrow s(\varphi (a))=s(\varphi (b\rightarrow a))\rightarrow s(\varphi (a))=\tilde{s}(b\rightarrow a)\rightarrow \tilde{s}(a)$. So $\tilde{s}$ is a type II state. Let $s_1:A\rightarrow L$ be a type I state (respectively a type II state). By applying Proposition \[p3.6\], (\[p3.6(1)\]) (respectively Proposition \[p3.9\], (\[p3.9(3)\])), we obtain, for all $a\in A$, $s_1(a)=s_1(\varphi (a))$. Then, if $s:{\rm Reg}(A)\rightarrow L$ is a type I state (respectively a type II state), it follows that $\tilde{s}:A\rightarrow L$ is the unique type I state (respectively the unique type II state) such that $\tilde{s}\mid _{{\rm Reg}(A)}=s$. \[r5.2\] In the following, let $A$ be an arbitrary residuated lattice. On the set $A$ we introduce the binary operation $\oplus $ by: for all $a,b\in A$, $a\oplus b=\neg \, a\rightarrow \neg \, \neg \, b=\neg \, b\rightarrow \neg \, \neg \, a$ (see Lemma \[l2.3\], (\[l2.3(4)\]) and (\[l2.3(2)\])). [[@kuhr], Lemma 3.6.2]{} For all $a,b,c\in A$, we have: 1. \[l5.3(1)\] $a\oplus 0=\neg \, \neg \, a$; 2. \[l5.3(2)\] $a\oplus 1=a$; 3. \[l5.3(3)\] $\oplus $ is associative and commutative; 4. \[l5.3(4)\] if $a\leq b$ then $a\oplus c\leq b\oplus c $; 5. \[l5.3(5)\] $a\vee b\leq a\oplus b$; 6. \[l5.3(6)\] $a\oplus b=\neg \, \neg \, (a\oplus b)=\neg \, \neg \, a\oplus \neg \, \neg \, b$. \[l5.3\] For all $a,b\in A$, we denote $a\perp b$ iff $\neg \, \neg \, a\leq \neg \, b$ iff $\neg \, \neg \, b\leq \neg \, a$ (see Lemma \[l2.3\], (\[l2.3(3)\]) and (\[l2.3(2)\])). A [Riečan state on $A$]{} is a function $m:A\rightarrow [0,1]$ such that $m(1)=1$ and, for all $a,b\in A$ with $a\perp b$, $m(a\oplus b)=m(a)+m(b)$. [[@gg], [@dvura], [@lcciu]]{} If $m$ is a Riečan state on $A$, then: 1. \[l5.4(1)\] for all $a\in A$, $m(\neg \, a)=1-m(a)$; 2. \[l5.4(2)\] $m(0)=0$; 3. \[l5.4(3)\] $m$ is order-preserving. \[l5.4\] Riečan states on pseudo-BL-algebras have been defined in [@gg], by generalizing a notion of state on BL-algebras that had been introduced by Riečan in [@br2]. Later, Riečan states on more general structures have been studied ([@dvura2], [@lcciu], [@kuhr], [@turmer]). In what follows we shall extend the notion of Riečan state to the context of this paper and we shall point out the relation between the notion we shall obtain and generalized Bosbach states. In the following, let $A$ and $L$ be residuated lattices. A function $m:A\rightarrow L$ is called a [generalized Riečan state]{} iff the following conditions are verified, for all $a,b\in A$: \(a) $m(1)=1$; \(b) if $a\perp b$, then $m(a)\perp m(b)$; \(c) if $a\perp b$, then $m(a\oplus b)=m(a)\oplus m(b)$. \[d5.5\] Let $m:A\rightarrow L$ be a generalized Riečan state. Then, for all $a,b\in A$, we have: 1. \[p5.6(1)\] $\neg \, \neg \, m(\neg \, a)=\neg \, m(a)$; if $L$ is involutive, then $m(\neg \, a)=\neg \, m(a)$ and $m(\neg \, \neg \, a)=m(a)$; 2. \[p5.6(2)\] $m(0)=0$; 3. \[p5.6(3)\] if $b\leq a$, then $\neg \, m(a)\leq \neg \, m(b)$; if $L$ is involutive and $b\leq a$, then $m(b)\leq m(a)$. \[p5.6\] Let $a,b\in A$. (\[p5.6(1)\]) Obviously, $a\perp \neg \, a$, so $m(a)\perp m(\neg \, a)$, that is $\neg \, \neg \, m(\neg \, a)\leq \neg \, m(a)$. Also, by Lemma \[l2.2\], (\[l2.2(3)\]) and Lemma \[l2.3\], (\[l2.3(2)\]), $1=m(1)=m(a\oplus \neg \, a)=m(a)\oplus m(\neg \, a)=\neg \, m(a)\rightarrow \neg \, \neg \, m(\neg \, a)$, thus $\neg \, m(a)\leq \neg \, \neg \, m(\neg \, a)$. Hence $\neg \, \neg \, m(\neg \, a)=\neg \, m(a)$. (\[p5.6(2)\]) Set $a=0$ in (\[p5.6(1)\]) and apply Lemma \[l2.3\], (\[0neg1\]). (\[p5.6(3)\]) By Lemma \[l2.3\], (\[l2.3(3)\]), if $b\leq a$ then $b\perp \neg \, a$, so $m(b)\perp m(\neg \, a)$, thus, by (\[p5.6(1)\]), $\neg \, m(a)=\neg \, \neg \, m(\neg \, a)\leq \neg \, m(b)$. The next proposition shows that, in the case when $L$ is the standard MV-algebra $[0,1]$, Riečan states coincide with generalized Riečan states. Let $m:A\rightarrow [0,1]$ be an arbitrary function. We consider on $[0,1]$ the standard MV-algebra structure. Then: $m$ is a Riečan state iff $m$ is a generalized Riečan state. \[p5.7\] Let $a,b\in A$. Assume that $m$ is a Riečan state. If $a\perp b$ then $\neg \, \neg \, a\leq \neg \, b$, so, by Lemma \[l5.4\], (\[l5.4(1)\]) and (\[l5.4(3)\]), $\neg \, \neg \, m(a)=m(a)=m(\neg \, \neg \, a)\leq m(\neg \, b)=1-m(b)=\neg \, m(b)$. Hence $m(a)\perp m(b)$. Thus $m$ is a generalized Riečan state. Now assume that $m$ is a generalized Riečan state. If $a\perp b$ then $m(a)\perp m(b)$, so $m(a\oplus b)=m(a)\oplus m(b)=m(a)+m(b)$. Thus $m$ is a Riečan state. Any order-preserving type I state is a generalized Riečan state. \[p5.8\] Let $s:A\rightarrow L$ be an order-preserving type I state and $a,b\in A$ with $a\perp b$. Then $\neg \, \neg \, a\leq \neg \, b$, so, by Proposition \[p3.6\], (\[p3.6(1)\]) and the fact that $s$ is order-preserving, $\neg \, \neg \, s(a)=s(\neg \, \neg \, a)\leq s(\neg \, b)=\neg \, s(b)$. Hence $s(a)\perp s(b)$. By Proposition \[p3.3\], (\[p3.3(2)\]) and Proposition \[p3.6\], (\[p3.6(1)\]), $s(a\oplus b)=s(\neg \, b\rightarrow \neg \, \neg \, a)=s(\neg \, b)\rightarrow s(\neg \, \neg \, a)=\neg \, s(b)\rightarrow \neg \, \neg \, s(a)=s(a)\oplus s(b)$. So $s$ is a generalized Riečan state. Obviously, if $A$ has Glivenko property and $m:A\rightarrow L$ is a generalized Riečan state, then $m\mid _{{\rm Reg}(A)}:{\rm Reg}(A)\rightarrow L$ is a generalized Riečan state. Assume that $A$ has Glivenko property and $L$ is involutive. Then any generalized Riečan state $m:A\rightarrow L$ is an order-preserving type I state. \[p5.9\] Let $m:A\rightarrow L$ be a generalized Riečan state and $a,b\in A$ such that $b\leq a$. We show that $m(a\rightarrow b)=m(a)\rightarrow m(b)$. By Lemma \[l2.3\], (\[l2.3(3)\]), since $b\leq a$, we have that $b\perp \neg \, a$, so $m(b)\perp m(\neg \, a)$. We notice that $\neg \, a\oplus b=\neg \, b\rightarrow \neg \, \neg \, \neg \, a=\neg \, b\rightarrow \neg \, a=a\rightarrow \neg \, \neg \, b$, by Lemma \[l2.3\], (\[l2.3(2)\]) and (\[l2.3(5)\]). Since $A$ has Glivenko property and by Lemma \[l2.3\], (\[l2.3(5)\]) and (\[l2.3(2)\]), $\neg \, \neg \, (a\rightarrow b)=a\rightarrow \neg \, \neg \, b=\neg \, b\rightarrow \neg \, a=\neg \, b\rightarrow \neg \, \neg \, \neg \, a=\neg \, a\oplus b$. By Proposition \[p5.6\], (\[p5.6(1)\]) and the fact that $L$ is involutive, $m(a\rightarrow b)=m(\neg \, \neg \, (a\rightarrow b))=m(\neg \, a\oplus b)=m(\neg \, a)\oplus m(b)=\neg \, m(a)\oplus m(b)=m(a)\rightarrow m(b)$. So $m$ is an order-preserving type I state. If $A$ has Glivenko property and $L$ is involutive, then, by Propositions \[p5.8\] and \[p5.9\], order-preserving type I states $s:A\rightarrow L$ coincide with generalized Riečan states $s:A\rightarrow L$. In particular, if $A$ has Glivenko property and $L$ is the standard MV-algebra, then Bosbach states $s:A\rightarrow L$ coincide with Riečan states $s:A\rightarrow L$ (see [@dvura2], [@turmer]). Not all generalized Riečan states are type I or type II states. We consider the residuated lattice $A$ from Remark \[r3.6\]. The generalized Riečan states $m:A\rightarrow A$ are the following: $x$ $0$ $a$ $b$ $c$ $d$ $1$ ---------- ----- ----- ----- ----- ----- ----- $s_1(x)$ $0$ $a$ $0$ $1$ $a$ $1$ $m_1(x)$ $0$ $a$ $0$ $1$ $1$ $1$ $m_2(x)$ $0$ $a$ $b$ $c$ $c$ $1$ $s_2(x)$ $0$ $a$ $b$ $c$ $d$ $1$ $m_3(x)$ $0$ $a$ $c$ $b$ $b$ $1$ $m_4(x)$ $0$ $a$ $1$ $0$ $0$ $1$ $m_5(x)$ $0$ $1$ $0$ $1$ $a$ $1$ $s_3(x)$ $0$ $1$ $0$ $1$ $1$ $1$ $s_4(x)$ $0$ $1$ $b$ $c$ $c$ $1$ $m_6(x)$ $0$ $1$ $b$ $c$ $d$ $1$ $s_5(x)$ $0$ $1$ $c$ $b$ $b$ $1$ $s_6(x)$ $0$ $1$ $1$ $0$ $0$ $1$ As mentioned in Remark \[r3.6\], the type I states from $A$ to $A$ are $s_i$, with $i\in \overline{1,6}$, and the type II states from $A$ to $A$ are $s_i$, with $i\in \overline{3,6}$. Out of the generalized Riečan states $m_i$, with $i\in \overline{1,6}$, none is a type I or a type II state. If $A$ is involutive and $s:A\rightarrow L$ is a generalized Riečan state such that, for all $a\in A$, $s(\neg \, a)=\neg \, s(a)$, then $s$ is an order-preserving type I state. Let $A$ and $s$ be as in the hypothesis and let $a,b\in A$ such that $b\leq a$. Since $A$ is involutive, it follows that $b=\neg \, \neg \, b$ and $a=\neg \, \neg \, a=\neg \, c$, with $c=\neg \, a$. Thus $\neg \, \neg \, b\leq \neg \, c$, that is $b\perp c$, hence $s(b\oplus c)=s(b)\oplus s(c)$, that is $s(\neg \, c\rightarrow \neg \, \neg \, b)= \neg \, s(c)\rightarrow \neg \, \neg \, s(b)$, that is $s(a\rightarrow b)=s(\neg \, c)\rightarrow s(\neg \, \neg \, b)$, that is $s(a\rightarrow b)=s(a)\rightarrow s(b)$. So, by Proposition \[p3.3\], (\[p3.3(2)\]), $s$ is a type I state. It remains to show that $s(b)\leq s(a)$, which will allow us to conclude that $s$ is order-preserving. We saw that $b\perp c$; it follows that $s(b)\perp s(c)$, which means that $\neg \, \neg \, s(b)\leq \neg \, s(c)$, that is $s(\neg \, \neg \, b)\leq s(\neg \, c)$, that is $s(b)\leq s(a)$. If $A$ is involutive and $s:A\rightarrow L$ is both a generalized Riečan state and a type II state, then $s$ is an order-preserving type I state. Similarity Convergences and Continuity of States {#convergente} ================================================ The similarity convergence in residuated lattices has been defined in [@ggap] based on the biresiduum. In the particular case of MV-algebras, it is dual to the order-convergence, a notion that is defined starting from the distance in MV-algebras. For non-involutive residuated lattices, this duality is not kept, but most part of a good convergence theory (for example, type Cauchy completions) can be obtained. Starting from the similarity convergence, in this section we introduce three notions of continuity of a generalized Bosbach state and we study the relation between them. If $L$ is a residuated lattice and $E:X^2\rightarrow L$ is an $L$-similarity relation on a nonempty set $X$ ([@ggap]), then the similarity convergence on $L$ allows us to define a convergence on $X$ (called $E$-convergence). To an order-preserving type I state $s:A\rightarrow L$ we associate an $L$-similarity relation $\rho_s:A^2\rightarrow L$. The $\rho_s$-convergence is kept by the residuated lattice operations of $A$. Next, working on the $\rho_s$-Cauchy sequences of $A$, we generalize to the context of this paper an important construction from [@ioanal3]: the metric completion of an MV-algebra. Until mentioned otherwise, let $X$ be a nonempty set and $L$ a residuated lattice. We recall from [@beloh] that an $L$-binary relation on $X$, that is a function $E:X^2\rightarrow L$, is called an [$L$-similarity relation on $X$]{} (or an [$L$-equivalence on $X$]{}) iff, for all $a,b,c\in X$: $E(a,a)=1$, $E(a,b)=E(b,a)$ and $E(a,b)\odot E(b,c)\leq E(a,c)$. An $L$-similarity relation $E$ on $X$ is called an [$L$-equality on $X$]{} iff, for all $a,b\in X$, $E(a,b)=1$ implies $a=b$. By Lemma \[lA\], (\[lA(1)\]), (\[lA(2)\]) and (\[lA(3)\]), $d_L:L^2\rightarrow L$ is an L-equality on $L$. The fact that a sequence $(c_n)_{n\geq 0}\subseteq L$ is increasing is denoted $(c_n)_{n\geq 0}\uparrow $. The sequence $(c_n)_{n\geq 0}$ is said to be [increasing towards $c\in L$]{} iff $(c_n)_{n\geq 0}\uparrow $ and $\bigvee _{n\geq 0}c_n=c$; this is denoted by $(c_n)_{n\geq 0}\uparrow c$. The fact that a sequence $(c_n)_{n\geq 0}\subseteq L$ is decreasing is denoted $(c_n)_{n\geq 0}\downarrow $. The sequence $(c_n)_{n\geq 0}$ is said to be [decreasing towards $c\in L$]{} iff $(c_n)_{n\geq 0}\downarrow $ and $\bigwedge _{n\geq 0}c_n=c$; this is denoted by $(c_n)_{n\geq 0}\downarrow c$. A sequence $(a_n)_{n\geq 0}\subseteq L$ is said to be [similarity convergent]{} (or, in brief, [convergent]{}) [towards $a\in L$]{} iff there exists a sequence $(c_n)_{n\geq 0}\subseteq L$ such that $(c_n)_{n\geq 0}\uparrow 1$ and, for all $n\in \N $, $c_n\leq d_L(a_n,a)$; this is denoted by $\lim _{n\rightarrow \infty }a_n=a$ and $a$ is called the [limit of $(a_n)_{n\geq 0}$]{}. By [@ggap Remark 3.7, (i)], the limit of a convergent sequence in a residuated lattice is unique. Obviously, if, for all $n\in \N $, $a_n=\alpha \in L$, then $\lim _{n\rightarrow \infty }a_n=\alpha $. Also, it is obvious that, if $k\in \N $, $a\in L$ and $(b_n)_{n\geq 0}\subseteq L$ such that, for all $n\geq k$, $b_n=a_n$, then: $\lim _{n\rightarrow \infty }a_n=a$ iff $\lim _{n\rightarrow \infty }b_n=a$, as we may take in the definition of the similarity convergence $c_n=0$ for all $n<k$. The sequence $(a_n)_{n\geq 0}\subseteq L$ is said to be [similarity Cauchy]{} (or, in brief, [Cauchy]{}) iff $\lim _{n,m\rightarrow \infty }d_L(a_n,a_m)=1$, where, naturally, for all $(l_{n,m})_{n,m\geq 0}\subseteq L$, we set $\lim _{n,m\rightarrow \infty }l_{n,m}=\lim _{n\rightarrow \infty }\lim _{m\rightarrow \infty }l_{n,m}$. Any convergent sequence is Cauchy, as shown in [@ggap]. $L$ is said to be [Cauchy-complete]{} iff in $L$ any Cauchy sequence is convergent. In [@ggap], a sequence $(a_n)_{n\geq 0}\subseteq L$ is defined to be [similarity Cauchy]{} iff there exists a sequence $(c_n)_{n\geq 0}\subseteq L$ such that $(c_n)_{n\geq 0}\uparrow 1$ and, for all $n,p\in \N $, $c_n\leq d_L(a_n,a_{n+p})$. This is equivalent to our definition, as, for all $(l_n)_{n\geq 0}\subseteq L$, we have, by the definitions above: $\lim _{n\rightarrow \infty }l_n=1$ iff there exists $(c_n)_{n\geq 0}\subseteq L$ such that $(c_n)_{n\geq 0}\uparrow 1$ and, for all $n\in \N $, $c_n\leq d_L(l_n,1)$ iff there exists $(c_n)_{n\geq 0}\subseteq L$ such that $(c_n)_{n\geq 0}\uparrow 1$ and, for all $n\in \N $, $c_n\leq l_n$, because $d_L(l_n,1)=l_n$ by Lemma \[l2.2\], (\[l2.2(1)\]) and (\[l2.2(2)\]). [[@ggap]]{} Let $(a_n)_{n\geq 0},(b_n)_{n\geq 0}\subseteq L$ and $a,b\in L$. If $\lim _{n\rightarrow \infty }a_n=a$ and $\lim _{n\rightarrow \infty }b_n=b$, then $\lim _{n\rightarrow \infty }(a_n\circ b_n)=a\circ b$ for each $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$. Thus $\lim _{n\rightarrow \infty }\neg \, a_n=\neg \, a$ and, if $a_n\leq b_n$ for all $n\in \N $ (or for all $n\geq k\in \N $), then $a\leq b$. \[l6.1\] [[@ggap]]{} Let $(a_n)_{n\geq 0}\subseteq L$ and $a\in L$. If $(a_n)_{n\geq 0}\uparrow a$ or $(a_n)_{n\geq 0}\downarrow a$ then $\lim _{n\rightarrow \infty }a_n=a$. \[l6.2\] A sequence $(a_n)_{n\geq 0}\subseteq X$ is said to be [$E$-convergent towards $a\in X$]{} iff $\lim _{n\rightarrow \infty }E(a_n,a)=1$; this is denoted by $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a$. $(a_n)_{n\geq 0}$ is said to be [$E$-Cauchy]{} iff $\lim _{n,m\rightarrow \infty }E(a_n,a_m)=1$. Assume that $E:X^2\rightarrow L$ is an $L$-equality and let $(a_n)_{n\geq 0}\subseteq X$, $a,a^{\prime }\in X$. If $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a$ and $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a^{\prime }$ then $a=a^{\prime }$. \[l6.3\] Assume that $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a$ and $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a^{\prime }$, that is $\lim _{n\rightarrow \infty }E(a_n,a)=1$ and $\lim _{n\rightarrow \infty }E(a_n,a^{\prime })=1$, thus, by Lemma \[l6.1\], $\lim _{n\rightarrow \infty }(E(a_n,a)\odot E(a_n,a^{\prime }))=1\odot 1=1$. But, for all $n\in \N $, $E(a_n,a)\odot E(a_n,a^{\prime })\leq E(a,a^{\prime })$, thus $E(a,a^{\prime })=1$ by Lemma \[l6.1\], so $a=a^{\prime }$. If $E$ is an $L$-equality, then any $E$-convergent sequence is $E$-Cauchy. \[l6.4\] Let $(a_n)_{n\geq 0}\subseteq X$ and $a\in X$ such that $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a$, that is $\lim _{n\rightarrow \infty }E(a_n,a)=1$. Then, by Lemma \[l6.1\], $\lim _{n,m\rightarrow \infty }E(a_n,a_m)\geq \lim _{n,m\rightarrow \infty }(E(a_n,a)\odot E(a_m,a))=1\odot 1=1$, therefore $(a_n)_{n\geq 0}$ is $E$-Cauchy. Until mentioned otherwise, let $A$ and $L$ be two residuated lattices and $E:A^2\rightarrow L$ an $L$-similarity relation. If $E:A^2\rightarrow L$ is an $L$-equality and any $E$-Cauchy sequence is $E$-convergent, then the residuated lattice $A$ is said to be [$E$-complete]{}. For any function $s:A\rightarrow L$, we denote by $\rho _s:A^2\rightarrow L$ the function defined by: for all $a,b\in A$, $\rho _s(a,b)=s(d_A(a,b))$. Let $s:A\rightarrow L$ be an order-preserving type I state. Then, for all $a,b,x,y\in A$, we have: 1. \[l6.5(1)\] $\rho _s(a,b)\leq \rho _s(\neg \, a,\neg \, b)$; 2. \[l6.5(2)\] $\rho _s(a,b)\odot \rho _s(x,y)\leq \rho _s(a\circ x,b\circ y)$, for each $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$; 3. \[l6.5(3)\] $\rho _s(a,b)\leq d_L(s(a),s(b))$; 4. \[l6.5(4)\] if $a$ and $b$ are comparable, then: $\rho _s(a,b)=d_L(s(a),s(b))$; 5. \[l6.5(5)\] $\rho _s(a,x)\odot \rho _s(b,y)\leq d_L(\rho _s(a,b),\rho _s(x,y))$. \[l6.5\] (\[l6.5(1)\]) By Lemma \[lA\], (\[lA(4)\]) and the fact that $s$ is order-preserving, $\rho _s(a,b)=s(d_A(a,b))\leq s(d_A(\neg \, a,\neg \, b))=\rho _s(\neg \, a,\neg \, b)$. (\[l6.5(2)\]) Let $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$. By Proposition \[p3.8\], (\[p3.8(1)\]), Lemma \[lA\], (\[lA(5)\]) and the fact that $s$ is order-preserving, $\rho _s(a,b)\odot \rho _s(x,y)=s(d_A(a,b))\odot s(d_A(x,y))\leq s(d_A(a,b)\odot d_A(x,y))\leq s(d_A(a\circ x,b\circ y))=\rho _s(a\circ x,b\circ y)$. (\[l6.5(3)\]) By Proposition \[p3.8\], (\[p3.8(5)\]). (\[l6.5(4)\]) Assume, for instance, that $b\leq a$. Then, by Lemma \[l2.2\], (\[l2.2(3)\]) and Proposition \[p3.3\], (\[p3.3(2)\]), $\rho _s(a,b)=s(d_A(a,b))=s(a\rightarrow b)=s(a)\rightarrow s(b)=d_L(s(a),s(b))$. (\[l6.5(5)\]) By Proposition \[p3.8\], (\[p3.8(6)\]). If $s:A\rightarrow L$ is an order-preserving type I state, then $\rho _s$ is an $L$-similarity relation on $A$. \[p6.6\] By Proposition \[p3.8\], (\[p3.8(1)\]), Lemma \[lA\], (\[lA(5)\]) and the fact that $s$ is order-preserving, $\rho _s(a,b)\odot \rho _s(b,c)=s(d_A(a,b))\odot s(d_A(b,c))\leq s(d_A(a,b)\odot d_A(b,c))\leq s(d_A(a,c))=\rho _s(a,c)$. If $s:A\rightarrow L$ is a generalized Bosbach state or a Riečan state, then we will say that $s$ is [faithful]{} iff, for all $a\in A$, $s(a)=1$ implies $a=1$. By Lemma \[lA\], (\[lA(1)\]), if $s:A\rightarrow L$ is a faithful order-preserving type I state, then $\rho _s$ is an $L$-equality on $A$. \[r6.7\] Let $s:A\rightarrow L$ be a faithful order-preserving type I state, $(a_n)_{n\geq 0},(b_n)_{n\geq 0}\subseteq A$ and $a,b\in A$. If $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$ and $b_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }b$, then $a_n\circ b_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a\circ b$ for each $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$. From this and the definitions of $\neg \, $ and $\leq $, it follows that $\neg \, a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }\neg \, a$ and, if $a_n\leq b_n$ for all $n\in \N $ (or for all $n\geq k\in \N $), then $a\leq b$. \[l6.8\] Apply Lemma \[l6.1\] and Lemma \[l6.5\], (\[l6.5(2)\]). Let $s:A\rightarrow L$ be an arbitrary function and $a\in A$. Then $s$ is said to be: - [$\uparrow $-continuous in $a$]{} iff, for any sequence $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\uparrow a$, we have $\lim _{n\rightarrow \infty }s(a_n)=s(a)$; - [$\downarrow $-continuous in $a$]{} iff, for any sequence $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\downarrow a$, we have $\lim _{n\rightarrow \infty }s(a_n)=s(a)$; - [continuous in $a$]{} iff it is $\uparrow $-continuous in $a$ and $\downarrow $-continuous in $a$. $s$ is said to be [$\uparrow $-continuous]{} (respectively [$\downarrow $-continuous]{}, [continuous]{}) iff it is $\uparrow $-continuous (respectively $\downarrow $-continuous, continuous) in any $a\in A$. Assume that $L$ is involutive and let $s:A\rightarrow L$ be a type I state and $a\in A$. If $s$ is $\downarrow $-continuous in $a$ then it is also $\uparrow $-continuous in $a$. Thus, if $s$ is $\downarrow $-continuous then it is also $\uparrow $-continuous. \[p6.9\] Assume that $s$ is $\downarrow $-continuous and let $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\uparrow a$, that is $a_n\uparrow $ and $\bigvee _{n\in \N }a_n=a$. Then, by Lemma \[l2.3\], (\[l2.3(3)\]) and Lemma \[l2.2\], (\[l2.2(9)\]), $(\neg \, a_n)_{n\geq 0}\downarrow $ and $\bigwedge _{n\geq 0}(\neg \, a_n)=\neg \, (\bigvee _{n\geq 0}a_n)=\neg \, a$, thus $\neg \, a_n\downarrow \neg \, a$. By Lemma \[l6.1\], Proposition \[p3.6\], (\[p3.6(1)\]) and the fact that $L$ is involutive, $\neg \, \lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }\neg \, s(a_n)=\lim _{n\rightarrow \infty }s(\neg \, a_n)=s(\neg \, a)=\neg \, s(a)$, hence $\lim _{n\rightarrow \infty }s(a_n)=s(a)$. Therefore $s$ is $\uparrow $-continuous. Let $s:A\rightarrow L$ be a type II state and $a\in A$. If $s$ is $\downarrow $-continuous in $a$ then it is also $\uparrow $-continuous in $a$. Thus, if $s$ is $\downarrow $-continuous then it is also $\uparrow $-continuous. \[p6.10\] Assume that $s$ is $\downarrow $-continuous and let $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\uparrow a$. Then, by the proof of Proposition \[p6.9\], $\neg \, a_n\downarrow \neg \, a$, hence $\lim _{n\rightarrow \infty }s(\neg \, a_n)=\neg \, a$. By Proposition \[p3.9\], (\[p3.9(3)\]) and (\[p3.9(2)\]), and Lemma \[l6.1\], $\lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }s(\neg \, \neg \, a_n)=\lim _{n\rightarrow \infty }\neg \, s(\neg \, a_n)=\neg \, \lim _{n\rightarrow \infty }s(\neg \, a_n)=\neg \, s(\neg \, a)=s(a)$. Therefore $s$ is $\uparrow $-continuous. Let $A$ be an MV-algebra and $s:A\rightarrow L$ an order-preserving type I state. Let us consider the following statements: 1. \[p6.12(1)\] $s$ is $\uparrow $-continuous in $1$; 2. \[p6.12(2)\] $s$ is $\uparrow $-continuous; 3. \[p6.12(3)\] $s$ is $\downarrow $-continuous in $0$; 4. \[p6.12(4)\] $s$ is $\downarrow $-continuous; 5. \[p6.12(5)\] $s$ is continuous. Then (\[p6.12(2)\])$\Leftrightarrow $(\[p6.12(1)\])$\Rightarrow $(\[p6.12(4)\])$\Rightarrow $(\[p6.12(3)\]). If $L$ is involutive then (\[p6.12(1)\]) iff (\[p6.12(2)\]) iff (\[p6.12(3)\]) iff (\[p6.12(4)\]) iff (\[p6.12(5)\]). \[p6.12\] First let us prove that (\[p6.12(1)\]) iff (\[p6.12(2)\]). The converse implication is trivial. For the direct implication, let us assume that $s$ is $\uparrow $-continuous in $1$. Let $a\in A$ and $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\uparrow a$, hence, for all $n\in \N $, $a_n\leq a$, which implies that $d_A(a_n,a)=a\rightarrow a_n$, by Lemma \[l2.2\], (\[l2.2(3)\]). Thus, by Lemma \[l2.2\], (\[l2.2(4)\]), $(d_A(a_n,a))_{n\geq 0}\uparrow $. Moreover, $\bigvee _{n\geq 0}d_A(a_n,a)=\bigvee _{n\geq 0}(a\rightarrow a_n)=a\rightarrow (\bigvee _{n\geq 0}a_n)=a\rightarrow a=1$, by Lemma \[mvlema\], (\[mvvee\]), and Lemma \[l2.2\], (\[l2.2(3)\]). Thus $(d_A(a_n,a))_{n\geq 0}\uparrow 1$. By Lemma \[l6.1\], the fact that, for all $n\in \N $, $a_n\leq a$, and Lemma \[l6.5\], (\[l6.5(4)\]), it follows that $d_L(\lim _{n\rightarrow \infty }s(a_n),s(a))=\lim _{n\rightarrow \infty }d_L(s(a_n),s(a))=\lim _{n\rightarrow \infty }s(d_A(a_n,a))=s(1)=1$. Hence $\lim _{n\rightarrow \infty }s(a_n)=s(a)$, therefore $s$ is $\uparrow $-continuous in $a$. Now let us prove that (\[p6.12(1)\]) implies (\[p6.12(4)\]). Thus let us assume that $s$ is $\uparrow $-continuous in $1$. Let $a\in A$ and $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\downarrow a$, hence, for all $n\in \N $, $a_n\geq a$, which implies that $d_A(a_n,a)=a_n\rightarrow a$, by Lemma \[l2.2\], (\[l2.2(3)\]). Thus, by Lemma \[l2.2\], (\[l2.2(4)\]), $(d_A(a_n,a))_{n\geq 0}\uparrow $. Moreover, by Lemma \[mvlema\], (\[mvwedge\]). $\bigvee _{n\geq 0}d_A(a_n,a)=\bigvee _{n\geq 0}(a_n\rightarrow a)=(\bigwedge _{n\geq 0}a_n)\rightarrow a=a\rightarrow a=1$, by Lemma \[mvlema\], (\[mvwedge\]), and Lemma \[l2.2\], (\[l2.2(3)\]). Thus $(d_A(a_n,a))_{n\geq 0}\uparrow 1$. By Lemma \[l6.1\], the fact that, for all $n\in \N $, $a_n\geq a$, and Lemma \[l6.5\], (\[l6.5(4)\]), it follows that $d_L(\lim _{n\rightarrow \infty }s(a_n),s(a))=\lim _{n\rightarrow \infty }d_L(s(a_n),s(a))=\lim _{n\rightarrow \infty }s(d_A(a_n,a))=s(1)=1$. Hence $\lim _{n\rightarrow \infty }s(a_n)=s(a)$, therefore $s$ is $\downarrow $-continuous in $a$. Trivially (\[p6.12(4)\]) implies (\[p6.12(3)\]). Now let us assume that $L$ is involutive. For proving the equivalences in the enunciation it remains to show that (\[p6.12(3)\]) implies (\[p6.12(1)\]). Thus, let us assume that $s$ is $\downarrow $-continuous in $0$ and let $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\uparrow 1$. Then $a_n\uparrow $, thus $\neg \, a_n\downarrow $, by Lemma \[l2.2\], (\[l2.2(4)\]). Moreover, by Lemma \[l2.2\], (\[l2.2(9)\]) and Lemma \[l2.3\], (\[0neg1\]), $\bigwedge _{n\geq 0}\neg \, a_n=\neg \, (\bigvee _{n\geq 0}a_n)=\neg \, 1=0$. So $\neg \, a_n\downarrow 0$, hence $\lim _{n\rightarrow \infty }s(\neg \, a_n)=s(0)=0$. By Lemma \[p3.6\], (\[p3.6(1)\]), and Lemma \[l6.1\], $\lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }\neg \, \neg \, s(a_n)=\lim _{n\rightarrow \infty }\neg \, s(\neg \, a_n)=\neg \, (\lim _{n\rightarrow \infty }s(\neg \, a_n))=\neg \, 0=1=s(1)$. Hence $s$ is $\uparrow $-continuous in $1$. Let $E:A^2\rightarrow L$ be an $L$-similarity relation and $s:A\rightarrow L$ an arbitrary function. We say that $s$ is [$E$-continuous in $a\in A$]{} iff, for all $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\stackrel{\textstyle E}{\textstyle \rightarrow }a$, we have $\lim _{n\rightarrow \infty }s(a_n)=s(a)$. We say that $s$ is [$E$-continuous]{} iff it is $E$-continuous in any $a\in A$. Actually, these definitions are valid for the residuated lattice $A$ replaced by an arbitrary nonempty set $X$, but we shall not work with them in this general case. Any order-preserving type I state $s:A\rightarrow L$ is $\rho _s$-continuous. \[p6.11\] Let $s:A\rightarrow L$ be an order-preserving type I state, $a\in A$ and $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$, that is $\lim _{n\rightarrow \infty }\rho _s(a_n,a)=1$. By Lemma \[l6.5\], (\[l6.5(3)\]), for all $n\in \N $, $\rho _s(a_n,a)\leq d_L(s(a_n),s(a))$. By Lemma \[l6.1\] and Lemma \[lA\], (\[lA(1)\]), it follows that $1=\lim _{n\rightarrow \infty }d_L(s(a_n),s(a))=d_L(\lim _{n\rightarrow \infty }s(a_n),s(a))$, hence $\lim _{n\rightarrow \infty }s(a_n)=s(a)$, thus $s$ is $\rho _s$-continuous in $a$. A residuated lattice $A$ is said to be [$\sigma $-complete]{} iff any sequence $(a_n)_{n\geq 0}\subseteq A$ has a supremum and an infimum in $A$. Notice that: $A$ is [$\sigma $-complete]{} iff any increasing sequence in $A$ has a supremum in $A$ and any decreasing sequence in $A$ has an infimum in $A$. This is easily shown, because, if the latter is verified, then, for any $(a_n)_{n\geq 0}\subseteq A$, if we consider the increasing sequence $(\bigvee _{k=0}^{n}a_k)_{n\geq 0}$, that has a supremum by the hypothesis, and the decreasing sequence $(\bigwedge _{k=0}^{n}a_k)_{n\geq 0}$, that has an infimum by the hypothesis, then $\bigvee _{n\geq 0}(\bigvee _{k=0}^{n}a_k)=\bigvee _{n\geq 0}a_n$ and $\bigwedge _{n\geq 0}(\bigwedge _{k=0}^{n}a_k)=\bigwedge _{n\geq 0}a_n$, which can easily be shown by the definition of the supremum and that of the infimum. Let $s:A\rightarrow L$ be a faithful order-preserving type I state, $A$ be $\rho _s$-complete and $L$ be $\sigma $-complete. Then $A$ is $\sigma $-complete and $s$ is $\uparrow $-continuous in $1$. \[p6.14\] By Remark \[r6.7\], $\rho _s$ is an $L$-equality on $A$. Let $(a_n)_{n\geq 0}\subseteq A$ be such that $(a_n)_{n\geq 0}\uparrow $. Since $s$ is order-preserving, it follows that $(s(a_n))_{n\geq 0}\uparrow $ in $L$. Since $L$ is $\sigma $-complete, there exists $\bigvee _{n\geq 0}s(a_n)$ in $L$, thus $(s(a_n))_{n\geq 0}\uparrow \bigvee _{n\geq 0}s(a_n)$, therefore, by Lemma \[l6.2\], $(s(a_n))_{n\geq 0}$ is convergent in $L$, hence $(s(a_n))_{n\geq 0}$ is Cauchy. By Lemma \[l6.5\], (\[l6.5(4)\]), for all $n,m\in \N $, $\rho _s(a_n,a_m)=d_L(s(a_n),s(a_m))$, thus $\lim _{n,m\rightarrow \infty }\rho _s(a_n,a_m)=\lim _{n,m\rightarrow \infty }d_L(s(a_n),s(a_m))=1$, so $(a_n)_{n\geq 0}$ is $\rho _s$-Cauchy. But $A$ is $\rho _s$-complete, therefore there exists $a\in A$ such that $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$. Let $k\in \N $, arbitrary but fixed. By Lemma \[l6.8\], $a_n\vee a_k\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a\vee a_k$. Since $(a_n)_{n\geq 0}\uparrow $, we have that, for all $n\geq k$, $a_n\vee a_k=a_n$, and, since $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$, we may conclude that $(a_n\vee a_k)_{n\geq 0}\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$. By Lemma \[l6.3\], it follows that $a\vee a_k=a$, that is $a_k\leq a$. Thus $a_n\leq a$ for all $n\in \N $. Now let $b\in A$ such that, for all $n\in \N $, $a_n\leq b$, that is $a_n\vee b=b$. By Lemma \[l6.8\], it follows that $a_n\vee b\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a\vee b$, that is $b\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a\vee b$, that is $b=a\vee b$, thus $a\leq b$. Hence $\bigvee _{n\geq 0}a_n=a$. Analogously one can prove that any decreasing sequence in $A$ has an infimum in $A$. Therefore $A$ is $\sigma $-complete. It remains to show that $s$ is $\uparrow $-continuous in $1$. Let $(a_n)_{n\geq 0}\subseteq A$ such that $a_n\uparrow 1$. By the above, there exists $a\in A$ such that $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$ and $a_n\leq a$ for all $n\in \N $, thus $1=\bigvee _{n\geq 0}a_n\leq a$, hence $a=1$. So $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }1$, that is $\lim _{n\rightarrow \infty }\rho _s(a_n,1)=1$. But, for all $n\in \N $, $\rho _s(a_n,1)=s(d_A(a_n,1))=s(a_n)$, as Lemma \[l2.2\], (\[l2.2(1)\]) and (\[l2.2(2)\]), shows. So $\lim _{n\rightarrow \infty }s(a_n)=1=s(1)$, hence $s$ is $\uparrow $-continuous in $1$. Let $A$ be an MV-algebra, $L$ a $\sigma $-complete involutive residuated lattice and $s:A\rightarrow L$ a faithful order-preserving type I state such that $A$ is $\rho _s$-complete. Then, by Propositions \[p6.12\] and \[p6.14\], $s$ is continuous. This way, Theorem 3.7 from [@ioanal] becomes a particular case of Proposition \[p6.14\]. \[r6.15\] In [@ioanal], the author defines and studies the [metric completion]{} of an MV-algebra endowed with an MV-state. This is a version for MV-algebras of the metric completion of an $l$-group with a state (see [@good]). In the following, we shall analyse the way in which this construction can be generalized to the case of a residuated lattice $A$ endowed with an order-preserving type I state. Throughout the rest of this section, $A$ and $L$ will be two residuated lattices such that $L$ is Cauchy-complete and $s:A\rightarrow L$ will be an order-preserving type I state. By Proposition \[p6.6\], $\rho _s$ is an $L$-similarity relation on $A$. Let us denote by ${\cal C}_s(A)$ the set of the $\rho _s$-Cauchy sequences in $A$ and let us define on ${\cal C}_s(A)$ the following binary operations: for all $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$, we define: for all $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$, $\underline{a}\circ \underline{b}=(a_n\circ b_n)_{n\geq 0}\in {\cal C}_s(A)$, because, by Lemma \[l6.5\], (\[l6.5(2)\]) and Lemma \[l6.1\], $\lim _{n,m\rightarrow \infty }\rho _s(a_n\circ b_n,a_m\circ b_m)\geq (\lim _{n,m\rightarrow \infty }\rho _s(a_n,a_m))\odot (\lim _{n,m\rightarrow \infty }\rho _s(b_n,b_m))=1$, thus $\lim _{n,m\rightarrow \infty }\rho _s(a_n\circ b_n,a_m\circ b_m)=1$, so $(a_n\circ b_n)_{n\geq 0}$ is a $\rho _s$-Cauchy sequence in $A$. We denote $\underline{0}=(0)_{n\geq 0},\underline{1}=(1)_{n\geq 0}\in {\cal C}_s(A)$, as all constant sequences in $A$ are obviously $\rho _s$-Cauchy (see Lemma \[lA\], (\[lA(1)\])). It is immediate that $({\cal C}_s(A),\vee ,\wedge ,\odot ,\rightarrow ,\underline{0},\underline{1})$ is a residuated lattice, whose biresiduum is $\leftrightarrow $ and whose negation is: for all $\underline{a}=(a_n)_{n\geq 0}\in {\cal C}_s(A)$, $\neg \, \underline{a}=\underline{a}\rightarrow \underline{0}=(a_n\rightarrow 0)_{n\geq 0}=(\neg \, a_n)_{n\geq 0}\in {\cal C}_s(A)$, as $\underline{a}\rightarrow \underline{0}\in {\cal C}_s(A)$. Let $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$. By Lemma \[l6.5\], (\[l6.5(5)\]) and Lemma \[l6.1\], for all $n,m\in \N $, $\rho _s(a_n,a_m)\odot \rho _s(b_n,b_m)\leq d_L(\rho _s(a_n,b_n),\rho _s(a_m,b_m))$, hence $\lim _{n,m\rightarrow \infty }d_L(\rho _s(a_n,b_n),\rho _s(a_m,b_m))=1$, thus the sequence $(\rho _s(a_n,b_n))_{n\geq 0}\subseteq L$ is Cauchy and hence convergent, since $L$ is Cauchy-complete. Let us define on ${\cal C}_s(A)$ the following binary relation: $\sim \subseteq {\cal C}_s(A)\times {\cal C}_s(A)$, defined by: for all $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$, $\underline{a}\sim \underline{b}$ iff $\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=1$. $\rho _s$ is an $L$-similarity relation on $A$, hence, by applying Lemma \[l6.1\], we obtain that $\sim $ is an equivalence relation on ${\cal C}_s(A)$. Let us consider the quotient set $\tilde{A}_s:={\cal C}_s(A)/_{\sim }=\{\tilde{\underline{a}}\|\underline{a}\in {\cal C}_s(A)\}$, where we denoted by $\tilde{\underline{a}}$ the equivalence class of a sequence $\underline{a}\in {\cal C}_s(A)$ with respect to $\sim $. Let us define on $\tilde{A}_s$ the following binary operations: for all $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$, we define: for all $\underline{a},\underline{b}\in {\cal C}_s(A)$, $\tilde{\underline{a}}\circ \tilde{\underline{b}}=\widetilde{\underline{a}\circ \underline{b}}\in \tilde{A}_s$. Let us prove that all of these operations are well defined. Let $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow ,\leftrightarrow \}$ and let $\underline{a}=(a_n)_{n\geq 0},\underline{a^{\prime }}=(a^{\prime }_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0},\underline{b^{\prime }}=(b^{\prime }_n)_{n\geq 0}\in {\cal C}_s(A)$ such that $\underline{a}\sim \underline{a^{\prime }}$ and $\underline{b}\sim \underline{b^{\prime }}$, that is: $\lim _{n\rightarrow \infty }\rho _s(a_n,a^{\prime }_n)=\lim _{n\rightarrow \infty }\rho _s(b_n,b^{\prime }_n)=1$. By Lemma \[l6.5\], (\[l6.5(2)\]) and Lemma \[l6.1\], it follows that $\lim _{n\rightarrow \infty }\rho _s(a_n\circ b_n,a^{\prime }_n\circ b^{\prime }_n)=1$, that is $\underline{a}\circ \underline{b}\sim \underline{a^{\prime }}\circ \underline{b^{\prime }}$, that is $\widetilde{\underline{a}\circ \underline{b}}=\widetilde{\underline{a^{\prime }}\circ \underline{b^{\prime }}}$. So $\circ $ is well defined. Thus $\sim $ has become a congruence relation on the residuated lattice $({\cal C}_s(A),\vee ,\wedge ,\odot ,\rightarrow ,\underline{0},\underline{1})$, and the fact that residuated lattices form an equational class ensures us that $(\tilde{A}_s,\vee ,\wedge ,\odot ,\rightarrow ,\tilde{\underline{0}},\tilde{\underline{1}})$ is a residuated lattice, whose biresiduum is obviously $\leftrightarrow $ and whose negation is: for all $\underline{a}\in {\cal C}_s(A)$, $\neg \, \tilde{\underline{a}}=\tilde{\underline{a}}\rightarrow \tilde{\underline{0}}=\widetilde{\underline{a}\rightarrow \underline{0}}=\widetilde{\neg \, \underline{a}}\in \tilde{A}_s$. If $L$ is involutive then $\tilde{A}_s$ is involutive. \[l6.16\] By Lemma \[l2.3\], (\[l2.3(2)\]), Proposition \[p3.3\], (\[p3.3(2)\]), Proposition \[p3.6\], (\[p3.6(1)\]), the fact that $L$ is involutive and Lemma \[l2.2\], (\[l2.2(3)\]), for all $a\in A$, $s(\neg \, \neg \, a\rightarrow a)=s(\neg \, \neg \, a)\rightarrow s(a)=\neg \, \neg \, s(a)\rightarrow s(a)=s(a)\rightarrow s(a)=1$ and thus $\rho _s(a,\neg \, \neg \, a)=s(d_A(a,\neg \, \neg \, a))=s(\neg \, \neg \, a\rightarrow a)=1$. Thus, for all $a\in A$, $\rho _s(a,\neg \, \neg \, a)=s(d_A(a,\neg \, \neg \, a))=s()$. Let $\underline{a}=(a_n)_{n\geq 0}\in {\cal C}_s(A)$ and let us consider the sequence $\neg \, \neg \, \underline{a}=(\neg \, \neg \, a_n)_{n\geq 0}\in {\cal C}_s(A)$. For all $n\in \N $, $\rho _s(a_n,\neg \, \neg \, a_n)=1$, hence $\neg \, \neg \, \underline{a}\sim \underline{a}$, that is $\widetilde{\neg \, \neg \, \underline{a}}=\tilde{\underline{a}}$, that is $\neg \, \neg \, \tilde{\underline{a}}=\tilde{\underline{a}}$. For all $a\in A$, let us denote in this paragraph the constant sequence $\underline{a}=(a)_{n\geq 0}\in {\cal C}_s(A)$. The function $\psi _s:A\rightarrow {\cal C}_s(A)$, defined by $\psi _s(a)=\underline{a}$ for all $a\in A$, is obviously an injective residuated lattice morphism. By composing the canonical projection from ${\cal C}_s(A)$ to the quotient residuated lattice $\tilde{A}_s$ with the morphism $\psi _s$, we obtain the residuated lattice morphism $\varphi _s:A\rightarrow \tilde{A}_s$, defined by $\varphi _s(a)=\tilde{\underline{a}}$ for all $a\in A$. Let $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0},\underline{c}=(c_n)_{n\geq 0},\underline{d}=(d_n)_{n\geq 0}\in {\cal C}_s(A)$. If $\underline{a}\sim \underline{c}$ and $\underline{b}\sim \underline{d}$, then $\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=\lim _{n\rightarrow \infty }\rho _s(c_n,d_n)$. \[l6.17\] By the fact that $\rho _s$ is an $L$-similarity relation on $A$ and Lemma \[l2.2\], (\[l2.2nenum2\]), we have that: for all $n\in \N $, $\rho _s(c_n,a_n)\odot \rho _s(a_n,b_n)\odot \rho _s(b_n,d_n)\leq \rho _s(c_n,d_n)$. By Lemma \[lA\], (\[lA(2)\]) and Lemma \[l6.1\], it follows that $1\odot (\lim _{n\rightarrow \infty }\rho _s(a_n,b_n))\odot 1\leq \lim _{n\rightarrow \infty }\rho _s(b_n,d_n)$, hence $\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)\leq \lim _{n\rightarrow \infty }\rho _s(c_n,d_n)$. The converse inequality results in a similar way. By Lemma \[l6.17\], we can define the function $\tilde{\rho _s}:\tilde{A}_s\times \tilde{A}_s\rightarrow L$, by: for all $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$, $\tilde{\rho _s}(\tilde{\underline{a}},\tilde{\underline{b}})=\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)$. $\tilde{\rho _s}$ is an $L$-similarity relation on $\tilde{A}_s$. \[p6.18\] It is immediate that $\tilde{\rho _s}$ is reflexive and symmetric. In order to prove that it is transitive, let us consider $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0},\underline{c}=(c_n)_{n\geq 0}\in {\cal C}_s(A)$. Then, by the fact that $\rho _s$ is an $L$-similarity relation on $A$, it follows that, for all $n\in \N $, $\rho _s(a_n,b_n)\odot \rho _s(b_n,c_n)\leq \rho _s(a_n,c_n)$. By applying Lemma \[l6.1\], we obtain: $\tilde{\rho _s}(\tilde{\underline{a}},\tilde{\underline{b}})\odot \tilde{\rho _s}(\tilde{\underline{b}},\tilde{\underline{c}})=\lim _{n\rightarrow \infty }(\rho _s(a_n,b_n)\odot \rho _s(b_n,c_n))\leq \tilde{\rho _s}(\tilde{\underline{a}},\tilde{\underline{c}})$. Let $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$. If $\underline{a}\sim \underline{b}$, then $\lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }s(b_n)$. \[l6.19\] By Lemma \[l6.5\], (\[l6.5(3)\]), for all $n\in \N $, $\rho _s(a_n,b_n)\leq d_L(s(a_n),s(b_n))$. By Lemma \[l6.1\] and the fact that $\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=1$, we have: $d_L(\lim _{n\rightarrow \infty }s(a_n),\lim _{n\rightarrow \infty }s(b_n))=\lim _{n\rightarrow \infty }d_L(s(a_n),s(b_n))=1$. By Lemma \[lA\], (\[lA(1)\]), we get: $\lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }s(b_n)$. Lemma \[l6.19\] allows us to define the function $\tilde{s}:\tilde{A}_s\rightarrow L$, for all $\underline{a}=(a_n)_{n\geq 0}\in {\cal C}_s(A)$, $\tilde{s}(\tilde{\underline{a}})=\lim _{n\rightarrow \infty }s(a_n)$. $\tilde{s}$ is a faithful order-preserving type I state. \[p6.20\] Obviously, $\tilde{s}(\tilde{\underline{0}})=0$ and $\tilde{s}(\tilde{\underline{1}})=1$. By Lemma \[l6.1\], $\tilde{s}$ is an order-preserving function. Now let $\underline{a}=(a_n)_{n\geq 0},(b_n)_{n\geq 0}\in {\cal C}_s(A)$. Then, by Proposition \[p3.3\], (\[p3.3(3)\]) and Lemma \[l6.1\], $\tilde{s}(\tilde{\underline{a}}\rightarrow \tilde{\underline{b}})=\lim _{n\rightarrow \infty }s(a_n\rightarrow b_n)=\lim _{n\rightarrow \infty }(s(a_n)\rightarrow s(a_n\wedge b_n))=(\lim _{n\rightarrow \infty }s(a_n))\rightarrow (\lim _{n\rightarrow \infty }s(a_n\wedge b_n))=\tilde{s}(\tilde{\underline{a}})\rightarrow \tilde{s}(\widetilde{\underline{a}\wedge \underline{b}})=\tilde{s}(\tilde{\underline{a}})\rightarrow \tilde{s}(\tilde{\underline{a}}\wedge \tilde{\underline{b}})$. Thus, by Proposition \[p3.3\], (\[p3.3(3)\]), $\tilde{s}$ is a type I state. If $\tilde{s}(\tilde{\underline{a}})=1$, then, by Lemma \[l2.2\], (\[l2.2(1)\]) and (\[l2.2(2)\]), $\lim _{n\rightarrow \infty }\rho _s(a_n,1)=\lim _{n\rightarrow \infty }s(d_A(a_n,1))=\lim _{n\rightarrow \infty }s(a_n)=1$, so $\underline{a}\sim \underline{1}$, that is $\tilde{\underline{a}}=\tilde{\underline{1}}$. Hence $\tilde{s}$ is faithful. The following theorem collects the main properties of $\tilde{A}_s$, $\tilde{\rho _s}$ and $\tilde{s}$. Let $A$ and $L$ be two residuated lattices, such that $L$ is Cauchy-complete, and $s:A\rightarrow L$ an order-preserving type I state. Then: 1. \[t6.2(1)\] $\tilde{A}_s$ is a residuated lattice; if $L$ is involutive then $\tilde{A}_s$ is also involutive; 2. \[t6.2(2)\] $\tilde{s}$ is a faithful order-preserving type I state; 3. \[t6.2(4)\] $\varphi _s$ is a residuated lattice morphism and $\tilde{s}\circ \varphi _s=s$; 4. \[t6.2(5)\] $\varphi _s$ is injective iff $s$ is faithful; 5. \[t6.2(6)\] $\tilde{\rho _s}=\rho _{\tilde{s}}$; 6. \[t6.2(7)\] for any $(a_n)_{n\geq 0}\subseteq A$ and $a\in A$, if $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$, then $\varphi _s(a_n)\stackrel{\textstyle \tilde{\rho _s}}{\textstyle \rightarrow }\varphi _s(a)$; 7. \[t6.2(8)\] for any residuated lattice $C$, any faithful order-preserving type I state $m:C\rightarrow L$ such that $C$ is $\rho _m$-complete, and any residuated lattice morphism $f:A\rightarrow C$ such that $m\circ f=s$, there exists a residuated lattice morphism $\tilde{f}:\tilde{A}_s\rightarrow C$ such that $m\circ \tilde{f}=\tilde{s}$ and $\tilde{f}\circ \varphi _s=f$. \[t6.2\] (\[t6.2(1)\]) This is Lemma \[l6.16\]. (\[t6.2(2)\]) This is Proposition \[p6.20\]. (\[t6.2(4)\]) We know that $\varphi _s$ is a residuated lattice morphism. Let $a\in A$ and $\underline{a}=(a)_{n\geq 0}$. $(\tilde{s}\circ \varphi _s)(a)=\tilde{s}(\tilde{\underline{a}})=\lim _{n\rightarrow \infty }s(a)=s(a)$. Thus $\tilde{s}\circ \varphi _s=s$. (\[t6.2(5)\]) Let $a\in A$ and $\underline{a}=(a)_{n\geq 0}\in {\cal C}_s(A)$. We have the equivalences: $a\in {\rm Ker}(\varphi _s)$ iff $\varphi _s(a)=\tilde{\underline{1}}$ iff $\tilde{\underline{a}}=\tilde{\underline{1}}$ iff $\lim _{n\rightarrow \infty }\rho _s(a,1)=1$ iff $\lim _{n\rightarrow \infty }s(a)=1$ iff $s(a)=1$, by Lemma \[l2.2\], (\[l2.2(1)\]) and (\[l2.2(2)\]). Hence: $\varphi _s$ is injective iff ${\rm Ker}(\varphi _s)=\{1\}$ iff the fact that $s(a)=1$ implies $a=1$ iff $s$ is faithful. (\[t6.2(6)\]) $\tilde{\rho _s},\rho _{\tilde{s}}:\tilde{A}_s\times \tilde{A}_s\rightarrow L$. For all $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$, we have the following equalities: $\rho _{\tilde{s}}(\tilde{\underline{a}},\tilde{\underline{b}})=\tilde{s}(d_{\tilde{A}_s}(\tilde{\underline{a}},\tilde{\underline{b}}))=\tilde{s}((d_A(a_n,b_n))_{n\geq 0})=\lim _{n\rightarrow \infty }s(d_A(a_n,b_n))=\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=\tilde{\rho _s}(\tilde{\underline{a}},\tilde{\underline{b}})$. Thus $\tilde{\rho _s}=\rho _{\tilde{s}}$. (\[t6.2(7)\]) Let $\underline{x}=(a_n)_{n\geq 0}\subseteq A$ and $a\in A$, such that $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$, that is $\lim _{n\rightarrow \infty }\rho _s(a_n,a)=1$. Let us denote $\underline{a}=(a)_{n\geq 0}$. For all $n\in \N $, $\tilde{\rho _s}(\varphi _s(a_n),\varphi _s(a))=\rho _{\tilde{s}}(\varphi _s(a_n),\varphi _s(a))=\tilde{s}(d_{\tilde{A}_s}(\varphi _s(a_n),\varphi _s(a)))=\tilde{s}(d_{\tilde{A}_s}(\tilde{\underline{x}},\tilde{\underline{a}}))=\tilde{s}((d_A(a_n,a))_{n\geq 0})=\lim _{n\rightarrow \infty }s(d_A(a_n,a))=\lim _{n\rightarrow \infty }\rho _s(a_n,a)=1$. Hence $\lim _{n\rightarrow \infty }\tilde{\rho _s}(\varphi _s(a_n),\varphi _s(a))=1$, that is $\varphi _s(a_n)\stackrel{\textstyle \tilde{\rho _s}}{\textstyle \rightarrow }\varphi _s(a)$. (\[t6.2(8)\]) Let $C$, $m$ and $f$ be like in the enunciation. Then, by Remark \[r6.7\], $\rho _m$ is an $L$-equality on $C$. We shall denote by $\approx $ the congruence on ${\cal C}_m(C)$ defined in the same way as $\sim $ on ${\cal C}_s(A)$. Let $(a_n)_{n\geq 0}\in {\cal C}_s(A)$, arbitrary but fixed, so $\lim _{n,k\rightarrow \infty }\rho _s(a_n,a_k)=1$. For all $n,k\in \N $, since $f$ is a residuated lattice morphism, we have: $\rho _m(f(a_n),f(a_k))=m(d_C(f(a_n),f(a_k)))=m(f(d_A(a_n,a_k)))=s(d_A(a_n,a_k))=\rho _s(a_n,a_k)$. Thus $\lim _{n,k\rightarrow \infty }\rho _m(f(a_n),f(a_k))=1$, that is $(f(a_n))_{n\geq 0}\in {\cal C}_m(C)$, so, since $C$ is $\rho _m$-complete, there exists $c\in C$ such that $f(a_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }c$. This element $c$ of $C$ is unique, as Lemma \[l6.3\] shows. We set $\tilde{f}(\widetilde{(a_n)_{n\geq 0}})=c$. Let us prove that $\tilde{f}$ is well defined. Let $(a_n)_{n\geq 0},(b_n)_{n\geq 0}\in {\cal C}_s(A)$, such that $(a_n)_{n\geq 0}\sim (b_n)_{n\geq 0}$. By the above, there exist $c,d\in C$ such that $f(a_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }c$ and $f(b_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }d$. We have to prove that $c=d$. The fact that $f(a_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }c$ is equivalent to $\lim _{n\rightarrow \infty }\rho _m(f(a_n),c)=1$, that is $(f(a_n))_{n\geq 0}\approx (c)_{n\geq 0}$ (the constant sequence). Analogously, $(f(b_n))_{n\geq 0}\approx (d)_{n\geq 0}$. By Lemma \[l6.19\], $\lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }s(b_n)$, that is $\lim _{n\rightarrow \infty }m(f(a_n))=\lim _{n\rightarrow \infty }m(f(b_n))$. By the fact that $f$ is a residuated lattice morphism and $(a_n)_{n\geq 0}\sim (b_n)_{n\geq 0}$, it follows that $\lim _{n\rightarrow \infty }\rho _m(f(a_n),f(b_n))=\lim _{n\rightarrow \infty }m(d_C(f(a_n),f(b_n)))=\lim _{n\rightarrow \infty }m(f(d_A(a_n,b_n)))=\lim _{n\rightarrow \infty }s(d_A(a_n,b_n))=\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=1$, hence $(f(a_n))_{n\geq 0}\approx (f(b_n))_{n\geq 0}$. By the symmetry and the transitivity of $\approx $, it follows that $(c)_{n\geq 0}\approx (d)_{n\geq 0}$, thus $1=\lim _{n\rightarrow \infty }\rho _m(c,d)=\rho _m(c,d)=m(d_C(c,d))$. By the fact that $m$ is faithful and by Lemma \[lA\], (\[lA(1)\]), it results that $c=d$, therefore $\tilde{f}$ is well defined. Let us prove that $\tilde{f}$ defined this way is a residuated lattice morphism. It is trivial that $\tilde{f}(\tilde{\underline{0}})=0$ and $\tilde{f}(\tilde{\underline{1}})=1$. Now let $\circ \in \{\vee ,\wedge ,\odot ,\rightarrow \}$ and $(a_n)_{n\geq 0},(b_n)_{n\geq 0}\in {\cal C}_s(A)$. By the above, there exist $c,d\in C$ such that $f(a_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }c$ and $f(b_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }d$, and $\tilde{f}(\widetilde{(a_n)_{n\geq 0}})=c$ and $\tilde{f}(\widetilde{(b_n)_{n\geq 0}})=d$. By Lemma \[l6.8\] and the fact that $f$ is a residuated lattice morphism, we have: $f(a_n\circ b_n)=f(a_n)\circ f(b_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }c\circ d$, thus $\tilde{f}(\widetilde{(a_n)_{n\geq 0}}\circ \widetilde{(b_n)_{n\geq 0}})=\tilde{f}(\widetilde{(a_n\circ b_n)_{n\geq 0}})=c\circ d=\tilde{f}(\widetilde{(a_n)_{n\geq 0}})\circ \tilde{f}(\widetilde{(b_n)_{n\geq 0}})$. So $\tilde{f}$ is a residuated lattice morphism. For all $a\in A$, the constant sequence $(f(a))_{n\geq 0}\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }f(a)\in C$, thus $\tilde{f}(\varphi _s(a))=\tilde{f}(\widetilde{(a)_{n\geq 0}})=f(a)$. So $\tilde{f}\circ \varphi _s=f$. Now let $(a_n)_{n\geq 0}\in {\cal C}_s(A)$. By the above, there exists $c\in C$ such that $f(a_n)\stackrel{\textstyle \rho _m}{\textstyle \rightarrow }c$, so $\tilde{f}(\widetilde{(a_n)_{n\geq 0}})=c$. As above, one can show that $(f(a_n))_{n\geq 0}\approx (c)_{n\geq 0}$, thus, by Lemma \[l6.19\], $\tilde{s}(\widetilde{(a_n)_{n\geq 0}})=\lim _{n\rightarrow \infty }s(a_n)=\lim _{n\rightarrow \infty }m(f(a_n))=\lim _{n\rightarrow \infty }m(c)=m(c)=(m\circ \tilde{f})(\widetilde{(a_n)_{n\geq 0}})$. Hence $m\circ \tilde{f}=\tilde{s}$. Prove that the morphism $\tilde{f}:\tilde{A}_s\rightarrow C$ from Theorem \[t6.2\], (\[t6.2(8)\]) is unique. Now let us analyse the construction of $\tilde{A}_s$ for the particular case when $L=([0,1],\max ,\min ,\odot _L,\rightarrow _L)$ is the standard MV-algebra, which is Cauchy-complete, as one can easily deduce from the fact that, if $d$ is the Euclidean distance in $\R $ restricted to $[0,1]\times [0,1]$, then $([0,1],d)$ is a complete metric space, and from the computation: for all $x,y\in L=[0,1]$, $d_L(x,y)=\min \{\min \{1,1-x+y\},\min \{1,1-y+x\}\}=\min \{1,1-x+y,1-y+x\}=\min \{1-x+y,1-y+x\}=1-\max \{x-y,y-x\}=1-\|x-y\|=1-d(x,y)$ (the deduction can be made in a similar manner to the one below that shows that $(A,\delta _s)$ is a complete pseudo-metric space iff $A$ is $\rho _s$-complete). We are still in the framework: $A$ is a residuated lattice and $s:A\rightarrow L$ is an order-preserving type I state, thus, in this case, $s:A\rightarrow [0,1]$ is a Bosbach state, as Remark \[r3.7\] shows. Let us define the function $\delta _s:A^2\rightarrow [0,1]$, for all $a,b\in A$, $\delta _s(a,b)=1-\rho _s(a,b)$. A function $\delta _t$ can be defined in this way for any Bosbach state $t$ on any residuated lattice. $\delta _s$ is a pseudo-metric on $A$. Indeed, by Proposition \[p6.6\], for all $a,b,c\in A$, $\rho _s(a,b)\odot _L\rho _s(b,c)\leq \rho _s(a,c)$, thus $(1-\delta _s(a,b))\odot _L(1-\delta _s(b,c))\leq 1-\delta _s(a,c)$, that is $\max \{0,1-\delta _s(a,b)-\delta _s(b,c)\}\leq 1-\delta _s(a,c)$, therefore $1-\delta _s(a,b)-\delta _s(b,c)\leq 1-\delta _s(a,c)$, that is $\delta _s(a,c)\leq \delta _s(a,b)+\delta _s(b,c)$. Moreover, by Lemma \[lA\], (\[lA(1)\]), it follows that $\delta _s$ is a metric on $A$ iff $s$ is faithful. This is valid for any residuated lattice $A$ and any Bosbach state $s$ on $A$. \[r6.22\] The remark above shows that $(A,\delta _s)$ is a pseudo-metric space, thus we can construct its metric completion. In order to accomplish this, let us notice that: a sequence $(a_n)_{n\geq 0}$ in the pseudo-metric space $(A,\delta _s)$ converges towards $a\in A$ (in the pseudo-metric sense) iff $\lim _{n\rightarrow \infty }\delta _s(a_n,a)=0$, that is $\lim _{n\rightarrow \infty }\rho _s(a_n,a)=1$, that is $a_n\stackrel{\textstyle \rho _s}{\textstyle \rightarrow }a$. Also, a sequence $(a_n)_{n\geq 0}\subseteq A$ is Cauchy in the pseudo-metric space $(A,\delta _s)$ iff $\lim _{n,m\rightarrow \infty }\delta _s(a_n,a_m)=0$ iff $\lim _{n,m\rightarrow \infty }\rho _s(a_n,a_m)=1$ iff $(a_n)_{n\geq 0}$ is $\rho _s$-Cauchy. It follows that $(A,\delta _s)$ is a complete pseudo-metric space iff $A$ is $\rho _s$-complete, and this is valid for an arbitrary residuated lattice $A$ and an arbitrary Bosbach state $s:A\rightarrow [0,1]$. It also follows that, with the definition above, ${\cal C}_s(A)$ is equal to the set of the Cauchy sequences of the pseudo-metric space $(A,\delta _s)$ and the binary relation $\sim $ on ${\cal C}_s(A)$ satisfies: for all $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$, $\underline{a}\sim \underline{b}$ iff $\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=1$ iff $\lim _{n\rightarrow \infty }\delta _s(a_n,b_n)=0$. With the notations above, let us define the function $\tilde{\delta _s}:\tilde{A}_s\times \tilde{A}_s\rightarrow L$, for all $\underline{a}=(a_n)_{n\geq 0},\underline{b}=(b_n)_{n\geq 0}\in {\cal C}_s(A)$, $\tilde{\delta _s}(\tilde{\underline{a}},\tilde{\underline{b}})=\lim _{n\rightarrow \infty }\delta _s(a_n,b_n)=1-\lim _{n\rightarrow \infty }\rho _s(a_n,b_n)=1-\tilde{\rho _s}(\tilde{\underline{a}},\tilde{\underline{b}})$; $\tilde{\delta _s}$ is well defined because $\tilde{\rho _s}$ is well defined. By Proposition \[p6.18\], $\tilde{\rho _s}$ is an $L$-similarity relation on $\tilde{A}_s$. Moreover, by Theorem \[t6.2\], (\[t6.2(6)\]), $\tilde{\delta _s}=1-\tilde{\rho _s}=1-\rho _{\tilde{s}}=\delta _{\tilde{s}}$ with the notation above Remark \[r6.22\], hence, by Remark \[r6.22\] and since $\tilde{s}$ is a faithful Bosbach state by Proposition \[p6.20\], it follows that $\tilde{\delta _s}=\delta _{\tilde{s}}$ is a metric on $\tilde{A}_s$. The usual construction from the theory of metric spaces identifies $(\tilde{A}_s,\tilde{\delta _s}=\delta _{\tilde{s}})$ to be the metric completion of $(A,\delta _s)$. The universality property of the metric completion ensures us that, for any Cauchy-complete metric space $C$ and any isometry $f:A\rightarrow C$, there exists a unique isometry $\tilde{f}:\tilde{A}_s\rightarrow C$ such that $\tilde{f}\circ \varphi _s=f$. We can translate this as the theorem below, by relying on the following lemma. Let $A_1$ and $A_2$ be two residuated lattices, $s_1:A_1\rightarrow [0,1]$ and $s_2:A_2\rightarrow [0,1]$ Bosbach states and $h:A_1\rightarrow A_2$ a morphism of residuated lattices. Then: $s_2\circ h=s_1$ iff $h$ is an isometry between the pseudo-metric spaces $(A_1,\delta _{s_1})$ and $(A_2,\delta _{s_2})$. We shall use the fact that $h$ is a residuated lattice morphism and thus it preserves the biresiduum. “$\Rightarrow $“: Assume that $s_2\circ h=s_1$ and let $a,b\in A_1$. $\delta _{s_2}(h(a),h(b))=1-\rho _{s_2}(h(a),h(b))=1-s_2(d_{A_2}(h(a),h(b)))=1-s_2(h(d_{A_1}(a,b)))=1-s_1(d_{A_1}(a,b))=1-\rho _{s_1}(a,b)=\delta _{s_1}(a,b)$. Hence $h$ is an isometry. “$\Leftarrow $“: Assume that $h$ is an isometry, that is, for all $a,b\in A_1$, $\delta _{s_2}(h(a),h(b))=\delta _{s_1}(a,b)$. Let $a\in A_1$. By Lemma \[l2.2\], (\[l2.2(1)\]) and (\[l2.2(2)\]), $s_2(h(a))=s_2(h(d_{A_1}(a,1)))=s_2(d_{A_2}(h(a),h(1)))=\rho _{s_2}(h(a),h(1))=1-\delta _{s_2}(h(a),h(1))=1-\delta _{s_1}(a,1)=\rho _{s_1}(a,1)=s_1(d_{A_1}(a,1))=s_1(a)$. Hence $s_2\circ h=s_1$. For any residuated lattice $C$, any faithful Bosbach state $m:C\rightarrow [0,1]$ such that $(C,\delta _m)$ is a Cauchy-complete metric space, and any residuated lattice morphism $f:A\rightarrow C$ such that $m\circ f=s$, there exists a unique residuated lattice morphism $\tilde{f}:\tilde{A}_s\rightarrow C$ such that $m\circ \tilde{f}=\tilde{s}$ and $\tilde{f}\circ \varphi _s=f$. \[t01\] By an observation above, the fact that $(C,\delta _m)$ is Cauchy-complete is equivalent to the fact that $C$ is $\rho _m$-complete and hence the unique morphism $\tilde{f}$ from Theorem \[t01\] is none other than the morphism constructed in the proof of Theorem \[t6.2\], (\[t6.2(8)\]). $\tilde{A}_s$ is $\rho _{\tilde{s}}$-complete, $\sigma $-complete and involutive, and $\tilde{s}$ is $\uparrow $-continuous in $1$. If $A$ is an MV-algebra, then $\tilde{s}$ is continuous. By the above, $(\tilde{A}_s,\delta _{\tilde{s}})$ is a complete metric space, thus $\tilde{s}$ is a faithful Bosbach state and $\tilde{A}_s$ is a $\rho _{\tilde{s}}$-complete residuated lattice. Since $[0,1]$ with the natural order is $\sigma $-complete, it follows by Proposition \[p6.14\] that $A$ is $\sigma $-complete and $s$ is $\uparrow $-continuous in $1$. If $A$ is an MV-algebra, then obviously $\tilde{A}_s$ is an MV-algebra. $[0,1]$ is involutive, as any MV-algebra is. By Remark \[r6.15\], it follows that $\tilde{s}$ is continuous. Adopting a denomination from [@ioanal3], we shall call $\tilde{A}_s$ [the $s$-completion of $A$]{}. Final Remarks {#finalremarks} ============= In this section we will sketch two ways in which we can relate generalized Bosbach states to monoidal t-norm-based logics and we will formulate some open problems. \(I) The probabilistic logic FP(Ł$_n$,Ł) studied in [@flago], [@fla1] is a formal description of a way of reasoning on the probability of fuzzy events through the infinite-valued Łukasiewicz logic Ł. In [@flago], [@fla1] the authors admit the hypothesis that fuzzy events follow the rules of the finite-valued Łukasiewicz logic Ł$_n$. We shall sketch now a more general context for developping some logics similar to FP(Ł$_n$,Ł). Let ${\cal C}_1$ and ${\cal C}_2$ be two schematic extensions of the MTL logic ([@fla1]). The probabilistic logic ${\rm FP}({\cal C}_1,{\cal C}_2)$ is based on the following hypotheses: - the events are structured by the logic ${\cal C}_1$; - the evaluation of the probability of the events is made in conformity to the logic ${\cal C}_2$. The language of the logic ${\rm FP}({\cal C}_1,{\cal C}_2)$ is constructed by starting from a numerable set of propositional variables $V=\{p_1,p_2,\ldots ,p_k,\ldots \}$, the truth constant $\perp $, the connectives $\vee ,\wedge ,\rightarrow ,\& $ and a symbol $P$ (for the modality “probably“). The formulas of ${\rm FP}({\cal C}_1,{\cal C}_2)$ are defined in two steps: - the set $F_m(V)$ of the [non-modal]{} formulas is exactly the set of the formulas of ${\cal C}_1$ (the non-modal formulas will be denoted $\varphi ,\ \psi ,\ldots $); - the atomic modal formulas are of the form $P(\varphi )$, with $\varphi \in F_m(V)$; the set $MF_m(V)$ of the modal formulas is constructed inductively, starting from the atomic modal formulas and using the connectives $\vee ,\wedge ,\rightarrow ,\& $ and the truth constant $\perp $. ${\rm FP}({\cal C}_1,{\cal C}_2)$ has the following axioms: - the axioms of ${\cal C}_1$ for non-modal formulas; - the axioms of ${\cal C}_2$ for modal formulas; - the following axioms for the modality $P$: (A1) $P(\varphi \rightarrow \psi )\rightarrow (P(\varphi )\rightarrow P(\psi ))$ (A2) $P(\varphi \rightarrow \psi )\rightarrow (P(\varphi )\rightarrow P(\varphi \wedge \psi ))$ ${\rm FP}({\cal C}_1,{\cal C}_2)$ has two deduction rules: - the modus ponens rule (for modal and non-modal formulas); - the necessity rule: from $\varphi $ derive $P(\varphi )$. Remark \[r3.26\] shows that the logic FP(Ł$_n$,Ł) can be obtained from ${\rm FP}({\cal C}_1,{\cal C}_2)$ by setting ${\cal C}_1=$Ł$_n$ and ${\cal C}_2=$Ł. Define a semantics corresponding to the logic ${\rm FP}({\cal C}_1,{\cal C}_2)$ (by extending the notions of weak probabilistic Kripke model and strong probabilistic Kripke model from [@flago], [@fla1]) and prove the weak and strong completeness theorems for ${\rm FP}({\cal C}_1,{\cal C}_2)$. \[o7.1\] \(II) Let ${\cal C}$ be a schematic extension of MTL and ${\cal C}_{\forall }$ be the predicate logic associated to ${\cal C}$ (see [@haj], [@[a]]). We shall denote by $E$ the set of the sentences of ${\cal C}_{\forall }$ and by $E/_{\sim }=\{\hat{\varphi }\|\varphi \in E\}$ the Lindenbaum-Tarski algebra of ${\cal C}_{\forall }$. $E/_{\sim }$ is an MTL-algebra that also verifies the algebraic form of the axioms specific to ${\cal C}_{\forall }$. Let $D$ be a subset of $E$ such that: - $D$ contains all the formal theorems of ${\cal C}_{\forall }$; - $D$ is closed with respect to the connectives $\vee ,\wedge ,\rightarrow ,\& $ and $D$ contains the truth constant $\perp $. Then $D/_{\sim }=\{\hat{\varphi }\|\varphi \in D\}$ is a subalgebra of $E/_{\sim }$ (in particular, $D/_{\sim }$ is an MTL-algebra). We consider on $[0,1]$ the structure of MTL-algebra induced by a left-continuous t-norm ([@beloh]). A function $\mu :D\rightarrow [0,1]$ is called a [logical probability]{} on $D$ iff, for all $\varphi ,\psi \in D$: (P1) if $\vdash \varphi $ then $\mu (\varphi )=1$; (P2) $\mu (\varphi \rightarrow \psi )\rightarrow (\mu (\varphi )\rightarrow \mu (\psi ))=1$; (P3) $\mu (\varphi \rightarrow \psi )=\mu (\varphi )\rightarrow \mu (\varphi \wedge \psi )$. \[d7.2\] Let $\mu :D\rightarrow [0,1]$ be a logical probability and $\varphi ,\psi \in D$. Then: 1. \[l7.3(1)\] if $\vdash \varphi \rightarrow \psi $ then $\mu (\varphi )\leq \mu (\psi )$; 2. \[l7.3(2)\] if $\vdash \varphi \leftrightarrow \psi $ then $\mu (\varphi )=\mu (\psi )$. \[l7.3\] By Lemma \[l7.3\], (\[l7.3(2)\]), we can define a function $\tilde{\mu }:D/_{\sim }\rightarrow [0,1]$ by $\tilde{\mu }(\hat{\varphi })=\mu (\varphi )$ for all $\varphi \in D$. It immediately follows that $\tilde{\mu }$ is an order-preserving type I state on the residuated lattice $D/_{\sim }$. Let $U$ be a set of new constants and ${\cal C}_{\forall }(U)$ be the language obtained from ${\cal C}_{\forall }$ by adjoining the constants from $U$. We denote by $E(U)$ the set of the constants of ${\cal C}_{\forall }(U)$. We fixe a set of constants $U$ and a logical probability $m:E(U)\rightarrow [0,1]$. We shall introduce two conditions on the pair $(U,m)$: $(G\exists )$ for any formula $\phi (x)$ of ${\cal C}_{\forall }(U)$, $m(\exists x\phi (x))=\sup \{m(\bigvee _{i=1}^{n}\phi (a_i))\|n\in \N ^{},a_1,\ldots ,a_n\in U\}$; $(G\forall )$ for any formula $\phi (x)$ of ${\cal C}_{\forall }(U)$, $m(\forall x\phi (x))=\inf \{m(\bigwedge _{i=1}^{n}\phi (a_i))\|n\in \N ^{},a_1,\ldots ,a_n\in U\}$. $(G\exists )$ and $(G\forall )$ are similar to the Gaifman conditions on the probabilities defined in classical first-order logic ([@[b]]). A [probabilistic structure]{} on ${\cal C}_{\forall }$ is a pair $(U,m)$ that satisfies $(G\exists )$ and $(G\forall )$. A probabilistic structure $(U,m)$ is a [probabilistic model of a logical probability $\mu :D\rightarrow [0,1]$]{} iff $m\mid _{D}=\mu $. Prove for some schematic extensions ${\cal C}$ of the MTL logic the following completeness theorem: any logical probability $\mu $ admits a probabilistic model. \[o7.4\] In the case of classical first-order logic, the enunciation above is Gaifman‘s completeness theorem ([@[b]]). If ${\cal C}$ is the infinite-valued Łukasiewicz logic Ł, then such a completeness theorem is valid ([@[c]]). [99]{} R. Balbes, P. Dwinger, [Distributive Lattices]{}, University of Missouri Press, Columbia, Missouri (1974). R. Bělohl${\rm \acute{a}}$vek, [Fuzzy Relational Systems. Foundations and Principles]{}, Kluwer Academic/Plenum Publishers, New York/Boston/Dordrecht/London/Moscow (2002). R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, [Algebraic Foundations of Many-valued Reasoning]{}, Trends in Logic-Studia Logica Library 7, Kluwer Academic Publishers, Dordrecht (2000). R. Cignoli, A. Torrens, Free Algebras in Varieties of Glivenko MTL-algebras Satisfying the Equation $2(x^{2})=(2x)^{2}$, [Studia Logica]{} 83 (2006), 157-181. P. Cintula, P. ${\rm H\acute{a}jek}$, Triangular Norm Based Predicate Fuzzy Logics, [Fuzzy Sets and Systems]{} 161, No. 3 (2010), 311-346. L. C. Ciungu, [Algebraic Models for Multiple-valued Logics]{}, Ph. D. thesis, Bucharest University, Bucharest (2007). L. C. Ciungu, Bosbach and Riečan States on Residuated Lattices, [J. Appl. Func. Analysis]{} 2 (2008), 175-188. L. C. Ciungu, A. Dvurečenskij, M. Hyčko, State BL-algebras, to appear in [Soft Computing - A Fusion of Foundations, Methodologies and Applications]{}, published online: 11 March 2010. I. Cuculescu, O. Onicescu, [Probability Theory on Boolean Algebras of Events]{}, Editura Academiei Române, Bucharest (1976). A. Di Nola, A. Dvurečenskij, State-morphism MV-algebras, [Ann. Pure Appl. Logic]{} 161 (2009), 161-173. A. Dvurečenskij, States on Pseudo-MV-algebras, [Studia Logica]{} 68 (2001), 301-329. A. Dvurečenskij, J. Rachunek, Probabilistic Averaging in Bounded Rl-monoids, [Semigroup Forum]{} 72 (2006), 191-206. A. Dvurečenskij, J. Rachunek, On Riečan and Bosbach States for Bounded Non-commutative Rl-monoids, [Mathematica Slovaca]{} 56, No. 5 (2006), 487-500. F. Esteva, L. Godo, Monoidal t-norm-based Logic: towards a Logic for Left-continuous t-norms, [Fuzzy Sets and Systems]{} 124, No. 3 (2001), 271-288. T. Flaminio, L. Godo, A Logic for Reasoning about the Probability of Fuzzy Events, [Fuzzy Sets and Systems]{} 158, No. 6 (2007), 625-638. T. Flaminio, Strong Non-standard Completeness for Fuzzy Logic, [Soft Computing]{} 12 (2008), 321-333. T. Flaminio, F. Montagna, MV-algebras with Internal States and Fuzzy Logics, [Inter. J. Approx. Reasoning]{} 50 (2009), 138-152. D. H. Fremlin, Measure Theory, available from the author‘s site at Sussex University (1995). H. Gaifman, Concerning Measures in First Order Calculi, [Israel J. Math.]{} 2, No. 1 (1964), 1-18. N. Galatos, P. Jipsen, T. Kowalski, H. Ono, [Residuated Lattices: An Algebraic Glimpse at Substructural Logics]{}, Studies in Logic and The Foundations of Mathematics 151, Elsevier, Amsterdam/Boston/Heidelberg/London/New York/Oxford/Paris/San Diego/San Francisco/Singapore/Sydney/Tokyo (2007). G. Georgescu, Bosbach States on Fuzzy Structures, [Soft Computing]{} 8 (2004), 217-230. G. Georgescu, A. Popescu, Similarity Convergence in Residuated Structures, [Logic Journal of IGPL]{} 13, No. 4 (2005), 389-413. G. Georgescu, States on Polyadic MV-algebras, [Studia Logica]{} 94 (2010), 231-243. K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation, [Providence, R. I.: Amer. Math. Soc.]{} (1986). P. ${\rm H\acute{a}jek}$, [Metamathematics of Fuzzy Logic]{}, Trends in Logic-Studia Logica, Kluwer Academic Publishers, Dordrecht/Boston/London (1998). A. Iorgulescu, [Algebras of Logic as BCK Algebras]{}, Editura ASE, Bucharest (2008). T. Kowalski, H. Ono, [Residuated Lattices: An Algebraic Glimpse at Logics without Contraction]{}, manuscript (2000). J. ${\rm K\ddot{u}hr}$, [Pseudo-BCK-algebras and Related Structures]{}, Univ. ${\rm Palech\acute{e}kov}$, Olomouci (2007). I. Leuştean, [Contributions to the Theory of MV-algebras]{}, Ed. Universitară, Bucharest (2009). I. Leuştean, Metric Completions of MV-algebras with States: An Approach to Stochastic Independence, [Journal of Logic and Computation Advance Access]{}, published online on March 10, 2009. I. Leuştean, Tensor Products of Probability MV-algebras, to appear in [Journal of Multiple-valued Logic and Soft Computing]{}. D. Mundici, Averaging the Truth Value in Łukasiewicz Sentential Logic, [Studia Logica]{} 55 (1995), 113-127. D. Piciu, [Algebras of Fuzzy Logic]{}, Editura Universitaria Craiova, Craiova (2007). B. Riečan, T. Neubrun, [Integral, Measure and Ordering]{}, Kluwer (1997). B. Riečan, On the Probability on BL-algebras, [Acta Math. Nitra]{} 4 (2000), 3-13. B. Riečan, D. Mundici, Probability in MV-algebras, in: E. Pap, [Handbook of Measure Theory]{}, North-Holland (2002), 869-909. E. Turunen, [Mathematics behind Fuzzy Logic]{}, Advances in Soft Computing, Physica-Verlag, Heidelberg (1999). E. Turunen, J. Mertanen, States on Semi-divisible Residuated Lattices, [Soft Computing]{} 12 (2008), 353-357.
--- abstract: 'We introduce the Virgo Consortium’s [EAGLE]{} project, a suite of hydrodynamical simulations that follow the formation of galaxies and supermassive black holes in cosmologically representative volumes of a standard $\Lambda$CDM universe. We discuss the limitations of such simulations in light of their finite resolution and poorly constrained subgrid physics, and how these affect their predictive power. One major improvement is our treatment of feedback from massive stars and AGN in which thermal energy is injected into the gas without the need to turn off cooling or decouple hydrodynamical forces, allowing winds to develop without predetermined speed or mass loading factors. Because the feedback efficiencies cannot be predicted from first principles, we calibrate them to the present-day galaxy stellar mass function and the amplitude of the galaxy-central black hole mass relation, also taking galaxy sizes into account. The observed galaxy stellar mass function is reproduced to $\la 0.2$ dex over the full resolved mass range, $10^8 < M_\ast/{{{\rm M}_\odot}}\la 10^{11}$, a level of agreement close to that attained by semi-analytic models, and unprecedented for hydrodynamical simulations. We compare our results to a representative set of low-redshift observables not considered in the calibration, and find good agreement with the observed galaxy specific star formation rates, passive fractions, Tully-Fisher relation, total stellar luminosities of galaxy clusters, and column density distributions of intergalactic and . While the mass-metallicity relations for gas and stars are consistent with observations for $M_\ast \ga 10^9\,{{{\rm M}_\odot}}$ ($M_\ast \ga 10^{10}\,{{{\rm M}_\odot}}$ at intermediate resolution), they are insufficiently steep at lower masses. For the reference model the gas fractions and temperatures are too high for clusters of galaxies, but for galaxy groups these discrepancies can be resolved by adopting a higher heating temperature in the subgrid prescription for AGN feedback. The [EAGLE]{} simulation suite, which also includes physics variations and higher-resolution zoomed-in volumes described elsewhere, constitutes a valuable new resource for studies of galaxy formation.' author: - \| Joop Schaye,$^1$[^1] Robert A. Crain,$^1$ Richard G. Bower,$^2$ Michelle Furlong,$^2$ Matthieu Schaller,$^2$ Tom Theuns,$^{2,3}$ Claudio Dalla Vecchia,$^{4,5}$ Carlos S. Frenk,$^2$ I. G. McCarthy,$^6$ John C. Helly,$^2$ Adrian Jenkins,$^2$ Y. M. Rosas-Guevara,$^2$ Simon D. M. White,$^7$ Maarten Baes,$^8$ C. M. Booth,$^{1,9}$ Peter Camps,$^8$ Julio F. Navarro,$^{10}$ Yan Qu,$^2$ Alireza Rahmati,$^7$ Till Sawala,$^2$ Peter A. Thomas,$^{11}$ James Trayford$^2$\ $^1$ Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands\ $^2$ Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK\ $^3$ Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium\ $^4$ Instituto de Astrofísica de Canarias, C/ Vía Láctea s/n,38205 La Laguna, Tenerife, Spain\ $^5$ Departamento de Astrofśica, Universidad de La Laguna, Av. del Astrofísico Franciso Sánchez s/n, 38206 La Laguna, Tenerife, Spain\ $^6$ Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\ $^7$ Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany\ $^8$ Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium\ $^9$ Department of Astronomy & Astrophysics, The University of Chicago, Chicago, IL 60637, USA\ $^{10}$ Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada\ $^{11}$ Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK bibliography: - 'ms.bib' title: 'The [EAGLE]{} project: Simulating the evolution and assembly of galaxies and their environments' --- cosmology: theory – galaxies: formation – galaxies: evolution Introduction ============ Cosmological simulations have greatly improved our understanding of the physics of galaxy formation and are widely used to guide the interpretation of observations and the design of new observational campaigns and instruments. Simulations enable astronomers to “turn the knobs” much as experimental physicists are able to in the laboratory. While such numerical experiments can be valuable even if the simulations fail to reproduce observations, in general our confidence in the conclusions drawn from simulations, and the number of applications they can be used for, increases with the level of agreement between the best-fit model and the observations. For many years the overall agreement between hydrodynamical simulations and observations of galaxies was poor. Most simulations produced galaxy mass functions with the wrong shape and normalisation, the galaxies were too massive and too compact, and the stars formed too early. Star formation in high-mass galaxies was not quenched and the models could not simultaneously reproduce the stellar masses and the thermodynamic properties of the gas in groups and clusters (e.g. @Scannapieco2012Aquila and references therein). Driven in part by the failure of hydrodynamical simulations to reproduce key observations, semi-analytic and halo-based models have become the tools of choice for detailed comparisons between galaxy surveys and theory (see @Baugh2006SAMReview and @Cooray2002HaloModelReview for reviews). Thanks to their flexibility and relatively modest computational expense, these approaches have proven valuable for many purposes. Examples include the interpretation of observations of galaxies within the context of the cold dark matter framework, relating galaxy populations at different redshifts, the creation of mock galaxy catalogues to investigate selection effects or to translate measurements of galaxy clustering into information concerning the occupation of dark matter haloes by galaxies. However, hydrodynamical simulations have a number of important advantages over these other approaches. The risk that a poor or invalid approximation may lead to over-confidence in an extrapolation, interpretation or application of the model is potentially smaller, because they do not need to make as many simplifying assumptions. Although the subgrid models employed by current hydrodynamical simulations often resemble the ingredients of semi-analytic models, there are important parts of the problem for which subgrid models are no longer required. Since hydrodynamical simulations evolve the dark matter and baryonic components self-consistently, they automatically include the back-reaction of the baryons on the collisionless matter, both inside and outside of haloes. The higher resolution description of the baryonic component provided by hydrodynamical simulations also enables one to ask more detailed questions and to compare with many more observables. Cosmological hydrodynamical simulations can be used to model galaxies and the intergalactic medium (IGM) simultaneously, including the interface between the two, which may well be critical to understanding the fuelling and feedback cycles of galaxies. The agreement between hydrodynamical simulations of galaxy formation and observations has improved significantly in recent years. Simulations of the diffuse IGM already broadly reproduced quasar absorption line observations of the [Ly$\alpha$]{} forest two decades ago [e.g. @Cen1994LyaForest; @Zhang1995LyaForest; @Hernquist1996LyaForest; @Theuns1998LyaForest; @Dave1999LyaForest]. The agreement is sufficiently good that comparisons between theory and observation can be used to measure cosmological and physical parameters [e.g. @Croft1998LyaPowerSpectrum; @Schaye2000IGMTemp; @Viel2004LyaPowerSpectrum; @McDonald2005LyaPowerSpectrum]. More recently, simulations that have been re-processed using radiative transfer of ionizing radiation have succeeded in matching key properties of the high-column density absorbers [e.g. @Pontzen2008DLAs; @Altay2011HICDDF; @McQuinn2011LLS; @Rahmati2013HICDDF]. Reproducing observations of galaxies and the gas in clusters of galaxies has proven to be more difficult than matching observations of the low-density IGM, but several groups have now independently succeeded in producing disc galaxies with more realistic sizes and masses (e.g. @Governato2004Disk [@Governato2010Disk; @Okamoto2005Disks; @Agertz2011Disk; @Guedes2011Eris; @McCarthy2012RotSize; @Brook2012MagicDisks; @Stinson2013MAGICC; @Munshi2013StellarHaloMass; @Aumer2013Disks; @Hopkins2013FIRE; @Vogelsberger2013IllustrisModel; @Vogelsberger2014Illustris; @Marinacci2014Disks]). For the thermodynamic properties of groups and clusters of galaxies the progress has also been rapid [e.g. @Puchwein2008AGN; @McCarthy2010AGN; @Fabjan2010AGN; @LeBrun2014CosmoOWLS]. The improvement in the realism of the simulated galaxies has been accompanied by better agreement between simulations and observations of the metals in circumgalactic and intergalactic gas [e.g. @Stinson2012CGM; @Oppenheimer2012IGMMetals], which suggests that a more appropriate description of galactic winds may have been responsible for much of the progress. Indeed, the key to the increase in the realism of the simulated galaxies has been the use of subgrid models for feedback from star formation that are more effective in generating galactic winds and, at the high-mass end, the inclusion of subgrid models for feedback from active galactic nuclei (AGN). The improvement in the resolution afforded by increases in computing power and code efficiency has also been important, but perhaps mostly because higher resolution has helped to make the implemented feedback more efficient by reducing spurious, numerical radiative losses. Improvements in the numerical techniques to solve the hydrodynamics have also been made [e.g. @Price:2008kx; @Springel2010Arepo; @Read:2010uq; @Saitoh:2013uq; @Hopkins:2013lr] and may even be critical for particular applications [e.g. @Agertz:2007fk; @Bauer2012SubsonicTurbulence], but overall their effect appears to be small compared to reasonable variations in subgrid models for feedback processes [@Scannapieco2012Aquila]. Here we present the [EAGLE]{} project[^2], which stands for Evolution and Assembly of GaLaxies and their Environments. [EAGLE]{} consists of a suite of cosmological, hydrodynamical simulations of a standard $\Lambda$CDM universe. The main models were run in volumes of 25 to 100 comoving Mpc (cMpc) on a side and employ a resolution that is sufficient to marginally resolve the Jeans scales in the warm ($T\sim 10^4\,{{\rm K}}$) interstellar medium (ISM). The simulations use state-of-the-art numerical techniques and subgrid models for radiative cooling, star formation, stellar mass loss and metal enrichment, energy feedback from star formation, gas accretion onto, and mergers of, supermassive black holes (BHs), and AGN feedback. The efficiency of the stellar feedback and the BH accretion were calibrated to broadly match the observed $z\sim 0$ galaxy stellar mass function (GSMF) subject to the constraint that the galaxy sizes must also be reasonable, while the efficiency of the AGN feedback was calibrated to the observed relation between stellar mass and BH mass. The goal was to reproduce these observables using, in our opinion, simpler and more natural prescriptions for feedback than used in previous work with similar objectives. By “simpler” and “more natural”, which are obviously subjective terms, we mean the following. Apart from stellar mass loss, we employ only one type of stellar feedback, which captures the collective effects of processes such as stellar winds, radiation pressure on dust grains, and supernovae. These and other feedback mechanisms are often implemented individually, but we believe they cannot be properly distinguished at the resolution of $10^2$–$10^3\,{{\rm pc}}$ that is currently typical for simulations that sample a representative volume of the universe. Similarly, we employ only one type of AGN feedback (as opposed to e.g. both a “radio” and “quasar” mode). Contrary to most previous work, stellar (and AGN) feedback is injected in thermal form without turning off radiative cooling and without turning off hydrodynamical forces. Hence, galactic winds are generated without specifying a wind direction, velocity, mass loading factor, or metal mass loading factor. We also do not need to boost the BH Bondi-Hoyle accretion rates by an ad-hoc factor. Finally, the amount of feedback energy (and momentum) that is injected per unit stellar mass depends on local gas properties rather than on non-local or non-baryonic properties such as the dark matter velocity dispersion or halo mass. The [EAGLE]{} suite includes many simulations that will be presented elsewhere. It includes higher-resolution simulations that zoom into individual galaxies or galaxy groups [e.g. @Sawala2014EagleZooms]. It also includes variations in the numerical techniques [@Schaller2014EagleSPH] and in the subgrid models [@Crain2014EagleModels] that can be used to test the robustness of the predictions and to isolate the effects of individual processes. This paper is organised as follows. We begin in §\[sec:general\] with a discussion of the use and pitfalls of cosmological hydrodynamical simulations in light of the critical role played by subgrid processes. We focus in particular on the implications for the interpretation and the predictive power of the simulations, and the role of numerical convergence. In §\[sec:simulations\] we describe the simulations and our definition of a galaxy. This section also briefly discusses the numerical techniques and subgrid physics. The subgrid models are discussed in depth in §\[sec:subgrid\]; readers not interested in the details may wish to skip this section. In §\[sec:cal\_obs\] we show the results for observables that were considered in the calibration of the subgrid models, namely the $z\sim 0$ GSMF, the related relation between stellar mass and halo mass, galaxy sizes, and the relations between BH mass and stellar mass. We also consider the importance of the choice of aperture used to measure stellar masses and investigate both weak and strong convergence (terms that are defined in §\[sec:general\]). In §\[sec:otherobs\] we present a diverse and representative set of predictions that were not used for the calibration, including specific star formation rates and passive fractions, the Tully-Fisher relation, the mass-metallicity relations, various properties of the intracluster medium, and the column density distributions of intergalactic metals. All results presented here are for $z\sim 0$. We defer an investigation of the evolution to @Furlong2014EagleEvolution and other future papers. We summarize and discuss our conclusions in §\[sec:summary\]. Finally, our implementation of the hydrodynamics and our method for generating the initial conditions are summarized in Appendices \[app:hydro\] and \[app:ics\], respectively. Implications of the critical role of subgrid models for feedback {#sec:general} ================================================================ In this section we discuss what, in our view, the consequences of our reliance on subgrid models for feedback are for the predictive power of the simulations (§\[sec:need\]) and for the role of numerical convergence (§\[sec:conv\_discussion\]). The need for calibration {#sec:need} ------------------------ Because the recent improvement in the match between simulated and observed galaxies can, for the most part, be attributed to the implementation of more effective subgrid models for feedback, the success of the hydrodynamical simulations is subject to two important caveats that are more commonly associated with semi-analytic models. First, while it is clear that effective feedback is required, the simulations can only provide limited insight into the nature and source of the feedback processes. For example, suppose that the implemented subgrid model for supernovae is too inefficient because, for numerical reasons, too much of the energy is radiated away, too much of the momentum cancels out, or the energy/momentum are coupled to the gas at the wrong scale. If we were unaware of such numerical problems, then we might erroneously conclude that additional feedback processes such as radiation pressure are required. The converse is, of course, also possible: the implemented feedback can also be too efficient, for example because the subgrid model underestimates the actual radiative losses. The risk of misinterpretation is real, because it can be shown that many simulations underestimate the effectiveness of feedback due to excessive radiative losses [e.g. @DallaVecchia2012Winds], which themselves are caused by a lack of resolution and insufficiently realistic modelling of the ISM. Second, the ab initio predictive power of the simulations is currently limited when it comes to the properties of galaxies. If the efficiency of the feedback processes depends on subgrid prescriptions that may not be good approximations to the outcome of unresolved processes, or if the outcome depends on resolution, then the true efficiencies cannot be predicted from first principles. Note that the use of subgrid models does not in itself remove predictive power. If the physical processes that operate below the resolution limit and their connection with the physical conditions on larger scales are fully understood and can be modelled or observed, then it may be possible to create a subgrid model that is sufficiently realistic to retain full predictive power. However, this is currently not the case for feedback from star formation and AGN. As we shall explain below, this implies that simulations that appeal to a subgrid prescription for the generation of outflows are unable to predict the stellar masses of galaxies. Similarly, for galaxies whose evolution is controlled by AGN feedback, such simulations cannot predict the masses of their central BHs. To illustrate this, it is helpful to consider a simple model. Let us assume that galaxy evolution is self-regulated, in the sense that galaxies tend to evolve towards a quasi-equilibrium state in which the gas outflow rate balances the difference between the gas inflow rate and the rate at which gas is locked up in stars and BHs. The mean rate of inflow (e.g. in the form of cold streams) evolves with redshift and tracks the accretion rate of dark matter onto haloes, which is determined by the cosmological initial conditions. For simplicity, let us further assume that the outflow rate is large compared to the rate at which the gas is locked up. Although our conclusions do not depend on the validity of this last assumption, it simplifies the arguments because it implies that the outflow rate balances the inflow rate, when averaged over appropriate length and time scales. Note that the observed low efficiency of galaxy formation (see Fig. \[fig:eta\] in §\[sec:eta\]) suggests that this may actually be a reasonable approximation, particularly for low-mass galaxies. This toy model is obviously incorrect in detail. For example, it ignores the re-accretion of matter ejected by winds, the recycling of stellar mass loss, and the interaction of outflows and inflows. However, recent numerical experiments and analytic models provide some support for the general idea [e.g. @Finlator2008MZ; @Schaye2010OWLS; @Booth2010DMHaloesBHs; @Dave2012EquilModel; @Haas2013OwlsI; @Haas2013OwlsII; @Feldmann2013SFLaw; @Dekel2013ToyModel; @Altay2013DLAs; @Lilly2013EquilModel; @Sanchez2014Review]. This idea in itself is certainly not new and follows from the existence of a feedback loop [e.g. @White1991GF], as can be seen as follows. If the inflow rate exceeds the outflow rate, then the gas fraction will increase and this will in turn increase the star formation rate (and/or, on a smaller scale, the BH accretion rate) and hence also the outflow rate. If, on the other hand, the outflow rate exceeds the inflow rate, then the gas fraction will decrease and this will in turn decrease the star formation rate (and/or the BH accretion rate) and hence also the outflow rate. In this self-regulated picture of galaxy evolution the outflow rate is determined by the inflow rate. Hence, the outflow rate is not determined by the efficiency of the implemented feedback. Therefore, if the outflow is driven by feedback from star formation, then the star formation rate will adjust until the outflow rate balances the inflow rate, irrespective of the (nonzero) feedback efficiency. However, the star formation rate for which this balance is achieved, and hence also ultimately the stellar mass, do depend on the efficiency of the implemented feedback. If the true feedback efficiency cannot be predicted, then neither can the stellar mass. Similarly, if the outflow rate is driven by AGN feedback, then the BH accretion rate will adjust until the outflow rate balances the inflow rate (again averaged over appropriate length and time scales). The BH accretion rate, and hence the BH mass, for which this balance is achieved depend on the efficiency of the implemented feedback, which has to be assumed. According to this toy model, which appears to be a reasonable description of the evolution of simulated galaxies, the stellar and BH masses are thus determined by the efficiencies of the (subgrid) implementations for stellar and AGN feedback, respectively. The simulations therefore need to be calibrated to produce the correct stellar and BH masses. Moreover, if the true efficiency varies systematically with the physical conditions on a scale resolved by the simulations, then the implemented subgrid efficiency would also have to be a function of the local physical conditions in order to produce the correct mass functions of galaxies and BHs. A similar story applies to the gas fractions of galaxies or, more precisely, for the amount of gas above the assumed star formation threshold, even if the simulations have been calibrated to produce the correct GSMF. We can see this as follows. If the outflow rate is determined by the inflow rate, then it is not determined by the assumed subgrid star formation law. Hence, if we modify the star formation law,[^3] then the mean outflow rate should remain unchanged. And if the outflow rate remains unchanged, then so must the star formation rate because for a fixed feedback efficiency the star formation rate will adjust to the rate required for outflows to balance inflows. If the star formation rate is independent of the star formation law, then the galaxies must adjust the amount of star-forming gas that they contain when the star formation law is changed. Hence, to predict the correct amount of star-forming gas, we need to calibrate the subgrid model for star formation to the observed star formation law. Fortunately, the star formation law is relatively well characterised observationally on the $\sim 10^2 - 10^3\,{{\rm pc}}$ scales resolved by large-volume simulations, although there are important unanswered questions, e.g. regarding the dependence on metallicity. Ultimately the star formation law must be predicted by simulations and will probably depend on the true efficiency of feedback processes within the ISM, but resolving such processes is not yet possible in simulations of cosmological volumes. It is not obvious how the efficiency of feedback from star formation should be calibrated. We could choose to calibrate to observations of outflow rates relative to star formation rates. However, those outflow rates are highly uncertain and may be affected by AGN feedback. It is also unclear on what scale the outflow rate should be calibrated. In addition, the outflow velocity and the wind mass loading may be individually important. Moreover, unless the interaction of the wind with the circumgalactic medium is modelled correctly and resolved, then obtaining a correct outflow rate on the scale used for the calibration does not necessarily imply that it is also correct for the other scales that matter. We choose to calibrate the feedback efficiency using the observed present-day GSMF, as is also common practice for semi-analytic models. We do this mostly because it is relatively well constrained observationally and because obtaining the correct stellar mass - halo mass relation, and hence the correct GSMF if the cosmological initial conditions are known, is a pre-condition for many applications of cosmological simulations. For example, the physical properties of the circumgalactic medium (CGM) are likely sensitive to the halo mass, but because halo mass is difficult to measure, observations and simulations of the CGM are typically compared for galaxies of the same stellar mass. One may wonder what the point of hydrodynamical simulations (or, indeed, semi-analytic models) is if they cannot predict stellar masses or BH masses. This is a valid question for which there are several answers. One is that the simulations can still make predictions for observables that were not used for the calibration, and we will present such predictions in §\[sec:otherobs\] and in subsequent papers. However, which observables are unrelated is not always unambiguous. One way to proceed, and an excellent way to learn about the physics of galaxy formation, is to run multiple simulations with varying subgrid models. It is particularly useful to have multiple prescriptions calibrated to the same observables. [EAGLE]{} comprises many variations, including several that reproduce the $z\sim 0$ GSMF through different means [@Crain2014EagleModels]. A second answer is that making good use of simulations of galaxy formation does not necessarily mean making quantitative predictions for observables of the galaxy population. We can use the simulations to gain insight into physical processes, to explore possible scenarios, and to make qualitative predictions. How does gas get into galaxies? What factors control the size of galaxies? What is the origin of scatter in galaxy scaling relations? What is the potential effect of outflows on cosmology using weak gravitational lensing or the [Ly$\alpha$]{} forest? The list of interesting questions is nearly endless. A third answer is that cosmological, hydrodynamical simulations can make robust, quantitative predictions for more diffuse components, such as the low-density IGM and perhaps the outer parts of clusters of galaxies. A fourth answer is that calibrated simulations can be useful to guide the interpretation and planning of observations, as the use of semi-analytic and halo models has clearly demonstrated. In this respect hydrodynamical simulations can provide more detailed information on both the galaxies and their gaseous environments. Numerical convergence {#sec:conv_discussion} --------------------- The need to calibrate the efficiency of the feedback and the associated limits on the predictive power of the simulations call the role of numerical convergence into question. The conventional point of view is that subgrid models should be designed to yield numerically converged predictions. Convergence is clearly a necessary condition for predictive power. However, we have just concluded that current simulations cannot, in any case, make ab initio predictions for some of the most fundamental observables of the galaxy population. While it is obvious that we should demand convergence for predictions that are relatively robust to the choice of subgrid model, e.g. the statistics of the [Ly$\alpha$]{} forest, it is less obvious that the same is required for observables that depend strongly and directly on the efficiency of the subgrid feedback. One could argue that, instead, we only need convergence after recalibration of the subgrid model. We will call this “weak convergence”, as opposed to the “strong convergence” that is obtained if the results do not change with resolution when the model is held fixed. If only weak convergence is required, then the demands placed on the subgrid model are much reduced, which has two advantages: First, we can take better advantage of increases in resolution. The subgrid scale can now move along with the resolution limit, so we can potentially model the physics more faithfully if we adopt higher resolution. A second advantage of demanding only weak convergence is that we do not have to make the sacrifices that are required to improve the strong convergence and that might have undesirable consequences. We will provide three examples of compromises that are commonly made. Simulations that sample a representative volume currently lack the resolution and the physics to predict the radiative losses to which outflows are subject within the ISM. Strong convergence can nevertheless be achieved if these losses are somehow removed altogether, for example, by temporarily turning off radiative cooling and calibrating the criterion for switching it back on [e.g. @Gerritsen1997PhD; @Stinson2006Winds]. However, it is then unclear for which gas the cooling should be switched off. Only the gas elements into which the subgrid feedback was directly injected? Or also the surrounding gas that is subsequently shock-heated? Other ways to circumvent radiative losses in the ISM are to generate the outflow outside the galaxy or to turn off the hydrodynamic interaction between the wind and the ISM [e.g. @Springel2003Multiphase; @Oppenheimer2006Wpot; @Oppenheimer2010GSMF; @Puchwein2013GSMF; @Vogelsberger2013IllustrisModel; @Vogelsberger2014Illustris]. This is a valid choice, but one that eliminates the possibility of capturing any aspect of the feedback other than mass loss, such as puffing up of discs, blowing holes, driving turbulence, collimating outflows, ejecting gas clouds, generating small-scale galactic fountains, etc. Furthermore, it necessarily introduces new parameters that control where the outflow is generated and when the hydrodynamics is turned back on. These parameters may directly affect results of interest, including the state of gas around galaxies, and may also re-introduce resolution effects. A potential solution to this problem is to never re-couple and hence to evaluate all wind interactions using a subgrid model, even outside the galaxies, as is done in semi-analytic models. However, bypassing radiative losses in the ISM is not by itself sufficient to achieve strong convergence. In addition, the feedback must not depend on physical conditions in the ISM since those are unlikely to be converged. Instead, one can make the feedback depend on properties defined by the dark matter, such as its local velocity dispersion or halo mass [e.g. @Oppenheimer2006Wpot; @Okamoto2010Sats; @Oppenheimer2010GSMF; @Puchwein2013GSMF; @Vogelsberger2013IllustrisModel; @Vogelsberger2014Illustris], which are generally better converged than the properties of the gas. As was the case for turning off cooling or hydrodynamic forces, this choice makes the simulations less “hydrodynamical”, moving them in the direction of more phenomenological approaches, and it also introduces new problems. How do we treat satellite galaxies given that their subhalo mass and dark matter velocity dispersion are affected by the host halo? Or worse, what about star clusters or tidal dwarf galaxies that are not hosted by dark matter haloes? In practice, however, the distinction between weak and strong convergence is often unclear. One may surmise that keeping the physical model fixed is equivalent to keeping the code and subgrid parameters fixed (apart from the numerical parameters controlling the resolution), but this is not necessarily the case because of the reliance on subgrid prescriptions and the inability to resolve the first generations of stars and BHs. For typical subgrid prescriptions, the energy, the mass, and the momentum involved in individual feedback events, and the number or intermittency of feedback events do not all remain fixed when the resolution is changed. Any such changes could affect the efficiency of the feedback. Consider, for example, a star-forming region and assume that feedback energy from young stars is distributed locally at every time step. If the resolution is increased, then the time step and the particle mass will become smaller. If the total star formation rate remains the same, then the feedback energy that is injected per time step will be smaller because of the decrease in the time step. If the gas mass also remains the same, then the temperature increase per time step will be smaller. A lower post-feedback temperature often leads to larger thermal losses. If, instead, the subgrid model specifies the temperature jump (or wind velocity), then the post-feedback temperature will remain the same when the resolution is increased, but the number of heating events will increase because the same amount of feedback energy has to be distributed over lower-mass particles. There is no guarantee that more frequent, lower-energy events drive the same outflows as less frequent, higher-energy events. Moreover, for cosmological initial conditions, higher resolution implies resolving smaller haloes, and hence tracing the progenitors of present-day galaxies to higher redshifts. If these progenitors drive winds, then this may impact the subsequent evolution. In §\[sec:gsmf\] we investigate both the weak and strong convergence of our simulations, focusing on the GSMF. We test the weak convergence for a wide variety of predictions in sections \[sec:cal\_obs\] and \[sec:otherobs\]. Simulations {#sec:simulations} =========== Cosmological parameter Value ------------------------------------------- --------- $\Omega_{\rm m}$ 0.307 $\Omega_\Lambda$ 0.693 $\Omega_{\rm b}$ 0.04825 $h \equiv H_0$/(100 kms$^{-1}$Mpc$^{-1})$ 0.6777 $\sigma_8$ 0.8288 $ n_{\rm s}$ 0.9611 $Y$ 0.248 : The cosmological parameters used for the [EAGLE]{} simulations: $\Omega_{\rm m}$, $\Omega_\Lambda$, and $\Omega_{\rm b}$ are the average densities of matter, dark energy and baryonic matter in units of the critical density at redshift zero; $H_0$ is the Hubble parameter, $\sigma_8$ is the square root of the linear variance of the matter distribution when smoothed with a top-hat filter of radius $8~{h^{-1}\,{\rm cMpc}}$, $n_{\rm s}$ is the scalar power-law index of the power spectrum of primordial adiabatic perturbations, and $Y$ is the primordial abundance of helium. \[tbl:cosmo\_params\] [EAGLE]{} was run using a modified version of the $N$-Body Tree-PM smoothed particle hydrodynamics (SPH) code <span style="font-variant:small-caps;">gadget</span> 3, which was last described in @Springel2005Gadget2. The main modifications are the formulation of SPH, the time stepping and, most importantly, the subgrid physics. The subgrid physics used in [EAGLE]{} is based on that developed for [OWLS]{} [@Schaye2010OWLS], and used also in [GIMIC]{} [@Crain2009GIMIC] and [cosmo-OWLS]{} [@LeBrun2014CosmoOWLS]. We include element-by-element radiative cooling for 11 elements, star formation, stellar mass loss, energy feedback from star formation, gas accretion onto and mergers of supermassive black holes (BHs), and AGN feedback. As we will detail in §\[sec:subgrid\], we made a number of changes with respect to [OWLS]{}. The most important changes concern the implementations of energy feedback from star formation (which is now thermal rather than kinetic), the accretion of gas onto BHs (which now accounts for angular momentum), and the star formation law (which now depends on metallicity). In the simulations presented here the amount of feedback energy that is injected per unit stellar mass decreases with the metallicity and increases with the gas density. It is bounded between one third and three times the energy provided by supernovae and, on average, it is about equal to that amount. The metallicity dependence is motivated by the fact that we expect greater (unresolved) thermal losses when the metallicity exceeds $\sim 10^{-1}\,{\rm Z}_\odot$, the value for which metal-line cooling becomes important. The density dependence compensates for spurious, numerical radiative losses which, as expected, are still present at our resolution even though they are greatly reduced by the use of the stochastic prescription of @DallaVecchia2012Winds. The simulations were calibrated against observational data by running a series of high-resolution 12.5 cMpc and intermediate resolution 25 cMpc test runs with somewhat different dependencies on metallicity and particularly density. From the models that predicted reasonable physical sizes for disc galaxies, we selected the one that best fit the $z\sim 0$ GSMF. For more details on the subgrid model for energy feedback from star formation we refer the reader to §\[sec:snii\]. As described in more detail in Appendix \[app:hydro\], we make use of the conservative pressure-entropy formulation of SPH derived by @Hopkins:2013lr, the artificial viscosity switch from @Cullen:2010qy, an artificial conduction switch similar to that of @Price:2008kx, the $C^2$ @Wendland:1995 kernel and the time step limiters of @Durier:2012fj. We will refer to these numerical methods collectively as “Anarchy”. Anarchy will be described in more detail by Dalla Vecchia (in preparation), who also demonstrates its good performance on standard hydrodynamical tests (see @Hu2014SPHGal for tests of a similar set of methods). In @Schaller2014EagleSPH we will show the relevance of the new hydrodynamical techniques and time stepping scheme for the results of the [EAGLE]{} simulations. Although the Anarchy implementation yields dramatic improvements in the performance on some standard hydrodynamical tests as compared to the original implementation of the hydrodynamics in <span style="font-variant:small-caps;">gadget</span> 3, we generally find that the impact on the results of the cosmological simulations is small compared to those resulting from reasonable variations in the subgrid physics (see also @Scannapieco2012Aquila). ----------- ---------------- ---------- ----------------------- ----------------------- ---------------------- ----------------------- -- Name $L$ $N$ $m_{\rm g}$ $m_{\rm dm}$ $\epsilon_{\rm com}$ $\epsilon_{\rm prop}$ (comoving Mpc) (${{{\rm M}_\odot}}$) (${{{\rm M}_\odot}}$) (comoving kpc) (proper kpc) L025N0376 25 $376^3$ $1.81\times 10^6$ $9.70\times 10^6$ 2.66 0.70 L025N0752 25 $752^3$ $2.26\times 10^5$ $1.21\times 10^6$ 1.33 0.35 L050N0752 50 $752^3$ $1.81\times 10^6$ $9.70\times 10^6$ 2.66 0.70 L100N1504 100 $1504^3$ $1.81\times 10^6$ $9.70\times 10^6$ 2.66 0.70 ----------- ---------------- ---------- ----------------------- ----------------------- ---------------------- ----------------------- -- The values of the cosmological parameters used for the [EAGLE]{} simulations are taken from the most recent Planck results [@PlanckI Table 9] and are listed in Table \[tbl:cosmo\_params\]. A transfer function with these parameters was generated using CAMB [@CAMB version Jan\_12]. The linear matter power spectrum was generated by multiplying a power-law primordial power spectrum with an index of $n_{\rm s} = 0.9611$ by the square of the dark matter transfer function evaluated at redshift zero[^4]. Particles arranged in a glass-like initial configuration were displaced according to 2nd-order Lagrangian perturbation theory using the method of @Jenkins20102lpt and the public Gaussian white noise field Panphasia [@Jenkins2013ICs; @Jenkins2013Panphasia]. The methods used to generate the initial conditions are described in detail in Appendix \[app:ics\]. Table \[tbl:sims\] lists box sizes and resolutions of the main [EAGLE]{} simulations. All simulations were run to redshift $z=0$. Note that contrary to convention, box sizes, particles masses and gravitational softening lengths are not quoted in units of $h^{-1}$. The gravitational softening was kept fixed in comoving units down to $z=2.8$ and in proper units thereafter. We will refer to simulations with the same mass and spatial resolution as L100N1504 as intermediate resolution runs and to simulations with the same resolution as L025N0752 as high-resolution runs. Particle properties were recorded for 29 snapshots between redshifts 20 and 0. In addition, we saved a reduced set of particle properties (“snipshots”) at 400 redshifts between 20 and 0. The largest simulation, L100N1504, took about 4.5 M CPU hours to reach $z=0$ on a machine with 32 TB of memory, with the [EAGLE]{} subgrid physics typically taking less than 25 per cent of the CPU time. The resolution of [EAGLE]{} suffices to marginally resolve the Jeans scales in the warm ISM. The Jeans mass and length for a cloud with gas fraction, $f_{\rm g}$, are, respectively, $M_{\rm J} \approx 1 \times 10^7\,{{{\rm M}_\odot}}\, f_{\rm g}^{3/2} (n_{\rm H} /10^{-1}\,{{\rm cm}}^{-3})^{-1/2} (T / 10^4\,{{\rm K}})^{3/2}$ and $L_{\rm J} \approx 2~{{\rm kpc}}~f_{\rm g}^{1/2} (n_{\rm H} / 10^{-1}\,{{\rm cm}}^{-3})^{-1/2} (T / 10^4\,{{\rm K}})^{1/2}$, where $n_{\rm H}$ and $T$ are the total hydrogen number density and the temperature, respectively. These Jeans scales can be compared to the gas particle masses and maximum proper gravitational softening lengths listed in columns 4 and 7 of Table \[tbl:sims\]. Simulations with the same subgrid physics and numerical techniques as used for L100N1504 were carried out for all box sizes (12.5 – 100 cMpc) and particles numbers ($188^3$ – $1504^3$). We will refer to this physical model as the reference model and will indicate the corresponding simulations with the prefix “Ref-” (e.g. Ref-L100N1504). As detailed in §\[sec:subgrid\], we re-ran the high-resolution simulations with recalibrated parameter values for the subgrid stellar and AGN feedback to improve the match to the observed $z\sim 0$ GSMF. We will use the prefix “Recal-” when referring to the simulations with this alternative set of subgrid parameters (e.g. Recal-L025N0752). Note that in terms of weak convergence, Ref-L100N1504 is more similar to model Recal-L025N0752 than to model Ref-L025N0752 (see §\[sec:conv\_discussion\] for a discussion of weak and strong convergence). In addition, we repeated the L050N0752 run with adjusted AGN parameters in order to further improve the agreement with observations for high-mass galaxies. We will refer to this model with the prefix “AGNdT9”. Table \[tbl:subgridpars\] summarizes the values of the four subgrid parameters that vary between the models presented here. @Crain2014EagleModels and @Schaller2014EagleSPH will present the remaining [EAGLE]{} simulations, which concern variations in the subgrid physics and the numerical techniques, respectively. Finally, @Sawala2014EagleZooms present very high-resolution zoomed simulations of Local Group like systems run with the [EAGLE]{} code and a physical model that is nearly identical to the one used for the Ref-L100N1504 model described here. -------- --------------------- ----------- -------------------- ---------------------- Prefix $n_{\rm H,0}$ $n_n$ $C_{\rm visc}$ $\Delta T_{\rm AGN}$ (${{\rm cm}}^{-3}$) (K) Ref 0.67 $2/\ln10$ $2\pi$ $10^{8.5}$ Recal 0.25 $1/\ln10$ $2\pi \times 10^3$ $10^9$ AGNdT9 0.67 $2/\ln10$ $2\pi \times 10^2$ $10^9$ -------- --------------------- ----------- -------------------- ---------------------- : Values of the subgrid parameters that vary between the models presented here. The parameters $n_{\rm H,0}$ and $n_n$ control, respectively, the characteristic density and the power-law slope of the density dependence of the energy feedback from star formation (see equation \[eq:f(Z,n)\] in §\[sec:calibration\]). The parameter $C_{\rm visc}$ controls the sensitivity of the BH accretion rate to the angular momentum of the gas (see equation \[eq:mdotaccr\] in §\[sec:bh\_accretion\]) and $\Delta T_{\rm AGN}$ is the temperature increase of the gas during AGN feedback (see §\[sec:AGNfeedback\]). []{data-label="tbl:subgridpars"} Figure \[fig:zoom\] illustrates the large dynamic range of [EAGLE]{}. It shows the large-scale gas distribution in a thick slice through the $z=0$ output of the Ref-L100N1504 run, colour-coded by the gas temperature. The insets zoom in on an individual galaxy. The first zoom shows the gas, but the last zoom shows the stellar light after accounting for dust extinction. This image was created using three monochromatic radiative transfer simulations with the code [[skirt]{}]{} [@Baes2011SKIRT] at the effective wavelengths of the Sloan Digital Sky Survey (SDSS) u, g & r filters. Dust extinction is implemented using the metal distribution predicted by the simulations and assuming that 30 per cent of the metal mass is locked up in dust grains. Only material within a spherical aperture with a radius of 30 pkpc is included in the radiative transfer calculation. More examples of [[skirt]{}]{} images of galaxies are shown in Figure \[fig:morphology\], in the form of a Hubble sequence. This figure illustrates the wide range of morphologies present in [EAGLE]{}. Note that @Vogelsberger2014IllustrisNature showed a similar figure for their Illustris simulation. In future work we will investigate how morphology correlates with other galaxy properties. More images, as well as videos, can be found on the [EAGLE]{} web sites at Leiden, <http://eagle.strw.leidenuniv.nl/>, and Durham, <http://icc.dur.ac.uk/Eagle/>. We define galaxies as gravitationally bound subhaloes identified by the <span style="font-variant:small-caps;">subfind</span> algorithm [@Springel2001Subfind; @Dolag2009Substructure]. The procedure consists of three main steps. First we find haloes by running the Friends-of-Friends (FoF; @Davis1985FoF) algorithm on the dark matter particles with linking length 0.2 times the mean interparticle separation. Gas and star particles are assigned to the same, if any, FoF halo as their nearest dark matter particles. Second, <span style="font-variant:small-caps;">subfind</span> defines substructure candidates by identifying overdense regions within the FoF halo that are bounded by saddle points in the density distribution. Note that whereas FoF considers only dark matter particles, <span style="font-variant:small-caps;">subfind</span> uses all particle types within the FoF halo. Third, particles that are not gravitationally bound to the substructure are removed and the resulting substructures are referred to as subhaloes. Finally, we merged subhaloes separated by less than the minimum of 3 pkpc and the stellar half-mass radius. This last step removes a very small number of very low-mass subhaloes whose mass is dominated by a single particle such as a supermassive BH. For each FoF halo we define the subhalo that contains the particle with the lowest value of the gravitational potential to be the central galaxy while any remaining subhaloes are classified as satellite galaxies. The position of each galaxy is defined to be the location of the particle belonging to the subhalo for which the gravitational potential is minimum. The stellar mass of a galaxy is defined to be the sum of the masses of all star particles that belong to the corresponding subhalo and that are within a 3-D aperture with radius 30 pkpc. Unless stated otherwise, other galaxy properties, such as the star formation rate, metallicity, and half-mass radius, are also computed using only particles within the 3-D aperture. In §\[sec:aperture\] we show that this aperture gives a nearly identical GSMF as the 2-D Petrosian apertures that are frequently used in observational studies. We find the effect of the aperture to be negligible for $M_\ast < 10^{11}\,{{{\rm M}_\odot}}$ for all galaxy properties that we consider. However, for more massive galaxies the aperture reduces the stellar masses somewhat by cutting out intracluster light. For example, at a stellar mass $M_\ast = 10^{11}\,{{{\rm M}_\odot}}$ as measured using a 30 pkpc aperture, the median subhalo stellar mass is 0.1 dex higher (see §\[sec:aperture\] for the effect on the GSMF). Without the aperture, metallicities are slightly lower and half-mass radii are slightly larger for $M_\ast > 10^{11}\,{{{\rm M}_\odot}}$, but the effect on the star formation rate is negligible. Subgrid physics {#sec:subgrid} =============== In this section we provide a thorough description and motivation for the subgrid physics implemented in [EAGLE]{}: radiative cooling (§\[sec:cooling\]), reionisation (§\[sec:reionisation\]), star formation (§\[sec:sf\]), stellar mass loss and metal enrichment (§\[sec:chemo\]), energy feedback from star formation (§\[sec:snii\]), and supermassive black holes and AGN feedback (§\[sec:BHs\]). These subsections can be read separately. Readers who are mainly interested in the results may skip this section. Radiative cooling {#sec:cooling} ----------------- Radiative cooling and photoheating are implemented element-by-element following @Wiersma2009Cooling, including all 11 elements that they found to be important: H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe. @Wiersma2009Cooling used <span style="font-variant:small-caps;">cloudy</span> version[^5] 07.02 [@Ferland1998Cloudy] to tabulate the rates as a function of density, temperature, and redshift assuming the gas to be in ionisation equilibrium and exposed to the cosmic microwave background (CMB) and the @Haardt2001UVB model for the evolving UV/X-ray background from galaxies and quasars. By computing the rates element-by-element, we account not only for variations in the metallicity, but also for variations in the relative abundances of the elements. We caution that our assumption of ionisation equilibrium and the neglect of local sources of ionizing radiation may cause us to overestimate the cooling rate in certain situations, e.g. in gas that is cooling rapidly [e.g. @Oppenheimer2013Nonequil] or that has recently been exposed to radiation from a local AGN [@Oppenheimer2013AGNFossils]. We have also chosen to ignore self-shielding, which may cause us to underestimate the cooling rates in dense gas. While we could have accounted for this effect, e.g. using the fitting formula of @Rahmati2013HICDDF, we opted against doing so because there are other complicating factors. Self-shielding is only expected to play a role for $n_{\rm H} > 10^{-2}\,{{\rm cm}}^{-3}$ and $T\la 10^4\,{{\rm K}}$ [e.g. @Rahmati2013HICDDF], but at such high densities the radiation from local stellar sources, which we neglect here, is expected to be at least as important as the background radiation [e.g. @Schaye2001MaxHI; @Rahmati2013LocalSources]. Reionization {#sec:reionisation} ------------ Hydrogen reionization is implemented by turning on the time-dependent, spatially-uniform ionizing background from @Haardt2001UVB. This is done at redshift $z=11.5$, consistent with the optical depth measurements from @PlanckI. At higher redshifts we use net cooling rates for gas exposed to the CMB and the photo-dissociating background obtained by cutting the $z = 9$ @Haardt2001UVB spectrum above 1 Ryd. To account for the boost in the photoheating rates during reionization relative to the optically thin rates assumed here, we inject 2 eV per proton mass. This ensures that the photoionised gas is quickly heated to $\sim 10^4\,{{\rm K}}$. For H this is done instantaneously, but for the extra heat is distributed in redshift with a Gaussian centred on $z=3.5$ of width $\sigma(z)=0.5$. @Wiersma2009Chemo showed that this choice results in broad agreement with the thermal history of the intergalactic gas as measured by @Schaye2000IGMTemp. Star formation {#sec:sf} -------------- Star formation is implemented following @Schaye2008SF, but with the metallicity-dependent density threshold of @Schaye2004SF and a different temperature threshold, as detailed below. Contrary to standard practice, we take the star formation rate to depend on pressure rather than density. As demonstrated by @Schaye2008SF, this has two important advantages. First, under the assumption that the gas is self-gravitating, we can rewrite the observed Kennicutt-Schmidt star formation law [@Kennicutt1998Law], $\dot{\Sigma}_\ast = A (\Sigma_{\rm g}/1~{{{\rm M}_\odot}}\,{{\rm pc}}^{-2})^n$, as a pressure law: $$\dot{m}_\ast = m_{\rm g} A \left (1~{{{\rm M}_\odot}}\,{{\rm pc}}^{-2}\right )^{-n} \left ({\gamma \over G} f_{\rm g} P\right )^{(n-1)/2}, \label{eq:sflaw}$$ where $m_{\rm g}$ is the gas particle mass, $\gamma=5/3$ is the ratio of specific heats, $G$ is the gravitational constant, $f_{\rm g}$ is the mass fraction in gas (assumed to be unity), and $P$ is the total pressure. Hence, the free parameters $A$ and $n$ are determined by observations of the gas and star formation rate surface densities of galaxies and no tuning is necessary. Second, if we impose an equation of state, $P=P_{\rm eos}(\rho)$, then the observed Kennicutt-Schmidt star formation law will still be reproduced without having to change the star formation parameters. In contrast, if star formation is implemented using a volume density rather than a pressure law, then the predicted Kennicutt-Schmidt law will depend on the thickness of the disc and thus on the equation of state of the star forming gas. Hence, in that case the star formation law not only has to be calibrated, it has to be recalibrated if the imposed equation of state is changed. In practice, this is rarely done. Equation (\[eq:sflaw\]) is implemented stochastically. The probability that a gas particle is converted into a collisionless star particle during a time step $\Delta t$ is $\min(\dot{m}_\ast \Delta t/m_{\rm g},1)$. We use $A=1.515\times10^{-4}~{{\rm M}_\odot\,{\rm yr}^{-1}\,{\rm kpc}^{-2}}$ and $n=1.4$, where we have decreased the amplitude by a factor 1.65 relative to the value used by @Kennicutt1998Law because we use a Chabrier rather than a Salpeter stellar initial mass function (IMF). We increase $n$ to 2 for $n_{\rm H} > 10^3\,{{\rm cm}}^{-3}$, because there is some evidence for a steepening at high densities [e.g. @Liu2011KSLaw; @Genzel2010SFLaw], but this does not have a significant effect on the results since only $\sim 1$% of the stars form at such high densities in our simulations. Star formation is observed to occur in cold ($T\ll 10^4\,{{\rm K}}$), molecular gas. Because simulations of large cosmological volumes, such as ours, lack the resolution and the physics to model the cold, interstellar gas phase, it is appropriate to impose a star formation threshold at the density above which a cold phase is expected to form. In [OWLS]{} we used a constant threshold of $n_{\rm H}^* = 10^{-1}\,{{\rm cm}}^{-3}$, which was motivated by theoretical considerations and yields a critical gas surface density $\sim 10~{{{\rm M}_\odot}}\,{{\rm pc}}^{-2}$ [@Schaye2004SF; @Schaye2008SF]. The critical volume density, $n_{\rm H} = 0.1~{{\rm cm}}^{-3}$, is also similar to the value used in other work of comparable resolution [e.g. @Springel2003Multiphase; @Vogelsberger2013IllustrisModel]. Here we instead use the metallicity-dependent density threshold of @Schaye2004SF as implemented in [OWLS]{} model “SFTHRESZ” (eq. 4 of @Schaye2010OWLS; equations 19 and 24 of @Schaye2004SF), $$\label{eq:sfthresz} n_{\rm H}^(Z)=10^{-1}\,{{\rm cm}}^{-3} \left ({Z \over 0.002}\right )^{-0.64},$$ where $Z$ is the gas metallicity (i.e. the fraction of the gas mass in elements heavier than helium). In the code the threshold is evaluated as a mass density rather than a total hydrogen number density. To prevent an additional dependence on the hydrogen mass fraction (beyond that implied by equation \[eq:sfthresz\]), we convert $n_{\rm H}$ into a mass density assuming the initial hydrogen mass fraction, $X=0.752$. Because the @Schaye2004SF relation diverges at low metallicities, we impose an upper limit of $n_{\rm H}^=10~{{\rm cm}}^{-3}$. To prevent star formation in low overdensity gas at very high redshift, we also require the gas density to exceed 57.7 times the cosmic mean, but the results are insensitive to this value. The metallicity dependence accounts for the fact that the transition from a warm, neutral to a cold, molecular phase occurs at lower densities and pressures if the metallicity, and hence also the dust-to-gas ratio, is higher. The phase transition shifts to lower pressures if the metallicity is increased due to the higher formation rate of molecular hydrogen, the increased cooling due to metals and the increased shielding by dust [e.g. @Schaye2001MaxHI; @Schaye2004SF; @Pelupessy2006H2; @Krumholz2008HIH2; @Gnedin2009H2form; @Richings2014Shielding]. Our metallicity-dependent density threshold causes the critical gas surface density below which the Kennicutt-Schmidt law steepens to decrease with increasing metallicity. Because our simulations do not model the cold gas phase, we impose a temperature floor, $T_{\rm eos}(\rho_{\rm g})$, corresponding to the equation of state $P_{\rm eos}\propto \rho_{\rm g}^{4/3}$, normalised to[^6] $T_{\rm eos} = 8\times 10^3\,{{\rm K}}$ at $n_{\rm H} = 10^{-1}\,{{\rm cm}}^{-3}$, a temperature that is typical for the warm ISM [e.g. @Richings2014Shielding]. The slope of $4/3$ guarantees that the Jeans mass, and the ratio of the Jeans length to the SPH kernel, are independent of the density, which prevents spurious fragmentation due to the finite resolution [@Schaye2008SF; @Robertson2008SFlaw]. Following @DallaVecchia2012Winds, gas is eligible to form stars if $\log_{10} T < \log_{10}T_{\rm eos} + 0.5$ and $n_{\rm H} > n_{\rm H}^$, where $n_{\rm H}^$ depends on metallicity as specified above. Because of the existence of a temperature floor, the temperature of star forming (i.e. interstellar) gas in the simulation merely reflects the effective pressure imposed on the unresolved, multiphase ISM, which may in reality be dominated by turbulent rather than thermal pressure. If the temperature of this gas needs to be specified, e.g. when computing neutral hydrogen fractions in post-processing, then one should assume a value based on physical considerations rather than use the formal simulation temperatures at face value. In addition to the minimum pressure corresponding to the equation of state with slope $4/3$, we impose a temperature floor of 8000 K for densities $n_{\rm H}>10^{-5}\,{{\rm cm}}^{-3}$ in order to prevent very metal-rich particles from cooling to temperatures characteristic of cold, interstellar gas. This constant temperature floor was not used in [OWLS]{} and is unimportant for our results. We impose it because we do not wish to include a cold interstellar phase since we do not model all the physical processes that are needed to describe it. We only impose this limit for densities $n_{\rm H}>10^{-5}\,{{\rm cm}}^{-3}$, because we should not prevent the existence of cold, adiabatically cooled, intergalactic gas, which our algorithms can model accurately. Stellar mass loss and type Ia supernovae {#sec:chemo} ---------------------------------------- Star particles are treated as simple stellar populations (SSPs) with a @Chabrier2003IMF IMF in the range $0.1-100~{{{\rm M}_\odot}}$. The implementation of stellar mass loss is based on @Wiersma2009Chemo. At each time step[^7] and for each stellar particle, we compute which stellar masses reach the end of the main sequence phase using the metallicity-dependent lifetimes of @Portinari1998Chemo. The fraction of the initial particle mass reaching this evolutionary stage is used, together with the initial elemental abundances, to compute the mass of each element that is lost through winds from AGB stars, winds from massive stars, and core collapse supernovae using the nucleosynthetic yields from @Marigo2001AGByields and @Portinari1998Chemo. The elements H, He, C, N, O, Ne, Mg, Si, and Fe are tracked individually, while for Ca and S we assume fixed mass ratios relative to Si of 0.094 and 0.605, respectively [@Wiersma2009Chemo]. In addition, we compute the mass and energy lost through supernovae of type Ia. The mass lost by star particles is distributed among the neighbouring SPH particles using the SPH kernel, but setting the mass of the gas particles equal to the constant initial value, $m_{\rm g}$. Each SPH neighbour $k$ that is separated by a distance $r_k$ from a star particle with smoothing length $h$ then receives a fraction $\frac{m_{\rm g}}{\rho_k} W(r_k,h)/\Sigma_i \frac{m_{\rm g}}{\rho_i}W(r_i,h)$ of the mass lost during the time step, where $W$ is the SPH kernel and the sum is over all SPH neighbours. To speed up the calculation, we use only 48 neighbours for stellar mass loss rather than the 58 neighbours used for the SPH. In @Wiersma2009Chemo and [OWLS]{} we used the current gas particle masses rather than the constant, initial gas particle mass when computing the weights. The problem with that approach is that gas particles that are more massive than their neighbours, due to having received more mass lost by stars, carry more weight and therefore become even more massive relative to their neighbours. We found that this runaway process can cause a very small fraction of particles to end up with masses that far exceed the initial particle mass. The fraction of very massive particles is always small, because massive particles are typically also metal rich and relatively quickly converted into star particles. Nevertheless, it is still undesirable to preferentially direct the lost mass to relatively massive gas particles. We therefore removed this bias by using the fixed initial particle mass rather than the current particle mass, effectively taking the dependence on gas particle mass out of the equation for the distribution of stellar mass loss. We also account for the transfer of momentum and energy associated with the transfer of mass from star to gas particles. We refer here to the momentum and energy related to the difference in velocity between the star particle and the receiving gas particles, in addition to that associated with the mass loss process itself (e.g. winds or supernovae). We assume that winds from AGB stars have a velocity of $10~{{\rm km}\,{\rm s}^{-1}}$ [@Bergeat2005AGBMassLoss]. After adjusting the velocities of the receiving gas particles to conserve momentum, energy conservation is achieved by adjusting their entropies. Momentum and energy transfer may, for example, play a role if the differential velocity between the stellar and gas components is similar to or greater than the sound speed of the gas, although we should keep in mind that the change in the mass of a gas particle during a cooling time is typically small. As in @Wiersma2009Chemo, the abundances used to evaluate the radiative cooling rates are computed as the ratio of the mass density of an element to the total gas density, where both are calculated using the SPH formalism. Star particles inherit their parent gas particles’ kernel-smoothed abundances[^8] and we use those to compute their lifetimes and yields. The use of SPH-smoothed abundances, rather than the mass fractions of the elements stored in each particle, is consistent with the SPH formalism. It helps to alleviate the symptoms of the lack of metal mixing that occurs when metals are fixed to particles. However, as discussed in @Wiersma2009Chemo, it does not solve the problem that SPH may underestimate metal mixing. The implementation of diffusion can be used to increase the mixing [e.g. @Greif2009Mixing; @Shen2010IGMEnrichment], but we have opted not to do this because the effective diffusion coefficients that are appropriate for the ISM and IGM remain unknown. The rate of supernovae of type Ia (SNIa) per unit initial stellar mass is given by, $$\dot{N}_{\rm SNIa} = \nu \frac{e^{-t/\tau}}{\tau}, \label{eq:snia}$$ where $\nu$ is the total number of SNIa per unit initial stellar mass and $\exp(-t/\tau)/\tau$ is a normalised, empirical delay time distribution function. We set $\tau = 2$ Gyr and $\nu = 2\times 10^{-3}\,{{{\rm M}_\odot}}^{-1}$. Figure \[fig:snia\] shows that these choices yield broad agreement with the observed evolution of the SNIa rate density for the intermediate resolution simulations, although the AGNdT9-L050N0752 may overestimate the rate by $\sim 30$ per cent for lookback times of 4–7 Gyr. The high-resolution model, Recal-L025N0752, is consistent with the observations at all times. At each time step for which the mass loss is evaluated, star particles transfer the mass and energy associated with SNIa ejecta to their neighbours. We use the SNIa yields of the W7 model of @Thielemann2003SNIayields. Energy feedback from SNIa is implemented identically as for prompt stellar feedback using the stochastic thermal feedback model of @DallaVecchia2012Winds summarized in §\[sec:snii\], using $\Delta T = 10^{7.5}\,{{\rm K}}$ and $10^{51}\,{{\rm erg}}$ per SNIa. Energy feedback from star formation {#sec:snii} ----------------------------------- Stars can inject energy and momentum into the ISM through stellar winds, radiation, and supernovae. These processes are particularly important for massive and hence short-lived stars. If star formation is sufficiently vigorous, the associated feedback can drive large-scale galactic outflows [e.g. @Veilleux2005WindsReview]. Cosmological, hydrodynamical simulations have traditionally struggled to make stellar feedback as efficient as is required to match observed galaxy masses, sizes, outflow rates and other data. If the energy is injected thermally, it tends to be quickly radiated away rather than to drive a wind [e.g. @Katz1996TreeSPH]. This “overcooling” problem is typically attributed to a lack of numerical resolution. If the simulation does not contain dense, cold clouds, then the star formation is not sufficiently clumpy and the feedback energy is distributed too smoothly. Moreover, since in reality cold clouds contain a large fraction of the mass of the ISM, in simulations without a cold interstellar phase the density of the warm, diffuse phase, and hence its cooling rate, is overestimated. While these factors may well contribute to the problem, @DallaVecchia2012Winds [see also @DallaVecchia2008Winds, @Creasey2011Overcooling and @Keller2014Winds] argued that the fact that the energy is distributed over too much mass may be a more fundamental issue. For a standard IMF there is $\sim 1$ supernova per 100 ${{{\rm M}_\odot}}$ of SSP mass and, in reality, all the associated mechanical energy is initially deposited in a few solar masses of ejecta, leading to very high initial temperatures (e.g. $\sim 2\times 10^8\,{{\rm K}}$ if $10^{51}\,{{\rm erg}}$ is deposited in $10~{{{\rm M}_\odot}}$ of gas). In contrast, in SPH simulations that distribute the energy produced by a star particle over its SPH neighbours, the ratio of the heated mass to the mass of the SSP will be much greater than unity. The mismatch in the mass ratio implies that the maximum temperature of the directly heated gas is far lower than in reality, and hence that its radiative cooling time is much too short. Because the mass ratio of SPH to star particles is independent of resolution, to first order this problem is independent of resolution. At second order, higher resolution does help, because the thermal feedback can be effective in generating an outflow if the cooling time is large compared with the sound crossing time across a resolution element, and the latter decreases with increasing resolution (but only as $m_{\rm g}^{1/3}$). Thus, subgrid models are needed to generate galactic winds in large-volume cosmological simulations. Three types of prescriptions are widely used: injecting energy in kinetic form [e.g. @Navarro1993KineticFeedback; @Springel2003Multiphase; @DallaVecchia2008Winds; @Dubois2008Winds] often in combination with temporarily disabling hydrodynamical forces acting on wind particles [e.g. @Springel2003Multiphase; @Okamoto2005Disks; @Oppenheimer2006Wpot], temporarily turning off radiative cooling [e.g. @Gerritsen1997PhD; @Stinson2006Winds], and explicitly decoupling different thermal phases (also within single particles) [e.g. @Marri2003Multiphase; @Scannapieco2006Multiphase; @Murante2010Multiphase; @Keller2014Winds]. Here we follow @DallaVecchia2012Winds [see also @Kay2003XrayGroups] and opt for a different type of solution: stochastic thermal feedback. By making the feedback stochastic, we can control the amount of energy per feedback event even if we fix the mean energy injected per unit mass of stars formed. We specify the temperature jump of gas particles receiving feedback energy, $\Delta T$, and use the fraction of the total amount of energy from core collapse supernovae per unit stellar mass that is injected on average, $f_{\rm th}$, to set the probability that an SPH neighbour of a young star particle is heated. We perform this operation only once, when the stellar particle has reached the age $3\times 10^7\,{{\rm yr}}$, which corresponds to the maximum lifetime of stars that explode as core collapse supernovae. The value $f_{\rm th}=1$ corresponds to an expectation value for the injected energy of $8.73\times 10^{15}\,{{\rm erg}}\,{{\rm g}}^{-1}$ of stellar mass formed, which corresponds to the energy available from core collapse supernovae for a Chabrier IMF if we assume $10^{51}\,{{\rm erg}}$ per supernova and that stars with mass $6-100~{{{\rm M}_\odot}}$ explode ($6-8~{{{\rm M}_\odot}}$ stars explode as electron capture supernovae in models with convective overshoot; e.g. @Chiosi1992Overshoot). If $\Delta T$ is sufficiently high, then the initial (spurious, numerical) thermal losses will be small and we can control the overall efficiency of the feedback using $f_{\rm th}$. This freedom is justified, because there will be physical radiative losses in reality that we cannot predict accurately for the ISM. Moreover, because the true radiative losses likely depend on the physical conditions, we may choose to vary $f_{\rm th}$ with the relevant, local properties of the gas. By considering the ratio of the cooling time to the sound crossing time across a resolution element, @DallaVecchia2012Winds derive the maximum density for which the thermal feedback can be efficient (their equation 18), $${n_{{\rm H},t_{\rm c}}}\sim 10~{{\rm cm}}^{-3} \left (\frac{T}{10^{7.5}\,{{\rm K}}}\right )^{3/2} \left (\frac{m_{\rm g}}{10^6\,{{{\rm M}_\odot}}}\right )^{-1/2}, \label{eq:nhtc}$$ where $T> \Delta T$ is the temperature after the energy injection and we use $\Delta T = 10^{7.5}\,{{\rm K}}$. This expression assumes that the radiative cooling rate is dominated by free-free emission and will thus significantly overestimate the value of ${n_{{\rm H},t_{\rm c}}}$ when line cooling dominates, i.e. for $T \ll 10^7\,{{\rm K}}$. In our simulations some stars do, in fact, form in gas that far exceeds the critical value ${n_{{\rm H},t_{\rm c}}}$, particularly in massive galaxies. Although the density of the gas in which the stars inject their energy will generally be lower than that of the gas from which the star particle formed, since the star particles move relative to the gas during the $3\times 10^7\,{{\rm yr}}$ delay between star formation and feedback, this does mean that for stars forming at high gas densities the radiative losses may well exceed those that would occur in a simulation that has the resolution and the physics required to resolve the small-scale structure of the ISM. As we calibrate the total amount of energy that is injected per unit stellar mass to achieve a good match to the observed GSMF, this implies that we may overestimate the required amount of feedback energy. At the high-mass end AGN feedback controls the efficiency of galaxy formation in our simulations. If the radiative losses from stellar feedback are overestimated, then this could potentially cause us to overestimate the required efficiency of AGN feedback. The critical density, ${n_{{\rm H},t_{\rm c}}}$, increases with the numerical resolution, but also with the temperature jump, $\Delta T$. We could therefore reduce the initial thermal losses by increasing $\Delta T$. However, for a fixed amount of energy per unit stellar mass, i.e. for a fixed value of $f_{\rm th}$, the probability that a particular star particle generates feedback is inversely proportional to $\Delta T$. @DallaVecchia2012Winds show that, for the case of equal mass particles, the expectation value for the number of heated gas particles per star particle is (their equation 8) $$\left < N_{\rm heat}\right > \approx 1.3 f_{\rm th} \left (\frac{\Delta T}{10^{7.5}\,{{\rm K}}}\right )^{-1}$$ for our Chabrier IMF and only accounting for supernova energy (assuming that supernovae associated with stars in the range 6-100 ${{{\rm M}_\odot}}$ each yield $10^{51}\,{{\rm erg}}$). Hence, using $\Delta T \gg 10^{7.5}\,{{\rm K}}$ or $f_{\rm th} \ll 1$ would imply that most star particles do not inject any energy from core collapse supernovae into their surroundings, which may lead to poor sampling of the feedback cycle. We therefore keep the temperature jump set to $\Delta T = 10^{7.5}\,{{\rm K}}$. Although the stochastic implementation enables efficient thermal feedback without the need to turn off cooling, the thermal losses are unlikely to be converged with numerical resolution for simulations such as [EAGLE]{}. Hence, recalibration of $f_{\rm th}$ may be necessary when the resolution is changed. ### Dependence on local gas properties {#sec:calibration} We expect the true thermal losses in the ISM to increase when the metallicity becomes sufficiently high for metal-line cooling to become important. For temperatures of $10^5\,{{\rm K}}< T < 10^7{{\rm K}}$ this happens when $Z \ga 10^{-1}\,{{\rm Z}_\odot}$ [e.g. @Wiersma2009Cooling]. Although the exact dependence on metallicity cannot be predicted without full knowledge of the physical conditions in the ISM, we can capture the expected, qualitative transition from cooling losses dominated by H and He to losses dominated by metals by making $f_{\rm th}$ a function of metallicity, $$f_{\rm th} = f_{\rm th,min} + \frac{f_{\rm th,max} - f_{\rm th,min}} {1 + \left (\frac{Z}{0.1{{\rm Z}_\odot}}\right )^{n_Z}}, \label{eq:f(Z)}$$ where ${{\rm Z}_\odot}= 0.0127$ is the solar metallicity and $n_Z>0$. Note that $f_{\rm th}$ asymptotes to $f_{\rm th,max}$ and $f_{\rm th,min}$ for $Z \ll 0.1{{\rm Z}_\odot}$ and $Z\gg 0.1{{\rm Z}_\odot}$, respectively. Since metallicity decreases with redshift at fixed stellar mass, this physically motivated metallicity dependence tends to make feedback relatively more efficient at high redshift. As we show in @Crain2014EagleModels, this leads to good agreement with the observed, present-day GSMF. In fact, @Crain2014EagleModels show that using a constant $f_{\rm th} =1$ appears to yield even better agreement with the low-redshift mass function, but we keep the metallicity dependence because it is physically motivated: we do expect larger radiative losses for $Z \gg 0.1Z_\odot$ than for $Z \ll 0.1Z_\odot$. If we were only interested in the GSMF, then equation (\[eq:f(Z)\]) (or $f_{\rm th} =1$) would suffice. However, we find that pure metallicity dependence results in galaxies that are too compact, which indicates that the feedback is too inefficient at high gas densities. As discussed above, this is not unexpected given the resolution of our simulations. Indeed, we found that increasing the resolution reduces the problem. We therefore found it desirable to compensate for the excessive initial, thermal losses at high densities by adding a density dependence to $f_{\rm th}$: $$f_{\rm th} = f_{\rm th,min} + \frac{f_{\rm th,max} - f_{\rm th,min}} {1 + \left (\frac{Z}{0.1{{\rm Z}_\odot}}\right )^{n_Z} \left (\frac{n_{\rm H,birth}}{n_{{\rm H},0}}\right )^{-n_n}}, \label{eq:f(Z,n)}$$ where $n_{\rm H,birth}$ is the density inherited by the star particle, i.e. the density of its parent gas particle at the time it was converted into a stellar particle. Hence, $f_{\rm th}$ increases with density at fixed metallicity, while still respecting the original asymptotic values. We use $n_Z = n_n = 2/\ln10$. The seemingly unnatural value $2/\ln10 \approx 0.87$ of the exponent is a leftover from an equivalent, but more complicated expression that was originally used in the code. Using the round number 1 instead of 0.87 would have worked equally well. We use $n_{{\rm H},0} = 0.67~{{\rm cm}}^{-3}$, a value that was chosen after comparing a few test simulations to the observed present-day GSMF and galaxy sizes. The higher resolution simulation Recal-L025N0752 instead uses $n_{{\rm H},0} = 0.25~{{\rm cm}}^{-3}$ and a power-law exponent for the density term of $-1/\ln10$ rather than $-2/\ln10$ (see Table \[tbl:subgridpars\]), which we found gives better agreement with the GSMF. Note that a density dependence of $f_{\rm th}$ may also have a physical interpretation. For example, higher mean densities on $10^2-10^3\,{{\rm pc}}$ scales may result in more clustered star formation, which may reduce thermal losses. However, we stress that our primary motivation was to counteract the excessive thermal losses in the high-density ISM that can be attributed to our limited resolution. We use the asymptotic values $f_{\rm th,max}=3$ and $f_{\rm th,min}=0.3$, where the high asymptote $f_{\rm th,max}$ is reached at low metallicity and high density, and vice versa for the low asymptote. As discussed in @Crain2014EagleModels, where we present variations on the reference model, the choice of the high asymptote is the more important one. Using a value of $f_{\rm th,max}$ greater than unity enables us to reproduce the GSMF down to lower masses. Values of $f_{\rm th}$ greater than unity can be motivated on physical grounds by appealing to other sources of energy than supernovae, e.g. stellar winds, radiation pressure, or cosmic rays, or if supernovae yield more energy per unit mass than assumed here (e.g. in case of a top-heavy IMF). However, we believe that a more appropriate motivation is again the need to compensate for the finite numerical resolution. Galaxies containing few star particles tend to have too high stellar fractions [e.g. @Haas2013OwlsI], which can be understood as follows. The first generations of stars can only form once the halo is resolved with a sufficient number of particles to sample the high-density gas that is eligible to form stars. We do not have sufficient resolution to resolve the smallest galaxies that are expected to form in the real Universe. Hence, the progenitors of the galaxies in the simulations started forming stars, and hence driving winds, too late. As a consequence, our galaxies start with too high gas fractions and initially form stars too efficiently. As the galaxies grow substantially larger than our resolution limit, this initial error becomes progressively less important. Using a higher value of $f_{\rm th,max}$ counteracts this sampling effect as it makes the feedback from the first generations of stars that form more efficient. The mean and median values of $f_{\rm th}$ that were used for the feedback from the stars present at $z=0.1$ in Ref-L100N1504 are 1.06 and 0.70, respectively. For Recal-L025N0752 these values are 1.07 and 0.93. Hence, averaged over the entire simulation, the total amount of energy is similar to that expected from supernovae alone. A more detailed discussion of the effects of changing the functional form of $f_{\rm th}$ is presented in @Crain2014EagleModels. In that work we also present models in which $f_{\rm th}$ is constant or depends on halo mass or dark matter velocity dispersion. Black holes and feedback from AGN {#sec:BHs} --------------------------------- In our simulations feedback from accreting, supermassive black holes (BHs) quenches star formation in massive galaxies, shapes the gas profiles in the inner parts of their host haloes, and regulates the growth of the BHs. Models often make a distinction between “quasar-” and “radio-mode” BH feedback [e.g. @Croton2006SA; @Bower2006SA; @Sijacki2007AGN], where the former occurs when the BH is accreting efficiently and comes in the form of a hot, nuclear wind, while the radio mode operates when the accretion rate is low compared to the Eddington rate and the energy is injected in the form of relativistic jets. Because cosmological simulations lack the resolution to properly distinguish these two feedback modes and because we want to limit the number of feedback channels to the minimum required to match the observations of interest, we choose to implement only a single mode of AGN feedback with a fixed efficiency. The energy is injected thermally at the location of the BH at a rate that is proportional to the gas accretion rate. Our implementation may therefore be closest to the process referred to as quasar-mode feedback. For [OWLS]{} we found that this method led to excellent agreement with both optical and detailed X-ray observations of groups and clusters [@McCarthy2010AGN; @McCarthy2011AGN; @LeBrun2014CosmoOWLS]. Our implementation consists of two parts: i) prescriptions for seeding low-mass galaxies with central BHs and for their growth via gas accretion and merging (we neglect any growth by accretion of stars and dark matter); ii) a prescription for the injection of feedback energy. Our method for the growth of BHs is based on the one introduced by @Springel2005AGN and modified by @Booth2009AGN and @Rosas2013BHs, while our method for AGN feedback is close to the one described in @Booth2009AGN. Below we summarize the main ingredients and discuss the changes to the methods that we made for [EAGLE]{}. ### BH seeds The BHs ending up in galactic centres may have originated from the direct collapse of (the inner parts of) metal-free dwarf galaxies, from the remnants of very massive, metal-free stars, or from runaway collisions of stars and/or stellar mass BHs (see e.g. @Kocsis2013BHReview for a recent review). As none of these processes can be resolved in our simulations, we follow @Springel2005AGN and place BH seeds at the centre of every halo with total mass greater than $10^{10}\,{{{\rm M}_\odot}}/h$ that does not already contain a BH. For this purpose, we regularly run the friends-of-friends (FoF) finder with linking length 0.2 on the dark matter distribution. This is done at times spaced logarithmically in the expansion factor $a$ such that $\Delta a = 0.005a$. The gas particle with the highest density is converted into a collisionless BH particle with subgrid BH mass $m_{\rm BH} = 10^5\,{{{\rm M}_\odot}}/h$. The use of a subgrid BH mass is necessary because the seed BH mass is small compared with the particle mass, at least for our default resolution. Calculations of BH properties such as its accretion rate are functions of $m_{\rm BH}$, whereas gravitational interactions are computed using the BH particle mass. When the subgrid BH mass exceeds the particle mass, it is allowed to stochastically accrete neighbouring SPH particles such that BH particle and subgrid masses grow in step. Since the simulations cannot model the dynamical friction acting on BHs with masses $\la m_{\rm g}$, we force BHs with mass $< 100 m_{\rm g}$ to migrate towards the position of the minimum of the gravitational potential in the halo. At each time step the BH is moved to the location of the particle that has the lowest gravitational potential of all the neighbouring particles whose velocity relative to the BH is smaller than $0.25 c_{\rm s}$, where $c_{\rm s}$ is the speed of sound, and whose distance is smaller than three gravitational softening lengths. These two conditions prevent BHs in gas poor haloes from jumping to nearby satellites. ### Gas accretion {#sec:bh_accretion} The rate at which BHs accrete gas depends on the mass of the BH, the local density and temperature, the velocity of the BH relative to the ambient gas, and the angular momentum of the gas with respect to the BH. Specifically, the gas accretion rate, $\dot{m}_{\rm accr}$, is given by the minimum of the Eddington rate, $$\dot{m}_{\rm Edd} = \frac{4\pi G m_{\rm BH} m_{\rm p}}{\epsilon_{\rm r} \sigma_{\rm T} c},$$ and $$\dot{m}_{\rm accr} = \dot{m}_{\rm Bondi} \times \min\left (C_{\rm visc}^{-1}(c_{\rm s}/V_\phi)^3,1 \right ), \label{eq:mdotaccr}$$ where $\dot{m}_{\rm Bondi}$ is the Bondi-Hoyle ([-@Bondi1944Accr]) rate for spherically symmetric accretion, $$\dot{m}_{\rm Bondi} = \frac{4\pi G^2 m_{\rm BH}^2 \rho}{(c_{\rm s}^2 + v^2)^{3/2}}.$$ Here $m_{\rm p}$ is the proton mass, $\sigma_{\rm T}$ the Thomson cross section, $c$ the speed of light, $\epsilon_{\rm r}=0.1$ the radiative efficiency of the accretion disc, and $v$ the relative velocity of the BH and the gas. Finally, $V_\phi$ is the rotation speed of the gas around the BH computed using equation (16) of @Rosas2013BHs and $C_{\rm visc}$ is a free parameter related to the viscosity of the (subgrid) accretion disc. The mass growth rate of the BH is given by $$\dot{m}_{\rm BH} = (1-\epsilon_{\rm r}) \dot{m}_{\rm accr}.$$ The factor $(c_{\rm s}/V_\phi)^3/C_{\rm visc}$ by which the Bondi rate is multiplied in equation (\[eq:mdotaccr\]) is equivalent to the ratio of the Bondi and the viscous time scales (see @Rosas2013BHs). We set $C_{\rm visc} = 2\pi$ for Ref-L100N1504, but increase the value of $C_{\rm visc}$ by a factor $10^3$ for the recalibrated high-resolution model, Recal-L025N0752, and by a factor $10^2$ for AGNdT9-L050N0752 (see Table \[tbl:subgridpars\]). Since the critical ratio of $V_\phi/c_{\rm s}$ above which angular momentum is assumed to reduce the accretion rate scales with $C_{\rm visc}^{-1/3}$, angular momentum is relatively more important in the recalibrated simulations, delaying the onset of quenching by AGN to larger BH masses. As demonstrated by @Rosas2013BHs, the results are only weakly dependent on $C_{\rm visc}$ because the ratio of $V_\phi/c_{\rm s}$ above which the accretion rate is suppressed, which scales as $C_{\rm visc}^{-1/3}$, is more important than the actual suppression factor, which scales as $C_{\rm visc}$. Our prescription for gas accretion differs from previous work in two respects. First, the Bondi rate is not multiplied by a large, ad-hoc factor, $\alpha$. @Springel2005AGN used $\alpha=100$ while [OWLS]{} and @Rosas2013BHs used a density dependent factor that asymptoted to unity below the star formation threshold. Although the use of $\alpha$ can be justified if the simulations underestimate the gas density or overestimate the temperature near the Bondi radius, the correct value cannot be predicted by the simulations. We found that at the resolution of [EAGLE]{}, we do not need to boost the Bondi-Hoyle rate for the BH growth to become self-regulated. Hence, we were able to reduce the number of free parameters by eliminating $\alpha$. Second, we use the heuristic correction of @Rosas2013BHs to account for the fact that the accretion rate will be lower for gas with more angular momentum (because the accretion is generally not spherically symmetric as assumed in the Bondi model, but proceeds through an accretion disc). ### BH mergers BHs are merged if they are separated by a distance that is smaller than both the smoothing kernel of the BH, $h_{\rm BH}$, and three gravitational softening lengths, and if their relative velocity is smaller than the circular velocity at the distance $h_{\rm BH}$, $v_{\rm rel} < \sqrt{G m_{\rm BH}/h_{\rm BH}}$, where $h_{\rm BH}$ and $m_{\rm BH}$ are, respectively, the smoothing length and subgrid mass of the most massive BH in the pair. The limit on the allowed relative velocity prevents BHs from merging during the initial stages of galaxy mergers. ### AGN feedback {#sec:AGNfeedback} AGN feedback is implemented thermally and stochastically, in a manner analogous to energy feedback from star formation. The energy injection rate is $\epsilon_{\rm f} \epsilon_{\rm r} \dot{m}_{\rm accr} c^2$, where $\epsilon_{\rm f} = 0.15$ is the fraction of the radiated energy that is coupled to the ISM. As was the case for the stellar feedback efficiency, $f_{\rm th}$, the value of $\epsilon_{\rm f}$ must be chosen by calibrating to observations, in this case the normalisation of the relation between BH mass and stellar mass. As demonstrated and explained by @Booth2010DMHaloesBHs [see also @Booth2009AGN], the value of $\epsilon_{\rm f}$ only affects the BH masses, which are inversely proportional to $\epsilon_{\rm f}$. In particular, the outflow rate generated by the AGN and hence also the factor by which the star formation is reduced, are highly insensitive to $\epsilon_{\rm f}$ provided it is nonzero. This can be explained by self-regulation: the BH accretion rate adjusts until the rate at which energy is injected is sufficient for outflows to balance inflows. We use the same value for the AGN efficiency as in [OWLS]{}, $\epsilon_{\rm f}=0.15$ and $\epsilon_{\rm r}=0.1$, which implies that a fraction $\epsilon_{\rm f}\epsilon_{\rm r}=0.015$ of the accreted rest mass energy is returned to the local ISM. As was the case for stellar feedback, the required value will depend on the radiative losses in the ISM, which may depend on the resolution and the precise manner in which the energy is injected. We do not implement a dependence on metallicity, because metals are not expected to dominate the radiative losses at the high temperatures associated with AGN feedback. As shown in Figure \[fig:bh\], a constant value of $\epsilon_{\rm f}=0.15$ yields broad agreement with observations of the relation between BH mass and stellar mass. Each BH carries a “reservoir” of feedback energy, $E_{\rm BH}$. After each time step $\Delta t$, we add $\epsilon_{\rm f} \epsilon_{\rm r} \dot{m}_{\rm accr} c^2 \Delta t$ to this reservoir. If the BH has stored sufficient energy to heat at least $n_{\rm heat}$ particles of mass $m_{\rm g}$, then the BH is allowed to stochastically heat each of its SPH neighbours by increasing their temperature by $\Delta T_{\rm AGN}$. For each neighbour the heating probability is $$P = \frac{E_{\rm BH} }{ \Delta\epsilon_{\rm AGN} N_{\rm ngb} \left <m_{\rm g}\right > },$$ where $\Delta\epsilon_{\rm AGN}$ is the change in internal energy per unit mass corresponding to the temperature increase, $\Delta T_{\rm AGN}$ (we convert the parameter $\Delta T_{\rm AGN}$ into $\Delta\epsilon_{\rm AGN}$ assuming a fully ionised gas with primordial composition), $N_{\rm ngb}$ is the number of gas neighbours of the BH and $\left <m_{\rm g}\right >$ is their mean mass. We then reduce $E_{\rm BH}$ by the expectation value for the injected energy. We use $n_{\rm heat}=1$ and limit the time step of the BHs such that we expect[^9] $P<0.3$ (see §\[sec:injection\]). The most important parameter for the AGN feedback is the temperature increase $\Delta T_{\rm AGN}$. Larger values will make individual feedback events more energetic, generally resulting in smaller radiative losses in the ISM. However, larger values will also make the feedback more intermittent. We set $\Delta T_{\rm AGN} = 10^{8.5}\,{{\rm K}}$ in the L100N1504 reference model, but use $10^9\,{{\rm K}}$ for our recalibrated high-resolution model Recal-L025N0752 and model AGNdT9-L050N0752 (see Table \[tbl:subgridpars\]). These temperatures exceed the value of $10^8\,{{\rm K}}$ used in [OWLS]{} and the $\Delta T = 10^{7.5}\,{{\rm K}}$ that we use for stellar feedback. As can be seen from equation (\[eq:nhtc\]), the critical density above which the feedback energy is expected to be radiated away increases with the value of $\Delta T$. Because the density of the ambient gas around the BH tends to increase with resolution, we found that we need to increase $\Delta T$ when increasing the resolution. Similarly, because the gas density around the BH often reaches values that are much higher than is typical for star-forming gas, we require higher temperature jumps for AGN feedback than for stellar feedback. Comparison with observables considered during the calibration of the feedback {#sec:cal_obs} ============================================================================= In this section we will compare the main [EAGLE]{} simulations to $z\sim 0$ observations of the GSMF, the related stellar mass - halo mass relation, galaxy sizes, and the relation between BH mass and stellar mass. Since these observables were considered during the calibration of the subgrid models for feedback, we cannot consider the [EAGLE]{} results reported in this section to be “predictions”. However, note that we had no control over the slope of the $M_{\rm BH}-M_\ast$ relation and that galaxy sizes were only used to rule out strongly discrepant models (i.e. models without a density dependence of the energy feedback from star formation). The galaxy stellar mass function {#sec:gsmf} -------------------------------- Figure \[fig:gsmf\] shows the $z=0.1$ galaxy stellar mass function (GSMF) from [EAGLE]{}. The dark blue curve shows Ref-L100N1504, the green curve shows the high-resolution simulation Recal-L025N0752, and the red curve shows AGNdT9-L050N0752. Recall that AGNdT9-L050N0752 employs a higher heating temperature for AGN feedback than the reference model, which makes the feedback more efficient. While this is unimportant for the GSMF, we will see in §\[sec:groups\] that it offers a significant improvement for the intracluster medium. At the high-mass end the curves switch from a solid to a dashed line style where there are fewer than 10 objects per (0.2 dex) stellar mass bin. At the low-mass end the curves become dotted when the stellar mass falls below that corresponding to 100 baryonic particles, where sampling effects associated with the limited resolution become important, as can be seen by comparing the intermediate- and high-resolution simulations. The GSMF of the high-resolution simulation Recal-L025N0752 is noisier because the box size is too small to provide a representative sample. Note that the main problem is not Poisson noise due to the small number of objects per bin, but the small number of large-scale modes that modulate the local number density of galaxies of various masses. Indeed, Fig. \[fig:gsmf\_conv\] shows that the GSMF of Recal-L025N0752 has the same wiggles as that of Ref-L025N0376, which uses the same box size and, apart from the change in resolution, the same initial conditions. The wiggles that are present for Ref-L025N0376 are absent for model Ref-L100N1504, even though these two simulations use identical resolutions and (subgrid) parameter values. This confirms that the wiggles in the GSMF of Recal-L025N0752 are caused by the small size of its simulation volume. We will therefore focus on the larger volume simulations when comparing the simulated and observed GSMFs. The simulation results are compared with observations from the Galaxy And Mass Assembly (GAMA) survey (@Baldry2012GSMF; open circles) and from SDSS (@Li2009GSMF; filled circles). For the intermediate-resolution simulations the galaxy number densities agree with the observations to $\la 0.2$ dex over the full mass range for which the resolution and box size are adequate, i.e. from $2\times 10^8\,{{{\rm M}_\odot}}$ to over $10^{11}\,{{{\rm M}_\odot}}$ (slightly below $10^{11}\,{{{\rm M}_\odot}}$ for Recal-L025N0752). The observed shape of the GSMF is thus reproduced well. At fixed number density, the differences in stellar mass between the simulations and observations are smaller than 0.3 dex for Ref-L100N1504 and AGNdT9-L050N0752. Given that even for a fixed IMF, uncertainties in the stellar evolution models used to infer stellar masses are $\sim 0.3$ dex [e.g. @Conroy2009SPSSUncertainty; @Behroozi2010Uncertainties; @Pforr2012SPSSUncertainty; @Mitchell2013MassUncertainty], there is perhaps little point in trying to improve the agreement between the models and the data further. The subgrid models for energy feedback from star formation and for BH accretion have been calibrated to make the simulated GSMF fit the observed one, so the excellent agreement with the data cannot be considered a successful prediction. However, success was by no means guaranteed given that the computational expense of hydrodynamical simulations severely limits the number of test runs that can be performed and, more importantly, because the freedom built into the model is rather limited. For example, while the mass scale above which AGN feedback becomes dominant is sensitive to the parameter $C_{\rm visc}$ of the subgrid model for BH accretion (see equation \[eq:mdotaccr\] in §\[sec:bh\_accretion\]), the efficiency of the AGN feedback was calibrated to the observed relation between BH mass and stellar mass and does not affect the shape of the GSMF [@Booth2009AGN; @Booth2010DMHaloesBHs]. Figure \[fig:gsmf\_other\] shows that the level of correspondence between the data and [EAGLE]{} is close to that attained for semi-analytic models (left panel) and is unprecedented for large, hydrodynamical simulations (right panel). As can be seen from the right panel, even though @Oppenheimer2010GSMF, @Puchwein2013GSMF, and Illustris [@Vogelsberger2014IllustrisNature; @Genel2014IllustrisEvolution] all adjusted their subgrid feedback models to try to match the data, the fits to the data are substantially less good than for [EAGLE]{}. In particular, their models all produce mass functions that are too steep below the “knee” of the Schechter function and too shallow for larger masses. It is worth noting that each of these three groups implemented the feedback from star formation kinetically, scaled the wind velocity with the velocity dispersion of the dark matter, determined the dependence of the wind mass loading on the dark matter velocity dispersion by assuming a constant wind energy, and temporarily turned off the hydrodynamical forces on wind particles to allow them to escape the galaxies. This contrasts with [EAGLE]{}, where the feedback was implemented thermally rather than kinetically, the feedback energy varied with local gas properties, and the hydrodynamical forces were never turned off. Hence, contrary to the other models shown, [EAGLE]{}’s subgrid model does not impose any particular wind velocity or mass loading or any dependence on dark matter or halo properties. The injected energy does depend on the local metallicity and gas density, but the relation between the outflow properties and the energy injected at the star formation site is an outcome of the simulation. @Crain2014EagleModels will show that while varying the feedback energy with local gas properties is necessary to obtain reasonable galaxy sizes, the $z\sim 0$ GSMF is actually also reproduced by the [EAGLE]{} model that injects a constant energy per unit stellar mass (equal to the energy from supernovae) without any calibration. While the excellent fit to the low-$z$ GSMF is encouraging, the success of the model can only be judged by comparing to a wide range of observables and redshifts, particularly those that were not considered during the calibration. We will consider a diverse selection of observables in §\[sec:otherobs\] and will investigate their evolution in @Furlong2014EagleEvolution and other future papers. ### Effect of the choice of aperture {#sec:aperture} For the simulations we chose to define a galaxy’s stellar mass as the sum of the mass of the stars that are part of a gravitationally bound subhalo and that are contained within a 3-D aperture of radius 30 proper kiloparsec (see §\[sec:simulations\]). Figure \[fig:gsmf\_aperture\] shows the effect of the choice of aperture for Ref-L100N1504. For $M_\ast < 10^{11}\,{{{\rm M}_\odot}}$ the results are insensitive to the aperture, provided it is $\ga 30$ pkpc. However, for $M_\ast > 10^{11}\,{{{\rm M}_\odot}}$ the aperture does become important, with larger apertures giving larger masses. The same is true for the observations, as can be seen by comparing the data from @Li2009GSMF with the re-analysis of SDSS data by @Bernardi2013GSMF (open triangles in Fig. \[fig:gsmf\_aperture\]). @Baldry2012GSMF and @Li2009GSMF are in good agreement, but @Bernardi2013GSMF find a much shallower bright-end slope than previous analyses. For $M_\ast > 10^{11}\,{{{\rm M}_\odot}}$ @Bernardi2013GSMF attribute substantially more mass to galaxies than @Li2009GSMF and @Baldry2012GSMF. Part of the difference is due to the assumed mass-to-light ratios (even though all studies assume a Chabrier IMF) and the way in which the background is subtracted (see e.g. @Bernardi2013GSMF and @Kravtsov2014GalaxyHalo for discussion). Most of the difference between @Li2009GSMF and @Bernardi2013GSMF can probably be attributed to the way in which a galaxy’s light is measured. @Li2009GSMF integrate the light within a 2-D aperture of size twice the Petrosian radius, defined to be the radius at which the mean local surface brightness is 0.2 times the mean internal surface brightness. @Bernardi2013GSMF on the other hand, estimate the total amount of light by integrating Sérsic plus exponential profile fits. Hence, the @Bernardi2013GSMF mass function potentially includes intracluster light and the discrepancy between different authors is related to the fact that it is unclear where cD galaxies end. @Baldry2012GSMF integrate single Sérsic fits to the light profiles, which we would expect includes less intracluster light than the Sérsic plus exponential fits of @Bernardi2013GSMF but more than the Petrosian apertures of @Li2009GSMF. However, @Bernardi2013GSMF find that the high-mass end of the @Baldry2012GSMF mass function is affected by their redshift cut ($z<0.06$). We believe the @Baldry2012GSMF and @Li2009GSMF data to be the most suitable for comparison to our results, since our definition of a galaxy excludes intracluster light. For @Li2009GSMF this is confirmed by our finding that a 3-D aperture of 30 pkpc gives nearly identical results to a 2-D Petrosian cut, as can be seen from Figure \[fig:gsmf\_aperture\]. Thus, for masses $> 10^{11}\,{{{\rm M}_\odot}}$ comparisons of the GSMF with observations would benefit from mimicking the particular way in which the mass is estimated for real data. This would, however, have to be done separately for each survey. For our present purposes this is unnecessary, also because our simulation volume is in any case too small to study the GSMF at masses $\gg 10^{11}\,{{{\rm M}_\odot}}$. ### Numerical convergence {#sec:convergence} The left panel of Figure \[fig:gsmf\_conv\] compares the GSMFs for model Ref-L025N0376, which has the same resolution as the largest [EAGLE]{} volume Ref-L100N1504, and the higher-resolution model Ref-L025N0752. The two Ref-L025 simulations use identical subgrid parameters, but the mass and spatial resolution differ by factors of 8 and 2, respectively. In §\[sec:conv\_discussion\] we termed a comparison between models with identical parameters a “strong convergence test”. Below $10^9\,{{{\rm M}_\odot}}$ the mass function is substantially flatter in the high-resolution model. However, at $M_\ast \sim 10^9\,{{{\rm M}_\odot}}$ its GSMF is up to 0.4 dex higher than for the fiducial resolution, leading to disagreement with the data. The largest discrepancy is the stellar mass corresponding to a number density of $\sim 2\times 10^{-2}\,{\rm cMpc}^{-3}$, which is about an order of magnitude higher than observed. The thin curves in Figure \[fig:gsmf\_conv\] show the strong convergence test of @Vogelsberger2013IllustrisModel using the galaxy formation model that was also used for Illustris. Clearly, the strong convergence is similarly poor. This is somewhat surprising, since Illustris uses a subgrid model for feedback from star formation that was designed to give good strong convergence. In particular, the parameters of the subgrid wind model vary with the velocity dispersion of the dark matter rather than with the properties of the gas and hydrodynamical interactions between the wind and the ISM are not modelled. That the strong convergence is not particularly good for [EAGLE]{} is unsurprising for the reasons discussed in §\[sec:conv\_discussion\] and §\[sec:snii\]. For $M_\ast < 2\times 10^8\,{{{\rm M}_\odot}}$ galaxies in Ref-L025N0376 contain fewer than 100 star particles, which is insufficient to properly sample the feedback from star formation in the context of [EAGLE]{}’s subgrid model. Because the feedback can be modelled down to lower masses in Ref-L025N0752, galaxies with $M_\ast \sim 10^9\,{{{\rm M}_\odot}}$ have had systematically different histories than galaxies of a similar mass in Ref-L025N0376. In addition, higher resolution enables the gas density distribution to be populated by particles up to higher densities, where our fiducial implementation of thermal feedback becomes inefficient (equation \[eq:nhtc\] in §\[sec:snii\]). In §\[sec:conv\_discussion\] we argued that hydrodynamical simulations such as [EAGLE]{} should recalibrate the efficiency of the subgrid feedback when the resolution is changed substantially. In general, keeping the subgrid parameters fixed does not imply that the physical model remains unchanged, since the energy, mass or intermittency associated with the feedback events changes. Moreover, the efficiency of the feedback cannot, in any case, be predicted from first principles, even if the convergence were perfect. Recal-L025N0752 is our recalibrated high-resolution simulation. As detailed in §\[sec:calibration\] and Table \[tbl:subgridpars\], the dependence of the feedback energy per unit stellar mass on the gas density is somewhat different between the different resolutions. However, the mean values of $f_{\rm th}$, which is equal to the expectation value of the amount of injected energy in units of the energy available from core collapse supernovae, are nearly identical: 1.06 at intermediate resolution (for stars formed at $z>0.1$ in Ref-L100N1504) and 1.07 at high resolution (for stars formed at $z>0.1$ in Recal-L025N0752). The asymptotic maximum of $f_{\rm th}$, reached at low metallicity and low gas density, is 3 in both cases. As detailed in §\[sec:bh\_accretion\] and Table \[tbl:subgridpars\], Recal-L025N0752 also uses a different value for the parameter that controls the importance of angular momentum in suppressing accretion onto BHs, making the accretion rate more sensitive to the angular momentum of the accreting gas. Without this change, AGN feedback would become important at too low masses. Finally, the high-resolution model uses a higher AGN feedback temperature, $\Delta T_{\rm AGN} = 10^9\,{{\rm K}}$ rather than $10^{8.5}\,{{\rm K}}$, which helps to suppress the increase in the cooling losses that would otherwise occur due to the higher gas densities that are resolved in the higher resolution model. Without this change the AGN feedback would be insufficiently effective. The right panel of Figure \[fig:gsmf\_conv\] shows a “weak convergence test”, i.e. a comparison of the GSMFs of the calibrated intermediate resolution model Ref-L025N0376 and the recalibrated high-resolution model Recal-L025N0752. The two curves show some of the same bumps and wiggles, because the initial conditions used for the two simulations share the same large-scale modes. In the mass range for which galaxies in the intermediate-resolution model are resolved with more than 100 star particles ($M_\ast > 2\times 10^8\,{{{\rm M}_\odot}}$) the difference in the galaxy number density is smaller than 0.2 dex. We conclude that the weak convergence is good. The relation between stellar mass and halo mass {#sec:eta} ----------------------------------------------- The GSMF can be thought of as a convolution between the mass function of dark matter haloes and a function describing the galaxy content of the haloes as a function of their mass. The halo mass function can be predicted accurately when the cosmology is known, but the galaxy content of haloes is very sensitive to the baryonic processes involved in the formation of galaxies. As modelling galaxy formation is [EAGLE]{}’s primary goal, it is of interest to compare the relation between stellar mass and halo mass in the simulations to the relation inferred from observations. Because the subgrid model for feedback was calibrated to fit the $z\sim 0$ GSMF, the relation between stellar and halo mass can hardly be considered a prediction. We therefore discuss this relation in this section, even though we did not calibrate the simulations to fit the relation inferred from observations. Figure \[fig:eta\] shows the “galaxy formation efficiency”, $(M_\ast / M_{200}) / (\Omega_{\rm b}/\Omega_{\rm m})$, for central galaxies as a function of either the mass of their host halo (left panel) or their stellar mass (right panel). Here the halo mass, $M_{200}$, is defined as the total mass contained within the virial radius $R_{200}$, defined to be the radius within which the mean internal density is 200 times the critical density, $3H^2/8\pi G$, centred on the dark matter particle of the corresponding FoF halo with the minimum gravitational potential (see §\[sec:simulations\]). If the baryon fraction in the halo were equal to the cosmic average of $\Omega_{\rm b}/\Omega_{\rm m} \approx 0.16$, then an efficiency of unity would indicate that the stellar mass accounts for all the halo’s share of baryons. We focus on central galaxies because the strong tidal stripping to which satellite haloes are subject obscures the underlying relation between galaxy formation efficiency and halo mass. The simulation clearly shows that galaxy formation is most efficient in haloes with mass $\sim 10^{12}\,{{{\rm M}_\odot}}$, as has been found by many others. In fact, it would be more appropriate to say that this is the mass where galaxy formation is “least inefficient” as the efficiency is only $\sim 10$% at the peak. The efficiency is sharply peaked at a stellar mass of $\sim 10^{10.4}\,{{{\rm M}_\odot}}$, which corresponds to the onset of the knee in the GSMF (Fig. \[fig:gsmf\]). As is the case for most models of galaxy formation, in [EAGLE]{} the sharp reduction at lower masses is mostly due to stellar feedback, while the drop off at higher masses can in part be attributed to inefficient cooling, but is mostly caused by AGN feedback. Although halo masses can be measured observationally, e.g. from gravitational lensing or satellite kinematics, the errors are still relatively large and it is difficult to disentangle central and satellite galaxies. In Figure \[fig:eta\] we therefore compare with results obtained through the abundance matching technique. In its most basic form abundance matching relates central galaxies to haloes by matching the observed GSMF to the halo mass function predicted from a collisionless simulation, assuming that the stellar masses of galaxies increase monotonically with the masses of their host haloes [e.g. @Vale2004AbundanceMatching]. Modern versions allow for scatter and evolution, and assume that the masses of satellite galaxies are set at the last time they were centrals. Figure \[fig:eta\] compares [EAGLE]{} to the abundance matching results of @Behroozi2013AbundMatching and @Moster2013AbundMatching. Note that the abundance matching studies assumed the WMAP7 cosmology, whereas we assume the Planck cosmology. For [EAGLE]{} we use the total mass of the halo in the hydrodynamical simulation, whereas abundance matching studies use collisionless simulations. Because feedback processes reduce halo masses, we expect $M_{\rm 200}$ to be biased high by $\sim 10$% for the abundance matching results [e.g. @Sawala2013HaloMass; @Cui2014HaloMass; @Velliscig2014HaloMass; @Cusworth2014ClusterMass; @Martizzi2014ClusterMass; @Sawala2014EagleZooms; @Vogelsberger2014Illustris], but this effect is small compared to the dynamic range shown[^10]. Beyond the peak the results become increasingly sensitive to the aperture used to measure the galaxy’s light. For example, @Kravtsov2014GalaxyHalo show that using the @Bernardi2013GSMF GSMF as input increases the efficiency by $\sim 0.5$ dex at $M_{\rm 200} = 10^{14}\,{{{\rm M}_\odot}}$ relative to the values of @Behroozi2013AbundMatching and @Moster2013AbundMatching. However, as discussed in §\[sec:aperture\], our use of a fixed 30 pkpc aperture means that comparison to @Bernardi2013GSMF is inappropriate at the high-mass end. In §\[sec:groups\] we will show that a more robust comparison with observations of the total stellar content of massive galaxies reveals good agreement with [EAGLE]{}. The convergence with resolution is good and the galaxy formation efficiency in [EAGLE]{} is very close to that inferred from abundance matching. This was of course to be expected, given the good convergence and the good agreement with the observations for the GSMF. The peak efficiency is 0.1–0.2 dex lower in [EAGLE]{} and is reached at a slightly ($\sim 0.2$ dex) higher stellar mass, which is consistent with the fact that [EAGLE]{} slightly undershoots the observed GSMF at the knee (see Fig. \[fig:gsmf\]). Galaxy sizes {#sec:sizes} ------------ The parameters of the subgrid model for feedback from star formation and AGN were calibrated to observations of the $z\sim 0$ GSMF. The parameter that controls the importance of the angular momentum of the gas in suppressing BH accretion was set to a value for which AGN feedback causes the GSMF to turn over at a mass similar to what is observed. As will be shown in @Crain2014EagleModels, we found that for [EAGLE]{}, calibration of the stellar feedback is actually unnecessary to reproduce the GSMF. Fixing the amount of energy injected per unit stellar mass to that available in the form of core collapse supernovae, i.e. $f_{\rm th}=1$, works well, as does the physically motivated dependence on the gas metallicity that we use (eq. \[eq:f(Z)\]). However, such models produce galaxies that are far too compact because of excessive radiative losses at high gas densities, and we can show analytically that these spurious cooling losses are caused by our limited numerical resolution (see §\[sec:snii\]). We consider it reassuring that the breakdown of the subgrid model for feedback from star formation at high density is understood and leads to a clear conflict with observations. On the other hand, the fact that such an unrealistic model has no trouble matching the observed GSMF emphasizes the importance of comparing to a wide range of observables. To counteract the numerical radiative losses occurring at high gas densities, we introduced a dependence of the feedback energy from star formation on the gas density, while keeping both the maximum and mean amounts of energy reasonable (see §\[sec:calibration\]). Although we could not afford the computational expense of calibrating the models to fit both the $z\sim 0$ GSMF and the size distribution in detail, we did reject models that produced galaxies that were far too small. As a consequence of this strategy, the $z\sim 0$ galaxy sizes cannot be regarded as true predictions. Figure \[fig:sizes\] plots the median value of the half-mass radius, $R_{50}$, i.e. the radius that encloses 50 per cent of the stellar mass in projection, as a function of galaxy stellar mass. The half-mass radii were determined by fitting Sérsic laws to the projected, azimuthally averaged surface density profiles, as in @McCarthy2012RotSize. Following @Shen2003Sizes, only galaxies with Sérsic index $n_{\rm s}<2.5$ are included. For Ref-L1001504, 94% of the galaxies with more than 600 star particles have $n_{\rm s}<2.5$. The high-resolution Recal-L025N0752 agrees very well with the intermediate-resolution models for $M_\ast > 10^{9}\,{{{\rm M}_\odot}}$, which corresponds to about 600 star particles for the intermediate-resolution runs. For this mass the median $R_{50}$ is about three and a half times the maximum gravitational softening length (see Table \[tbl:sims\]). Hence, we take the stellar mass $600 m_{\rm g}$ as the minimum value for which we can measure half-mass radii. We thus require six times more stellar particles to measure sizes than we need to measure mass. The simulations are compared to data from SDSS [@Shen2003Sizes] and GAMA [@Baldry2012GSMF]. Note that the observations fit surface brightness profiles and provide half-light radii rather than half-mass radii, so the comparison with the models is only fair if the stellar mass-to-light ratio does not vary strongly with radius. As mentioned above, @Shen2003Sizes select galaxies with $n_{\rm s}<2.5$, as we have done here. @Baldry2012GSMF on the other hand present results separately for red and blue galaxies, finding that the latter are $\sim 0.2$ dex more extended at fixed stellar mass. @Shen2003Sizes use Petrosian apertures, which we expect to yield results similar to the 3-D apertures of 30 pkpc that we use for the simulations (see §\[sec:aperture\]). For $M_\ast \gg 10^8\,{{{\rm M}_\odot}}$ @Shen2003Sizes agree better with the @Baldry2012GSMF results for red galaxies, even though $n_{\rm s} < 2.5$ should pick out more disky and hence bluer galaxies. The differences between the two data sets are indicative of the level of correspondence between independent measurements of observed galaxy sizes. For $10^9 < M_\ast/{{{\rm M}_\odot}}< 10^{10}$ the simulation results fall in between those of @Baldry2012GSMF for red and blue galaxies. For $M_\ast< 10^9\,{{{\rm M}_\odot}}$ and $M_\ast> 10^{10}\,{{{\rm M}_\odot}}$ the simulations agree very well with the sizes of blue and red galaxies, respectively. At $10^{11}\,{{{\rm M}_\odot}}$ the red sample of @Baldry2012GSMF gives sizes that are about 0.1–0.2 dex larger than found for both the simulations and the data from @Shen2003Sizes. This difference may be due to the fact that @Shen2003Sizes use Petrosian sizes, whereas @Baldry2012GSMF do not. Indeed, if we do not impose any 3-D aperture, then the simulation curve follows the results of the red sample nearly exactly for $M_\ast \ga 10^{11}\,{{{\rm M}_\odot}}$, while the sizes of lower-mass galaxies remain unchanged (not shown). The agreement with @Shen2003Sizes is excellent: the difference with the simulations is $\le 0.1$ dex for all models and for the full range of stellar mass. For $M_\ast > 10^{10}\,{{{\rm M}_\odot}}$ the scatter in the sizes of the simulated galaxies is similar to the observed dispersion, but at lower masses it appears to be smaller. This could be due to a lack of resolution or some other deficiency in the simulations or halo finder, but it could also be due to observational errors or to the fact that we have ignored variations in the stellar mass-to-light ratio and dust extinction. The relation between BH mass and stellar mass {#sec:magorrian} --------------------------------------------- Figure \[fig:bh\] shows the mass of the central supermassive BH as a function of the galaxy’s stellar mass. The simulation results are compared with the compilation of observations from @McConnell2013BH. The observed stellar mass was obtained by extrapolating a fit to the mass profile of the bulge inferred from kinematic data. Because the observed galaxies were selected to be early-type, the bulge likely dominates the stellar mass, at least for the massive systems. The three [EAGLE]{} simulations give nearly identical results, indicating good convergence. For $M_\ast \ll 10^{10}\,{{{\rm M}_\odot}}$ the BH mass asymptotes to $10^5\,{{{\rm M}_\odot}}/h$, which is the mass of the seed BHs that are inserted into FoF haloes with mass $>10^{10}\,{{{\rm M}_\odot}}/h$ that do not already contain BHs. As can be seen from Fig. \[fig:eta\], a halo mass of $10^{10}\,{{{\rm M}_\odot}}$ corresponds to $M_\ast\sim 10^8\,{{{\rm M}_\odot}}$. Above $M_\ast \sim 10^{10}\,{{{\rm M}_\odot}}$ the relation between BH mass and stellar mass steepens, but it quickly flattens off to a relation that agrees very well with the observations for $M_\ast \ga 10^{11}\,{{{\rm M}_\odot}}$. The rapid growth of the BHs between $M_\ast = 10^{10}$ and $10^{11}\,{{{\rm M}_\odot}}$ coincides with the steepening of the GSMF (compare Fig. \[fig:gsmf\]) and the sharp increase in the fraction of galaxies that are passive (right panel of Fig. \[fig:ssfr\]). This is understandable, as the AGN feedback associated with the rapid BH growth quenches star formation. The agreement with the observations is good, although the observed scatter is larger. In terms of the normalisation of the $M_{\rm BH}$-$M_\ast$ relation the good agreement is perhaps not a surprise. The normalisation is determined by the assumed efficiency of the AGN feedback, $\epsilon_{\rm f}\epsilon_{\rm r}$, i.e. the amount of energy that is injected per unit of accreted mass [e.g. @Booth2009AGN; @Booth2010DMHaloesBHs] . We used the same value ($\epsilon_{\rm f}\epsilon_{\rm r} = 0.015$) as was used for [OWLS]{} and [cosmo-OWLS]{}, which @Booth2009AGN and @LeBrun2014CosmoOWLS found to give agreement with the observed $M_{\rm BH}$-$M_\ast$ relation. Fig. \[fig:bh\] shows that this efficiency also works for [EAGLE]{}, even though the mass resolution of [EAGLE]{} is nearly two orders of magnitude better than for [OWLS]{} and about 3 orders of magnitude better than for [cosmo-OWLS]{}. Note, however, that we used higher AGN heating temperatures than the $\Delta T_{\rm AGN} = 10^8\,{{\rm K}}$ that was used in [OWLS]{} (see Table \[tbl:subgridpars\]). It would clearly be desirable to extend the comparison to observations to lower masses, but in this regime a more careful analysis is required. This is because of the importance of systematic and selection effects for the observations [e.g. @Lauer2007BHBias; @Schulze2011BHBias] and because a bulge-to-disc decomposition would be necessary for the simulations since most low-mass galaxies are disky. The same issues likely also affect the comparison of the scatter. Comparison with other observations {#sec:otherobs} ================================== In this section we will compare the results of [EAGLE]{} to a diverse set of low-$z$ observations of galaxies, galaxy clusters, and the IGM. The results reported in this section were not used to calibrate the subgrid models for feedback and can therefore be considered predictions that can be used as independent consistency checks. During the testing phase, we did look at earlier, more basic versions of some of the plots shown here, so most of the predictions cannot be considered blind. However, we have not adjusted any model parameters to improve the results shown in this section. There are two exceptions to the above statements. First, we plotted the metal column density distributions (§\[sec:cddf\]) for the first time after the simulations had finished, so this was a truly blind prediction. Second, the discrepancy between the gas fraction in clusters predicted by Ref-L100N1504 and inferred from X-ray observations that will be discussed in §\[sec:groups\] was the motivation for running model AGNdT9-L050N0752. This model represents an educated guess in terms of the modifications to the subgrid AGN feedback, because we could only afford to calibrate models using volumes of 25 cMpc on a side, which are too small to contain clusters of galaxies. The observables presented in this section were not selected because the models reproduce them accurately. They were selected because they give a broad overview of the $z\sim 0$ [EAGLE]{} universe, because we had the tools to compute them, and because we are currently not preparing separate papers on them. Future papers will present more observables as well as results for higher redshifts. Specific star formation rates and passive fractions {#sec:ssfr} --------------------------------------------------- The left panel of Figure \[fig:ssfr\] shows the specific star formation rate (SSFR), $\dot{M}_\ast/M_\ast$, of actively star-forming galaxies as a function of stellar mass. Here, galaxies are classified to be star-forming if the ${\rm SSFR} > 0.01~{{\rm Gyr}}^{-1}$, which is indicated by the horizontal, dashed line in the left panel. The higher and lower diagonal lines in the left panel indicate the SSFR corresponding to 10 star-forming gas particles (assuming a gas density of $n_{\rm H}=10^{-1}\,{{\rm cm}}^{-3}$, the star formation threshold that we impose at the metallicity $Z=0.002$) at intermediate and high resolution, respectively. To the left of these curves resolution effects become important, which we indicate by using dotted lines. In particular, the increase in the SSFR at low stellar mass that is clearly visible for the intermediate resolution simulations is a numerical effect: the curves trace lines of constant numbers of star-forming particles. Compared with the intermediate-resolution models, the high-resolution simulation Recal-L025N0752 predicts slightly higher SSFRs. The difference is 0.2 dex at $M_\ast = 10^9\,{{{\rm M}_\odot}}$ and less than 0.1 dex above $10^{10}\,{{{\rm M}_\odot}}$. The models are compared with observations from @Bauer2013ssfr, who measured the SSFRs of $\sim 73,000$ galaxies from the GAMA survey using spectroscopic H$\alpha$ measurements and dust corrections based on Balmer decrements. The intermediate-resolution simulations agree with the data at the high-mass end, but underpredict the SSFR at low masses, reaching a maximum discrepancy of 0.3 – 0.4 dex at $10^9\,{{{\rm M}_\odot}}$. The high-resolution model also underpredicts the SSFR, but the discrepancy is less than 0.2 dex. These differences are comparable to the systematic uncertainty in the data. For example, even for a fixed IMF the systematic uncertainty in the stellar mass, which shifts the data parallel to the diagonal lines, is $\sim 0.3$ dex [@Conroy2009SPSSUncertainty; @Behroozi2010Uncertainties; @Pforr2012SPSSUncertainty; @Mitchell2013MassUncertainty] and the systematic error in the star formation rate, which shifts the data vertically, is likely to be at least as large [e.g. @Moustakas2006SfrIndicators]. The scatter in the simulations is $\sim 50$% smaller than observed, but the observed scatter includes measurement and systematic uncertainties. The right panel of Figure \[fig:ssfr\] shows the fraction of galaxies that are passive as a function of stellar mass. For the simulations we classify galaxies as passive if they have ${\rm SSFR} < 0.01~{{\rm Gyr}}^{-1}$, but the observational papers use somewhat different and varying criteria. We leave a more precise comparison for future work, e.g. using colours and accounting for dust extinction for the simulated galaxies. At low stellar masses the curves become dashed where there are, on average, fewer than 10 star-forming gas particles in a galaxy with ${\rm SSFR} = 0.01~{{\rm Gyr}}^{-1}$. These parts of the curves are unreliable and the upturn of the passive fraction at low mass is thus due to the limited resolution of the simulations. This interpretation is confirmed by the fact that the upturn shifts to eight times lower masses if the particle mass is decreased by a factor of eight, switching from the intermediate resolution Ref-L100N1504 to the high-resolution Recal-L025N0752. For $M_\ast \gg 10^9\,{{{\rm M}_\odot}}$, where the simulations are close to converged, both the simulations and the observations show a strong increase of the passive fraction with mass, from $\sim 10$ per cent at $10^9\,{{{\rm M}_\odot}}$ to $\sim 90$ per cent at $10^{11.5}\,{{{\rm M}_\odot}}$. Relative to the data, the simulation curves are shifted towards higher stellar masses by about 0.3 dex. This difference is similar to the systematic uncertainty in the observed stellar masses. We also find shifts of similar magnitudes if we vary the critical SSFR below which simulated galaxies are classified as passive by a factor of two. We conclude that in the regime where the simulations can be trusted, the predicted SSFRs and passive fractions are slightly lower than the observations but agree with them to within the expected (systematic) errors. Tully-Fisher relation {#sec:tf} --------------------- Figure \[fig:tf\] shows the relation between the maximum of the rotation curve and stellar mass for disc galaxies, i.e. a close relative of the Tully-Fisher relation [@Tully1977TF]. For the simulations we classify galaxies with Sérsic index $n_{\rm s}<2.5$ as late-type, as we did when considering galaxy sizes (§\[sec:sizes\]). We use circular velocities ($v_{\rm c}=\sqrt{GM(<r)/r}$) rather than trying to estimate rotation velocities, since the latter become noisy for galaxies that are not resolved with many particles. The data points with $1\sigma$ error bars correspond to the set of homogenised observations of disc galaxies compiled by @Avila-Reese2008TF and the grey line indicates the median. The stellar masses have been reduced by 0.15 dex, which is necessary to convert to a Chabrier IMF (Avila-Reese, private communication). In addition, following @McCarthy2012RotSize and @Dutton2011TF, we applied a small correction to the stellar masses using the expression given in the appendix of @Li2009GSMF to improve the consistency with those derived from more accurate five-band SDSS data. All simulations track each other very closely, implying excellent numerical convergence. The simulations are in excellent agreement with the data. Over the mass range $10^9 \la M_\ast/{{{\rm M}_\odot}}<10^{11}$ the difference in velocity between the models and the data compiled by @Avila-Reese2008TF is less than 0.03 dex, which is smaller than the 0.1 dex $1\sigma$ error on the fit to the observations. At higher masses, which are only probed by Ref-L100N1504, the difference with the observations increases, reaching 0.12 dex at $M_\ast = 10^{11.3}\,{{{\rm M}_\odot}}$. However, most of these very massive galaxies do not look disky and would probably not be selected by @Avila-Reese2008TF. Note that we have not attempted to analyse the simulations and the data in the same manner, because this would go beyond the scope of the current study. As mentioned above, we use maximum circular velocities, whereas the observations are based on maximum gas rotation velocities, which may show more scatter if the orbits are not all circular. In addition, the observations probe only the inner parts of the halo, whereas we consider the entire halo. @McCarthy2012RotSize found that for the [GIMIC]{} simulations the maximum circular velocities are nearly always reached within two effective radii for $M_\ast \ga 10^{9.5}\,{{{\rm M}_\odot}}$, and should therefore be easily accessible to the observations, but it is possible that for smaller masses the observations underestimate the maximum rotation velocity. Mass-metallicity relations {#sec:Zm} -------------------------- The left panel of Figure \[fig:mz\] shows the metallicity of the ISM, which we take to be star-forming gas for the simulations, as a function of stellar mass. For both the intermediate- and the high-resolution models the gas metallicity increases with stellar mass and flattens off for $M_\ast > 10^{10}\,{{{\rm M}_\odot}}$. However, the high-resolution simulation, Recal-L025N0752, predicts systematically lower metallicities. For $M_\ast \ga 10^{10}\,{{{\rm M}_\odot}}$ the difference is less than 0.15 dex, but it increases with decreasing mass, reaching a maximum of 0.4 dex at $M_\ast \sim 10^{8.5}\,{{{\rm M}_\odot}}$. Because there is no clear mass below which the two resolutions diverge, it is unclear where to put the resolution limit and we therefore have not dotted any part of the curves. Interestingly, model Ref-L025N0752 (not shown) yields a mass-metallicity relation that agrees better with Ref-L100N1504 than the prediction of Recal-L025N0752 does, particularly for $M_\ast < 10^9\,{{{\rm M}_\odot}}$. The high-resolution run again predicts lower metallicities than the intermediate-resolution version, but the maximum difference is smaller than 0.2 dex. For $M_\ast < 10^{7.5}\,{{{\rm M}_\odot}}$ the metallicity is actually lower at intermediate resolution than at high resolution. Hence, for the mass-metallicity relation the strong convergence is considerably better than one might infer from the comparison of Ref-L025N0752 and Recal-L025N0752. Recall that the latter was recalibrated to fit the GSMF, which meant the efficiency of feedback had to be increased relative to the reference model, particularly at $M_\ast \sim 10^9\,{{{\rm M}_\odot}}$ (see Fig. \[fig:gsmf\_conv\]). Apparently, the stronger outflows in Recal-L025N0752 reduce the metallicity of the ISM. Thus, the “strong convergence” is better than the “weak convergence”. This is possible because in this case the weak convergence test compares simulations that were each calibrated to fit the GSMF, not the mass-metallicity relation. The two sets of observations that are shown in the left panel of Figure \[fig:mz\] are both derived from SDSS data. @Tremonti2004Zgas estimated the metallicity statistically based on theoretical model fits to various strong emission lines, while @Zahid2014Zgas derived metallicities using the R23 strong line method as calibrated by @Kobulnicky2004Zgas. The two studies do not agree with each other. In particular, while @Tremonti2004Zgas and @Zahid2014Zgas agree at $M_\ast \sim 10^{11}\,{{{\rm M}_\odot}}$, the former find a steeper relation than the latter, resulting in metallicities that are about 0.2 dex lower for $10^9 - 10^{10}\,{{{\rm M}_\odot}}$. The difference is due to the uncertain calibration of the emission-line diagnostics. In fact, as shown by @Kewley2008ZCalibration, the systematic uncertainty is even larger than suggested by this plot. For example, the empirical calibration of @Pilyugin2005ZCalibration yields a metallicity that is 0.75 dex lower than that of @Tremonti2004Zgas at $10^{11}\,{{{\rm M}_\odot}}$ and an almost flat relation with stellar mass, dropping by only 0.2 dex when the stellar mass decreases to $10^9\,{{{\rm M}_\odot}}$. Besides the calibration issues, the gas phase abundance likely underestimates the total metallicity of the ISM because a non-negligible fraction of the metals may condense onto dust-grains [e.g. @Dwek1998Dust; @Mattsson2012Dust]. Finally, the systematic uncertainty in the stellar mass, for a fixed IMF, is about 0.3 dex [e.g. @Conroy2009SPSSUncertainty]. The metallicities predicted by the simulations are also subject to significant systematic uncertainties unrelated to the galaxy formation physics. Even for a fixed IMF, the nucleosynthetic yields are uncertain at the factor of two level [e.g. @Wiersma2009Chemo]. However, we choose not to simply re-scale the simulation metallicities within this uncertainty because that would make them inconsistent with the radiative cooling rates used during the simulation. Given the large systematic uncertainties in both the normalisation and the shape of the observed mass-metallicity relation, and the systematic uncertainties in the yields adopted in the simulations, care needs to be taken when comparing the models and the data. We will nevertheless proceed to make such a comparison. The median mass-metallicity relations predicted by the intermediate-resolution simulations agree with @Zahid2014Zgas to better than 0.2 dex at all masses and to better than 0.1 dex for $M_\ast > 10^{9.5}\,{{{\rm M}_\odot}}$, but the observed relation is steeper at lower masses. The predicted scatter is larger than observed by @Tremonti2004Zgas, particularly for the highest masses. The scatter in the gas metallicity of these massive objects is large in the simulations because they typically contain very few star-forming gas particles. This causes strong sampling effects and large variations in time following AGN outbursts. The median metallicity predicted by the high-resolution model Recal-L025N0752 matches @Tremonti2004Zgas to better than 0.2 dex over the full mass range covered by both the simulation and the observations ($10^{8.5} < M_\ast/{{{\rm M}_\odot}}< 10^{11}$) and to better than 0.1 dex for $M_\ast > 10^{9.2}\,{{{\rm M}_\odot}}$. Apparently, the increase in the efficiency of energy feedback from star formation that is required to make the GSMF fit the observations (and which was implemented by changing the density dependence of the efficiency, see §\[sec:calibration\]), simultaneously decreases the metallicity of the ISM of low-mass galaxies to the values observed by @Tremonti2004Zgas. The predicted relations between stellar metallicity and mass are shown in the right panel of Figure \[fig:mz\] and compared with observations from SDSS from @Gallazzi2005Zstars and for dwarf galaxies from @Kirby2013Zstars. The trends and differences largely parallel those seen for the gas-phase abundances in the left panel. For $M_\ast \ga 10^9\,{{{\rm M}_\odot}}$ simulation Recal-L025N0752 is relatively close to the data, but at lower masses all models predict higher metallicities than observed by @Kirby2013Zstars. As was the case for the gas metallicity, the (strong) convergence is actually much better than suggested by this figure. For $M_\ast > 10^{7.5}\,{{{\rm M}_\odot}}$ simulation Ref-L025N0752 (not shown) predicts a stellar metallicity that is lower, but within 0.1 dex of the metallicity predicted by Ref-L100N1504. Model AGNdT9-L050N0752 predicts slightly higher metallicities than Ref-L100N1504 for $M\gg 10^{10}\,{{{\rm M}_\odot}}$, which agrees better with the data. The main difference between the conclusions that can be drawn from the gas and stellar metallicities concerns the scatter. While the scatter in the gas phase abundances was overestimated in the simulations, the scatter in the stellar abundances appears to be strongly underestimated. However, it would be surprising for the scatter in the observed stellar metallicity to be so much larger than the observed scatter in the gas phase metallicity, which suggests that the scatter in the observed stellar metallicities may be dominated by errors. Indeed, while the mean relation from the CALIFA integral field survey is close to that of @Gallazzi2005Zstars, the scatter is about a factor of two smaller [@Gonzalez2014CalifaZM]. X-ray observations of the intracluster medium {#sec:groups} --------------------------------------------- In this section we will consider some parameters that are commonly measured from X-ray observations of the intragroup and intracluster gas. The comparison to observations is more like-for-like than in previous sections, because all simulation results are derived by applying observational analysis techniques to virtual X-ray observations of the simulations. Simulation Recal-L025N0752 is not considered here because the simulation box is too small to produce clusters of galaxies. The methods used to generate the plots are identical to those employed for [cosmo-OWLS]{} in @LeBrun2014CosmoOWLS and we refer the reader to §2.2 of that paper for details. Briefly, gas density, temperature and metallicity profiles are determined by fitting single temperature, single metallicity “Astrophysical Plasma Emission Code” (APEC) [@Smith2001APEC] models to synthetic Chandra X-ray spectra in three-dimensional radial bins centred on the minimum of the gravitational potential in the halo. Mass profiles are obtained by fitting the functions proposed by @Vikhlinin2006 to the density and temperature profiles and assuming hydrostatic equilibrium. We then determine the radius within which the mean internal density equals 500 times the critical density, $R_{500, {\rm hse}}$ , and the corresponding spherical overdensity mass,$M_{500, {\rm hse}}$. We will use the subscript “hse” to indicate that the quantity has been inferred from virtual observations under the assumption of hydrostatic equilibrium (which holds only approximately, see @LeBrun2014CosmoOWLS and references therein). Mean X-ray temperatures and elemental abundances within $R_{500, {\rm hse}}$ are determined by fitting APEC models to a single radial bin. We include all $z=0$ haloes with FoF mass $ > 10^{12.5}\,{{{\rm M}_\odot}}$ but plot only results for haloes with $M_{500, {\rm hse}} > 10^{13}\,{{{\rm M}_\odot}}$ for which the correspondence between $M_{\rm 500}$ and $M_{500, {\rm hse}}$ is good for most objects, except that $M_{500, {\rm hse}}$ is systematically biased low by $\sim 20$ per cent [see Fig. B1 of @LeBrun2014CosmoOWLS]. Figure \[fig:Lopt\_groups\] shows the (Cousins) $I$-band luminosity within $R_{500,{\rm hse}}$ as a function of $M_{500,{\rm hse}}$. Each point corresponds to a single simulated or observed object. The predicted luminosity-mass relation matches the observations very well. As the $I$-band luminosity is a proxy for stellar mass and the simulations were calibrated to the observed GSMF, this may at first sight not be surprising. However, the high-mass tail of the GSMF was not calibrated to any observations, because the test simulations were too small to contain such rare objects. Moreover, here we plot the total luminosity within $R_{500}$, a radius that exceeds the aperture used for the GSMF by more than an order of magnitude. Hence, the results shown here include contributions from satellites and the intracluster light, both for the observations and simulations. Figure \[fig:fgas\_groups\] shows the gas mass fraction, $M_{\rm gas,500,hse}/M_{\rm 500,hse}$ as a function of mass $M_{\rm 500,hse}$. Because the gas mass is derived from the (virtual) X-ray data, it only correctly accounts for gas that has a temperature similar to that of the gas that dominates the X-ray emission. For the reference model the gas mass inferred from X-ray observations, under the assumption of hydrostatic equilibrium, is about 0.2 dex higher than observed, except perhaps for the two most massive objects. @LeBrun2014CosmoOWLS have shown that the gas fraction is particularly sensitive to the temperature to which the AGN heat the surrounding gas in our subgrid prescription for AGN feedback. In particular, higher heating temperatures, which correspond to more energetic but less frequent bursts, eject the gas more effectively, yielding lower gas fractions. This was the motivation for running model AGNdT9-L050N0752, which uses a heating temperature $\Delta T_{\rm AGN}$ of $10^9\,{{\rm K}}$, compared with $10^{8.5}\,{{\rm K}}$ for the reference model. Before running this model, we used a 25 cMpc version to (approximately) recalibrate the BH accretion model so as to maintain the good match with the GSMF, in particular the location of the knee. We could, however, not afford to run multiple 50 cMpc models and could therefore not calibrate to observations of groups of galaxies. As can be seen from Figure \[fig:fgas\_groups\], contrary to model Ref-L100N1504, model AGNdT9-L050N0752 does appear to reproduce the observations of group gas fractions. That is, for $M_{500,{\rm hse}} < 10^{13.5}\,{{{\rm M}_\odot}}$ the simulation points agree with an extrapolation of the observations for high-mass systems. There is a strong hint that the gas fraction may again become too high for more massive clusters, although with only 1 object with $M_{500,{\rm hse}}> 10^{13.5}\,{{{\rm M}_\odot}}$ it is hard to judge the significance of this deviation. @LeBrun2014CosmoOWLS found that the [cosmo-OWLS]{} simulations, which use $2\times 1024^3$ particles in $400~{h^{-1}\,{\rm cMpc}}$ volumes, reproduce these and many other observations of groups and clusters over the full mass range of $10^{13}-10^{15}\,{{{\rm M}_\odot}}$ for $\Delta T_{\rm AGN} = 10^8\,{{\rm K}}$. This may seem surprising given that [EAGLE]{} requires higher values of $\Delta T_{\rm AGN}$. Note, however, that because the particle mass in [cosmo-OWLS]{} is more than 3 orders of magnitudes larger than for [EAGLE]{}, the energy in individual AGN feedback events in [cosmo-OWLS]{} is still much larger than that in AGNdT9-L050N0752. Figure \[fig:Lx\_groups\] shows the X-ray luminosity in the 0.5–2.0 keV band as a function of the temperature measured from the (virtual) X-ray data. For the reference model the agreement with the observations is reasonably good at low temperatures (the lack of simulated points with $L \ll 10^{42}\,{{\rm erg}}\,{{\rm s}}^{-1}$ is due to the fact that we only selected systems with $M_{500,{\rm hse}}> 10^{13}\,{{{\rm M}_\odot}}$), but the predicted luminosity is about a factor of three too high above 1 keV. Model AGNdT9-L050N0752 appears to match the data well, but more objects with $k_{\rm B}T> 1~{\rm keV}$ are needed to better assess the degree of correspondence. Column density distributions of intergalactic metals {#sec:cddf} ---------------------------------------------------- The galactic outflows that we invoke to reproduce observations of galaxies also disperse heavy elements into the IGM. Furthermore, the winds shock-heat the gas, which may, in turn, change its ionisation balance. Hence, it is interesting to compare the predicted distribution of intergalactic metal ions to the observations. This is a strong test for the model, since the subgrid feedback was only calibrated to match the stellar properties of galaxies. Figure \[fig:cddf\] compares the predicted column density distribution functions (CDDFs) of (left panel) and (right panel) with measurements derived from quasar absorption line observations, mainly from the Hubble Space Telescope (HST). Note that this prediction was completely blind. The CDDF is conventionally defined as the number of absorbers per unit column density, $N$, and per unit absorption distance, $dX$. The number of absorbers per unit absorption distance is obtained from the quantity that is actually observed, the number of absorbers per unit redshift, via $dX = dz (H_0/H(z))(1+z)^2$. The redshift ranges of the observations vary and are indicated in the legend. All observations are for $z<1$ and most for much lower redshift. For clarity we only show the simulation results for our $z=0.27$ snapshots. However, limiting the comparison to $z=0.27$ does not affect our conclusions because the evolution is weak. For the simulations we compute ion fractions for each gas particle using <span style="font-variant:small-caps;">cloudy</span> photoionisation models, assuming the gas is in ionisation equilibrium and exposed to the @Haardt2001UVB model for the UV/X-ray background from galaxies and quasars. We then obtain the CDDF by projecting the simulation cube onto a 2-D grid and applying SPH interpolation to compute the ion column density in each cell. We use a grid cell size of 10 ckpc, which is sufficiently small to obtain convergence, and have verified that projection effects are negligible by comparing results obtained from simulations using different box sizes. Observationally, the CDDF is obtained by decomposing the identified absorption features into Voigt profiles and grouping those into systems using criteria that differ between observers and that are not always well defined. We intend to mimic the observational procedures more closely in future work. From Figure \[fig:cddf\] it is clear that the differences between different sets of observations exceed the reported statistical uncertainties, suggesting the presence of significant systematic errors. Particularly for , the analysis of COS spectra by @Danforth2014LowzAbs yields systematically more absorbers than the earlier analyses of STIS/FUSE/GHRS data by @Danforth2008LowzAbs, @Thom2008OVI, and @Tripp2008OVI. As discussed in §\[sec:Zm\], even for a fixed IMF the nucleosynthetic yields are uncertain at the factor of two level [e.g. @Wiersma2009Chemo]. This suggests that we are free to rescale the metal column densities, i.e. to shift the curves in Figure \[fig:cddf\] horizontally by up to 0.3 dex. However, doing so would break the self-consistency of the simulations as the metal abundances determine the cooling rates. The simulation predictions generally agree well with the data, falling in between the different sets of observations, both for and . The simulations appear to produce too few ultra-strong absorbers, i.e. systems with column densities $\sim 10^{15}\,{{\rm cm}}^{-2}$. However, the frequency of these extremely rare systems is particularly sensitive to systematics and hence requires a more careful comparison. For the difference between Ref-L100N1504 and Recal-L025N0752 is substantial for $N_{{\hbox{\scriptsize O\,{\tiny VI}}}} \sim 10^{14}\,{{\rm cm}}^{-2}$ with the high-resolution model yielding up to a factor of 3 more absorbers. However, this does not lead to any disagreement with the data as all simulations fall in between the different sets of observations. Recall that in low-mass galaxies feedback from star formation is more effective in the recalibrated, high-resolution model Recal-L025N0752 than in the reference model. It is interesting that while this boost in the feedback efficiency decreases the metallicity of the ISM (Fig. \[fig:mz\]), it boosts the abundance of metal ions in the IGM. It is tempting to conclude that the more effective feedback transports more metals from galaxies into the IGM. However, whether this is the case is not clear from the results presented here due to the importance of ionisation corrections. In future work we will compare with high-redshift data and with absorption line observations of the gas around galaxies of known mass. For now, we are encouraged by the fact that a model that was calibrated to the GSMF and galaxy sizes, also yields good agreement with observations of intergalactic metals. Summary and discussion {#sec:summary} ====================== We have introduced the [EAGLE]{} project, where [EAGLE]{} stands for “Evolution and Assembly of GaLaxies and their Environments”. [EAGLE]{} consists of a suite of large, hydrodynamical cosmological simulations. In this introductory paper we have focused on a set of simulations for which the subgrid parameters for feedback were calibrated to match the observed $z\sim 0$ galaxy stellar mass function (GSMF), subject to the constraint that the galaxy sizes must also be reasonable. @Crain2014EagleModels will present models in which the subgrid physics is varied. The largest [EAGLE]{} simulation, Ref-L100N1504, uses nearly 7 billion ($2\times 1504^3$) particles in a 100 cMpc box. This corresponds to an initial baryonic particle mass of $1.8\times 10^6\,{{{\rm M}_\odot}}$ and a force resolution of 0.7 proper kpc (smaller at high redshift), which we refer to as “intermediate resolution”. The resolution was chosen to marginally resolve the Jeans scales in the warm ($T\sim 10^4\,{{\rm K}}$) ISM. The high-resolution model, Recal-L025N0752, has eight times better mass resolution and two times better spatial resolution, thus resolving a galaxy like the Milky Way with $\sim 10^6$ particles. The simulations were run with the code <span style="font-variant:small-caps;">gadget</span> 3, but with a modified implementation of SPH, the time stepping, and the subgrid models. The simulations include subgrid prescriptions for (element-by-element) radiative cooling, star formation, stellar evolution and mass loss, energy feedback from star formation, the growth of supermassive BHs, and AGN feedback. The prescription for star formation accounts for the observation that stars form from molecular clouds and that the -H$_2$ transition depends on metallicity. The subgrid model for accretion onto BHs accounts for the fact that angular momentum suppresses the accretion rate. The most critical parts of the model are the implementations of energy feedback from star formation and AGN. We argued that present-day simulations of representative volumes cannot predict the efficiency of the feedback processes from first principles because of their reliance on subgrid models, because of spurious radiative losses due to the limited resolution, and because they lack the resolution and do not include all the physics necessary to model the structure of the ISM. We discussed some of the implications of the inability to predict the efficiency of the feedback from first principles. We argued that current cosmological simulations can predict neither BH nor stellar masses, which implies that the subgrid models for feedback need to be calibrated to observations. Another consequence is that it is difficult to distinguish different physical feedback mechanisms that operate nearly simultaneously, such as winds driven by supernovae and radiation pressure. Furthermore, unless one can demonstrate that the model does not suffer from overcooling due to limited numerical resolution, one cannot conclude that there is a need for a new, physical feedback process just because the implemented feedback is insufficiently effective. Because the spurious radiative losses depend on the resolution, one may have to recalibrate when the resolution is changed. We termed this “weak convergence” as opposed to the “strong convergence” that corresponds to the same physical model giving consistent results at different resolutions. However, we argued that most subgrid models for feedback effectively change with resolution even if the subgrid parameters are kept constant. The quest for strong convergence of simulations that lack the resolution to model the ISM has led to significant sacrifices, which generally involve disabling aspects of the hydrodynamics during feedback. Examples include temporarily turning off radiative cooling, temporarily turning off hydrodynamical forces, and making the feedback efficiency dependent on dark matter velocity dispersion rather than on local properties of the gas. However, until the cooling losses can be predicted, even fully converged simulations will be unable to predict stellar and BH masses from first principles. We therefore prefer to minimize the sacrifices and to opt for weak convergence. Nevertheless, we demonstrated that the strong convergence of our model is reasonably good (Fig. \[fig:gsmf\_conv\]). Motivated by the above considerations, we chose to keep our subgrid models for feedback as simple as possible. We employ only one type of stellar feedback and hence we do not distinguish between stellar winds, radiation pressure, and core collapse supernovae. Similarly, we include only one type of AGN feedback and therefore do not implement separate “quasar” and “radio modes”. We find that a more complex approach is not required to match observational data. We implement both feedback from star formation and AGN thermally using the stochastic prescription of @DallaVecchia2012Winds. By injecting the energy stochastically rather than at every time step, we can specify both the temperature jump of the heated gas and the expectation value for the amount of energy that is injected. This enables us to better mimic the physical conditions associated with observed feedback processes, in particular the high heating temperatures that suppress the initial radiative losses, than would otherwise be possible given the limited resolution of the simulations. The velocities and mass loading factors of galactic winds are thus not imposed, but are an outcome of the simulation. The temperature jump associated with feedback events is chosen to balance the need to minimize both the initial, radiative losses (which are largely numerical) and the time between feedback events (to allow for self-regulation). The probability of heating events then needs to be calibrated by comparing the simulation results for some observable to real data. The subgrid efficiency of the AGN feedback, i.e. the expectation value for the amount of energy that is injected into the ISM per unit of accreted gas mass, is constant and was chosen to match the normalisation of the observed relation between the masses of galaxies and their central supermassive BHs. This parameter is, however, unimportant for observables other than the masses of BHs. The subgrid efficiency of the feedback from star formation, $f_{\rm th}$, i.e. the expectation value for the amount of energy that is injected into the ISM in units of the energy available from core collapse supernovae, was chosen to reproduce the observed GSMF for $M_\ast < 10^{10.5}\,{{{\rm M}_\odot}}$, i.e. below the knee of the Schechter function. Finally, the value of the parameter that controls the sensitivity of the BH accretion rate to the angular momentum of the surrounding gas was adjusted to make the mass function turn over at the onset of the exponential drop of the observed GSMF. We made $f_{\rm th}$ a function of both metallicity and density. We use a physically motivated metallicity dependence with $f_{\rm th}$ dropping when the metallicity is increased from values $\ll 0.1Z_\odot$ to $\gg 0.1 Z_\odot$. This reduction in the efficiency is meant to capture the increase in radiative losses that is expected when metal-line cooling becomes important, which happens for $Z > 0.1 Z_\odot$ at the temperatures relevant for gas shock-heated in galactic winds [e.g. @Wiersma2009Cooling]. While a constant value of $f_{\rm th}=1$, or a pure metallicity dependence, each give an excellent fit to the GSMF, they result in galaxies that are far too compact [@Crain2014EagleModels]. This happens because, at the resolution of [EAGLE]{}, the stochastic implementation for stellar feedback is still subject to numerical radiative losses at high gas densities, as we demonstrated analytically. To compensate for these spurious losses, we increase $f_{\rm th}$ at high gas densities. However, $f_{\rm th}$ never exceeds 3 and the mean value is smaller than 1.1. We compared [EAGLE]{} to a diverse set of observations of the low-redshift Universe, carefully distinguishing between observations that were considered during the calibration (the GSMF and thus also the directly related $M_\ast - M_{200}$ relation, galaxy sizes, and the $M_{\rm BH} - M_\ast$ relation) and those that were not. We came to the following conclusions: - The observed GSMF is reproduced over the range $10^8 < M_\ast/{{{\rm M}_\odot}}\la 10^{11}$. At fixed mass, the difference in number density relative to the data is $\la 0.2$ dex. At fixed number density, the difference in mass is smaller than 0.3 dex (Fig. \[fig:gsmf\]). Even for a fixed IMF, this discrepancy is comparable to the systematic uncertainty in the observed masses due to stellar evolution alone. This level of agreement with the data is close to that obtained by semi-analytic models and is unprecedented for hydrodynamical simulations (Fig. \[fig:gsmf\_other\]). - Three-dimensional apertures of 30 proper kpc, which we used throughout the paper, give results close to the Petrosian masses that are often used for observations, e.g. by SDSS. For $M_\ast > 10^{11}\,{{{\rm M}_\odot}}$ larger apertures yield higher masses (Fig. \[fig:gsmf\_aperture\]). - The stellar mass - halo mass relation for central galaxies is close to that inferred from abundance matching. The efficiency of galaxy formation, $M_\ast/M_{200}$, peaks at the halo mass $M_{200} \sim 10^{12}\,{{{\rm M}_\odot}}$ and at the stellar mass $M_\ast \sim 10^{10.4}\,{{{\rm M}_\odot}}$ (Fig. \[fig:eta\]). - Disc galaxy sizes are well matched to the observations. Over the full range of stellar mass, $10^8 < M_\ast/{{{\rm M}_\odot}}< 10^{11.5}$, the median stellar half-mass radii of late-type galaxies agree with the observed half-light radii to within 0.1 dex (Fig. \[fig:sizes\]). - The median relation between BH mass and stellar mass agrees with the observations, but the scatter in the model is smaller than observed. The simulations predict that galaxies with total stellar masses of $10^9-10^{10}\,{{{\rm M}_\odot}}$ typically host BHs with masses that fall below the extrapolation of the high-mass power-law relation (Fig. \[fig:bh\]). - The predicted relation between the median specific star formation rate ($\dot{M}_\ast/M_\ast$; SSFR) and stellar mass for star-forming galaxies, i.e. the “main sequence of star formation”, agrees with the observations to within 0.2 dex over the observed range of $10^9 < M_\ast/{{{\rm M}_\odot}}<10^{11}$ at high-resolution and to within 0.35 dex at intermediate resolution (Fig. \[fig:ssfr\], left panel). - The predicted fraction of galaxies that are passive, which we define as SSFR $< 10^{-2}\,{{\rm Gyr}}^{-1}$ for the simulations, increases sharply with stellar mass between $10^{10}$ and $10^{11.5}\,{{{\rm M}_\odot}}$, in agreement with the observations (Fig. \[fig:ssfr\], right panel). - The predicted median relation between the maximum of the rotation curve and stellar mass of late-type galaxies, i.e. a close analogue of the Tully-Fisher relation, agrees with the observations to better than 0.03 dex over the observed mass range of $10^9 \la M_\ast/{{{\rm M}_\odot}}<10^{11}$ (Fig. \[fig:tf\]). - The relations between ISM metallicity and stellar mass and between stellar metallicity and stellar mass are predicted to flatten with stellar mass. For the gas the predicted median metallicities agree with the observed values to within 0.1 dex for $M_\ast > 10^{9.5}\,{{{\rm M}_\odot}}$ at intermediate resolution and down to the lowest observed mass, $M_\ast \sim 10^{8.5}\,{{{\rm M}_\odot}}$, at high resolution. At lower masses the predicted relations are less steep than extrapolations of the observed trends. For the stellar metallicities the discrepancies are larger. For $M_\ast > 10^{10}\,{{{\rm M}_\odot}}$ all simulations agree with the data to better than 0.2 dex, but the difference increases with decreasing mass. At $M_\ast \sim 10^8\,{{{\rm M}_\odot}}$ the stellar metallicities in the intermediate- and high-resolution simulations are higher than observed by about 0.7 and 0.3 dex, respectively. - For the mass-metallicity relations the strong convergence is significantly better than the weak convergence, i.e. simulations that keep the subgrid parameters fixed converge better with numerical resolution than simulations for which the feedback is (re)calibrated to the $z\sim 0$ GSMF at each resolution. Hence, the increase in the efficiency of the feedback from star formation that was applied at high resolution in order to match the observed GSMF, simultaneously steepens the $Z(M_\ast)$ relations, improving the agreement with the data. - A comparison to observations of groups and clusters of galaxies with $M_{500,{\rm hse}} > 10^{13}\,{{{\rm M}_\odot}}$, where the subscript “hse” indicates that the quantity was estimated from virtual observations under the assumption of hydrostatic equilibrium, revealed that: - The predicted relation between the total $I$-band light within $R_{500,{\rm hse}}$ and $M_{500,{\rm hse}}$ agrees with the data. Note that this includes contributions from satellites and intracluster light (Fig. \[fig:Lopt\_groups\]). - The gas mass fractions, $M_{\rm gas,500,hse}/M_{\rm 500,hse}$, are overestimated by about 0.2 dex in the reference model. For $M_{\rm 500,hse} < 10^{13.5}\,{{{\rm M}_\odot}}$ this can be remedied by increasing the subgrid AGN heating temperature, as implemented in model AGNdT9-L050N0752. At higher masses this change may be insufficient, although larger simulation volumes are needed to confirm this (Fig. \[fig:fgas\_groups\]). - The reference model predicts soft X-ray luminosities that are about 0.5 dex higher than observed for clusters with spectroscopic temperatures $\sim 1$ keV. However, model AGNdT9-L050N0752 is consistent with the observations (Fig. \[fig:Lx\_groups\]). - The column density distributions of intergalactic and are in good agreement with the data, falling in between the results obtained by different surveys (Fig. \[fig:cddf\]). Hence, in the resolved mass range, which spans $10^9 \la M_\ast/{{{\rm M}_\odot}}\la 10^{11}$ for some observables and $10^8 \la M_\ast/{{{\rm M}_\odot}}\la 10^{11}$ for others, [EAGLE]{} agrees with a diverse set of low-redshift observations of galaxies. At the same time, [EAGLE]{} reproduces some key observations of intergalactic metals. The only discrepancies found in this work that substantially exceed observational uncertainties concern the gas and stellar metallicities of dwarf galaxies, which are too high, and the predictions of the reference model for X-ray observations of the intracluster medium. The metallicity problem is only substantial at intermediate resolution, so it is possible that it can be resolved simply by increasing the resolution further. We already demonstrated that the problem with groups of galaxies can be remedied by increasing the heating temperature used in the subgrid model for AGN feedback, as implemented in model AGNdT9-L050N0752, without compromising the successes of the reference model. However, larger volumes are needed to judge whether the increase in the heating temperature that was implemented in this model suffices to obtain agreement with the data for massive ($M_{500} \ga 10^{14}\,{{{\rm M}_\odot}}$) clusters of galaxies. In future papers we will test many more predictions of [EAGLE]{}. Although we will undoubtedly uncover problems, so far we have no reason to believe that the results shown here are unrepresentative. We will show that the success of [EAGLE]{} extends to other areas that have in the past proven to be challenging for hydrodynamical simulations, such as the bimodal distribution of galaxies in colour-magnitude diagrams. We will also demonstrate that the relatively good agreement with the data is not limited to low redshift. In addition to further exploring the models that have been presented here, we plan to use the larger suite of physical models presented in @Crain2014EagleModels to gain insight into the physical processes underlying the observed phenomena. Finally, we have already begun to carry out higher-resolution resimulations of individual structures [e.g. @Sawala2014EagleZooms; @Sawala2014ChosenFew] with the code used for [EAGLE]{}. Although the relatively good agreement between [EAGLE]{} and the observations, as well as that between other recent, hydrodynamical simulations of representative volumes and the data [e.g. @Vogelsberger2014IllustrisNature], is encouraging, we should keep in mind that we have not attempted to model many of the physical processes that may be important for the formation and evolution of galaxies. For example, [EAGLE]{} does not include a cold interstellar gas phase, radiation transport, magnetohydrodynamics, cosmic rays, conduction, or non-equilibrium chemistry, and [EAGLE]{} does not distinguish between different forms of energy feedback from star formation and between different forms of AGN feedback. We argued that at present there are good reasons for such omissions, but many of those arguments would no longer apply if the numerical resolution were increased by several orders of magnitude. While it will take some time for simulations of representative volumes to attain the resolution that is required to model the cold ISM, simulations of individual objects can already do much better. Ultimately, simulations should be able to predict the efficiencies of the most important feedback processes and hence to predict, rather than calibrate to, the GSMF. We hope that [EAGLE]{} will prove to be a useful resource for the community.[^11] The agreement with observations is sufficiently good for the simulations to be used in ways that have so far been reserved for semi-analytic models of galaxy formation. At the same time, because hydrodynamical simulations provide much more detailed 3-D information, make fewer simplifying assumptions, and simultaneously model the galaxies and the IGM, [EAGLE]{} enables one to ask many questions that are difficult to investigate with semi-analytic models. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Volker Springel for sharing his codes and for his advice and discussions. We gratefully acknowledge discussions with Jarle Brinchmann, Shy Genel, Justin Read, Debora Sijacki. We are also thankful to Martin Bourne and Laura Sales for their contributions to the initial phase of the project, Amandine Le Brun for her help with the X-ray plotting routines, Peter Draper and Lydia Heck for their help with the computing resources in Durham, and to Wojciech Hellwing for help with computing in Poland. We are also grateful to all the people working on the analysis of the [EAGLE]{} simulations and would like to thank the anonymous referee for a constructive report. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. We also gratefully acknowledge PRACE for awarding us access to the resource Curie based in France at Très Grand Centre de Calcul. This work was sponsored by the Dutch National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from the Netherlands Organization for Scientific Research (NWO) and by the HPC Infrastructure for Grand Challenges of Science and Engineering Project, co-financed by the European Regional Development Fund under the Innovative Economy Operational Programme and conducted at the Institute for Mathematical and Computational Modelling at University of Warsaw. The research was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreements 278594-GasAroundGalaxies, GA 267291 Cosmiway, and 321334 dustygal, the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (\[AP P7/08 CHARM\]), the National Science Foundation under Grant No. NSF PHY11-25915, the UK Science and Technology Facilities Council (grant numbers ST/F001166/1 and ST/I000976/1), Rolling and Consolodating Grants to the ICC, Marie Curie Reintegration Grant PERG06-GA-2009-256573. Hydrodynamics {#app:hydro} ============= Recently, much effort has been directed at solving a well-known issue with the standard SPH implementation: multi-valued particle pressure and large artificial viscosity causing unphysical surface tension at contact discontinuities [for a detailed description of the problem see, e.g. @Agertz:2007fk]. This surface tension impedes the development of hydrodynamical instabilities resulting in poor mixing of gas phases, which could in principle compromise simulations of galaxy formation [e.g. @Sijacki2012Hydro; @Nelson2013GasAccretion]. Several solutions have been suggested in order to smooth the pressure at contact discontinuities [e.g. @Ritchie:2001fk; @Price:2008kx; @Read:2010uq; @Saitoh:2013uq; @Hopkins:2013lr], and to reduce the artificial viscosity away from shocks [e.g. @Morris:1997fj; @Cullen:2010qy]. As described in more detail below, we employ the fully conservative SPH formulation derived by @Hopkins:2013lr, of which the solutions suggested by @Ritchie:2001fk, @Read:2010uq and @Saitoh:2013uq are special cases. We use the artificial viscosity switch from @Cullen:2010qy and a switch for artificial conduction similar to that of @Price:2008kx. We apply the time step limiters of @Durier:2012fj. We adopt the $C^2$ @Wendland:1995 kernel with $N_{\rm ngb} = 58$ neighbours. This kernel inhibits particle pairing [@Dehnen:2012uq] and the number of neighbours was chosen to give an effective resolution that is close to that of the cubic spline kernel with 48 neighbours that was used in [OWLS]{}. The methods used here are collectively referred to as “Anarchy” and will be described in more detail in Dalla Vecchia (in preparation), who also demonstrates its performance on standard hydrodynamical tests. In @Schaller2014EagleSPH we compare the results of [EAGLE]{} cosmological simulations with different hydrodynamics and time stepping schemes. Consistent with previous work [e.g. @Scannapieco2012Aquila], we find that our results are generally substantially less sensitive to changes in the hydrodynamical techniques than to reasonable variations in the subgrid physics. SPH {#sec:sph} --- Following [@Hopkins:2013lr], the generalised equation of motion is $$m_i\frac{{\rm d}\mathbf{v}_i}{{\rm d}t}=-\sum_{j=1}^N x_i x_j\left[ \frac{P_i}{y_i^2}f_{ij}\nabla_i W_{ij}(h_i) + \frac{P_j}{y_j^2}f_{ji}\nabla_i W_{ij}(h_j) \right], \label{eqmot}$$ where $m_i$, $\mathbf{v}_i$ and $P_i$ are the particle mass, velocity and pressure, respectively; $W_{ij}$ is the SPH kernel; $h_i$ is the SPH smoothing length; $f_{ij}$ is the correction term for variable smoothing lengths (the so-called “grad-$h$” term), given by $$f_{ij} = 1 - \frac{\tilde{x}_j}{x_j}\left(\frac{h_i}{n_D \tilde{y}_i}\frac{\partial y_i}{\partial h_i}\right) \left[1 + \frac{h_i}{n_D \tilde{y}_i}\frac{\partial \tilde{y}_i}{\partial h_i}\right]^{-1}, \label{fij}$$ where $n_{\rm D}$ is the number of spatial dimensions. In the above equations, $\tilde{x}_i$ and its SPH smoothed value, $\tilde{y}_i=\sum_j \tilde{x}_j W_{ij}(h_i)$, define the particle volume, $\tilde{V}_i=\tilde{x}_i/\tilde{y}_i$. The particle smoothing length is defined by the relation $$\frac{4\pi}{3} h_i^3=N_{\rm ngb} \tilde{V}_i ,$$ where $N_{\rm ngb}$ is the number of neighbouring particles.[^12] In our implementation we chose $\tilde{x}_i=m_i$ and $\tilde{y}_i\equiv\rho_i=\sum_j m_j W_{ij}(h_i)$, the SPH particle density. The remaining quantities, $x_i$ and $y_i=\sum_j x_j W_{ij}(h_i)$, define the “thermodynamical volume”, and can be chosen in order to obtain a smooth representation of the pressure. Since we follow the evolution of the gas pseudo entropy, $A \equiv P/\rho^\gamma$, the natural choice is then $x_i=m_i A_i^{1/\gamma}$ and $y_i\equiv \bar{P}_i^{1/\gamma}=\sum_{j=1}^N m_j A_j^{1/\gamma} W_{ij}(h_i)$ as suggested by [@Read:2010uq]. With this definition, the weighted pressure, $\bar{P}_i$, is now single-valued and varies smoothly through contact discontinuities. In practice, it is convenient to define a weighted density that can be used in the conversion between thermodynamical quantities (entropy, internal energy, temperature) and that can be predicted for inactive particles. We define the weighted density by writing the entropic function, $P=A\rho^{\gamma}$, as follows: $$\bar{P}_i = A_i \left(\frac{1}{A_i^{1/\gamma}}\sum_{j=1}^N m_j A_j^{1/\gamma} W_{ij}(h_i)\right)^{\gamma} = A_i \bar{\rho}_i^{\gamma}.$$ Note that this definition of the density is the only one that is consistent with the definition of the pressure [@Read:2010uq]. The formulation of the SPH equation in terms of the pressure and entropy thus introduces the notion of a weighted density $\bar\rho_i$. Despite having the units of a density, this quantity should not be confused with the physical density $\rho_i = \sum_j m_j W_{ij}(h_i)$. The weighted density should be thought of as an intermediate quantity required for the calculation of other thermodynamics quantities and for the SPH equation of motion. As a consequence, both densities must be used in the subgrid recipes. If the model requires a density (cooling, enrichment), then we use the physical density $\rho_i$. On the other hand, if the quantity of interest is the pressure or the temperature, then we use the weighted density $\bar\rho_i$ for consistency with the SPH equations. Finally, equation (\[eqmot\]) can be written as $$\begin{aligned} \frac{{\rm d}\mathbf{v}_i}{{\rm d}t}&=&-\sum_{j=1}^N m_j \left[ \frac{A_j^{1/\gamma}}{A_i^{1/\gamma}}\frac{\bar{P}_i}{\bar{\rho}_i^2}f_{ij}\nabla_i W_{ij}(h_i) ~ + \right . \nonumber \\ && \left . \frac{A_i^{1/\gamma}}{A_j^{1/\gamma}}\frac{\bar{P}_j}{\bar{\rho}_j^2}f_{ji}\nabla_i W_{ij}(h_j) \right] , \label{eqmot3}\end{aligned}$$ where the grad-$h$ terms are (see equation \[fij\]): $$f_{ij} = 1 - \frac{1}{A_j^{1/\gamma}}\left(\frac{h_i}{n_D \rho_i}\frac{\partial\bar{P}_i^{1/\gamma}}{\partial h_i}\right) \left[1 + \frac{h_i}{n_D \rho_i}\frac{\partial \rho_i}{\partial h_i}\right]^{-1}.$$ ### Injection of feedback energy {#sec:injection} When the equations of SPH are formulated using the pressure and entropy as main variables, particles do not carry a numerical field for their internal energy. This quantity has to be computed as a weighted sum over the particle neighbours in the same way as the density is computed in other formulations of SPH. Energy from feedback events can hence not be implemented by simply increasing the internal energy of the particle by some amount $\Delta u$. Furthermore, because the weighted density, $\bar{\rho}_i$, and the entropic function, $A_i$, of a particle are coupled, a naïve change of $A_i$ during energy injection would be incorrect as the corresponding weighted density would also change, making the total thermal energy of the gas (across all particles in the simulation volume) change by an amount different from $\Delta u$. In Anarchy this problem is partially solved by performing a series of iterations during which $A_i$ and $\bar\rho_i$ are changed until the two quantities have converged: $$\begin{aligned} A_{i,n+1} &=& {(\gamma-1) (u_{\rm old} +\Delta u) \over \bar\rho_{i,n}^{\gamma-1}} , \nonumber\\ \bar\rho_{i,n+1} &=& {\bar\rho_{i,n}\,A_{n}^{1/\gamma}-m_iW(0)A_{i,n}^{1/\gamma}+m_iW(0)A_{i,n+1}^{1/\gamma}\over A_{i,n+1}^{1/\gamma}}, \end{aligned}$$ where $m_i$ is the mass of particle $i$ and $W$ is the kernel function. This approximation is valid for reasonable values of $\Delta u$ and is crucial for injecting thermal feedback in the gas phase. For high thermal jumps with more than one particle being heated, as can for example occur for our AGN feedback scheme, the approximation provided by these iterations is not sufficiently accurate to properly conserve energy. We hence limit the amount of energy that can be injected in the gas phase by AGN in a single event by limiting the heating probability to $0.3$ (effectively limiting the number of particles being heated at the same time in a given neighbourhood) for which tests show that the correct amount of energy is distributed to the gas. Artificial viscosity {#sec:viscosity} -------------------- SPH requires artificial viscosity to capture shocks. The artificial viscosity switch has been implemented following @Cullen:2010qy. Their algorithm enables a precise detection of shocks and avoids excessive viscosity in pure shear flows. As in @Cullen:2010qy, particles are assigned individual values of the viscosity coefficient, $\alpha_{{\rm v},i}$. This is recomputed at every time step $n$, and if it exceeds the value at the previous step, $\alpha_{{\rm v},i}^n>\alpha_{{\rm v},i}^{n-1}$, the viscosity coefficient is set to $\min{(\alpha_{{\rm v},i}^n, \alpha_{\rm v,max})}$. If $\alpha_{{\rm v},i}^n\leq\alpha_{{\rm v},i}^{n-1}$, the viscosity coefficient decays towards $\alpha_{{\rm v},i}^n$ on a time scale proportional to the particle’s sound-crossing time, $\tau_i=h_i/(0.1 c_i)$: $$\alpha_{{\rm v},i} = \alpha_{{\rm v},i}^n + (\alpha_{{\rm v},i} - \alpha_{{\rm v},i}^n)\,e^{-\Delta t/\tau_i}\,,$$ and limiting the minimum allowed value, $\alpha_{{\rm v},i}\geq\alpha_{\rm v,min}\geq 0$. We adopt $\alpha_{\rm v,min}=0.05$ in order to facilitate particle ordering, and allow the coefficient to range up to $\alpha_{\rm v,max}=2$. We found that if the number of neighbours is sufficiently large ($\sim 10^2$), the calculation of the velocity divergence in <span style="font-variant:small-caps;">gadget</span> is sufficiently accurate for standard hydrodynamical tests. Therefore, we did not implement any expensive matrix calculation of the velocity divergence [@Cullen:2010qy; @Read:2012qy; @Hu2014SPHGal]. Entropy diffusion {#sec:diffusion} ----------------- SPH is by construction non-diffusive. However, some diffusion mechanism is required during mixing of gas phases in order to mimic thermal conduction. We do not attempt to model physical diffusion; the implemented diffusion is purely numerical. We also do not implement diffusion to solve numerical problems at contact discontinuities; these are solved by the adopted SPH scheme. The thermal energy, $u$, is diffused according to the following equation [e.g. @Monaghan:1997mz; @Price:2008kx], $$\frac{{\rm d}u_i}{{\rm d}t}=\sum_{j=1}^N\alpha_{{\rm d},ij} v_{{\rm d},ij} \frac{m_j }{\rho_{ij}}\left(u_i - u_j\right)\nabla_i W_{ij}(h_i,h_j) ,$$ where $v_{{\rm d},ij}={\rm max}(c_i + c_j + \mathbf{v}_{ij}\cdot\mathbf{r}_{ij}/r_{ij},0)$, and the diffusion coefficient, $\alpha_{{\rm d},ij}$, density and kernel derivative are averages among particle pairs. The purely numerical switch, similar to the one of [@Price:2008kx], is triggered by the spatial second derivative of the internal energy $$\dot{\alpha}_{{\rm d},i}=\beta\frac{h_i \nabla^2_i u_i}{\sqrt{u_i}} ,$$ where the growth speed of $\alpha_{{\rm d},i}$ can be tuned through the coefficient $\beta$. We adopt $\beta=0.01$. With this choice, diffusion is mild and there is no need of any further limiter in the presence of gravity. Finally, the diffusion coefficient evolves with time as $$\alpha_{{\rm d},i}(t+\Delta t) = \alpha_{{\rm d},i}(t) - \left(\frac{\alpha_{{\rm d},i}(t) - \alpha_{{\rm d,min}}}{\tau_i} - \dot{\alpha}_{{\rm d},i}\right)\Delta t ,$$ where the decay time scale, $\tau_i$, is the same as employed in the artificial viscosity, and $\alpha_{{\rm d,min}}=0$. We set the maximum allowed value to $\alpha_{{\rm d,max}}=1$, but this is unimportant because $\alpha_{{\rm d},i}\ll 1$ even for large discontinuities in the internal energy. Time stepping {#sec:timestepping} ------------- The accuracy of the time integration is increased by using a time-step limiter [e.g. @Saitoh:2013uq]. We adopted the solution of [@Durier:2012fj] which ensures that sudden changes in the particle internal energy, e.g. caused by feedback, are promptly captured and propagated to neighbouring particles by shortening their time step and by activating them. We set the maximum ratio of neighbouring particles’ time steps to four. Generation of the initial conditions {#app:ics} ==================================== We have made two types of initial conditions: dark matter only with all particles the same mass, and dark matter with gas. The dark matter with gas simulations are created starting from a corresponding dark matter only simulation so we first describe how the dark matter only initial conditions were made. Building dark matter only initial conditions -------------------------------------------- The initial conditions are created in three steps. Firstly, a particle load, representing an unperturbed homogeneous periodic universe in a 3-torus is produced. Secondly, a realisation of a Gaussian random density field with the appropriate linear power spectrum is created over the 3-torus. Thirdly the displacements and velocities, consistent with the pure growing mode of gravitational instability, are calculated from the Gaussian realisation and applied to the particle load producing the initial conditions. The unperturbed particle loads for the dark matter only initial conditions have a glass-like particle distribution produced by applying the method first described in [@White1994LesHouches]. This method, available as an option in the <span style="font-variant:small-caps;">gadget-2</span> code [@Springel2005Gadget2], was applied, with periodic boundary conditions, to make a “primitive” cubic glass distribution with $47^3$ particles. The particle loads required for each of the [EAGLE]{} initial conditions were built by tiling this primitive cubic glass file $n$ times in each of the three principal coordinate directions across a larger cubic 3-torus, giving particle loads with a glass distribution with $(47n)^3$ particles. The dark matter only initial conditions were generated using the [[ic\_2lpt\_gen]{}]{} code using the method described in [@Jenkins20102lpt] to create second order Lagrangian perturbation theory (2lpt) resimulation initial conditions. The [[ic\_2lpt\_gen]{}]{} code outputs Zeldovich initial conditions plus a “2lpt mass” for each particle. The [EAGLE]{} version of <span style="font-variant:small-caps;">gadget</span> 3 is then used to solve a Poisson equation sourced by the 2lpt masses placed at their unperturbed positions. The solution of this Poisson equation yields second-order Lagrangian growing mode displacements and velocities for each particle. Adding these to the Zeldovich displacements and velocities of all the particles produces the final 2lpt initial conditions. The 2lpt masses can then be discarded and the usual equations of motion are solved by integrating the initial conditions forward in time. ---------- ------------------------------------------------------------------------ Box size Phase Descriptor (cMpc) 6.25 \[Panph1,L19,(40044,38524,52597),S3,CH2062909610,EAGLE\_L0006\_VOL1\] 12.5 \[Panph1,L18,(34546,48586,31987),S3,CH1284484552,EAGLE\_L0012\_VOL1\] 25 \[Panph1,L17,(22872,9140,6502),S3,CH1193192352,EAGLE\_L0025\_VOL1\] 50 \[Panph1,L16,(9358,44124,48606),S3,CH1323953302,EAGLE\_L0050\_VOL1\] 100 \[Panph1,L16,(31250,23438,39063),S12,CH1050187043,EAGLE\_L0100\_VOL1\] 200 \[Panph1,L16,(27398,55228,10498),S3,CH664747129,EAGLE\_L0200\_VOL1\] 400 \[Panph1,L16,(11324,24834,60541),S3,CH846509636,EAGLE\_L0400\_VOL1\] 800 \[Panph1,L16,(65448,27937,42773),S3,CH773405482,EAGLE\_L0800\_VOL1\] 1600 \[Panph1,L15,(18083,14638,23364),S3,CH1829653368,EAGLE\_L1600\_VOL1\] 3200 \[Panph1,L14,(2152,5744,757),S3,CH1814785143,EAGLE\_L3200\_VOL1\] 6400 \[Panph1,L13,(3868,2093,2715),S3,CH1320830929,EAGLE\_L6400\_VOL1\] ---------- ------------------------------------------------------------------------ \[tbl:eaglephases\] Choice of phases ---------------- Generating a Gaussian random field requires choosing a set of random phases. For the [EAGLE]{} simulations we take these phases from Panphasia which is a public multiscale Gaussian white noise field [@Jenkins2013ICs; @Jenkins2013Panphasia]. Using Panphasia provides a simple way to publish the linear phases that define the [EAGLE]{} volumes. Table \[tbl:eaglephases\] lists the “phase descriptors” which define the location of the phase information of each volume within the much larger Panphasia field [@Jenkins2013ICs]. These phase descriptors define the phases on all scales and uniquely determine the phases not only for the simulations published here, but for any possible zoom simulation of any subregion of these volumes, and at any resolution (down to sub-Earth mass resolution if needed) in the future. In principle sufficient information is provided in this paper to enable anyone to re-run these simulations, or to resimulate objects identified from the [EAGLE]{} database. The information required is provided by the combination of the phase descriptors, the cosmological parameters and the linear matter power spectrum, and for the volumes themselves the details of how the particle load was constructed. Particle indexing ----------------- To make it possible to trace particles easily between the initial conditions and snapshots, each particle in the initial conditions was given a unique 42-bit integer index. The index was generated by assigning each particle a location on a space-filling Peano-Hilbert curve defined with a resolution of 14 bits per Cartesian coordinate over the simulation volume. The location for each particle was determined from its unperturbed position in the particle load. The particle index therefore encodes a Lagrangian position for the particle. Using a 42-bit index allows the Lagrangian position to be determined to a cubic cell of side length 1/16384 of the box size. This is small compared to the interparticle separations of particles in the initial conditions, which means that each particle has a unique index. The primitive $47^3$ glass file and routines to calculate the Peano-Hilbert indices are available at <http://eagle.strw.leidenuniv.nl/>. Making the full initial conditions ---------------------------------- The initial conditions for the hydrodynamical simulations are generated from the dark matter only sets of initial conditions. Each dark matter particle is replaced with a pair of particles consisting of a dark matter particle and gas particle with a combined mass equal to that of the original dark matter particle. The ratio of the gas and dark matter particles is equal to $\Omega_{\rm baryon}/(\Omega_{\rm matter}-\Omega_{\rm baryon})$. These particle pairs are positioned so that the centre of mass of the pair corresponds to the position of the original particle in the dark matter only initial conditions. The particle pairs are aligned with the (1,1,1) coordinate direction and the gas particle is positioned in the (1,1,1) direction relative to its corresponding dark matter particle. The magnitude of the displacement between the pair is chosen so that an initial cubic grid with mean density in the dark matter only initial conditions would transform into a body-centred cubic grid with dark matter (gas) particles at the centres of cubic cells made of gas (dark matter) particles. For the hydrodynamical simulations the index of the dark matter particles is taken to be exactly twice that of the corresponding index in the dark matter only initial conditions. The index of the gas particle is chosen to be one more than its corresponding dark matter particle. Thus, all dark matter particles have even indices, and all gas particles odd indices. [^1]: E-mail: schaye@strw.leidenuniv.nl [^2]: [EAGLE]{} is a project of the Virgo consortium for cosmological supercomputer simulations. [^3]: The argument breaks down if the gas consumption time scale becomes longer than the Hubble time. [^4]: The CAMB input parameter file and the linear power spectrum are available at <http://eagle.strw.leidenuniv.nl/>. [^5]: Note that [OWLS]{} used tables based on version 05.07. [^6]: For the purpose of imposing temperature floors, $T_{\rm eos}(\rho_{\rm g})$ is converted into an entropy assuming a fixed mean molecular weight of 1.2285, which corresponds to an atomic, primordial gas. Other conversions in the code use the actual mean molecular weight and hydrogen abundance, but we keep them fixed here to prevent particles with different abundances from following different effective equations of state. [^7]: To reduce the computational cost associated with neighbour finding for stars, we implement the enrichment every 10 gravitational time steps for star particles older than 0.1 Gyr; for the high-resolution run, Recal-L025N0752, this is further reduced to once every 100 time steps for star particles older than 1 Gyr. We have verified that our results are unaffected by this reduction in the sampling of stellar mass loss from older SSPs. [^8]: Note that this implies that metal mass is only approximately conserved. However, @Wiersma2009Chemo demonstrated that the error in the total metal mass is negligible even for simulations that are much smaller than [EAGLE]{}. [^9]: Because the expected probability is based on the accretion rate in the previous time step, limiting the BH time step does not guarantee that $P<0.3$. If the probability exceeds 0.3, then we limit it to 0.3 and store the unused energy in $E_{\rm BH}$. [^10]: For $M_{200} \ll 10^{10}\,{{{\rm M}_\odot}}$ the systematic errors in the abundance matching results are likely to be much greater because only a small fraction of such low-mass haloes may host galaxies [@Sawala2013HaloMass; @Sawala2014EagleZooms]. [^11]: We intend to make the simulation output public in due course, starting with galaxy properties, which we will make available using the SQL interface that was also used for the Millennium simulation [@Lemson2006Database; @Springel2005Millennium]. Details will be provided on the [EAGLE]{} web site, see <http://eagle.strw.leidenuniv.nl/>. [^12]: Note that the number of neighbours, $N_{\rm ngb}$, is a parameter and not the actual number of particles within the kernel.
--- abstract: 'We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the $p$–Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of $L^p$– and $L^2$–Sobolev spaces.' author: - 'H. Gimperlein, M. Maischak, E. Schrohe, E. P. Stephan' title: 'Adaptive FE–BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction' --- Introduction {#sec:intro} ============ Consider the following transmission problem on a bounded Lipschitz domain ${\Omega }\subset {{\mathbb{R}^n}}$: $$\begin{aligned} -\mathrm{div} \left( \varrho(\|\nabla u_1\|) \nabla u_1\right)= f \quad \text{in $\Omega$,} - \Delta u_2 =&0 \quad \text{in $\Omega^c$,} \nonumber\\ \varrho(\|\nabla u_1\|) \partial_\nu u_1-\partial_\nu u_2= t_0 \quad \text{on $\partial \Omega$,} u_1-u_2= &u_0\quad \text{on $\Gamma_t$,} \nonumber\\ -\varrho(\|\nabla u_1\|) \partial_\nu u_1 (u_0+u_2-u_1) + g\|(u_0+u_2-u_1)\|=&0, \label{diff}\\ \left\| \varrho(\|\nabla u_1\|) \partial_\nu u_1\right\| \leq g \quad \text{on $\Gamma_s$.}\nonumber\\ u_2(x)=\left\{\begin{array}{l@{}l} a+o(1)&,n=2 \\ \mathcal{O}(\|x\|^{2-n})&,n>2 \end{array}\right.\nonumber.&\end{aligned}$$ Here $\varrho(t)$ denotes a function $\varrho(x, t)\in C(\overline{\Omega} \times (0,\infty))$ satisfying $$0 \leq \varrho(t) \leq \varrho^* [t^\delta (1+t)^{1-\delta}]^{p-2},$$ $$\|\varrho(t)t-\varrho(s)s\| \leq \varrho^* [(t+s)^\delta (1+t+s)^{1-\delta}]^{p-2}\|t-s\|$$ and $$\varrho(t)t-\varrho(s)s \geq \varrho_* [(t+s)^\delta (1+t+s)^{1-\delta}]^{p-2}(t-s)$$ for all $t \geq s >0$ uniformly in $x \in \Omega$ ($\delta \in [0,1]$, $\varrho_, \varrho^ >0$). The interface ${{\partial \Omega} }= \overline{{{\Gamma_s}}\cup {{\Gamma_t}}}$ is divided into the disjoint components ${{\Gamma_s}}$ and ${{\Gamma_t}}\neq \emptyset$, and the data belong to the following spaces: $$f\in L^{p'}(\Omega),\ u_0 \in W^{\frac{1}{2},2}({{\partial \Omega} }),\ t_0 \in W^{-\frac{1}{2},2}({{\partial \Omega} }), \ g \in L^\infty({{\Gamma_s}}), \ a \in {\mathbb{R}}.$$ As usual, the normal derivatives are understood in terms of a Green’s formula, and it is convenient to set $a=0$ for $n>2$. In two dimensions one further condition is required to enforce uniqueness: $$\label{uniq} \int_\Omega f + \langle t_0, 1 \rangle=0.$$ We are looking for weak solutions $(u_1,u_2) \in W^{1,p}({\Omega }) \times W^{1,2}_{loc}({ {\Omega^c} })$ when $p \geq 2$. A typical example is given by $\varrho(t)=[t^\delta (1+t)^{1-\delta}]^{p-2}$, $\delta \in [0,1]$, with the $p$–Laplacian corresponding to the maximally degenerate case $\delta =1$. In this article we use layer potentials for the Laplace equation on ${ {\Omega^c} }$ to reduce the system to a uniquely solvable variational problem on $W^{1,p}({\Omega }) \times W^{\frac{1}{2},2}_0({{\Gamma_s}})$. The main idea of our theoretical analysis is simple: Because the traces of $W^{1,p}({\Omega })$–functions are continuously embedded into $W^{\frac{1}{2},2}({{\partial \Omega} })$ for $p \geq 2$, the quadratic form $\langle S u, u\rangle$ associated to the Steklov–Poincaré operator is accessible to Hilbert space methods whenever it is defined. In this slightly weaker setting, Friedrichs’ inequality (Prop. \[poinfried\]) allows to recover control over the $L^p$–norms in the interior, and as a consequence the full variational functional associated to the above equations is coercive in $W^{1,p}({\Omega })$. In the numerical part we present a model problem, which shows singularities resulting from the given boundary data, as well as from the change of boundary conditions, leading to a suboptimal convergence rate for uniform mesh refinements. We also present a Uzawa solver to deal with the variational inequality. With the help of a Korn inequality (Prop. \[korn\]), our method easily carries over to transmission problems in nonlinear elasticity, e.g. Hencky materials in $\Omega$ coupled to the Lamé equation in $\Omega^c$. A generalization to certain nonconvex energy functionals will be discussed elsewhere [@micro]. The outline of the article is as follows: Section \[sec:pre\] recalls some properties of $L^p$-Sobolev spaces and introduce a family of quasinorms adapted to the considered class of operators. In the following section \[formulation\] we introduce the boundary integral operators and derive our variational formulation. Section \[sec:uniq\] is dedicated to the existence and uniqueness of our model problem. The discretization of our problem is derived in section \[sec:disc\], as well as the a-priori error estimates. In section \[sec:post\] our a-posteriori error estimator is presented and its reliability proven. Finally, in section \[sec:num\] we present the Uzawa-solver and two numerical examples, clearly underlining our theoretical results. Preliminaries {#sec:pre} ============= Let ${\Omega }$ be an open subset of ${{\mathbb{R}^n}}$ with Lipschitz boundary ${{\partial \Omega} }$. Set $p'=\frac{p}{p-1}$ whenever $p \in (1,\infty)$. The Sobolev spaces $W_{(0)}^{k,p}({\Omega })$, $k \in {\mathbb{N}}_0$, are the completion of $C_{(c)}^\infty({\Omega })$ with respect to the norm $\\|u\\|_{W^{k,p}({\Omega })} = \\|u\\|_{k,p} = \\|u\\|_p + \sum_{\|\gamma\|=k} \\|\partial^\gamma u\\|_p$. The second term in the norm will be denoted by $\|u\|_{W^{1,p}({\Omega })} = \|u\|_{k,p}$. Let $W_0^{-k,p'}({\Omega }) = \left(W^{k,p}({\Omega })\right)'$ and $W^{-k,p'}({\Omega }) = \left(W_0^{k,p}({\Omega })\right)'$. $W^{1-\frac{1}{p},p}({{\partial \Omega} })$ denotes the space of traces of $W^{1,p}({\Omega })$–functions on the boundary. It coincides with the Besov space $B^{1-\frac{1}{p}}_{p,p}({{\partial \Omega} })$ as obtained by real interpolation of Sobolev spaces [@tri], and one may define $W^{s,p}({{\partial \Omega} })=B^{s}_{p,p}({{\partial \Omega} })$ for $s\in (-1,1)$. \[sobremarks\] We are going to need the following properties for bounded ${{\partial \Omega} }$ [@tri]:\ a) All the above spaces are reflexive and $\left(W^{s,p}({{\partial \Omega} })\right)'=W^{-s,p'}({{\partial \Omega} })$.\ b) For $p=2$ they coincide with the Sobolev spaces $H^s$.\ c) $W^{1-\frac{1}{p},p}({{\partial \Omega} }) \hookrightarrow W^{\frac{1}{2},2}({{\partial \Omega} })$ for $p \geq 2$.\ d) If ${{\partial \Omega} }$ is smooth, pseudodifferential operators of order $m$ with symbol in the Hörmander class $S^m_{1,0}({{\partial \Omega} })$ map $W^{s,p}({{\partial \Omega} })$ continuously to $W^{s-m,p}({{\partial \Omega} })$. For Lipschitz ${{\partial \Omega} }$, at least the first–order Steklov–Poincaré operator $S$ of the Laplacian on ${ {\Omega^c} }$ is continuous between $W^{\frac{1}{2},2}({{\partial \Omega} })$ and $W^{-\frac{1}{2},2}({{\partial \Omega} })$ [@cos].\ e) Points a) to d) imply that the quadratic form $\langle S u, u \rangle$ associated to $S$ is well-defined on $W^{1-\frac{1}{p},p}({{\partial \Omega} })$ if $p \geq 2$. $S$ being elliptic, the form cannot be defined for $p<2$ even if ${{\partial \Omega} }$ is smooth. Uniform monotony will be shown using a variant of Friedrichs’ inequality. \[poinfried\] Assume ${\Omega }$ is bounded and that $\Gamma \subset {{\partial \Omega} }$ has positive $(n-1)$–dimensional measure. Then there is a $C>0$ such that $$\\|u\\|_p \leq C( \\|\nabla u\\|_p + \\|u\|_{\Gamma}\\|_{L^1(\Gamma)}) \quad \text{for all $u \in W^{1,p}({\Omega })$}.$$ We apply an interpolation argument to the well-known Friedrichs’ inequality $$\\|u-u_{\Omega }\\|_p \leq C \\|\nabla u\\|_p, \qquad u_{\Omega }= \frac{1}{\|{\Omega }\|}\int_{\Omega }u,$$ on $W^{1,p}({\Omega })$ (see e.g. [@nec]). Let $L : W^{1,p}({\Omega }) \to L^p({\Omega })$ be the rank–$1$ operator $Lu = \frac{1}{\|\Gamma\|}\int_\Gamma u\|_\Gamma$ and $I$ the inclusion of $W^{1,p}({\Omega })$ into $L^p({\Omega })$. Then $I-L:W^{1,p}({\Omega }) \to L^p({\Omega })$ is bounded and $$\\|u-Lu\\|_p=\\|(I-L)(u-u_{\Omega })\\|_p\leq\\|I-L\\| \\|u-u_{\Omega }\\|_{1,p} \leq C \\|\nabla u\\|_p$$ for all $u \in W^{1,p}({\Omega })$. The assertion follows. Let $\omega(x,y) = (\|x\|+\|y\|)^{\delta}(1+\|x\|+\|y\|)^{1-\delta}$, $0 \leq \delta \leq 1$. In addition to the above norms, the following family of quasi–norms will prove useful: For $v,w \in W^{1,p}({\Omega })$ and $k \in {\mathbb{N}}_0$, define $$\|v\|_{(k,w,p)} = \left(\int_{\Omega }\omega(\nabla w, D^k v)^{p-2} \|D^k v\|^2\right)^{\frac{1}{2}},$$ where $\|D^k v\|^2 = \sum_{\|\gamma\| = k} \|\partial^\gamma v\|^2$. \[quasiremark1\] a) If $p\geq 2$, the $(1,w,p)$–quasi–norm can be estimated from above and below by suitable powers of the $W^{1,p}$–seminorm [@el]: $$\|v\|_{1,p}^p \leq \|v\|_{(1,w,p)}^2 \leq C(\|v\|_{1,p}, \|w\|_{1,p}) \|v\|_{1,p}^2.$$ b) In the nondegenerate case $\delta=0$, we have $\|v\|_{1,2}^2 \leq \|v\|_{(1,w,p)}^2$.\ c) The following inequality is useful for computations with quasi–norms: $$\lambda \mu \leq \max\{\varepsilon^{-1}, \varepsilon^{1/(1-p)}\} (a^{p-1} + \lambda)^{p'-2} \lambda^2 + \varepsilon(a+\mu)^{p-2}\mu^2$$ for $\lambda, \mu, a \geq 0$ and $\varepsilon >0$. The results of this paper easily generalize to the systems of equations describing certain inelastic materials. In this case, Lemma \[poinfried\] has to be replaced by the following Korn inequality: \[korn\] Assume ${\Omega }\subset {\mathbb{R}}^n$ is a bounded Lipschitz domain and $\Gamma \subset {{\partial \Omega} }$ has positive $(n-1)$–dimensional measure. Then there is a $C>0$ such that $$\\|u\\|_{1,p} \leq C( \\|\varepsilon(u)\\|_p + \\|u\|_{\Gamma}\\|_{L^1(\Gamma)}) \quad \text{for all $u \in (W^{1,p}({\Omega }))^n$}.$$ The $L^p$–version $\\|u\\|_{1,p} \leq C( \\|\varepsilon(u)\\|_p + \\|u\\|_{p})$ of Korn’s inequality is well-known (see e.g. [@korn]). Assume the assertion was false. Then $\\|\varepsilon(u_n)\\|_p + \\|u_n\|_{\Gamma}\\|_{L^1(\Gamma)} \leq \frac{1}{n}$ for some sequence in $W^{1,p}({\Omega })$ normalized to $\\|u_n\\|_{1,p}=1$. By the compactness of $W^{1,p}({\Omega }) \hookrightarrow L^p({\Omega })$, we may assume $u_n$ to converge in $L^p({\Omega })$. The cited variant of Korn’s inequality shows that $u_n$ is even Cauchy in $W^{1,p}({\Omega })$, hence converges to some $u_0$ with $\\|\varepsilon(u_0)\\|_p = \\|u_0\|_{\Gamma}\\|_{L^1(\Gamma)}=0$. The kernel of $\varepsilon$ consists of skew–symmetric affine transformations $A x + b$, $A = -A^T$. As $\dim \mathrm{ker}\, A \equiv n \,\, \mathrm{mod}\,2$, $u_0$ cannot vanish on all of the ($n-1$–dimensional) $\Gamma$ unless $u_0=0$. Contradiction to $\\|u_0\\|_{1,p}=1$. Variational Formulation and Reduction to ${{\partial \Omega} }$ {#formulation} =============================================================== We continue to use the notation from the Introduction and mainly follow [@mast]. Fix some $p\geq2$ and, for $q(t)= \int_0^t s \varrho(s)\ \mathrm{d}s$, let $G(u) = \int_{\Omega }q(\|\nabla u\|)$ with derivative $$DG(u,v)=\langle G' u, v\rangle = \int_{\Omega }\varrho(\|\nabla u\|) \nabla u \nabla v \qquad \text{($u, v \in W^{1,p}({\Omega })$)}$$ and $j(v) = \int_{{{\Gamma_s}}} g \|v\|$, $v \in L^1({{\Gamma_s}})$. $G$ is known to be strictly convex and $G' : W^{1,p}({\Omega }) \to \left(W^{1,p}({\Omega })\right)'$ bounded and uniformly monotone, hence coercive, with respect to the seminorm $\| \cdot \|_{1,p}$: There is some $\alpha_G>0$ such that for all $u, v \in W^{1,p}({\Omega })$ $$\langle G' u - G' v, u-v\rangle \geq \alpha_G \|u-v\|_{1,p}^p \quad \text{and} \quad \lim_{\|u\|_{1,p} \to \infty} \frac{\langle G' u, u\rangle}{\|u\|_{1,p}} = \infty.$$ The naive variational formulation of the transmission problem (\[diff\]) minimizes the functional $$\Phi(u_1, u_2) = G(u_1) + \frac{1}{2} \int_{ {\Omega^c} }\|\nabla u_2\|^2-\int_{\Omega }f u_1 - \langle t_0, u_2\|_{{\partial \Omega} }\rangle + j((u_2-u_1+u_0)\|_{{\Gamma_s}})$$ over a suitable convex set. Minimizing $\Phi$ over the nonempty, closed and convex subset $$C = \{(u_1,u_2)\in W^{1,p}({\Omega }) \times W^{1,2}_{loc}({ {\Omega^c} }) : (u_1-u_2)\|_{{\Gamma_t}}= u_0, \, u_2 \in \mathcal{L}_2\},$$ $$\mathcal{L}_2 = \{v \in W^{1,2}_{loc}({ {\Omega^c} }): \Delta v = 0\, \text{in $W^{-1,2}({ {\Omega^c} })$ $+$ radiation condition at $\infty$}\},$$ is equivalent to the system (\[diff\]) in the sense of distributions if $\varrho\in C^1(\overline{\Omega} \times (0,\infty))$. $C$ is apparently convex. A similar argument as in Remarks 2 and 4 of [@cag] shows that $C$ is closed and nonempty. The proof there almost exclusively involves the exterior problem in $\mathcal{L}_2$ and only requires basic measure theoretic properties of $W^{1,2}({\Omega })$, which also hold for $W^{1,p}({\Omega })$. Finally, repeat the computations of [@mast] to obtain equivalence with (\[diff\]). To reduce the exterior problem to the boundary, we are going to need the layer potentials $$\begin{aligned} \mathcal{V} \phi(x) &=& -\frac{1}{\pi} \int_{{\partial \Omega} }\phi(x')\ \log\|x-x'\|\ dx',\\ \mathcal{K} \phi(x) &=& -\frac{1}{\pi} \int_{{\partial \Omega} }\phi(x')\ \partial_{\nu_{x'}} \log\|x-x'\| \ dx',\\ \mathcal{K}'\phi(x) &=& -\frac{1}{\pi} \int_{{\partial \Omega} }\phi(x')\ \partial_{\nu_x} \log\|x-x'\|\ dx',\\ \mathcal{W} \phi(x) &=& \frac{1}{\pi}\ \partial_{\nu_x}\int_{{\partial \Omega} }\phi(x')\ \partial_{\nu_{x'}} \log\|x-x'\| \ dx'\end{aligned}$$ associated to the Laplace equation on ${ {\Omega^c} }$. They extend from $C^\infty({{\partial \Omega} })$ to a bounded map $\begin{pmatrix} -\mathcal{K} & \mathcal{V}\\ \mathcal{W} & \mathcal{K}' \end{pmatrix}$ on the Sobolev space $ W^{\frac{1}{2},2}({{\partial \Omega} }) \times W^{-\frac{1}{2},2}({{\partial \Omega} })$. If the capacity of ${{\partial \Omega} }$ is less than $1$, which can always be achieved by scaling, $\mathcal{V}$ and $\mathcal{W}$ considered as operators on $W^{-\frac{1}{2},2}({{\partial \Omega} })$ are selfadjoint, $\mathcal{V}$ is positive and $\mathcal{W}$ non-negative. Similarly, the Steklov-Poincaré operator $$S = \mathcal{W}+(1-\mathcal{K}')\mathcal{V}^{-1}(1-\mathcal{K}): W^{\frac{1}{2},2}({{\partial \Omega} }) \subset W^{-\frac{1}{2},2}({{\partial \Omega} }) \to W^{-\frac{1}{2},2}({{\partial \Omega} })$$ defines a positive and selfadjoint operator (pseudodifferential of order $1$, if ${{\partial \Omega} }$ is smooth) with the main property $$\partial_\nu u_2\|_{{\partial \Omega} }= - S (u_2\|_{{\partial \Omega} }-a)$$ for solutions $u_2 \in \mathcal{L}_2$ of the Laplace equation on ${ {\Omega^c} }$. By Remark \[sobremarks\] e), $S$ gives rise to a coercive and symmetric bilinear form $\langle S u , u\rangle$ on $W^{\frac{1}{2},2}({{\partial \Omega} })$ and, in particular, a pairing on the traces of $W^{1,p}({\Omega })$ if and only if $p \geq 2$.\ Using the weak definition of $\partial_\nu\|_{{\partial \Omega} }$, $S$ reduces the integral over ${ {\Omega^c} }$ in $\Phi$ to the boundary: $$\int_{ {\Omega^c} }\|\nabla u_2\|^2 = - \langle \partial_\nu u_2\|_{{\partial \Omega} }, u_2\|_{{\partial \Omega} }\rangle = \langle S (u_2\|_{{\partial \Omega} }-a), u_2\|_{{\partial \Omega} }\rangle \quad \text{for $u_2 \in \mathcal{L}_2$.}$$ Easy manipulations allow to substitute $u_2$ by a function $v$ on ${{\Gamma_s}}$ (cf. [@mast]): Let $${{\widetilde{W}^{\frac{1}{2},2}({{\Gamma_s}})}}= \{u \in W^{\frac{1}{2},2}({{\partial \Omega} }) : \mathrm{supp}\ u \subset \bar{\Gamma}_s\},\quad X^p = W^{1,p}({\Omega }) \times {{\widetilde{W}^{\frac{1}{2},2}({{\Gamma_s}})}}$$ and $(u,v) = (u_1 -c, u_0+u_2\|_{{\partial \Omega} }-u_1\|_{{\partial \Omega} }) \in X^p$ for a suitable $c \in {\mathbb{R}}$. Collecting the data–dependent terms in $$\lambda(u,v) = \langle t_0 +S u_0, u\|_{{\partial \Omega} }+ v\rangle + \int_{\Omega }f u$$ leads to $$\Phi(u_1,u_2) = G(u) + \frac{1}{2} \langle S(u\|_{{\partial \Omega} }+v),u\|_{{\partial \Omega} }+v\rangle - \lambda(u,v) +j(v) + \frac{1}{2} \langle Su_0, u_0\rangle + \langle t_0, u_0\rangle.$$ The first three terms on the right hand side will be called $J(u,v)$. Minimizing $\Phi$ over $C$ is equivalent to minimizing $J + j$ over the nonempty closed convex set $D = \{(u,v) \in X^p : \langle S(u\|_{{\partial \Omega} }+v-u_0), 1\rangle = 0 \,\, \text{if $n=2$}\}$ As in [@mast]. The main additional observation here is that the substitution $v = u_0+u_2\|_{{\partial \Omega} }-u_1\|_{{\partial \Omega} }$ indeed defines an element of ${{\widetilde{W}^{\frac{1}{2},2}({{\Gamma_s}})}}$, because $u_0, u_2\|_{{\partial \Omega} }\in W^{\frac{1}{2},2}({{\partial \Omega} })$, $u_1\|_{{\partial \Omega} }\in W^{1-\frac{1}{p},p}({{\partial \Omega} }) \subset W^{\frac{1}{2},2}({{\partial \Omega} })$ by Remark \[sobremarks\] and $v\|_{{\Gamma_t}}= 0$, if $(u_1, u_2) \in C$. Existence and Uniqueness {#sec:uniq} ======================== Minimization of $J+j$ over $D$ translates into the following variational inequality: Find $(\hat u, \hat v) \in X^p$ such that $$\langle G'\hat u, u-\hat u \rangle + \langle S(\hat u\|_{{\partial \Omega} }+\hat v), (u-\hat u)\|_{{\partial \Omega} }+ v-\hat v\rangle + j(v)-j(\hat v) \geq \lambda(u-\hat u, v-\hat v)$$ for all $(u,v) \in X^p$. Note that $D$ has been replaced by $X^p$.\ We now prove the crucial monotony estimate: \[monotony\] The operator in the variational inequality is uniformly monotone on $X^p$. There exists an $\alpha = \alpha(C) > 0$ such that for all $\\|u,v\\|_X, \\|\hat u,\hat v\\|_X <C$ $$\begin{aligned} &\alpha (\\|u-\hat u\\|^p_{W^{1,p}({\Omega })} + \\|v-\hat v\\|^p_{\widetilde W^{\frac{1}{2},2}({{\Gamma_s}})}) \leq \langle G' \hat u - G' u, \hat u - u\rangle \\ & \hspace{2.5cm}+\ \langle S((\hat u-u)\|_{{\partial \Omega} }+\hat v-v), (\hat u-u)\|_{{\partial \Omega} }+\hat v-v\rangle.\end{aligned}$$ Recall the monotony estimate for $G'$ from Section \[formulation\]: $$\langle G' \hat u - G' u, \hat u - u\rangle \geq \alpha_G \|\hat u-u\|_{1,p}^p.$$ The triangle inequality and convexity of $x^p$ imply $$\begin{aligned} \\|\hat v-v\\|^p_{\widetilde W^{\frac{1}{2},2}({{\Gamma_s}})} &\leq& (\\|(\hat u-u)\|_{{\Gamma_s}}+\hat v-v\\|_{ W^{\frac{1}{2},2}({{\Gamma_s}})}+\\|(\hat u-u)\|_{{\Gamma_s}}\\|_{ W^{\frac{1}{2},2}({{\Gamma_s}})})^p\\ & \leq & 2^{p-1} \ (\\|(\hat u-u)\|_{{\Gamma_s}}+\hat v-v\\|^p_{ W^{\frac{1}{2},2}({{\Gamma_s}})}+\\|(\hat u-u)\|_{{\Gamma_s}}\\|^p_{ W^{\frac{1}{2},2}({{\Gamma_s}})}).\end{aligned}$$ Using $W^{1-\frac{1}{p},p}({{\Gamma_s}}) \hookrightarrow W^{\frac{1}{2},2}({{\Gamma_s}})$ as well as the boundedness of the trace operator, $$2^{1-p}\ \\|\hat v-v\\|^p_{\widetilde W^{\frac{1}{2},2}({{\Gamma_s}})} - \beta\ \\|\hat u-u\\|^p_{W^{1,p}({\Omega })} \leq \\|(\hat u-u)\|_{{\Gamma_s}}+\hat v-v\\|^p_{ W^{\frac{1}{2},2}({{\Gamma_s}})}$$ follows for some $\beta\geq 1$. Let $$K = \{(u,v,\hat u,\hat v) \in X^p\times X^p : \\|(\hat u-u)\|_{{\partial \Omega} }+\hat v-v\\|_{W^{\frac{1}{2},2}({{\partial \Omega} })} < 2 \beta C\}$$ and $0<\varepsilon<\beta^{-1}$. Since $S$ is positive definite on $W^{\frac{1}{2},2}({{\partial \Omega} })$, we obtain from Friedrichs’ inequality for $(u,v,\hat u, \hat v) \in K$ or, in particular, if $\\|u,v\\|_X, \\|\hat u,\hat v\\|_X <C$: $$\begin{aligned} &\langle G' \hat u - G' u, \hat u - u\rangle + \langle S((\hat u-u)\|_{{\partial \Omega} }+\hat v-v), (\hat u-u)\|_{{\partial \Omega} }+\hat v-v\rangle\\ &\gtrsim \|\hat u-u\|_{1,p}^p + \\|(\hat u-u)\|_{{\partial \Omega} }+\hat v-v\\|^2_{W^{\frac{1}{2},2}({{\partial \Omega} })}\\ &\gtrsim \|\hat u-u\|_{1,p}^p + \\|(\hat u-u)\|_{{\partial \Omega} }+\hat v-v\\|^p_{W^{\frac{1}{2},2}({{\partial \Omega} })}\\ &\gtrsim \|\hat u-u\|_{1,p}^p + \varepsilon\ \\|(\hat u-u)\|_{{\Gamma_s}}+\hat v-v\\|^p_{ W^{\frac{1}{2},2}({{\Gamma_s}})} + \\|(\hat u-u)\|_{{\Gamma_t}}\\|^p_{W^{\frac{1}{2},2}({{\Gamma_t}})}\\ &\gtrsim \\|\hat u-u\\|_{W^{1,p}({\Omega })}^p + \varepsilon\ \\|(\hat u-u)\|_{{\Gamma_s}}+\hat v-v\\|^p_{ W^{\frac{1}{2},2}({{\Gamma_s}})}\\ &\gtrsim (1-\varepsilon \beta)\ \\|\hat u-u\\|_{W^{1,p}({\Omega })}^p + 2^{1-p}{\varepsilon}\ \\|\hat v-v\\|^p_{\widetilde W^{\frac{1}{2},2}({{\Gamma_s}})}.\end{aligned}$$ Uniform monotony on all of $X^p$ is shown similarly, but on the unbounded complement $(X^p\times X^p) \setminus K$ the exponents $p$ on the left hand side have to be replaced by $2$. \[exuniq\] The variational inequality is equivalent to the transmission problem (\[diff\]) and has a unique solution. We repeat the computations in [@mast] to get the equivalence with the minimization of $J+j$ over $D$, and hence with (\[diff\]). Existence and uniqueness follow from Lemma \[monotony\], e.g. by applying [@zei], Proposition 32.36. Discretization and Error Analysis {#sec:disc} ================================= In order to avoid using $S=\mathcal{W}+(1-\mathcal{K}')\mathcal{V}^{-1}(1-\mathcal{K})$ explicitly, the numerical implementation involves a variant of the variational inequality $$\langle G'\hat u, u-\hat u \rangle + \langle S(\hat u\|_{{\partial \Omega} }+\hat v), (u-\hat u)\|_{{\partial \Omega} }+ v-\hat v\rangle + j(v)-j(\hat v) \geq \lambda(u-\hat u, v-\hat v)$$ in terms of the layer potentials. Our a posteriori analysis is therefore based on the following equivalent problem: Find $(\hat u,\hat v, \hat \phi) \in X^p \times W^{-\frac{1}{2},2}({{\partial \Omega} })=: Y^p$, such that $$\begin{aligned} &\langle G'\hat u, u-\hat u \rangle +\langle \mathcal{W}(\hat u\|_{{\partial \Omega} }+\hat v) + (\mathcal{K}'-1) \hat \phi, (u-\hat u)\|_{{\partial \Omega} }+v-\hat v\rangle\\ &\hspace{0.8cm}+ j(v)-j(\hat v)\ \geq \ \langle t_0 + \mathcal{W} u_0, (u-\hat u)\|_{{\partial \Omega} }+v-\hat v\rangle + \int_{\Omega }f (u-\hat u), \\ &\langle \phi, \mathcal{V} \hat \phi + (1-\mathcal{K})(\hat u\|_{{\partial \Omega} }+\hat v)\rangle = \langle \phi, (1-\mathcal{K}) u_0\rangle\end{aligned}$$ for all $(u,v,\phi) \in Y^p$. More concisely, $$B(\hat u, \hat v, \hat \phi; u-\hat u, v-\hat v, \phi-\hat \phi)+ j(v)-j(\hat v) \geq \Lambda(u-\hat u, v-\hat v, \phi-\hat \phi)$$ with $$\begin{aligned} B(u, v, \phi; \bar u, \bar v, \bar \phi)&=&\langle G'u, \bar u \rangle +\langle \mathcal{W}(u\|_{{\partial \Omega} }+v) + (\mathcal{K}'-1) \phi, \bar u\|_{{\partial \Omega} }+\bar v\rangle\\ &&\quad+\langle \bar \phi, \mathcal{V} \phi + (1-\mathcal{K})(u\|_{{\partial \Omega} }+v)\rangle,\\ \Lambda(u, v, \phi) &=& \langle t_0 + \mathcal{W} u_0, u\|_{{\partial \Omega} }+v\rangle + \int_{\Omega }f u+\langle \phi, (1-\mathcal{K}) u_0\rangle.\end{aligned}$$ The more detailed a priori and a posteriori error analysis requires a few basic properties of the quasi–norms [@el]. \[quasiremark2\] a) The continuity and coercivity estimates can be sharpened: For all $u,v \in {{W^{1,p}({\Omega })}}$ $$\langle G' u - G' v, u-v\rangle\lesssim \|u-v\|_{(1,u,p)}^2 \lesssim \langle G' u - G' v, u-v\rangle.$$ b) There is $\theta>0$ such that for all $\varepsilon \in (0,\infty)$ and all $u,v,w \in {{W^{1,p}({\Omega })}}$ $$\|\langle G' u - G' v, w\rangle\| \lesssim \varepsilon \|u-v\|_{(1,u,p)}^2 + \varepsilon^{-\theta} \|w\|^2_{(1,u,p)}.$$ \[Bcoercive\] For all $(\hat u,\hat v,\hat \phi), (u,v,\phi) \in Y^p$ we have $$\begin{aligned} & \|\hat u - u\|_{(1,\hat u, p)}^2 + \\|(\hat u - u)\|_{{\partial \Omega} }+ \hat v - v\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\eta\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ & \lesssim \|\hat u - u\|_{(1,\hat u, p)}^2 + \\|(\hat u - u)\|_{{\partial \Omega} }+ \hat v - v\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\hat \phi- \phi\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2\\ & \lesssim B(\hat u,\hat v,\hat \phi;\hat u- u, \hat v- v, \eta) - B(u,v,\phi;\hat u- u, \hat v- v, \eta),\end{aligned}$$ where $2 \eta = \hat \phi -\phi + V^{-1}(1-K)((\hat u-u)\|_{{\partial \Omega} }+ \hat v-v)$. The right hand side of the identity $$\begin{aligned} & B(\hat u,\hat v,\hat \phi;\hat u- u, \hat v- v, \eta) - B(u,v,\phi;\hat u- u, \hat v- v, \eta)\\ &= \langle G'\hat u - G'u, \hat u - u\rangle + \textstyle{\frac{1}{2}} \langle \mathcal{W}((\hat u - u)\|_{{\partial \Omega} }+ \hat v - v),(\hat u - u)\|_{{\partial \Omega} }+ \hat v - v)\rangle \\ &\hphantom{=} + \textstyle{\frac{1}{2}} \langle S((\hat u - u)\|_{{\partial \Omega} }+ \hat v - v),(\hat u - u)\|_{{\partial \Omega} }+ \hat v - v)\rangle+ \textstyle{\frac{1}{2}}\langle \mathcal{V}(\hat \phi - \phi), \hat \phi - \phi \rangle.\end{aligned}$$ is, up to a constant, larger than $\\|\hat u - u,\hat v - v,\hat \phi-\phi\\|^2_{(\hat u, Y^p)}$. Furthermore, $$\\|\eta\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}\lesssim \\|\hat \phi - \phi\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}+\\|(\hat u - u)\|_{{\partial \Omega} }+ \hat v - v\\|_{W^{\frac{1}{2},2}({{\partial \Omega} })}.$$ Let $\{\mathcal{T}_h\}_{h\in I}$ a regular triangulation of ${\Omega }$ into disjoint open regular triangles $K$, so that $\overline{{\Omega }} = \bigcup_{K \in \mathcal{T}_h} K$. Each element has at most one edge on ${{\partial \Omega} }$, and the closures of any two of them share at most a single vertex or edge. Let $h_K$ denote the diameter of $K \in \mathcal{T}_h$ and $\rho_K$ the diameter of the largest inscribed ball. We assume that $1 \leq \max_{K \in \mathcal{T}_h} \frac{h_K}{\rho_K} \leq R$ independent of $h$ and that $h = \max_{K\in \mathcal{T}_h} h_K$. $\mathcal{E}_h$ is going to be the set of all edges of the triangles in $\mathcal{T}_h$, $D$ the set of nodes. Associated to $\mathcal{T}_h$ is the space ${{W_h^{1,p}({\Omega })}}\subset {{W^{1,p}({\Omega })}}$ of functions whose restrictions to any $K \in \mathcal{T}_h$ are linear. ${{\partial \Omega} }$ is triangulated by $\{l \in \mathcal{E}_h : l \subset {{\partial \Omega} }\}$. ${{W_h^{\frac{1}{2},2}({{\partial \Omega} })}}$ denotes the corresponding space of piecewise linear functions, and ${{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}$ the subspace of those supported on ${{\Gamma_s}}$. Finally, ${{W_h^{-\frac{1}{2},2}({{\partial \Omega} })}}\subset {{W^{-\frac{1}{2},2}({{\partial \Omega} })}}$. We denote by $i_h: {{W_h^{1,p}({\Omega })}}\hookrightarrow {{W^{1,p}({\Omega })}}$, $j_h : {{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}\hookrightarrow {{\widetilde{W}^{\frac{1}{2},2}({{\Gamma_s}})}}$ and $k_h: {{W_h^{-\frac{1}{2},2}({{\partial \Omega} })}}\hookrightarrow {{W^{-\frac{1}{2},2}({{\partial \Omega} })}}$ the canonical inclusion maps. Set $X^p_h = {{W_h^{1,p}({\Omega })}}\times {{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}$, We denote by $i_h: {{W_h^{1,p}({\Omega })}}\hookrightarrow {{W^{1,p}({\Omega })}}$, $j_h : {{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}\hookrightarrow {{\widetilde{W}^{\frac{1}{2},2}({{\Gamma_s}})}}$ and $k_h: {{W_h^{-\frac{1}{2},2}({{\partial \Omega} })}}\hookrightarrow {{W^{-\frac{1}{2},2}({{\partial \Omega} })}}$ the canonical inclusion maps. Set $X^p_h = {{W_h^{1,p}({\Omega })}}\times {{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}$, $$S_h = \frac12( W+(I-K')k_h(k_h^ V k_h)^{-1} k_h^(I-K))$$ and $$\lambda_h(u_h,v_h) =\langle t_0+ S_h u_0, u\|_{{\partial \Omega} }+ v\rangle + \int_{\Omega }f u_h.$$ As is well–known, there exists $h_0>0$ such that the approximate Steklov–Poincaré operator $S_h$ is coercive uniformly in $h<h_0$, i.e. $\langle S_h u_h, u_h\rangle \geq \alpha_S \\|u_h\\|_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}}^2$ with $\alpha_S$ independent of $h$. The discretized variational inequality reads as follows: Find $(\hat u_h, \hat v_h, \hat \phi_h) \in Y^p_h$ such that $$B(\hat u_h, \hat v_h, \hat \phi_h; u_h-\hat u_h, v_h-\hat v_h, \phi_h-\hat \phi_h)+ j(v_h)-j(\hat v_h) \geq \Lambda(u_h-\hat u_h, v_h-\hat v_h, \phi_h-\hat \phi_h)$$ for all $(u_h, v_h, \phi_h) \in Y^p_h$. Repeating the arguments from the previous section, one obtains a unique solution to the discretized variational inequality. \[apriori\] Let $(\hat u, \hat v, \hat \phi) \in Y^p$, $(\hat u_h, \hat v_h, \hat \phi_h) \in Y^p_h$ be the solutions of the continuous resp. discretized variational problem. The following a priori bound for the error holds uniformly in $h<h_0$: $$\begin{aligned} &\\|\hat u - \hat u_h, \hat v - \hat v_h, \hat \phi - \hat \phi_h\\|_{Y^p}^p\\ &\lesssim \|\hat u - \hat u_h\|_{(1,\hat u, p)}^2 + \\|(\hat u - \hat u_h)\|_{{\partial \Omega} }+ \hat v - \hat v_h\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\hat \phi- \hat \phi_h\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2\\ &\lesssim \inf_{(u_h,v_h,\phi_h) \in Y^p_h} \\|\hat u - u_h, \hat v - v_h, \hat \phi - \phi_h\\|_{Y^p}^2 + \\|\hat v-v_h\\|_{L^2({{\Gamma_s}})} .\end{aligned}$$ Let $(u,v,\phi)\in Y^p$, $(u_h,v_h,\phi_h)\in Y^p_h$. Lemma \[Bcoercive\] and the variational inequality imply $$\begin{aligned} &\|\hat u - \hat u_h\|_{(1,\hat u, p)}^2 + \\|(\hat u - \hat u_h)\|_{{\partial \Omega} }+ \hat v - \hat v_h\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\hat \phi- \hat \phi_h\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2\\ &\lesssim B(\hat u, \hat v, \hat \phi; \hat u - \hat u_h, \hat v - \hat v_h, \hat \phi-\hat \phi_h) - B(\hat u_h, \hat v_h, \hat \phi_h; \hat u - \hat u_h, \hat v - \hat v_h, \hat \phi-\hat \phi_h)\\ & \lesssim B(\hat u, \hat v, \hat \phi; u, v, \phi) - \Lambda(u-\hat u, v-\hat v, \phi-\hat\phi) + j(v) - j(\hat v)\\ & \hphantom{\lesssim} +B(\hat u_h, \hat v_h, \hat \phi_h; u_h,v_h,\phi_h) -\Lambda(u_h-\hat u_h, v_h-\hat v_h, \phi_h-\hat \phi_h) + j(v_h) - j(\hat v_h)\\ & \hphantom{\lesssim} - B(\hat u_h, \hat v_h, \hat \phi_h; \hat u, \hat v, \hat \phi) - B(\hat u, \hat v, \hat \phi; \hat u_h, \hat v_h, \hat \phi_h)\end{aligned}$$ Setting $(u,v,\phi)=(\hat u_h, \hat v_h, \hat \phi_h)$ and adding $0$, the right hand side turns into $$\begin{aligned} & B(\hat u, \hat v, \hat \phi; u_h-\hat u, v_h-\hat v, \phi_h-\hat\phi) - \Lambda(u_h-\hat u, v_h-\hat v, \phi_h-\hat\phi) + j(v_h) - j(\hat v)\\ & \quad +B(\hat u, \hat v, \hat \phi; \hat u-u_h, \hat v-v_h, \hat\phi-\phi_h)-B(\hat u_h, \hat v_h, \hat \phi_h; \hat u-u_h,\hat v-v_h,\hat \phi-\phi_h).\end{aligned}$$ We first consider the friction terms: $$j(v_h) - j(\hat v) = \int_{{\Gamma_s}}g(\|v_h\| - \|\hat v\|) \leq \int_{{\Gamma_s}}g(\|v_h - \hat v\|) \leq \\|g\\|_{L^2({{\Gamma_s}})}\\|v_h - \hat v\\|_{L^2({{\Gamma_s}})}.$$ The last two terms are bounded using Remark \[quasiremark2\]b and Cauchy-Schwarz: $$\begin{aligned} \langle G'\hat u - G'\hat u_h, \hat u - u_h \rangle &\lesssim& \varepsilon \|\hat u-\hat u_h\|_{(1,\hat u,p)}^2 + \varepsilon^{-\theta} \|\hat u-u_h\|^2_{(1,\hat u,p)},\\ & \lesssim & \varepsilon \|\hat u_h - \hat u\|_{(1,\hat u, p)}^2 + \varepsilon^{-\theta} C(\|\hat u\|_{1,p}, \|u_h\|_{1,p}) \|u_h-\hat u\|_{1,p}^2\end{aligned}$$ for sufficiently small $\varepsilon >0$. We may replace $C(\|\hat u\|_{1,p}, \|u_h\|_{1,p})$ by an honest constant noting that the coercivity of our functional gives an a priori bound on $\\|\hat u\\|_{{{W^{1,p}({\Omega })}}}$ and that we can restrict to those $u_h$ satisfying $\\|u_h\\|_{{{W^{1,p}({\Omega })}}} \leq 2 \\|\hat u\\|_{{{W^{1,p}({\Omega })}}}$. Moreover, $$\begin{aligned} & \langle \mathcal{W}((\hat u - \hat u_h)\|_{{\partial \Omega} }+ \hat v - \hat v_h) + (1-\mathcal{K}') (\hat \phi-\hat \phi_h), (\hat u - u_h)\|_{{\partial \Omega} }+ \hat v - v_h\rangle\\ & \lesssim \varepsilon \\|(\hat u - \hat u_h\|_{{\partial \Omega} }+ \hat v - \hat v_h\\|^2_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}} +\varepsilon \\|\hat \phi-\hat \phi_h\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ &\hphantom{\lesssim} + \varepsilon^{-1} \\|\hat u - u_h)\\|^2_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}} + \varepsilon^{-1} \\|\hat v - v_h\\|^2_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}},\end{aligned}$$ and $$\begin{aligned} &\langle \hat \phi - \phi_h, \mathcal{V}(\hat \phi - \hat \phi_h) + (1-\mathcal{K})((\hat u - \hat u_h)\|_{{\partial \Omega} }+ \hat v - \hat v_h)\rangle\\ &\lesssim \varepsilon^{-1}\\|\hat \phi - \phi_h\\|^2_{W^{-\frac{1}{2},2}({{\partial \Omega} })} +\varepsilon \\|\hat \phi - \hat \phi_h\\|^2_{W^{-\frac{1}{2},2}({{\partial \Omega} })}+ \varepsilon \\|(\hat u - \hat u_h)\|_{{\partial \Omega} }+ \hat v - \hat v_h\\|^2_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}}.\end{aligned}$$ Substituting $(u,v, \phi) = (u_h, \hat v,0)$ and $(u,v,\phi)=(2 \hat u - u_h, \hat v, 0)$ into the variational inequality on $Y^p$ and using that also the $\phi$ part is really an equality, the remaining two terms reduce to $$\begin{aligned} &\langle-t_0 - \mathcal{W} u_0+ \mathcal{W}(\hat u\|_{{\partial \Omega} }+\hat v) + (\mathcal{K}'-1) \hat \phi, v_h - \hat v\rangle \\ &= - \langle t_0 - S(\hat u\|_{{\partial \Omega} }+ \hat v-u_0), v_h - \hat v\rangle \\ &= - \langle \varrho(\|\nabla u\|) \partial_\nu u, v_h - \hat v\rangle \ \leq\ \\|g\\|_{L^2({{\Gamma_s}})}\\|v_h - \hat v\\|_{L^2({{\Gamma_s}})}.\end{aligned}$$ Applying these various estimates to the terms of the right hand side, the assertion follows from $$\\|\hat u - \hat u_h,\hat v - \hat v_h, \hat \phi - \hat \phi_h\\|_{Y^p}^p \lesssim \|\hat u - \hat u_h\|_{(1,\hat u, p)}^2 + \\|(\hat u - \hat u_h)\|_{{\partial \Omega} }+ \hat v - \hat v_h\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2+\\|\hat \phi - \hat \phi_h\\|^2_{W^{-\frac{1}{2},2}({{\partial \Omega} })}$$ as in Lemma \[monotony\]. In the nondegenerate case $\delta = 0$, we essentially recover the estimates for uniformly elliptic operators from [@cag; @mast]. \[ellapriori\] For $\delta = 0$, we obtain $$\\|\hat u - \hat u_h, \hat v - \hat v_h,\hat \phi - \hat \phi_h\\|_{Y^2}^2\ \lesssim \ \inf_{(u_h,v_h, \phi_h) \in Y^p_h} \\|\hat u - u_h, \hat v - v_h, \hat \phi - \phi_h\\|_{Y^p}^2 + \\|\hat v-v_h\\|_{L^2({{\Gamma_s}})}$$ uniformly in $h<h_0$ Use \[quasiremark1\]b) to estimate $\|\hat u_h-\hat u\|_{(1,\hat u,p)}$ in Theorem \[apriori\] from below. A posteriori error estimate {#sec:post} =========================== Denote by $$(e, \tilde e, \epsilon) = (\hat u - \hat u_h, \hat v - \hat v_h, \hat \phi - \hat \phi_h) \in Y^p$$ the error of the Galerkin approximation, and let $2\nu=\epsilon+\mathcal{V}^{-1}(1-\mathcal{K})(e\|_{{\partial \Omega} }+\tilde e)$. Our basic a posteriori estimate is the following. \[abstractapost\] For all $(e_h, \tilde e_h, \nu_h) \in Y^p_h$ $$\begin{aligned} &\|e\|_{(1,\hat u, p)}^2 + \\|e\|_{{\partial \Omega} }+ \tilde e\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\epsilon\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ &\lesssim \Lambda(e - e_h,\tilde e - \tilde e_h, \nu-\nu_h) + j(\tilde e_h +\hat v_h) - j(\hat v) \\ &\hphantom{\lesssim}-B(\hat u_h, \hat v_h, \hat \phi_h; e -e_h, \tilde e - \tilde e_h, \nu-\nu_h)\\ &= \int_{\Omega }f (e - e_h)-\langle G'\hat u_h, e - e_h\rangle+\int_{{\Gamma_s}}g (\|\tilde e_h +\hat v_h\| - \|\tilde e +\hat v_h\|)\\ &\hphantom{\lesssim} -\langle \nu-\nu_h, \mathcal{V} \hat \phi_h + (1-\mathcal{K})(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0)\rangle\\ &\hphantom{\lesssim}+\langle t_0-\mathcal{W}(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0) - (\mathcal{K}'-1) \hat \phi_h, (e -e_h)\|_{{\partial \Omega} }+\tilde e - \tilde e_h\rangle.\end{aligned}$$ Lemma \[Bcoercive\], the continuous and the discretized variational inequality imply $$\begin{aligned} &\|\hat u - u\|_{(1,\hat u, p)}^2 + \\|(\hat u - u)\|_{{\partial \Omega} }+ \hat v - v\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\hat \phi- \phi\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ & \lesssim B(\hat u,\hat v,\hat \phi;\hat u- \hat u_h, \hat v- \hat v_h, \nu) - B(\hat u_h,\hat v_h,\hat \phi_h;\hat u- \hat u_h, \hat v- \hat v_h, \nu)\\ &\lesssim \Lambda(\hat u- \hat u_h, \hat v- \hat v_h, \nu) +j(\hat v_h)-j(\hat v)-B(\hat u_h,\hat v_h,\hat \phi_h;\hat u- \hat u_h, \hat v- \hat v_h, \nu)\\ & \lesssim \Lambda(\hat u - \hat u_h - (u_h-\hat u_h),\hat v - \hat v_h - (v_h-\hat v_h), \nu-\nu_h) + j(v_h)-j(\hat v) \\ &\hphantom{\lesssim} - B(\hat u_h, \hat v_h, \hat \phi_h; \hat u - \hat u_h -(u_h-\hat u_h), \hat v - \hat v_h - (v_h-\hat v_h), \nu-\nu_h).\end{aligned}$$ Note that the variational inequalities are identities when restricted to the $\phi$-variable. The claim follows by setting $e_h = u_h - \hat u_h$ and $\tilde e_h = v_h - \hat v_h$. Simplifying the right hand side along the lines of [@cly] leads to a gradient recovery scheme in the interior with a residual type estimator on the boundary. With a straight forward modification of [@ly], also a method purely based on residual type estimates could be justified. For $1< p<\infty$ and $0 \leq \delta \leq 1$, define $$G_{p,\delta}(x,y) = \|y\|^2 \omega(x,y)^{p-2}=\|y\|^2[(\|x\|+\|y\|)^{\delta}(1+\|x\|+\|y\|)^{1-\delta}]^{p-2}$$ whenever $\|x\|+\|y\|>0$ and $0$ otherwise. As in [@cly], our analysis will be based on the following consequences of the monotony and convexity properties of $G_{p,\delta}$. \[gpoinc\] Assume that ${\Omega }$ is connected. Let $q$ be a continuous linear form on $W^{1,p}({\Omega })$ with ${\mathbb{R}}\cap \ker q = \{0\}$, where ${\mathbb{R}}$ is identified with the space of constant functions on ${\Omega }$. Then for any $1<p<\infty$ there exists $C_P = C_P(p,q,{\Omega })>0$ such that for all $a\geq 0$ and $u \in W^{1,p}({\Omega })$, $$\int_{\Omega }G_{p,\delta}(a,u) \leq C_P \left(G_{p,\delta}(a, q(u)) + \int_{\Omega }G_{p,\delta}(a,\|\nabla u\|)\right).$$ Cf. [@cly], Lemma 4.1 and its generalization in Remark 4.3. \[combilemma\] For any $d, k \in {\mathbb{N}}$ there is $C_\Sigma=C_\Sigma(p,d,k)>0$ such that for all $a_1, a_2, \dots , a_k \in {\mathbb{R}}^d$ $$\sum_{j=1}^k\sum_{l=1}^{j-1} G_{p,\delta}(a_j, a_j-a_l) \lesssim C_\Sigma \sum_{j=1}^{k-1} \min_{1 \leq m \leq k} G_{p,\delta}(a_m, a_{j+1} - a_j).$$ Cf. [@cly], Lemma 4.2 and its generalization in Remark 4.3. Even though Lemma \[gradlemma\] and Lemma \[flemma\] hold for any $1<p<\infty$ with minor modifications of the proofs (see [@cly] for a similar discussion), we will from now on concentrate on the range $2 \leq p <\infty$ relevant to our transmission problem. Let $z \in D$ be a node of the triangulation $\mathcal{T}_h$ and $\varphi_z \in {{W_h^{1,p}({\Omega })}}$ the associated nodal basis function. Let $\omega_z = \{x \in {\Omega }: \varphi_z(x) >0\}$ be the interior of the support of $\varphi_z$. The interpolation operator $\pi : {{W^{1,p}({\Omega })}}\to {{W_h^{1,p}({\Omega })}}$ is defined as $$\pi u = \sum_{z \in D} u_z \varphi_z, \qquad u_z = \int_{\Omega }\varphi_z u / \int_{\Omega }\varphi_z.$$ \[gradlemma\] Let $\mathcal{E}_h^z = \{l \in \mathcal{E}_h: l=\bar K_i \cap \bar K_j \text{ for some } K_i, K_j\subset \omega_z\}$. Given $u_h \in {{W_h^{1,p}({\Omega })}}$, let $\left[\partial_{\nu_\mathcal{E}} u_h\right]_l$ denote the jump of the normal derivative across the inner edge $l$ of the triangulation. Then, if $v \in {{W^{1,p}({\Omega })}}$ and $K \in \mathcal{T}_h$, the following estimate holds: $$\begin{aligned} &\int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(v-\pi v)) + \int_K G_{p,\delta}(\nabla u_h, \nabla(v-\pi v)) \\ &\lesssim\sum_{z \in D \cap \bar K} \left(\int_{\omega_z} G_{p,\delta}(\nabla u_h, \nabla v) + \sum_{l \in \mathcal{E}_h^z} \min_{\bar K' \cap l \neq \emptyset} \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_{K'}, \left[\partial_{\nu_\mathcal{E}} u_h\right]_l)\right).\end{aligned}$$ The proof is a modification of [@cly], Lemma 4.3. Concerning the first term on the left hand side, the convexity of $G_{p,\delta}$ in its second argument (a “triangle inequality”) and enlarging the domain of integration leads to $$\begin{aligned} \int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(v-\pi v)) & = & \int_K G_{p,\delta}(\nabla u_h, \sum_{z \in D \cap \bar K}h_K^{-1}(v- v_z)\varphi_z)\\ & \lesssim & \sum_{z \in D \cap \bar K} \int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(v- v_z)\varphi_z)\\ & \leq & \sum_{z \in D \cap \bar K} \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_K, h_K^{-1}(v- v_z)\varphi_z).\end{aligned}$$ As $G_{p,\delta}(\nabla u_h\|_K, \cdot)$ is increasing and $\|\varphi_z\|\leq 1$, Lemma \[gpoinc\] with $q(u) = \int_{\omega_z} \varphi_z u$ implies $$\begin{aligned} \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_K, h_K^{-1}(v- v_z)\varphi_z) &\leq & \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_K, h_K^{-1}(v- v_z)) \nonumber\\ &\leq& C_P \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_K, \nabla(v- v_z)) \label{helpfulestimate1}\\ & = & C_P \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_K, \nabla v) \nonumber\end{aligned}$$ for every term in the sum over $z \in D \cap \bar K$. To replace the constant $\nabla u_h\|_K$ by $\nabla u_h$, we repeatedly apply the usual triangle inequality and the convexity of $G_{p,\delta}$ to obtain $$\begin{aligned} &G_{p,\delta}(\nabla u_h\|_K, \nabla v) \\ &\leq G_{p,\delta}(\nabla u_h\|_K, \|\nabla v\| + \|\nabla u_h\|_K-\nabla u_h\|)\\ &= (\|\nabla v\| + \|\nabla u_h\|_K-\nabla u_h\|)^2 (\|\nabla u_h\|_K\|+\|\nabla v\| + \|\nabla u_h\|_K-\nabla u_h\|)^{\delta(p-2)}\\ &\hphantom{=} \times (1+\|\nabla u_h\|_K\|+\|\nabla v\| + \|\nabla u_h\|_K-\nabla u_h\|)^{(1-\delta)(p-2)}\\ &\leq (\|\nabla v\| + \|\nabla u_h\|_K-\nabla u_h\|)^2 (\|\nabla v\| + 2(\|\nabla u_h\|+\|\nabla u_h\|_K-\nabla u_h\|))^{\delta(p-2)}\\ &\hphantom{=} \times (1+\|\nabla v\| + 2(\|\nabla u_h\|+\|\nabla u_h\|_K-\nabla u_h\|))^{(1-\delta)(p-2)}\\ &\lesssim G_{p,\delta}(\nabla u_h, \|\nabla v\| + \|\nabla u_h\|_K-\nabla u_h\|)\\ &\lesssim G_{p,\delta}(\nabla u_h, \nabla v) + G_{p,\delta}(\nabla u_h,\nabla u_h\|_K-\nabla u_h).\end{aligned}$$ Altogether $$\int_K G_{p,\delta}(\nabla u_h\|_K, h_K^{-1}(v-\pi v)) \lesssim \sum_{z \in D \cap \bar K} \int_{\omega_z}\left\{G_{p,\delta}(\nabla u_h, \nabla v) + G_{p,\delta}(\nabla u_h,\nabla u_h\|_K-\nabla u_h)\right\}.$$ Let $\overline{\omega}_z = \bar K_1 \cup \cdots \cup \bar K_k$. Applying Lemma \[combilemma\] with $a_j = \nabla u_h\|_{K_j}$, $1 \leq j \leq k$, leads to the asserted bound for the first term. For the proof, note that the conormal derivatives of the piecewise linear function $u_h$ are determined by its boundary values on the corresponding edge. But $u_h \in {{W_h^{1,p}({\Omega })}}\subset {{W^{1,p}({\Omega })}}$, so the restrictions from both sides have to coincide, and the conormal derivative does not jump: $a_j - a_{j-1} = [\partial_{\nu_\mathcal{E}} u_h\|_{\bar K_j \cap \bar K_{j-1}}]$. As for the second term, let $c = \frac 1 {\|K\|} \int_K v$. Because $$\begin{aligned} \int_{K} G_{p,\delta}(\nabla u_h, \nabla(v - \pi v)) \lesssim \int_K G_{p,\delta}(\nabla u_h, \nabla v) + \int_K G_{p,\delta}(\nabla u_h, \nabla (\pi v-c))\end{aligned}$$ by convexity and the triangle inequality, it only remains to consider the second term $\int_K G_{p,\delta}(\nabla u_h, \nabla (\pi v-c))$. The inverse estimate $$\|\nabla(\pi v - c)\| \lesssim \frac 1 {\|K\|} \int_K h_K^{-1} \|\pi v - c\|$$ for the affine function $\pi v - c$ and Jensen’s inequality show $$\begin{aligned} \int_K G_{p,\delta}(\nabla u_h, \nabla (\pi v-c)) & \lesssim& \int_K \frac 1 {\|K\|} \int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(\pi v-c))\\ &=& \int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(\pi v-c)).\end{aligned}$$ However, as before $$\int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(\pi v-c)) \lesssim \int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(v-\pi v)) + \int_K G_{p,\delta}(\nabla u_h, h_K^{-1}(v-c)),$$ and the first term has been considered in the first step of the proof. Lemma \[gpoinc\] with $q(u) = \int_K u$ also bounds the final term by $\int_K G_{p,\delta}(\nabla u_h, \nabla v)$. \[flemma\] For any $\varepsilon >0$, $u_h \in {{W_h^{1,p}({\Omega })}}$, $v \in {{W^{1,p}({\Omega })}}$ and $f \in L^{p'}({\Omega })$, $$\begin{aligned} \int_{\Omega }f (v - \pi v) &\leq& C \varepsilon \int_{\Omega }G_{p,\delta}(\nabla u_h, \nabla v)\\ & & + C(\varepsilon) \sum_{z \in D} \sum_{K \subset \overline{\omega}_z}\int_{K} G_{p',1}(\|\nabla u_h\|^{p-1}, h_K(f-f_K))\\ & & + C \varepsilon \sum_{z\in D}\sum_{l \in \mathcal{E}_h^z} \min_{\bar K' \cap l \neq \emptyset} \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_{K'}, \left[\partial_{\nu_\mathcal{E}} u_h\right]_l).\end{aligned}$$ Here, $f_K = \frac{1}{\|K\|} \int_{K} f$. If $f \in W^{1,p'}({\Omega })$, the second term may be replaced by $$C(\varepsilon) \sum_{z \in D} \sum_{K \subset \overline{\omega}_z}\int_{K} G_{p',1}(\|\nabla u_h\|^{p-1}, h_K^2\nabla f).$$ We adapt the proof of [@cly], Lemma 4.4. Let $\tilde{K} \subset \overline{\omega}_z$ such that $\|\nabla u_h\|_{\tilde{K}}\| = \max_{K'\subset \overline{\omega}_z} \|\nabla u_h\|_{K'}\|$. Applying the inequality from Remark \[quasiremark1\]c) for some $\varepsilon >0$ and $C(\varepsilon) = C_P \max\{\varepsilon^{-1}, \varepsilon^{1/(1-p)}\}$, $$\begin{aligned} \int_{\Omega }f (v - \pi v) &=& \sum_{z \in D} \sum_{K \subset \overline{\omega}_z}\int_{K} h_K (f-f_K) h_K^{-1} (v - v_z) \varphi_z\\ &\leq& C_P^{-1}\ C(\varepsilon)\sum_{z \in D} \sum_{K \subset \overline{\omega}_z}\int_{K}(\|\nabla u_h\|_{\tilde{K}}\|^{p-1} + h_K \|f-f_K\|)^{p'-2} h_K^2 \|f-f_K\|^2 \\ & &+ \varepsilon \sum_{z \in D}\sum_{K \subset \overline{\omega}_z}\int_{K}(\|\nabla u_h\|_{\tilde{K}}\|+h_K^{-1} \|v - v_z\| \varphi_z)^{p-2}h_K^{-2} \|v - v_z\|^2 \varphi_z^2\\ &\leq & C_P^{-1}\ C(\varepsilon)\sum_{z \in D}\sum_{K \subset \overline{\omega}_z}\int_{K}G_{p',1}(\|\nabla u_h\|_{\tilde{K}}\|^{p-1}, h_K (f-f_K)) \\ & &+ \varepsilon \sum_{z \in D}\sum_{K \subset \overline{\omega}_z}\int_{K} G_{p,\delta}(\nabla u_h\|_{\tilde{K}}, h_K^{-1} (v - v_z) \varphi_z),\end{aligned}$$ because $\sum_{K \subset \overline{\omega}_z}\int_{K} f_K (v - v_z) \varphi_z=0$. However, by our choice of $\tilde{K}$ and because $p'\leq 2$, $$\int_{K}G_{p',1}(\|\nabla u_h\|_{\tilde{K}}\|^{p-1}, h_K (f-f_K)) \leq \int_{K}G_{p',1}(\|\nabla u_h\|^{p-1}, h_K (f-f_K)).$$ If $f \in W^{1,p'}({\Omega })$, Lemma \[gpoinc\] with $q(u) = \int_{K} u$ gives: $$\int_{K}G_{p',1}(\|\nabla u_h\|^{p-1}, h_K (f-f_K)) \leq C_P \int_{K}G_{p',1}(\|\nabla u_h\|^{p-1}, h_K^2 \nabla f).$$ Concerning the $G_{p,\delta}$–term, equation (\[helpfulestimate1\]) in the proof of Lemma \[gradlemma\] shows that it is dominated by $\varepsilon \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_{\tilde{K}}, \nabla v)$, which in turn was bounded by $$\varepsilon \int_{\omega_z} G_{p,\delta}(\nabla u_h, \nabla v) + \varepsilon \sum_{l \in \mathcal{E}_h^z} \min_{\bar K' \cap l \neq \emptyset} \int_{\omega_z} G_{p,\delta}(\nabla u_h\|_{K'}, \left[\partial_{\nu_\mathcal{E}} u_h\right]_l).$$ In order to define the a posteriori estimator, we still need to introduce some notation. For any $z \in D$, denote by $K_{j,z} \in \mathcal{T}_h$, $1\leq j \leq N_z$, the triangles neighboring $z$ in the sense that $\overline{\omega}_z = \bigcup_{j=1}^{N_z} \bar K_{j,z}$. To each $K_{j,z}$ we associate a weight factor $\alpha_{j,z} \geq 0$ normalized to $\sum_{j=1}^{N_z} \alpha_{j,z} = 1$. Given $u_h \in {{W_h^{1,p}({\Omega })}}$, define the gradient recovery $$G_h u_h = \sum_{z \in D} (G_h v_h)(z)\ \varphi_z, \quad (G_h v_h)(z) = \sum_{j=1}^{N_z} \alpha_{j,z} \nabla u_h\|_{K_{j,z}}.$$ The following theorem states our reliable, but presumably not efficient a posteriori estimate. Let $f \in L^{p'}({\Omega })$ and denote by $(e, \tilde e, \epsilon)$ the error between the Galerkin solution $(\hat u_h, \hat v_h, \hat \phi_h) \in Y^p_h$ and the true solution $(\hat u , \hat v, \hat \phi) \in Y^p$. If ${{\Gamma_s}}\neq \emptyset$, assume that $\nabla \hat u\|_{{\Gamma_s}}\in L^p({{\Gamma_s}})$. Then $$\begin{aligned} \\|e, \tilde e, \epsilon\\|_{Y^p}^p &\lesssim& \|e\|_{(1,\hat u, p)}^2 + \\|e\|_{{\partial \Omega} }+ \tilde e\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\epsilon\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ &\lesssim& \eta_{gr}^2 + \eta_f^2 +\eta_S^2+\eta_\partial^2+\eta_g^2 ,\end{aligned}$$ where $$\begin{aligned} \eta_{gr}^2 &= \sum_{K \in \mathcal{T}_h} \int_{K}G_{p,\delta}(\nabla \hat u_h, \nabla\hat u_h - G_h \hat u_h),\\ \eta_f^2 &= \sum_{K \in \mathcal{T}_h} \int_{K} G_{p',1}(\|\nabla \hat u_h\|^{p-1}, h_K (f-f_K)),\\ \eta_S^2 &= \sum_{l \subset {{\partial \Omega} }} l\ \\|\partial_s \{\mathcal{V} \hat \phi_h + (1-\mathcal{K})(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0)\}\\|_{L^2(l)}^2\\ \eta_{\partial}^2&=\sum_{l \subset {{\partial \Omega} }} l\ \\|-\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h+t_0-\mathcal{W}(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0) - (\mathcal{K}'-1) \hat \phi_h\\|_{L^2(l)}^2\\ \eta_{g}^2 &=\sum_{l\subset{{\Gamma_s}}} l \\|\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\Gamma_s}}\\|^2_{L^2(l)}+\\|g\\|^2_{W^{-\frac{1}{2},2}({{\Gamma_s}})} $$ If $f \in W^{1,p'}({\Omega })$, we may replace $\eta_f^2$ by $\sum_{K \in \mathcal{T}_h} \int_{K} G_{p',1}(\|\nabla \hat u_h\|^{p-1}, h_K^2 \nabla f)$. From Lemma \[abstractapost\] we know that for all $(e_h, \tilde e_h, \nu_h) \in Y^p_h$ $$\begin{aligned} &\\|e, \tilde e, \epsilon\\|_{Y^p}^p \lesssim \|e\|_{(1,\hat u, p)}^2 + \\|e\|_{{\partial \Omega} }+ \tilde e\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\epsilon\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ &\lesssim \int_{\Omega }f (e - e_h)- \sum_{K \in \mathcal{T}_h} \int_{\partial K} \varrho(\nabla \hat u_h)\ \partial_\nu \hat u_h\|_{\partial K}\ (e-e_h)\\ &\hphantom{\lesssim} +\int_{{\Gamma_s}}g (\|\tilde e_h +\hat v_h\| - \|\hat v\|)-\langle \nu-\nu_h, \mathcal{V} \hat \phi_h + (1-\mathcal{K})(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0)\rangle\\ &\hphantom{\lesssim}+\langle t_0-\mathcal{W}(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0) - (\mathcal{K}'-1) \hat \phi_h, (e -e_h)\|_{{\partial \Omega} }+\tilde e - \tilde e_h\rangle,\end{aligned}$$ with $2\nu=\epsilon+\mathcal{V}^{-1}(1-\mathcal{K})(e\|_{{\partial \Omega} }+\tilde e)$. The first two terms are mainly going to give the gradient recovery in the interior, the fourth term the error $\eta_S$ of constructing the Steklov-Poincaré operator, while the remaining terms add up to $\eta_\partial$. Concerning the first term: $$\begin{aligned} \int_{\Omega }f(e - e_h) &\lesssim& \varepsilon \sum_{K \in \mathcal{T}_h} \int_{K} G_{p,\delta}(\nabla \hat u_h, \nabla e)\\ & & + C(\varepsilon)\sum_{K \in \mathcal{T}_h} \int_{K} G_{p',1}(\|\nabla \hat u_h\|^{p-1}, h_z (f-f_z))\\ & & + \varepsilon \sum_{z\in D}\sum_{l \in \mathcal{E}_h^z} \min_{\bar K' \cap l \neq \emptyset} \int_{\omega_z} G_{p,\delta}(\nabla \hat u_h\|_{K'}, \left[\partial_{\nu_\mathcal{E}} u_h\right]_l)\\ &\lesssim& \varepsilon \|e\|_{(1,\hat u, p)}^2+C(\varepsilon)\ \eta_f^2 + \varepsilon \sum_{z\in D}\sum_{l \in \mathcal{E}_h^z} \min_{\bar K' \cap l \neq \emptyset} \int_{\omega_z} G_{p,\delta}(\nabla \hat u_h\|_{K'}, \left[\partial_{\nu_\mathcal{E}} u_h\right]_l). $$ $G_h \hat u_h$ is continuous across any interior edge $l$, so that $[\partial_{\nu}\hat u_h]_l = [\partial_{\nu}\hat u_h - G_h \hat u_h]_l$ and $$\min_{\bar K' \cap l \ne \emptyset} \int_{\omega_z} G_{p,\delta}(\nabla \hat u_h\|_{K'}, [\partial_{\nu}\hat u_h - G_hu_h]_l) \lesssim \int_{\omega_z} G_{p,\delta}(\nabla \hat u_h, \nabla \hat u_h - G_h \hat u_h).$$ Therefore, $$\begin{aligned} \int_{\Omega }f(e - e_h) &\lesssim& \varepsilon \|e\|_{(1,\hat u, p)}^2+C(\varepsilon) \eta_f^2 + \varepsilon \sum_{z\in D}\sum_{l \in \mathcal{E}_h^z} \int_{\omega_z} G_{p,\delta}(\nabla \hat u_h, [\partial_{\nu}\hat u_h- G_h \hat u_h]_l)\\ &\lesssim& \varepsilon \|e\|_{(1,\hat u, p)}^2+C(\varepsilon) \eta_f^2 +\varepsilon \sum_{K \in \mathcal{T}_h} \int_{K} G_{p,\delta}(\nabla \hat u_h, \nabla \hat u_h - G_h \hat u_h)\\ &=&\varepsilon \|e\|_{(1,\hat u, p)}^2+C(\varepsilon) \eta_f^2 +\varepsilon \eta_{gr}^2.\end{aligned}$$ Concerning the second term, let $$A_l = \varrho(\nabla \hat u_h\|_{K_{l,1}})\ \partial_{\nu} \hat u_h\|_{K_{l,1}} - \varrho(\nabla \hat u_h\|_{K_{l,2}})\ \partial_{\nu} \hat u_h\|_{K_{l,2}},$$ where again $l \subset \bar K_{l,1} \cap \bar K_{l,2}$, and the unit normal $\nu$ points outward of $K_{l,1}$. Therefore $$\begin{aligned} - \langle G'\hat u_h, e - \pi e\rangle &=& - \sum_{K \in \mathcal{T}_h} \int_{\partial K} \varrho(\nabla \hat u_h)\ \partial_\nu \hat u_h\|_{\partial K}\ (e-\pi e) \\ & = & - \sum_{l \not\subset {{\partial \Omega} }} \int_l A_l (e-\pi e) - \sum_{l \subset {{\partial \Omega} }} \int_l \varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_l\ (e-\pi e).\end{aligned}$$ Repeating the analysis of [@cly], Theorem 5.1, with the help of Lemma \[gradlemma\] gives $$- \sum_{l \not\subset {{\partial \Omega} }} \int_l A_l (e-\pi e) \lesssim \eta_{gr}^2 + \varepsilon (\|e\|_{(1, \hat u_h, p)}^2 + \eta_{gr}^2).$$ Thus $$\begin{aligned} \\|e, \tilde e, \epsilon\\|_{Y^p}^p &\lesssim & \|e\|_{(1,\hat u, p)}^2 + \\|e\|_{{\partial \Omega} }+ \tilde e\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \\|\epsilon\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2 \\ &\lesssim& \eta_f^2+\varepsilon (\eta_{gr}^2 + \|e\|_{(1,\hat u, p)}^2)+ \eta_{gr}^2 + \varepsilon (\|e\|_{(1, \hat u_h, p)}^2 + \eta_{gr}^2)\\ & & +\int_{{\Gamma_s}}\left\{-\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\Gamma_s}}(\tilde e_h - \tilde e)+g (\|\tilde e_h +\hat v_h\| - \|\tilde e +\hat v_h\|)\right\} \\ & & - \int_{{\partial \Omega} }\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\partial \Omega} }\ ((e-\pi e)\|_{{\partial \Omega} }+\tilde e - \tilde e_h)\\ &&+\langle t_0-\mathcal{W}(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0) - (\mathcal{K}'-1) \hat \phi_h, (e -\pi e)\|_{{\partial \Omega} }+\tilde e - \tilde e_h\rangle \\ & & -\langle \nu-\nu_h, \mathcal{V} \hat \phi_h + (1-\mathcal{K})(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0)\rangle. $$ We bound the second, third + fourth as well as the final line individually. Cauchy-Schwarz and Young’s inequality allow to estimate the last term by $$\varepsilon \\|e\|_{{\partial \Omega} }+ \tilde e\\|_{{W^{\frac{1}{2},2}({{\partial \Omega} })}}^2 + \varepsilon \\|\epsilon\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })}^2+\varepsilon^{-1}\ \\|\mathcal{V} \hat \phi_h + (1-\mathcal{K})(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0)\\|_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}}^2,$$ and the latter by $\eta_S^2$ (cf. [@cast]). The third and fourth lines are estimated by (cf. [@cast]) $$\\|-\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h+t_0-\mathcal{W}(\hat u_h\|_{{\partial \Omega} }+\hat v_h-u_0) - (\mathcal{K}'-1) \hat \phi_h\\|_{W^{-\frac{1}{2},2}({{\partial \Omega} })} \\|(e-\pi e)\|_{{\partial \Omega} }+\tilde e \\|_{{{W^{\frac{1}{2},2}({{\partial \Omega} })}}}$$ which lead to $\eta_{\partial}$, where we have choosen $\tilde e_h=0$, i.e. $v_h=\hat v_h$. Finally, using the triangle inequality, the second line is simplified as follows: $$\begin{aligned} &\int_{{\Gamma_s}}\{-\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\Gamma_s}}(\tilde e_h - \tilde e)+g (\|\tilde e_h +\hat v_h\| - \|\tilde e +\hat v_h\|)\}\\ &\leq\int_{{\Gamma_s}}\{-\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\Gamma_s}}(\tilde e_h - \tilde e)+g\|\tilde e_h -\tilde e \|\}\\ &=\int_{{\Gamma_s}}\{\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\Gamma_s}}\tilde e +g\|\tilde e \|\}\\ &\leq \\|\varrho(\nabla \hat u_h)\ \partial_{\nu} \hat u_h\|_{{\Gamma_s}}\\|_{W^{-\frac{1}{2},2}({{\Gamma_s}})}\\|\tilde e\\|_{W^{\frac{1}{2},2}({{\Gamma_s}})} +\\|g\\|_{W^{-\frac{1}{2},2}({{\Gamma_s}})}\\|\tilde e\\|_{W^{\frac{1}{2},2}({{\Gamma_s}})} .\end{aligned}$$ We may use the Cauchy-Schwartz inequality and the inverse inequality, leading to $\eta_g$. Numerical results {#sec:num} ================= With the subset $\Lambda_h$ of ${{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}$ given by $$\Lambda_h=\{\sigma_h\in {{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}\,:\,\|\sigma_h(x)\|\leq 1\mbox{ a.e. on } \Gamma_s\},$$ we can define an Uzawa algorithm for solving the variational inequality analogously to [@mast]. In order to introduce this algorithm, let $P_\Lambda$ be the projection of ${{\widetilde{W}_h^{\frac{1}{2},2}({{\Gamma_s}})}}$ onto $\Lambda_h$, i.e. for every nodal point of the mesh ${\mathcal{T}}_h\|_{\Gamma_s}$ holds $\delta\mapsto P_\Lambda(\delta)=\sup\{-1,\inf(1,\delta)\}$. \ 1. Choose $\sigma^0_h\in\Lambda_h$. 2. For $n=0,1,2,\ldots$ find $(u^n_h,v^n_h)\in X^p_h$ such that $$\label{equ:uzca} \langle G' u^n_h, u_h\rangle + \langle S_h(u^n_h\|_{{\partial \Omega} }+v^n_h), u_h\|_{{\partial \Omega} }+ v_h\rangle +\int_{\Gamma_s} g\sigma^n_hv_h\,ds =\lambda_h(u_h,v_h) $$ for all $(u_h,v_h)\in X^p_h$. 3. Set $$\label{equ:uzs} \sigma^{n+1}_h=P_\Lambda(\sigma^n_h+\rho g v^n_h),$$ where $\rho>0$ is a sufficiently small parameter that will be specified later. 4. Repeat with 2. until a convergence criterion is satisfied. In our first example the model problem is defined on the L-shape with $\Omega=[-\frac14,\frac14]^2\backslash[0,\frac14]^2$, $\Omega^c={\mathbb{R}}^2\backslash\Omega$. The friction part of the interface is $\Gamma_s=\overline{(-\frac14,-\frac14)(\frac14,-\frac14)} \cup\overline{(-\frac14,-\frac14)(-\frac14,\frac14)}$, see Figure \[fig:geo\]. In this example we choose $\varrho(t)=(\varepsilon+t)^{p-2}$, with $p=3$ and $\varepsilon=0.00001$. Our volume and boundary data are given by $f=0$ and $u_0=r^{2/3}\sin\frac23(\varphi-\frac\pi2)$, $t_0=\partial_\nu u_0\|_{{\partial \Omega} }$. The friction parameter is $g=0.5$, leading to slip conditions on the interface. We have applied the Uzawa algorithm as introduced above with the damping parameter $\rho=25$ to solve the variational inequality. The nonlinear variational problem in the Uzawa algorithm is then solved by Newton’s method in every Uzawa-iteration step. In Table \[tab:uni\] we give the degrees of freedom, the value $J_h(\hat u_h,\hat v_h)$ and the error measured with the help of $J$, i.e. $\delta J=J_h(\hat u_h,\hat v_h)-J(\hat u,\hat v)$, where we have obtained the value $J(\hat u,\hat v)$ by extrapolation of $J_h(\hat u_h,\hat v_h)$. Due to the slip condition, we need only a few Uzawa steps. But as a consequence of the degeneration of the system matrix, due to the nonlinearity, the iteration numbers for the MINRES solver, applied to the linearized system, are very high, leading to large computation times. The convergence rate $\alpha_J$ is suboptimal, due to the presence of singularities, in the boundary data as well, as due to the change of boundary conditions. (2.4,2.4)(-1.2,-1.2) (-1,-1)[(1,0)[2]{}]{} (-1,-1)[(0,1)[2]{}]{} ( 1,-1)[(0,1)[1]{}]{} (-1, 1)[(1,0)[1]{}]{} ( 0, 0)[(1,0)[1]{}]{} ( 0, 0)[(0,1)[1]{}]{} (-0.3,-0.3) (0.3,0.3) (1.1,-0.5) (-1.15,0.0) (-1.1,-1.1)[(1,0)[2.1]{}]{} (-1.1,-1.1)[(0,1)[1.0]{}]{} (-1.1, 0.2)[(0,1)[0.8]{}]{} (-1.15,1.0)[(1,0)[0.1]{}]{} (1.0,-1.15)[(0,1)[0.1]{}]{} DOF $J_h(\hat u_h,\hat v_h)$ $\delta J$ $\alpha_{J}$ $It_{\rm Uzawa}$ $\tau(s)$ ------- -------------------------- ------------ -------------- ------------------ ----------- 28 -0.511609 0.017249 — 2 0.190 80 -0.517938 0.010920 -0.435 2 0.640 256 -0.521857 0.007001 -0.382 2 2.440 896 -0.524293 0.004566 -0.341 2 11.05 3328 -0.525841 0.003017 -0.316 2 61.85 12800 -0.526865 0.001993 -0.308 2 437.5 50176 -0.527571 0.001287 -0.320 2 4218. : \[tab:uni\] Convergence rates and Uzawa steps for uniform meshes (Example 1) In our second example we use the same model geometry as before (see Fig. \[fig:geo\]). Here we choose the friction boundary $\Gamma_s=\emptyset$. Therefore our model problem reduces to a non-linear p-Laplacian FEM-BEM coupling problem, where we can prescribe the solution. In this example we choose $\varrho(t)=(\varepsilon+t)^{p-2}$, with $p=3$ and $\varepsilon=0.00001$. We prescribe the solution by $u_1=r^{2/3}\sin\frac23(\varphi-\frac\pi2)$ and $u_2=0$. Then the boundary data $u_0,t_0$ and volume data $f$ are given by $u_0=u_1\|_\Gamma$, $t_0=\varrho(\|\nabla u_1\|)\partial_\nu u_1$ and $f=-{\mathop{\rm div}\nolimits}(\varrho(\|\nabla u_1\|)\nabla u_1)$. In the following we give errors in the $\\|\cdot\\|_{W^{1,p}(\Omega)}$ norm and in the quasinorm $\|u-u_h\|_Q=\\|u-u_h\\|_{(1,u_h,p)}$. In Tab. \[tab:h4\] we give the errors, convergence rates, number of Newton iterations $It_{Newton}$ and the computing time for the uniform h-version with rectangles. We observe that the convergence rate in the quasi-norm $\|\cdot\|_{Q}$ is better than in the $\\|\cdot\\|_{W^{1,3}(\Omega)}$-norm. The number of Newton iterations appears to be bounded. In Tab. \[tab:h3\] for the uniform h-version with triangles, we give the errors, convergence rates, error estimator $\eta$, efficiency indices $\delta_u/\eta$ for the $\\|\cdot\\|_{W^{1,3}(\Omega)}$-norm and $\delta_q/\eta$ for the $\|\cdot\|_{Q}$-norm, number of Newton iterations and the computing time. Again, here we observe that the convergence rate in the quasi-norm $\|\cdot\|_{Q}$ is better than in the $\\|\cdot\\|_{W^{1,3}(\Omega)}$-norm and the number of Newton iterations is bounded. The efficiency index $\delta_u/\eta$ appears to be constant, whereas the efficiency index $\delta_q/\eta$ appears to be decreasing. Tab. \[tab:adap\] gives the corresponding numbers for the adaptive version, using a blue-green refining strategy for triangles and refining the 10% elements with the largest indicators. Here we observe that the convergence rates for both norms are very similar and that both efficiency indices are bounded. Figure \[fig:L2\] give the errors for all methods in the $\\|\cdot\\|_{W^{1,3}(\Omega)}$-norm and the $\|\cdot\|_{Q}$ quasi-norm together with the error indicators for the uniform and adaptive methods. Figure \[fig:meshes\] presents the sequence of meshes generated by the adaptive refinement strategy. We clearly observe the refinement towards the reentrant corner with the singularity of the solution. DOF $\\|u-u_h\\|_{1,3}$ $\alpha$ $\|u-u_h\|_{Q}$ $\alpha$ $It_{Newton}$ $\tau(s)$ -------- ------------------- ---------- --------------- ---------- --------------- ----------- 21 0.1711499 — 0.1293512 — 22 0.224 65 0.1308635 -0.238 0.0860870 -0.360 22 0.424 225 0.1039326 -0.186 0.0612225 -0.274 23 1.668 833 0.0826578 -0.175 0.0438478 -0.255 23 6.804 3201 0.0657091 -0.170 0.0314280 -0.247 23 27.28 12545 0.0522196 -0.168 0.0225589 -0.243 24 120.8 49665 0.0414910 -0.167 0.0162319 -0.239 24 560.1 197633 0.0329617 -0.167 0.0117169 -0.236 24 2678. : \[tab:h4\]Errors, convergence rates (Example 2, uniform mesh with rectangles) DOF $\\|u-u_h\\|_{1,3}$ $\alpha$ $\|u-u_h\|_{Q}$ $\alpha$ $\eta$ $\delta_u/\eta$ $\delta_q/\eta$ $It_{New}$ $\tau(s)$ -------- ------------------- ---------- --------------- ---------- -------- ----------------- ----------------- ------------ ----------- 21 0.1945908 — 0.1510064 — 1.027 0.190 0.147 22 0.620 65 0.1535874 -0.209 0.1081632 -0.295 0.690 0.223 0.157 22 2.212 225 0.1219287 -0.186 0.0774765 -0.269 0.516 0.236 0.150 22 8.617 833 0.0969249 -0.175 0.0555005 -0.255 0.394 0.246 0.141 23 36.00 3201 0.0770270 -0.171 0.0396882 -0.249 0.304 0.253 0.131 23 144.2 12545 0.0611994 -0.168 0.0283778 -0.246 0.236 0.260 0.120 24 608.7 49665 0.0486160 -0.167 0.0203130 -0.243 0.184 0.265 0.111 24 2530. 197633 0.0386151 -0.167 0.0145686 -0.241 0.144 0.269 0.102 24 11000 : \[tab:h3\]Errors, onvergence rates, estimator $\eta$, reliability $\delta_u/\eta$ and $\delta_q/\eta$ (Example 2, uniform mesh with triangles) DOF $\\|u-u_h\\|_{1,3}$ $\alpha$ $\|u-u_h\|_{Q}$ $\alpha$ $\eta$ $\delta_u/\eta$ $\delta_q/\eta$ $It_{New}$ $\tau(s)$ ------- ------------------- ---------- --------------- ---------- -------- ----------------- ----------------- ------------ ----------- 21 0.1945908 — 0.1510064 — 1.027 0.190 0.147 22 0.196 32 0.1602214 -0.461 0.1205155 -0.535 0.804 0.199 0.150 22 0.332 54 0.1275298 -0.436 0.0918131 -0.520 0.603 0.212 0.152 22 0.648 93 0.1019990 -0.411 0.0699054 -0.501 0.442 0.231 0.158 22 1.132 152 0.0821754 -0.440 0.0540462 -0.524 0.325 0.253 0.166 23 2.000 249 0.0679251 -0.386 0.0449420 -0.374 0.246 0.276 0.183 23 3.352 400 0.0558447 -0.413 0.0369614 -0.412 0.190 0.294 0.194 23 5.700 625 0.0439784 -0.535 0.0277857 -0.639 0.148 0.297 0.188 24 9.896 986 0.0352491 -0.485 0.0217361 -0.539 0.116 0.305 0.188 24 17.45 1528 0.0279287 -0.531 0.0167409 -0.596 0.091 0.308 0.184 25 31.16 2322 0.0222760 -0.540 0.0129489 -0.614 0.071 0.312 0.181 25 53.98 3620 0.0177640 -0.510 0.0102552 -0.525 0.056 0.316 0.182 25 106.7 5544 0.0142059 -0.524 0.0080233 -0.576 0.044 0.320 0.181 25 205.3 8449 0.0112965 -0.544 0.0063426 -0.558 0.035 0.322 0.181 26 422.4 12810 0.0090396 -0.536 0.0050706 -0.538 0.028 0.325 0.183 26 1060. 19222 0.0072288 -0.551 0.0040370 -0.562 0.022 0.329 0.184 26 2400. 29006 0.0057984 -0.536 0.0032478 -0.529 0.018 0.333 0.186 27 5460. 43593 0.0046615 -0.536 0.0026230 -0.524 0.014 0.337 0.190 27 13000 : \[tab:adap\]p-Laplacian (adaptive), convergence rates, estimator $\eta$, reliability $\delta_u/\eta$ and $\delta_q/\eta$ \ \ \ [99]{} C. Carstensen, J. Gwinner, FEM and BEM coupling for a nonlinear transmission problem with Signorini contact, SIAM J. Numer. Anal. 34* (1997) 1845–1864. C. Carstensen, W. Liu, N. Yan, A posteriori FE error control for $p$–Laplacian by gradient recovery in quasi–norm, Math. Comp. 75, 256 (2006) 1599–1616. C. Carstensen, E. P. Stephan, Adaptive coupling of boundary elements and finite elements, RAIRO Modél. Math. Anal. Numér. 29 (1995) 779–817. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988) 613–626. R. G. Durán, M. A. Muschietti, The Korn inequality for Jones domains, Electron. J. Differential Equations 127 (2004) 1–10. C. Ebmeyer, W. B. Liu, Quasi–norm interpolation error estimates for the piecewise linear finite element approximation of $p$–Laplacian problems, Numer. Math. 100 (2005) 233–258. H. Gimperlein, M. Maischak, E. Schrohe, E. P. Stephan, FE–BE coupling for a transmission problem involving microstructure, in preparation (2008). W. Liu, N. Yan, Quasi-norm local error estimators for $p$-Laplacian, SIAM J. Numer. Anal. 39 (2001) 100–127. M. Maischak, E. P. Stephan, A FEM–BEM coupling method for a nonlinear transmission problem modelling Coulomb friction contact, Comput. Methods Appl. Mech. Engrg. 194 (2005) 453–466. J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Teubner, Leipzig, 1983. H. Triebel, Theory of Function Spaces I/II, Birkhäuser, Basel, 1983/1992. E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer, New York, 1990.
--- abstract: 'Recent measurements of Si IV/C IV ratios in the high-redshift Ly$\alpha$ forest (Songaila & Cowie, AJ, 112, 335 \[1996a\]; Savaglio et al., A&A, in press \[1997\]) have opened a new window on chemical enrichment and the first generations of stars. However, the derivation of accurate Si/C abundances requires reliable ionization corrections, which are strongly dependent on the spectral shape of the metagalactic ionizing background and on the “local effects” of hot stars in nearby galaxies. Recent models have assumed power-law quasar ionizing backgrounds plus a decrement at 4 Ryd to account for He II attenuation in intervening clouds. However, we show that realistic ionizing backgrounds based on cosmological radiative transfer models produce more complex ionizing spectra between 1–5 Ryd that are critical to interpreting ions of Si and C. We also make a preliminary investigation of the effects of He II ionization front non-overlap. Because the attenuation and re-emission by intervening clouds enhance Si IV relative to C IV, the observed high Si IV/C IV ratios do [not]{} require an unrealistic Si overproduction \[Si/C $\geq 3$ (Si/C)$_{\odot}$\]. If the ionizing spectrum is dominated by “local effects” from massive stars, even larger Si IV/C IV ratios are possible. However, unless stellar radiation dominates quasars by more than a factor of 10, we confirm the evidence for some Si overproduction by massive stars; values Si/C $\approx 2$ (Si/C)$_{\odot}$ fit the measurements better than solar abundances. Ultimately, an adequate interpretation of the ratios of C IV, Si IV, and C II may require hot, collisionally ionized gas in a multiphase medium.' author: - 'Mark L. Giroux and J. Michael Shull' title: 'The Influence of the Photoionizing Radiation Spectrum on Metal - Line Ratios in Ly$\alpha$ Forest Clouds' --- Introduction ============ The detection of metals associated with the “high” column density \[N(H I) $> 10^{14.5}$ cm$^{-2}$\] Ly$\alpha$ forest of absorbers in quasar spectra (Cowie et al. 1995; Tytler et al. 1995) provides new challenges to the understanding of the nature and origin of these absorbers. The contamination by metals must now be accounted for in models for cloud evolution. It is also possible to perform a detailed analysis of the thermodynamic and ionization state of these absorbers, which have previously been probed only by their hydrogen Ly$\alpha$ absorption and the integrated effect of their He II Ly$\alpha$ absorption (Giroux, Fardal, & Shull 1995). If the clouds are photoionized, the detection of C II, C IV and Si IV may constrain the shape of the metagalactic radiation spectrum and indicate the epoch of dramatic changes in the ionizing radiation field at $z > 3$ (Songaila & Cowie 1996a; Savaglio et al. 1997). The multiple C IV lines associated with a single H I absorption complex (Songaila & Cowie 1996a) belies a simple picture of a homogeneous cloud. These absorbers, like Lyman Limit systems, may better be interpreted with multiphase models (Giroux, Sutherland, & Shull 1994; Petitjean, Riediger, & Rauch 1996; Haehnelt, Rauch, & Steinmetz 1997; Hellsten et al. 1997) that may include hot, collisionally ionized gas. We defer such interpretation to a later study, focussing here on the tradition of analyzing metal-line ratios with single-phase photoionization models. In this way, we can isolate the effects of the incident spectrum on species with ionization potentials in the range 1–5 Rydbergs (e.g., C III, C IV, Si III, Si IV). For example, Songaila & Cowie (1996a) argue that the observed increase in the Si IV/C IV ratio at $z > 3.1$ suggests that the He III regions around ionizing sources may not have overlapped by that epoch, cutting off all radiation with $h \nu > 54.4$ eV. Here, we primarily consider models for the averaged metagalactic background in which He II ionization fronts have overlapped before $z \approx 4$. We do consider a limiting case in which no photons above 4 Ryd are present in the background, as well as a case where we restore the much less heavily attenuated x-ray background. In the past decade, metal-line ratios in Lyman Limit ($N_{HI} {}^>_\sim 10^{17}$ cm$^{-2}$) absorbers have been used to probe the shape of the ionizing spectrum, and to estimate the intrinsic metallicity of the absorbers (cf. Bergeron & Stasinska 1986; Sargent, Boksenberg & Steidel 1988; Steidel, Sargent &, Boksenberg 1988; Steidel & Sargent 1989; Bergeron & Ikeuchi 1990; Miralda-Escudé & Ostriker 1990; Donahue & Shull 1991; Madau 1992). With the advent of new spectrographs on large telescopes, the lower $N_{HI}$ absorbers can be exploited in the same way. These absorbers hold a special interest, because the source of their metal enrichment is still unclear. While the metal lines in Lyman limit absorbers have been associated with the haloes of bright galaxies, the number of Ly$\alpha$ forest absorbers greatly exceeds the corresponding number of bright galaxies, and the Ly$\alpha$ absorbers do not seem to show the same clustering properties as galaxies. As yet, metal lines have only been associated with “high-column” absorbers \[N(H I) $>10^{14.5}$ cm$^{-2}$\], which numerically are a small fraction of the Ly$\alpha$ forest. It is still possible that these clouds may represent an overlap of the high-column end of the pristine Ly$\alpha$ forest clouds and the low-column end of the metal enriched clouds associated with galactic haloes. As a result some, if not all, of the metal-line absorbers may still be associated with the galaxies that contain the stars responsible for their metals. For example, over a Hubble time at $z = 3$, galactic outflows at $300 V_{300}$ km s$^{-1}$ will transport heavy elements a distance $$d \leq (300~{\rm kpc}) V_{300} h^{-1} \left[ \frac {1+z}{4} \right]^{-3/2}$$ for a Friedmann universe with $H_0 = (100~{\rm km~s}^{-1})h$ and $\Omega_0 = 1$. In practice, the period of heavy element injection may have lasted only $10^8$ yrs, and the metals could move distances of only 30-50 kpc from their sources (bright galaxies, dwarf galaxies, globular clusters). As Madau & Shull (1996) have emphasized, metal enrichment of the Ly$\alpha$ absorbers implies a substantial Lyman continuum (LyC) emission accompanying the star formation at $z {}^>_\sim 3.5$. Thus, a large fraction of the metagalactic background at high redshift could be due to massive stars, a point also made by Giroux & Shapiro (1996) and Savaglio et al. (1997). Furthermore, the absorbers with metal lines may be sufficiently close to the local sources that the incident radiation field is dominated by local hot-star radiation. In contrast, if the metal enrichment of the Ly$\alpha$ forest arises from Population III stars at $z > 10$ (Couchman 1985; Ostriker & Gnedin 1996), the stellar ionizing radiation will be greatly attenuated by intergalactic absorption, and the metagalactic background at $z = 3$ will be dominated by the harder spectrum of quasars. Because metal-line systems would then be less likely to be associated with nearby sources of stellar radiation, their absorption-line ratios would be more representative of the metagalactic ionizing background. Preliminary analyses (Songaila & Cowie 1996a; Savaglio et al. 1997; Haehnelt 1997) of the Ly$\alpha$ forest clouds have assumed photoionization by simple power-law ionizing spectra. The only way these models account for filtering of the ionizing source spectrum by intervening clouds is to include a decrement at $\nu_{HeII}$ (4 Ryd). More realistic models include several additional effects. First, when absorption due to intervening clouds is included, the background spectrum just above the H I and He II ionization edges shows a flatter power law than that of the sources (Miralda-Escudé & Ostriker 1990; Madau & Meiksin 1994; Giroux et al. 1995). Second, Haardt & Madau (1996) and Fardal, Giroux, & Shull (1997) show, using cosmological radiative transfer, that cloud emission by recombination to the H I and He II ground states and He II Ly$\alpha$ and 2-photon radiation alters the extent of the decrement at $\nu_{HeII}$ and changes the shape of the spectrum between 1 and 5 Rydbergs. Including a more realistic spectral shape will alter the relative populations of metal ions with edges in the 1 to 5 Ryd range even if the ionizing sources are all assumed to be quasars with intrinsic power-law ionizing spectra. In this paper, we consider the extent to which these alterations of the assumed shape of the incident radiation field affect the interpretation of the metallicity of these absorbers and the nature of the photoionizing sources. We use the data sets of Songaila & Cowie (1996a) and Savaglio et al. (1997), which include Si IV and C IV lines associated with higher column density Ly$\alpha$ forest clouds. Songaila & Cowie (1996a) provide an important subsample for which C II lines are also observed. The Metagalactic Radiation Background ===================================== Ionizing Sources ---------------- Assumptions about the nature of the ionizing sources of the metagalactic background are uncertain, since the intrinsic EUV spectra of quasars are not well defined. For simplicity, we assume that all quasars have constant power-law spectra $F_\nu \propto \nu^{-\alpha_s}$. Our choice of spectral index, $\alpha_s = 1.8$, is consistent with HST observations of the EUV spectrum of quasars (Zheng et al. 1997) for radio-quiet quasars at $z = 1.5-2.0$. As discussed by Fardal et al. (1997), if the ionizing background (1-5 Ryd) is dominated by these quasars, the He II Ly$\alpha$ absorption optical depth $\tau_{HeII} \approx 1$ at $z = 3$. For clarity, we denote the cloud-filtered, quasar-dominated mean intensity by $J_{Q,f}(\nu)$. The characteristic spectrum of stellar sources in starburst galaxies is also not well known. Most models agree (e.g., Bruzual & Charlot 1993) that few photons are present with frequency above 54.4 eV, and that the spectrum falls off above $h\nu \approx$ 45 eV. Recent line-blanketed, non-LTE hot-star models (Gabler et al. 1992; Sellmaier et al. 1996; Schaerer & de Koter 1997) vary considerably in the amount of radiation in the He I continuum ($24.6 < h\nu < 54.4$ eV) they predict, compared to the LTE line-blanketed models of Kurucz (1992). An ongoing uncertainty, for our purposes, is the lack of low-metallicity stellar models, which are more appropriate for high-redshift starburst galaxies. For the present work, we use a starburst spectrum (Sutherland & Shull 1997) discussed in the context of the ionizing spectrum of metal-forming galaxies by Madau & Shull (1996) \[see their Fig. 3\]. In brief, it is a time integrated spectrum of a Gaussian starburst of width $2 \sigma = $ 4 Myr containing $5000 M_\odot $ of stars between 8 and 85 $M_\odot$ with IMF slope $\Gamma = 1.6$ where $dN(>M)/dM \propto M^{-\Gamma}$ and $N(>M)$ is the number of stars with mass greater than $M$. Estimates of the mean metagalactic ionizing intensity at $z > 2$, based on the statistical diminution in the number of Ly$\alpha$ forest lines close to quasars (the proximity effect), indicate a specific intensity at the hydrogen Lyman limit, $\nu_H$, of $J_{-21} = 0.7-1.5$ where $J_{-21}$ is in units of $10^{-21}$ergs cm$^{-2}$ s$^{-1}$ Hz$^{-1}$ sr$^{-1}$ (Cooke, Espey, & Carswell 1997). If all Ly$\alpha$ absorbers possess a metallicity of $Z = 0.01 Z_\odot$ by redshift $z \approx 3.5$, the stellar ionizing photons associated with the production of the metals may be comparable to the ionizing photons produced by quasars (Madau & Shull 1996; Giroux & Shapiro 1996). If these ionizing photons are produced at higher redshift (e.g., Population III stars at $z > 5$) they will be highly attenuated by IGM absorption at $z > 4$. A further uncertainty is the fraction, $\langle f_{\rm esc} \rangle$, of ionizing photons that successfully escape from the galaxy in which the stars reside (Dove & Shull 1994). It does seem possible, though, that stellar sources could be an important contributor to the metagalactic background, which could dominate in regions local to starburst galaxies. For example, the total Lyman continuum production rate of the Milky Way was recently inferred from COBE observations of \[N II\] 205 $\mu$m emission to be $3.5\times 10^{53}$ photons s$^{-1}$ (Bennett et al. 1994), and most starburst galaxies exceed the value $S = (10^{54})S_{54}$ photons s$^{-1}$. Assuming that a substantial fraction, $\langle f_{\rm esc} \rangle \geq 0.1$, of these photons escape these starburst galaxies, the mean photoionization rate of hydrogen due to the galactic source may exceed $J_{-21} = 1$ if the absorber lies within a distance $$R_{\rm cr} = (88~{\rm kpc}) \; S_{54}^{1/2} J_{-21}^{-1/2} \; \langle f_{\rm esc} \rangle ^{1/2} \left[ \frac {\alpha_s +3}{4.8} \right]^{1/2} \; .$$ Here, we have assumed the QSO spectra in the range 1–5 Ryd to be power laws with spectral index $\alpha_s \approx 1.8$. In this paper, we account for a stellar contribution to the mean intensity falling upon our absorbers in a global way. We assume that up to $2/3$ of the ionizing emissivity from primary sources is stellar, as opposed to secondary sources due to recombination radiation from clouds. The remainder of the primary emissivity is then from AGN. We then compute the radiative transfer of primary and secondary sources through the IGM to derive the metagalactic ionizing background, $J_{SQ,f}(\nu)$. The mean intensities $J_{Q,f}$ (pure QSO sources) and $J_{SQ,f}$ (stars + QSOs) therefore bracket our estimates of the contribution of stellar sources to the uniform metagalactic background, $J_{MB}(\nu)$. However, even if the metagalactic ionizing background is quasar dominated, the absorber may primarily see a [local]{} unfiltered source of stellar photons, $J_{L} (\nu)$. We consider models which assume $J_{L} / J_{MB} = 5, 10, 20$ at $\nu = \nu_H$ for $J_{MB} = J_{Q,f}$ and $J_{L} / J_{MB} = 3, 5, 10, 20$ for $J_{MB} = J_{SQ,f}$. He II Ionization Fronts ----------------------- Within a picture of ionizing sources turning on in a neutral medium, Miralda-Escudé & Rees (1994) made the important suggestion that the He II I-front which propagates away from sources with a sufficiently soft ionizing spectrum can lag the hydrogen I-front. Madau & Meiksin (1994) showed that, if $\alpha_s = 1.9$, the overlap of He II I-fronts could be delayed to as late as $z=3$. As a result, both Jakobsen et al. (1994) and Songaila & Cowie (1996) suggested that, at $z = 3.1$, He II I-fronts around the dominant ionizing sources may not have overlapped. Although a full treatment of overlapping He II I-fronts is beyond the scope of this paper, we can make a simple estimate of the ratio of I-front velocities for H I and He II. From the flux-limiting equations governing I-front propagation (Shapiro & Giroux 1987; Donahue & Shull 1987), the velocity relative to the local Hubble flow is given by $$\left[ \frac {dr_i}{dt} - H r_i \right] = \frac {S_i} {4 \pi r_i^2 n_i} \; ,$$ where $i$ refers either to the H I ionizing front (1 Ryd continuum) or to the He II ionizing front (4 Ryd continuum). (The He I ionizing front is assumed to be coincident with the H I front, and we neglect recombinations and attenuation within the ionized zone.) Here, $S_i$ (photons s$^{-1}$) represents the photon production rates of the source in the H I and He II continua, $n_i$ is the density of H I or He II, and $r_i$ is the front distance measured from the source. The propagation speeds of the H I and He II ionization fronts will be governed by the “flux-to-density” ratio, $(S_i/n_i r_i^2$), at 1 and 4 Ryd. For a QSO spectrum with power-law index $\alpha_s$, one finds that $S_{\rm HeII}/S_{\rm HI} = 4^{-\alpha_s}$. The primordial helium abundance by mass has been estimated at values $Y_P=0.231 \pm 0.006$ (Skillman & Kennicutt 1993) and $Y_P = 0.232 \pm 0.003$ (Olive & Steigman 1995), while Copi et al. (1994) have suggested a $2\sigma$ concordance range of $Y_P = 0.221-0.243$ from theoretical models of Big-Bang nucleosynthesis. If helium has a cosmological abundance, $n_{\rm He} / n_{\rm H} = 0.0785$ by number ($Y = 0.239$ by mass), the flux-to-density ratios at a fixed radius $r_i$ are equal for $$\alpha_s = - \frac {\ln [n_{\rm He}/n_{\rm H}]} {\ln 4} \approx 1.84 \; .$$ At this critical spectral index, the H I and He II ionization fronts will coincide. Interestingly, this critical index is very close to the mean index found by Zheng et al. (1997) for radio-quiet quasars. For AGN with harder spectra ($\alpha_s < 1.84$), the He II ionization front will precede the H I front by a small amount, equivalent to a few optical depths in the 4 Ryd continuum. However, for those AGN with softer spectra ($\alpha_s > 1.84$), the He II front will propagate at a lower speed and lie well within the H I front. For example, Zheng et al. (1997) found that radio-loud quasars had a mean index $\alpha_s \approx 2.2$. In this case, the regions outside the He II fronts will see no 4 Ryd continuum radiation. We therefore consider a limiting case in which all photons above 4 Ryd are removed from the metagalactic background spectrum. We also consider an intermediate case, arising from the fact that quasars emit radiation at least into the x-ray range of energies, so that high-energy photons will not be strongly attenuated, even in regions where helium is entirely in the He II ionization stage. In the Ly$\alpha$ clouds, we estimate the effect of the presence of these high-energy photons by presuming that H II regions have overlapped by $z=4$, and that the Ly$\alpha$ forest clouds, whose distribution in $N_{\rm HI}$ is known (see §2.2), have a maximal He II/H I ratio. In interpreting He II Gunn-Peterson measurements, several groups (Madau & Meiksin 1994; Giroux et al. 1995) find that the ratio, $\eta$ = N(He II)/N(H I), must be 50–100 in order to explain the $\lambda < 304$ Å absorption as He II line opacity in the H I Ly$\alpha$ forest. This large ratio arises because He II is more difficult to photoionize than H I. The H neutral fraction $f_{HI}$ is always less than $(n_{\rm He}/n_{\rm H}) \eta^{-1}$, where $\eta =$ N(He II)/N(H I) within an absorbing cloud. Thus, the maximal ratio $\eta_{max}$ is bounded by limits on the baryon density from Big Bang nucleosynthesis (BBN), $\Omega_b h^2 \le 0.024$ (Copi et al. 1994). At $z=3.4$, if only absorbers with $N_{HI} < 10^{15}$ cm$^{-2}$ are considered to contribute to $\Omega_{HI}$, we find $\eta_{max} < 5000$. This limit produces a local continuum optical depth at 4 Ryd, $d\tau / dz > 180$ at $z =3.4$, falling to $4$ by $\nu = 25\nu_H$, the same continuum optical depth as that at 1 Ryd. As a result, for this model with non-overlapping He II I-fronts, we only remove photons between 4 and 25 Ryd, retaining metagalactic radiation background for $\nu > 25 \nu_H$. This is a more conservative estimate than if all He is assumed to be in He II and distributed uniformly. Cloud Opacity/Emission ---------------------- For a random spatial distribution, the local continuum optical depth of the intervening absorbers is given by (Paresce, McKee, & Bowyer 1980) $${ {d\tau(\nu)} \over {dz} } = \int^\infty_0 {\partial^2N_c \over {\partial N_{H I} \partial z} } {[1 - e^{-N_{HI}\sigma_{eff}(\nu)}]} \; dN_{H I} \; ,$$ where, for clouds composed of H and He, $$\sigma_{eff}(\nu) = \sigma_{HI}(\nu) + \left[ {N_{HeI} \over N_{HI}} \right] \sigma_{HeI}(\nu) + \left[ {N_{HeII} \over N_{HI}} \right] \sigma_{HeII}(\nu) \; ,$$ and ${{\partial^2 N_c} \over {\partial N_{HI} \partial z}}$ is the distribution of absorbers in column density and redshift. This distribution has often been parameterized by a function of the form $${{\partial^2 N_c} \over {\partial N_{HI} \partial z}} = A {(1+z)}^\gamma N_{HI}^{-\beta} \;$$ for $N_l$(H I) $\leq N_{HI} ({\rm cm}^{-2}) \leq N_u$(H I). Using new line lists from [Keck]{} data, Fardal et al. (1997) have completed a review of the statistics available on Ly$\alpha$ absorbers, Lyman Limit Systems, and damped Ly$\alpha$ systems. They suggest that the distribution of absorbers is best represented by a broken power law in $N_{HI}$. If $${{\partial^2 N_c} \over {\partial N_{17} \partial z}} = A_i {(1+z)}^\gamma N_{17}^{-\beta_i} \; ,$$ where $N_{17}= N_{HI} / (10^{17}$ cm$^{-2})$, $\gamma = 2.585$, and $\beta_i$ and $A_i$ represent coefficients appropriate for a given range in $N_{17}$ (see Table 2). We adopt their Model 2 for this paper. Use of either of their other two models does not affect our conclusions. Just as absorption due to intervening clouds will strongly alter the background radiation spectrum, the re-emission of ionizing radiation from the clouds will also affect the radiation (Haardt & Madau 1996; Fardal et al. 1997). Recombinations in the clouds produce continuum radiation (H I Lyc, He II Lyc, He II Bac, and He II 2-photon) as well as He II Ly$\alpha$ ($\lambda$304) line radiation. Accounting properly for the Lyc and Bac radiation is critical to the determination of the level of the mean intensity above 4 Ryd. While the line and $2$-photon radiation are not important for that, they strongly alter the spectrum between 1 and 3 Ryd and affect the populations of ions with thresholds in that range (see Table 1). A complete discussion of the treatment of reemitted radiation used in computing the filtered spectra is given in Fardal et al. (1997). Ly$\alpha$ Forest Cloud Models ============================== As stated previously, we neglect the possibility that the absorbers consist of more than one thermal and ionization phase to concentrate on the effects of the shape of the photoionizing radiation spectrum. We model the Ly$\alpha$ forest absorbers using the photoionization code CLOUDY version 90 (Ferland 1996). The absorber clouds are modelled as plane-parallel slabs of constant density, illuminated on both sides by the ionizing radiation field. We adopt a fiducial column density N(H I) $=10^{15}$ cm$^{-2}$ and assume a metallicity $Z = 10^{-2} Z_\odot$. We adopt the spectral shapes for the incident radiation discussed in §2.1 and vary the strength of the incident radiation through the ionization parameter, $U = n_\gamma / n_H$, where $n_\gamma$ and $n_H$ are the number densities of ionizing photons and hydrogen respectively. Each curve in Figs. 2, 4, and 6 represents a set of photoionization models paired with the corresponding spectrum shown in Figs. 1, 3, and 5, respectively. Although we focus this paper on the effect of the spectral shape, we have also explored the effect of varying the parameters assumed above. For the spectral shapes considered here, there is little change in the ratios (C II/C IV is reduced by $\sim 10\%$ and Si IV/C IV is raised $\sim 10\%$) if N(H I) is increased by a factor of 10, as long as the temperatures for the clouds are assumed to be consistent with the thermal equilibrium solution of the CLOUDY model. An increase (or decrease) by a factor of 3 in the assumed metallicity has a slightly larger effect. The C II/C IV fraction is reduced $20-30\%$ with a factor of 3 reduction in metallicity and raised $8-15\%$ by a factor of 3 increase in metallicity. The ratio Si IV/C IV is raised as much as $10\%$ and reduced by $6\%$ or less with the same changes in metallicity. Constraints on the H I columns of the clouds preclude large reductions in the assumed density for one-phase photoionization models, to avoid excessive length scales for the clouds and to ensure that contributions to the baryon density from these N(H I) ${{}^>_\sim}10^{15}$ cm$^{-2}$ absorbers do not exceed BBN limits. We normally allow the temperature to be that solved for by CLOUDY, assuming thermal equilibrium between photoelectric heating and radiative cooling. If the clouds are actually cooler, for example if they have expended some energy in expansion, or if they retain thermal memory of earlier epochs when Compton cooling off the microwave background was important, the ratios Si IV/C IV vs. C II/C IV may be increased by as much as a factor of 2 at low values of C II/C IV (high values of U). If, as suggested by Haehnelt, Rauch, & Steinmetz (1997), the temperatures of the clouds are larger than that expected from photoionization thermal equilibrium, the Si IV fraction is more temperature-sensitive than the C IV fraction, and Si IV/C IV is decreased further relative to C II/C IV. These higher temperatures, suggested as a way to increase the apparent thickness of the clouds, also imply much smaller metallicities, as the C IV fraction is decreased much less compared to the H I fraction by higher temperatures. For example, adopting the $J_{Q,f}$ spectrum and assuming $U = 10^{-2}$, we find that increasing the cloud temperature from 25,000 K to 70,000 K decreases the C IV fraction by less than 20% but reduces the H I fraction (and inferred metallicity) by a factor of 8. There is a further effect if the assumed temperature of the cloud is increased to the point that collisional ionization of H I becomes important. In this case, He II/H I may be larger than that assuming pure photoionization only, N(He II) may exceed $1 / \sigma_{HeII}$, and the cloud may become self-shielding. Even if this is not the case for a cloud with N(H I) $= 10^{15}$ cm$^{-2}$, this becomes increasingly likely for greater values of N(H I), so that radiative transfer within the individual clouds becomes important. In Fig. 7 we show the results of models that explore some of these temperature effects but defer a complete discussion to a later paper. Results ======= The Metagalactic Radiation Background ------------------------------------- Figure 2 summarizes the results of our photoionization models which assume different sources of photoionizing radiation for the metagalactic background. The solid and dotted curves both assume photoionization by sources that possess the same power-law spectral shape ($\alpha_s = 1.8$) but differ in the treatment of the filtering of their radiation by intervening clouds (see Fig. 1). Models incorporating a pure-AGN spectrum, $J_{Q,f}$, indicate that the Si IV/C IV ratio may be enhanced by almost a factor of 2 over the simpler model for the absorbers with C II/C IV ${}^>_\sim 0.1$. Even in this regime, Si is still overabundant relative to C, as previously argued by Songaila & Cowie (1996a). However, a factor of two enhancement may be sufficient. If the absorbers are photoionized, an increasing C II/C IV ratio is directly related to a decreasing ionization parameter (Figure 2). At high C II/C IV (low U) there are few high-energy photons available to ionize the higher ionization stages of silicon. As a result, the enhanced ionization of Si III by He II Ly$\alpha$ photons emitted by clouds is enough to increase the ratio of Si IV/C IV over that of the broken power-law models. Compared to more realistic calculations with cloud re-emission, models for the spectral shape of the metagalactic background with a simple break at 4 Ryd underestimate the number of high-energy photons in the metagalactic background. At low C II/C IV (high U), the Si IV fraction is strongly depleted by the increased amount of high-energy photons in the realistic spectrum. A metagalactic ionizing background dominated by stellar sources, $J_{MB} = J_{SQ,f}$, increases the Si IV/C IV ratios at C II/C IV $>$ 0.1 by $20-50\%$ over that of the $J_{MB} = J_{Q,f}$ case. Once again, using a more realistic spectrum increases the Si IV/C IV ratios by almost a factor of 2 compared to spectra with simple attenuation of He II ionizing photons. As in the pure AGN case, the simple representation overestimates the corresponding intensity at higher energies. To be compatible with observed Si IV/C IV ratios if C II/C IV $>$ 0.1, the relative abundance of Si must be increased by approximately a factor of 2. In all cases, the large Si IV/C IV ratios in absorbers with C II/C IV $<$ 0.1 are difficult to explain in the context of single-phase models unless Si/C is enhanced by an order of magnitude. Chemical evolution models with Si/C $\geq 3$ (Si/C)$_{\odot}$ are unrealistic, even if the nucleosynthesis is dominated by massive stars (Woosley & Weaver 1995). A Photoionizing Background Dominated by Local Sources ----------------------------------------------------- One resolution to the puzzle of large Si IV/C IV at small C II/C IV may be that the ionizing radiation incident on the absorbers is dominated by stellar sources of radiation. As Fig. 4 shows, if the ionizing spectrum is entirely stellar, very large ratios of Si IV/C IV are always possible due to the steep dropoff in radiation more energetic than 45 eV. In practice, however, as Figs. 3 and 4 show, if the metagalactic background is primarily due to quasars ($J_{MB} = J_{Q,f}$), even if $J_{L}/J_{MB} = 20$, high Si IV/C IV ratios at low C II/C IV require Si enhancements of a factor of 10. If the metagalactic background is dominated by stellar sources ($J_{MB} = J_{SQ,f}$), large enhancements of Si may be unnecessary (see Figs. 5 and 6). For many absorbers with C II/C IV $>$ 0.1, no overabundance of Si is necessary if $J_{L}/J_{MB} = 3-10$. At low C II/C IV, $J_{L}/J_{MB}$ must exceed $10$ if enhancements in Si exceeding 2 are to be avoided. As our order of magnitude estimates in §2.1 indicate, this would require that the absorbers lie within 30 kpc of starburst galaxies. However, this situation is quite feasible, given realistic constraints on absorber size and metal transport distances. Non-Equilibrium Temperatures and Non-Uniform Radiation Fields ------------------------------------------------------------- As Figs. 7a and 7b show, if the temperatures of the clouds are cooler than the photoionization thermal equilibrium temperatures (Ferrara & Giallongo 1997; Zhang et al. 1997), much higher ratios of Si IV/C IV are possible for low C II/C IV (high U) clouds. These high U clouds are precisely the clouds for which the photoionization thermal equilibrium temperature is highest ($T \approx 40,000$ K). The models denoted by the dotted curves in Figs. 7a and 7b assume $T=15,000$ K, which may not be compatible with the observed line widths, and which are cooler than is usually assumed for Ly$\alpha$ forest clouds. If higher temperatures are associated with the clouds, for example the dashed curves in Figs. 7a and 7b, which assume $T = 50,000$ K, it is more difficult to account for the ratios in a one-phase model. Another way to enhance Si IV/C IV at low C II/C IV, as suggested by Songaila & Cowie (1996a) and Savaglio et al. (1997), is to assume that no photons above 4 Ryd are present in the incident ionizing spectrum. This is the limiting case of a cloud embedded in a region where He II I-fronts have not yet overlapped. The Si IV/C IV ratios in Fig. 8 are not as high as those for the limiting case when all incident radiation is starburst radiation, since radiation between 45 eV and 54 eV is included, which will ionize Si IV to Si V. This effect is seen in comparing models which assume pure AGN and starburst/AGN sources in Fig. 8. As Fig. 8 also shows, a careful treatment of the propagation of these He II I-fronts is necessary, particularly because most AGN possess x-ray emission with spectra that flatten to approximately $\nu^{-1}$ above 0.3 keV. We have shown that including high-energy radiation with E $>$ 25 Ryd is sufficient to reduce Si IV/C IV ratios substantially at low C II/C IV. Discussion ========== From the results of our one-phase photoionization models we draw the following conclusions: 1\. [Ionizing Radiation Field]{}.– The metagalactic radiation field is likely to include both power-law (AGN) and stellar (hot-star) components. The Ly$\alpha$ forest absorbers with metal lines may also experience a local radiation field from starburst galaxies within 50–100 kpc. 2\. [Si/C Overabundance]{}.– For plausible mixtures of stellar and AGN spectra in the metagalactic background, our photoionization models produce enhanced Si IV/C IV ratios, consistent with high-$z$ absorbers with Si/C $\approx 2$(Si/C)$_{\odot}$ at low ionization (C II/C IV $>$ 0.1). For absorbers with C II/C IV $<$ 0.05, it is difficult to account for the high values of Si IV/C IV unless Si/C $>$ 10(Si/C)$_{\odot}$, an unrealistically large value for massive-star nucleosynthesis. These systems may include photoionization from local stellar sources as well as hot, collisionally ionized gas. 3\. [Local Ionizing Sources]{}.– If the radiation field incident on the absorbers is dominated by a nearby starburst galaxy, the Si IV/C IV ratios are further enhanced. If C II/C IV $>$ 0.1, no Si/C overabundance is necessary to explain Si IV/C IV ratios if absorbers are within about 40 kpc of a starburst galaxy [and]{} the background is dominated by stellar sources. Even if the metagalactic background is dominated by AGNs, close proximity to a starburst galaxy may reduce the needed Si/C overabundance to a factor 1.5. When C II/C IV $^<_\sim 0.05$, it remains very difficult to account for the high values of Si IV/C IV with photoionized models, although models with locally dominated radiation may only require Si/C enhancements of a factor of 2-3. 4\. [Non-Overlapping He II I-Fronts]{}.– Around quasars whose ionizing continua have steep spectral indices ($\alpha_s > 1.84$), the He II I-fronts will lie within the H I I-fronts. If many absorbers lie in regions where all photons above 4 Ryd are attenuated (Songaila & Cowie 1996a), we obtain good agreement with almost all measurements of Si IV/C IV if Si/C is enhanced by a factor of 2–3. However, including even a small contribution of higher energy photons in the background increases the needed Si/C overabundance to an order of magnitude for absorbers with the lowest C II/C IV ratios. 5\. [Temperature Effects]{}.– If the absorbers are cooler than expected for thermal equilibrium between photoelectric heating and radiative cooling, the Si IV/C IV ratio is increased for a given C II/C IV ratio. At low C II/C IV, this enhances Si IV/C IV by a factor of 5 in our photoionization models. Conversely, if the absorbers are hotter, Si IV/C IV is lower for a given C II/C IV ratio. While we have described several processes that might increase the Si IV/C IV ratio, an important additional constraint on these possibilities is the fact that the high ratios are preferentially found at $z > 3.1$. Recent observations (Boksenberg 1997) challenge this interpretation, by finding high Si IV/C IV ratios in absorbers at $z = 2-3$. If, however, Si IV/C IV rises above $z > 3.1$, this argues for a time dependence to whatever process increases this ratio. It is this property, as well as the increased He II absorption toward Q0302-003 at $z=3.28$ (Jakobsen et al. 1994), that Songaila & Cowie (1996a) use to support their suggestion that the absorbers lie in regions where He II I-fronts have not overlapped at $z > 3.1$. From a higher resolution HST/GHRS spectrum of Q0302-003, Hogan, Anderson, & Rugers (1997) find evidence for residual transmitted flux below the He II edge. Their 95% confidence upper limit, $\tau_{HeII} \le 3$, makes it less necessary to propose that Q0302-003 lies in a region where He II I-fronts have not overlapped, but does not preclude the suggestion that such regions existed at $z>3.1$. If this is the case, radiation outside of He III regions would largely limit the range in ionization stages for carbon to II-IV, and for silicon to II-V. In AGN, however, the large fluxes of unattenuated photons above 45 eV can ionize Si IV and C III, so that high observed Si IV/C IV ratios would be difficult to explain for large U (small C II/C IV). As a result, the spectrum below 4 Ryd must be dominated by stellar sources. This may be difficult to achieve with the known quasar luminosity functions at $z > 3.5$. In addition, soft x-ray radiation from quasars, which is less attenuated by intervening clouds even in the low He III porosity case, may again make high Si IV/C IV, low C II/C IV absorbers difficult to understand if they are solely photoionized by a metagalactic background. New or improved measurements of He II absorption may soon indicate whether He II I-fronts have overlapped well before $z = 3.1$. In that case, a sharp change in the shape of the metagalactic radiation field may be less plausible. Still, in general, the trend is likely to be more attenuation of photons above the 4 Ryd limit with increasing redshift. Other time dependent effects that enhance the ratio of Si IV/C IV are possible. Higher abundances of Si relative to C are associated with metal yields from the most massive stars. However, in a flat universe with $h=0.75$, the age of the universe exceeds $10^9$ years by $z = 3.1$, time enough for lower mass stars to enrich the gas with carbon. For example, if multiple supernovae eject metal-enriched gas into the IGM, the additional $4 \times 10^{8}$ years between $z=3.5$ and $z=2.5$ may make it more likely that the gas surrounding the high mass stars has been enriched with carbon from a previous episode of star formation. The temperature effects we discuss in §4 may also have a time dependence. If the IGM has been uniformly enriched by a much earlier episode of Population III star formation, and if the Ly$\alpha$ forest clouds with metals have their origin in growing overdensities in the IGM, then the absorbers observed at higher $z$ may have formed at an earlier epoch. If Compton cooling off the cosmic microwave background was a dominant coolant at this epoch (Miralda-Escudé & Rees 1994), the cloud may have retained memory of this lower temperature. This effect is probably not large, since there is not a significant evolution in the observed line widths. Alternatively, one may resolve the puzzle of the high Si IV/C IV, low C II/C IV absorbers by relaxing the single-phase model which we used to explore the effects of radiative transfer. As we have emphasized in a previous paper (Giroux et al. 1994), the production of heavy elements in QSO absorption systems is naturally accompanied by hot gas due to supernovae and hot-star winds. We thank Mark Fardal for the calculated metagalactic background radiation spectra. We also thank G. Ferland for discussions about updated versions of CLOUDY, and L. Cowie and A. Songaila for providing a summary of measurements and upper limits on metal column densities in advance of publication. This work was supported by the Astrophysical Theory Program at the University of Colorado (NASA grant NAGW-766). [ll]{} C II, III, IV & 24.38, 47.89, 64.49 N I, II, III, IV & 14.53, 29.60, 47.45, 77.47 O I, II, III & 13.62, 35.12, 54.94 Ne I, II, III & 21.57, 40.96, 63.46 Si II, III, IV, V& 16.35, 33.49, 45.14, 166.77 Al II, III & 18.83, 28.45 S II, III, IV, V & 23.33, 34.83, 47.3, 72.68 Fe II, III, IV & 16.19, 30.65, 54.8 [lll]{} $10^{-5}-10^{-3}$ & 0.132 & 1.41 $10^{-3}-0.02$ & $6.31\times 10^{-3}$ & 1.85 $0.02-10^{5}$ & $2.9\times 10^{-2}$ & 1.46 Bennett, C. L., Fixsen, D. J., Hinshaw, G., Mather, J. C., Moseley, S. H., Wright, E. L., Eplee, R. E., Gales, J., Hewagama, T., Isaacman, R. B., Shafer, R. A., & Turpie, K. 1994, , 434, 587 Bergeron, J., & Stasinska, G. 1986, , 169, 1 Bergeron, J., & Ikeuchi, S. 1990, , 235, 8 Boksenberg, A. 1997, private communication Bruzual, G., & Charlot, S. 1993, , 405, 538 Cooke, A. J., Espey, B., & Carswell, B. 1997, preprint, astro-ph/9609100 Copi, C. J., Schramm, D. N., & Turner, M. S. 1994, Science, 267, 192 Couchman, H, M. P. 1985, , 214, 137 Cowie, L. L., Songaila, A., Kim, T.-S., & Hu, E. M. 1995, , 109, 1522 Donahue, M., & Shull, J. M. 1987, , 323, L13 Donahue, M., & Shull, J. M. 1991, , 383, 511 Dove, J. B., & Shull, J. M. 1994, , 430, 222 Fardal, M. A., Giroux, M. L., & Shull, J. M. 1997, , submitted Ferland, G. J. 1996, Hazy, a Brief Introduction to Cloudy, University of Kentucky Department of Physics and Astronomy Internal Report Ferrara, A. & Giallongo, E. 1997, preprint Gabler, R., Gabler, A., Kudritzki, R. P., & Mendez, R. H. 1992, , 265, 656 Giroux, M. L., Fardal, M. A., & Shull, J. M. 1995, , 451, 477 Giroux, M. L., & Shapiro, P. R. 1996, , 102, 191 Giroux, M. L., Sutherland, R. S., & Shull, J. M. 1994, , 435, L97 Haardt, F., & Madau, P. 1996, , 461, 20 Haehnelt, M. G. 1997, preprint, astro-ph/9608009 Haehnelt, M. G., Rauch, M., & Steinmetz, M. 1997, preprint, astro-ph/9608133 Hellsten, U., Davé, R., Hernquist, L., Weinberg, D. H., & Katz, N. 1997, preprint, astro-ph/9701043 Hogan, C. J., Anderson, S. F., & Rugers, M. H. 1997, , in press Jakobsen, P., Boksenberg, A., Deharveng, J. M., Greenfield, P., Jedrzejewski, R., & Paresce, F. 1994, , 370, 35 Kurucz, R. L. 1992, in IAU Symp. 149, The Stellar Populations in Galaxies, ed. B. Barbuy & A. Renzini (Dordrecht: Kluwer), 225 Madau, P., & Meiksin, A. 1994, , 433, L53 Madau, P., & Shull, J. M. 1996, , 457, 551 Madau, P. 1992, , 389, L1 Miralda-Escudé, J., & Ostriker 1990, , 350, 1 Miralda-Escudé, J., & Ostriker 1992, , 392, 15 Miralda-Escudé, J., & Rees, M. J. 1994, , 266, 343 Olive, K. A., & Steigman, G. 1995, , 97, 49 Ostriker, J. P., & Gnedin, N. Y. 1996, , 472, L63 Paresce, F., Mckee, C. F., & Bowyer, S. 1980, , 240, 387 Petitjean, P., Riediger, R., & Rauch, M. 1996, , 307, 417 Press, W. H., & Rybicki, G. B. 1993, , 418, 585 (PR) Sargent, W. L. W., Boksenberg, A., & Steidel, C. C. 1988, , 68, 539 Sargent, W. L. W., Steidel, C. C., & Boksenberg, A. 1989, , 69, 703 (SSB) Savaglio, S., Cristiani, S., D’Odorico, S., Fontana, A., Giallongo, E., & Molaro, P. 1997, astro-ph/9606063 Schaerer, D., & de Koter, A. 1996, preprint, astro-ph/9611068 Sellmaier, F. H., Yamamoto, T., Pauldrach, A. W. A., & Rubin, R. H. 1996, , 305, L37 Skillman, E. D., & Kennicutt, R. C. 1993, , 411, 655 Shapiro, P. S., & Giroux, M. 1987, , 321, L107 Songaila, A., & Cowie, L. L. 1996a, , 112, 335 Songaila, A., & Cowie, L. L. 1996b, personal communication Steidel, C. C., Sargent, W. L. W., & Boksenberg, A. 1988, , 333, 5 Steidel, C. C., & Sargent, W. L. W. 1989, , 343, L33 Sutherland, R. S., & Shull, J. M. 1997, in preparation Woosley, S. E., & Weaver, T. W. 1995, ApJS, 101, 181 Zhang, Y., Anninos, P., Norman, M. L., & Meiksin, A. 1997, preprint Zheng, W., Kriss, G. A., Telfer, R. C., Grimes, J. P., & Davidsen, A. F., 1997, , in press
--- abstract: 'In this manuscript we review recent developments in the numerical simulations of bipartite SU($N$) spin models by quantum Monte Carlo (QMC) methods. We provide an account of a large family of newly discovered sign-problem free spin models which can be simulated in their ground states on large lattices, containing $O(10^5)$ spins, using the stochastic series expansion method with efficient loop algorithms. One of the most important applications so far of these Hamiltonians are to unbiased studies of quantum criticality between Néel and valence bond phases in two dimensions – a summary of this body of work is provided. The article concludes with an overview of the current status of and outlook for future studies of the “designer” Hamiltonians.' address: - 'Department of Physics & Astronomy, University of Kentucky, Lexington KY 40506-0055' - 'Department of Physics & Astronomy, California State University, Sacramento CA 95819' author: - 'Ribhu K. Kaul' - 'Matthew S. Block' bibliography: - '/Users/rkk/Physics/PAPERS/BIB/career.bib' - '/Users/rkk/Physics/PAPERS/BIB/rev\_bib.bib' title: 'Numerical studies of various Néel-VBS transitions in SU($N$) anti-ferromagnets' --- Overview ======== The study of ground states of lattice models of quantum spins has become a major field in condensed matter physics [@balents2010:spliq]. Despite their simplicity these models can be extremely hard to study theoretically, due in part to the rich variety of ground states that they can host. Recent years have seen a dramatic increase in the use of numerical methods to study the ground states of quantum spin models. Of particular interest are “unbiased” numerical methods, which solve for physical properties of model systems with numerical errors that can be controlled and estimated in a reliable fashion. Quantum Monte Carlo occupies a special place among the unbiased methods because when applicable, it is the only technique that is able to access the large systems sizes required for reliable extrapolation to the thermodynamic limit, especially in the proximity of critical phenomena [@kaul2013:qmc]. The simplest ground state that can arise in a quantum spin system is a magnetic ground state, where the expectation value of the spin on a site is finite, causing the spin to effectively “point” in a certain direction, thus spontaneously breaking the global symmetry associated with the rotation of spins. Such magnetic states are well known to arise in the low-energy space of classical spin models and their appearance in quantum models can be understood from semi-classical arguments. The focus of recent numerical studies of spin models has been in large part on the nature of the [non-magnetic phases]{} that arise at $T=0$ due to quantum fluctuations and the quantum phase transitions separating magnetic and non-magnetic phases. The non-magnetic phases may either break another symmetry, most often a lattice translational symmetry in which case they are called “solids” or be completely symmetric in both the lattice and spin symmetries in which case they are called “liquids.” The central questions of interest to numerical studies are: What is the precise characterization of the non-magnetic phases, both of the “solid” and “liquid” type that are found in simple microscopic models? How can the phase transitions between the magnetic and non-magnetic phases observed in numerical simulations be understood in terms of long wavelength quantum field theories? While these general questions have received copious attention from a multitude of complementary approaches in different contexts over the last three decades [@kaul2013:qmc; @Stoudenmire12; @sachdev1999:qpt; @laflorencie2004:ed], in this review we will outline how new “designer” Hamiltonians with SU($N$) symmetry have contributed to answers concerning phase transitions between SU($N$) symmetry breaking magnetic phases and lattice translational symmetry breaking valence-bond solid states, for which an exotic direct continuous transition – “deconfined criticality” – has been proposed [@senthil2004:science; @senthil2004:deconf_long; @senthil2005:jpsj]. For a fuller appreciation of this short review, a familiarity with the deconfined theory is recommended. In the interest of space this is not provided here, the interested reader is encouraged to refer to the original literature. In Section \[sec:models\] we briefly review the discovery of a large family of “designer” SU($N$) models that do not suffer from the sign problem. In Sec \[sec:pd\_deconf\] we describe the phase diagrams obtained for the Hamiltonians that have been studied so far and the nature of the critical points that arise between different phases. Finally, in Section \[sec:outlook\] we conclude with an outlook and directions for future work. SU($N$) Hamiltonians, loop models and the sign problem {#sec:models} ====================================================== In order to use unbiased quantum Monte-Carlo techniques efficiently, one needs to identify models that do not suffer from the sign problem. It is well known that Hamiltonians that satisfy Marshall’s sign condition are also sign-problem free. So a basic questions of central importance to the numerical simulations of quantum spin models is: What is the family of spin models that satisfies Marshall’s sign criteria? In order to make this question concrete, we specialize our considerations to bipartite models with a specific representation of SU($N$) symmetry, originally introduced to condensed matter physics by Affleck [@affleck1985:lgN]. This realization of SU($N$) symmetry requires spins on one sub-lattice to transform in the fundamental representation of SU($N$) and spins on the other sub-lattice to transform in the conjugate to fundamental representation. We note here for $N=2$, since the fundamental and conjugate to fundamental representations are identical, this realization of SU(2) symmetry gives rise to the familiar Heisenberg-like models. The standard quantum-classical mapping allows us to rewrite the quantum statistical mechanics of $d$-dimensional Hamiltonians as a classical statistical mechanics problem in $d+1$-dimensions, where the extra dimension is of extent $\beta = 1/T$. The stochastic series expansion (SSE) is an elegant and well-documented method to execute this step [@sandvik2010:vietri]. When carried out and as discussed in detail [@kaul2014:design], the Affleck SU($N$) spin models on bipartite lattices can be mapped to [oriented tightly packed loop models with $N$ colors]{} in one higher dimension. In order to carry out Monte Carlo sampling we require these configurations to have positive weights. In the language of loop models it is straightforward to systematically write down all possible interactions that keep the weights of the loop configurations positive. When the quantum-classical mapping is run backwards, the interactions in the classical loop model correspond to terms in quantum Hamiltonians that are Marshall positive. This picture allows one to systematically write down a large class of SU($N$) spin Hamiltonians that are Marshall positive. We note parenthetically that the Marshall positivity of these “designer” Hamiltonians is not obvious when viewed directly in the spin language, and that the “designer” models include all previously known Marshall positive spin models as particular cases. As a concrete example of the results, let us discuss the familiar $N=2$ case. Here the spins $\vec S$ on either sub-lattice can be written in terms of Pauli matrices in the usual way. Previously known sign-problem-free Hamiltonians include (with $J_1,J_2,Q>0$) [@sandvik2007:deconf], $$\begin{aligned} \label{eq:j1j2Q} H_{J_1} &=& J_1 {\vec S^{A}_1} \cdot {\vec S^{B}_1} \nonumber\\ H_{J_2} &=& - J_2{\vec S^{A}_1} \cdot {\vec S^{A}_2} \nonumber\\ H_{Q} &=& - Q\left ( {\vec S^{A}_1} \cdot {\vec S^{B}_1} -\frac{1}{4}\right ) \left ( {\vec S^{A}_2} \cdot {\vec S^{B}_2} -\frac{1}{4}\right )\end{aligned}$$ where the superscript $A,B$ indicates the sublattice the spin lives on and the subscript indicates the different spins on a given sublattice. Thus the $J_1$ interaction is defined on two spins, one of which lives on the A sublattice and the other on the B sublattice, the $J_2$ interaction is defined on two spins on the same sublattice, and the $Q$ interaction is defined on four spins, two on the A sublattice and two on the B sublattice. As an application of the new strategy sketched above, a new linearly independent four-spin interaction can be shown to be Marshall positive, $$\begin{aligned} H_{R} &=& R\left ( {\vec S^{A}_1} \cdot {\vec S^{A}_2} -\frac{1}{4}\right ) \left ( {\vec S^{B}_1} \cdot {\vec S^{B}_2} -\frac{1}{4}\right )\nonumber \\ &-& R\left ( {\vec S^{A}_1} \cdot {\vec S^{B}_1} -\frac{1}{4}\right ) \left ( {\vec S^{A}_2} \cdot {\vec S^{B}_2} -\frac{1}{4}\right ) - R\left ( {\vec S^{A}_1} \cdot {\vec S^{B}_2} -\frac{1}{4}\right ) \left ( {\vec S^{A}_1} \cdot {\vec S^{B}_2} -\frac{1}{4}\right )\end{aligned}$$ It is straightforward to prove the Marshall positivity of the above interaction by direct evaluation of its matrix elements when $R>0$, even though its positivity is not apparent from a naive inspection of the term. We note following the strategy described in Ref. [@kaul2014:design], all the Marshall-positive interactions involving an arbitrary large number of $A$ and $B$ spins and with any $N$ (of SU($N$)) can be systematically written down. Also from the way the interactions are written above it is clear that they can be written on any bipartite lattice in any dimension, and can be made to preserve the lattice symmetry through an appropriate summation over the entire lattice. Detailed reviews and pedagogical introductions to the kind of QMC algorithms that are used to simulate the spin models introduced in this Section may be found in [@sandvik2010:vietri; @kaul2013:qmc]. ![\[fig:vbsp\]Summary of valence-bond patterns that have been accessed in numerical simulations of two-dimensional bipartite SU($N$) designer Hamiltonians. The quantum phase transition into each of these phases from SU($N$) symmetry breaking Néel phases is the subject of this review. The thick red bonds when put in an SU($N$) singlet state provide a cartoon state for the symmetry breaking. (a-d) are the “columnar” patterns for which the theory of a continuous deconfined criticality can hold for sufficiently large $N$. The value of $q$ is the degeneracy of the VBS coverings in these states. (e) is a four-fold degenerate VBS state on a bilayer to which there is a first order transition. (f,g) are three- and four-fold staggered VBS states into which there are also first order transitions.](vbs_patterns.pdf){width="40pc"} Phase Diagrams & Quantum Criticality {#sec:pd_deconf} ==================================== The only two phases found so far in the numerical simulations of the bipartite SU($N$) sign-free models discussed in Sec. \[sec:models\] are Néel and “valence-bond” phases (VBS). By “valence-bond” phases we mean phases that are smoothly connected to a product state of two-site valence bond coverings (see Fig. \[fig:vbsp\] for cartoons of such coverings). The numerical study of the Néel and VBS phases of the Affleck SU($N$) models was initiated early on [@Santoro99; @harada2003:sun]. However, the most intriguing aspect of the phase diagram, the critical point, was first accessed with the introduction of the four-site $Q$ interaction \[eq:j1j2Q\] for the $N=2$ case [@sandvik2007:deconf]. While the pioneering study was carried out for $N=2$ on the square lattice, subsequent work has carried out the study for a large range of $N$ and for a variety of bipartite lattices and interactions. In order to summarize the studies in “conceptual” rather than historical order, we need one technical result from the deconfined theory. The deconfined theory predicts that for bipartite SU($N$) magnets in two dimensions, the Néel-VBS critical point is described by a [nc-$\mathbb{CP}^{N-1}$]{} critical field theory, only for certain “columnar” VBS states. In actual lattice realization, whether in numerical simulations or real materials, a perturbation to the [nc-$\mathbb{CP}^{N-1}$]{} theory, $\lambda_q$ (for the experts, it is the fugacity of $q$-monopoles in the gauge field) is present. The central difference between the various bipartite lattices is the integer $q$, which can be intuitively understood as the minimum degree of degeneracy of the VBS phase on the particular bipartite lattice under consideration. In order for the deconfined critical point to exist $\lambda_q$ must be [irrelevant]{} at the monopole-free fixed point. If it is relevant one expects a first order transition or a continuous transition in a universality class different from the [nc-$\mathbb{CP}^{N-1}$]{} universality. Let us begin with the case of the bilayer lattice. The simplest non-magnetic state in the bilayer system is a non-degenerate ($q=1$) rung singlet state (see Fig. \[fig:vbsp\](a)). For $N=2$ there is beautiful evidence that supports the theoretical expectation [@chakravarty1988:qaf] that there is a continuous critical point between Néel and rung-singlet state in the $2+1$-dimensional O(3) universality class [@wang2006:bilayer]. For $N\geq 4$ numerical simulations have found evidence for a first order transition, which is expected from Landau theory [@kaul2012:bilayer]. The case $N=3$ appears to have either a very weakly first order or a continuous transition. The field theoretical scenario explaining a possible continuous $N=3$ transition has been nicely summarized in Ref. [@nahum2013:long]. Other numerical work has studied the Néel-VBS transition on the rectangular ($q=2$) [@block2013:fate], honeycomb ($q=3$) [@block2013:fate; @pujari2013:hc; @harada2013:deconf] and square ($q=4$) [@sandvik2007:deconf; @lou2009:sun; @kaul2012:j1j2; @kaul2011:su34] lattices for a range of $N$ by using a judicious choice of the designer couplings defined in Section \[sec:models\]. See Fig. \[fig:vbsp\](b-d) for cartoons of the VBS states. Both first order transitions for small $N$ and continuous transitions for large $N$ have been identified. The critical value of $N$ at which the transition turns continuous decreases as $q$ increases, as expected theoretically. The appearance of continuous and first-order transitions occur in a systematic way that can be attributed to whether the lattice anisotropy in the form of $\lambda_q$ is relevant or irrelevant at the [nc-$\mathbb{CP}^{N-1}$]{} fixed point. The results may be summarized in Table \[tab:qN\], which shows whether the deconfined critical point between an SU($N$) magnet and $q$-degenerate VBS state (when it is the minimum degeneracy on a particular lattice) is stable. We note that in the bilayer model ($q=1$) the $R$ is expected to turn to $I$ at a finite value of $N$ – analytic estimates from large-$N$ theories suggest this happens around $N\approx 25$ [@murthy1990:mono]. However, such large values of $N$ have not been accessed numerically yet. For the values of $N$ for which there is a continuous transition, the critical exponents should not depend on the bipartite lattice geometry, since $\lambda_q$ is irrelevant if there is a continuous transition. Numerical measurements for the anomalous dimension of the Néel and VBS fields are consistent with this expectation and are consistent with results from the analytic $1/N$ expansion as shown in Fig. \[fig:eta\] $N=\infty, 1/N$ $I$ $I$ $I$ $I$ $\dots$ $I$ nc-$\mathbb{CP}^{N-1}$ ----------------- ------- ------- ------- ------- --------- ------------ ------------------------ $\dots$ $$&$$ $$&$$ $$&$$ $$ $N=10$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{9}$ $N=9$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{8}$ $N=8$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{7}$ $N=7$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{6}$ $N=6$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{5}$ $N=5$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{4}$ $N=4$ $R$ $I$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{3}$ $N=3$ $R$ $R$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{2}$ $N=2$ $R$ $R$ $I$ $I$ $$ $I$ nc-$\mathbb{CP}^{1}$ $N=1$ $R$ $R$ $R$ $I$ $$ $I$ $XY$ $N=0$ $R$ $R$ $R$ $R$ $$ $R$ photon $$ $q=1$ $q=2$ $q=3$ $q=4$ $\dots$ $q=\infty$ $$ : Table showing the inferred relevance ($R$) or irrelevance ($I$) of $q$-monopoles at the [nc-$\mathbb{CP}^{N-1}$]{} fixed point, which various studies summarized in the text have allowed us to complete. Numerical simulations of the Néel-VBS transition in the models discussed here only allow studies for $N\geq 2$. The entries with $R$ correspond to an unstable fixed point, and $I$ to a stable fixed point allowing for deconfined criticality. At some currently unknown critical value of $N>10$, the $q=1$ case switches from $R$ to $I$. Adapted from Ref. [@block2013:fate] []{data-label="tab:qN"} Having discussed the cases of a “columnar” VBS state to which the theory of deconfined criticality applies, it is also of interest to numerically study the Néel-VBS transition in situations where the deconfined criticality does [not]{} apply, as a non-trivial check on the theory. As we shall see in the examples below there are a number of different reasons why this can happen: \(1) In the bilayer geometry, consider the phase transition between a four-fold degenerate columnar VBS (c-VBS), see Fig. \[fig:vbsp\](e), and the SU($N$) Néel state. The c-VBS on the bilayer breaks exactly the same symmetries as the single-layer c-VBS state (see Fig. \[fig:vbsp\](d)) and thus one might conclude naively based on “Landau” theory that they are described by the same critical phenomena. However, it is well known that in the bilayer geometry the Berry phases that are crucial for deconfined criticality cancel between the layers, eliminating the possibility of an exotic continuous transition. Consistent with this expectation, numerical studies have found that the c-VBS-Néel transition in the bilayer is first order [@kaul2012:bilayer]. \(2) Back to the single-layer case, one can tune designer coupling to favor spontaneous symmetry breaking different from the “columnar” states for which the original proposal of deconfined criticality was made [@senthil2004:science]. Indeed, studies have been carried out for the transition between Néel and “staggered” VBS for SU(2), on the square lattice by tuning a “designer” six-spin interaction [@sen2010:first] and on the honeycomb by tuning a four-spin interaction [@banerjee2011:sthc]. The theory of a continuous deconfined critical point does not generalize to the staggered VBS case, and in the absence of any plausible alternative one expects a restoration of “Landau” theory and hence a first-order transition. Consistently, both studies find clear evidence for first-order transitions between the Néel and staggered VBS phases. \(3) There have been two studies of the Néel-VBS transition in designer Hamiltonians in [three- dimensional]{} bipartite SU($N$) systems thus far. Both studies, which are on cubic lattices but with distinct Hamiltonians, find first-order transitions between Néel and columnar VBS and no evidence for any new intervening phases [@beach2007:cubic; @block2012:cubic]. The absence of a direct continuous transition in $3+1$ dimensions is again consistent with the deconfined criticality scenario since various aspects of the [$\mathbb{CP}^{N-1}$]{} field theory are specific to $2+1$ dimensions and are known to be invalid in $3+1$ dimensions. ![\[fig:eta\] Comparison of anomalous dimensions of Néel and VBS operators in the case of continuous transitions for $q=2,3$ and $4$. (a) Anomalous dimension of the Néel order parameter as a function of $1/N$. (b) Anomalous dimension of the VBS order parameter as a function of $1/N$. The gray squares are the results of a square lattice study ($q=4$) [@lou2009:sun; @kaul2012:j1j2]. The blue circles are results from the honeycomb lattice ($q=3$) and the green diamonds are results from the rectangular lattice ($q=2$). The red line is the $1/N$ expansion. The universality of the exponents with respect to $q$ is a direct consequence of the irrelevance of $\lambda_q$. The agreement of the exponents with the $1/N$ computation is strong evidence for the emergence of the [nc-$\mathbb{CP}^{N-1}$]{} universality at the Néel-VBS transition. Adapted from Ref. [@block2013:fate].](eta.pdf){width="\columnwidth"} Outlook {#sec:outlook} ======= In the previous section, we have summarized how the deconfined criticality scenario and the [$\mathbb{CP}^{N-1}$]{} universality, convincingly explain unbiased numerical studies of the SU($N$) designer Hamiltonians in a variety of studies carried out by different groups, which have probed various distinct aspects of the deconfined criticality scenario. One concern is that the transition could be weakly first order and the numerical studies, which are necessarily limited to finite size systems, may not have accessed sizes large enough to detect a first-order transition [@kuklov2008:first]. If this is indeed the case, the transition must be so weakly first order that the correlation length exceeds the sizes of the largest lattices on which many of the designer Hamiltonians have been simulated, since no direct evidence for first-order behavior has been found in the cases where a continuous transition has been claimed. This makes the discussion of the presence of such a weakly first-order transition somewhat academic. Nevertheless, there are well established corrections to scaling observed in the numerical studies of the Néel-VBS transition in the designer Hamiltonians [@sandvik2010:logs; @banerjee2010:log; @kaul2011:su34; @harada2013:deconf]. The origin of these corrections to the asymptotic behavior is not completely understood. Three options might be considered: (a) the transition is described completely by deconfined criticality; the corrections to scaling are just the usual corrections to asymptotic behavior that arise from deviants from the true fixed point due to irrelevant operators, finite size, etc. and will vanish in the thermodynamic limit; (b) the transition is weakly first order and no non-compact [$\mathbb{CP}^{N-1}$]{} fixed point exists; the deviation from scaling is claimed as incipient first order behavior; (c) there is something fundamentally new in the scaling behavior of deconfined critical points that needs to be understood; the “corrections” are part of the true asymptotic behavior and their existence can be understood from a field theoretic argument that has so far been overlooked. Currently, it is not possible to rule out any of the three options categorically. Given the large body of evidence presented in Section \[sec:pd\_deconf\] that appears consistent with the deconfined scenario and the lack of direct evidence for a first-order transition, Occum’s razor suggests that (a) is the most likely explanation. The corrections to scaling arising due to a irrelevant operator with a small exponent [@bartosch2013:frg]. As a theoretical possibility, option (c) is the most exciting and is an interesting area for further field theoretic work [@sandvik2011:spinon; @nogueira2012:logs]. Likewise, some positive theoretical reasoning that supports the existence of a first-order transition and explains why it is so weak, or direct evidence numerical evidence for a first-order transition, would strengthen the case for option (b). It is also possible that the transition is first order for small-$N$ and becomes continuous only for some finite value of $N>2$. Numerical studies on the designer Hamiltonians do not see a dramatic difference in the simulations between $N=2$ and $N> 2$ for the cases where a continuous transition is found. It would be of interest to extend the studies of the $N=2$ [nc-$\mathbb{CP}^{N-1}$]{} field theory on a lattice, to $N>2$, which have not been carried out [@kuklov2008:first; @motrunich2008:cp1]. Beyond the study of deconfined criticality, an as yet unanswered question is what other phases can be accessed in the family of sign-free SU($N$) spin models presented in Sec. \[sec:models\]. The only two phases found so far in these models are Néel and VBS phases, with the VBS phases being the simplest possible, i.e., they are each connected without phase transition to a cartoon state that is simply a direct product of two-site SU($N$) singlets. There is no evidence for plaquette VBS states in the designer models, despite many approximate studies favoring such a state in similar models. A related unanswered question is whether the SU($N$) “designer” Hamiltonians can host a spin liquid, and if so how to design such models. We note that Marshall positive models with simpler U(1) symmetries have been shown convincingly to host $Z_2$ spin liquids [@isakov2011:tee; @isakov2006:sl], but no such model with SU($N$) symmetry is known yet. Finally, studies of the designer models with internal symmetries different from SU($N$), studies of their phase diagrams with quenched disorder, and studies of their spectral properties and doping with a small concentration of holes all provide exciting directions for future research. The authors would like to acknowledge Jon D’Emidio, Roger Melko, and Anders Sandvik for collaboration on related work and NSF DMR-1056536 for partial financial support. References {#references .unnumbered} ==========
--- abstract: 'This is the first in a series of papers studying the astrophysics and cosmology of massive, dynamically relaxed galaxy clusters. Here we present a new, automated method for identifying relaxed clusters based on their morphologies in X-ray imaging data. While broadly similar to others in the literature, the morphological quantities that we measure are specifically designed to provide a fair basis for comparison across a range of data quality and cluster redshifts, to be robust against missing data due to point-source masks and gaps between detectors, and to avoid strong assumptions about the cosmological background and cluster masses. Based on three morphological indicators – Symmetry, Peakiness and Alignment – we develop the SPA criterion for relaxation. This analysis was applied to a large sample of cluster observations from the [[Chandra]{}]{} and ROSAT archives. Of the 361 clusters which received the SPA treatment, 57 (16 per cent) were subsequently found to be relaxed according to our criterion. We compare our measurements to similar estimators in the literature, as well as projected ellipticity and other image measures, and comment on trends in the relaxed cluster fraction with redshift, temperature, and survey selection method. Code implementing our morphological analysis will be made available on the web.[^1]' author: - \| Adam B. Mantz,[$^\arabic{chicago}$]{}$^,$[$^\arabic{kicp}$]{}[^2] Steven W. Allen,[$^\arabic{kipac}$]{}$^,$[$^\arabic{stanford}$]{}$^,$[$^\arabic{slac}$]{} R. Glenn Morris,[$^\arabic{kipac}$]{}$^,$[$^\arabic{slac}$]{} Robert W. Schmidt,[$^\arabic{heidelberg}$]{}\ Anja von der Linden,[$^\arabic{kipac}$]{}$^,$[$^\arabic{stanford}$]{}$^,$[$^\arabic{dark}$]{} Ondrej Urban[$^\arabic{kipac}$]{}$^,$[$^\arabic{stanford}$]{} date: 'Accepted 2015 January 29. Received 2015 January 27; in original form 2014 November 13' title: \| Cosmology and Astrophysics from Relaxed Galaxy Clusters I:\ Sample Selection --- Introduction ============ Dynamically relaxed clusters of galaxies play a special role in investigations of cluster astrophysics and cosmology. While a variety of non-equilibrium processes taking place in the intracluster medium (ICM) are of astrophysical interest, it is only in the most regular systems that the large-scale, three-dimensional properties of the ICM can be studied in detail with minimal systematic uncertainties due to projection. In addition, the masses of relaxed clusters can be estimated with high precision and minimal bias. As a result, relaxed clusters have featured in a number of prominent studies of cluster astrophysics, scaling relations and cosmology [@Allen0110610; @Allen0205007; @Allen0405340; @Allen0706.0033; @Schmidt0405374; @Rapetti0409574; @Rapetti0710.0440; @Vikhlinin0412306; @Vikhlinin0507092; @Vikhlinin0805.2207; @Vikhlinin0812.2720; @Arnaud0709.1561; @Schmidt0610038; @Mantz0909.3098; @Mantz0909.3099]. High-resolution X-ray imaging data provide a powerful tool to assess the dynamical state of the ICM. The X-rays produced by hot clusters are primarily a combination of bremsstrahlung and line emission. Because the ICM is optically thin, X-ray data carry information about the gas at all radii, albeit in projection. Furthermore, the two-body nature of bremsstrahlung emission results in local density fluctuations producing an exaggerated contrast in surface brightness. This property has enabled studies of a variety of astrophysical features in the regions of clusters where the gas density is relatively high, including shocks and cold fronts (e.g. @Markevitch0001269 [@Markevitch0110468; @Markevitch0412451; @Vikhlinin0008496]; see @Markevitch0701821 for a review), gas sloshing (e.g. @Ascasibar0603246 [@Roediger1007.4209; @ZuHone1108.4427; @Johnson1106.3489; @Simionescu1208.2990; @Paterno-Mahler1306.3520]), cavities (e.g. @Fabian0007456 [@Fabian0306036; @Fabian0510476; @McNamara0001402; @Forman0312576; @Forman0604583; @Hlavacek-Larrondo1110.0489]; see also reviews by @McNamara0001402 and @Fabian1204.4114), and the cool, dense cores found in some clusters (e.g. @Fabian1994MNRAS.267..779F [@White9707269; @Peres9805122; @Peterson0512549]). In cluster outskirts, gas clumping (unresolved inhomogeneities) is implicated by excess X-ray brightness observed by ROSAT and [[Suzaku]{}]{}, although the very low density and emissivity of the gas at large radii makes these observations comparably difficult (e.g. @Simionescu1102.2429 [@Urban1102.2430; @Urban1307.3592; @Walker1205.2276; @Walker1203.0486]). The increase in surface brightness provided by cool cores significantly biases X-ray searches in favor of finding relaxed clusters. While this is an advantage in some sense, the redshift-dependent selection bias imposed by an X-ray flux limit complicates efforts to estimate the degree of relaxation of the cluster population as a whole, and particularly its evolution with time (e.g. @Vikhlinin0611438 [@Santos1008.0754]). At redshifts $z{\ {\raise-.75ex\hbox{$\buildrel>\over\sim$}}\ }0.5$, the bulk of high-resolution X-ray observations of clusters currently target systems discovered through the Sunyaev-Zel’dovich (SZ) effect or other means (e.g. association with a quasar). Within these data sets, some clusters with cool cores have been identified [@Allen0101162; @Siemiginowska1008.1739; @McDonald1208.2962; @Semler1208.3368], but constructing a complete picture of relaxed systems within the evolving cluster population remains challenging. While a number of studies have identified relaxed clusters “by eye,” others have proposed quantitative measurements of image features to assess dynamical state. These generally fall into two categories: those which attempt to measure bulk asymmetry on intermediate scales (e.g. @Mohr1993ApJ...413..492 [@Buote9502002; @Jeltema0501360; @Nurgaliev1309.7044; @Rasia1211.7040]), and those which attempt to assess the presence or development of a cool core (e.g. @Vikhlinin0611438 [@Santos0802.1445; @Mantz2009PhDT........18M; @Bohringer0912.4667]).[^3] Automated algorithms based on such simple measurements are inevitably limited compared to visual classification, but their reproducibility, objectivity and particularly their straightforward applicability to data sets from large follow-up programs make them appealing. This series of papers explores what can be learned by exploiting the most massive, relaxed galaxy clusters. Subsequent papers focus on cosmological constraints from measurements of the gas mass fraction in relaxed clusters ([Paper II]{}, @Mantz1402.6212), thermodynamic profiles and scaling relations of the ICM ([Paper III]{}, Mantz et al., in prep), and the calibration of X-ray hydrostatic mass estimates using weak gravitational lensing ([Paper IV]{}, Applegate et al., in prep). Here we present a new, automatic method for identifying relaxed clusters based on X-ray imaging data, and apply it in a comprehensive search of the [[Chandra]{}]{} archive, in order to produce a suitable sample for this work. Our approach broadly follows others in the literature, but with particular emphasis on wide applicability (across a range in redshift and image depth), robustness against missing data (point source masks and unexposed parts of the focal plane), and independence from cosmological assumptions. For example, these considerations lead us to forgo measurements in the literature which explicitly assume the angular diameter distance to a cluster (i.e. the conversion of angle to metric distance) or the cluster mass (or a radius linked to the mass), or which involve centroids (highly dependent on the treatment of missing data). In [Section]{} \[sec:data\], we describe in detail the reduction of the [[Chandra]{}]{} and ROSAT X-ray data, which are also used in our subsequent papers. [Section]{} \[sec:approach\] provides a broad overview of our approach to measuring the X-ray morphologies of clusters, and [Section]{} \[sec:procedure\] presents the procedure in detail. We discuss the resulting measurements, compare them to other work in the literature, and devise a criterion for relaxation in [Section]{} \[sec:results\]. [Section]{} \[sec:conclusion\] summarizes our findings. Where cosmological calculations are necessary, we adopt a flat [$\Lambda$CDM]{} model with Hubble parameter $H_0=70{{\ensuremath{\mathrm{\, km}}}}{{\ensuremath{\mathrm{\, s}}}}^{-1}{{\ensuremath{\mathrm{\, Mpc}}}}^{-1}$ and matter density with respect to critical ${{\ensuremath{\Omega_{\mathrm{m}}}}}=0.3$. Data {#sec:data} ==== For this work, we analyzed data for a large sample of galaxy clusters which have archival [[Chandra]{}]{} observations (as of 1 February, 2013). Clusters were selected from the following sources: 1. The ROSAT Brightest Cluster Sample (BCS; @Ebeling1998MNRAS.301..881E), with a minimum 0.1–2.4[[$\mathrm{\, keV}$]{}]{} luminosity of $2.5{\ensuremath{\times 10^{44}}}{{\ensuremath{\mathrm{\, erg}}}}{{\ensuremath{\mathrm{\, s}}}}^{-1}$. 2. The ROSAT-ESO Flux Limited X-ray (REFLEX) cluster sample [@Bohringer0405546], with the same luminosity threshold. 3. The Clusters In the Zone of Avoidance (CIZA) sample [@Ebeling2002ApJ...580..774; @Kocevski0512321], with the same luminosity threshold. 4. The MAssive Cluster Survey [@Ebeling0009101; @Ebeling0703394; @Ebeling1004.4683]. 5. The 400 Square Degree ROSAT survey (400d; @Burenin0610739). 6. The South Pole Telescope (SPT) SZ cluster survey (@Bleem1409.0850). 7. The cluster sample of @Allen0706.0033 [, hereafter ]. Our reduction of the [[Chandra]{}]{} data is described below in [Section]{} \[sec:chandra\]. The imaged field of view prohibits the use of [[Chandra]{}]{} data alone for morphological studies of very nearby clusters (redshifts $z{\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ }0.05$, in practice). For a small number of clusters at low redshifts, we have therefore analyzed ROSAT Positional Sensitive Proportional Counter (PSPC) data, as described in [Section]{} \[sec:rosat\]. Due to the low resolution of PSPC, additional caveats apply to these results, as discussed in [Section]{} \[sec:rosatres\]. In total, we reduced and analyzed data for 361 clusters. [Table]{}s \[tab:data\] and \[tab:rosatdata\] list the clusters and observations employed here. Reduction of [[Chandra]{}]{} Data {#sec:chandra} ----------------------------------- We used version 4.4 of the [[Chandra]{}]{} software analysis suite, [<span style="font-variant:small-caps;">ciao</span>]{},[^4] and version 4.4.10 of the [[Chandra]{}]{} calibration database, [<span style="font-variant:small-caps;">caldb</span>]{},[^5] throughout this work. Subsequent changes to the calibration are not expected to significantly influence the imaging analysis presented here. In order to ensure a uniform data reduction, and to obtain the benefits of calibration updates, all data were re-reduced to create new events files. Starting from the data products in the [[Chandra]{}]{} archive, the data were processed using the method outlined in the “ACIS \[Advanced CCD Imaging Spectrometer\] Data Preparation” [[Chandra]{}]{} analysis guide.[^6] The regenerated level-2 events files were screened for periods of high background by filtering their light curves using the [<span style="font-variant:small-caps;">lc\_clean</span>]{} tool. In detail, we begin by selecting a CCD to use for cleaning. Normally, this is the S1 chip for ACIS-S exposures, and the I0 or I2 chip for ACIS-I exposures. There are, however, many exceptions to this, dependent on the specific configuration of each observation. In some ACIS-S exposures the S1 chip is not active, and in these cases we use a relatively source-free area of the S3 chip where possible. For some ACIS-I exposures of low-redshift clusters, where the cluster fills some or all of the detector, we use the S2 chip. We visually inspect the chip and mask out any sources of astrophysical emission (point sources, cluster emission, etc.), and any bad pixels, cosmic rays etc. that were not removed during the reduction phase. We then produce a light curve, using the same parameters as were used to make the [[Chandra]{}]{} blank-sky background data sets,[^7] i.e. for front-illuminated (FI) CCDs the energy range 0.3–12[[$\mathrm{\, keV}$]{}]{}, and a time bin of 259.28s; and for back-illuminated (BI) CCDs the energy range 2.5–6[[$\mathrm{\, keV}$]{}]{} (S1 chip) or 2.5–7[[$\mathrm{\, keV}$]{}]{} (S3 chip), and a time bin of 1037.12s. (The different sets of parameters are motivated by the different sensitivities of the FI and BI chips to background flaring.) We then apply the [<span style="font-variant:small-caps;">lc\_clean</span>]{} tool with default settings: initial mean calculated using $3 \sigma$ clipping, followed by removal of intervals where the count rate is more than a factor of 1.2 different from the mean. In all cases, we visually inspected the resulting light-curves and checked that they were reasonable. The automatic clipping algorithm is sometimes misled by periods of exceptionally high background flaring. In cases like these, we manually exclude the time period corresponding to the flare, and/or manually set the initial mean to the correct quiescent level. For every exposure, we carry out this process for at least two CCDs, and check that they give consistent results. If both FI and BI chips are active, we always examine at least one of each type. Since the BI chips have a higher sensitivity to flares, the BI good-time interval (GTI) is generally applicable to the FI chips as well, but in a few cases we use separate GTIs for the FI and BI chips. As a final safety precaution, we check that the mean level of the light curve after filtering is reasonable, since there are sometimes extended periods of high background which are difficult to detect in short exposures. These values are shown in the left panel of [Figure]{} \[fig:meanctr\], as a function of the date of the observation. Values are per-CCD, corrected for any fraction of the chip area that was excluded. ![image](mean_ctr.pdf) ![image](bgscale.pdf) \[fig:bgscale\] The overall trend as a function of time (high at the start of the mission, before $\sim$2001, then fairly flat from 2001–2003, then rising until $\sim$2010, then declining again) is representative of the evolution of the [[Chandra]{}]{} background, which is influenced by the solar cycle. In addition to this overall shape, FAINT-mode exposures tend to have a higher rate than VFAINT-mode exposures from the same epoch. For some very extended, low-redshift clusters where there is essentially no region of the detector free from cluster emission, the rates are somewhat elevated due to cluster contamination (these are excluded from the figure). In these cases, all we can do is check that the light curve looks reasonable, and that excluding larger fractions of the chip in the direction of the cluster center reduces the normalized rate. On the basis of these checks, we exclude a minority of obsids from further analysis, generally because they are either extensively flared or suspected to be affected by flares, and only represent a small fraction of the data that exist for the target in question (these are noted in [Table]{} \[tab:data\]). Finally, any non-cluster sources in the analyzed fields were masked out by visual inspection of the cleaned events files. To account for possible variations between the blank-sky background exposures and the science exposures, we normalized the blank-sky files using the high-energy count rates, which should measure the overall level of the particle background. Specifically, we apply a multiplicative factor derived from the ratio of the 9.5–12[[$\mathrm{\, keV}$]{}]{} count rates in the science and blank-sky files. (Note that in background period A, and in a small number of science exposures, only events up to 10[[$\mathrm{\, keV}$]{}]{} were telemetered.) These scaling factors typically lie in the range 0.8–1.2, as shown in the right panel of [Figure]{} \[fig:bgscale\]. We find some evidence for chip-to-chip, and indeed node-to-node, variations in the scaling factors, but there are no clear trends. For detailed spectral analysis in subsequent papers, we use per-CCD scaling factors; here, for our basic imaging analysis, we take the more straightforward approach of adopting a single mean scaling per observation for all FI or BI chips. Note that in background epochs A–C, the blank-sky events files are in FAINT mode. In order to use these blank-sky files with science exposures, the science events files must also be processed in FAINT mode, i.e. the VFAINT correction cannot be applied even if available (resulting in somewhat noisier data than would otherwise be the case; see [Figure]{} \[fig:bgscale\]). Such exposures are indicated by “V\” in [Table]{} \[tab:data\]. Reduction of ROSAT Data {#sec:rosat} ----------------------- The ROSAT PSPC observations were reduced using the Extended Source Analysis Software package of @Snowden1994ApJ...424..714. In short, we identify good time intervals using a master veto threshold of 170counts${{\ensuremath{\mathrm{\, s}}}}^{-1}$, to exclude times of anomalously high particle background rates, and a time delay of 15s, to remove the events at the beginning of each observation before the detector high voltage achieved its nominal level. We create light curves for each the seven standard ROSAT bands, and compute a list of nominal scattered solar X-ray (SSX) background count rates, under the assumption that the residual atmosphere along the line of sight is optically thin. The solar X-ray spectrum is modelled as a two temperature thermal plasma, with individual temperatures of $10^{5.7}$K and $10^{6.2}$K. By inspecting the light curves of the SSX background count rates, we identify and exclude periods of intense SSX contamination. In the remaining time intervals, we model the X-ray background in the nominal energy bands of 0.7–0.9, 0.9–1.3 and 1.3–2.0keV (standard ROSAT bands R5–R7), using the standard assumption that the background consists of a cosmic component, the calibrated particle background, a SSX component and a possible long-term enhancement (where required). These models are used to generate background count rate maps. Note that these background maps are not equivalent to the blank-sky maps available for [[Chandra]{}]{}, since they do not account for the astrophysical background; this leads to small differences in our analysis of the ROSAT images in [Section]{} \[sec:boot\]. General Approach {#sec:approach} ================ Preliminaries ------------- Our procedure for characterizing the morphology of galaxy clusters, detailed in the next section, is guided by a few broad principles. (1) It should provide a fair basis to compare clusters spanning a wide range of redshift and mass, and using data of variable quality. Thus, very nearby clusters should not be penalized because we can discern detailed structure within them that would not be resolved at higher redshift. The most crucial step to achieving this is identifying comparable regions of different clusters, which is described in [Section]{} \[sec:sbradii\]. Additionally, because the gaps between [[Chandra]{}]{} CCDs generally mask part of the cluster emission at redshifts ${\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ }0.25$, we avoid the use of centroids and other quantities which assume complete images. (2) As much as possible, the algorithm should be insensitive to the prevalence of Poisson noise, to avoid unfairly penalizing clusters imaged with shallow exposures or located at high redshifts. Integral to meeting this requirement is the robust estimation of measurement uncertainties, which we address by bootstrapping the input photon images, as detailed in [Section]{} \[sec:boot\]. (3) Since the main purpose of this work is to identify a relaxed cluster sample to use for cosmological studies, it is also advantageous to avoid strong assumptions about either the mass (or virial radii) of the clusters, or the background cosmology. The particular quantities that we calculate from the cluster images are designed to measure the features on which subjective determinations of relaxation are generally based. In general terms, these are: 1. the sharpness of the peak in surface brightness. 2. the shifting of isophotes with respect to one another (i.e. the appearance of sloshing). 3. the distance between the center of symmetry on large scales (a low brightness isophote) and small scales (e.g. the cool core, if any). [Section]{}s \[sec:peakiness\] and \[sec:ellipses\] provide more complete details of the measurements, which are carefully designed to respect the “fair comparison” requirement above. In practice, this suite of three relatively simple calculations performs well, and the close connection between the measurements and visible features aids their interpretation. The particular thresholds for the measured values that we adopt to identify relaxed clusters are roughly placed with reference to prior, subjective decisions. Once in place, however, the thresholds are applied without regard to any subjective determinations. We assess the performance of the algorithm both by whether its decisions are subjectively reasonable, and, more pertinently, by comparing the measured intrinsic scatter of the gas mass fraction for the new relaxed sample with the subjectively identified sample of ; this comparison was made only after* the new sample was finalized. As described in [Section]{} \[sec:fgascompare\], the algorithmically identified sample has a somewhat smaller intrinsic scatter than the sample. Although it is beyond the scope of this work, testing our algorithm against mock X-ray images of simulated galaxy clusters can potentially provide further refinements. Standardizing Cluster Surface Brightness {#sec:sbradii} ---------------------------------------- Outside of their central regions, the surface brightness profiles of galaxy clusters are approximately self-similar (e.g. @Vikhlinin0507092 [@Croston0801.3430]). This raises the possibility of identifying characteristic radii that are comparable across clusters via the surface brightness. To that end, we motivate a redshift- and temperature-dependent scaling of surface brightness based on the self-similar model of @Kaiser1986MNRAS.222..323K [, see also @Santos0802.1445]. The average surface brightness within a circular aperture of angular radius $\theta$, corresponding to physical radius $r=\theta\,{{\ensuremath{d_{\mathrm{A}}}}}(z)$, for a cluster with redshift $z$ and angular diameter distance ${{\ensuremath{d_{\mathrm{A}}}}}(z)$, is $$S = \frac{F}{\pi\theta^2} \propto \frac{K(z,T,{{\ensuremath{N_{\mathrm{H}}}}}) L}{(1+z)^4r^2},$$ where $F$ and $L$ are, respectively, the observer-frame flux and rest-frame bolometric luminosity of the cluster. Here the coefficient $K$ accounts for the redshift- and temperature-dependent K-correction from bolometric flux to flux in the observed energy band, as well as any Galactic absorption (equivalent absorbing hydrogen column density, [[$N_{\mathrm{H}}$]{}]{}). For self-similar profiles, this proportionality also holds for surface brightness at a given characteristic radius, $r_\Delta$, defined in terms of the cluster mass and the critical density of the Universe; $M(<r_\Delta)=(4/3)\pi \Delta {{\ensuremath{\rho_{\mathrm{cr}}}}}(z) r_\Delta^3$. Using scalings from the Kaiser model, $$\begin{aligned} \label{eq:simscaling} L & \propto & T^2 E(z), \\ r_\Delta & \propto & \frac{T^{1/2}}{E(z)}, \nonumber\end{aligned}$$ where $E(z) = H(z)/H_0$ is the normalized Hubble parameter, we can eliminate $L$ and $r_\Delta$ in favor of the ICM temperature, $T$. This yields the relation $S(r_\Delta) \propto f_S$, with $$\label{eq:sbscal} f_S \equiv K(z,T,{{\ensuremath{N_{\mathrm{H}}}}}) \frac{E(z)^3}{(1+z)^4} \left(\frac{kT}{\mathrm{keV}}\right) ~\mathrm{photons}{{\ensuremath{\mathrm{\, Ms}}}}^{-1}{{\ensuremath{\mathrm{\, cm}}}}^{-2}\,(0.984\,\mathrm{arcsec})^{-2},$$ where we have assigned units which are convenient for the analysis of [[Chandra]{}]{} data (see [Section]{} \[sec:boot\]). Following the argument above, surface brightness levels corresponding to constant multiples of $f_S$ should correspond to approximately the same values of $\Delta$ across all clusters, provided they fall in the self-similar part of the profile. With this rescaling, it becomes possible to identify approximately corresponding regions of clusters with different masses and redshifts, without explicitly assuming the angular diameter distance to each or a prescription for estimating some scale radius $r_\Delta$ (equivalently $M_\Delta$). There is an implicit assumption of cosmological parameters necessary to evaluate $E(z)$, but this sensitivity is relatively mild. As input, we need only the redshifts of clusters, column densities for their positions on the sky, and rough temperature estimates for them.[^8] As an a posteriori check of how reasonable this scaling is, [Figure]{} \[fig:allprofiles\] shows surface brightness profiles from our analysis of [[Chandra]{}]{} data ([Section]{} \[sec:sbprofile\]). The surface brightness values are background-subtracted and shown in units of $f_S$, and the radial coordinate is scaled by $E(z)/\sqrt{T}$ according to [Equation]{} \[eq:simscaling\].[^9] The intrinsic scatter among profiles is significant at small radii, tightening to a self-similar profile at large radii. The clusters that are ultimately identified as relaxed in this work form a particularly tight locus. [ ![ Scaled, background subtracted surface brightness profiles from our [[Chandra]{}]{} analysis. The scaling factors follow from the self-similar model and are given in [Equation]{}s \[eq:simscaling\]–\[eq:sbscal\]. Clusters that are ultimately categorized as relaxed in this work are shown in blue, and others in red. Measurement errors on the individual profiles are not shown (but see [Figure]{} \[fig:sbprof\]). ](sb_profiles.pdf "fig:") ]{} \[fig:allprofiles\] Procedure {#sec:procedure} ========= This section describes in detail our procedure for measuring morphological indicators and their uncertainties. Data Preparation and Bootstrapping {#sec:boot} ---------------------------------- For the [[Chandra]{}]{} observations, images in the 0.6–2.0[[$\mathrm{\, keV}$]{}]{} band are extracted from both the cleaned science and blank-sky event files, and are binned by a factor of two (obtaining $\approx1$ arcsec resolution). An appropriate exposure map is generated for the same energy range. Off-chip pixels and pixels contaminated by point sources are flagged in the science images. These files, along with the blank-sky normalization factor and its statistical error, serve as input to our morphological algorithm. All the steps described below are performed on 1000 bootstrap realizations of each observation. We bootstrap the science and blank-sky images at the level of individual counts; that is, the pixel locations of each detected photon in the original image are listed (with repetition, as appropriate), and photons are added to pixels of the bootstrap image by sampling from this list with replacement. For each bootstrap iteration, we also sample a new value of the blank-sky normalization factor, based on its statistical uncertainty. To estimate statistical signal-to-noise throughout the analysis, we keep track of the variance in various quantities, beginning with the counts in the images. We assign the statistical variance $N+1$ to each pixel of the science and blank-sky images, where $N$ is the number of counts in the corresponding pixel. This choice is motivated by the fact that the Bayesian posterior for the average number of counts in an equal-length exposure, based on the observed counts $N$, has variance $N+1$;[^10] furthermore, it neatly avoids the pathological assignment of zero uncertainty to pixels with zero counts. Note that our final uncertainties are entirely characterized by the bootstrap procedure; the error maps described here only provide approximate signal-to-noise estimates for, e.g., the surface brightness profile fitting and adaptive smoothing steps below. The blank-sky image is rescaled according to the normalization factor and subtracted from the science image (recall that each of these ingredients is bootstrapped), propagating the variance of the background-subtracted image straightforwardly. The result is divided by the exposure map, assuming no uncertainty in the latter, to create a flat-fielded image. At this stage, it is possible to straightforwardly combine images from multiple observations of a cluster by the same telescope. Finally, we convert the brightness images to intensity in units of photons${{\ensuremath{\mathrm{\, Ms}}}}^{-1}{{\ensuremath{\mathrm{\, cm}}}}^{-2}\,(0.984\,\mathrm{arcsec})^{-2}$.[^11] For ROSAT observations, our procedure differs in a few details. The ROSAT images cover the 0.7–2.0[[$\mathrm{\, keV}$]{}]{} energy band and have the native PSPC resolution of 14.9 arcsec. Since there are no blank-sky fields, we subtract the ROSAT particle background rate maps from the images after converting the latter to count rates but before flat fielding (since the particle background is not vignetted). A spatially constant residual background level, accounting for unresolved astrophysical sources, is fit and subtracted at a later stage (see [Section]{} \[sec:sbprofile\]). Center Finding and Surface Brightness Profiling {#sec:sbprofile} ----------------------------------------------- A global center for each cluster is defined by computing the median photon location in an iteratively shrinking aperture. Beginning with the entire image, the center is defined as $(\tilde{x},\tilde{y})$, where $\tilde{x}$ ($\tilde{y}$) is calculated by summing the image over columns (rows), shifting the resulting one-dimensional array to be non-negative, and computing the median of the resulting discrete function of $x$ ($y$). A new image is extracted, centered on $(\tilde{x},\tilde{y})$ but with dimensions smaller by a factor of $2/3$ (or more if the edge of the image is encountered), and the procedure is repeated until a minimum aperture size of 40 pixels square has been reached and the center is static. In practice, this median center compromises between two widely used alternatives, the brightest pixel and the centroid. In clusters having a cool core that is offset from the center of emission on larger scales, the median center tends to be located within the cool core, although not necessarily at its center or brightest point. Like the centroid, it does respond to a degree to the weight of emission in the fainter regions of the cluster. However, the median center is much less biased by the presence of masked regions than the centroid, to the extent that “filling in” masked regions and gaps between detectors is generally unnecessary. Compared to simply choosing the brightest pixel, the median procedure has the clear advantage that it is less susceptible to Poisson noise or mistakenly unmasked point-source emission. An azimuthally averaged surface brightness profile about the median center is calculated in annuli which are adaptively chosen to provide a signal-to-noise ratio $>2$ (with a single, signal-to-noise $<2$ annulus covering the largest imaged radii). A $\beta$ model [@Cavaliere76] plus constant background level are then fitted to the radially outermost half of the profile,[^12] and the best-fitting constant is subtracted from the image and surface brightness profile. When brightness levels are compared to the surface brightness profile in the following sections, we compare to the measured profile at radii where it was constrained with the target signal-to-noise, and to the $\beta$ model at larger radii. Similarly, when random values are drawn to be consistent with the profile at a given radius, we scatter them according to the measurement uncertainty for the appropriate annulus, or the outermost annulus in the case of extrapolation. Following the argument in [Section]{} \[sec:sbradii\], we define a set of characteristic surface brightness levels in our adopted units, $$\label{eq:sblevels} S_j = 0.002 \times 10^{0.28j} f_S,$$ where $j=0,1,\ldots,5$. The number of levels and the range in surface brightness that they span were chosen empirically to provide good performance for the measurement of our morphological estimates (described in the following sub-sections) over a wide range of data quality and cluster redshifts. These scaled surface brightness levels are shown along with example profiles for Abell1835 (which has a cool, bright core) and Abell2163 (which has a flat core) in the left panel of [Figure]{} \[fig:sbprof\]. ![image](sb_fig.pdf) ![image](a1835.pdf) ![image](a2163.pdf) \[fig:ellipses\] Surface Brightness Peakiness {#sec:peakiness} ---------------------------- The presence of a core of bright, relatively cool, X-ray emitting gas in the center of a cluster is a common signature of dynamically relaxed systems [@Fabian1994MNRAS.267..779F; @Peterson0512549]. The formation of these features is expected, and to some extent observed, to be disrupted by major mergers [@Burns0708.1954; @Henning0903.4184; @Million0910.0025; @Rossetti1106.4563; @Skory1211.3117; @Ichinohe1410.1955]. Thus, while cool cores are not necessarily completely destroyed by major mergers once formed, requiring the presence of a core should provide an efficient way to reject unrelaxed clusters. Although measuring a temperature decrement in the center of a cluster is relatively involved, detecting the presence of a central brightness enhancement is straightforward. Consequently, simple measurements of the sharpness of the peak in surface brightness at cluster centers have been widely employed as a proxy for the presence of cool cores. Various measurements of peak strength have been introduced. @Vikhlinin0611438 used the logarithmic slope of the gas density profile at a radius of $0.04\,r_{500}$. @Santos0802.1445 advocate using the ratio of fluxes contained in two metric apertures; flux ratios in apertures linked to $r_{500}$ have also been employed (e.g. @Mantz2009PhDT........18M [@Bohringer0912.4667]). For the present work, the explicit reliance of each of these approaches on metric distances (i.e. on an assumed angular diameter distance) or scale radii ($r_{500}$) is a disadvantage. Instead, we introduce a measurement which relies only on the scaled surface brightness profile in the region where it is typically very well constrained, as follows. First, we determine the angular radius, $\theta_5$, where the measured surface brightness profile is equal to $S_5$, as defined in the previous section; if the profile never exceeds this value, then the radius bounding the innermost bin of the surface brightness profile is used. We then calculate the average surface brightness at distances $\leq \theta_5$ from the global center of [Section]{} \[sec:sbprofile\] in units of $f_S$, assigning to each masked pixel in this region a random value based on the surface brightness profile and its uncertainty at the appropriate radius. (This calculation is statistically equivalent to taking the area-weighted average of the surface brightness profile at radii $\leq\theta_5$.) This average, scaled central surface brightness, $\bar{S}(\theta\leq\theta_5)/f_S$, shows an overall downward trend with redshift across the data set, as seen in [Figure]{} \[fig:censbav\]. This is expected; qualitatively similar trends have been reported in measurements of surface brightness “concentration” (@Santos0802.1445; see also @Santos1008.0754 [@McDonald1305.2915]), which our measurements are closely related to (see [Section]{} \[sec:otherx\]). Physically, this increase of brightness with time, particularly at the high central brightness end, presumably corresponds to non-self-similar evolution in the development of cool cores in relaxed clusters. Since our procedure is intended to select morphologically relaxed clusters at any redshift, we include a redshift weighting, which in the absence of precise predictions from hydrodynamical simulations, we assume to be linear.[^13] Taking the logarithm for convenience, the surface brightness peakiness, $p$, is thus defined as $$\label{eq:peakiness} p = \log_{10} \left[ (1+z) \frac{\bar{S}(\theta \leq \theta_5)}{f_S} \right].$$ To the extent that cluster surface brightness profiles are self-similar at radii greater than $\theta_5$, this quantity contains as much information as the ratio of flux in small and large apertures, while being measured more precisely. The particular value of $S_5$ ([Equation]{} \[eq:sblevels\]) was chosen for exactly this purpose; the divergence of the surface brightness profiles of Abell1835 (bright core) and Abell2163 (non-bright core) at radii $<\theta_5$ seen in [Figure]{} \[fig:sbprof\] is typical (see also [Figure]{} \[fig:allprofiles\]). A more extreme contrast can be seen in [Figure]{} \[fig:extremep\], which compares the clusters with the lowest and highest values of $p$ from our analysis. [ ![ Average central surface brightness in scaled units as a function of redshift from our [[Chandra]{}]{} analysis. Clusters that we ultimately classify as relaxed ([Section]{} \[sec:morph\_criterion\]) are shown as blue circles, and others as red crosses. A net decreasing trend can be seen, qualitatively in agreement with observations based on similar surface brightness measurements [@Santos0802.1445; @Santos1008.0754; @McDonald1305.2915]. Our peakiness measure incorporates a $1+z$ weighting to approximately compensate for this evolution in core brightness; the dashed line corresponds to the constant-peakiness threshold used to define the relaxed sample in [Section]{} \[sec:morph\_criterion\]. ](sbav_z.pdf "fig:") ]{} \[fig:censbav\] ![image](spt2344-4242.pdf) ![image](a2255.pdf) ![image](peak_cbar.pdf)\ ![image](spt2344-4242_surf.png) ![image](a2255_surf.png) Elliptical Isophote Fitting and Statistics {#sec:ellipses} ------------------------------------------ Our other morphological measurements aim to quantify the two-dimensional structure of clusters. Here again we avoid algorithms which assume complete imaging coverage, such as the centroid variance [@Mohr1993ApJ...413..492] and various other measures of substructure and asymmetry (e.g. @Nurgaliev1309.7044 [@Rasia1211.7040]), as masked point sources or the gaps between adjacent CCDs often impinge on cluster images in practice. (Indeed, [Figure]{}s \[fig:ellipses\], \[fig:extremep\], \[fig:extremesa\] and \[fig:triangle\] all provide examples of this.) Instead, our approach fits elliptical shapes to the 5 isophotes defined by the brightness levels in [Equation]{} \[eq:sblevels\]. This analysis does not use the “filled-in” image introduced in [Section]{} \[sec:peakiness\], since azimuthal symmetry is assumed in the production of those images. Instead, to reduce Poisson noise, we apply an adaptive boxcar smoothing algorithm to the original flat-fielded image, with a maximum kernel radius of 10 pixels and target signal-to-noise of two, enforcing that pixels masked in the original image remain masked in the final product. To prevent very distant pixels with large noise fluctuations from influencing our results, these smoothed images are cropped beyond the radius corresponding to $0.1 S_0$. We then identify pixels in the smoothed image with values in each of the 5 brightness ranges (isophotes) $S_j<S<S_{j+1}$. An elliptical shape is fit to each of these isophotes, where the fit minimizes the sum of absolute distances from the ellipse to each pixel in the isophote along the line passing through the pixel and the ellipse center. To automatically catch cases where the ellipse fit is suspect, we compute the following two quantities. The first, [[$f_{\mathrm{el}}$]{}]{}, is straightforwardly the fraction of the ellipse which falls on unmasked pixels; this is useful for identifying cases where the ellipse fit should not be trusted because most of the true azimuthal extent of the isophote was not imaged. The second quantity is ${{\ensuremath{\Gamma_{\mathrm{el}}}}}= \left\langle e^{i\phi} \right\rangle$, where $\phi$ is the angle between the major axis of the ellipse and a ray from the ellipse center to a given pixel, and the average is over pixels in the corresponding isophote. This statistic measures how balanced the distribution of isophote pixels is with respect to the fitted ellipse center, and efficiently finds cases where the best-fitting ellipse simply passes as closely as possible to a very non-elliptical distribution of pixels.[^14] For a given isophote and bootstrap iteration, if ${{\ensuremath{f_{\mathrm{el}}}}}<0.5$ or either the real or imaginary part of [[$\Gamma_{\mathrm{el}}$]{}]{} has magnitude $>0.4$, the fit is considered to have failed, and the isophote is discarded. In addition, no fit is attempted for isophotes where the lower end of the brightness range lies in the outer portion of the surface brightness profile (where the target signal-to-noise was not achieved), for isophotes where the upper end of the brightness range is greater than the central point in the surface brightness profile, or for isophotes consisting of $<100$ pixels. For an isophote to contribute to the final set of statistics for a cluster, we require it to be successfully fit in $>3/4$ of bootstrap iterations. In [Paper II]{}, mass profiles are derived for a sample of 40 relaxed clusters identified in the present work. Histograms of the mean of the semi-major and semi-minor axes in units of $r_{2500}$ are shown for these clusters in [Figure]{} \[fig:isoph\_radii\]. As expected, the isophotes in units of $f_S$ broadly map onto comparable radii in units of $r_{2500}$. [ ![ Histograms of the average of the semi-major and semi-minor axes of ellipses corresponding to the five isophotes used in our analysis, as a fraction of $r_{2500}$ (shown as lines, for clarity). Only 40 clusters which are classified as highly relaxed here and for which we can reliably determined mass profiles (hence $r_{2500}$; see [Paper II]{}) are used here. The cluster region used in our isophote analysis typically spans radii $0.2{\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ }r/r_{2500}{\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ }1$. ](fig_isophote_x2500.pdf "fig:") ]{} \[fig:isoph\_radii\] From this set of ellipses, we calculate two statistics, which we refer to as alignment, $a$, and symmetry, $s$. These are defined to have the same sense as the peakiness, i.e. more positive (negative) values being typical of more (less) relaxed clusters. The alignment is defined as $$a = -\log_{10}\left[\frac{1}{{{\ensuremath{N_{\mathrm{el}}}}}-1} \sum_{j=1}^{{{\ensuremath{N_{\mathrm{el}}}}}-1} \frac{\delta_{j,j+1}}{\langle b\rangle_{j,j+1}}\right],$$ where [[$N_{\mathrm{el}}$]{}]{} is the number of ellipses and the sum is over pairs of “adjacent” ellipses, i.e. those corresponding to progressively higher surface brightness. Here $\delta_{j,j+1}$ is the distance between the centers of two ellipses, and $\langle b\rangle_{j,j+1}$ is the average of the four ellipse axis lengths (major and minor axes of both ellipses). The symmetry statistic is $$s = -\log_{10} \left[ \frac{1}{{{\ensuremath{N_{\mathrm{el}}}}}} \sum_{j=1}^{{{\ensuremath{N_{\mathrm{el}}}}}} \frac{\delta_{j,\mathrm{c}}}{\langle b\rangle_j} \right],$$ where $\delta_{j,\mathrm{c}}$ is the distance between the center of the $j$th ellipse and the global center identified in [Section]{} \[sec:sbprofile\], and $\langle b\rangle_j$ is the average of the major and minor axes of the ellipse. These quantities provide complementary measurements of cluster substructure. The alignment is sensitive to shifts in the center of emission at the relatively large scales probed by our set of isophotes, whereas the symmetry parameter measures the overall agreement of those isophotes with the global center. Note that, by design, the brightness range covered by this analysis does not extend to the brightest (spatially central) regions of cool core clusters (left panel of [Figure]{} \[fig:sbprof\]), where complex, non-elliptical features such as cavities and small-scale sloshing are ubiquitous, even in more globally relaxed clusters. [Figure]{} \[fig:ellipses\] shows smoothed images and isophote ellipse fits to the unmodified (i.e. not bootstrapped) data for the example clusters A1835 and A2163, which respectively have relatively high and low values of both alignment and symmetry. Clusters representing even more extreme values of $a$ and $s$ are on display in [Figure]{} \[fig:extremesa\]. ![image](a2204.pdf) ![image](a115.pdf) ![image](macs0417-1154.pdf) Results {#sec:results} ======= The procedure of [Section]{} \[sec:procedure\] was applied to obtain morphological statistics from 1000 bootstrap simulations of the clusters identified in [Section]{} \[sec:data\]. Results are tabulated in [Table]{}s \[tab:morph\] and \[tab:allresults\]. For the [[Chandra]{}]{} sample, these are also shown in [Figure]{} \[fig:morph\_cuts\]. We note that there are cases where our morphology code fails outright. For example, for flat-core (low $p$) clusters in very shallow images, we are sometimes unable to constrain even two isophote ellipses, which is necessary for the calculation of alignment; however, in these cases, it is generally still possible to measure peakiness. The great majority of these can be classified as unrelaxed according to the criterion introduced in [Section]{} \[sec:morph\_criterion\] based solely on peakiness. In yet lower signal-to-noise data, it is sometimes impossible to obtain meaningful constraints on the surface brightness profile, and thus even peakiness cannot be measured. Subjectively speaking, this small minority of clusters appears unambiguously unrelaxed, and we classify them as such. ![image](peak_symm.pdf) ![image](peak_align.pdf) ![image](symm_align.pdf) Note that there is a strong correlation between symmetry and alignment ([Figure]{} \[fig:morph\_cuts\]), by virtue of their similar definitions in terms of isophote properties. Somewhat weaker correlations exists between symmetry or alignment on one hand and peakiness on the other; these presumably reflect the role of mergers in either destroying or preventing the formation of cool cores. Comparison with Other X-ray Morphology Statistics {#sec:otherx} ------------------------------------------------- To provide some context, we now compare our morphological statistics to typical estimators used in the literature. Specifically, we have chosen the surface brightness concentration parameter of @Santos0802.1445 and the centroid variance [@Mohr1993ApJ...413..492], defined by $$\begin{aligned} {{\ensuremath{c_{\mathrm{SB}}}}}&=& \frac{F(r<40{{\ensuremath{\mathrm{\, kpc}}}})}{F(r<400{{\ensuremath{\mathrm{\, kpc}}}})}, \\ w^2 &=& \frac{1}{r_{500}^2} \mathrm{Var}(\Delta), \nonumber\end{aligned}$$ where we estimate $r_{500}$ from the temperature–mass relation of @Mantz0909.3099. The distances $\Delta$ are calculated between our global centers and the centroids of emission in our “filled-in” images within apertures of radius $(0.1j)r_{500}$ ($j=1,2,\ldots,10$) about the global centers. We additionally compute the power ratio $P_3/P_0$ [@Buote9502002], again using the filled-in images. We compare our morphological statistics to these alternatives in [Figure]{} \[fig:morph\_lit\]. Not surprisingly, peakiness correlates most strongly with [[$c_{\mathrm{SB}}$]{}]{},[^15] while both alignment and symmetry anti-correlate strongly with centroid variance. The power ratio correlates less well with our statistics. While there are important differences, it is clear that our statistics measure similar image features to these other quantities. In fact, the cuts in $s$, $p$ and $a$ that we use to define a relaxed sample in [Section]{} \[sec:morph\_criterion\], which were determined before we had even calculated [[$c_{\mathrm{SB}}$]{}]{} and $w$, correspond surprisingly well to the cuts used by @Santos0802.1445 and @Bohringer0912.4667 to define strong cool cores and low centroid variance, respectively. Note, however, that our final selection appears to be somewhat more conservative than these cuts on [[$c_{\mathrm{SB}}$]{}]{} and $w$ would be, as one might generically expect given the use of a third, non-degenerate measurement in our selection. ![image](peak_csb.pdf) ![image](align_cv.pdf) ![image](symm_cv.pdf) Comparison with BCG/X-ray Offsets {#sec:bcgs} --------------------------------- A simple metric that has been used to try to distinguish between relaxed and unrelaxed clusters is the distance in projection between the center of the X-ray emission and the location of the brightest cluster galaxy (BCG). This approach is potentially appealing because in principle the X-ray data need not be deep enough to provide peakiness measurements, let alone the more challenging alignment and symmetry measurements. This may be the case for, e.g., X-ray snapshots of distant SZ- or IR-selected clusters, whose X-ray brightness is not well known prior to the observations. At the same time, optical or IR imaging is still commonly used to confirm the presence of a galaxy overdensity at the location of a candidate cluster, and to study the properties of cluster galaxies, and so a BCG identification may be readily available. Where available, we use the BCGs identified in the Weighing the Giants project (54 clusters; @von-der-Linden1208.0597) or for the SPT survey (18 clusters in common with our sample; @Song1207.4369). For the remaining clusters, we query the DR7 and DR10 catalog and imaging databases of the Sloan Digital Sky Survey[^16] (SDSS; @Abazajian0812.0649 [@Ahn1307.7735]), which provides BCGs for an additional 123 clusters. The clusters considered here span a wide redshift range, and several are known to have central galaxies bluer than the red sequence (e.g. @Crawford9903057), making simple algorithmic identification schemes difficult to implement. We therefore verify each BCG candidate by eye, considering galaxies up to 1Mpc from the X-ray center. For each cluster, the initial BCG candidates are taken as the brightest objects likely to be elliptical galaxies (in the SDSS [Galaxies]{} catalog, with concentration $R_{90}/R_{50}>2.3$, and where a de Vaucouleur profile is a better fit than an exponential) within two apertures (50kpc and 500kpc) from the X-ray center. For 73 clusters, the two apertures select the same galaxy; in 69 clusters, it also passes visual verification (in the remaining 4 clusters the initial candidate is a foreground galaxy). For 38 clusters, the two apertures select different BCG candidates; in 21 (17) clusters, we select the candidate within 50kpc (500kpc). For 12 clusters, the BCG is not one of these two candidates for a variety of reasons (e.g. nearby BCGs are de-blended into several detections). In total, this yields 195 BCG positions. ![image](bcg_morph_peak.pdf) ![image](bcg_morph_alig.pdf) ![image](bcg_morph_symm.pdf) [Figure]{} \[fig:morph\_bcg\] shows the projected distance between these BCG locations and the global X-ray centers defined in [Section]{} \[sec:sbprofile\] versus the corresponding measurements of X-ray symmetry, peakiness and alignment. Note that a large fraction of the $<10{{\ensuremath{\mathrm{\, kpc}}}}$ offsets translate to $<1''$ in angular distance (i.e. less than the resolution of our X-ray images), and so are uncertain in detail. (Conversely, offsets $>10{{\ensuremath{\mathrm{\, kpc}}}}$ are resolved, i.e. $>1''$, for the entire data set.) Nevertheless, there is a clear correlation between the BCG/X-ray offset and peakiness, while in contrast there is not such a pronounced trend between the offset and either alignment or symmetry. This makes physical sense, since merger activity generically should produce BCG/X-ray offsets as well as a reduction in peakiness at some level. At the same time, while the offsets for clusters that we ultimately classify as morphologically relaxed ([Section]{} \[sec:morph\_criterion\]) are generally small, there is a range in offsets, reaching 24[[$\mathrm{\, kpc}$]{}]{} in the most extreme case.[^17] This scatter has a natural explanation in sloshing of the ICM due to merger events; the small-scale displacement of the ICM from the precise center of the gravitational potential may persist for Gyr, even as the effect on X-ray emission on the larger scales probed by the symmetry and alignment measurements is muted [@ZuHone1108.4427]. Based on the distributions in [Figure]{} \[fig:morph\_bcg\], it is not clear that measurements of the BCG offset contribute much in addition to the full set of X-ray morphological measurements, particularly peakiness. On the other hand, given BCG locations and relatively poor X-ray data – sufficient to find an X-ray center, but not to measure even peakiness, e.g. from a shallow survey – a suitable cut on the BCG offset clearly would eliminate a large fraction of unrelaxed clusters. Comparison with Radio Halo/Relic Samples {#sec:radiohalos} ---------------------------------------- Radio halos, low surface brightness synchrotron emission located in the central regions of clusters, have been associated with merging activity, although not all merging clusters display radio halos (see @Feretti1205.1919 and references therein). [Figure]{} \[fig:morph\_halos\] shows our morphological measurements for clusters with detected radio halos [@Feretti1205.1919; @Cassano1306.4379]. Also shown are clusters for which strong upper limits have been placed on the radio power without detecting a halo [@Cassano1306.4379]. The radio halo clusters are uniformly unrelaxed according to our X-ray morphological analysis ([Section]{} \[sec:morph\_criterion\]), while the clusters with only upper limits split between being relaxed and unrelaxed. These trends are consistent with previous work comparing the incidence of radio halos with other morphological estimators, namely power ratios, surface brightness concentration and/or centroid variance [@Buote0104211; @Cassano1008.3624]. Similarly, all the clusters in our analysis which host radio relics (emission localized to cluster outskirts) according to the compilation of @Feretti1205.1919 are found to be unrelaxed. ![image](peak_symm_radioHalos.pdf) ![image](peak_alig_radioHalos.pdf) ![image](symm_alig_radioHalos.pdf) The SPA Criterion for Relaxation {#sec:morph_criterion} -------------------------------- An interesting extension of this work would be to test our morphological statistics against the actual dynamical state of simulated clusters using mock X-ray images, as in @Bohringer0912.4667 and @Meneghetti1404.1384, although we note that overcooling in simulations has historically limited the applicability of this approach. For the moment, we are concerned only with selecting the most morphologically relaxed group of clusters, rather than clusters that meet a specific criterion in terms of non-thermal support. We therefore use the subjective determinations of as a broad guide for identifying the ranges of $p$, $a$ and $s$ corresponding to the most relaxed clusters. Note that the selection, though subjective, has previously survived “double-blind” tests; i.e., the same clusters were independently selected as the most relaxed by multiple viewers, with cluster identities hidden. The advantage of this work is that it provides a practical and evenhanded way to compare a large number of clusters, putting the selection in a wider context. [Figure]{} \[fig:morph\_A08\] shows the distribution of peakiness, alignment and symmetry for the large sample of analyzed clusters as purple ‘$\times$’ symbols, with clusters from shown as green triangles. Clearly, the morphological statistics introduced above are related to the subjective determinations used by . At the same time, within the context of the large, homogeneously analyzed sample, it is clear that not all of the clusters belong to a well defined locus in the most relaxed corner of parameter space. Introducing cuts based on our morphology measurements may thus produce a more rigorously defined relaxed sample. ![image](peak_symm_A08.pdf) ![image](peak_align_A08.pdf) Motivated by the distributions in [Figure]{} \[fig:morph\_A08\], we introduce the Symmetry-Peakiness-Alignment (SPA) criterion for cluster relaxation. Namely, we define simple cuts in these three parameters, as depicted in [Figure]{} \[fig:morph\_cuts\]: $s>0.87$, $p>-0.82$, and $a>1.00$ ([Figure]{} \[fig:morph\_cuts\]).[^18] We categorize a cluster as relaxed if $>50$ per cent of the $s$-$p$-$a$ triplets from the cluster’s bootstrap analysis simultaneously satisfy all three of these cuts.[^19] [Table]{} \[tab:morph\] lists whether each cluster was classified as relaxed. Our intent is to generate a conservative (i.e. as pure as possible) sample of relaxed clusters, even at the expense of excluding some legitimately relaxed systems; however, for convenience, we will use the term “unrelaxed” to refer to clusters that do not meet the SPA criterion. We compare the resulting selection to similarly motivated samples in the literature in [Section]{}s \[sec:fgascompare\] and \[sec:othersel\], below. [Figure]{} \[fig:triangle\] shows the SPA cuts in relation to the bootstrap confidence regions associated with three example clusters, Abell1413, MACSJ0744.8+3927 and RXJ0331.1$-$2100, along with smoothed images. Each of these clusters is classified as unrelaxed due to only one of the SPA criteria (i.e., each would be classified as relaxed if only two of the cuts were applied to the bootstrap distributions). Specifically, the emission from Abell1413 is very regular, but not strongly peaked; MACSJ0744.8+3927 has a strong peak and acceptable alignment, but fails the symmetry requirement; and RXJ0331.1$-$2100 has acceptable peakiness and symmetry, but low alignment. ![image](trianglefig.pdf){width="\textwidth"} Differences from the Sample {#sec:fgascompare} --------------------------- One motivation for this work is to identify a relaxed cluster sample to be used for cosmological studies of the gas mass fraction, as in and [Paper II]{}. The cosmological sample must meet additional criteria to those discussed here, regarding the cluster temperature and data quality (see [Paper II]{} for details). Nevertheless, we note here the differences between the two cosmology samples which are due to morphological considerations. Specifically, Abell1795, Abell1413, Abell963, Abell2390, Abell611, Zw3146, Abell2537, MACSJ0329.7$-$0212, MACSJ0744.9+3927,\ MS1137.5+6625, and CLJ1226.9+3332 were used in but are excluded from the sample used in [Paper II]{} (henceforth [SPA$_\mathrm{c}$]{}) by the present analysis. (This analysis adds an equal number of clusters to the [SPA$_\mathrm{c}$]{} sample, on the basis of data taken since 2008.) A gallery of clusters in the [SPA$_\mathrm{c}$]{} sample appears in [Figure]{} \[fig:gallery\]. ![image](gallery/a2029.pdf) ![image](gallery/a478.pdf) ![image](gallery/rxj1524.pdf) ![image](gallery/pks0745.pdf) ![image](gallery/a2204.pdf) ![image](gallery/rxj0439.pdf) ![image](gallery/z2701.pdf) ![image](gallery/rxj1504.pdf) ![image](gallery/z2089.pdf) ![image](gallery/rxj2129.pdf) ![image](gallery/rxj1459-1811.pdf) ![image](gallery/a1835.pdf) ![image](gallery/a3444.pdf) ![image](gallery/ms2137.pdf) ![image](gallery/macs0242.pdf) ![image](gallery/macs1427-25.pdf) ![image](gallery/macs2229.pdf) ![image](gallery/macs0947.pdf) ![image](gallery/macs1931.pdf) ![image](gallery/macs1115.pdf) ![image](gallery/macs1532.pdf) ![image](gallery/macs0150-1005.pdf) ![image](gallery/macs0011.pdf) ![image](gallery/macs1720.pdf) ![image](gallery/macs0429.pdf) ![image](gallery/macs0159.pdf) ![image](gallery/macs2046.pdf) ![image](gallery/i09104.pdf) ![image](gallery/macs1359.pdf) ![image](gallery/rxj1347.pdf) ![image](gallery/3c295.pdf) ![image](gallery/macs1621.pdf) ![image](gallery/macs1427+44.pdf) ![image](gallery/macs1423.pdf) ![image](gallery/spt2331-5051.pdf) ![image](gallery/spt2344-4242.pdf) ![image](gallery/spt0000-5748.pdf) ![image](gallery/spt2043-5035.pdf) ![image](gallery/cl1415.pdf) ![image](gallery/3c186.pdf) The intrinsic scatter in cluster gas mass fractions, [[$f_{\mathrm{gas}}$]{}]{}, is a useful metric for determining the effect of our more stringent morphological criteria compared to . To the extent that dynamical state is the main difference between the [SPA$_\mathrm{c}$]{} and samples, the intrinsic scatter in [[$f_{\mathrm{gas}}$]{}]{} can be interpreted as a surrogate for scatter in non-thermal support, since other systematics affecting the [[$f_{\mathrm{gas}}$]{}]{} measurements should be roughly equivalent across the two samples. We use the gas mass fraction measured in a spherical shell at radii $0.8<r/r_{2500}<1.2$, as discussed in detail in [Paper II]{}, and compare the intrinsic scatter of [[$f_{\mathrm{gas}}$]{}]{} for the [SPA$_\mathrm{c}$]{} sample to that of [SPA$_\mathrm{c}$]{} plus the clusters which were included in but are classified as unrelaxed on morphological grounds in this work. Marginalizing over a complete model, including cosmological terms appropriate for non-flat [$\Lambda$CDM]{} models and various astrophysical and calibration nuisance parameters (see [Paper II]{}), yields intrinsic scatters of $7.4\pm2.3$ and $13.5\pm2.4$ per cent for these two samples. We conclude that adopting the more stringent selection criteria motivated by our morphological analysis results in a quantitatively more relaxed cluster sample. The smaller intrinsic scatter of the [SPA$_\mathrm{c}$]{} sample translates directly into tighter cosmological constraints on dark energy parameters ([Paper II]{}). Note that this check was performed a posteriori, and did not influence the construction of the [SPA$_\mathrm{c}$]{} sample itself. Caveats Regarding ROSAT Observations {#sec:rosatres} ------------------------------------ Image resolution potentially affects many stages of our morphology analysis. Low resolution generically results in flatter surface brightness peaks, rounder isophotes, and a diminished sensitivity to structure that would otherwise influence the global center and isophote centers. These limitations should be kept in mind when interpreting our results based on ROSAT PSPC data, although their effect should be negligible for the largest, most nearby clusters such as Perseus and Coma. For 17 clusters spanning redshifts $0.04<z<0.1$, we directly compared the SPA values obtained from ROSAT and [[Chandra]{}]{}. As expected, the peakiness values from ROSAT are lower, although only by $\sim0.04\pm0.03$ (mean and intrinsic scatter). Alignment and symmetry values are higher by $0.08\pm0.23$ and $0.11\pm0.18$, respectively. Somewhat surprisingly, there is no clear trend with redshift over the range probed (i.e. as a function of how well resolved the clusters are), although in the cases of alignment and symmetry a trend could easily be lost in the scatter. Among the 24 clusters for which we only use ROSAT data, only three are classified as relaxed: Abell133, Abell780 and Perseus. Each of these meets the SPA criteria with sufficient margin that the above scatter should not affect this determination. Comparison with Other X-ray Image-Based Samples {#sec:othersel} ----------------------------------------------- For reference, we show in [Figure]{} \[fig:morph\_other\] the morphological quantities from our analysis for clusters which have been selected by broadly similar criteria to ours, specifically subsets of the Cluster Lensing And Supernova survey with Hubble (CLASH; @Postman1106.3328), the Local Cluster Substructure Survey (LoCuSS; @Martino1406.6831), and the Canadian Cluster Comparison Project (CCCP; @Mahdavi1210.3689). Significantly, only in the case of CLASH is the cluster selection explicitly described as targeting relaxed systems (in this case, a majority are selected from ). The LoCuSS and CCCP clusters considered here are instead selected based on a single measurement, respectively the centroid variance and central entropy. While only $\sim50$ per cent of the clusters selected in these independent samples typically meet our SPA criterion, they clearly are close to relaxation (by our metric) compared with the cluster population as a whole, as one would expect. (The most obvious outlier in [Figure]{} \[fig:morph\_other\] is Abell 115, selected in CCCP due to the cool core in the northern sub-cluster.) ![image](peak_symm_comp.pdf) ![image](peak_alig_comp.pdf) Additional X-ray Morphological Statistics ----------------------------------------- In this section, we consider three additional morphological quantities which are potentially of interest, but which do not inform our criterion for relaxation. Each of these is a function of the elliptical isophote model fits described in [Section]{} \[sec:ellipses\], namely (1) their mean ellipticity, (2) the change of ellipticity with brightness, and (3) the change of position angle with brightness. The latter two cases we quantify with a “slope” obtained by regressing ellipticity or position angle against the index of the isophotes, which is effectively the logarithm of the surface brightness ([Equation]{} \[eq:sblevels\]). [Figure]{} \[fig:ellippa\] compares histograms of relaxed and unrelaxed clusters for these three quantities. ![image](ellipticity_mean_hist.pdf) ![image](ellipticity_slope_hist.pdf) ![image](pa_slope_hist.pdf)\ ![image](ellipticity_slope_spa2.pdf) \[fig:elslope\] While the lowest mean ellipticity clusters are relaxed, and the highest unrelaxed, the two distributions overlap considerably. In particular, the excess density of the relaxed distribution at the lowest ellipticities corresponds to only 3 clusters. At large ellipticities, the heavy tail seen in the unrelaxed cluster distribution consists of messy mergers rather than simple, prolate ellipsoids seen in the plane of the sky, and is thus not replicated in the relaxed sample. Discounting this tail, we thus see no evidence that the SPA selection of relaxed clusters is particularly biased towards lower than typical projected ellipticities, i.e. clusters likely to be elongated along the line of sight as opposed to in the plane of the sky. This is by construction, since our morphological estimators do not penalize clusters for having ellipsoidal rather than circular shapes in projection. For all clusters, the mean ellipticity is 0.22, with an intrinsic (Gaussian) scatter of 0.08. The distributions of ellipticity slope and position angle slope peak near zero for both relaxed and unrelaxed clusters, but are more sharply peaked for relaxed clusters. The difference is particularly evident for the ellipticity slope, which for unrelaxed clusters is asymmetric and has a heavy tail towards positive values (larger ellipticity at smaller radius/greater brightness). The ellipticity slope is plotted against each of the SPA measurements in [Figure]{} \[fig:elslope\], which shows that the clusters with the lowest alignment and symmetry also tend to have large absolute values of the ellipticity slope. This is intuitive, as all three indicators should be sensitive to the effects of ongoing merger activity on cluster emission. Trends with Redshift, Temperature and Parent Sample {#sec:trends} --------------------------------------------------- The fraction of clusters that are relaxed as a function of mass and redshift has important implications for cluster cosmology, in addition to astrophysical significance. In this section, we consider four subsets of the data set, defined according to how they were originally selected: from the X-ray flux-limited ROSAT All-Sky Survey (RASS),[^20] the 400d ROSAT survey, the SPT-SZ cluster survey, and the [[Planck]{}]{} Early SZ sample [@Planck1303.5089]. Here we remove from consideration the 400d detections at $z<0.35$, for which [[Chandra]{}]{} follow-up is neither extensive nor systematic. For the [[Planck]{}]{} sample, we consider only the 30 most significant SZ detections in terms of signal-to-noise, all of which were previously known in our source X-ray catalogs. The resulting sample is thus well represented in our data set, while nevertheless being SZ rather than X-ray selected. To good approximation, this [[Planck]{}]{} sample, and the [[Chandra]{}]{} follow-up of SPT clusters, can be considered fair selections of SZ signal-to-noise limited surveys, with the effective mass limit of the [[Planck]{}]{} sample being somewhat higher. The distribution of each of these samples in redshift and temperature is shown in [Figure]{} \[fig:morph\_samples\]. Note that in this section we use only clusters where our temperatures are based on spectral measurements, as opposed to being estimated using an X-ray luminosity–temperature relation. [ ![ The redshift–temperature distribution of four differently selected cluster populations within our data set: those detected in the X-ray flux-limited ROSAT All-Sky Survey (blue, open circles), the smaller 400 square degree ROSAT survey (black triangles), the SZ-selected SPT cluster survey (red, filled circles), and an SZ-selected subset of the [[Planck]{}]{} Early SZ catalog (green crosses). ](morph_samples_z_kT.pdf "fig:") ]{} \[fig:morph\_samples\] In principle, X-ray selected samples should be biased in favor of detecting strongly peaked clusters, due to the enhanced X-ray surface brightness that this implies, and we therefore expect the yield of relaxed clusters to be higher than in other samples. In contrast, SZ selection is not directly dependent on any of the X-ray surface brightness features we have measured. Merging could plausibly affect the SZ detectability of a cluster: in most cases we expect a decrease in the SZ signal for a given mass, since the ICM takes some time to reach its post-merger virial temperature, but the generation of a strong shock could significantly if briefly boost the SZ signal from a merging cluster. A variety of hydrodynamical simulations indicate that the net bias of SZ samples due to mergers should be relatively small [@Yang1010.0249; @Rasia1012.4027; @Battaglia1109.3709; @Krause1107.5740], although the dependence of these predictions on complex gas physics is such that they must be treated with caution. The uncertain effect of X-ray and SZ selection biases, as well as the relatively large statistical uncertainties, should be kept in mind throughout the following discussion. With that caveat in mind, [Figure]{} \[fig:relaxed\_z\_kT\] shows, for each cluster sample, the redshift and temperature dependence of three quantities: the fraction of relaxed clusters, the fraction of peaky clusters (satisfying our cut in peakiness, irrespective of symmetry or alignment), and the fraction of “undisturbed” clusters (satisfying cuts in symmetry and alignment, irrespective of peakiness). Horizontal bars in the figure show the bins in $z$ or $kT$, points the relaxed, peaky or undisturbed fraction in each bin, and vertical bars the corresponding 68.3 per cent confidence intervals.[^21] In choosing the bins, we have endeavored to make the results for different samples as straightforward to compare as possible, while still having a statistically useful number of clusters in each bin.[^22] ![image](fractions_z_kT.pdf) Due to selection effects, we expect the X-ray samples to contain a larger fraction of peaky clusters than SZ samples at any redshift or temperature. In fact, since there is also a correlation between peakiness and both symmetry and alignment, this preference should also hold for the undisturbed and relaxed fractions. For the RASS sample this is indeed the case; the relaxed, peaky and undisturbed fractions uniformly exceed those of SZ samples. They are, in addition, approximately constant as a function of both redshift and mass (with the possible exception of the peaky fraction as a function of $z$). Overall, the relaxed cluster fraction of RASS is 29 per cent. However, the situation is markedly different for the 400d sample, which in all respects appears more similar to the SZ samples (below) than to the RASS sample. In particular, the fraction of peaky clusters in the 400d sample is significantly smaller than in RASS, as has been remarked on previously [@Vikhlinin0611438; @Mantz2009PhDT........18M; @Santos1008.0754]. We find no relaxed clusters in the 400d sample. Note that, while the RASS and 400d samples are essentially disjoint in the X-ray luminosity–redshift plane (e.g. @Allen1103.4829), they do overlap in both redshift and temperature (a more reliable tracer of mass than luminosity; see the right panel of [Figure]{} \[fig:relaxed\_z\_kT\]). The level of disagreement between the two X-ray samples suggests two possible explanations: either the relaxed cluster fraction drops precipitously at relatively high redshifts and low masses, or the selection properties of the two samples are significantly different. For example, wavelet-based detection algorithms designed to automatically reject point-like sources, which the 400d sample employs, could plausibly be biased against finding peaky clusters near the flux limit [@Santos1008.0754]. Taking the SPT and [[Planck]{}]{} samples together, the relaxed cluster fraction in SZ samples is consistent with being constant with redshift; this behavior is similar to the RASS sample, but the SZ relaxed fraction is lower (8.5 per cent overall). The SZ relaxed fraction is consistent with RASS at high temperatures, $kT {\ {\raise-.75ex\hbox{$\buildrel>\over\sim$}}\ }10{{\ensuremath{\mathrm{\, keV}}}}$, but appears to decrease down to zero for cooler clusters, $kT {\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ }6{{\ensuremath{\mathrm{\, keV}}}}$. As a function of temperature, the peaky and undisturbed fractions behave similarly, increasing from ${\ {\raise-.75ex\hbox{$\buildrel<\over\sim$}}\ }0.1$ at low temperatures to values comparable to the RASS sample at ${\ {\raise-.75ex\hbox{$\buildrel>\over\sim$}}\ }10{{\ensuremath{\mathrm{\, keV}}}}$. In contrast, their trends with redshift differ; the peaky fraction is consistent with a constant, while the undisturbed fraction decreases with $z$. The latter is, however, largely an artifact of the observed $kT$ dependence combined with the differing redshift–temperature distributions of the [[Planck]{}]{} and SPT samples. Restricting the SPT sample to $kT>6{{\ensuremath{\mathrm{\, keV}}}}$ (i.e. to the range spanned by the [[Planck]{}]{} clusters) increases its undisturbed fraction to 26 per cent, reducing the evidence of a trend with redshift, while not significantly changing the picture for the peaky fraction. Both the absolute value of the SZ peaky fraction (14 per cent overall) and its constant behavior with redshift are consistent with the predictions of hydrodynamical simulations [@Burns0708.1954; @Planelles0906.4024]. However, the same simulations predict a decreasing cool-core fraction with cluster mass, which contradicts the increasing fraction of peaky clusters with temperature observed for the SZ sample. The increase in the undisturbed fraction with temperature, and its decrease with redshift (if real), are also seemingly in contradiction with simulations, which predict a mildly decreasing relaxed fraction (increasing fraction of merging clusters) as a function of mass and a constant merging fraction with redshift [@Planelles0906.4024; @Fakhouri1001.2304]. Note, however, that these simulations contain relatively few clusters in the mass range of our data set, and generally combine these into a single bin of masses ${\ {\raise-.75ex\hbox{$\buildrel>\over\sim$}}\ }10^{14}{\ensuremath{\, M_{\odot}}}$. Hence, the simulation results reflect trends with mass between cluster and group scales, not necessarily within the mass range probed by our data. A strong SZ selection bias favoring mergers, though contrary to expectations, could account for the lack of relaxed clusters at low temperatures in our SZ sample. However, the close agreement of the SZ and 400d results poses a problem for this explanation, since it would need the 400d X-ray selection to be similarly biased in favor of mergers. A simpler scenario is simply that the 400d selection is not biased towards finding strongly peaked clusters, as speculated above, and thus finds clusters morphologically similar to SZ searches. Note that, according to this picture, the lack of cool cores in the 400d sample compared to RASS is not due to its higher redshift coverage (as suggested by @Vikhlinin0611438), but rather its lower mass range in combination with different selection effects. Assuming that the temperature trends seen in the SZ sample are indeed real, they have potentially interesting implications for cool core formation and survival. Specifically, the increasing peaky fraction implies that cool core disruption is more efficient in less massive halos. There are several known examples of cool cores being destroyed by ram pressure stripping as they oscillate (slosh) about the bottom of the cluster potential following a merger (@Markevitch0001269 [@Mazzotta0102291; @Million0910.0025; @Ehlert1010.0253; @Ichinohe1410.1955], Canning et al., in prep.), a process also observed in hydrodynamic simulations (e.g. @Burns0708.1954 [@ZuHone1108.4427]). Hence a possible explanation is that mergers with the necessary mass ratio and impact parameter to destroy a hosted cool core via sloshing are relatively less common for the most massive clusters, despite these clusters having a larger merger rate overall; this would be qualitatively consistent with the larger undisturbed fraction we observe for the most massive clusters. Since cool core development is manifestly a non-self-similar phenomenon, it may also be the case that cool cores formed in more massive clusters are intrinsically more resilient to ram pressure stripping by the ambient ICM. Regardless of the reasons underlying the observed trends, we can make some broad statements about the best strategy for finding new relaxed clusters. Overall, the greatest yield of relaxed clusters can be obtained from an all-sky X-ray survey with greater sensitivity than RASS (such as eROSITA; @Predehl1001.2502), provided that the cluster detection algorithm does not reject peaky cool-core clusters. Assuming optical/IR follow-up observations exist, a first cut for selecting relaxed clusters can be made using the X-ray/BCG position offset in all cases. For a fraction of the discovered clusters, it should be possible to make additional, preliminary cuts from the X-ray survey data based on peakiness alone or, for the brightest systems, using the full suite of SPA measurements (adjusting appropriately for image resolution). However, the similarity of the RASS and SZ relaxed fractions at high temperatures strongly suggests that targeted X-ray snapshots of the most significant detections in SZ surveys would be an efficient complement for finding relaxed clusters, particularly at high redshifts where X-ray survey data suffer more from cosmological dimming. Summary {#sec:conclusion} ======= We have presented a new suite of image measurements used to assess the X-ray morphology of galaxy clusters. These estimators are designed to provide a fair basis for comparison over a wide range in redshift, to avoid strong assumptions regarding the background cosmology and cluster scaling relations, and to be as robust as possible against incomplete images (due to CCD gaps, point-source masks, etc.). The three statistics we use respectively probe the [peakiness]{} of the cluster surface brightness profile, the degree of [alignment]{} between isophotes at intermediate radii, and the [symmetry]{} of those isophotes with respect to a globally determined center. Uncertainties are propagated faithfully by bootstrap sampling the original images and varying the background normalization. These measurements were performed for a sample of 361 galaxy clusters, selected from several X-ray and SZ cluster surveys, using a combination of archival [[Chandra]{}]{} and ROSAT observations. There are clear correlations between the new measurements and more traditional X-ray estimators, indicating that they are sensitive to similar features, as expected. Intuitively, our peakiness measure also correlates clearly with the metric distance separating the X-ray center and the BCG. Motivated by trends in the data and comparison with the earlier relaxed cluster sample of , we define a requirement for a cluster to be considered morphologically relaxed in terms of the symmetry, peakiness and alignment measurements. The fraction of relaxed clusters identified this way is strongly dependent on the selection of the parent sample. We find a higher relaxed fraction in clusters selected from the RASS compared with SZ samples (respectively 0.29 and 0.085), as expected due to the strong dependence of X-ray detectability on surface brightness peakiness. Furthermore, the relaxed fraction in RASS is consistent with being constant with both redshift and ICM temperature, whereas an increasing trend with temperature is observed in the SZ-selected sample. The relaxed sample identified here, with some refinements based on cluster temperature and data quality, is used to derive cosmological constraints from cluster gas mass fractions in [Paper II]{}. As described in that work, significant improvements in dark energy constraints using this method will require the efficient identification and follow-up of relaxed clusters discovered in new cluster surveys. The algorithms introduced here provide a widely useful tool for identifying relaxed systems in new data, and for quantifying the morphological states of cluster samples in general. Acknowledgments {#acknowledgments .unnumbered} =============== ABM was supported by National Science Foundation grants AST-0838187 and AST-1140019. We acknowledge support from the U.S. Department of Energy under contract number DE-AC02-76SF00515, and from the National Aeronautics and Space Administration (NASA) through Chandra Award Numbers GO8-9118X and TM1-12010X, issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. [2]{} K. N. et al., 2009, [](http://adsabs.harvard.edu/abs/2009ApJS..182..543A) C. P. et al., 2014, [](http://adsabs.harvard.edu/abs/2014ApJS..211...17A) S., [Schmidt]{} R., [Fabian]{} A., 2002, [](http://adsabs.harvard.edu/abs/2002MNRAS.334L..11A) S. W., [Evrard]{} A. E., [Mantz]{} A. B., 2011, [](http://adsabs.harvard.edu/abs/2011ARA%26A..49..409A) S. W., [Rapetti]{} D. A., [Schmidt]{} R. W., [Ebeling]{} H., [Morris]{} R. G., [Fabian]{} A. C., 2008, [](http://adsabs.harvard.edu/abs/2008MNRAS.383..879A) S. W., [Schmidt]{} R. W., [Ebeling]{} H., [Fabian]{} A. C., [van Speybroeck]{} L., 2004, [](http://adsabs.harvard.edu/abs/2004MNRAS.353..457A) S. W., [Schmidt]{} R. W., [Fabian]{} A. C., 2001, [](http://adsabs.harvard.edu/abs/2001MNRAS.328L..37A) S. W. et al., 2001, [](http://adsabs.harvard.edu/abs/2001MNRAS.324..842A) K. A., [Fabian]{} A. C., [Eales]{} S. A., [Jones]{} C., [Forman]{} W., 1984, [](http://adsabs.harvard.edu/abs/1984MNRAS.211..981A) M., [Pointecouteau]{} E., [Pratt]{} G. W., 2007, [](http://adsabs.harvard.edu/abs/2007A%26A...474L..37A) Y., [Markevitch]{} M., 2006, [](http://adsabs.harvard.edu/abs/2006ApJ...650..102A) N., [Bond]{} J. R., [Pfrommer]{} C., [Sievers]{} J. L., 2012, [](http://adsabs.harvard.edu/abs/2012ApJ...758...74B) L. E. et al., 2015, [](http://adsabs.harvard.edu/abs/2015ApJS..216...27B) H. et al., 2010, [](http://adsabs.harvard.edu/abs/2010A%26A...514A..32B) H. et al., 2004, [](http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2004A%26A...425..367B&db_key=AST) D. A., 2001, [](http://adsabs.harvard.edu/abs/2001ApJ...553L..15B) D. A., [Tsai]{} J. C., 1995, [](http://adsabs.harvard.edu/abs/1995ApJ...452..522B) R. A., [Vikhlinin]{} A., [Hornstrup]{} A., [Ebeling]{} H., [Quintana]{} H., [Mescheryakov]{} A., 2007, [](http://adsabs.harvard.edu/abs/2007ApJS..172..561B) J. O., [Hallman]{} E. J., [Gantner]{} B., [Motl]{} P. M., [Norman]{} M. L., 2008, [](http://adsabs.harvard.edu/abs/2008ApJ...675.1125B) R. et al., 2013, [](http://adsabs.harvard.edu/abs/2013ApJ...777..141C) R., [Ettori]{} S., [Giacintucci]{} S., [Brunetti]{} G., [Markevitch]{} M., [Venturi]{} T., [Gitti]{} M., 2010, [](http://adsabs.harvard.edu/abs/2010ApJ...721L..82C) A., [Fusco-Femiano]{} R., 1976, [](http://adsabs.harvard.edu/abs/1976A%26A....49..137C) C. S., [Allen]{} S. W., [Ebeling]{} H., [Edge]{} A. C., [Fabian]{} A. C., 1999, [](http://adsabs.harvard.edu/abs/1999MNRAS.306..857C) J. H. et al., 2008, [](http://adsabs.harvard.edu/abs/2008A%26A...487..431C) H., [Barrett]{} E., [Donovan]{} D., [Ma]{} C.-J., [Edge]{} A. C., [van Speybroeck]{} L., 2007, [](http://adsabs.harvard.edu/abs/2007ApJ...661L..33E) H., [Edge]{} A. C., [Bohringer]{} H., [Allen]{} S. W., [Crawford]{} C. S., [Fabian]{} A. C., [Voges]{} W., [Huchra]{} J. P., 1998, [](http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1998MNRAS.301..881E&db_key=AST) H., [Edge]{} A. C., [Henry]{} J. P., 2001, [](http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2001ApJ...553..668E&db_key=AST) H., [Edge]{} A. C., [Mantz]{} A., [Barrett]{} E., [Henry]{} J. P., [Ma]{} C. J., [van Speybroeck]{} L., 2010, [](http://adsabs.harvard.edu/abs/2010MNRAS.407...83E) H., [Mullis]{} C. R., [Tully]{} R. B., 2002, [](http://adsabs.harvard.edu/abs/2002ApJ...580..774E) S. et al., 2011, [](http://adsabs.harvard.edu/abs/2011MNRAS.411.1641E) A. C., 2012, [](http://adsabs.harvard.edu/abs/2012ARA%26A..50..455F) A. C., [Crawford]{} C. S., [Edge]{} A. C., [Mushotzky]{} R. F., 1994, [](http://adsabs.harvard.edu/abs/1994MNRAS.267..779F) A. C., [Sanders]{} J. S., [Allen]{} S. W., [Crawford]{} C. S., [Iwasawa]{} K., [Johnstone]{} R. M., [Schmidt]{} R. W., [Taylor]{} G. B., 2003, [](http://adsabs.harvard.edu/abs/2003MNRAS.344L..43F) A. C. et al., 2000, [](http://adsabs.harvard.edu/abs/2000MNRAS.318L..65F) A. C., [Sanders]{} J. S., [Taylor]{} G. B., [Allen]{} S. W., [Crawford]{} C. S., [Johnstone]{} R. M., [Iwasawa]{} K., 2006, [](http://adsabs.harvard.edu/abs/2006MNRAS.366..417F) O., [Ma]{} C.-P., [Boylan-Kolchin]{} M., 2010, [](http://adsabs.harvard.edu/abs/2010MNRAS.406.2267F) L., [Giovannini]{} G., [Govoni]{} F., [Murgia]{} M., 2012, [](http://adsabs.harvard.edu/abs/2012A%26ARv..20...54F) W., [Bechtold]{} J., [Blair]{} W., [Giacconi]{} R., [van Speybroeck]{} L., [Jones]{} C., 1981, [](http://adsabs.harvard.edu/abs/1981ApJ...243L.133F) W. et al., 2007, [](http://adsabs.harvard.edu/abs/2007ApJ...665.1057F) W. et al., 2005, [](http://adsabs.harvard.edu/abs/2005ApJ...635..894F) J. W., [Gantner]{} B., [Burns]{} J. O., [Hallman]{} E. J., 2009, [](http://adsabs.harvard.edu/abs/2009ApJ...697.1597H) J., [Fabian]{} A. C., [Edge]{} A. C., [Ebeling]{} H., [Sanders]{} J. S., [Hogan]{} M. T., [Taylor]{} G. B., 2012, [](http://adsabs.harvard.edu/abs/2012MNRAS.421.1360H) Y., [Werner]{} N., [Simionescu]{} A., [Allen]{} S. W., [Canning]{} R. E. A., [Ehlert]{} S., [Mernier]{} F., [Takahashi]{} T., 2014, [](http://arxiv.org/abs/1410.1955) T. E., [Canizares]{} C. R., [Bautz]{} M. W., [Buote]{} D. A., 2005, [](http://adsabs.harvard.edu/abs/2005ApJ...624..606J) R. E., [Zuhone]{} J., [Jones]{} C., [Forman]{} W. R., [Markevitch]{} M., 2012, [](http://adsabs.harvard.edu/abs/2012ApJ...751...95J) N., 1986, [](http://adsabs.harvard.edu/abs/1986MNRAS.222..323K) D. D., [Ebeling]{} H., [Mullis]{} C. R., [Tully]{} R. B., 2007, [](http://adsabs.harvard.edu/abs/2007ApJ...662..224K) E., [Pierpaoli]{} E., [Dolag]{} K., [Borgani]{} S., 2012, [](http://adsabs.harvard.edu/abs/2012MNRAS.419.1766K) A., [Hoekstra]{} H., [Babul]{} A., [Bildfell]{} C., [Jeltema]{} T., [Henry]{} J. P., 2013, [](http://adsabs.harvard.edu/abs/2013ApJ...767..116M) A., 2009, Ph.D. thesis, Stanford University, [](http://adsabs.harvard.edu/abs/2009PhDT........18M) A., [Allen]{} S. W., [Ebeling]{} H., [Rapetti]{} D., [Drlica-Wagner]{} A., 2010b, [](http://adsabs.harvard.edu/abs/2010MNRAS.406.1773M) A., [Allen]{} S. W., [Rapetti]{} D., [Ebeling]{} H., 2010a, [](http://adsabs.harvard.edu/abs/2010MNRAS.406.1759M) A. B., [Allen]{} S. W., [Morris]{} R. G., [Rapetti]{} D. A., [Applegate]{} D. E., [Kelly]{} P. L., [von der Linden]{} A., [Schmidt]{} R. W., 2014, [](http://adsabs.harvard.edu/abs/2014MNRAS.440.2077M) M., [Gonzalez]{} A. H., [David]{} L., [Vikhlinin]{} A., [Murray]{} S., [Forman]{} W., [Jones]{} C., [Tucker]{} W., 2002, [](http://adsabs.harvard.edu/abs/2002ApJ...567L..27M) M., [Govoni]{} F., [Brunetti]{} G., [Jerius]{} D., 2005, [](http://adsabs.harvard.edu/abs/2005ApJ...627..733M) M. et al., 2000, [](http://adsabs.harvard.edu/abs/2000ApJ...541..542M) M., [Vikhlinin]{} A., 2007, [](http://adsabs.harvard.edu/abs/2007PhR...443....1M) R., [Mazzotta]{} P., [Bourdin]{} H., [Smith]{} G. P., [Bartalucci]{} I., [Marrone]{} D. P., [Finoguenov]{} A., [Okabe]{} N., 2014, [](http://adsabs.harvard.edu/abs/2014MNRAS.443.2342M) P., [Markevitch]{} M., [Vikhlinin]{} A., [Forman]{} W. R., [David]{} L. P., [van Speybroeck]{} L., 2001, [](http://adsabs.harvard.edu/abs/2001ApJ...555..205M) M. et al., 2012, [](http://adsabs.harvard.edu/abs/2012Natur.488..349M) M. et al., 2013, [](http://adsabs.harvard.edu/abs/2013ApJ...774...23M) B. R. et al., 2000, [](http://adsabs.harvard.edu/abs/2000ApJ...534L.135M) M. et al., 2014, [](http://adsabs.harvard.edu/abs/2014ApJ...797...34M) E. T., [Allen]{} S. W., [Werner]{} N., [Taylor]{} G. B., 2010, [](http://adsabs.harvard.edu/abs/2010MNRAS.405.1624M) J. J., [Fabricant]{} D. G., [Geller]{} M. J., 1993, [](http://adsabs.harvard.edu/abs/1993ApJ...413..492M) D., [McDonald]{} M., [Benson]{} B. A., [Miller]{} E. D., [Stubbs]{} C. W., [Vikhlinin]{} A., 2013, [](http://adsabs.harvard.edu/abs/2013ApJ...779..112N) R., [Blanton]{} E. L., [Randall]{} S. W., [Clarke]{} T. E., 2013, [](http://adsabs.harvard.edu/abs/2013ApJ...773..114P) C. B., [Fabian]{} A. C., [Edge]{} A. C., [Allen]{} S. W., [Johnstone]{} R. M., [White]{} D. A., 1998, [](http://adsabs.harvard.edu/abs/1998MNRAS.298..416P) J. R., [Fabian]{} A. C., 2006, [](http://adsabs.harvard.edu/abs/2006PhR...427....1P) , 2014, [](http://adsabs.harvard.edu/abs/2014A%26A...571A..29P) S., [Quilis]{} V., 2009, [](http://adsabs.harvard.edu/abs/2009MNRAS.399..410P) M. et al., 2012, [](http://adsabs.harvard.edu/abs/2012ApJS..199...25P) P. et al., 2010, [](http://adsabs.harvard.edu/abs/2010SPIE.7732E..23P) D., [Allen]{} S. W., [Mantz]{} A., 2008, [](http://adsabs.harvard.edu/abs/2008MNRAS.388.1265R) D., [Allen]{} S. W., [Weller]{} J., 2005, [](http://adsabs.harvard.edu/abs/2005MNRAS.360..555R) E., [Mazzotta]{} P., [Evrard]{} A., [Markevitch]{} M., [Dolag]{} K., [Meneghetti]{} M., 2011, [](http://adsabs.harvard.edu/abs/2011ApJ...729...45R) E., [Meneghetti]{} M., [Ettori]{} S., 2013, [](http://adsabs.harvard.edu/abs/2013AstRv...8a..40R) E., [Br[ü]{}ggen]{} M., [Simionescu]{} A., [B[ö]{}hringer]{} H., [Churazov]{} E., [Forman]{} W. R., 2011, [](http://adsabs.harvard.edu/abs/2011MNRAS.413.2057R) M., [Eckert]{} D., [Cavalleri]{} B. M., [Molendi]{} S., [Gastaldello]{} F., [Ghizzardi]{} S., 2011, [](http://adsabs.harvard.edu/abs/2011A%26A...532A.123R) J. S., [Rosati]{} P., [Tozzi]{} P., [B[ö]{}hringer]{} H., [Ettori]{} S., [Bignamini]{} A., 2008, [](http://adsabs.harvard.edu/abs/2008A%26A...483...35S) J. S., [Tozzi]{} P., [Rosati]{} P., [B[ö]{}hringer]{} H., 2010, [](http://adsabs.harvard.edu/abs/2010A%26A...521A..64S) R. W., [Allen]{} S. W., 2007, [](http://adsabs.harvard.edu/abs/2007MNRAS.379..209S) R. W., Allen S. W., [Fabian]{} A. C., 2004, [](http://adsabs.harvard.edu/abs/2004MNRAS.352.1413S) D. R. et al., 2012, [](http://adsabs.harvard.edu/abs/2012ApJ...761..183S) A., [Burke]{} D. J., [Aldcroft]{} T. L., [Worrall]{} D. M., [Allen]{} S., [Bechtold]{} J., [Clarke]{} T., [Cheung]{} C. C., 2010, [](http://adsabs.harvard.edu/abs/2010ApJ...722..102S) A. et al., 2011, [](http://adsabs.harvard.edu/abs/2011Sci...331.1576S) A. et al., 2012, [](http://adsabs.harvard.edu/abs/2012ApJ...757..182S) S., [Hallman]{} E., [Burns]{} J. O., [Skillman]{} S. W., [O’Shea]{} B. W., [Smith]{} B. D., 2013, [](http://adsabs.harvard.edu/abs/2013ApJ...763...38S) S. L., [McCammon]{} D., [Burrows]{} D. N., [Mendenhall]{} J. A., 1994, [](http://adsabs.harvard.edu/abs/1994ApJ...424..714S) J. et al., 2012, [](http://adsabs.harvard.edu/abs/2012ApJ...761...22S) O. et al., 2014, [](http://adsabs.harvard.edu/abs/2014MNRAS.437.3939U) O., [Werner]{} N., [Simionescu]{} A., [Allen]{} S. W., [B[ö]{}hringer]{} H., 2011, [](http://adsabs.harvard.edu/abs/2011MNRAS.414.2101U) A., [Burenin]{} R., [Forman]{} W. R., [Jones]{} C., [Hornstrup]{} A., [Murray]{} S. S., [Quintana]{} H., 2007, [](http://adsabs.harvard.edu/abs/2007hvcg.conf...48V) A. et al., 2009a, [](http://adsabs.harvard.edu/abs/2009ApJ...692.1033V) A., [Kravtsov]{} A., [Forman]{} W., [Jones]{} C., [Markevitch]{} M., [Murray]{} S. S., [Van Speybroeck]{} L., 2006, [](http://adsabs.harvard.edu/abs/2006ApJ...640..691V) A. et al., 2009b, [](http://adsabs.harvard.edu/abs/2009ApJ...692.1060V) A., [Markevitch]{} M., [Murray]{} S. S., 2001, [](http://adsabs.harvard.edu/abs/2001ApJ...551..160V) A., [Markevitch]{} M., [Murray]{} S. S., [Jones]{} C., [Forman]{} W., [Van Speybroeck]{} L., 2005, [](http://adsabs.harvard.edu/abs/2005ApJ...628..655V) von der [Linden]{} A. et al., 2014, [](http://adsabs.harvard.edu/abs/2014MNRAS.439....2V) S. A., [Fabian]{} A. C., [Sanders]{} J. S., [George]{} M. R., 2012a, [](http://adsabs.harvard.edu/abs/2012MNRAS.424.1826W) S. A., [Fabian]{} A. C., [Sanders]{} J. S., [George]{} M. R., [Tawara]{} Y., 2012b, [](http://adsabs.harvard.edu/abs/2012MNRAS.422.3503W) D. A., [Jones]{} C., [Forman]{} W., 1997, [](http://adsabs.harvard.edu/abs/1997MNRAS.292..419W) H.-Y. K., [Bhattacharya]{} S., [Ricker]{} P. M., 2010, [](http://adsabs.harvard.edu/abs/2010ApJ...725.1124Y) J. A., [Markevitch]{} M., [Lee]{} D., 2011, [](http://adsabs.harvard.edu/abs/2011ApJ...743...16Z) X-ray Data ========== [Table]{}s \[tab:data\] and \[tab:rosatdata\] provides details of the [[Chandra]{}]{} and ROSAT observations employed here. Complete Results ================ [Table]{} \[tab:allresults\] extends the listing of results in [Table]{} \[tab:morph\] to the entire data set. [^1]: <http://www.slac.stanford.edu/~amantz/work/morph14/> [^2]: Corresponding author e-mail: <amantz@kicp.uchicago.edu> [^3]: Cool cores are generally thought to correlate strongly with relaxation, although there exist notable examples of merging clusters containing remnant cool cores of gas, for example Abell115 [@Forman1981ApJ...243L.133F], CygnusA [@Arnaud1984MNRAS.211..981A], and the Bullet Cluster, 1E0657$-$56 [@Markevitch0110468]. [^4]: <http://cxc.harvard.edu/ciao/> [^5]: <http://cxc.harvard.edu/caldb/> [^6]: <http://cxc.harvard.edu/ciao/guides/acis_data.html> [^7]: <http://cxc.cfa.harvard.edu/ciao/threads/acisbackground/> [^8]: Approximate temperatures are sufficient, since the product $K(z,T,{{\ensuremath{N_{\mathrm{H}}}}})\,kT {\ {\raise-.75ex\hbox{$\buildrel\propto\over\sim$}}\ }T^{1/2}$ (for temperatures characteristic of the clusters in our data set, for which there is negligible line emission at soft energies) varies slowly with $kT$. [^9]: Note that the conversion of angular to metric distance introduces an additional cosmology-dependent factor of ${{\ensuremath{d_{\mathrm{A}}}}}(z)$ in the radial coordinates. [^10]: More specifically, $N+1$ is the variance when the prior is chosen to be a flat Gamma distribution such that the posterior is maximized at $N$ (shape parameter $k=1$ and rate parameter $\beta=0$). [^11]: The particular choice of units here is purely for convenience, as it makes the intensity of a typical cluster center of order unity, and simplifies the case of $2\times2$ binned [[Chandra]{}]{} images. [^12]: Blindly fitting the outermost half of the profiles works well in general for [[Chandra]{}]{} data, where the blank-sky background subtraction is typically approximately correct. Given the adaptive binning of the profile, the outer half tends to span the power-law tail of the cluster and any residual foreground/background, e.g. Galactic contamination which is not included in the blank-sky maps. In rare cases where the residual constant term has very high signal-to-noise (so that the outer half of the adaptively binned profile excludes too much of the cluster), we manually choose the radial range of the profile to use in this step. [^13]: We have not attempted to fine-tune the redshift dependence further, since the motivation for doing so is questionable and since it would provide an opportunity to tailor the high-redshift content of our final relaxed sample (potentially biasing the cosmological results of [Paper II]{}). However, a posteriori, it is interesting to note that the $1+z$ weighting results in an approximately constant fraction of peaky clusters with redshift ([Section]{} \[sec:trends\]), seemingly in good agreement with the constant cool-core fraction predicted from simulations [@Burns0708.1954]. This is encouraging, as it suggests we are selecting dynamically similar clusters at each redshift. [^14]: Note that [[$\Gamma_{\mathrm{el}}$]{}]{} is similar to the displacement between the ellipse center and the isophote centroid, under the assumption that all pixels in the isophote have exactly the same brightness. [^15]: Consequently, we can also conclude that peakiness correlates with other cool-core indicators, such as central cooling time, which have been observed to correlate with [[$c_{\mathrm{SB}}$]{}]{} [@Santos0802.1445]. [^16]: Querying both databases is advantageous since bright galaxies are masked in the DR10 catalog processing. [^17]: The relaxed cluster with the largest BCG/X-ray offset (24kpc) is MACSJ1311.0$-$0311. This cluster fails the additional cuts required for inclusion in our cosmology sample, although for reasons of data quality rather than morphology ([Paper II]{}). The other relaxed clusters all have BCG/X-ray offsets $<14$kpc. [^18]: A posteriori, these cuts appear well matched to thresholds in surface brightness concentration and centroid variance, respectively used by @Santos0802.1445 and @Bohringer0912.4667, as noted in [Section]{} \[sec:otherx\]. [^19]: There is a straightforward degeneracy between the location of the cuts themselves and the fraction of passing bootstrap samples required for to be classified as relaxed. While essentially the same selection could be obtained with an ostensibly stricter threshold (given slightly shifted cuts), the 50 per cent threshold is convenient because it makes plots of the bootstrap mean for each cluster simpler to interpret (e.g. [Figure]{}s \[fig:morph\_cuts\]–\[fig:morph\_A08\]). Note, however, that this 50 per cent criterion is not identical to only requiring the bootstrap mean to satisfy all three cuts, even assuming a symmetric bootstrap distribution. [^20]: Strictly speaking, the BCS, REFLEX, CIZA and MACS samples, which we collectively call RASS here, were also constructed using different methods to detect cluster emission. However, particularly given the exhaustive optical follow-up and confirmation employed for the RASS samples, these differences are relatively minor. [^21]: We adopt a uniform prior between 0 and 1 on the fraction of relaxed (or peaky or undisturbed) clusters in a given redshift or temperature bin. With this choice, for a bin where $x$ clusters are found to be relaxed and $y$ unrelaxed, the posterior for the relaxed fraction is the Beta distribution with shape parameters $x+1$ and $y+1$. [^22]: In practice, we aimed to have $\geq10$ clusters in each bin. Matching the approximate redshift and temperature binning across samples sometimes resulted in there being significantly more, $\sim70$ in the case of the most populated bin. The exception is the highest-$kT$ bin for the 400d sample, which contains only 1 cluster.
--- author: - \| Yafeng Zhang and Donatello Telesca\ Department of Biostatistics, University of California Los Angeles,\ Los Angeles, California, U.S.A. bibliography: - 'jointModel.bib' title: \| Joint Clustering and Registration\ of Functional Data --- Abstract Curve registration and clustering are fundamental tools in the analysis of functional data. While several methods have been developed and explored for either task individually, limited work has been done to infer functional clusters and register curves simultaneously. We propose a hierarchical model for joint curve clustering and registration. Our proposal combines a Dirichlet process mixture model for clustering of common shapes, with a reproducing kernel representation of phase variability for registration. We show how inference can be carried out applying standard posterior simulation algorithms and compare our method to several alternatives in both engineered data and a benchmark analysis of the Berkeley growth data. We conclude our investigation with an application to time course gene expression. .3in [Keywords:]{} Curve registration; Dirichlet process, Functional data clustering; Time course microarray data. Introduction ============ Functional data is often characterized by both shape and phase variability. A typical example where these two sources of variation are clearly identified and interpreted are data arising from the study of human growth. Panel (a) and (b) of Figure \[growthVelocity\] shows growth velocity curves of 39 boys and 54 girls from the Berkeley Growth Study [@Tudden:Sny:1954]. An overall pattern is observed that growth velocity decelerates to zero from infancy to adulthood, with some subtle acceleration-deceleration pulses during late childhood and a prominent pubertal growth spurt. In this setting, phase variability is identified as variation in the timing of subject-specific growth. Explicit consideration of phase variability is necessary in order to obtain consistent estimation of typical growth patterns. The formal analytical treatment of this problem has a long history in Statistics and Engineering. Initial contributions focused on curve alignment (registration) via dynamic time warping [@Sakoe:Chiba:1978; @Wang:Gass:alig:1997; @Wang:Gass:sync:1999] or landmark registration [@Gass:Knei:sear:1995]. Model-based alternatives represent subject-specific profiles as a parametric transformation of a common smooth regression function, evaluated over random functionals of time [@Lawt:Sylv:Magg:self:1972; @Knei:Gass:conv:1988]. Several of these methods involve a transformation of both the $x$ and $y$ axes, essentially defining the mean profile for curve $i$ as $f_i(x)=b_i + a_i m\big(\mu_i(x)\big) $, where $\mu_i(x)$ is a monotone transformation function accounting for phase variability. In longitudinal settings, @Brum:Lind:self:2004 introduced a mixed effect formulation of these models, formally accounting for dependence within subject. Similary, @Telesca:Inoue:2007 proposed a Bayesian hierarchical curve registration (BHCR) model allowing for posterior inference on both the shape function $m(\cdot)$ and transformation functions $\mu_i(x)$. Whereas these considerations are valid for any function argument $x$, it is most natural to think of $x$ as a time scale. In the following, we will therefore focus on the case of functional data observed over time. Besides technical differences, these models of curve registration share a fundamental assumption, implying that all observed functional profiles are generated through semi-parametric transformations of a common shape $m(\cdot)$. While this assumption is likely to be warranted in standard applications, the increasing popularity of these methods for the analysis of more general data classes [@TelescaEtal2009; @TelescaErosheva:2012] motivates a methodological extension, conceiving the possible existence of shape-invariant subgroups, with group shapes $m_1(\cdot),\ldots,m_k(\cdot)$. We are not the first to recognize a need for combing clustering and registration. Stepwise procedures, where first curves are registered and then clustered according to a chosen heuristic, have been explored in several applications [@LiuMuller2004; @TangMuller2009; @SlaetsEtal2012]. Joint clustering and registration procedures have been discussed by @GaffneySmyth2005 and @LiuYang2009. While stepwise procedures are likely to provide suboptimal estimation, available joint clustering and registration techniques have only been developed under the assumption of linear shape invariant time transformations, where $\mu_i(t) = \alpha_it + \beta_i$. Furthermore, model complexity, conceived as the number of clusters, is only treated as a nuisance parameter and fixed in a post-hock fashion via BIC or pBIC [@ChouReichl1999]. .1in We extend the BHCR model of @Telesca:Inoue:2007 to allow for shape-subgroups. Our proposal is based on a reproducing kernel (B-spline) representation of both shape and time transformation functions. To relax the homomorphic assumption, we define a non-parametric prior over shape functionals via a Dirichlet process (DP) mixture [@Ferguson1973; @Antoniak1974; @Quintana2006]. Clustering is achieved implicitly and is interpreted in terms of shape similarities. The number of clusters is subject to direct estimation and inferences account fully for this layer of uncertainty, without the need for post-hock adjustments. Furthermore, we show how posterior simulation remains straightforward via a simple extension to standard Metropolis within Gibbs MCMC transitions. Following @LiuMuller2004, we show how this modeling approach is particularly useful for the analysis of time-course gene expression. While it is known that co-expressed genes are likely to be co-regulated, various regulation mechanisms, such as feedback loops and regulation cascades, may warp the timing of expression for genes involved in the same process or regulatory pathway [@Weber2007]. It is therefore desirable to have a model that can assign genes with similar, yet time-warped, expression profiles to the same cluster [@QianEtal2001; @Qin2006]. In other words, it is important to have a model that is phase-variation tolerant when defining curve subgroups. The remainder of this article is organized as follows. In section \[modelFormulation\], we describe the the sampling model and priors. A posterior simulation strategy via MCMC is described in Section \[Posterior\]. In section \[simulation\], we apply the joint model to simulated datasets and compare it with single-purpose models: a clustering only model and a registration only model. In section \[growthData\] we apply the model to the Berkeley Growth Study data. In section \[fibroResponse\], we apply the model to time course microarray data of response of human fibroblasts to serum stimulation. Finally in section \[discussion\], we conclude the paper with a critical discussion. Model Formulation {#modelFormulation} ================= Sampling Model {#modelDescription} -------------- Let $y_i(t)$ denote the observation of curve $i$ at time $t$, where $i=1, \ldots, N$ and $t \in T = [t_1, t_n]$. The sampling model is specified as follows: $$\label{modelStage1} y_i(t)=c_i+a_i m_i\{\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i), {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i\} + \epsilon_{i}(t),$$ where $\epsilon_{i}(t)\sim \mathrm{N}(0, 1/\tau_i)$ and $\tau_i$ is the precision parameter. In formula (\[modelStage1\]), $\mu_i()$ is the curve specific time transformation function, characterizing the latent time scale of curve $i$, and $m_i()$ is the curve specific shape function. The apparent lack of identifiability between $\mu_i()$ and $m_i()$ will be resolved in §\[Priors\] by specifying a random probability functional prior for $m_i()$, implicitly producing functional clusters. To achieve flexible modeling of both time transformation and shape functions, we use B-splines [@deBoor:1978]. We model $\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)={\mbox{\boldmath $\!B\!$ \unboldmath}}^T_{\mu}(t){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$, where ${\mbox{\boldmath $\!B\!$ \unboldmath}}_{\mu}(t)$ is the B-spline basis vector at time $t$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ is the curve specific basis coefficient vector. $\mu_i()$ is a monotone function mapping the sampling time interval $T$ to the interval $\mathcal{T}=[t_1-\Delta, t_n+\Delta]$, with expansion constraint $\Delta \geq 0$ to allow the time scale to be transformed outside the observed sampling time interval $T$. To ensure monotonicity and function image boundaries, we impose the following constraints $$\label{phiConstraint} (t_1-\Delta)\leq \phi_{i1} < \ldots < \phi_{iq} < \phi_{i(q+1)} < \ldots < \phi_{iQ} \leq (t_n+\Delta).$$ We model shape functions as $m_i\{\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i), {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i\}={\mbox{\boldmath $\!B\!$ \unboldmath}}^T_m\{\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)\} {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ where ${\mbox{\boldmath $\!B\!$ \unboldmath}}_m(\cdot)$ is a B-spline basis vector and ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ is the curve specific basis coefficient vector. No constraints are usually imposed on ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$, unless specific shapes are preferred a-priori (see for example, [@TelescaErosheva:2012]). We note that the stochastic functionals $m_i()$ and $\mu_i()$ are now fully described by the distributions of ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ respectively. Identifiability of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ is ensured by modeling ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ as a Dirichlet process mixture. In this setting, realizations of ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ are discrete with probability one, with $K<N$ unique component vectors ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^$, ($k=1,\ldots,K$). These component vectors, in turn, define cluster specific shape functions $m_k^()$, to which member curves are aligned through $\mu_i(t)^{-1}$. Details are discussed in the following section. Prior Model {#Priors} ----------- We assume that shape function parameters ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and precisions $\tau_i$ to follow a Dirichlet process mixture prior. Let $G_0()$ be a base distribution absolutely continuous with respect to the Lebesque measure on $\mathbb{R}^p\cup \mathbb{R}^+$ and $\delta({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$ a Dirac mass at $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$. Using a predictive Pòlya urn scheme [@BlackwellMacQueen1973], we specify the prior distribution as follows: $$\label{DPprior} {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i\|{\mbox{\boldmath $\!\theta\!$ \unboldmath}}_{-i}, \tau_{-i} \sim \frac{\alpha}{(\alpha+N-1)} G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)+\frac{1}{(\alpha+N-1)}\sum_{j\neq i}\delta({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j),$$ where $-i=\{j: j\neq i\}$ is the set all the indices other than $i$ and $\alpha$ is the weight parameter of the Dirichlet process model. This prior generates the shape ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and error precision $\tau_i$ for curve $i$, from a mixture involving a random draw from the base density $G_0()$ or the point mass $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$’s, $j \neq i$. Realizations from the prior in (\[DPprior\]) define a discrete distribution, implying ties among $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$’s, $i=1,\ldots,N$. These ties are naturally interpreted as clusters among the $N$ curves, namely, curve $i$ and $j$ belong to the same cluster if $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_j, \tau_j)$. As a result, only $K < N$ unique values are observed, each of which is associated with a cluster and is denoted by $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^, \tau_k^)$, $k=1, \ldots, K$. In this setting, we can re-express formula (\[DPprior\]) as: $$\label{DPprior2} {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i\|{\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^, \tau_k^{} \sim \frac{\alpha}{(\alpha+N-1)} G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)+\frac{1}{(\alpha+N-1)}\sum_{k=1}^{K_{-i}}n_{k(-i)}\delta({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^, \tau_k^{}),$$ where $n_{k(-i)}$ is the size of cluster $k$ and $K_{-i}$ is number of clusters when curve $i$ is excluded. The representation above implies that a complete sample of $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$, $(i=1,\ldots,N)$ is in one to one correspondence with a set of unique values, $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^, \tau_k^)$, $(k=1,\ldots,K)$, through cluster labels ${\mbox{\boldmath $\!s\!$ \unboldmath}}=(s_1, \ldots, s_N)$. Specifically, $s_i=k$ if $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^, \tau_k^{})$ and $s_i=K_{-i}+1$ if $({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$ is a new sample from $G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)$, which means curve $i$ forms a new cluster of its own. As a result, the number of clusters $K$ is also determined implicitly. We note that if we omit the time transformation modeling with $\mu_i(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)$ and use time $t$ directly in the shape functions $m_i(t, {\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i)$, our model reduces to standard functional clustering via Dirichlet process mixtures. .2in We assume that the base DP mixture density factors as $G_0({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i, \tau_i)=p({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i \mid \tau_i)p(\tau_i)$, where ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i \mid \tau_i \sim \mathrm{N}\left({\mbox{\boldmath $\!0\!$ \unboldmath}}, (\tau_{\theta}\tau_i{\mbox{\boldmath $\!\Sigma\!$ \unboldmath}})^{-1}\right)$ and $\tau_i \sim \mathrm{Ga}(a, b)$, a Gamma distribution with mean $a/b$. The specific form of the precision matrix $\Sigma$ is determined by a second-order shrinkage process: $\theta_{ip}-\theta_{i(p-1)}=$ $\theta_{i(p-1)}-\theta_{i(p-2)}+\xi_{ip}$ with $\xi_{ip} \sim \mathrm{N}\big(0, 1/(\tau_{\theta}\tau_i)\big)$ ($p=1,\ldots,P$) where $P$ is the dimension of ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i$ and $\theta_{i0}=\theta_{i(-1)}=0$ [@Lang:Brez:baye:2004]. In this setting, the product $\tau_{\theta}\tau_i$ can be interpreted as a smoothing parameter for curve $i$. Similarly, we also use a penalized B-spline prior on the time transformation function parameters ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$. In particular, letting ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_0$ be the vector associated with identity transformation so that $\mu(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_0)=t$, we assume $\phi_{iq}-\phi_{0q}=$ $\phi_{i(q-1)}-\phi_{0(q-1)}+\nu_{iq}$ with $\nu_{iq} \sim \mathrm{N}(0, 1/\tau_{\phi})$ ($q=1,\ldots, Q$) where $Q$ is dimension of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ and $\phi_{i0}=0$, implying ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i \sim \mathrm{N}\left({\mbox{\boldmath $\!\phi\!$ \unboldmath}}_0, (\tau_{\phi}\Omega)^{-1}\right)$. In the foregoing, $\Omega$ is deterministic and $\tau_{\phi}$ is interpreted as a smoothing parameter. Following @TelescaEtal2009, when modeling cluster specific common shape functions, we let the number of spline knots equal to the number of sampling time points. For the curve specific time transformation functions structural smoothness is imposed by their monotonicity (\[phiConstraint\]), suggesting parsimony in the choice of the number of knots. In many application contexts, $1$ to $4$ equally spaced interior knots allow for enough flexibility in the representation of time transformation. .2in For ease of computation, we complete our model with priors and hyperpriors following principles of conditional conjugacy. Specifically, curve specific mean levels parameters are specified as $c_i \sim \mathrm{N}(c_0, 1/\tau_c)$ and curve specific amplitude parameters $a_i \sim \mathrm{N}(a_0, 1/\tau_a)I(a_i>0)$. The assumption of strictly positive amplitudes is appropriate if synchronous but negatively correlated curves are to be clustered separately. Removing positivity restrictions will imply clustering of synchronous profiles. We complete our prior specifications assuming $c_0 \sim \mathrm{N}(0, 1/\tau_{c_0})$, $a_0 \sim \mathrm{N}(1, 1/\tau_{a_0})$, $\tau_a \sim \mathrm{Ga}(a_a, b_a)$ and $\tau_c \sim \mathrm{Ga}(a_c, b_c)$. Smoothing parameters priors are specified as $\tau_{\theta} \sim \mathrm{Ga}(a_{\theta}, b_{\theta})$ and $\tau_{\phi} \sim \mathrm{Ga}(a_{\phi}, b_{\phi})$. Finally, the weight parameter of the Dirichlet process mixture is defined as $\alpha \sim \mathrm{Ga}(a_{\alpha}, b_{\alpha})$. Posterior Inference {#Posterior} =================== Posterior Simulation {#MCMC} -------------------- Markov Chain Monte Carlo simulation from the posterior distribution is conceptually straightforward and obtained as a simple sequence of Metropolis-Hastings within Gibbs transitions. For ease of notation, we let ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i^t, \tau_i)^t$ and ${\mbox{\boldmath $\!y\!$ \unboldmath}}_i=(y_i(t_1), \ldots, y_i(t_n))^t$. Without loss of generality, we also assume that curves are of the same length $n$. The proposed Markov transition sequence is implemented by: (i) sampling $({\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!s\!$ \unboldmath}}_i)$ given all other parameters, (ii) resampling ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}^_k=({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^{t}, \tau_k^{})^t$ given cluster indicators ${\mbox{\boldmath $\!s\!$ \unboldmath}}_i$ and all other parameters and (iii) sampling $\alpha$ and remaining parameters from their full conditional posteriors. We outline details as follows. .1in (i) [Sampling $({\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i,\,{\mbox{\boldmath $\!s\!$ \unboldmath}}_i)$.]{} The full conditional posterior of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ is a Dirichlet process mixture with updated mixing probabilities and components [@Escobar1994; @WestEtal1994]: $$\label{eta} {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, {\mbox{\boldmath $\!y\!$ \unboldmath}}_i \sim \frac{q_{i0}G_i({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, \,{\mbox{\boldmath $\!y\!$ \unboldmath}}_i) +\sum\limits_{k=1}^{K_{-i}}q_{ik}\delta({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^)} {q_{i0}+\sum\limits_{k=1}^{K_{-i}}q_{ik}},$$ where $q_{i0}=\alpha\int f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\phi_i\!$ \unboldmath}}, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)\mathrm{d}G_0({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)$ is $\alpha$ times the marginal likelihood of ${\mbox{\boldmath $\!y\!$ \unboldmath}}_i$, $G_i({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, \,{\mbox{\boldmath $\!y\!$ \unboldmath}}_i) \propto f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\phi_i\!$ \unboldmath}}, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)G_0({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i) $ is the full conditional density of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ and $q_{ik}=n_{k(-i)}f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^)$ is the product of cluster size $n_{k(-i)}$ and the likelihood associated with ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i={\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^$. To improve mixing rates, we combine the sampling of ${\mbox{\boldmath $\!s\!$ \unboldmath}}_i$, ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$. Specifically, we use $K_{-i}$ copies of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$, ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^1, \ldots, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{K_{-i}}$, one for each cluster. We update ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^k$, assuming ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i={\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^$, with the Metropolis-Hastings algorithm as in @Telesca:Inoue:2007, so that the appropriate time transformation is found for curve $i$ to be registered with the common shape function of cluster $k$. We calculate each $q_{ik}$ $(k =1, \ldots, K_{-i})$ in (\[eta\]) using the corresponding ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^$ and ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^k$. When calculating $q_{i0}$, we use the value of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ from the previous iteration of the Gibbs sampler. Specifically, let ${\mbox{\boldmath $\!B\!$ \unboldmath}}_i={\mbox{\boldmath $\!B\!$ \unboldmath}}_m\{u_i({\mbox{\boldmath $\!t\!$ \unboldmath}}, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i)\} = {\mbox{\boldmath $\!B\!$ \unboldmath}}_m\{{\mbox{\boldmath $\!B\!$ \unboldmath}}_{\mu}^T({\mbox{\boldmath $\!t\!$ \unboldmath}}){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i\}$ and ${\mbox{\boldmath $\!c\!$ \unboldmath}}_i=c_i{\mbox{\boldmath $\!1\!$ \unboldmath}}$, and define the following summaries: ${\mbox{\boldmath $\!E\!$ \unboldmath}}_i=a_i^2{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T{\mbox{\boldmath $\!B\!$ \unboldmath}}_i+\tau_{\theta}{\mbox{\boldmath $\!\Sigma\!$ \unboldmath}}$, ${\mbox{\boldmath $\!\mu\!$ \unboldmath}}_i=a_i{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)$, $a'_i=\frac{n}{2}+a$ and $b'_i=\frac{1}{2}({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)^T ({\mbox{\boldmath $\!I\!$ \unboldmath}}-a_i^2{\mbox{\boldmath $\!B\!$ \unboldmath}}_i{\mbox{\boldmath $\!E\!$ \unboldmath}}_i^{-1}{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T)({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i) +b$. To sample ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ from its full conditional (\[eta\]), we follow the procedure below: 1. Sample cluster membership $s_i$ which takes values on $K_{-i}+1, 1, \ldots, K_{-i}$ with probabilities proportional to $q_{i0}, q_{i1}, \ldots, q_{iK_{-i}}$. 2. If $s_i=K_{-i}+1$, we keep ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ unchanged. Curve $i$ forms a new cluster, and a draw of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i$ from $G_i({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i \mid {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i,{\mbox{\boldmath $\!y\!$ \unboldmath}}_i)$ is obtained by first sampling $\tau_i \sim \mathrm{Ga}(a'_i, b'_i)$ and then sampling ${\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i \mid \tau_i \sim \mathrm{N}({\mbox{\boldmath $\!E\!$ \unboldmath}}_i^{-1}{\mbox{\boldmath $\!\mu\!$ \unboldmath}}_i ,(\tau_i{\mbox{\boldmath $\!E\!$ \unboldmath}}_i)^{-1})$. If $s_i=k$, we use the corresponding ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^k$ as a draw of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i$ and let ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i={\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^$. .2in (ii) [Resampling ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}^_k$ given cluster indicators ${\mbox{\boldmath $\!s\!$ \unboldmath}}_i$.]{} After a sample of ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}^T=({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_1^T, \ldots, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_N^T)$ and ${\mbox{\boldmath $\!s\!$ \unboldmath}}=(s_1, \ldots, s_N)^T$ is generated, to improve mixing rates, we update each ${\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^$ from its full conditional $G_k({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^\mid {\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k}) \propto \prod_{i\in S_k}\!f({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\mid {\mbox{\boldmath $\!\phi_i\!$ \unboldmath}}, {\mbox{\boldmath $\!\eta\!$ \unboldmath}}_i)G_0({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^)$, where $S_k=\{i:s_i=k\}$ is the set of curves in cluster $k$ [@MacEachernMuller1998]. Furthermore, $G_k({\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^\mid {\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k}) \propto p({\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^\mid\tau_k^,{\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k})p(\tau_k^\mid{\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k})$, s.t. $$\label{etak} \begin{split} &{\mbox{\boldmath $\!\eta\!$ \unboldmath}}_k^\mid {\mbox{\boldmath $\!y\!$ \unboldmath}}_{i\in S_k} \sim \mathrm{N}({\mbox{\boldmath $\!E\!$ \unboldmath}}_k^{-1}{\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k, (\tau_k^{\mbox{\boldmath $\!E\!$ \unboldmath}}_k)^{-1})\\ &\times \mathrm{Ga}\left(\frac{1}{2}\sum_{i\in S_k}\!\!n_i+a, \frac{1}{2}\left(\sum_{i\in S_k}({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)^T({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i) - {\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k^T{\mbox{\boldmath $\!E\!$ \unboldmath}}_k^{-1}{\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k\right)+b\right), \end{split}$$ where ${\mbox{\boldmath $\!E\!$ \unboldmath}}_k=\sum_{i\in S_k} a_i^2{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T{\mbox{\boldmath $\!B\!$ \unboldmath}}_i + \tau_{\theta}{\mbox{\boldmath $\!\Sigma\!$ \unboldmath}}$ and ${\mbox{\boldmath $\!\mu\!$ \unboldmath}}_k=\sum_{i\in S_k}a_i{\mbox{\boldmath $\!B\!$ \unboldmath}}_i^T({\mbox{\boldmath $\!y\!$ \unboldmath}}_i-{\mbox{\boldmath $\!c\!$ \unboldmath}}_i)$. .2in (iii) [Sampling $\alpha$ and all hyper parameters.]{} To develop the full conditional of $\alpha$, we note that $p(K\|\alpha,N)\propto N!\alpha^K\frac{\Gamma(\alpha)}{\Gamma(\alpha+N)}$ [@Antoniak1974]. Following [@West1992], we define an auxiliary random quantity $x\mid \alpha \sim \mathrm{B}(\alpha+1, N)$ and a mixing probability $\pi_x$: $$\frac{\pi_x}{1-\pi_x}=\frac{a_{\alpha}+K-1}{N(b_{\alpha}-log(x))}.$$ Conditioning on $x$, it is easily shown that the full conditional distribution of $\alpha$ is a mixture of gamma densities. Specifically, $$\label{alpha3} \begin{split} &\alpha\mid x, K \sim\\ &\pi_x \mathrm{Ga}(a_{\alpha}+K, b_{\alpha}-log(x)) + (1-\pi_x)\mathrm{Ga}(a_{\alpha}+K-1, b_{\alpha}-log(x)) \end{split}$$ The rest of the model parameters are simulated directly from their full conditional posterior distributions. Detailed results are reported in Web Appendix A. Posterior Inference {#inference} ------------------- We base our inference on MCMC samples from the posterior distribution of the model parameters. Inference for functional quantities is obtained by post-processing these finite-dimensional posterior samples. To get a point estimate of the clustering structure we use the maximum a-posterior (MAP) clustering. Given $M$ posterior samples of ${\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{(j)}$, $(j=1, \ldots, M)$, posterior samples of the time transformation function $\mu_i(t)$ at any time point $t \in T$ can be calculated as: $$\label{muPosterior} \mu_i^{(j)}(t)=\mu_i^{(j)}(t, {\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{(j)})={\mbox{\boldmath $\!B\!$ \unboldmath}}_{\mu}^T(t){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i^{(j)}.$$ Here, the posterior mean function $\hat{\mu}_i(t)=\frac{1}{M}\sum_{j=1}^M\mu_i^{(j)}(t)$ provides an point estimate of $\mu_i(t)$, and curves are registered on the transformed time scales $\hat{\mu}_i(t)$ within each cluster. Similar estimators are defined for cluster-specific shape functions: $$\label{clusterShapeFunction} m_k(t)=c_0+a_0{\mbox{\boldmath $\!B\!$ \unboldmath}}^T_m(t){\mbox{\boldmath $\!\theta\!$ \unboldmath}}_k^, \;\;\;\; (k=1,\ldots,K);$$ and curve specific profiles: $$\label{curveShapeFunction} m_i(\mu_i(t))=c_i+a_i{\mbox{\boldmath $\!B\!$ \unboldmath}}^T_m({\mbox{\boldmath $\!B\!$ \unboldmath}}^T_{\mu}(t){\mbox{\boldmath $\!\phi\!$ \unboldmath}}_i){\mbox{\boldmath $\!\theta\!$ \unboldmath}}_i.$$ Point-wise credible intervals and functional bands are easily obtained as empirical quantiles. Alternatively, the simultaneous credible band for a function $f(\cdot)$ can be obtained as described in @CrainiceanuEtal2007 and @Telesca:Inoue:2007. To assess the model fit, we use the Conditional Predictive Ordinate (CPO) [@geisser1979predictive; @Pettie1990]. The theoretical CPO for curve $i$ is defined as $$\label{CPO} p({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\|{\mbox{\boldmath $\!y\!$ \unboldmath}}_{-i}) = \int p({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\|\Theta)p(\Theta\|{\mbox{\boldmath $\!y\!$ \unboldmath}}_{-i})d\Theta,$$ where $\Theta$ denotes the collection of all model parameters. A Monte Carlo estimate, based on posterior draws is defined as: $$\label{CPOMCMC} \mathrm{CPO}_i= \left\{M^{-1}\sum_{j=1}^M p({\mbox{\boldmath $\!y\!$ \unboldmath}}_i\|\Theta^{(j)})^{-1}\right\}^{-1}.$$ Overall model fit is assessed using the log pseudo marginal likelihood (LPML), computed as: $$\label{LPML} \mathrm{LPML}=\sum_{i=1}^N \log(\mathrm{CPO}_i).$$ A Monte Carlo Study of Engineered Data {#simulation} ====================================== We carry out a simulation study aimed at assessing the merits of joint clustering and registration and comparing the performance of our modeling strategy to common clustering techniques. We consider 100 datasets, each consisting of, $45$ curves in $4$ clusters. Each curve $i$ is generated as: $y_i(t)=c_i+a_if_k(\mu_i(t))+\epsilon_{it}$, (if $s_i=k$), with $c_i \sim \mathrm{N}(0, \sigma=0.3)$, $a_i \sim \mathrm{N}(1, \sigma=0.3)I(a_i>0)$ and $\epsilon_{it} \sim \mathrm{N}(0, \sigma=0.3)$. We simulate realizations at $21$ equidistant time points within interval $T=[0,20]$. We use the following cluster specific shape functions: $f_1(t)=\cos(t/4)+\sin(t/4)$, $f_2(t)=\cos(t/8)$, $f_3(t)=\sin(t/2)$ and $f_4(t)=0$. Cluster $4$ serves as a noise cluster with no signal. Finally, time transformations $\mu_i(t)$ are generated as a monotone linear combination of B-spline basis, defined by one interior knot at $t=10$. We fit our model overparametrizing functional forms and fix $31$ equidistant interior knots between $-5$ to $25$ to model common shapes spline bases, and $3$ interior knots at $(5, 10, 15)$ to model time transformation spline bases. Precisions and the Dirichelet mixture weight $\alpha$ are assigned diffuse $\mathrm{Ga}(0.01,0.01)$ priors, (mean=1, variance=100). .1in To assess the joint model’s ability to simultaneous cluster and register curves, we compare the model with a registration only (BHCR) model and the clustering only model as described in section \[Priors\]. For both the clustering only model and the registration only model, $20K$ iterations is run for the MCMC with the first $10K$ as burn-in. We also compare our model with model-based clustering (MCLUST) [@FraleyRaftery2002a] and functional clustering (FCM) [@JamesSugar2003]. Figure \[simulCurv\] shows the results from one of the simulations. Panel (a) shows $45$ curves color-coded by cluster membership. Panel (b) shows estimated individuals curves clustered and registered within each cluster, with superimposed cluster-specific shape functions (solid black). Panel (c) shows posterior expected cluster-specific shape functions (black) against the simulation truth (gray). The model is able to accurately recover cluster specific shapes. Panel (d)-(f) show results for three individual curves, each from one of the first three clusters. Posterior estimates of individual curves (solid black) are close to the simulation truth (solid gray) and $95\%$ simultaneous credible bands achieve calibrated coverage. Also shown are profile estimates from the the registration only model (dotdash) and the clustering only model (dotted). As expected, since the registration only model assumes all the curves share a common shape function, cluster-specific functional features are confounded and model fits tend to exhibit spurious features. The clustering only model is inherently highly flexible, as small sub-clusters are allowed to form and fit specific profiles. However, since there tend to be only few curves in each cluster, the loss of information results in noisier estimates. ![\[simulCurv\]Simulation study: assessing model fit. (a) Simulated unregistered curves in $4$ clusters shown in different colors from one sample dataset. (b) Estimated individuals curves clustered and registered within each cluster, superimposed by the posterior cluster specific common shape functions (solid black). (c) Estimated cluster specific common shape functions (black) and simulation truth (gray). (c)-(f) Three individual curves from cluster 1, 2 and 3, circles indicate the data points for each curve. Estimated individual curves and the true curves are shown in solid black and gray, respectively. $95\%$ simultaneous credible bands are shown as dashed lines. Estimated individual curves from the registration only model and the clustering only model are shown as the dotdash and dotted lines respectively.](simulCurv.eps){width="130mm"} .1in We also compared our joint model with MCLUST and FCM in terms of both curve estimation accuracy and clustering accuracy. We apply MCLUST and FCM on the same 100 datasets, allowing for up to 10 clusters. Figure \[simulClust\] summarizes comparison results. Panel (a) shows boxplots of the log pseudo marginal likelihood (LPML) comparing the three Bayesian models. In panel (b) we show boxplots of the simulation mean squared error (MSE) between the estimated and true individual curves for all five models. The joint model exhibits best performance in terms of MSE, and the three Bayesian model perform better than MCLUST and FCM. To compare the clustering performance we used adjusted Rand index [@HubertArabie1985]. Panel (c) shows the boxplots of adjusted Rand indices for the four clustering models. The joint model leads to much higher indices, when compared to the other models considered. The clustering only model does as well as MCLUST and considerably better than FCM. Panel (d) shows a bar plot for the number of clusters identified by the four models. Out of the 100 datasets, the joint model identifies 4 clusters in 38 datasets and 5 clusters in 33 datasets. FCM also does well in identifying the correct cluster numbers, specifically it identifies 4 clusters in 46 datasets and 3 clusters in 27 datasets. The clustering only model and MCLUST tend to overestimate the number of clusters. ![\[simulClust\]Simulation study: clustering comparison. (a) Boxplots of log pseudo marginal likelihood (LPML) over the 100 datasets for the joint model (JM), the clustering only model (CO) and the registration only model (RO). (b) Boxplots of MSE over all the estimated curves by the five models, JM, CO, RO, MCLUST (MC) and FCM (FC). (c) Boxplots of adjusted Rand indices for the four clustering models, JM, CO, MC and FC. (d) Bar plots of cluster numbers identified by the four clustering models, JM (black), CO (dark gray), MC (light gray) and FC (white), in the 100 datasets. ](simulClust.eps){width="130mm"} .1in We repeated the joint clustering and registration analysis under several prior specifications, in order to assess sensitivity. While the formal task is daunting, due to the large number of parameters in the model, we have found that reasonable variations in prior choice has little impact on final inference, detailed results are reported in Web Appendix A. Clearly, different considerations may apply under different sample size scenarios. A Cluster Analysis of the Berkeley Growth Data {#growthData} ============================================== We apply the proposed model to the well known Berkeley Growth data and compare it with the clustering only model, registration only model, MCLUST and FCM. As discussed in Section \[introduction\], the Berkeley Growth Study [@Tudden:Sny:1954] recorded the height of 39 boys and 54 girls for 27 time points between age 2 to 18, with one measurement a year before age 9 and two measurements a year after. To construct the growth velocity curves from the original growth curves, a smoothing spline model was fitted to each growth curve, and the first degree derivatives were obtained from the model and used in our comparisons. In Figure \[growthVelocity\](a) and (b), the growth velocity curves of boys (blue) and girls (pink) are plotted against age with superimposed cross-section means (black). Within each sex, curves have similarities in shape, while each curve shows individual time and amplitude variation. As pointed out by @Rams:Li:curv:1998 and @Gerv:Gass:self:2004, failing to account for time variability, produces inconsistent estimates of sex-specific growth velocities. Our analysis is non-standard, as we use sex as a hidden label to assess clustering performance. While illustrative, this exercise finds justification in the fact that sex is expected to explain a large portion of variation in adolescent growth patterns. Shape functions basis are constructed fixing $\Delta=7$ and placing 27 equidistant interior knots between $-3$ to $23$. To model time transformation functions, we place four interior knots at $(5.2, 8.2, 11.6, 14.8)$ and partition the interval $T=[2, 18]$ into five subintervals. Priors on precisions and mixture weight are set as in Sec. \[simulation\]. Our inferences are based on $20K$ MCMC iterations, with $10K$ burn-in. ![\[growthVelocity\]Growth velocity data analysis. (a) and (b) Individual unregistered growth velocity curves for 39 boys (blue dashed) and 54 girls (pink dashed): cross-sectional means in solid-black. (c) Registered curves in the 1st cluster: 34 boys (blue dashed) and 11 girls (pick dashed). Estimated common shapes are indicated in (solid-black), MCLUST (red), FCM (green) and the cross sectional mean in (dashed-black). (d) Registered curves in the 2nd cluster: 43 girls (pink dashed) and 5 boys (blue dashed). Common shape functions as in (c). (e) and (f) Estimated curve specific time transformation functions for the two clusters: boys (dashed-blue) and girls (dashed-pink). ](growthVelocity.eps){width="130mm"} The model identifies two clusters, seemingly discriminative according to sex.If we label the first cluster as the “boy” cluster and the second cluster as the “girl” cluster, then $43$ out of $54$ girls are clustered correctly and $34$ out of $39$ boys are clustered correctly. The overall classification accuracy is $83\%$. Estimated time transformation functions, common shape functions and registered curves are shown in Figure \[growthVelocity\]. Panel (c) and (d) show the registered curves for the $2$ clusters, superimposed by their corresponding common shape functions from the joint model (black solid), MCLUST (red), FCM (green) and the cross sectional mean curves (black dashed). Individual curves are colored by their true gender information, blue for boys and pink for girls. Therefore, pink curves in panel (c) and blue curves in panel (d) show the misclassified cases. Panel (e) and (f) show the estimated curve specific time transformation functions for the two clusters. We compared the joint model with the clustering only model, the registration only model, MCLUST and FCM, and the results are shown in Figure \[growthCompPlot\]. Panel (a) shows the boxplots of CPO of the $93$ individual growth curves by the three Bayesian models. It shows that the joint model fits the data best, followed by the registration only model and the clustering only model. When the curves are not too dramatically different, the registration only model can fit the data accurately by finding a common shape function representing all the curves well. As a comparable measure of model fit we compute the squared error (SE) between each curve and its fitted profiles.Panel (b) shows the boxplots of the SE over all the growth curves for the five models. The joint model gives the smallest SE, and the three Bayesian models fit the data better than MCLUST and FCM in terms of SE. Panel (c) and (d) show the model fitting results of two individual curves of a boy and a girl. ![\[growthCompPlot\]Growth velocity data model comparison. (a) Boxplots of log CPO of the $93$ individual growth curves by the joint model (JM), the clustering only model (CO) and the registration only model (RO). (b) Boxplots of the squared error (SE) over all the growth curves for JM, CO, RO, MCLUST (MC) and FCM (FC). (c) Model fitting results of the growth velocity curve of a boy (blue line with circles) by JM (black solid), CO (black dashed), RO (black dotdash), MCLUST (red) and FCM (green). (d)Model fitting results of the growth velocity curve of a girl (pick line with circles) by JM (black solid), CO (black dashed), RO (black dotdash), MCLUST (red) and FCM (green). ](growthCompPlot.eps){width="130mm"} Interpreting sex as a clustering lable, we compare the joint model, the clustering only model, MCLUST and FCM using adjusted Rand indices (RIs). We find the following: FMC (RI = 0.61), MCLUST (RI = 0.47), JM (RI = 0.43) and CO (RI = 0.10). By this measure FCM and MCLUST seem to outperform our joint clustering and registrations model, with FCM giving the best clustering results. We note that when fitting MCLUST, we set the candidates of cluster numbers to be between 2 to 10, because when 1 is included as a candidate, MCLUST chooses it as the optimal cluster number, which leads to an adjusted Rand index of 0. On the other hand, as shown in panel (a) and (b) in Figure \[growthVelocity\], FMC and MCLUST seem to provide unsatisfactory estimates of cluster specific shape functions, which the joint clustering and registration model estimates consistently with the findings of @Rams:Gass:1995 and @Telesca:Inoue:2007, supporting the existence of the mid growth spurts. Response of Human Fibroblasts to Serum {#fibroResponse} ====================================== In this section, we apply the joint model to time course expression data of the response of human fibroblasts to serum in a microarray experiment of $8613$ genes [@IyerEtal1999]. For human fibroblasts to proliferate in culture, they require growth factors provided by fetal bovine serum (FBS). In their study, after inducing primary cultured human fibroblasts to enter a quiescent state by serum deprivation for $48$ hours, the authors stimulated fibroblasts by adding medium containing $10\%$ FBS. A microarray experiment was then conducted to measure temporal gene expression levels at $12$ time points, from $15$ minutes to $24$ hours after serum stimulation. Furthermore, they selected $517$ genes with substantial time course expression change in response to serum and formed clusters using K-means clustering [@EisenEtal1998]. In our analysis, we consider a subset of $78$ genes, since they are associated with clear biological function categories as described in the original paper, and this provides a standard for us to validate the biological relevance of the clustered identified by our model. We use the same prior setup as in previous sections. To model shape functions we use a maximum expansion constraint $\Delta=6$ and place interior knots at the sampling time points and $5$ equidistant points in two intervals from $-5$ to $-1$ and from $25$ to $29$ respectively. To estimate the time transformation functions, we place four interior knots at $(0.5, 2, 8, 16)$ in the sampling interval $T=[0, 24]$. Our inferences are based on $20K$ MCMC iterations, with $10K$ burn-in. ![\[geneCurves\]Human fibroblast gene expression analysis*. (a) Unregistered time coures expression curves for 78 genes selected from a microarray experiment of Human fibroblasts’ response to serum. (b) Registered expression curves forming $4$ clusters colored by green, red, blue and pink. (c) Thirty five genes in cluster $1$ superimposed by the cluster specific common shape function (solid black). (d) Nine genes in cluster $2$. (e) Five genes in cluster $3$. (f) Twenty nice genes in cluster $4$.](geneClusters.eps){width="130mm"} Panel (a) of Figure \[geneCurves\] shows the unregistered temporal expression curves of the 78 genes selected from the microarray experiment of human fibroblasts’ response to serum. Panel (b) shows the registered expression curves which are clustered into $4$ groups. Panel (c)-(f) show the $4$ clusters of registered expression curves separately, superimposed by their cluster specific common shape functions. Cluster Size Typical Genes Functions --------- ------ --------------------------- ------------------------------ 1 35 PCNA, Cyclin A, Cyclin B1 Cell cycle and proliferation CDC2, CDC28 kinase 2 9 LBR Cell cycle and proliferation 3 5 PAI1, PLAUR, ID3 Coagulation and hemostasis Transcription factors 4 29 MINOR, JUNB, CPBP Signal transduction TIGF, SGK, NET1 Transcriptional factors : \[geneTable\] Clusters of genes and their biological functions As shown in Figure \[geneCurves\] (c), genes in cluster $1$ are down-regulated at first and reach their lowest expression levels between $4$ and $12$ hours after serum stimulation. They begin to express about $16$ hours after the serum treatment, which is also the time when the stimulated fibroblasts replicate their DNA and reenter into the cell-division cycle. Several genes in cluster $1$ are known to be involved in mediating cell cycle and proliferation, for instance, PCNA, Cyclin A, Cyclin B1, CDC2 and CDC28 kinase, as shown in Table \[geneTable\]. Cluster $2$ in Figure \[geneCurves\](d) shows similar expression pattern to cluster $1$, except they expression level are lower than those in cluster $1$ through the time window. Genes in cluster $2$ are also involved in cell cycle and proliferation, such as LBR. Figure \[geneCurves\](e) shows that genes in cluster $3$ respond immediately to serum stimulation, reach their expression peaks around 10 hours later and remain induced towards the end. They are known to be transcription factors and involved in coagulation and hemostasis because of fibroblasts’ role in clot remodeling. Typical genes include PAI1, PLAUR and ID3. As shown in Figure \[geneCurves\](f), genes in cluster $4$ are also induced quickly by serum treatment, reach their peaks at about 2 hours, and then gradually return to a quiescent state. Several of the genes here are known to encode transcriptional factors and other proteins involved in signal transduction, such as MINOR, JUNB, CPBP, TIGF, SGK and NET1. Discussion ========== We propose a Bayesian hierarchical model for joint curve registration and clustering. Compared to previous methods, our proposal comes with several advantages. First, the model provides flexible nonlinear modeling for both components of variation. The Dirichlet process mixture prior over shape functionals strikes an automatic balance between complexity and parsimony. The implied posterior identifies subgroups of homomorphic curves, without the need to specify the number of clusters a priori. Finally, the increased model flexibility is still amenable to straightforward posterior simulation via MCMC, which provides exact inferences about a rich set of quantities of interest, without the need for simplifications or approximations. The proposed B-spline representation of both shape and time transformation functions requires the a priori specification of the number and placement of spline knots. Our experiences suggests that a set of knots reproducing the original sampling time points works well for shape functions and 1 to 4 equidistant interior knots are enough for time transformation functions, as they carry smoothing properties through monotonicity constraints. Our simulation study shows that the model is robust to different prior choices. We however maintain, that different considerations may apply to ultra-sparse or, conversely, ultra-dense data settings. The proposed modeling strategy has potentially broad applications to functional data analysis; especially when curve registration and clustering are of joint interest, as shown in our applications. In the first case study of the Berkeley Growth Data, our model is able to accurately separate growth curves into two clusters labelled by sex, and to correctly estimate the overall growth patterns for both sexes after registering curves in each cluster. In the second case study of time course expression data of human fibroblasts’ response to serum, our model identifies fours clusters of genes involved in distinct biological functions. The proposed estimator of the clustering structure is the MAP clustering. Because Dirichlet process mixtures fully account for stochasticity in the potential alternative assignment of individual profiles to functional groups, it is possible, in principle, that the clustering structure with the second highest posterior probability is only a little less probable than the MAP clustering, yet it provides quite a different grouping structure. We have not detected this type of phenomenon in our analyses. However, when it happens, some care is needed in summarizing complex posterior evidence. A possible alternative strategy to MAP is based on the estimation of a pairwise probability matrix whose elements are estimated probabilities that two curves are in the same cluster. Such a matrix can be easily generated by averaging the sampled association matrices from the MCMC output. Elements of an association matrix takes values $1$, if two corresponding curves are in the same cluster, and $0$ otherwise. Hierarchical clustering may be used subsequently as a way to explore grouping structures [@MedvedovicSivaganesan2002]. Alternatively, @Dahl2006 proposed a least squares clustering by selecting the sampled clustering which minimizes the sum of squared deviations of its association matrix from the pairwise probability matrix. Finally, when covariate information is available, the proposed model is easily extended to include a dependent Dirichlet process prior, using covariates to inform clustering. Supplementary Materials {#supplementary-materials .unnumbered} ======================= Supplementary information is available from the authors. \[lastpage\]
--- abstract: 'Using an algorithm for simulating equilibrium configurations, we study a fluctuating helical polymer either (i) contained in a cylindrical pore or (ii) wound around a cylindrical rod. We work in the regime where both the contour length and the persistence length of the helical polymer are much larger than the diameter of the cylinder. In case (i) we calculate the free energy of confinement and interpret it in terms of a worm-like chain in a pore with an effective diameter that depends on the parameters of the helix. In case (ii) we consider the possibility that one end of the helical polymer escapes from the rod and wanders away. The average numbers of turns at which the helix escapes or intersects the rod are measured in the simulations, as a function of the pitch $p_0$. The behavior for large and small $p_0$ is explained with simple scaling arguments.' author: - 'A. Lamura$^{1,2}$, T. W. Burkhardt$^{2,3}$, and G. Gompper$^{2}$' title: Helical polymer in cylindrical confining geometries --- Introduction ============ In this paper we study some of the equilibrium statistical properties of a confined helical or ribbon-like polymer. The cases of (i) a polymer contained in a cylindrical pore and (ii) a polymer wound around a cylindrical rod are considered. Some motivation is provided by the following observations: Biological polymers differ from synthetic polymers in that they are semi-flexible, with a persistence length much larger than the monomer size, and usually have a helical structure. This is well known for DNA, but F-actin also has a double-helical structure, while microtubuli are helical cylinders. The diameter of these biological polymers is in the range of 1 to 25 nm. Polymeric helical structures are also found in self-assembling systems, consisting of either amphiphiles or peptides. In some cases the diameters and pitch lengths are much larger than for the biopolymers mentioned above. In amphiphilic systems the formation of helical ribbons has been observed in multicomponent mixtures of a bile salt or some other nonionic surfactant, phosphatidylcholine or a fatty acid, and a steroid analog of cholesterol [@chun93; @zast99]. The ribbons have typical diameters in the range of 5 to 20 $\mu$m, and pitch angles between 10 and $60^o$. Other examples are ethanolic/water solutions of diacetylenic phospholipids, in which the formation of hollow tubules of diameter 0.6 $\mu$m and typical lengths of 10 to 100 $\mu$m has been observed [@yage84; @schn93; @thom99; @pakh03]. Helically-coiled phospholipid-bilayer ribbons appear as metastable intermediates in the growth of these tubules. Other systems, which show spontaneous assembly of ribbons, are aqueous solutions of peptides [@zhan93; @agge97; @nyrk00a; @nyrk00b]. Depending on the solution conditions, the same peptide exists in different conformations, such as random coils, $\alpha$-helices, or $\beta$-sheets. At not too low peptide concentrations, the molecules self-assemble into long $\beta$-sheet structures which form twisted ribbons (with a straight central axis). The width of these ribbons is about 4 nm, and their length is of the order of 500 nm [@nyrk00a; @nyrk00b]. These ribbons can aggregate due to face-to-face attraction into twisted fibrils of a thickness of 8-10 nm. Interestingly, in a self-assembling system of gemini surfactants (two surfactant molecules covalently linked at their charged head group), the degree of twist and the pitch of the micrometer-scale ribbons has been found to be tunable by the introduction of opposite-handed chiral counterions [@oda99]. The confinement of polymers in cylindrical tubes is one of the classical problems in polymer physics. For biological polymers, such a confinement occurs, for example, when viral DNA of a bacteriophage squeezes through the narrow tail during DNA injection. Technological advances in the manipulation of single molecules in micro- and nanofluidic devices [@chou00; @gior01] has fueled interest in the structure and dynamics of biological polymers in confined geometries [@jend03]. Helical and twisted ribbons can be confined not only by external walls, but also by winding around each other, as in the fibril formation of twisted $\beta$-sheet peptides mentioned above. The simple model we consider, consisting of a helical wound around a thin cylinder, is a step in this direction but leaves out some important physical features, such as the face-to-face attraction in the fibrils. Free Energy of Confinement ========================== The free energy $\Delta F$ of confinement of a fluctuating polymer of contour length $\ell$ in a cylindrical pore of diameter $D$ is defined by $$\exp(-\Delta F/k_BT)=\frac{Z(D,\ell)} {Z(\infty,\ell)}=p(\ell)\;\label{freeenergy}$$ Here $Z(D,\ell)$ and $Z(\infty,\ell)$ are the partition functions of the polymer with one end fixed in the presence and absence of the cylindrical confining geometry, respectively. The quantity $\Delta F$ represents the work required to squeeze the polymer reversibly into the cylindrical pore. It may be evaluated in simulations by generating polymer configurations with one fixed end in an infinite volume with the Boltzmann probability, computing the fraction $p(\ell)$ of the configurations of arc length $\ell$ which lie entirely within a cylindrical domain of diameter $D$, and making use of Eq. (\[freeenergy\]) For a flexible, self-avoiding polymer with vanishing bending rigidity, $\Delta F$ is purely entropic. The confinement of such a polymer in a cylindrical pore is considered in Refs. [@degennes; @bg; @ps]. In the worm-like chain model of a semi-flexible polymer, the polymer is represented by an inextensible line or filament ${\bf r}(s)$ with contour length $\ell$ and elastic energy $$E_{\rm worm}=\frac{\kappa}{2}\int_0^\ell\left(\frac{d{\bf t}_3} {ds}\right)^2\thinspace ds\;.\label{wormlike}$$ Here $s$ specifies distance along the contour, ${\bf t}_3=d{\bf r}/ds$ is the unit tangent vector, $\kappa$ is the bending rigidity, and $P=\kappa/k_BT$ is the the persistence length. In the narrow-pore, long-polymer limit $D\ll P\ll \ell$, the polymer is almost a straight line, i.e. the angle between the tangent vector ${\bf t}_3$ and the $z$ axis or symmetry axis of the cylinder is a small quantity. In this case the right-hand side of Eq. (\[freeenergy\]) decays as $$p(\ell)\sim e^{-E_0\ell}\;\label{pell}$$ for large $\ell$, where $\exp(-E_0\thinspace dz)$ is the largest eigenvalue of the transfer matrix of a slice of the system of thickness $dz$. The quantity $E_0^{-1}$ represents a typical contour length at which the configurations intersect the pore wall. According to Eqs. (\[freeenergy\]) and (\[pell\]) the confinement free energy per unit length $\Delta f=\Delta F/\ell$ is given by $$\frac{\Delta f}{k_BT}=E_0(P,D)=\frac{A_\circ} {P^{1/3}D^{2/3}}\;,\label{circle}$$ where the dependence on $P$ and $D$ follows from simple scaling or dimensional arguments [@odijk; @dijkstra; @twb97]. Similarly, for a pore with a rectangular cross section with edges $L_1,L_2\ll P$, $$\frac{\Delta f}{k_BT}=E_0(P,L_1,L_2)=\frac{A_\Box} {P^{1/3}}\left(\frac{1}{L_1^{2/3}}+\frac{1} {L_2^{2/3}}\right)\;.\label{rectangle}$$ The quantities $A_\circ$ and $A_\Box$ on the right-hand sides of Eqs. (\[circle\]) and (\[rectangle\]) are dimensionless universal numbers $A_\circ$ and $A_\Box$, which are the same for all worm-like chains. The prediction $A_\Box=1.1036$ was obtained in Ref. [@twb97] by solving an integral equation numerically which arises in an exact analytical approach. Measuring the probability $p(\ell)$ in Eq. (\[pell\]) in simulations, fitting the large $\ell$ behavior with the exponential form (\[pell\]), and making use of Eqs. (\[circle\]) and (\[rectangle\]), Bicout and Burkhardt [@Bicout] estimated $$A_\circ=2.375\pm0.013\;,\quad\quad A_\Box=1.108\pm 0.013\;.\label{bicout}$$ An earlier estimate from simulations, $A_\circ=2.46\pm0.07$, was given by Dijkstra [et al.]{} [@dijkstra]. Helical Polymer Model ===================== In this paper we generalize the above results to helical polymers or chiral ribbons, which have spontaneous curvature and torsion. Again the polymer is replaced by a curve ${\bf r}(s)$ of fixed contour length ${\cal S}$. To each point on the line a right-handed triad of unit vectors ${\bf t}_1(s), {\bf t}_2(s),{\bf t}_3(s)$ is assigned, where ${\bf t}_3=d{\bf r}/ds$ is the tangent vector and ${\bf t}_1$, ${\bf t}_2$ correspond to principal axes of the polymer cross section. The rotation of the triad along the curve is governed by the generalized Frenet equations [@PanRabPRL00; @PanRabPRE00; @KKRabPRE02] $$\frac{d {\bf t}_i}{d s}={\boldsymbol\omega}\times{\bf t}_i\;,\quad {\boldsymbol\omega}={\bf t}_1\omega_1+{\bf t}_2\omega_2+{\bf t}_3\omega_3\;,\label{frenet1}$$ or $$\frac{d {\bf t}_i}{ds}=\sum_{j,k}\epsilon_{ijk}{\bf t}_j\omega_k\;.\label{frenet2}$$ The elastic energy is given by [@PanRabPRL00; @PanRabPRE00; @KKRabPRE02] $$E_{\rm helix} = \frac{1}{2} \sum_{j=1}^{3} b_j \int_0^{\cal S} ds \left[\omega_j(s) - \omega_{0j}(s)\right]^2\;, \label{energy}$$ where the coefficient $b_1$ and $b_2$ are bending rigidities along the principal axes of the cross section, and $b_3$ is the twist rigidity. The parameters $\omega_j(s)$ and $\omega_{0j}(s)$ determine the curvatures and torsions in the deformed and stress-free states of the polymer, respectively. Since the energy is quadratic in the deviations $\delta \omega_j = \omega_j-\omega_{0j}$, the distribution of $\delta \omega_j$ is Gaussian, with zero mean and second moment $$\langle\delta \omega_i(s) \delta \omega_j(s^{'})\rangle = \frac{k_BT}{b_i}\thinspace\delta_{ij}\thinspace\delta(s - s^{'})\;.\label{moments}$$ We restrict our attention to the case $\omega_{0j}(s)={\rm constant}$, corresponding to a helical polymer with spontaneous curvature and torsion but without spontaneous twist. In the absence of fluctuations, i.e. in the limit $b_1=b_2=b_3=\infty$, the Frenet equations are readily solved [@explain1], yielding $$\begin{aligned} {\bf r}(s)={\bf r}(0)&+&\frac{1}{\omega_0}\Biggl\{{\bf t}_3(0)\sin(\omega_0s)+{\bf e}(0)\thinspace\omega_{03}\left[s- \frac{\sin(\omega_0 s)}{\omega_0}\right] \nonumber\\ &+& {\bf e}(0)\times{\bf t}_3(0)\left[1-\cos(\omega_0 s)\right]\Biggr\}\;,\label{nofluc1}\end{aligned}$$ where $${\bf e}(s)={\bf t}_1(s)\frac{\omega_{01}}{\omega_0}+{\bf t}_2(s)\frac{\omega_{02}}{\omega_0}+ {\bf t}_3(s)\frac{\omega_{03}}{\omega_0}\;, \quad \omega_0=\left(\omega_{01}^2+\omega_{02}^2+\omega_{03}^2\right)^{1/2}\;.\label{nofluc2}$$ Equation (\[nofluc1\]) represents a helix with radius $r_0$ and pitch $p_0$, where $$r_0=\frac{(\omega_{01}^2+\omega_{02}^2)^{1/2}}{\omega_0^2}\;,\quad p_0=2\pi\frac{\omega_{03}}{\omega_0^2}\;,\label{radiuspitch}$$ winding around an axis pointing in the direction of the unit vector ${\bf e}(0)$. Including Gaussian fluctuations according to Eq. (\[moments\]), Panyukov and Rabin [@PanRabPRL00; @PanRabPRE00] showed that $$\langle{\bf t}_i(s)\cdot{\bf t}_j(0)\rangle=\left(e^{-{\bf \Gamma}\thinspace s}\right)_{ij}\;,$$ where ${\bf\Gamma}$ is the matrix with elements $$\Gamma_{ij}= \frac{1}{2}k_BT \left(\sum_{k}b_k^{-1}-b_i^{-1}\right)\delta_{ij}- \sum_k\epsilon_{ijk}\omega_{0k}\;.$$ The two-point correlation function of the unit vector ${\bf e}(s)$ in Eq. (\[nofluc2\]), which is directed along the axis of the helix, follows from this result. In the special case $b=b_1=b_2=b_3$ considered in our simulations, $$\langle{\bf e}(s)\cdot{\bf e}(0)\rangle=e^{-s/L_p}\;, \quad L_p=\frac{b}{k_BT}\;,\label{expdecay}$$ where $L_p$ is the persistence length. Simulations =========== Following Kats [et al.]{} [@KKRabPRE02], we replace the differential equations (\[frenet2\]) by the difference equations $$t_{i k}(s+ds)=\sum_{j}O_{ij}t_{j k}(s)\label{differenceeq}$$ in our simulations. Here $t_{i k}$ denotes the $k$-th component of ${\bf t}_i$ with respect to a fixed Cartesian coordinate system, $O$ is the orthogonal matrix $$O=\left(1+\frac{1}{2}A\thinspace ds\right)\left(1-\frac{1}{2} A\thinspace ds\right)^{-1}\;,\label{matrixO}$$ and $A$ is the antisymmetric matrix with elements $A_{ij}=\sum_k\epsilon_{ijk}\omega_k$. The difference equations are consistent with the Frenet equations (\[frenet2\]) to first order in $ds$, and the orthogonality of the matrix $O$ preserves the orthonormality of the ${\bf t}_i$ in the simulations. For simplicity we set $b=b_1=b_2=b_3$, corresponding to Eq. (\[expdecay\]). In accordance with Eq. (\[moments\]), the $\delta\omega_j(s)$ are chosen randomly from a Gaussian distribution with zero mean and standard deviation $(k_BT/b\thinspace ds)^{1/2}$, where $ds\ll L_p=b/k_BT$. Helical polymer in a cylindrical pore ===================================== We have determined the confinement free energy of a helical polymer fluctuating in a narrow cylindrical pore from simulations. Cylinders with both circular and square cross sections were considered, and we use the same symbol $D$ for the diameter and edge, respectively. The symmetry axis of the cylinder defines the $z$ axis of our fixed coordinate system. The helical polymer was generated step by step using the numerical procedure described in the preceding Section. The radius $r_0$ and pitch $p_0$ were set to desired values by choosing $\omega_{01}$, $\omega_{03}$ in Eq. (\[radiuspitch\]) appropriately, with $\omega_{02}=0$. The starting point was chosen randomly, apart from the requirements that the stress-free helix fit inside the cylinder, with its axis parallel to the $z$ axis. This is the case for the initial vectors ${\bf t}_1(0) = (\omega_{03}/\omega_0,0,\omega_{01}/\omega_0)$, ${\bf t}_2(0) = (0,1,0)$, and ${\bf t}_3(0) = (-\omega_{01}/\omega_0,0,\omega_{03}/\omega_0)$, which were used. The other simulation parameters were $ds=10^{-4}$, $D=1$, $L_p=b_1/k_B T = b_2/k_B T = b_3/k_B T = 8000$. Clearly $ds \ll D \ll L_p\thinspace$. The pore is narrow in comparison with the persistence length, and $ds$ is small in order to approximate the continuum model (\[frenet1\]). To obtain the free energy of confinement of the helical polymer, we proceeded as discussed below Eq. (\[freeenergy\]), generating many polymer configurations and computing the probability $P(n)$ that the polymer has not yet intersected the pore wall [@explain2] after $n$ steps of the algorithm. The determination of $P(n)$ was based on $50,000$ independent helices. For large $n$ an exponential decay $$P(n)\sim e^{-\lambda_0 n}\;,\label{expdecay2}$$ similar to the result (\[pell\]) for semi-flexible polymers, is expected. According to Eq. (\[freeenergy\]), the free energy of confinement per unit length along the axis of the helix is given by $$\frac{\Delta f}{k_BT}=\frac{\lambda_0}{\xi ds}\;,\quad \xi=\frac{p_0}{\ \left[p_0^2+(2\pi r_0)^2\right]^{1/2}}\;.\label{freeenergy2}$$ Here we use the relation $\ell=\xi s$ between the contour length $s$ and the corresponding length $\ell$ along the axis of the helix. The persistence length $L_p$, defined with respect to the contour length as in Eq. (\[expdecay\]), and the corresponding persistence length $P$, defined with respect to length along the axis of the helix, also satisfy $P=\xi L_p$. A helical polymer with persistence length $L_p$ in a pore with diameter $D\ll L_p$ has the same confinement free energy as a semi-flexible polymer with persistence length $P=\xi L_p$ in a pore with effective diameter $D_{\rm eff}$. To define $D_{\rm eff}$ quantitatively, we equate the free energies of confinement (\[circle\]) and (\[freeenergy2\]), obtaining $$\frac{A_\circ}{D_{\rm eff}^{2/3}}=\frac{(\xi L_p)^{1/3}}{\xi ds}\lambda_0\;,\quad \frac{A_\Box}{D_{\rm eff}^{2/3}}= \frac{(\xi L_p)^{1/3}}{2\xi ds}\lambda_0\;.\label{Deff}$$ The probability $P(n)$ for $r_0=p_0=0.3$, with the other simulation parameters noted above, is shown in Fig. 1. The data are in good agreement with the exponential decay (\[expdecay2\]), and the values of $\lambda_0$ are given in the figure caption. As in the case of a semi-flexible polymer [@Bicout], the curves $P(n)$ for the circular and square cylinders practically coincide when plotted versus $\lambda_0 n$ instead of $n$. For the exponential decay (\[expdecay2\]), the mean number of steps of the algorithm at which the polymer intersects the wall equals $\lambda_0^{-1}$, corresponding to $N_i=\xi ds/(p_0\lambda_0)$ turns of the helix. The values of $\lambda_0$ in the caption of Fig. 1 yield $N_i=8.0$ and $8.7$ for the circular and square cross sections. Since the number of turns before intersecting the wall is fairly large, the helix should be equivalent to a worm-like chain in a pore of width $D_{\rm eff}=D - 2 r_0$. To check the equivalence quantitatively, we use Eq. (\[Deff\]) with $D_{\rm eff}=D - 2 r_0$ and the values of $\lambda_0$ in the caption of Fig. 1 to predict the amplitudes $A_\circ$, $A_\Box$. This yields $$A_{\circ} = 2.45 \pm 0.05\;,\quad\quad A_{\Box} = 1.12 \pm 0.04\;,$$ in good agreement with the results (\[bicout\]) for semi-flexible polymers. We have also studied the dependence of $D_{\rm eff}$ on the polymer pitch $p_0$, keeping the radius $r_0$ and the persistence length $L_p$ constant. For small $p_0$ the polymer makes many turns before intersecting the wall and is equivalent to a semi-flexible polymer in a pore of diameter $D-2r_0$, as discussed in the preceding paragraph. In the limit $p_0 \to\infty$, the helical polymer does not make any turns before intersecting the wall and corresponds to a semi-flexible polymer in a pore of diameter $D$. As $p_0$ increases from $0$ to $\infty$, $D_{\rm eff}$ is expected to vary monotonically between these two limiting values. For various values of $p_0$ we have computed the probability $P(n)$ that a helical polymer with radius $r_0=0.3$, persistence length $L_p=8000$, and contour length $nds$ in a cylindrical pore with a circular cross section of diameter $D=1$ does not intersect the wall. The corresponding $\lambda_0$ was obtained from an exponential fit (\[expdecay2\]) for large $n$. Finally $D_{\rm eff}$ was calculated using Eq. (\[Deff\]) and the best estimate (\[bicout\]) for $A_\circ$. The results are shown in Fig. 2. The data do indeed interpolate between the expected limiting values $D-2r_0=0.4$ and $D=1$ for small and large $p_0$, respectively. The crossover region in Fig. 2, where $D_{\rm eff}$ varies most rapidly with $p_0$, is around $p_0\approx 40$, $D_{\rm eff}\approx 0.7$. According to Eqs. (\[pell\]), (\[circle\]), and (\[freeenergy2\]), these values of $p_0$ and $D_{\rm eff}$ correspond to $N_i=(\xi L_p)^{1/3}D_{\rm eff}^{2/3}/(A_\circ p_0)\approx 0.2$ turns of the helix before intersecting the wall. Helical polymer encircling a cylindrical rod ============================================ In this Section we consider a helical polymer wound around a long cylindrical rod with a circular cross section and diameter $D\ll L_p$. We study the possibility that the polymer generated in the simulation escapes from the rod as $n$ increases and wanders away. In the simulations the parameters $ds=10^{-4}$, $b_1/k_B T = b_2/k_B T = b_3/k_B T = L_p=8000$ were the same as in the preceding section. The diameter of the rod was $D=0.2$, and the radius of the helix was $r_0=0.3$. For these parameters $ds\ll D<2r_0\ll L_p\thinspace$. The starting point of the polymer was chosen randomly, apart from the requirements that the stress-free helix wind around the cylindrical rod without touching it, with the axis of the helix parallel to the rod. >From the simulation data we computed the probability $P(n)$ that after $n$ steps the polymer has not yet intersected the rod [@explain2]. Each curve $P(n)$ is based on $10,000$ independent helices. The results for three different values of the pitch $p_0$ are shown in Fig. 3. Unlike the case of a polymer in a cylindrical pore, shown in Fig. 1, $P(n)$ does not decay to zero as $n$ increases. Instead, above a characteristic value which depends on the pitch, the curve flattens and approaches a nonzero limiting value. This is because the polymer generated in the simulation sometimes escapes from the rod, due to a sufficiently large fluctuation, and wanders away as $n$ increases, without ever returning to intersect the rod. A simple theory of the escape, which suggests $P(n)=A+Be^{-C n}$, in qualitative consistency with Fig. 3, is given in the Appendix. We determined the average number of turns at which the helix escapes from the rod or intersects it by making two checks after each step of the growth algorithm: (i) If the distance of the endpoint ${\bf r}(s)$ from the axis of the rod is less than $D/2$, the polymer has intersected the rod. (ii) If the distance of the endpoint ${\bf r}_{\rm axis}(s)$ of the [axis]{} of the helix is greater than $r_0+D/2$, the circular cross section of the helix no longer encircles the rod, i.e. the helix has escaped. Geometrically ${\bf r}_{\rm axis}(s)$ is determined as follows: Since the unit vectors ${\bf t}_3(s)$ and ${\bf e}(s)$ are tangent to the helix and directed along its axis, respectively, ${\bf e}(s)\times{\bf t}_3(s)$ is directed perpendicularly from the point ${\bf r}(s)$ on the helix contour toward the corresponding point ${\bf r}_{\rm axis}(s)$ on the axis of the helix. Thus, $$\begin{aligned} {\bf r}_{\rm axis}(s)&=&{\bf r}(s)+r_0\thinspace \frac{{\bf e}(s)\times {\bf t}_3(s)}{\|{\bf e}(s)\times{\bf t}_3(s)\|}\nonumber\\ &=& {\bf r}(s)+{\bf t}_1(s) \frac{\omega_{02}}{\omega_0^2}-{\bf t}_2(s) \frac{\omega_{01}}{\omega_0^2}\;,\label{raxis}\end{aligned}$$ where we have used Eqs. (\[nofluc2\]) and (\[radiuspitch\]). In Fig. 4 the average numbers of turns $N_e^1$, $N_e^2$ at which the helix escapes from the rod and the average number of turns $N_i$ at which the helix intersects the rod are shown as functions of $p_0$. For each value of $p_0$, 10,000 independent configurations were generated. Each configuration was continued until it intersected the rod [@explain2] or the number of steps of the algorithm exceeded $5\times 10^6$, whichever came first. The average $N_e^1$ is based on all configurations which escape, independent of whether they return to intersect the rod or not. The quantity $N_e^2$ is the average value for only those configurations which escape and in $5\times 10^6$ steps of the algorithm still have not intersected the rod. For $p_0\leq 1$ the probability of the polymer escaping from the rod is so small that $N_e^{1,2}$ could not be determined reliably with configurations of $5\times 10^6$ steps. For larger $p_0$, the data for $N_e^1$ and $N_e^2$ practically coincide, indicating that the polymer rarely returns to intersect the cylinder once it has escaped. The data are in excellent agreement with $N_e^{1,2},N_i\sim p_0^{-1}$ for large $p_0$ and $N_i\sim p_0^{-2/3}$ for small $p_0$. These power laws may be understood as follows. The transverse fluctuations of the endpoint of the axis of an [unconfined]{} helical polymer of length $\ell$ and persistence length $P$ about the corresponding endpoint of the unstressed helix are readily calculated from Eq. (\[expdecay\]) and given by [@explain3] $$\langle r_\perp^2\rangle=\frac{2}{3}\thinspace \frac{\ell^3}{P} \label{fluc}$$ for $\ell\ll P$. Here both $\ell=\xi s$ and $P=\xi L_p$ are measured along the axis of the helix, as discussed below Eq. (\[freeenergy2\]). Equation (\[fluc\]) also applies to the worm-like chain. Apart from the factor $2/3$, Eq. (\[fluc\]) follows from simple scaling or dimensional arguments [@odijk; @dijkstra; @twb97]. The powers of $r_\perp$, $\ell$, and $P$ in Eq. (\[fluc\]) are also consistent with $ E_0 \ell\sim 1$ and $r_\perp\sim D$ in Eqs. (\[pell\]) and (\[circle\]). Replacing $r_\perp$ in Eq. (\[fluc\]) by $r_0+D/2=0.4$, as in the simulations, and solving for $N_e=\ell/p_0$ yields the estimate $$N_e=\frac{12}{p_0^{2/3}\left(p_0^2+3.6\right)^{1/6}}\;.\label{Ne}$$ for the number of turns of the helix at which the typical transverse displacement equals the value needed for escape from the rod. Roughly speaking, the polymer wrapped around the rod escapes in $N_e$ turns, as given by Eq. (\[Ne\]) for $N_e\stackrel{<}{\sim} 1$, i.e. $p_0\stackrel{>}{\sim} 12$. For smaller $p_0$, the typical transverse fluctuations in a single turn of the helix are too small for escape from the rod, and the helix is more likely to intersect the rod than to escape. Equation (\[Ne\]), which ignores this possibility, no longer applies. As noted above, for $p_0<1$ the escape probability is too small for a reliable determination of $N_e^{1,2}$ with configurations of $5\times 10^6$ steps. In the region $N_e\stackrel{<}{\sim} 1$, i.e. $p_0\stackrel{>}{\sim} 12$, Eq. (\[Ne\]), which corresponds to the solid curve in Fig. 4, is in good quantitative agreement with the simulation data for $N_e^{1,2}$. For large $p_0$, $N_e\approx 12/p_0$. The coefficient 12 is an order-of-magnitude estimate that happens to give a good fit to the simulation data, whereas the power law $N_e\sim p_0^{-1}$ for large $p_0$ is exact. An argument based on Eq. (\[fluc\]) similar to the one for $N_e$ predicts $N_i\sim p_0^{-1}$ for large $p_0$, in excellent agreement with Fig. 4. Note that the data points for $N_e^{1,2}$ and $N_i$ practically coincide. Since the possibility of escape is negligible for small $p_0$, the helical polymer is equivalent to a worm-like chain in a pore with diameter $D_{\rm eff}=2r_0-D$. To estimate the average number of turns $N_i$ at which the polymer intersects the rod, we replace $D$ by $D_{\rm eff}$ in Eqs. (\[pell\]) and (\[circle\]) and solve for the typical intersection length $\ell\approx E_0^{-1}$. This yields $\ell\sim P^{1/3}$. Substituting $r_\perp\sim D_{\rm eff}$ in Eq. (\[fluc\]) and solving for $\ell$ leads to the same result. According to the discussion below Eq. (\[freeenergy2\]), $P=\xi L_p\approx L_p(p_0/2\pi r_0)$ for $p_0\ll r_0$. Keeping track of the powers of $p_0$, we obtain the power law $N_i=\ell/p_0\sim p_0^{-2/3}$ for small $p_0$, in excellent agreement with the simulation data in Fig. 4 In Fig. 5 the fractions $f_e^1$, $f_e^2$, and $f_i$ of the 10,000 configurations which contribute to $N_e^1$, $N_e^2$, and $N_i$ in Fig. 4 are shown as functions of $p_0$. For $p_0\stackrel{<}{\sim}1$, $f_e^{1,2}\approx 0$ and $f_i\approx 1$, i.e. almost all the configurations intersect the cylinder and never escape. Around $p_0\approx 10$ the curves cross, and for larger $p_0$ the polymer is more likely to escape than to intersect the rod. Concluding Remarks ================== We have studied some statistical properties of a helical polymer in cylindrical restrictive geometries of diameter $D$, in the limit that the persistence length $P$ along the axis of the helix is large in comparison with $D$ and the radius $r_0$ of the helix. In this limit the helical polymer has much in common with the worm-like chain. We interpret the simulation data for the free energy of confinement in a cylindrical pore using the scaling form (\[circle\]) for a worm-like chain in a pore, with an effective diameter $D_{\rm eff}$ that is renormalized by the helical structure. As the pitch $p_0$ of the helix increases from $0$ to $\infty$, $D_{\rm eff}$ increases monotonically from $D-2r_0$ to $D$, as shown in Fig. 2. Thinking in terms of a worm-like chain also proves useful in connection with the escape of the helical polymer encircling a cylindrical rod. In the limit $P\gg r_0$ the transverse fluctuations of the axis of the helix are given by the same result (\[fluc\]) as for the worm-like chain. As $p_0$ increases, the typical transverse displacement in one turn of the helix also increases, resulting in a greater probability per turn of escape. We have used Eq. (\[fluc\]) to estimate the average number of turns at which the helix escapes from the rod or intersects it. The simulation data in Fig. 4 are in excellent agreement with the predicted asymptotic forms $N_e,N_i\sim p_0^{-1}$, $N_i\sim p_0^{-2/3}$ for large and small $p_0$, respectively. It would be interesting to include an attractive interaction between the rod and the polymer wound around it. In the fibril formation mentioned in the Introduction, the attraction is an essential ingredient. AL and TWB thank Gerhard Gompper and coworkers for hospitality at the Forschungszentrum Jülich. GG acknowledges the hospitality of the Aspen Center for Physics during the final stages of this work. Simple theory of escape of a polymer ==================================== Let us define $P^{ne}_N$ as the probability that the polymer has neither intersected the cylindrical surface of the rod nor escaped in the first $N$ turns of the helix and $P^e_N$ as the probability that it has not yet intersected the cylindrical surface but that that it has escaped. Treating each turn of the helix as statistically independent, we denote the probability that a polymer which has not yet escaped does escape in the next turn by q, and the probability that it neither escapes nor intersects the rod in the next turn by p. (The third possibility, that it intersects the rod in the next turn, has probability 1-q-p.) In addition we assume that once the polymer escapes, it never intersects the rod. These assumptions imply the recurrence relations $$\begin{aligned} &&P^{ne}_{N+1}=pP^{ne}_N\;,\label{recurr1}\\ &&P^e_{N+1}=qP^{ne}_N+P^e_N\;,\label{recurr2}\end{aligned}$$ with initial conditions $P^{ne}_0=1$, $P^e_0=0$. Writing down the first few iterates, it is easy to see that $$\begin{aligned} &&P^{ne}_N=p^N\;,\label{Pne}\\ &&P^e_N=q\thinspace \frac{1-p^N}{1-p}\;.\label{Pe}\end{aligned}$$ The probability $P_N=P^{ne}_N+P^e_N$ that the polymer has not yet intersected the rod after $N$ steps is analogous to $P(n)$ in Section V. From Eqs. (\[Pne\]) and (\[Pe\]) $$P_N=\frac{q}{1-p}+\frac{1-p-q}{1-p}\thinspace p^N\;.\label{decay}$$ Thus, as $N$ increases, $P_N$ decays exponentially from $P_0=1$ to $P_\infty=q/(1-p)$. The mean number of turns $N_e$ at which escape occurs is given by $$N_e=\frac{\sum_{N=1}^\infty N[P^e_N-P^e_{N-1}]}{\sum_{N=1}^\infty [P^e_N-P^e_{N-1}]}=\frac{1}{1-p}\;.$$ An analogous calculation for the mean number of turns $N_i$ at which the polymer insects the rod yields $N_i=N_e$. This theory is obviously an oversimplification, but the form (\[decay\]) of the decay, $P_N=A+Be^{-C N}$, is qualitatively consistent with Fig. 3. D. S. Chung, G. B. Benedek, F. M. Konikoff, and J. M. Donovan, Proc. Natl. Acad. Sci. USA [90]{}, 11341 (1993). Y. V. Zastavker, N. Asherie, A. Lomakin, J. Pande, J. M. Donovan, J. M. Schnur, and G. B. Benedek, Proc. Natl. Acad. Sci. USA [96]{}, 7883 (1999). P. Yager and P. E. Schoen, Mol. Cryst. Liq. Cryst. [106]{}, 371 (1984). J. M. Schnur, Science [262]{}, 1669 (1993). B. N. Thomas, C. M. Lindemann, and N. A. Clark, Phys. Rev. E [59]{}, 3040 (1999). S. Pakhomov, R. P. Hammer, B. K. Mishra, and B. N. Thomas, Proc. Natl. Acad. Sci. USA [100]{}, 3040 (2003). S. G. Zhang, T. Holmes, C. Lockshin, and A. Rich, Proc. Natl. Acad. Sci. USA [90]{}, 3334 (1993). A. Aggeli, M. Bell, N. Boden, J. N. Keen, P. F. Kowles, T. C. B. McLeish, M. Pitkeathly, and S. E. Radford, Nature [386]{}, 259 (1997). I. A. Nyrkova, A. N. Semenov, A. Aggeli, and N. Boden, Eur. Phys. J. B [17]{}, 481 (2000). I. A. Nyrkova, A. N. Semenov, A. Aggeli, M. Bell, N. Boden, and T. C. B McLeish, Eur. Phys. J. B [17]{}, 499 (2000). R. Oda, I. Huc, M. Schmutz, S. J. Candau, and F. C. MacKintosh, Nature [399]{}, 566 (1999). C.-F. Chou, R. H. Austin, O. Bakajin, J. O. Tegenfeldt, J. A. Castelino, S. S. Chan, E. C. Cox, H. Craighead, N. Darnton, T. Duke, J. Han, S. Turner, Electrophoresis [21]{}, 81 (2000). N. Giordano and J.-T. Cheng, J. Phys.: Condens. Matter [13]{}, R271 (2001) R. M. Jendrejack, E. T. Dimalanta, D. C. Schwartz, M. D. Graham, and J. J. [de Pablo]{}, Phys. Rev. Lett. [91]{}, 038102 (2003). P.-G. de Gennes, [Scaling Concepts in Polymer Physics]{} (Cornell University Press, Ithaca, 1979). T. W. Burkhardt and I. Guim, Phys. Rev. E [59]{}, 5833 (1999). P. Sotta, J. Chem. Phys. [112]{}, 1565 (2000). T. Odijk, Macromolecules [16]{}, 1340 (1983); [ibid.]{} [19]{}, 2313 (1986). M. Dijkstra, D. Frenkel, and H. N. W. Lekkerkerker, Physica A [19]{}, 374 (1993). T. W. Burkhardt, J. Phys. A [30]{}, L167 (1997). D. J. Bicout and T. W. Burkhardt, J. Phys. A [34]{}, 5745 (2001). S. Panyukov and Y. Rabin, Phys. Rev. Lett. [85]{}, 2404 (2000). S. Panyukov and Y. Rabin, Phys. Rev. E [62]{}, 7135 (2000). Y. Kats, D. A. Kessler, and Y. Rabin, Phys. Rev. E [65]{}, 020801(R) (2002). In the absence of fluctuations Eq. (\[frenet1\]) implies $d^2{\bf t}_3/d s^2={\boldsymbol\omega_0}\times\left({\boldsymbol\omega_0}\times{\bf t}_3\right)=-\omega_0^2{\bf t}_3+{\boldsymbol\omega_0}\omega_{03}$, which has the solution ${\bf t}_3(s)={\bf A}\cos(\omega_0s)+{\bf B}\sin(\omega_0s)+{\bf e}(0)\omega_{03}/\omega_0$. Choosing the integration constants ${\bf A}$, ${\bf B}$ to fit the initial conditions and integrating $d{\bf r}={\bf t}_3 ds$ yields Eq. (\[nofluc1\]). The cylindrical boundary is assumed to be an hard wall. In all but tangential intersections of the polymer with the wall, ${\bf t}_3=d{\bf r}/ds$ must change discontinuously, to avoid penetration. However, in the model of Section II the discontinuity costs an infinite bending energy and is also forbidden. In the simulations a polymer configuration that intersects the boundary at step $n$ of the algorithm should not be continued to larger $n$, since both penetration and reflection are forbidden. Tangential intersections with continuous ${\bf t}_1,{\bf t}_2,{\bf t}_3$ are allowed, but they represent a negligible fraction of all intersections. Equation (\[fluc\]) is obtained by writing ${\bf r}_\perp(s)=[{\bf I}- {\bf e}(0){\bf e}(0)]\cdot\int_0^s {\bf e}(s')\xi ds'$, where ${\bf I}$ is the unit dyadic, calculating the second moment $\langle r_\perp^2(s)\rangle$ with the help of Eq. (\[expdecay\]), and making use of $P=\xi L_p$, $\ell=\xi s$, as discussed below Eq. (\[freeenergy2\]). ![Probability $P(n)$ that a helical polymer with radius $r_0=0.3$, pitch $p_0=0.3$, and persistence length $L_p=8000$ in a cylindrical pore does not intersect the pore wall in the first $n$ steps of the algorithm. The full circles ($\bullet$) correspond to a pore with a circular cross section with diameter $D=1$ and the triangles ($\triangle$) to a square cross section with edge length $D=1$. Fitting the data to Eq. (\[expdecay2\]) for large $n$ yields $\lambda_0=6.57\times 10^{-6}$ and $6.01\times 10^{-6}$, respectively. The full line corresponds to the exact exponential decay $e^{-\lambda_0 n}$.[]{data-label="fig1"}](figure1.eps){width="70.00000%"} ![The effective diameter $D_{\rm eff}$ as a function of the pitch $p_0$ for a helical polymer with radius $r_0=0.3$ and persistence length $L_p=8000$ in a cylindrical pore with a circular cross section with diameter $D=1$. The data interpolate between the limiting values $D-2r_0=0.4$ and $D=1$ for small and large $p_0$, respectively.[]{data-label="fig2"}](figure2.eps){width="70.00000%"} ![Probability $P(n)$ that a helical polymer with radius $r_0=0.3$ and persistence length $L_p=8000$, wound at the fixed end around a cylindrical rod of diameter $D=0.2$ does not intersect the rod in the first $n$ steps of the algorithm. The pitch of the helix is $p_0=10$ ($\bullet$), 30 ($\triangle$), and 100 ($\circ$).[]{data-label="fig3"}](figure3.eps){width="70.00000%"} ![Average numbers of turns $N_e^1$ ($\triangle$), $N_e^2$ ($\bullet$) at which the helix escapes from the rod, and the average number of turns $N_i$ ($\circ$) at which the helix intersects the rod as a function of $p_0$. Here $N_e^1$ is based on all the configurations which escape, independent of whether they return to intersect the rod or not; $N_e^2$ is based on the configurations which escape and in $5\times 10^6$ steps of the algorithm do not return to intersect the rod. The dashed lines on the left and right have slopes -2/3 and -1, respectively, and the solid line shows the prediction (\[Ne\]). []{data-label="fig4"}](figure4.eps){width="70.00000%"} ![Fractions $f_e^1$ ($\triangle$), $f_e^2$ ($\bullet$), and $f_i$ ($\circ$) of the 10,000 configurations which contribute to $N_e^1$, $N_e^2$, and $N_i$ in Fig. 4, as a function of $p_0$.[]{data-label="fig5"}](figure5.eps){width="70.00000%"}
--- abstract: 'In this paper we characterize the strong reflecting property for $L$-cardinals for all $\omega_n$, characterize Harrington’s Principle $HP(L)$ and its generalization and discuss the relationship between the strong reflecting property for $L$-cardinals and Harrington’s Principle $HP(L)$.' address: 'Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany' author: - Yong Cheng title: 'The strong reflecting property and Harrington’s Principle' --- Introduction and preliminaries ============================== The notion of the strong reflecting property for $L$-cardinals is introduced in [@YongCheng1 Definition 2.8]. The motivation of introducing this notion is to force a set model of Harrington’s Principle, $HP(L)$ for short (cf. Definition \[def of HP\]), over higher order arithmetic (cf. Definition \[def of higher order ar\]). However the proof of The Main Theorem in [@YongCheng1] uses very little knowledge about the strong reflecting property for $L$-cardinals. In this paper, in Section 2 we develop the full theory of the strong reflecting property for $L$-cardinals and characterize $SRP^{L}(\omega_n)$ for $n\in\omega$ (see Proposition \[srp of omega first\], Proposition \[second strong reflecting property\], Theorem \[strength of omega two\] and Theorem \[characterization of srp above third level\]). We also generalize some results on $SRP^{L}(\gamma)$ to $SRP^{M}(\gamma)$ for other inner models $M$ (see Theorem \[characterizaton of zero sharp thm\] and Theorem \[big thm about u\]). In Section 3, we define the generalized Harrington’s Principle $HP(M)$ for any inner model $M$, give characterizations of $HP(M)$ for some well known inner models (see Theorem \[cited theorem from Woodin\] and \[HP for measurable cn\]) and show that, in some cases, this generalized principle fails (see Corollary \[Theorem for core model\] and Theorem \[HP for HOD\]). In Section 4, we discuss the relationship between the strong reflecting property for $L$-cardinals and Harrington’s Principle $HP(L)$. Our definitions and notations are standard. We refer to textbooks such as [@Jech], [@Higherinfinite] and [@Kunen1] for the definitions and notations we use. For the definition of admissible set and admissible ordinal, see [@Constructiblity]. For notions of large cardinals, see [@Higherinfinite]. Our notations about forcing are standard (cf. [@Jech] and [@JamesCummings]). For the theory of $0^{\sharp}$ see [@Constructiblity] and [@Jech]. Recall that $0^{\sharp}$ is the unique well founded remarkable $E.M.$ set, and $0^{\sharp}$ exists if and only if for some uncountable limit ordinal $\lambda, L_{\lambda}$ has an uncountable set of indiscernibles (cf. [@Constructiblity] and [@Jech]). For the theory of $0^{\dag}$ see [@Higherinfinite]. \[def of higher order ar\] ([@YongCheng1]) (i) $Z_{2}= ZFC^{-} +$ Any set is Countable.[^1] (ii) $Z_{3}= ZFC^{-} + \mathcal{P}(\omega)$ exists + Any set is of cardinality $\leq \beth_1$. (iii) $Z_4=ZFC^{-}+ \mathcal{P}(\mathcal{P}(\omega))$ exists + Any set is of cardinality $\leq \beth_2$. $Z_2, Z_3$ and $Z_4$ are the corresponding axiomatic systems for Second Order Arithmetic (SOA), Third Order Arithmetic and Fourth Order Arithmetic. Throughout this paper whenever we write $X\prec H_{\kappa}$ and $\gamma\in X$, $\bar{\gamma}$ always denotes the image of $\gamma$ under the transitive collapse of $X$. If $U$ is an ultrafilter on $\kappa$, we say that $U$ is countably complete if and only if whenever $Y\subseteq U$ is countable, we have that $\bigcap Y\neq\emptyset$. The distinction between $V$-cardinals and $L$-cardinals is present throughout the article. Whenever we write $\omega_n$ (for some $n$) without a superscript it is understood that we mean the $\omega_n$ of $V$. In this paper, $\kappa$-model is a model in the form $L[U]$ such that $\langle L[U], \in, U\rangle\models U$ is a normal ultrafilter over $\kappa$. Characterizations of the strong reflecting property for $L$-cardinals ===================================================================== In this section we develop the full theory of the strong reflecting property for $L$-cardinals and characterize $SRP^{L}(\omega_n)$ for $n\in\omega$. We also generalize some results on $SRP^{L}(\gamma)$ to $SRP^{M}(\gamma)$ for any inner model $M$. Recall that an inner model $M$ is $L$-like if $M$ is in the form $\langle L[\vec{E}], \in, \vec{E} \rangle$ where $\vec{E}$ is a coherent sequence of extenders; moreover, for an $L$-like inner model $M, M\|\theta$ is of the form $\langle J_{\theta}^{\vec{E}}, \in,\vec{E}\upharpoonright \theta, \varnothing\rangle$.[^2] \[convention\] Throughout, whenever we consider an inner model $M$ we assume that $M$ is $L$-like and has the property that $M\|\theta$ is definable in $H_{\theta}$ for any regular cardinal $\theta>\omega_2$.[^3] \[definition of strong reflecting property and weakly\] Let $\gamma\geq\omega_1$ be an $L$-cardinal. (i) $\gamma$ has the strong reflecting property for $L$-cardinals, denoted $SRP^{L}(\gamma)$, if and only if for some regular cardinal $\kappa> \gamma$, if $X\prec H_{\kappa}, \|X\|=\omega$ and $\gamma\in X$, then $\bar{\gamma}$ is an $L$-cardinal. (ii) $\gamma$ has the weak reflecting property for $L$-cardinals, denoted $WRP^{L}(\gamma)$, if and only if for some regular cardinal $\kappa> \gamma$, there is $X\prec H_{\kappa}$ such that $\|X\|=\omega, \gamma\in X$ and $\bar{\gamma}$ is an $L$-cardinal. \[reflecting\] Suppose $\gamma\geq\omega_1$ is an $L$-cardinal. Then the following are equivalent: (1) $SRP^{L}(\gamma)$. (2) For any regular cardinal $\kappa> \gamma$, if $X\prec H_{\kappa}, \|X\|=\omega$ and $\gamma\in X$, then $\bar{\gamma}$ is an $L$-cardinal. (3) For some regular cardinal $\kappa> \gamma, \{X\mid X\prec H_{\kappa}, \|X\|=\omega, \gamma\in X$ and $\bar{\gamma}$ is an $L$-cardinal} contains a club. (4) There exists $F:\gamma^{<\omega}\rightarrow\gamma$ such that if $X\subseteq\gamma$ is countable and closed under $F$,[^4] then $o.t.(X)$ is an $L$-cardinal. (5) For any regular cardinal $\kappa> \gamma, \{X\mid X\prec H_{\kappa}, \|X\|=\omega, \gamma\in X$ and $\bar{\gamma}$ is an $L$-cardinal} contains a club. Note that $(2)\Rightarrow (1), (1)\Rightarrow (3), (2)\Rightarrow (5)$ and $(5)\Rightarrow (3)$. It suffices to show that $(4)\Rightarrow (2)$ and $(3)\Rightarrow (4)$. For the proof see [@YongCheng1 Proposition 2.7]. Suppose $\gamma\geq\omega_1$ is an $L$-cardinal. Let $(1)^{\ast}, (2)^{\ast}, (3)^{\ast},(4)^{\ast}$ and $(5)^{\ast}$ respectively be the statements which replace “is an $L$-cardinal" with “is not an $L$-cardinal" in Definition \[definition of strong reflecting property and weakly\](i) and statements $(2), (3),(4)$ and $(5)$ in Proposition \[reflecting\]. The following corollary is an observation from the proof of Proposition \[reflecting\]. \[corollary about strong reflecting property\] $(1)^{\ast}\Leftrightarrow (2)^{\ast}\Leftrightarrow (3)^{\ast}\Leftrightarrow (4)^{\ast}\Leftrightarrow (5)^{\ast}$. \[equivalent forms of strong reflecting property\] Suppose $\gamma\geq\omega_1$ is an $L$-cardinal, $\kappa$ is regular and $\|\gamma\|=\kappa$. Then the following are equivalent: (a) $SRP^{L}(\gamma)$. (b) For any bijection $\pi: \kappa\rightarrow \gamma$, there exists a club $D\subseteq\kappa$ such that for any $\theta\in D$, $o.t.(\{\pi(\alpha)\mid\alpha< \theta\})$ is an $L$-cardinal. (c) For some bijection $\pi: \kappa\rightarrow \gamma$, there exists a club $D\subseteq\kappa$ such that for any $\theta\in D$, $o.t.(\{\pi(\alpha)\mid\alpha<\theta\})$ is an $L$-cardinal. The proof is essentially the same as the case $\kappa=\omega_1$ in [@YongCheng1 Proposition 2.9]. Let $(6)^{\ast}$ and $(7)^{\ast}$ respectively be the statement which replaces “is an $L$-cardinal" with “is not an $L$-cardinal" in Proposition \[equivalent forms of strong reflecting property\](b) and Proposition \[equivalent forms of strong reflecting property\](c). The following corollary is an observation from the proof of Proposition \[equivalent forms of strong reflecting property\]. \[corollary about weakly reflecting property\] Suppose $\gamma\geq\omega_1$ is an $L$-cardinal, $\kappa$ is regular and $\|\gamma\|=\kappa$. Then $(1)^{\ast}\Leftrightarrow (6)^{\ast}\Leftrightarrow (7)^{\ast}$. \[weakly reflecting\] Suppose $\gamma\geq\omega_1$ is an $L$-cardinal. Then the following are equivalent: (a) $WRP^{L}(\gamma)$. (b) For any regular cardinal $\kappa> \gamma$, there is $X\prec H_{\kappa}$ such that $\|X\|=\omega, \gamma\in X$ and $\bar{\gamma}$ is an $L$-cardinal. (c) For some regular cardinal $\kappa> \gamma, \{X\mid X\prec H_{\kappa}, \|X\|=\omega, \gamma\in X$ and $\bar{\gamma}$ is an $L$-cardinal} is stationary. (d) For any $F: \gamma^{<\omega}\rightarrow\gamma$, there exists $X\subseteq\gamma$ such that $X$ is countable, closed under $F$ and $o.t.(X)$ is an $L$-cardinal. (e) For any regular cardinal $\kappa> \gamma, \{X\mid X\prec H_{\kappa}, \|X\|=\omega, \gamma\in X$ and $\bar{\gamma}$ is an $L$-cardinal} is stationary. Note that $(e)\Rightarrow (c)$ and $(c)\Rightarrow (a)$. It suffices to show that $(a)\Rightarrow (d), (d)\Rightarrow (b)$ and $(b)\Rightarrow (e)$. $(a)\Rightarrow (d)$ follows from $(4)^{\ast}\Leftrightarrow (2)^{\ast}$ in Corollary \[corollary about strong reflecting property\]. $(d)\Rightarrow (b)$ follows from $(1)^{\ast}\Leftrightarrow (4)^{\ast}$ in Corollary \[corollary about strong reflecting property\]. $(b)\Rightarrow (e)$ follows from $(3)^{\ast}\Leftrightarrow (1)^{\ast}$ in Corollary \[corollary about strong reflecting property\]. Suppose $\gamma\geq\omega_1$ is an $L$-cardinal, $\kappa$ is regular and $\|\gamma\|=\kappa$. Then the following are equivalent: (1) $WRP^{L}(\gamma)$. (2) For some bijection $\pi: \kappa\rightarrow \gamma$, there exists a stationary $D\subseteq\kappa$ such that for any $\theta\in D$, $o.t.(\{\pi(\alpha)\mid\alpha<\theta\})$ is an $L$-cardinal. (3) For any bijection $\pi: \kappa\rightarrow \gamma$, there exists a stationary $D\subseteq\kappa$ such that for any $\theta\in D$, $o.t.(\{\pi(\alpha)\mid\alpha< \theta\})$ is an $L$-cardinal. Follows from Corollary \[corollary about weakly reflecting property\] and $(1)^{\ast}\Leftrightarrow (2)^{\ast}$ in Corollary \[corollary about strong reflecting property\]. The proof is standard and we omit the details. \[srp of omega first\] The following are equivalent: (1) $\omega_1$ is a limit cardinal in $L$. (2) $WRP^{L}(\omega_1)$. (3) $SRP^{L}(\omega_1)$. It suffices to show that $(1)\Rightarrow (3)$ and $(2)\Rightarrow (1)$ since $(3)\Rightarrow (2)$ is immediate. $(1)\Rightarrow (3)$ Suppose $\omega_1$ is a limit cardinal in $L$. Then $\{\alpha<\omega_1: \alpha$ is an $L$-cardinal} is a club. By Proposition \[equivalent forms of strong reflecting property\], $SRP^{L}(\omega_1)$ holds. $(2)\Rightarrow (1)$ Suppose $WRP^{L}(\omega_1)$ holds. Then $\{X\cap \omega_1\| X\prec H_{\omega_2}\wedge \|X\|=\omega\wedge o.t.(X\cap\omega_1)$ is an $L$-cardinal} is stationary in $\omega_1$. It is easy to see that for any $\alpha<\omega_1$ there is $\alpha<\beta<\omega_1$ such that $\beta$ is an $L$-cardinal. \[transive collapse of L cardinal\] Suppose $\gamma\geq\omega_1$ is an $L$-cardinal, $\kappa>\gamma$ is a regular cardinal and $SRP^{L}(\gamma)$ holds. If $Z\prec H_{\kappa}$, $\|Z\|\leq \omega_1$ and $\gamma\in Z$, then $\bar{\gamma}$ is an $L$-cardinal. Suppose $\bar{\gamma}$ is not an $L$-cardinal. Let $M$ be the transitive collapse of $Z$ and $\pi: M\prec H_{\kappa}$ be the inverse of the collapsing map. Take $Y\prec H_{\kappa}$ such that $\|Y\|=\omega$ and $M, \bar{\gamma} \in Y$. Note that $Y\models ``\bar{\gamma}$ is not an $L$-cardinal". Hence $\bar{\bar{\gamma}}$ is not an $L$-cardinal.[^5] Let $X=\pi``(Y\cap M)$. Since $\bar{\gamma}\in Y\cap M$ and $\pi(\bar{\gamma})=\gamma$, $\gamma\in X$. Note that $X\prec Z\prec H_{\kappa}$ and the image of $\gamma$ under the transitive collapse of $X$ is $\bar{\bar{\gamma}}$. By $SRP^{L}(\gamma)$, $\bar{\bar{\gamma}}$ is an $L$-cardinal. Contradiction. \[compare strong cardinal\] Suppose $\omega_1\leq\gamma_0<\gamma_1$ are $L$-cardinals. Then $SRP^{L}(\gamma_1)$ implies $SRP^{L}(\gamma_0)$ (respectively $WRP^{L}(\gamma_1)$ implies $WRP^{L}(\gamma_0)$). We only show the strong reflecting property case (the argument for the weak reflecting property case is similar). Let $\kappa>\gamma_1$ be a regular cardinal. It suffices to show if $X\prec H_{\kappa}, \|X\|=\omega$ and $\{\gamma_0, \gamma_1\}\subseteq X$, then $\bar{\gamma_0}$ is an $L$-cardinal. Note that $L_{\gamma_1}\models \gamma_0$ is a cardinal. Since $\gamma_1 \in X, L_{\gamma_1}\in X$. Since $\bar{L_{\gamma_1}}=L_{\bar{\gamma_1}}$ and $\bar{L_{\gamma_1}}\models \bar{\gamma_0}$ is a cardinal, $L_{\bar{\gamma_1}}\models \bar{\gamma_0}$ is a cardinal. By $SRP^{L}(\gamma_1)$, $\bar{\gamma_1}$ is an $L$-cardinal and hence $\bar{\gamma_0}$ is an $L$-cardinal. \[second strong reflecting property\] The following are equivalent: (1) $SRP^{L}(\omega_2)$. (2) $\omega_2$ is a limit cardinal in $L$ and for any $L$-cardinal $\omega_1\leq\gamma<\omega_2$, $SRP^{L}(\gamma)$ holds. (3) $\{\alpha<\omega_2\mid\alpha$ is an $L$-cardinal and $SRP^{L}(\alpha)$ holds} is unbounded in $\omega_2$. $(1)\Rightarrow (2)$ By Proposition \[compare strong cardinal\], it suffices to show $\omega_2$ is a limit cardinal in $L$. Let $\kappa>\omega_2$ be the regular cardinal that witnesses $SRP^{L}(\omega_2)$. Fix $\alpha<\omega_2$. Pick $Z\prec H_{\kappa}$ such that $\|Z\|=\omega_1$, $\alpha\subseteq Z$ and $\omega_2\in Z$. By Proposition \[transive collapse of L cardinal\], $\bar{\omega_2}$ is an $L$-cardinal. Note that $\alpha\leq \bar{\omega_2}<\omega_2$. $(2)\Rightarrow (1)$ Suppose $\kappa>\omega_2$ is a regular cardinal, $X\prec H_{\kappa}, \|X\|=\omega$ and $\omega_2\in X$. We show that $\bar{\omega_2}$ is an $L$-cardinal. Note that $\bar{\omega_2}=o.t.(X\cap \omega_2)$. Let $E=\{\gamma\mid \omega_1\leq\gamma<\omega_2 \wedge\gamma$ is an $L$-cardinal}. $E$ is definable in $H_{\kappa}$. Since $\omega_2$ is a limit cardinal in $L$, $E$ is cofinal in $\omega_2$ and hence $E\cap X$ is cofinal in $\omega_2\cap X$. For $\gamma\in E\cap X, \bar{\gamma}=o.t.(X\cap\gamma)$ and by $SRP^{L}(\gamma)$, $\bar{\gamma}$ is an $L$-cardinal. Note that $\bar{\omega_2}=sup(\{\bar{\gamma}\mid \gamma\in E\cap X\})$. Hence $\bar{\omega_2}$ is an $L$-cardinal. $(1)\Leftrightarrow (3)$ Follows from $(1)\Leftrightarrow (2)$ and Proposition \[compare strong cardinal\]. The notion of remarkable cardinal is introduced by Ralf Schindler in [@Schindler2]. Any remarkable cardinal is remarkable in $L$ (cf.[@Schindler2 Lemma 1.7]). ([@Schindler2]) (1) Let $\kappa$ be a cardinal, $G$ be $Col(\omega, <\kappa)$-generic over $V$, $\theta>\kappa$ be a regular cardinal and $X\in [H_{\theta}^{V[G]}]^{\omega}$. We say that $X$ condenses remarkably if $X=ran(\pi)$ for some elementary $\pi: (H_{\beta}^{V[G\cap H_{\alpha}^{V}]}, \in, H_{\beta}^{V}, G\cap H_{\alpha}^{V})\rightarrow (H_{\theta}^{V[G]}, \in, H_{\theta}^{V}, G)$ where $\alpha=crit(\pi)<\beta<\kappa$ and $\beta$ is a cardinal in $V$. (2) For regular cardinal $\theta>\kappa, \kappa$ is $\theta$-remarkable if and only if in $V^{Col(\omega, <\kappa)}, \{X\in [H_{\theta}]^{\omega}: X$ condenses remarkably} is stationary. We say that $\kappa$ is remarkable if $\kappa$ is $\theta$-remarkable for all regular cardinal $\theta>\kappa$. \[key lemma on relationship between remark and wfp\] ([@YongCheng1 Lemma 2.3])Suppose $\kappa$ is an $L$-cardinal. The following are equivalent: (1) $\kappa$ is remarkable in $L$; (2) If $\gamma\geq\kappa$ is an $L$-cardinal, $\theta>\gamma$ is a regular cardinal in $L$, then $\Vdash^{L}_{Col(\omega, <\kappa)} ``\{X\| X\prec L_{\check{\theta}}[\dot{G}], \|X\|=\omega$ and $o.t.(X\cap\check{\gamma})$ is an $L$-cardinal} is stationary". \[key coro of above lemma\] If $\kappa$ is remarkable in $L$ and $G$ is $Col(\omega, <\kappa)$-generic over $L$, then $L[G]\models WRP^{L}(\gamma)$ holds for any $L$-cardinal $\gamma\geq\kappa$. Follows from Lemma \[key lemma on relationship between remark and wfp\]. Fix some $L$-cardinal $\gamma\geq\omega_1$. $SRP^{L}(\gamma)$ is upward absolute (cf. [@YongCheng1 Proposition 2.11]).[^6] As a corollary, $WRP^{L}(\gamma)$ is downward absolute.[^7] So if $WRP^{L}(\gamma)$ holds, then $WRP^{L}(\gamma)$ holds in $L$. The converse is not true in general. \[new non propro\] Suppose $WRP^{L}(\kappa)$ holds where $\kappa\geq\omega_1$ is an $L$-cardinal. Then $L\models\omega_1$ is $\kappa^{+}$-remarkable and for any regular $\theta>\kappa$ in $L$, $L\models\omega_1$ is $\theta$-remarkable. $L\models WRP^{L}(\kappa)$ $\{X\| X\prec L_{\kappa^{+}}, \|X\|=\omega$ and $o.t.(X\cap\kappa)$ is an $L$-cardinal} is stationary in $L$ for any $L$-regular cardinal $\theta>\kappa, \{X\| X\prec L_{\theta}, \|X\|=\omega$ and $o.t.(X\cap\kappa)$ is an $L$-cardinal} is stationary in $L$. For $L$-regular cardinal $\theta>\kappa$, $L\models\omega_1$ is $\theta$-remarkable for any $G$ which is $Col(\omega, <\omega_1)$-generic over $L$, $L[G]\models \{X\in [L_{\theta}]^{\omega}\| X=ran(\pi), \pi: (L_{\beta}[G\upharpoonright\alpha],\in,L_{\beta},G\upharpoonright\alpha)\prec (L_{\theta}[G],\in, L_{\theta},G)$ where $\alpha=crit(\pi)<\beta<\omega_1$ and $\beta$ is an $L$-cardinal} is stationary. Note that $L\models WRP^{L}(\kappa)$ and $Col(\omega, <\omega_1)$ is stationary preserving. “For any $L$-cardinal $\gamma\geq\omega_1, WRP^{L}(\gamma)$ holds" is equiconsistent with $\omega_1$ is remarkable. Follows from Corollary \[key coro of above lemma\] and Proposition \[new non propro\]. \[strength of omega two\] (Set forcing)The following two theories are equiconsistent: (1) $SRP^{L}(\omega_2)$. (2) $ZFC\, +$ there exists a remarkable cardinal with a weakly inaccessible cardinal above it. We first show that the consistency of (2) implies the consistency of (1). Let $S=\{\omega_1\leq\alpha<\omega_2\mid \alpha$ is an $L$-cardinal}. Note that $SRP^{L}(\omega_2)$ is equivalent to $S$ being a club such that $SRP^{L}(\alpha)$ holds for any $\alpha\in S$. In [@YongCheng1 Section 3.1], assuming there exists a remarkable cardinal with a weakly inaccessible cardinal above it, we force a model $L[G,H]$ in which $S$ is a club and $SRP^{L}(\alpha)$ holds for any $\alpha\in S$. So $SRP^{L}(\omega_2)$ holds in $L[G,H]$. From [@YongCheng1 Section 3.2-3.4], if $S$ is a club and $SRP^{L}(\alpha)$ holds for any $\alpha\in S$, then we can force a model of $Z_3\, +\, HP(L)$ . So the consistency of (1) implies the consistency of $Z_3 + HP(L)$. By [@YongCheng2 Theorem 3.2], $Z_3\, +\, HP(L)$ implies $L\models ZFC\, +\, \omega_1^{V}$ is remarkable. By Proposition \[second strong reflecting property\], $\omega_2^{V}$ is inaccessible in $L$. So the consistency of (1) implies the consistency of (2). \[general definition of strong reflecting property and weakly\] Suppose $M$ is an inner model and $\gamma\geq\omega_1$ is an $M$-cardinal. We say that $\gamma$ has the strong reflecting property for $M$-cardinals, denoted $SRP^{M}(\gamma)$, if and only if for some regular cardinal $\kappa> \gamma$, if $X\prec H_{\kappa}, \|X\|=\omega$ and $\gamma\in X$, then $\bar{\gamma}$ is an $M$-cardinal. \[covering thm for mc\] Suppose $M$ is an inner model. We say that $M$ has the full covering property if for any set $X$ of ordinals, there is $Y\in M$ such that $X\subseteq Y$ and $\|Y\|=\|X\|+\omega_1$. We say that $M$ has the rigidity property if there is no nontrivial elementary embedding from $M$ to $M$. \[characterizaton of zero sharp thm\] Suppose $M$ is an inner model which satisfies Convention \[convention\] and has both the full covering and the rigidity property. Then, for every M-cardinal $\gamma>\omega_2, SRP^{M}(\gamma)$ fails. Suppose $SRP^{M}(\gamma)$ holds for some $\gamma>\omega_2$. Let $\kappa>\gamma$ be the witnessing regular cardinal for $SRP^{M}(\gamma)$. Build an elementary chain $\langle Z_{\alpha}\mid\alpha<\omega_1\rangle$ of submodels of $H_{\kappa}$ such that for all $\alpha<\beta<\omega_1$, $Z_{\alpha}\prec Z_{\beta}\prec H_{\kappa}$, $Z_{\alpha}\in Z_{\beta}$ , $\|Z_{\alpha}\|=\omega$ and $\{\gamma, \omega_2\} \subseteq Z_{0}$. Let $Z=\bigcup_{\alpha<\omega_1} Z_{\alpha}$. Then $\|Z\|=\omega_1$ and $Z\prec H_{\kappa}$. Let $\pi: N\cong Z\prec H_{\kappa}$ and $\pi_{\alpha}: N_{\alpha}\cong Z_{\alpha}\prec H_{\kappa}$ be the inverses of the collapsing maps. Let $j_{\alpha}: N_{\alpha}\prec N$ be the induced elementary embedding. Since $\omega_1\subseteq Z$, $crit(\pi)>\bar{\omega_1}$. Since $\omega_2\in Z$ and $\|Z\|=\omega_1, crit(\pi)\leq \bar{\omega_2}$. So $crit(\pi)=\bar{\omega_2}$. Note that Proposition \[transive collapse of L cardinal\] still holds if we replace $L$ with $M$. By $SRP^{M}(\gamma)$, $\bar{\gamma}$ is an $M$-cardinal. Since $M\|\bar{\gamma}$ is definable in $H_{\kappa}$, $\mathcal{P}(\bar{\omega_2})\cap M\subseteq M\|\bar{\gamma}\in N$ and $\mathcal{P}(\bar{\omega_2})\cap M\in N$. Define $U=\{X\subseteq \bar{\omega_2}\mid X\in M\wedge\bar{\omega_2}\in \pi(X)\}$. $U$ is an $M$-ultrafilter. For $\alpha<\omega_1$, the image of $Z_{\alpha}$ under the transitive collapse of $Z$ is $j_{\alpha}``N_{\alpha}$ and $j_{\alpha}``N_{\alpha}\in N$. \[a key lemma on U\] $U$ is countably complete. Suppose $Y\subseteq U$ and $Y$ is countable. We show that $\bigcap Y\neq\emptyset$. Since $Y\subseteq N$, take $\alpha<\omega_1$ large enough such that $Y\subseteq j_{\alpha}``N_{\alpha}$. Let $S=\mathcal{P}(\bar{\omega_2})\cap M\cap j_{\alpha}``N_{\alpha}$. Note that $S\in N$ and $N\models S$ is countable. Note that $H_{\kappa}\models ``M$ has the full covering property"[^8] and hence $N\models M$ has the full covering property. Fix $T\in N$ such that $T\subseteq \mathcal{P}(\bar{\omega_2})\cap M, T\supseteq S, T\in M$ and $N\models \|T\|=\omega_1$. Since $\bar{\omega_2}=crit(\pi)>\omega_1, \pi(T)=\pi``T$. Since $T\in N, \mathcal{P}(T)\cap M\in N$. $U\cap T\in N$. Since $\pi(T)=\pi``T\in M, \pi``(U\cap T)=\{\pi(A)\mid A\in T\wedge \bar{\omega_2}\in \pi(A)\}=\{B\in \pi(T)\mid \bar{\omega_2}\in B \}$ and $\pi``(U\cap T)\in M$. Note that $\mathcal{P}(\pi``T)\cap M=\pi``(\mathcal{P}(T)\cap M)$ since for all $D\in \mathcal{P}(T)\cap M, \pi(D)=\pi``D$. Since $\pi``(U\cap T)\in \mathcal{P}(\pi``T)\cap M$, $\pi``(U\cap T)=\pi(D)=\pi``D$ for some $D\in \mathcal{P}(T)\cap M\subseteq N$. So $U\cap T=D$ and hence $U\cap T\in N$. Note that $Y\subseteq j_{\alpha}``N_{\alpha}\cap \mathcal{P}(\bar{\omega_2})\cap M=S\subseteq T$. Since $Y\subseteq T\cap U$, to show that $\bigcap Y\neq\emptyset$, it suffices to show that $\bigcap (U\cap T)\neq\emptyset$. Note that $\bar{\omega_2}\in \bigcap\pi``(U\cap T)$ and $\pi(U\cap T)=\pi``(U\cap T)$. Then $\bigcap\pi``(U\cap T)=\bigcap\pi(U\cap T)=\pi(\bigcap (U\cap T))\neq\emptyset$. So $\bigcap (U\cap T)\neq\emptyset$. So we can build a nontrivial embedding from $M$ to $M$ which contradicts the rigidity property of $M$. \[characterization of srp above third level\] The following are equivalent: (i) $SRP^{L}(\gamma)$ holds for some $L$-cardinal $\gamma>\omega_2$. (ii) $0^{\sharp}$ exists. (iii) $SRP^{L}(\gamma)$ holds for every $L$-cardinal $\gamma\geq\omega_1$. $(i)\Rightarrow (ii)$ Assume $0^{\sharp}$ does not exist. Then $L$ satisfies all the conditions for $M$ in Theorem \[characterizaton of zero sharp thm\]. From the proof of Theorem \[characterizaton of zero sharp thm\] (replace $M$ with $L$), $SRP^{L}(\gamma)$ does not hold for any $L$ cardinal $\gamma>\omega_2$. $(ii) \Rightarrow (iii)$ Note that if $X\prec H_{\kappa}$ and $\gamma\in X$, then $\mathcal{M}(0^{\sharp}, \gamma+1)\in X$ and its image under the transitive collapse of $X$ is $\mathcal{M}(0^{\sharp}, \bar{\gamma}+1)$.[^9] Note that for $\alpha\in Ord, \mathcal{M}(0^{\sharp}, \alpha)\prec L$. So for $n\geq 3, SRP^{L}(\omega_n)$ is equivalent to $0^{\sharp}$ exists. We have characterized $SRP^{L}(\omega_n)$ for $n\geq 1$. Suppose $M$ is an inner model. For $M$-cardinal $\lambda$, let $SRP^{M}_{<\lambda}(\lambda)$ denote the statement: for some regular cardinal $\theta>\lambda$, if $X\prec H_{\theta},\|X\|<\lambda$ and $\lambda\in X$, then $\bar{\lambda}$ is an $M$-cardinal. \[covering thm for mc\] ([@CoveringLemma Theorem 1.3])Assume $0^{\dag}$ does not exist but there is an inner model with a measurable cardinal and $L[U]$ is chosen such that $\kappa=crit(U)$ is as small as possible. The one of the following holds: (a) For every set $X$ of ordinals, there is a set $Y\in L[U]$ such that $Y\supseteq X$ and $\|Y\|=\|X\| + \omega_1$; (b) There is a sequence $C\subseteq\kappa$, which is Prikry generic over $L[U]$, such that for all set $X$ of ordinals, there is a set $Y\in L[U,C]$ such that $Y\supseteq X$ and $\|Y\|=\|X\| + \omega_1$. \[fact on zero dagger\] ([@Higherinfinite 21.22 Exercise]) The following are equivalent: (1) $0^{\dag}$ exists. (2) There is a $\kappa$-model for some $\kappa$ and an elementary embedding from that model to itself with critical point greater than $\kappa$. \[big thm about u\] Suppose there is an inner model with a measurable cardinal and $L[U]$ is chosen such that $\kappa=crit(U)$ is as small as possible. Suppose $\lambda>\kappa^{+}$ is an $L[U]$-cardinal. Then $SRP^{L[U]}_{<\lambda}(\lambda)$ if and only if $0^{\dag}$ exists. $(\Rightarrow)$ We assume that $0^{\dag}$ does not exist and try to get a contradiction. By Fact \[covering thm for mc\], we need to discuss two cases. Case 1: Fact \[covering thm for mc\](a) holds. Let $\theta>\lambda$ be the witness regular cardinal for $SRP^{L[U]}_{<\lambda}(\lambda)$. Build an elementary chain $\langle Z_{\alpha}\mid\alpha<\kappa\rangle$ of submodels of $H_{\theta}$ such that for $\alpha<\beta<\kappa, Z_{\alpha}\prec Z_{\beta}\prec H_{\theta}, Z_{\alpha}\in Z_{\beta}, \|Z_{\alpha}\|=\kappa$ and $\{\kappa^{+}, \lambda\}\cup tr(\{U\})\subseteq Z_0$.[^10] Let $Z=\bigcup_{\alpha<\kappa} Z_{\alpha}$. Then $\|Z\|=\kappa$. Let $\pi: N\cong Z\prec H_{\theta}$ and $\pi_{\alpha}: N_{\alpha}\cong Z_{\alpha}\prec H_{\theta}$ be the inverses of the collapsing maps. Since $Z_{\alpha}\prec Z$, let $j_{\alpha}: N_{\alpha}\prec N$ be the induced embedding. Then $\pi_{\alpha}=\pi\circ j_{\alpha}$ and $N=\bigcup_{\alpha<\kappa} j_{\alpha}``N_{\alpha}$. Let $crit(\pi)=\eta$. Then $\eta>\kappa=\bar{\kappa}$ and since $\|Z\|=\kappa, \eta\leq \bar{\kappa^{+}}$. So $\eta=\bar{\kappa^{+}}<\bar{\lambda}$. By $SRP^{L[U]}_{<\lambda}(\lambda)$, $\bar{\lambda}$ is an $L[U]$-cardinal. Let $W=\{X\subseteq\eta\mid X\in L[U]$ and $\eta\in\pi(X)$}. Note that $U=\bar{U}\in N$ and $W\subseteq L_{\bar{\lambda}}[U]\subseteq N$. $W$ is $L[U]$-ultrafilter on $\eta$. Note that $Z\models ``\|Z_{\alpha}\|=\kappa$“ and the image of $Z_{\alpha}$ under the transitive collapse of $Z$ is $j_{\alpha}``N_{\alpha}$. So for $\alpha<\kappa, j_{\alpha}``N_{\alpha}\in N$ and $N\models ``\|j_{\alpha}``N_{\alpha}\|=\kappa$”. \[a key lemma on U\] $W$ is countably complete. Suppose $Y\subseteq W$ and $Y$ is countable. We show that $\bigcap Y\neq\emptyset$. Since $Y\subseteq N$, take $\alpha<\kappa$ large enough such that $Y\subseteq j_{\alpha}``N_{\alpha}$. Let $S=\mathcal{P}(\eta)\cap L[U]\cap j_{\alpha}``N_{\alpha}$. Note that $\mathcal{P}(\eta)\cap L[U]\in N$ and hence $S\in N$. $N\models \|S\|\leq\kappa$. Since Fact \[covering thm for mc\](a) holds in $H_{\theta}$ and $N\prec H_{\theta}$, Fact \[covering thm for mc\](a) holds in $N$. Take $T\in N$ such that $T\subseteq \mathcal{P}(\eta)\cap L[U], T\supseteq S, T\in L[U]$ and $N\models \|T\|\leq\kappa$. Since $\eta>\kappa, \pi(T)=\pi``T$. Let $\bar{T}=\{X\in T\mid \eta\in \pi(X)\}$. $\bar{T}\in N$. Since $N\models \|T\|\leq\kappa$, there is $h\in N$ such that $h: T\leftrightarrow \gamma$ for some $\gamma<\eta$. Then $\bar{T}=\{X\in T\mid \eta\in \pi``(h^{-1})(h(X))\}$. So $\bar{T}\in N$. Note that $\bigcap \bar{T}\neq\emptyset$ since $\pi(\bar{T})=\pi``\bar{T}$ and $\eta\in\bigcap \pi``\bar{T}=\bigcap \pi(\bar{T})=\pi(\bigcap \bar{T})$. Since $Y\subseteq S\subseteq T$ and $Y\subseteq W$, $Y\subseteq \bar{T}$ and hence $\bigcap Y\neq\emptyset$. So there exists a nontrivial elementary embedding $j: L[U]\prec L[U]$ with $crit(j)=\eta>\kappa$. By Fact \[fact on zero dagger\], $0^{\dag}$ exists. Contradiction. Case 2: Fact \[covering thm for mc\](b) holds. The proof is essentially the same as Case 1 with small modifications (for example, let $tr(\{U,C\})\subseteq Z_0$ and $W=\{X\subseteq\eta\mid X\in L[U,C]$ and $\eta\in\pi(X)$}). Since Priky forcing preserves all cardinals, $\bar{\lambda}$ is an $L[U,C]$-cardinal. As in Case 1, we can show that there exists a nontrivial elementary embedding $j: L[U,C]\prec L[U,C]$. Since $j(U,C)=(U,C), j\upharpoonright L[U]: L[U]\prec L[U]$. $crit(j\upharpoonright L[U])=\eta>\kappa$. So by Fact \[fact on zero dagger\], $0^{\dag}$ exists. Contradiction. $(\Leftarrow)$ Assume $0^{\dag}$ exists. Suppose $\theta>\lambda$ is regular, $X\prec H_{\theta}, \|X\|<\lambda$ and $\lambda\in X$. We show that $\bar{\lambda}$ is an $L[U]$-cardinal. Since $\lambda\in X$ and $0^{\dag}\in X, \mathcal{M}(0^{\dag},\omega, \lambda+1)\in X$.[^11] Note that for any $\alpha, \beta\in Ord, \mathcal{M}(0^{\dag},\alpha, \beta)\prec L[U]$. Since $\lambda$ is an $L[U]$-cardinal and $\lambda\in \mathcal{M}(0^{\dag},\omega, \lambda+1), \mathcal{M}(0^{\dag},\omega, \lambda+1)\models \lambda$ is a cardinal. Note that the image of $\mathcal{M}(0^{\dag},\omega, \lambda+1)$ under the transitive collapse of $X$ is $\mathcal{M}(0^{\dag}, \omega,\bar{\lambda}+1)$. So $\mathcal{M}(0^{\dag}, \omega,\bar{\lambda}+1)\models ``\bar{\lambda}$ is a cardinal". Since $\mathcal{M}(0^{\dag}, \omega,\bar{\lambda}+1)\prec L[U],\bar{\lambda}$ is an $L[U]$-cardinal. In [@Schindler3], Thoralf Räsch and Ralf Schindler introduced the condensation principle $\nabla_{\kappa}$: for any regular cardinal $\theta>\kappa, \{X\prec L_{\theta}\mid \|X\|<\kappa, X\cap\kappa\in\kappa$ and $L\models o.t.(X\cap \theta)$ is a cardinal} is stationary. The notion of the strong reflecting property for $L$-cardinals was introduced before the author knew about the work on $\nabla_{\kappa}$ in [@Schindler3]. The following theorem summarizes the strength of $\nabla_{\omega_n}$ for $n\in\omega$. \[thm about remarkble cn\] (1) ([@Schindler3 Theorem 2, 4])The following theories are equiconsistent: (a) $ZFC\,+\, \nabla_{\omega_1}$. (b) $ZFC\,+\, \nabla_{\omega_2}$. (c) $ZFC\,+$ there exists a remarkable cardinal. (2) [@Schindler3 Corollary 12]For $n\geq 3, \nabla_{\omega_n}$ is equivalent to $0^{\sharp}$ exists. Now we discuss the relationship between $SRP^{L}(\omega_n)$ and $\nabla_{\omega_n}$ for $n\in\omega$. By Theorem \[characterization of srp above third level\] and \[thm about remarkble cn\], for $n\geq 3, SRP^{L}(\omega_n)$ is equivalent to $\nabla_{\omega_n}$. If $\kappa$ is regular cardinal and $\nabla_{\kappa}$ holds, then $\kappa$ is remarkable in $L$ (cf. [@Schindler3 Lemma 7]). By Proposition \[srp of omega first\], $\nabla_{\omega_1}$ implies $SRP^{L}(\omega_1)$ which is strictly weaker. By Theorem \[strength of omega two\], $SRP^{L}(\omega_2)$ does not imply $\nabla_{\omega_2}$ since $\nabla_{\omega_2}$ implies $\omega_2$ is remarkable in $L$. By Theorem \[thm about remarkble cn\] and \[strength of omega two\], the strength of $SRP^{L}(\omega_2)$ is strictly stronger than $\nabla_{\omega_2}$. In Definition \[definition of strong reflecting property and weakly\], we only consider countable elementary submodels of $H_{\kappa}$. Similarly as $\nabla_{\kappa}$ we could also consider uncountable elementary submodels of $H_{\kappa}$. However this does not change the picture. Obviously, $SRP^{L}_{<\omega_1}(\omega_1)$ $SRP^{L}(\omega_1)$. By Proposition \[transive collapse of L cardinal\], $SRP^{L}_{<\omega_2}(\omega_2)$ $SRP^{L}(\omega_2)$. By Theorem \[characterizaton of zero sharp thm\], for $n\geq 3, SRP^{L}_{<\omega_n}(\omega_n)$ $0^{\sharp}$ exists $SRP^{L}(\omega_n)$. Harrington’s Principle $HP(L)$ and its generalization ===================================================== In this section, we define the generalized Harrington’s Principle $HP(M)$ for any inner model $M$. Considering various known examples of inner models we give particular characterizations of $HP(M)$, while we also show that in some cases this generalized principle fails. Recall that for limit ordinal $\alpha>\omega$, $\alpha$ is $x$-admissible if and only if there is no $\Sigma_1(L_{\alpha}[x])$ mapping from an ordinal $\delta<\alpha$ cofinally into $\alpha$ (see [@Constructiblity Lemma 7.2]). \[def of HP\] Suppose $M$ is an inner model. The Generalized Harrington’s Principle $HP(M)$ denotes the following statement: there is a real $x$ such that, for any ordinal $\alpha$, if $\alpha$ is $x$-admissible then $\alpha$ is an $M$-cardinal, i.e., $M\models \alpha$. $HP(L)$ denotes Harrington’s Principle. Harrington’s principle $HP(L)$ was isolated by Harrington in the proof of his celebrated theorem “$Det(\Sigma^1_1)$ implies $0^{\sharp}"$ in [@Harrington]. \[fact on zero sharp in devlin\] (Essentially [@Constructiblity])$(Z_4)$ $L_{\omega_2}$ has an uncountable set of indiscernibles if and only if $0^{\sharp}$ exists. \[cited theorem from Woodin\] $(Z_4)$ The following are equivalent:[^12] (1) $HP(L)$. (2) $L_{\omega_2}$ has an uncountable set of indiscernibles. (3) $0^{\sharp}$ exists. Note that in $Z_2, 0^{\sharp}$ implies $HP(L)$ since any $0^{\sharp}$-admissible ordinal is an $L$-cardinal. It suffices to show that $(1)\Rightarrow (2)$. Let $a$ be the witness real for $HP(L)$. We work in $L[a]$. Pick $\eta>\omega_2$ and $N$ such that $\eta$ is $a$-admissible, $N\prec L_{\eta}[a],\omega_2\in N, \|N\|=\omega_1$ and $N$ is closed under $\omega$-sequences. Let $j: L_{\theta}[a]\cong N\prec L_{\eta}[a]$ be the inverse of the collapsing map and $\kappa=crit(j)$. By $HP(L), \theta$ is an $L$-cardinal. Define $U=\{X\subseteq \kappa\mid X\in L\wedge\kappa\in j(X)\}$. Note that $(\kappa^{+})^{L}\leq\theta<\omega_2$ and $U\subseteq L_{\theta}$ is an $L$-ultrafilter on $\kappa$. Do the ultrapower construction for $\langle L_{\omega_2}, \in, U\rangle$. Since $L_{\theta}[a]$ is closed under $\omega$-sequences,$L_{\omega_2}/U$ is well founded and hence we get a nontrivial elementary embedding $e: L_{\omega_2}\prec L_{\omega_2}$ with $crit(e)=\kappa$. Now we show that there exists a club on $\omega_2$ of regular $L$-cardinals. Suppose $X\prec L_{\eta}[a], \omega_1\subseteq X$ and $\omega_2\in X$. The transitive collapse of $X$ is $L_{\bar{\eta}}[a]$ for some $\bar{\eta}$. Since $L_{\eta}\models \omega_2$ is a regular cardinal, $L_{\bar{\eta}}\models \bar{\omega_2}$ is a regular cardinal. By $HP(L)$, $\bar{\eta}$ is an $L$-cardinal and hence $\bar{\omega_2}$ is a regular $L$-cardinal. Since $\omega_1\subseteq X$, $\bar{\omega_2}=X\cap \omega_2$. We have shown that if $X\prec L_{\eta}[a], \omega_1\subseteq X$ and $\omega_2\in X$, then $X\cap \omega_2=\bar{\omega_2}$ is a regular $L$-cardinal. So there exists a club on $\omega_2$ of regular $L$-cardinals. Let $D$ be such a club such that $D\cap(\kappa+1)=\emptyset$. \[equlity claim\] For any $\alpha\in D, e(\alpha)=\alpha$. Suppose $\alpha\in D$ and $f\in L_{\omega_2}$ where $f: \kappa\rightarrow\alpha$. Since $\alpha>\kappa$ is a regular $L$-cardinal, $f$ is bounded by some $\eta<\alpha$. So $[f]<[c_{\eta}]$. Hence $e(\alpha)=lim_{\beta\rightarrow\alpha} e(\beta)$. If $\beta<\alpha$, then $\|e(\beta)\|\leq (\|\beta^{\kappa}\|)^{L}\leq\alpha$. So $e(\alpha)=\alpha$. We define a sequence $\langle C_\alpha: \alpha<\omega_1\rangle$ as follows. Let $C_{0}=D$. For any $\nu<\omega_1, C_{\nu+1}=\{\mu\in C_{\nu}\mid \mu$ is the $\mu$-th element of $C_{\nu}$ in the increasing enumeration of $C_{\nu}$}. If $\nu\leq\omega_1$ is a limit ordinal, $C_{\nu}=\bigcap_{\beta<\nu} C_{\beta}$. Note that $C_{\nu}$ is a club on $\omega_2$ for all $\nu\leq\omega_1$. By Claim \[equlity claim\], for $\nu\leq\omega_1, e\upharpoonright C_{\nu}=id$. Now we will find $\omega_1$-many indiscernibles for $(L_{\omega_2}, \in)$. The rest of the argument essentially follows from [@Jech Theorem 18.20]. For each $\nu<\omega_1$, let $M_{\nu}$ be the Skolem hull of $\kappa\cup C_{\nu}$ in $L_{\omega_2}$. The transitive collapse of $M_{\nu}$ is $L_{\omega_2}$. Let $i_{\nu}: L_{\omega_2}\cong M_{\nu}\prec L_{\omega_2}$ be the inverse of the collapsing map and $\kappa_{\nu}=i_{\nu}(\kappa)$. By [@Jech Lemma 18.24,18.25, 18.26], $\{\kappa_{\nu}\mid \nu<\omega_1\}$ is a set of indiscernibles for $L_{\omega_2}$.[^13] \[main theorem by chengyong\] ([@YongCheng2])$Z_3\, +\, HP(L)$ does not imply $0^{\sharp}$ exists. By a similar argument as in Theorem \[cited theorem from Woodin\] we can show from $Z_3\, +\, HP(L)$ that there exists a nontrivial elementary embedding $j: L_{\omega_1}\prec L_{\omega_1}$ and there is a club $C\subseteq \omega_1$ of regular $L$-cardinals. However, by Theorem \[main theorem by chengyong\], from these we can not prove in $Z_3$ that $0^{\sharp}$ exists. Note that Theorem 3.3 still holds if we replace the term “$L$-cardinal" with any large cardinal notion compatible with $L$ in the definition of $HP(L)$. This is because the Silver indiscernibles can have any large cardinal property compatible with $L$.[^14] \[fact on zero dagger two\] ([@Higherinfinite Theorem 21.15]) The following are equivalent: (1) $0^{\dag}$ exists. (2) For every uncountable cardinal $\kappa$ there is a $\kappa$-model and a double class $\langle X,Y\rangle$ of indiscernibles for it such that: $X\subseteq\kappa$ is closed unbounded, $Y\subseteq Ord\setminus(\kappa+1)$ is a closed unbounded class, $X\cup\{\kappa\}\cup Y$ contains every uncountable cardinal and the Skolem hull of $X\cup Y$ in the $\kappa$-model is again the model. \[condensation for L\[U\]\] ([@begininnermodel Lemma 1.7])Suppose that $A$ is a set, $X\prec L_{\alpha}[A]$ where $\alpha\in Ord\cup \{Ord\}$ and the transitive closure of $A\cap L_{\alpha}[A]$ is contained in $X$. Then $X\cong L_{\alpha^{\prime}}[A]$ for some $\alpha^{\prime}\leq\alpha$. \[fact on zero dagger lage cardinal\] (Folklore)Suppose $0^{\dag}$ exists, $L[U]$ is the unique $\kappa$-model and $\langle X,Y\rangle$ is the double class of indiscernibles for $L[U]$ as in Fact \[fact on zero dagger two\]. If $\alpha\leq\kappa$ is $0^{\dag}$-admissible, then $X$ is unbounded in $\alpha$, and if $\alpha>\kappa$ is $0^{\dag}$-admissible, then $Y$ is unbounded in $\alpha$.[^15] \[HP for measurable cn\] Suppose $\kappa$ is a measurable cardinal and $L[U]$ is the unique $\kappa$-model. Then $HP(L[U])$ if and only if $0^{\dag}$ exists. $(\Rightarrow)$ Let $x$ be the witness real for $HP(L[U])$. Pick $\lambda> 2^{\kappa}$ and $X$ such that $\lambda$ is $(x,U)$-admissible, $X\prec L_{\lambda}[U][x]$, $\|X\|=2^{\kappa}$, $X$ is closed under $\omega$-sequences and the transitive closure of $U\cap L_{\lambda}[U]$ is contained in $X$. By Fact \[condensation for L\[U\]\], the transitive collapse of $X$ is of the form $L_{\theta}[U][x]$. Let $j: L_{\theta}[U][x]\cong X\prec L_{\lambda}[U][x]$ be the inverse of the collapsing map and $\eta=crit(j)$. Note that $\eta>\kappa$. Since $\theta$ is $(x,U)$-admissible, by $HP(L[U])$, $\theta$ is an $L[U]$-cardinal. Define $\bar{U}=\{X\subseteq\eta\mid X\in L[U]$ and $\eta\in j(X)\}$. Since $(\eta^{+})^{L[U]}\leq\theta, \bar{U}\subseteq L_{\theta}[U]$. $\bar{U}$ is an $L[U]$-ultrafilter on $\eta$. Since $L_{\theta}[U][x]$ is closed under $\omega$-sequences, $\bar{U}$ is countably complete. So we can build a nontrivial embedding from $L[U]$ to $L[U]$ with critical point greater than $\kappa$. By Fact \[fact on zero dagger\], $0^{\dag}$ exists. $(\Leftarrow)$ Suppose $0^{\dag}$ exists and $\alpha$ is $0^{\dag}$-admissible. We show that $\alpha$ is an $L[U]$-cardinal. By Fact \[fact on zero dagger two\], let $\langle X,Y\rangle$ be the double class of indiscernibles for $L[U]$. If $\alpha\leq\kappa$, then by Fact \[fact on zero dagger lage cardinal\], $\alpha\in X$. If $\alpha>\kappa$, then by Fact \[fact on zero dagger lage cardinal\], $\alpha\in Y$. Trivially, elements of $X$ and $Y$ are $L[U]$-cardinals. \[embedding not exist\] ([@CoveringLemma], [@Steel])Suppose there is no inner model with one measurable cardinal and let $K$ be the corresponding core model. Then, $K$ has the rigidity property. \[Theorem for core model\] (1) Suppose $0^{\sharp}$ exists. Then $HP(L[0^{\sharp}])$ if and only if $(0^{\sharp})^{\sharp}$ exists. (2) Suppose there is no inner model with one measurable cardinal and that $K$ is the corresponding core model. Then $HP(K)$ does not hold. $(1)$ Follows from the proof of $``HP(L)\Leftrightarrow 0^{\sharp}$ exists". Note that if $\alpha$ is $(0^{\sharp})^{\sharp}$-admissible and $I$ is the class of Silver indiscernibles for $L[0^{\sharp}]$, then $I$ is unbounded in $\alpha$ and hence $\alpha\in I$. \(2) Note that $K=L[\mathcal{M}]$ where $\mathcal{M}$ is a class of mice. Suppose $HP(K)$ holds and $x$ is the witness real for $HP(K)$. Pick $\theta>\omega_2$ and $X$ such that $\theta$ is $(\mathcal{M}, x)$-admissible, $X\prec J_{\theta}[\mathcal{M}, x]$, $\omega_2\in X, \|X\|=\omega_1$ and $X$ is closed under $\omega$-sequences. Since $K\models GCH$, such an $X$ exists. By the condensation theorem for $K$, let $j: J_{\theta^{\prime}}[\mathcal{M}\upharpoonright \theta^{\prime}, x]\cong X \prec J_{\theta}[\mathcal{M}, x]$ be the inverse of the collapsing map. Let $\lambda=crit(j)$ and $U=\{X\subseteq\lambda\mid X\in K$ and $\lambda\in j(X)$}. Note that $\theta^{\prime}$ is a $K$-cardinal and $U$ is a countably complete $K$-ultrafilter on $\lambda$. So there is a nontrivial elementary embedding from $K$ to $K$ which contradicts Fact \[embedding not exist\]. From proof of Corollary \[Theorem for core model\](2), if $M$ is an $L$-like inner model, $M$ has the rigidity property and some proper form of condensation, and $M\models CH$, then $HP(M)$ does not hold. \[fact about HOD\] ([@Steel])$(AD^{L(R)})\quad \mathsf{HOD}^{L(R)}=L(P)$ for some $P\subseteq \Theta$ where $\Theta=\sup\{\alpha\mid \exists f\in L(R)(f:R\rightarrow\alpha$ is surjective$)\}$. It is an open question whether there exists a nontrivial elementary embedding from $\mathsf{HOD}$ to $\mathsf{HOD}$.[^16] However, the following fact shows that the answer to this question is negative for embeddings which are definable in $V$ from parameters. \[definable HOD\] ([@Hamkins Theorem 35])Do not assume $AC$. There is no nontrivial elementary embedding from $\mathsf{HOD}$ to $\mathsf{HOD}$ that is definable in $V$ from parameters. \[HP for HOD\] $(ZF+ AD^{L(R)})\quad \mathsf{HP(HOD)}$ does not hold. By Fact \[fact about HOD\], under $ZF + AD^{L(R)}, \mathsf{HOD}=L(P)$ for some $P\subseteq \Theta$. Suppose $\mathsf{HP(HOD)}$ holds. Then since $L(P)\models CH$, by a similar proof as in Corollary \[Theorem for core model\](2) we can show that there exists a nontrivial elementary embedding $j: L(P)\rightarrow L(P)$. Note that $j$ is definable in $V$ from parameters. i.e. there is a formula $\varphi$ and parameter $\vec{a}$ such that $j(x)=y$ if and only if $\varphi(x,y,\vec{a})$. This contradicts Fact \[definable HOD\]. Relationship between $HP(L)$ and the strong reflecting property for $L$-cardinals ================================================================================= In this section, we discuss the relationship between the strong reflecting property for $L$-cardinals and Harrington’s Principle $HP(L)$. \[compare first level thm\] (Set forcing) $SRP^{L}(\omega_1)$ implies $Con(Z_2\,+ \,HP(L))$. Suppose $SRP^{L}(\omega_1)$ holds and we want to build a model of $Z_2\,+ \,HP(L)$. By Proposition \[srp of omega first\], $\omega_1$ is limit cardinal in $L$. i.e. $\{\alpha<\omega_1\mid \alpha$ is an $L$-cardinal} is a club. Let $C=\{\omega\leq\alpha<\omega_{1}\mid \alpha$ is an $L$-cardinal and $L_{\alpha}\prec L_{\omega_1}\}$. Note that $C$ is a club. Let $$D=\{\gamma<\omega_1\mid (L_{\gamma}[C], C\cap\gamma)\prec (L_{\omega_1}[C], C)\}.$$ Note that $D\subseteq C$. Define $F: \omega^{\omega}\rightarrow \omega^{\omega}$ as follows: if $y\subseteq\omega$ codes $\gamma$, then $F(y)$ is a real which codes $(\beta, C\cap\beta)$ where $\beta$ is the least element of $D$ such that $\beta>\gamma$ (since $D$ is a club in $\omega_1$, such a $\beta$ exists); if $y$ does not code an ordinal, let $F(y)=\emptyset$. Let $\langle\delta_{\alpha}\mid \alpha<\omega_1\rangle$ be a pairwise almost disjoint set of reals such that $\delta_{\alpha}$ is the $<_{L[C]}$-least real which is almost disjoint from any member of $\{\delta_{\beta}\mid \beta<\alpha\}$ and $\langle\delta_{\nu}\mid \nu<\omega\rangle\in L_{\alpha}$ for every admissible ordinal $\alpha<\omega_1$. Let $\langle x_{\alpha}\mid \alpha<\omega_1\rangle$ be the enumeration of $\mathcal{P}(\omega)$ in $L[C]$ in the order of construction. Let $Z_F \subseteq\omega_1$ be defined as: $$Z_{F}=\{\alpha\cdot\omega+i\mid \alpha<\omega_1\wedge i\in F(x_{\alpha}) \}.$$ Now we do almost disjoint forcing to code $Z_{F}$ via $\langle\delta_{\alpha}\mid \alpha<\omega_1\rangle$. Then we get a real $x$ such that $\alpha\in Z_{F}\Leftrightarrow \|x\cap \delta_{\alpha}\|<\omega$. The forcing is $c.c.c$ and hence preserves all cardinals. Now we work in $L[x]$. Take the least $\theta$ such that $L_{\theta}[x]\models Z_{2}$. We will show that $L_{\theta}[x]\models HP(L)$. By absoluteness, it suffices to show that if $\alpha<\theta$ is $x$-admissible, then $\alpha$ is an $L$-cardinal. Fix some $x$-admissible $\alpha<\theta$ and let $$\gamma_0=\sup(\alpha\cap D).$$ If $\alpha\cap D=\emptyset$, let $\gamma_0=0$. Note that if $\gamma_0>0$, then $\gamma_0\in D$. We assume that $\gamma_0<\alpha$ and try to get a contradiction. Let $\alpha_0$ be the least admissible ordinal such that $\alpha_0>\gamma_0$. Since $\alpha$ is admissible, $\alpha_0\leq\alpha$. \[show that\] $C\cap\alpha_0=C\cap (\gamma_0+1).$ We show that $C\cap\alpha_0\subseteq C\cap (\gamma_0+1)$. Suppose $\gamma\in C\cap\alpha_0$ and $\gamma>\gamma_0$. Since $\gamma\in C, L_{\gamma}\prec L_{\omega_1}$. Since $\alpha_0$ is definable from $\gamma_0$, it follows that $\alpha_0$ is definable in $L_{\gamma}$. So $\alpha_0\leq\gamma$. Contradiction. By Claim \[show that\], $L_{\alpha_0}[C]=L_{\alpha_0}[C\cap\gamma_0]$. We need the following lemma to get that $L_{\gamma_0}[C\cap\gamma_0][x]=L_{\gamma_0}[x]$ in Claim \[key calim on countable\]. \[key claim in step\] $C\cap\gamma_0\in L_{\gamma_0+1}[x].$ We prove by induction that for any $\gamma\in D\cap\theta, C\cap\gamma\in L_{\gamma+1}[x]$. Fix $\gamma\in D\cap\theta$. Suppose for any $\gamma^{\prime}\in D\cap\gamma$, $C\cap\gamma^{\prime}\in L_{\gamma^{\prime}+1}[x]$. We show that $C\cap\gamma\in L_{\gamma+1}[x]$. Case 1: There is $\gamma^{\prime}\in D$ such that $\gamma$ is the least element of $D$ such that $\gamma>\gamma^{\prime}$. Let $\eta$ be the least admissible ordinal such that $\eta>\gamma^{\prime}$. By a similar argument as in Claim \[show that\], $C\cap\eta=C\cap(\gamma^{\prime}+1)$. From our definitions, for any $\beta<\eta$ we have: (1) $\langle x_{\xi}\mid\xi\in\beta\rangle\in L_{\eta}[C]=L_{\eta}[C\cap\gamma^{\prime}]$; (2) $\langle\delta_{\xi}\mid\xi\in\beta\rangle\in L_{\eta}[C]=L_{\eta}[C\cap\gamma^{\prime}]$; and (3) $\langle x_{\xi}\mid\xi\in\eta\rangle$ enumerates $\mathcal{P}(\omega)\cap L_{\eta}[C]=\mathcal{P}(\omega)\cap L_{\eta}[C\cap\gamma^{\prime}]$. Suppose $y\subseteq\omega$ and $y\in L_{\eta}[C\cap\gamma^{\prime}]$. Then $y=x_{\xi}$ for some $\xi<\eta$. Note that $\xi\cdot\omega+i<\eta$ for any $i<\omega$. Moreover, $i\in F(y)$ if and only if $\|x\cap\delta_{\xi\cdot\omega+i}\|<\omega$. So $F(y)\in L_{\eta}[C\cap\gamma^{\prime}][x]$. Hence we have shown that if $y\in\mathcal{P}(\omega)\cap L_{\eta}[C\cap\gamma^{\prime}]$, then $F(y)\in L_{\eta}[C\cap\gamma^{\prime}, x]$. \[first key calim on countable\] $L_{\eta}[C\cap\gamma^{\prime}]\models\gamma^{\prime}<\omega_1$. Suppose, towards a contradiction, that $$\label{key equation in sec two} \text{$\gamma^{\prime}=\omega_1^{L_{\eta}[C\cap\gamma^{\prime}]}$.}$$ Let $P$ be the almost disjoint forcing that codes $Z_{F}$ via the almost disjoint system $\langle\delta_\beta \mid \beta<\omega_1\rangle$.[^17] From our definitions of $C, F$ and $\langle x_{\alpha}\mid \alpha<\omega_1\rangle$, $P$ is a definable subset of $L_{\omega_1}[C]$. Standard argument gives that $P$ is $\omega_1$-c.c. in $L_{\omega_1}[C]$.[^18] Let $P^{\ast}=P\cap L_{\gamma^{\prime}}[C]$. Since $\gamma^{\prime}\in D$, $$\label{property of gamma zero} (L_{\gamma^{\prime}}[C], C\cap\gamma^{\prime})\prec (L_{\omega_1}[C], C).$$ Suppose $D^{\ast}\subseteq P^{\ast}$ is a maximal antichain with $D^{\ast}\in L_{\gamma^{\prime}}[C]$. Then by (\[property of gamma zero\]), $D^{\ast}$ is a maximal antichain in $P$. Since $L_{\omega_1}[C]\models D^{\ast}$ is at most countable, by (\[property of gamma zero\]), $L_{\gamma^{\prime}}[C]\models D^{\ast}$ is at most countable. So $P^{\ast}$ is $\omega_1$-c.c. in $L_{\gamma^{\prime}}[C]$. By (\[key equation in sec two\]), $$\label{equation on two} \text{$L_{\eta}[C\cap\gamma^{\prime}]\cap 2^{\omega}=L_{\gamma^{\prime}}[C\cap\gamma^{\prime}]\cap 2^{\omega}$.}$$ Since $P^{\ast}$ is $\omega_1$-c.c. in $L_{\gamma^{\prime}}[C]$, by (\[equation on two\]), $P^{\ast}$ is $\omega_1$-c.c in $L_{\eta}[C\cap\gamma^{\prime}]$. We show that $x$ is generic over $L_{\eta}[C\cap\gamma^{\prime}]$ for $P^{\ast}$. Let $Y\subseteq P^{\ast}$ be a maximal antichain with $Y\in L_{\eta}[C\cap\gamma^{\prime}]$. Since $P^{\ast}$ is $\omega_1$-c.c in $L_{\eta}[C\cap\gamma^{\prime}]$, by (\[key equation in sec two\]), $Y\in L_{\gamma^{\prime}}[C\cap\gamma^{\prime}]$. By (\[property of gamma zero\]), $Y$ is a maximal antichain in $P$. So the filter given by $x$ meets $Y$. Note that $\gamma^{\prime}=\omega_1^{L_{\eta}[C\cap\gamma^{\prime}]}=\omega_1^{L_{\eta}[C\cap\gamma^{\prime}][x]}$. Since $\gamma^{\prime}\in D$, by induction hypothesis $L_{\gamma^{\prime}}[C\cap\gamma^{\prime}, x]=L_{\gamma^{\prime}}[x]$. So $L_{\gamma^{\prime}}[x]\models Z_2$ which contradicts the minimality of $\theta$. Take $y\in L_{\eta}[C\cap\gamma^{\prime}]\cap\mathcal{P}(\omega)$ such that $y$ codes $\gamma^{\prime}$. So $F(y)$ codes $(\gamma, C\cap\gamma)$ and $F(y)\in L_{\eta}[C\cap\gamma^{\prime}, x]$. Then $F(y)$ is definable in $L_{\gamma}[C\cap\gamma^{\prime}, x]$. By induction hypothesis, $F(y)\in L_{\gamma+1}[x]$. Since $F(y)$ codes $C\cap\gamma$, $C\cap\gamma\in L_{\gamma+1}[x]$. Case 2: $\gamma$ is the least element of $D$. Take $y\in L_{\omega}[C]\cap\mathcal{P}(\omega)$ such that $y$ codes $0$. Then $y=x_0$. Since $\gamma$ is the least element of $D$ such that $\gamma>0$, $F(y)$ codes $C\cap\gamma$. Note that for any $\beta<\omega,\langle\delta_{\xi}\mid\xi\in\beta\rangle\in L_{\omega}[C]$ and $i\in F(y)$ if and only if $\|x\cap\delta_{i}\|$ is finite. So $F(y)$ is definable in $L_{\omega}[x,C]$. Since $C\cap \omega=\emptyset$, $F(y)\in L_{\gamma+1}[x]$. Since $F(y)$ codes $C\cap\gamma$, $C\cap\gamma\in L_{\gamma+1}[x]$. Case 3: $\gamma$ is a limit point of $D$. Then a standard argument gives that $C\cap\gamma\in L_{\gamma+1}[x]$ by induction hypothesis. Since $\gamma_0\in D\cap\theta$, we have $C\cap\gamma_0\in L_{\gamma_0+1}[x]$. \[key calim on countable\] $\gamma_0$ is countable in $L_{\alpha_0}[C\cap\gamma_0]$. The proof is essentially the same as Claim \[first key calim on countable\] (replace $\eta$ by $\alpha_0$ and $\gamma^{\prime}$ by $\gamma_0$). Suppose, towards a contradiction, that $\gamma_0=\omega_1^{L_{\alpha_0}[C\cap\gamma_0]}$. By the similar argument as Claim \[first key calim on countable\], we can show that $x$ is generic over $L_{\alpha_0}[C\cap\gamma_0]$ for $P^{\ast}=P\cap L_{\gamma_0}[C]$.[^19] Since $\gamma_0=\omega_1^{L_{\alpha_0}[C\cap\gamma_0]}=\omega_1^{L_{\alpha_0}[C\cap\gamma_0][x]}$ and by Lemma \[key claim in step\], $L_{\gamma_0}[C\cap\gamma_0][x]=L_{\gamma_0}[x]$, we have $L_{\gamma_0}[x]\models Z_2$ which contradicts the minimality of $\theta$. From our definitions, we have: $$\label{rule for adjs} \text{For $\eta<\alpha_0, \langle\delta_{\beta}: \beta<\eta\rangle\in L_{\alpha_0}[C]=L_{\alpha_0}[C\cap\gamma_0]$;}$$ $$\label{euma rule} \text{$\langle x_{\alpha}\mid \alpha<\alpha_0\rangle$ enumerates $\mathcal{P}(\omega)\cap L_{\alpha_0}[C]=\mathcal{P}(\omega)\cap L_{\alpha_0}[C\cap\gamma_0]$.}$$ \[countable claim\] If $y\in \mathcal{P}(\omega)\cap L_{\alpha_0}[C\cap\gamma_0]$, then $F(y)\in L_{\alpha_0}[x]$. Suppose $y\in \mathcal{P}(\omega)\cap L_{\alpha_0}[C\cap\gamma_0]$. By , $y=x_{\xi}$ for some $\xi<\alpha_0$. Note that for $\xi<\alpha_0, \xi\cdot\omega+i<\alpha_0$ for any $i\in\omega$. By the definition of $Z_{F}, i\in F(y)\Leftrightarrow \xi\cdot\omega+i\in Z_{F}\Leftrightarrow \|x\cap \delta_{\xi\cdot\omega+i}\|<\omega$. By , $F(y)\in L_{\alpha_0}[C\cap\gamma_0][x]$. Since $C\cap\gamma_0\in L_{\gamma_0+1}[x]$ by Lemma \[key claim in step\], we have $L_{\alpha_0}[C\cap\gamma_0][x]=L_{\alpha_0}[x]$. So $F(y)\in L_{\alpha_0}[x]$. By Claim \[key calim on countable\], there exists a real $y\in L_{\alpha_0}[C\cap\gamma_0]$ such that $y$ codes $\gamma_0$. Note that $F(y)$ codes $\gamma_1$ where $\gamma_1$ is the least element of $C$ such that $\gamma_1>\gamma_0$ and $(L_{\gamma_1}[C], C\cap\gamma_1)\prec (L_{\omega_1}[C], C)$. Since $F(y)$ codes $\gamma_1$ and $F(y)\in L_{\alpha_0}[x]$, $\gamma_1<\alpha_0$. Since $\gamma_1<\alpha$ and $(L_{\gamma_1}[C], C\cap\gamma_1)\prec (L_{\omega_1}[C], C)$, by the definition of $\gamma_0$, we have that $\gamma_1\leq\gamma_0$. Contradiction. So the assumption $\gamma_0<\alpha$ is false. Then $\gamma_0=\alpha$. So $\alpha \in C$ and hence $\alpha$ is an $L$-cardinal. We have shown that $L_{\theta}[x]\models Z_2\,+ \,HP(L)$. \[result on class forcing\] ([@YongCheng2 Theorem 3.1, 3.2]) (Class forcing) $Z_2\,+ \,HP(L)$ is equiconsistent with $ZFC$ and $Z_3\,+ \,HP(L)$ is equiconsistent with $ZFC\, +$ there exists a remarkable cardinal. (a) For $n\geq 3, SRP^{L}(\omega_n)$ is equivalent to $HP(L)$. (b) (Set forcing) $SRP^{L}(\omega_2)$ is strictly stronger than $Z_3\,+ \,HP(L)$. (c) (Set forcing) $SRP^{L}(\omega_1)$ is strictly stronger than $Z_2\,+ \,HP(L)$. \(a) follows from Theorem \[characterization of srp above third level\] and Theorem \[cited theorem from Woodin\]. (b) follows from Theorem \[strength of omega two\] and Theorem \[result on class forcing\]. (c) follows from Theorem \[compare first level thm\], Theorem \[result on class forcing\] and Proposition \[srp of omega first\]. [99]{} To appear in [Mathematical Logic Quarterly]{}. To appear in [The Journal of Symbolic Logic]{}. Iterated Forcing and Elementary Embeddings, Chapter 12 in [Handbook of Set Theory]{}, Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. Springer, Berlin, 1984. Constructibility and Class Forcing, Chapter 8 in [Handbook of Set Theory]{}, Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone, What is the theory $ZFC$ without Powerset? See http://arxiv.org/abs/1110.2430 Analytic determinacy and $0^{\sharp}$, [The Journal of Symbolic Logic,]{} 43(1978), 685-693. Springer, Berlin, 2003. Generalizations of the Kunen inconsistency, [Annals of Pure and Applied Logic,]{} vol. 163, iss. 12, pp. 1872-1890, December 2012. Akihiro Kanamori, [The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings]{}, Springer Monographs in Mathematics, Springer, Berlin, 2003, Second edition. North-Holland, Amsterdam, 1980. William J. Mitchell, Beginning Inner Model Theory, Chapter 17 in [Handbook of Set Theory]{}, Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. William J. Mitchell, The Covering Lemma, Chapter 18 in [Handbook of Set Theory]{}, Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. 44 (2005), pp.159-166. 66 (2001), pp. 1481-1492. An Outline of Inner Model Theory, Chapter 19 in [Handbook of Set Theory]{}, Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. [^1]: $ZFC^{-}$ denotes $ZFC$ with the Power Set Axiom deleted and Collection instead of Replacement. For the discussion of the theory $ZFC$ without power set, see [@VictoriaGitman]. [^2]: For the definition of coherent sequences of extenders $\vec{E}$, $J_{\alpha}^{\vec{E}}$ and $\vec{E}\upharpoonright \alpha$, see Section 2.2 in [@Steel]. [^3]: All known core models satisfy this convention. [^4]: In this paper, we say that $X$ is closed under $F$ if $F``X^{<\omega}\subseteq X$. [^5]: $\bar{\bar{\gamma}}$ is the image of $\bar{\gamma}$ under the transitive collapse of $Y$. [^6]: The key point is that the statement Proposition \[reflecting\](4) is upward absolute. [^7]: The key point is that the statement Proposition \[weakly reflecting\](d) is downward absolute. [^8]: Here we use that $M\|\theta$ is definable in $H_{\theta}$ for regular cardinal $\theta>\omega_2$. [^9]: $\mathcal{M}(0^{\sharp}, \alpha)$ is the unique transitive $(0^{\sharp},\alpha)$-model. For the notation, see [@Higherinfinite]. [^10]: In this article, $tr(X)$ stands for the transitive closure of $X$. [^11]: Note that $\mathcal{M}(0^{\dag},\omega, \alpha)$ is the unique transitive $(0^{\dag},\omega,\alpha)$-model. For the notation of $\mathcal{M}(0^{\dag},\omega, \alpha)$, see [@Higherinfinite]. [^12]: In [@YongCheng2], we define $0^{\sharp}$ as the minimal iterable mouse and prove in $Z_4$ that $HP(L)$ is equivalent to $0^{\sharp}$ exists. Theorem \[cited theorem from Woodin\] proves that these two definitions of $0^{\sharp}$ are equivalent in $Z_4$. [^13]: Note that the proof of [@Jech Theorem 18.20], as opposed to the proof of Theorem \[cited theorem from Woodin\] above, is not done in $Z_4$. [^14]: Examples of large cardinal notions compatible with $L$: inaccessible cardinal,reflecting cardinal, Mahlo cardinal, weakly compact, indescribable cardinal, unfoldable cardinal, subtle cardinal, ineffable cardinal, 1-iterable cardinal, remarkable cardinal, 2-iterable cardinal and $\omega$-Erd$\ddot{o}$s cardinal. [^15]: I would like to thank W.Hugh Woodin and Sy Friedman for pointing out this fact to me. The proof of this fact is essentially similar as the proof of the following standard fact: if $0^{\sharp}$ exists, $I$ is the class of Silver indiscernibles and $\alpha$ is $0^{\sharp}$-admissible, then $I$ is unbounded in $\alpha$ (see [@SyFriedman Theorem 4.3]). [^16]: The answer to this question is negative if $V=\mathsf{HOD}$.[@Hamkins Theorem 21] provides a very easy proof of the Kunen inconsistency in the case $V=\mathsf{HOD}$. [^17]: $P=[\omega]^{<\omega}\times [Z_{F}]^{<\omega}$. $(p,q)\leq (p^{\prime},q^{\prime})$ $p\supseteq p^{\prime}, q\supseteq q^{\prime}$ and $\forall\alpha\in q^{\prime}(p\cap \delta_{\alpha}\subseteq p^{\prime})$. [^18]: i.e. If $D\subseteq P$ is a maximal antichain with $D\in L_{\omega_1}[C]$, then $L_{\omega_1}[C]\models D$ is at most countable. [^19]: $P$ is the almost disjoint forcing that codes $Z_{F}$ via $\langle\delta_{\beta}\mid \beta<\omega_1\rangle$.
--- abstract: 'We show that if the propagating speed of gravitational waves (GWs) gradually diminishes during inflation, the power spectrum of primordial GWs will be strongly blue, while that of the primordial scalar perturbation may be unaffected. We also illustrate that such a scenario is actually a disformal dual to the superinflation, but it does not have the ghost instability. The blue tilt obtained is $0<n_T\lesssim 1$, which may significantly boost the stochastic GWs background at the frequency band of Advanced LIGO/Virgo, as well as the space-based detectors.' author: - 'Yong Cai$^{1}$[^1]' - 'Yu-Tong Wang$^{1}$[^2]' - 'Yun-Song Piao$^{1}$[^3]' title: Propagating speed of primordial gravitational waves and inflation --- Introduction ============ Recently, the LIGO Scientific Collaboration has observed a transient gravitational wave (GWs) signal with a significance in excess of 5.1$\sigma$ [@Abbott:2016blz], which is consistent with an event of the binary black hole coalescence. This discovery will be a scientific milestone for understanding our universe, if it is confirmed. It is speculated that the stochastic GWs background contributed by the incoherent superposition of all merging binaries in the universe might be higher than expected previously [@TheLIGOScientific:2016wyq], which is potentially measurable around $25$Hz by the Advanced LIGO/Virgo detectors operating at their projected final sensitivity. However, some cosmological sources may also contribute a stochastic background of GWs at the corresponding frequency band, such as cosmic strings [@Damour:2000wa] and cosmological phase transitions [@Kamionkowski:1993fg][@Dev:2016feu]. It is well known that the standard slow-roll inflation predicts a nearly flat spectrum of scalar perturbation, as well as primordial GWs [@Starobinsky:1979ty][@Rubakov:1982]. Recently, the BICEP2/Keck data, combined with the Planck data and the WMAP data, have put the constraint $r<0.09$ (95% C.L.) [@Array:2015xqh] on the amplitude of primordial GWs on large scale, or at ultra-low frequency, which corresponds to $\Omega_{gw}\sim 10^{-15}$, but there is no strong limit for its tilt $n_T$. Actually, as long as its spectrum is blue enough, the stochastic GWs background from primordial inflation is also not negligible at the frequency band of Advanced LIGO/Virgo. The slow-roll inflation model with $\epsilon = -{\dot H}/H^2\ll 1$ generally has $n_T=-2\epsilon < 0$. Thus $n_T>0$ requires either the superinflation [@Piao:2004tq][@Baldi:2005gk], also [@Liu:2013iha][@Cai:2015yza], which breaks the null energy condition (NEC), or an anisotropic stress source during inflation, e.g., the particle production [@Cook:2011sp][@Sorbo:2011ja][@Mukohyama:2014gba][@Namba:2015gja]. During the superinflation, the primordial GWs come from the amplification of vacuum tensor perturbations. However, since the almost scale-invariance of the scalar perturbation requires $\|\epsilon\|\sim 0.01 $, we generally have $\|n_T\|\sim {\cal O}(0.01)$ for the superinflation. Obtaining a blue GWs spectrum $n_T>0.1$ without the ghost instability while reserving a scale-invariant scalar spectrum with slightly red tilt is still a challenge for the inflation scenario[^4], see e.g.[@Wang:2014kqa] for comments. In Einstein gravity, the propagating speed $c_T$ of GWs is the same as the speed of light, thus can naturally be set as unity. Nevertheless, it might be modified when dealing with the extremely early universe, e.g., the low-energy effective string theory with higher-order corrections [@Met:1987][@Antoniadis:1993jc][@Kawai:1998ab][@Cartier:1999vk], see also [@Maeda:2004vm][@Nojiri:2015qyc]. Since the amplitude of the primordial GWs is determined by $c_T$ and the Hubble radius $\sim H^{-1}$, the running of $c_T$ will inevitably affect the power spectrum of primordial GWs (see also [@Giovannini:2015kfa] for the study from the point of view of the running of GWs’ refractive index $n$). It was found in [@Cai:2015ipa][@Cai:2015dta] that the oscillation of $c_T$ may leave some observable imprints in CMB B-mode polarization. The effect of the sound speed $c_S$ of scalar perturbation on the scalar spectrum has been investigated in e.g.[@Khoury:2008wj] [@Park:2012rh]. Here, we show that if the propagating speed $c_T$ of GWs gradually diminishes during inflation, the power spectrum of primordial GWs will be strongly blue, while the spectrum of scalar perturbation may be still that of slow-roll inflation. There is no the ghost instability. The blue tilt obtained is $0<n_T\lesssim 1$, which may significantly boost the stochastic GWs background within the window of Advanced LIGO, as well as those of the space-based detectors. Inflation and $c_T$ =================== The model {#model} --------- We follow the effective field theory of inflation [@Cheung:2007st], beginning with the Langrangian in unitary gauge S & = & [M\_p\^22]{}d\^4x , \[action\] where $M_p=1/\sqrt{8\pi G}$, $c_1(t)=2({\dot H}+3H^2)$, $c_2(t)=-2{\dot H}$, and a dot denotes the derivative with respect to cosmic time $t$. We will work in the inflation background with $0<\epsilon\ll 1$, which may be set by requiring $\|{\dot H}\|\ll H^2$ in (\[action1\]). The scalar perturbation at quadratic order is not affected by $\delta K_{\mu\nu}\delta K^{\mu\nu}-\delta K^2$, see Appendix \[scalar\], and also [@Creminelli:2014wna], so its spectrum is determined by slow-roll parameters. However, the quadratic action of tensor perturbation is altered as[^5] S\^[(2)]{}\_=dd\^3x [M\_p\^2a\^2c\_T\^[-2]{}8]{}, \[paction\]where $\tau=\int dt/a$, and $\gamma_{ij}$ satisfies $\gamma_{ii}=0$ and $\partial_i \gamma_{ij}=0$. The Fourier series of $\gamma_{ij}$ is \_[ij]{}(,)=e\^[-i]{} \_[=+,]{} \_(,) \^[()]{}\_[ij]{}(), in which $ \hat{\gamma}_{\lambda}(\tau,\mathbf{k})= \gamma_{\lambda}(\tau,k)a_{\lambda}(\mathbf{k}) +\gamma_{\lambda}^(\tau,-k)a_{\lambda}^{\dag}(-\mathbf{k})$, the polarization tensors $\epsilon_{ij}^{(\lambda)}(\mathbf{k})$ satisfy $k_{j}\epsilon_{ij}^{(\lambda)}(\mathbf{k})=0$, $\epsilon_{ii}^{(\lambda)}(\mathbf{k})=0$, and $\epsilon_{ij}^{(\lambda)}(\mathbf{k}) \epsilon_{ij}^{(\lambda^{\prime}) }(\mathbf{k})=\delta_{\lambda \lambda^{\prime} }$, $\epsilon_{ij}^{(\lambda) }(\mathbf{k})=\epsilon_{ij}^{(\lambda) }(-\mathbf{k})$, the annihilation and creation operators $a_{\lambda}(\mathbf{k})$ and $a^{\dag}_{\lambda}(\mathbf{k}^{\prime})$ satisfy $[ a_{\lambda}(\mathbf{k}),a_{\lambda^{\prime}}^{\dag}(\mathbf{k}^{\prime}) ]=\delta_{\lambda\lambda^{\prime}}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})$. The equation of motion for $u(\tau,k)$ is +(c\_T\^2k\^2- )u=0, \[eom1\] where (,k)= \_(,k) [z\_T]{}, z\_T= [aM\_p [ c\_T\^[-1]{} ]{}2]{}.\[zt\] Initially, the perturbations are deep inside the sound horizon, i.e., $c_T^2k^2 \gg \frac{d^2z_T/d\tau^2}{z_T}$, the initial state is the Bunch-Davies vacuum, thus $u\sim \frac{1}{\sqrt{2c_T k} }e^{-i c_T k\tau}$. The power spectrum of primordial GWs is P\_T=\_[=+,]{} \|\_ \|\^2= \|u \|\^2, aH/(c\_Tk) 1.\[pt\] The diminishment of $c_T$ may be regarded as c\_T= (-H\_[inf]{})\^p, \[cTn\]in which $p>0$, and $H_{inf}$ is the Hubble parameter during inflation, which is regarded as constant for simplicity. Additionally, Eq. (\[cTn\]) suggests ${{\dot c}_T\over H_{inf}c_T}=-p$. We set $dy=c_Td\tau$, thus Eq. (\[eom1\]) is rewritten as u\_[,yy]{}+(k\^2- )u=0, \[eom4\] where ${u}(y,k)= \gamma_{\lambda}(y,k) {z_T}$, $z_T= {aM_p { c_T^{-1/2} }\over 2}$ and the subscript ‘$,y$’ denotes $d/dy$. Note here $u(y,k)$ and $z_T$ are different from those in Eq. (\[eom1\]), but $\gamma_{\lambda}$ is still the same. The solution of Eq.(\[eom4\]) is u\_k(y)=[2]{}H\^[(1)]{}\_(-ky),\[cq002\] where H\^[(1)]{}\_(-ky)-i([2-ky]{})\^[()]{}, and $\nu=1+{1\over 2(1+p)}$. Thus the spectrum (\[pt\]) is P\_T=[4k\^3\^2 M\_P\^2]{} [c\_T\|u\|\^2a\^2]{}= [2\^[-p1+p]{}]{}\^2 ([12(1+p)]{}) [2H\_[inf]{}\^2\^2 M\_P\^2 c\_T]{} (-ky)\^[p1+p]{}, \[nT\] where $y={c_T\tau\over {1+p} }={-}{c_T\over {(1+p)}aH_{inf} }$. Therefore, n\_T=[p1+p]{} \[nTB\]is blue-tilt, which is $n_T\simeq p$ for $p\ll 1$ and $n_T\simeq 1$ for $p\gg 1$. Here, the running of $H_{inf}$ may contribute $-2\epsilon\sim -0.01$, which has been neglected. Thus, we obtain a blue-tilt GWs spectrum with $0<n_T\leqslant 1$. Here, both the scalar perturbation and the background are unaffected by additional operator (\[action\]). The background is set by (\[action1\]), which is the slow-roll inflation with $0<\epsilon\ll 1$, so the scalar spectrum is flat with a slightly red tilt, which is consistent with the observations. It is noticed that based on the effective field theory of inflation, the introducing of other operators may also result in the blue-tilt GWs spectrum [@Cannone:2014uqa][@Baumann:2015xxa], however, in [@Cannone:2014uqa] $n_T>0.1$ requires that the graviton have a large mass $m_{graviton}\simeq H_{inf}$, while in [@Baumann:2015xxa] $\|{{\dot c}_T\over H_{inf}c_T}\|\ll 1$ was implicitly assumed. It is well known that the blue-tilt GWs spectrum is the hallmark of the superinflation. Here, the scenario proposed is actually a disformal dual to the superinflation. We will discuss this issue in detail in Sec. \[app-disformal\]. The stochastic background of GWs {#twoB} -------------------------------- We will focus on the stochastic background of GWs from such a scenario of inflation. The present observations are still not able to put stringent constraints on $c_T$ at present (see, e.g., [@Amendola:2014wma][@Raveri:2014eea], also [@Moore:2001bv] for the constraint on the phase velocity and [@Blas:2016qmn] for the group velocity). Future observations may put more stringent constraints on $c_T$ [@Nishizawa:2014zna][@Nishizawa:2016kba]. However, we will not get involved in this issue too much and we will assume that $c_T(t)$ will return to $c_T=1$ at certain time before the end of inflation. Conventionally, one define $$\label{density} \Omega_{\text{gw}}(k, \tau_{0})=\frac{1}{\rho_{\text{c}}}\frac{d\rho_{\text{gw}}}{d\ln k}=\frac{k^{2}}{12 a^2_0H^2_0}P_{T}T^2(k,\tau_0)\,,$$ where $\rho_{\text{c}}=3H^{2}_0/\big(8\pi G\big)$, $\tau_{0}=1.41\times10^{4}$ Mpc, $a_0=1$, $H_0=67.8$ km s$^{-1}$ Mpc$^{-1}$, the reduced Hubble parameter $h=H/\big(100\, \text{km s}^{-1}\text{Mpc}^{-1}\big)$, and $\rho_{\text{gw}}$ is the energy density of relic GWs at present, so $\Omega_{gw}(k, \tau_{0})$ reflects the fraction of $\rho_{\text{gw}}$ per logarithmic frequency interval. The transfer function is [@Turner:1993vb][@Boyle:2005se][@Zhang:2006mja] T(k,\_[0]{})=, \[Tk\] where $k_{\text{eq}}=0.073\,\Omega_{\text{m}} h^{2}$ Mpc$^{-1}$ is that of the perturbation mode that entered the horizon at the equality of matter and radiation. We have neglected the effects of the neutrino free-streaming on $T(k,\tau_0)$ [@Weinberg:2003ur], which is actually negligible. The underlying assumption on the thermal history of the post-inflation universe is able to affect $T(k,\tau_{0})$ significantly, see e.g.[@Kuroyanagi:2014nba], but we will only focus on the simplest case described by Eq.(\[Tk\]). One generally parameterizes $P_T$ as P\_T = A\_T()\^[n\_T]{}, \[para1\] where $k_=0.01$ Mpc$^{-1}$ is the pivot scale. However, if $n_T>0.4$, one will have $P_T>1$ at high-frequency region ($f>10^5$Hz). The GWs with $P_T\sim 1$ will induce the same-order scalar perturbation at nonlinear order, e.g.[@Wang:2014kqa], which will result in the overproduction of primordial black holes at the corresponding scale, which is inconsistent with their abundance. The upper bound put by the production of primordial black holes is $P_T<0.4$ [@Nakama:2015nea]. In addition, the indirect upper bound given by the combination of CMB with lensing, BAO and BBN observations is $\Omega_{gw}<3.8\times 10^{-6}$ [@Pagano:2015hma], which also puts a strong constraint on $n_T$, i.e., $n_T<0.36$ at $95\%$ C.L. for $r=0.11$ [@Lasky:2015lej], otherwise $\Omega_{gw}$ at higher frequency will exceed this bound. However, in our scenario, $c_T(t)$ is assumed to return to unity at a certain time $t_c$ before the end of inflation, as has been mentioned. This means that the blue-tilt spectrum will acquire a cutoff around $k_c$, see Sec. \[app-cutoff\] for details, which may avoid the above constraints on $n_T$. We may parameterize the corresponding $P_T$ as P\_[T]{} = A\_T()\^[n\_T]{}, \[para2\] which is (\[para1\]) for $k\ll k_c$, and tends to a constant $A_T(\frac{k_c}{k_})^{n_T}$ for $k\gg k_c$. Though we will use (\[para1\]) and (\[para2\]) since we are mainly interested in the boosted blue-tilted spectrum, we should point out that $P_T$ will decrease at $k>k_c$ or $k\gg k_c$ (which may be out of the range we are interested in), if we assume that $c_T$ will increase back to unity. In such case, $P_T$ may be parameterized as P\_T= A\_T ()\^[n\_T]{} [ 11+([k]{}/[k\_c]{})\^[n\_[Tc]{}]{}]{}, \[para3\] where $n_{Tc}> n_T$, so that when $k\gg k_c$, $P_T=A_T ({k_c}/{k_})^{n_T}({k}/{k_c})^{n_T-n_{Tc}}$ has a red tilt. When $n_{Tc}=n_T$, (\[para3\]) is similar to (\[para2\]). We plot the stochastic background of our GWs in Fig.\[fig01\]. It is obvious that a blue-tilt primordial GWs with $n_T\gtrsim 0.4$ is able to contribute a large stochastic GWs background within the windows of Advanced LIGO/Virgo, which may be greater than the contribution from the incoherent superposition of all binary black hole coalescence. $n_T\gtrsim 0.4$ requires $p\gtrsim 2/3$ in (\[cTn\]), which suggests that the diminishment of $c_T$ in units of Hubble time is not too fast. It is also interesting to notice that if such a GWs background could be detected by Advanced LIGO/Virgo in upcoming observing runs, it will also be able to be detected by the space-based interferometers at a lower frequency band, such as eLISA, and China’s Taiji program in space, see Fig.\[fig02\], as well as the PTA, e.g.[@Lasky:2015lej][@Liu:2015psa]. ![The brown line is the stochastic GWs background from inflation with spectral index $n_T=0.45$ and tensor-to-scalar ratio $r=0.05$ at the CMB scale. O1, O2, and O5 curves, taken from [@TheLIGOScientific:2016wyq], are the current Advanced LIGO/Virgo sensitivity, the observing run (2016-2017) and (2020-2022) sensitivities at $1\sigma$ C.L., respectively. The blue curve is the GWs background generated by all binary black hole coalescence without excluding potentially resolvable binaries. []{data-label="fig01"}](ligo){width="55.00000%"} ![The green and the brown lines are the stochastic GWs backgrounds from inflation with $n_T=0.3$ in (\[para1\]) and $n_T=0.45$ in (\[para2\]), respectively. Both C1 and C4-lines are eLISA’s representative configurations given in [@Caprini:2015zlo]. The sensitivity curves of DECIGO and BBO are given in [@Kuroyanagi:2010mm]. The red dashed curve is Taiji’s sensitivity curve, see, e.g., [@Gao:2016tzv] for a preliminary report. Fig.\[fig01\] is actually the amplification of image at the frequency band 10-400 Hz in this figure.[]{data-label="fig02"}](omega){width="55.00000%"} Disformal dual to superinflation {#app-disformal} ================================= ![This sketch illustrates the evolutions of the primordial perturbations during inflation in our scenario. The brown line is $\sim 1/aH$. The blue line is $\sim c_T/aH$, which is the sound horizon of GWs. We assume that $c_T$ decrease to some value less than unit and begin to increase later, so that it could return to unity and both horizons coincides before or near the end of inflation. []{data-label="fig03"}](aH.eps){width="55.00000%"} The superinflation is the inflation with $\epsilon=-{\dot H}/H^2<0$, i.e. ${\dot H}>0$, which breaks the NEC. The model we proposed in Sec. \[model\], i.e., inflation with a diminishing $c_T=(-H_{inf}\tau)^p$, is actually disformally dual to superinflation. This can be inferred from the evolution of the GWs sound horizon. The perturbation mode outside the comoving sound horizon $1/(aH_{Per})$ of the perturbations[^6] (i.e., $k\ll a H_{Per}$) will freeze, while it will evolve inside $1/(aH_{Per})$. In inflation scenario, the spectrum of GWs generally has similar shape to that of the scalar perturbation, since both GWs and scalar perturbations have a comoving sound horizon $1/(aH_{Per})$ almost coincide with $1/(aH)$. Here, since the comoving sound horizon of GWs is $c_T/(aH)$, and its evolution is completely different from $1/(aH)$, the spectrum of GWs shows itself blue-tilt, see Fig.\[fig03\]. However, the tilt of $c_T/(aH)$-line in Fig.\[fig03\] is the same as that of the superinflation with $c_T=1$, see Fig.1 in [@Piao:2006jz]. This indicates the physical processes of horizon crossing of GWs modes are same in these two scenarios, thus will generate the same power spectra. In fact, these two scenarios can be connected by a disformal transformation. Bellow, we give the strict proof. We make a disformal redefinition of the metric [@Creminelli:2014wna] g\_c\_T\^[-1]{}. \[gmunu1\] with c\_T\^[1/2]{}dt,()c\_T\^[-1/2]{}a(t), \[tildea\]which makes (\[paction\]) become S\^[(2)]{}=d d\^3x [M\_p\^2\^28]{}\[newaction\] with ${\tilde c_T}=1$. Here, with $d\tilde{\tau}={d\tilde{t}/\tilde{a}}$, which implies =\^(-H\_[inf]{})\^p d=-(H\_[inf]{})\^p[(-)\^[p+1]{}p+1]{}, we have =c\_T\^[-1/2]{}a \~(-)\^[-[2+p2(1+p)]{}]{}. ==c\_T\^[-1/2]{}(H\_[inf]{}-)\~(-)\^[-[p2(1+p)]{}]{}. \[tildeH\]Thus the value of $\tilde H$ is gradually increasing. This suggests that after the disformal transformation the background is actually the superinflation with ${\tilde \epsilon}=-p/(2+p)$, which satisfies $-1\lesssim \tilde{\epsilon} <0$. The scenario with $\tilde{\epsilon}\ll -1$ is the slow expansion, which was implemented in [@Piao:2003ty]. The equation of motion for $u(\tilde{\tau},k)$ is +(k\^2- )u=0,\[eom2\] where ${u}(\tilde{\tau},k)= \gamma_{\lambda}(\tilde{\tau},k) {\tilde{z}_T}$ and $\tilde{z}_T= {\tilde{a}M_p/ 2}$. The initial state is still the Bunch-Davies vacuum $u\sim\frac{1}{\sqrt{2k} }e^{-ik\tilde{\tau}}$. The solution is u\_k()=[2]{} H\^[(1)]{}\_(-k), where H\^[(1)]{}\_(-k) -i([2-k]{})\^, and $\tilde{\nu}=1+{1\over2(1+p)}$. Thus the power spectrum is P\_T &= & \_[=+,]{} \|\_ \|\^2\ &=& [4k\^3\^2M\_p\^2\^2]{}(-k) [2\^[2+[11+p]{}]{}(-k)\^[2+[11+p]{}]{}]{}\ &=&[2\^2\^2 M\_p\^2]{} \^2([12(1+p)]{})(-k)\^[p1+p]{}\[Ptilde\]\ &=&[c\_Tk\^2\^3M\_p\^2]{}\^2([12(1+p)]{}) (kH\_[inf]{}\^p)\^[-[2+p1+p]{}]{}\ &=&[2H\_[inf]{}\^2\^2M\_p\^2c\_T]{} \^2([12(1+p)]{})(-ky)\^[p1+p]{}. This result is completely the same as Eq.(\[nT\]). When $p\ll1$, we have \^2([12(1+p)]{})1+0.27p+[O]{}(p\^2) in Eq.(\[Ptilde\]) and $\tilde{\epsilon }=-\frac{d\tilde{H}/d\tilde{t} }{\tilde{H}^2}\ll 1$. Thus with (\[Ptilde\]), we have P\_T=2[H]{}\^2/\^2 M\_P\^2, \[PT3\]i.e. Creminelli et.al’s result [@Creminelli:2014wna]. Actually, it is well known that the spectrum of GWs, as well as scalar perturbation, is independent of the disformal redefinition (\[gmunu1\]) of the metric [@Creminelli:2014wna][@Minamitsuji:2014waa]. An intuition argument for it is the comoving horizon of scalar perturbation = [1c\_T ]{}\~(-)\^[11+p]{}\~[1a H\_[inf]{}]{} i.e., the relation between the comoving wave number $k$ and the comoving sound horizon is not altered, where ${\tilde c}_s=1/c_T$ [@Creminelli:2014wna]. Conventionally, the superinflation breaks the NEC. Implementing the superinflation without the ghost instability is still a significant issue, e.g.[@Wang:2014kqa][@Cai:2014uka]. Here, we actually suggest such a superinflation scenario. It might be just a slow-roll inflation living in a disformal metric with $c_T$ gradually diminishing, however, if we see it with $c_T=1$, what we will feel is the superinflation. The violation of NEC in modified gravity does not necessarily mean ghost instability. Because the quadratic actions (\[paction\]) and (\[newaction\]) for the tensor (as well as those for scalar) are canonical, there is no ghost instability in both frames. Cutoff of blue spectrum {#app-cutoff} ======================= To avoid $P_T\sim 1$ at high frequency, we have to require that the diminishment of $c_T$ stop at a certain time $\tau_c$. Additionally, we assume that $c_T(t)$ will return to unity before the end of inflation, as in Sec. \[twoB\]. We assume that c\_T&=&(-H\_[inf]{})\^p\_c. We set $dy=c_Td\tau$. The solution of (\[eom4\]) is u\_2(y)=for $y>y_c$, and is $u_1(y)$ for $y<y_c$, which is actually (\[cq002\]), where $\nu=1+{1\over 2(1+p)}$, $y_c=c_{Tc}\tau_c$. When $-ky\ll1$, u\_2\|C\_1-C\_2\|. Thus the spectrum of primordial GWs is P\_T&=&[4k\^3\^2 M\_P\^2]{} [c\_T\|u\|\^2a\^2]{}= [2H\_[inf]{}\^2\^2M\_p\^2]{} f(p,y\_c,k), \[PT4\] where f(p,y\_c,k)&=&[4kc\_[Tc]{}]{}\|C\_1-C\_2\|\^2, and C\_1&=&-[i\^[3/2]{}16]{}, C\_2&=&[i\^[3/2]{}16]{}are set by the continuities of $u(y)$ and ${du/ dy}$ at $\tau_c$. We plot (\[PT4\]) in Fig.\[fig04\], and see that, although $P_T$ has a blue tilt, it is flat at a high frequency. We analytically calculate it as follows. ![$P_T/P_{T}^{inf}=f(p,y_c,k)$. The parameters of the magenta dashed and brown solid curves are $c_{Tc}=10^{-3}$ and $10^{-5}$, respectively, while we set $p=0.7$.[]{data-label="fig04"}](power-02.eps){width="55.00000%"} When $-ky_c\ll1$, C\_1&=&-2\^[-[4+5p2(1+P)]{}]{}e\^[iky\_c]{} [1]{}(-ky\_c)\^[-[6+5p2(1+p)]{}]{} [(3+2p2(1+p))1+p]{}\ &&, C\_2&=&2\^[-[4+5p2(1+P)]{}]{}e\^[-iky\_c]{}[1]{} (-ky\_c)\^[-[6+5p2(1+p)]{}]{}[(3+2p2(1+p))1+p]{}\ &&. We have f(p,y\_c,k)&=&[4kc\_[Tc]{}]{}\|C\_1-C\_2\|\^2\ &&[2\^[-[p1+p]{}]{}9(1+p)\^2c\_[Tc]{}]{}\^2([3+2p2(1+p)]{}) (6+5p)\^2(-ky\_c)\^[[p1+p]{}]{}. Thus the tilt $n_T={p\over 1+p}$, which is the same as (\[nT\]). When $-ky_c\gg1$, C\_1=e\^[[i4]{}-[i2]{}]{}, C\_2=e\^[[i4]{}-[i2]{}(+4ky\_c)]{}. We have f(p,y\_c,k) &&[1c\_[Tc]{}]{}. Thus the spectrum is flat. From the above result, we can infer that if $c_T$ slowly diminishes to a value less than unity during inflation and then increases back to unity before the end of inflation, $\Omega_{gw}$ could be strongly boosted at the frequency band of Advanced LIGO/Virgo. Discussion ========== In the inflation scenario, obtaining a blue GWs spectrum ($n_T>0.1$) without the ghost instability while reserving a scale-invariant scalar spectrum with a slightly red tilt is still a challenge. We find that if the propagating speed of GWs gradually diminishes during inflation, the power spectrum of primordial GWs will be strongly blue, while that of the scalar perturbation may be unaffected. It is well known that the blue-tilt GWs is the hallmark of superinflation [@Piao:2004tq][@Baldi:2005gk]. It may be implemented without ghost in G-inflation [@KYY], but it is difficult, however, to simultaneously give it a slightly red-tilt scalar spectrum [@Wang:2014kqa], see also [@Cai:2014uka]. Our scenario is actually a disformal dual to the superinflation, see Sec. \[app-disformal\]. In this duality, our background is actually a slow-roll inflation living in a disformal metric with $c_T$ gradually diminishing. However, if we see it with $c_T=1$, what we will feel is the superinflation, but there is no ghost instability. Thus our work might offer a far-sighted perspective on superinflation. The blue tilt obtained is $0<n_T\lesssim 1$, which may significantly boost the stochastic GWs background at the frequency band of Advanced LIGO/Virgo, as well as the space-based detectors. This indicates that the primordial GWs recording the origin of the universe may be potentially measurable by the corresponding experiments. To conclude, if a stochastic background of GWs is detected by Advanced LIGO/Virgo in the upcoming observing runs, it also possibly comes from the primordial inflation, and encodes the physics beyond GR at inflation scale. Acknowledgments This work is supported by NSFC, No. 11222546, 11575188, and the Strategic Priority Research Program of Chinese Academy of Sciences, No. XDA04000000. We thank Cong-Feng Qiao and Yun-Kau Lau for suggesting that we use the sensitivity curves of China’s Taiji program in space, which will appear in a full work report. Scalar perturbation {#scalar} =================== We work with the ADM metric $$\begin{aligned} g_{\mu\nu}=\left(\begin{array}{cc}N_kN^k-N^2&N_j\\N_i& h_{ij}\end{array}\right),~~ g^{\mu\nu}=\left(\begin{array}{cc}-N^{-2}&\frac{N^j}{N^2}\\ \frac{N^i}{N^2}& h^{ij}-\frac{N^iN^j}{N^2}\end{array}\right)\,,\end{aligned}$$ where $h_{ij}=a^2e^{2\zeta}(e^{\gamma})_{ij}$, and $\gamma_{ii}=0=\partial_i\gamma_{ij}$. Generally, $N=1+\alpha$ and $N_i=\partial_i\beta$ are set for the scalar perturbations . It is convenient to define the normal vector of 3-dimensional hypersurface $n_\mu=n_0{dt/dx^\mu}=(n_0,0,0,0)$ and $n^{\mu}=g^{\mu\nu}n_\nu$. Using the normalization $n_\mu n^\mu=-1$, one has $n_0=-N$, which suggests $n_\mu=(-N,0,0,0), n^\mu=(\frac{1}{N},\frac{N^i}{N})$, and the 3-dimensional induced metric, orthogonal to the normal vector, i.e., $H_{\mu\nu}n^\nu=0$, can be defined to be $H_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu$, $$\begin{aligned} H_{\mu\nu}=\left(\begin{array}{cc}N_kN^k&N_j\\N_i& h_{ij}\end{array}\right),~~ H^{\mu\nu}=\left(\begin{array}{cc}0&0\\ 0& h^{ij} \end{array}\right).\end{aligned}$$ The covariant derivative associated with $H_{\mu\nu}$ is $D_{\mu}$, which is applied to define the extrinsic curvature $K_{\mu\nu}$: K\_=[12N]{}(\_-D\_N\_-D\_N\_). We have & & K\_K\^-(K)\^2\ &=&[1(1+)\^2]{}{-6(-H)\^2 +4a\^[-2]{}e\^[-2]{}(-H)(\_i\_i+\_i\_i)\ && +a\^[-4]{}e\^[-4]{}}, where $\delta K_{\mu\nu}=K_{\mu\nu}-H_{\mu\nu}H$. Thus the quadratic action of scalar perturbation for (\[action1\]) and (\[action\]) is &&S\^[(2)]{}\_=dx\^4 M\_p\^2 { a\^3H\^2 \^2 - 27a\^3H\^2 \^2 + 9a\^3H\^2 \^2 - 18a\^3H\ && +a( )\^2 - 2 a\_i\_i -[1c\_T\^2]{} }. \[scalar1\]The constraints can be solved as &=&[H]{},\ \_i\_i&=&[c\_T\^2H]{}(a\^2H-\_i\_i). Inserting them into (\[scalar1\]), S\^[(2)]{}\_=dx\^4 [M\_p\^2]{} a\^3 is obtained. Therefore, the scalar perturbation is not affected by the operator $\delta K_{\mu\nu}\delta K^{\mu\nu}-(\delta K)^2$ at quadratic order. [99]{} B. P. Abbott [et al.]{} \[LIGO Scientific and Virgo Collaborations\], Phys. Rev. Lett. [116]{}, 6, 061102 (2016) \[arXiv:1602.03837 \[gr-qc\]\]. \[The LIGO Scientific and the Virgo Collaborations\], arXiv:1602.03847 \[gr-qc\]. T. Damour and A. Vilenkin, Phys. Rev. Lett. [85]{}, 3761 (2000) \[gr-qc/0004075\]; T. Damour and A. Vilenkin, Phys. Rev. D [71]{}, 063510 (2005) \[hep-th/0410222\]. M. Kamionkowski, A. Kosowsky and M. S. Turner, Phys. Rev. D [49]{}, 2837 (1994) \[astro-ph/9310044\]. P. S. B. Dev and A. Mazumdar, arXiv:1602.04203 \[hep-ph\]. A. A. Starobinsky, JETP Lett. [30]{}, 682 (1979). V. A. Rubakov, M. V. Sazhin and A. V. Veryaskin, Phys. Lett. [B115]{}, 189 (1982). P. A. R. Ade [et al.]{} \[BICEP2 and Keck Array Collaborations\], arXiv:1510.09217 \[astro-ph.CO\]. Y. S. Piao and Y. Z. Zhang, Phys. Rev. D [70]{}, 063513 (2004) \[astro-ph/0401231\]. M. Baldi, F. Finelli and S. Matarrese, Phys. Rev. D [72]{}, 083504 (2005) \[astro-ph/0505552\]. Z. G. Liu, Z. K. Guo and Y. S. Piao, Eur. Phys. J. C [74]{}, 8, 3006 (2014) \[arXiv:1311.1599 \[astro-ph.CO\]\]; Y. Cai, Y. T. Wang and Y. S. Piao, arXiv:1510.08716 \[astro-ph.CO\]. J. L. Cook, and L. Sorbo, Phys. Rev. D [85]{}, 023534 (2012) \[arXiv:1109.0022 \[astro-ph.CO\]\]. L. Sorbo, JCAP [1106]{}, 003 (2011) \[arXiv:1101.1525 \[astro-ph.CO\]\]. S. Mukohyama, R. Namba, M. Peloso and G. Shiu, arXiv:1405.0346 \[astro-ph.CO\]. R. Namba, M. Peloso, M. Shiraishi, L. Sorbo and C. Unal, arXiv:1509.07521 \[astro-ph.CO\]. R. Brustein, M. Gasperini, M. Giovannini and G. Veneziano, Phys. Lett. B [361]{}, 45 (1995) doi:10.1016/0370-2693(95)01128-D \[hep-th/9507017\]. M. Gasperini and G. Veneziano, Phys. Rept. [373]{}, 1 (2003) doi:10.1016/S0370-1573(02)00389-7 \[hep-th/0207130\]. M. Gasperini, “Elements of string cosmology” (Cambridge University Press, Cambridge UK, 2007). Y. Wang and W. Xue, JCAP [1410]{}, 10, 075 (2014) \[arXiv:1403.5817 \[astro-ph.CO\]\]. R. R. Metsaev and A. A. Tseytlin, Nucl. Phys. [B293]{}, 385 (1987). I. Antoniadis, J. Rizos and K. Tamvakis, Nucl. Phys. B [415]{}, 497 (1994) \[hep-th/9305025\]. S. Kawai, M. Sakagami and J. Soda, Phys. Lett. B [437]{}, 284 (1998) \[gr-qc/9802033\]; J. Soda, M. Sakagami and S. Kawai, gr-qc/9807056; S. Kawai and J. Soda, Phys. Lett. B [460]{}, 41 (1999) \[gr-qc/9903017\]. C. Cartier, E. J. Copeland and R. Madden, JHEP [0001]{}, 035 (2000) \[hep-th/9910169\]; C. Cartier, J. c. Hwang and E. J. Copeland, Phys. Rev. D [64]{}, 103504 (2001) \[astro-ph/0106197\]. K. i. Maeda and N. Ohta, Phys. Lett. B [597]{}, 400 (2004) \[hep-th/0405205\]; K. Akune, K. i. Maeda and N. Ohta, Phys. Rev. D [73]{}, 103506 (2006) \[hep-th/0602242\]. S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D [92]{}, 12, 124059 (2015) \[arXiv:1511.06776 \[gr-qc\]\]. S. Nojiri, S. D. Odintsov and V. K. Oikonomou, arXiv:1512.07223 \[gr-qc\]. M. Giovannini, Class. Quant. Grav. [33]{}, no. 12, 125002 (2016) doi:10.1088/0264-9381/33/12/125002 \[arXiv:1507.03456 \[astro-ph.CO\]\]. Y. Cai, Y. T. Wang and Y. S. Piao, JHEP [1602]{}, 059 (2016) \[arXiv:1508.07114 \[hep-th\]\]. Y. Cai, Y. T. Wang and Y. S. Piao, Phys. Rev. D [91]{}, 103001 (2015) \[arXiv:1501.06345 \[astro-ph.CO\]\]. J. Khoury and F. Piazza, JCAP [0907]{}, 026 (2009) \[arXiv:0811.3633 \[hep-th\]\]. M. Park and L. Sorbo, Phys. Rev. D [85]{}, 083520 (2012) \[arXiv:1201.2903 \[astro-ph.CO\]\]. C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JHEP [0803]{}, 014 (2008) \[arXiv:0709.0293 \[hep-th\]\]. P. Creminelli, J. Gleyzes, J. Nore$\tilde{\mathrm{n}}$a and F. Vernizzi, Phys. Rev. Lett. [113]{}, 23, 231301 (2014) \[arXiv:1407.8439 \[astro-ph.CO\]\]. D. Cannone, G. Tasinato and D. Wands, JCAP [1501]{}, 01, 029 (2015) \[arXiv:1409.6568 \[astro-ph.CO\]\]. D. Baumann, H. Lee and G. L. Pimentel, arXiv:1507.07250 \[hep-th\]. L. Amendola, G. Ballesteros and V. Pettorino, Phys. Rev. D [90]{}, 043009 (2014) \[arXiv:1405.7004 \[astro-ph.CO\]\]. M. Raveri, C. Baccigalupi, A. Silvestri and S. Y. Zhou, Phys. Rev. D [91]{}, 6, 061501 (2015) \[arXiv:1405.7974 \[astro-ph.CO\]\]. G. D. Moore and A. E. Nelson, JHEP [0109]{}, 023 (2001) doi:10.1088/1126-6708/2001/09/023 \[hep-ph/0106220\]. D. Blas, M. M. Ivanov, I. Sawicki and S. Sibiryakov, arXiv:1602.04188 \[gr-qc\]. A. Nishizawa and T. Nakamura, Phys. Rev. D [90]{}, no. 4, 044048 (2014) doi:10.1103/PhysRevD.90.044048 \[arXiv:1406.5544 \[gr-qc\]\]. A. Nishizawa, Phys. Rev. D [93]{}, no. 12, 124036 (2016) doi:10.1103/PhysRevD.93.124036 \[arXiv:1601.01072 \[gr-qc\]\]. M. S. Turner, M. J. White and J. E. Lidsey, Phys. Rev. D [48]{}, 4613 (1993) \[astro-ph/9306029\]. L. A. Boyle and P. J. Steinhardt, Phys. Rev. D [77]{}, 063504 (2008) \[astro-ph/0512014\]. Y. Zhang, X. Z. Er, T. Y. Xia, W. Zhao and H. X. Miao, Class. Quant. Grav. [23]{}, 3783 (2006) \[astro-ph/0604456\]. S. Weinberg, Phys. Rev. D [69]{}, 023503 (2004) \[astro-ph/0306304\]. S. Kuroyanagi, T. Takahashi and S. Yokoyama, JCAP [1502]{}, 003 (2015) \[arXiv:1407.4785 \[astro-ph.CO\]\]. T. Nakama and T. Suyama, Phys. Rev. D [92]{}, 12, 121304 (2015) \[arXiv:1506.05228 \[gr-qc\]\]. L. Pagano, L. Salvati and A. Melchiorri, arXiv:1508.02393 \[astro-ph.CO\]. P. D. Lasky [et al.]{}, arXiv:1511.05994 \[astro-ph.CO\]. X. J. Liu, W. Zhao, Y. Zhang and Z. H. Zhu, Phys. Rev. D [93]{}, 2, 024031 (2016); W. Zhao, Y. Zhang, X. P. You and Z. H. Zhu, Phys. Rev. D [87]{}, 12, 124012 (2013) \[arXiv:1303.6718 \[astro-ph.CO\]\]; M. L. Tong and Y. Zhang, Phys. Rev. D [80]{}, 084022 (2009) \[arXiv:0910.0325 \[gr-qc\]\]; C. Caprini [et al.]{}, arXiv:1512.06239 \[astro-ph.CO\]. S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D [83]{}, 043514 (2011) doi:10.1103/PhysRevD.83.043514 \[arXiv:1010.5246 \[astro-ph.CO\]\]. W. Gao [et al.]{}, Science China Physics, Mechanics and Astronomy 58.12 (2015) 1-41, \[arXiv:1601.07050 \[astro-ph.IM\]\]. Y. S. Piao, Phys. Rev. D [73]{}, 047302 (2006) \[gr-qc/0601115\]; Y. S. Piao and E. Zhou, Phys. Rev. D [68]{}, 083515 (2003) \[hep-th/0308080\]; Z. G. Liu, J. Zhang and Y. S. Piao, Phys. Rev. D [84]{}, 063508 (2011) \[arXiv:1105.5713\]; Y. Cai and Y. S. Piao, arXiv:1601.07031 \[hep-th\]. M. Minamitsuji, Phys. Lett. B [737]{}, 139 (2014) \[arXiv:1409.1566 \[astro-ph.CO\]\]. Y. F. Cai, J. O. Gong, S. Pi, E. N. Saridakis and S. Y. Wu, Nucl. Phys. B [900]{}, 517 (2015) \[arXiv:1412.7241 \[hep-th\]\]. T. Kobayashi, M. Yamaguchi and J. i. Yokoyama, Phys. Rev. Lett. [105]{}, 231302 (2010) \[arXiv:1008.0603 \[hep-th\]\]. [^1]: caiyong13@mails.ucas.ac.cn [^2]: wangyutong12@mails.ucas.ac.cn [^3]: yspiao@ucas.ac.cn [^4]: It is found that in the pre-big bang scenario (obtained in the context of string cosmology) the primordial GWs spectrum is blue [@Brustein:1995ah][@Gasperini:2002bn][@Gasperini:2007zz]. [^5]: In [@Giovannini:2015kfa], the author investigated the effect induced by the running of GWs’ refractive index $n(\tau)$, which is similar to that of $c_T$. But note that $c_T\neq 1/n$, as can be seen from the difference between Eq.(\[paction\]) here and the Eq.(2.8) in [@Giovannini:2015kfa]. [^6]: Here, $H_{Per}$ is defined as $H_{Per}={(z_T''/z_T)^{1/2}\over a c_T}$, where $z_T$ is given by Eq.(\[zt\]). For $c_T\sim (-\tau)^p$ and $a\sim(-\tau)^{-1}$, we have =\~[c\_TaH]{}.
--- abstract: '0.3in We present a combined measurement of the production cross section of $VZ$ ($V=W$ or $Z$) events in final states containing charged leptons (electrons or muons) or neutrinos, and heavy flavor jets, using data collected by the CDF and DØ detectors at the Fermilab Tevatron Collider. The analyzed samples of $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV correspond to integrated luminosities of – . Assuming the ratio of the production cross sections $\sigma(WZ)$ and $\sigma(ZZ)$ as predicted by the standard model, we measure the sum of the $WZ$ and $ZZ$ cross sections to be . This is consistent with the standard model prediction and corresponds to a significance of standard deviations above the background-only hypothesis.' author: - 'The TEVNPH Working Group[^1]' title: 'Combined CDF and measurement of $\bm{WZ}$ and $\bm{ZZ}$ production in final states with $\bm{b}$-tagged jets\' --- 0.5in Preliminary Results for the Moriond 2012 Conferences Introduction {#intro} ============ 0[[@dzWHl; @dzZHv; @dzZHl]]{} Studies on the production of $VV$ ($V=W,Z$) boson pairs provide an important test of the electroweak sector of the standard model (SM). In $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV, the next-to-leading order (NLO) SM cross sections for these processes are $\sigma(WW)=\wwnlo\pm\wwnloe$ pb, $\sigma(WZ)=\wznlo\pm\wznloe$ pb and $\sigma(ZZ)=\zznlo\pm\zznloe$ pb [@dibo]. These cross sections assume both $\gamma^{}$ and $Z^{\circ}$ components in the neutral current exchange and corresponding production of dilepton final states in the region 75 $\leq m_{\ell^+\ell^-} \leq$ 105 GeV/$c^2$. Measuring a significant departure in cross section or deviations in the predicted kinematic distributions would indicate the presence of anomalous gauge boson couplings [@bib:anocoups] or new particles in extensions of the SM [@bib:newphen]. The $VV$ production in $\pp$ collisions at the Fermilab Tevatron Collider has been observed in fully leptonic decay modes [@bib:leptonic] and in semi-leptonic decay modes [@bib:hadronic], where the combined $WW+WZ$ cross section was measured. Recently, the DØ experiment presented evidence for $WZ$ and $ZZ$ production in semileptonic decays with a $b$-tagged final state [@dzDibosonCombo]. The $WZ$ and $ZZ$ production cross sections, as well as their sum, were measured in final states where one of the $Z$ bosons decays into $\bb$ (although there is some signal contribution from $\wcs$, $\zcc$) and the other weak boson decays to charged leptons or neutrinos ($\wlv$, $\zvv$, or $\zll$, with $\ell=e,\mu$). In this note we report an improved measurement of the $WW+ZZ$ production cross section in such final states based on the combination of the results from [@dzDibosonCombo], with a corresponding new set of CDF analyses [@cdfDibosonCombo]. This analysis is relevant as a proving ground for the combined Tevatron search for a low-mass Higgs boson produced in association with a weak boson and decaying into a $\bb$ pair [@bib:higgs] since it shares the same selection criteria as well as analysis and combination techniques. Summary of Contributing Analyses {#analyses} ================================ This result is the combination of three CDF analyses and three analyses 0 outlined in Table \[tab:chans\]. These analyses utilize data corresponding to integrated luminosities ranging from to , collected with the CDF [@cdf] and [@dzero] detectors at the Fermilab Tevatron Collider, and they are organized into multiple sub-channels for each different configuration of final state particles. To facilitate proper combination of signals, the analyses from a given experiment are constructed to use mutually exclusive event selections. In the analyses , events containing an isolated electron or muon, and two or three jets are selected (exactly two jets in the case of the CDF analysis). The presence of a neutrino from the $W$ decay is inferred from a large imbalance of transverse momentum ($\met$). The analyses select events containing large $\met$ and two or three jets (exactly two jets in the case of the analysis). Finally, in the analyses events are required to contain two electrons or two muons and at least two jets. In the case of the CDF analysis, events with two or three jets are analyzed separately. In the and analyses as well as the CDF analysis, each lepton flavor of the $W/Z$ boson decay ($\ell=e,\mu$) is treated as an independent channel. In the case of the CDF analysis lepton types are separated into four different channels based on their purity and location within the detector. To ensure that event samples used for the different analyses do not overlap, the analyses reject events in which a second isolated electron or muon is identified, and the analyses reject events in which any isolated electrons or muons are identified. To isolate the $\zbb$ decays, algorithms for identifying jets consistent with the decay of a heavy-flavor quark are applied to the jets in each event candidate ($b$-tagging). All of the analyses, as well as the CDF and analyses, use multivariate discriminants based on sets of kinematic variables sensitive to displaced decay vertices and tracks within jets with large transverse impact parameters relative to the hard-scatter vertices. The algorithm is a boosted decision tree discriminant which builds upon the previously utilized neural network $b$-tagging tool [@bib:btagnn], while the CDF algorithm [@bib:HOBIT] is based on a neural network discriminant. In both cases, a spectrum of increasingly stringent $b$-tagging operating points is constructed through the use of progressively higher requirements on the minimum output of the $b$-tagging discriminant. The analyses are separated into two groups: a double-tag (DT) group in which two of the jets are $b$-tagged with a loose tag requirement ( and ) or one loose and one tight tag requirement (); and an orthogonal single-tag (ST) group in which only one jet has a loose ( and ) or tight () $b$-tag. A typical per-jet $b$ efficiency and fake rate for the loose (tight) $b$-tag selection is about 80% (50%) and 10% (0.5%), respectively. The corresponding efficiency for jets from $c$-quarks is 45% (12%). The and analyses also use the output of the $b$-tagging algorithm as an additional input to the discriminants used in the final signal extraction. Candidate events in the CDF and analyses are also separated into channels based on tight and loose tagging definitions. Events with two tight tags (TT), one tight and one loose tag (TL), two loose tags (LL), and a single tight tag (Tx) are used by both analyses. The CDF analysis also considers events with a single loose tag (Lx). A typical per-jet efficiency and fake rate for the CDF loose (tight) neural network $b$-tag selection is about 70% (45%) and 7% (0.6%), respectively. The CDF analysis utilizes a tight $b$-tagging algorithm [@bib:SecVtx] based on reconstruction of a displaced secondary vertex and a loose $b$-tagging algorithm [@bib:JetProb] that assigns a likelihood for the tracks within a jet to have originated from a displaced vertex. Based on the output of these algorithms events with two tight tags (SS) and those with one tight tag and one loose tag (SJ) are separated into independent analysis channels. The signal in all of the double-tag samples is expected to be primarily composed of events with $\zbb$ decays, with smaller contributions from $\zcc$ and $\wcs$ decays. In the single-tag samples, which are defined by less stringent requirements on the $b$-jet content of the event, the contributions from the three decay modes are comparable. Experiment Channel Luminosity () Reference ------------ --------------------------- --------------- ----------- CDF , TT/TL/Tx/LL/Lx, 2 jets 9.5 [@cdfWHl] CDF , SS/SJ, 2/3 jets 9.5 [@cdfZHv] CDF , TT/TL/Tx/LL, 2/3 jets 9.5 [@cdfZHl] , ST/DT, 2/3 jets 7.5 [@dzWHl] , ST/DT, 2 jets 8.4 [@dzZHv] , ST/DT, $\geq$ 2 jets 7.5 [@dzZHl] : \[tab:chans\]List of analysis channels and their corresponding integrated luminosities. See Sect. \[analyses\] for details ($\ell=e, \mu$). The primary background is from $W/Z$+jets, which is modeled with [@alpgen] by both CDF and DØ. The backgrounds from multijet production are measured from control samples in the data. At DØ the other backgrounds are generated with and [@singletop], with [@pythia] providing parton-showering and hadronization. At CDF most backgrounds from other SM processes are modeled using Monte Carlo samples. Background rates are normalized either to next-to-leading order (NLO) or higher-order theory calculations or to data control samples. The DØ and both experiment’s analyses normalize $W/Z$+jets backgrounds to data, whereas the the CDF and both experiment’s analyses normalize them to the predictions from . The fraction of the $W/Z$+jets in which the jets arise from heavy quarks ($b$ or $c$) is obtained from NLO calculations using [@mcfm] at DØ while at CDF the prediction from is corrected based on a data control region. The background from events is normalized to the approximate NNLO cross section [@ttbar_xsec]. The $s$-channel and $t$-channel cross sections for the production of single-top quarks are from approximate NNLO+NNLL calculations [@schan_top_xsec] and approximate NNNLO+NLL calculations [@tchan_top_xsec], respectively. The background from $WW$ events is normalized to NLO calculations from [@dibo]. All Monte Carlo samples are passed through detailed [geant]{}-based simulations [@geant] of the CDF and D0 detectors. The analyses use multivariate discriminants (MVA) based on decision trees as the final variables for extracting the $VZ$ signal from the backgrounds. These decision trees are trained to discriminate the $VZ$ signal from the backgrounds using the same set of discriminant variables as in the corresponding Higgs analyses. The CDF analyses follow the same strategy, using neural network-based discriminants instead for signal-to-background discrimination. Systematic Uncertainties ======================== Systematic uncertainties differ between experiments and analyses, and they affect the normalizations and the differential distributions (shapes) of the predicted signal and background templates in correlated ways. The combined result incorporates the sensitivity of predictions to values of nuisance parameters and takes into account correlations in these parameters both within each individual experiment and between experiments. The largest uncertainty contributions and their correlations between and within the two experiments are discussed here. Further details on the individual analyses are available in Refs. . ### Correlated Systematics between CDF and The uncertainties on measurements of the integrated luminosities are 5.9% (CDF) and 6.1% (). Of these values, 4% arises from the uncertainty on the inelastic $\pp$ scattering cross section, which is correlated between CDF and DØ. CDF and also share the assumed values and uncertainties on the cross sections for $WW$ production and top-quark production processes ( and single top). In most analyses determination of the multijet (“QCD”) background involves data control samples, and the methods used differ between CDF and DØ, and even between analyses within the collaborations. Therefore, there is no assumed correlation in the predicted rates of this background between analysis channels. Likewise, calibrations of quantities such as the fake lepton rate, $b$-tag efficiencies, and mistag rates are performed by each collaboration using independent data samples and different methods, and are treated as uncorrelated. Similarly, different techniques are used to estimate background rates for $W/Z$+heavy flavor backgrounds and the associated uncertainties are taken as uncorrelated. ### Correlated Systematic Uncertainties for CDF The dominant systematic uncertainties for the CDF analyses are shown in Appendix Tables \[tab:cdfsystwh1\] and \[tab:cdfsystwh2\] for the channels, in Table \[tab:cdfsystzhvv\] for the channels, and in Tables \[tab:cdfllbb1\] and \[tab:cdfllbb2\] for the channels. Each source induces a correlated uncertainty across all of CDF’s channels’ signal and background contributions which are sensitive to that source. The largest uncertainties on signal arise from measured $b$-tagging efficiencies, jet energy scale, and other Monte Carlo modeling. Shape dependencies of templates on jet energy scale, $b$-tagging, and gluon radiation (“ISR” and “FSR”) are taken into account for some analyses (see tables). Uncertainties on background event rates vary significantly for the different processes. The backgrounds with the largest systematic rate uncertainties are in general quite small. Such uncertainties are constrained through fits to the nuisance parameters and do not affect the result significantly. Since normalizations for the $W/Z$+heavy flavor backgrounds are obtained from data in the and analyses, the corresponding rate uncertainties associated with each analysis are treated as uncorrelated even within CDF. ### Correlated Systematic Uncertainties for The and analyses carry an uncertainty on the integrated luminosity of 6.1% [@lumi], while the overall normalization of the analysis is determined from the NNLO $Z/\gamma^$ cross section [@dyxsec] in data events near the peak of $\zll$ decays. The uncertainty from the identification and energy measurement of jets is $\sim$7%. The uncertainty arising from the $b$-tagging rate ranges from 1 to 10%. All analyses include uncertainties associated with lepton measurement and acceptances, which range from 1 to 9% depending on the final state. The largest contribution for all analyses is the theoretical uncertainty on the background cross sections at 7-20% depending on the analysis channel and specific background. The uncertainty on the expected multijet background is dominated by the statistics of the data sample from which it is estimated. Further details on the systematic uncertainties are given in Tables \[tab:d0systwh\]-\[tab:d0llbb1\]. All systematic uncertainties originating from a common source are taken to be 100% correlated, as detailed in Table \[tab:corr\]. Measurement of the $\bm{WZ+ZZ}$ Cross Section ============================================= The total $VZ$ cross section is determined from a maximum likelihood fit of the MVA distributions for the background and signal samples from the contributing analyses to the data. The cross section for the signal ($WZ+ZZ$) is a free parameter in the fit, but the ratio of the $WZ$ and $ZZ$ cross sections is fixed to the SM prediction. Events from $WW$ production are considered as a background. The fit is performed simultaneously on the distributions in all sub-channels. As a consistency check, we also determine the Bayesian posterior probability by integrating over the nuisance parameters. Here we report only the results from the maximum likelihood fit, but the results from the Bayesian method are consistent. ![image](dbFigs/tevDibosonCondensedFit_combo_Subtract.eps){height="0.2\textheight"} The combined fit for the total $VZ$ cross section distributions yields . This measurement is consistent with the NLO SM prediction of $\sigma(WW+WZ)=\vznlo\pm\vznloe$ pb [@dibo], as well as with the individual measurements from [@dzDibosonCombo], $\sigma(WW+WZ)=5.0 \pm 1.6$ pb, and from CDF [@cdfDibosonCombo], $\sigma(WW+WZ)=4.1 ^{+1.4}_{-1.3}$ pb. Based on the measured central value for the $VZ$ cross section and its uncertainties, the observed significance is estimated to be standard deviations (s.d.), while the expected significance is $\sim 4.8$ s.d. To visualize the sensitivity of the combined analysis, we calculate the expected signal over background ($s/b$) in each bin of the MVA distributions from the contributing analyses. Bins with similar $s/b$ are then combined to produce a single distribution, shown in Fig. \[fig:rfsub\]. The binning was chosen to keep the background fluctuations roughly of the same size as in the dijet mass distributions. Figure \[fig:mjj\] shows the distributions of the invariant mass of the dijet system, summed over all channels from CDF and DØ, after adjusting the signal and background predictions according to the results of the fit. Figure \[fig:mjj\_sub\] shows the background subtracted dijet mass distributions after the fit, demonstrating the presence of a hadronic resonance in the data consistent with the SM expectation, both in shape and normalization. ------------------------------------------------------ ------------------------------------------------------ ------------------------------------------------------ ![image](dbFigs/tevDBmjjST_combo.eps){width="2.4in"} ![image](dbFigs/tevDBmjjDT_combo.eps){width="2.4in"} ![image](dbFigs/tevDBmjjAT_combo.eps){width="2.4in"} [(a)]{} [(b)]{} [(c)]{} ------------------------------------------------------ ------------------------------------------------------ ------------------------------------------------------ --------------------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- ![image](dbFigs/tevDBmjjST_combo_Subtract.eps){width="2.4in"} ![image](dbFigs/tevDBmjjDT_combo_Subtract.eps){width="2.4in"} ![image](dbFigs/tevDBmjjAT_combo_Subtract.eps){width="2.4in"} [(a)]{} [(b)]{} [(c)]{} --------------------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- Summary ======= In summary, we combine analyses in the , , and ($\ell=e,~\mu$) final states from the CDF and DØ experiments to observe, with a significance of s.d., the production of $VZ$ ($V=W$ or $Z$) events. The analyzed samples correspond to to of $\pp$ collisions at $\sqrt{s}=1.96$ TeV. We measure the total cross section for $VZ$ production to be . This result demonstrates the ability of the Tevatron experiments to measure a SM production process with cross section of the same order magnitude as that expected for Higgs production from the same set of background-dominated final states containing two heavy-flavor jets used in our low mass Higgs searches. [99]{} J. M. Campbell and R. K. Ellis, Phys. Rev. D [60]{}, 113006 (1999). We used [MCFM]{} v6.0. Cross sections are computed using a choice of scale $\mu_0^2=M_V^2+p_T^2(V)$, where $V$ is the vector boson, and the MSTW2008 PDF set. K. Hagiwara, S. Ishihara, R. Szalapski, and D. Zeppenfeld, Phys. Rev. D [48]{} (1993). J. C. Pati and A. Salam, Phys. Rev. D [10]{}, 275 (1974); [11]{} 703(E) (1975);\ G. Altarelli, B. Mele, and M. Ruiz-Altaba, Z. Phys. C [45]{}, 109 (1989); [47]{}, 676(E) (1990);\ L. Randall and R. Sundrum, Phys. Rev. Lett. [83]{}, 3370 (1999);\ H. Davoudiasl, J. L. Hewett, and T. G. Rizzo, Phys. Rev. D [63]{}, 075004 (2001);\ H. He [et al.]{}, Phys. Rev. D [78]{}, 031701 (2008). T. Aaltonen [et al.]{} (CDF Collaboration), Phys. Rev. Lett. [104]{}, 201801 (2010);\ V. M. Abazov [et al.]{} (D0 Collaboration), Phys. Rev. Lett. [101]{}, 171803 (2008);\ V. M. Abazov [et al.]{} (D0 Collaboration), Phys. Lett. B [695]{}, 67 (2011);\ V. M. Abazov [et al.]{} (D0 Collaboration), Phys. Rev. D [84]{}, 011103 (2011);\ V. M. Abazov [et al.]{} (D0 Collaboration), arXiv:1201.5652 \[hep-ex\]. T. Aaltonen [et al.]{} (CDF Collaboration), Phys. Rev. Lett. [103]{}, 091803 (2009);\ T. Aaltonen [et al.]{} (CDF Collaboration), Phys. Rev. Lett. [104]{}, 101801 (2010);\ V. M. Abazov [et al.]{} (D0 Collaboration), arXiv:1112.0536 \[hep-ex\]. V. M. Abazov [et al.]{} (D0 Collaboration), Note 6260-CONF (2011). T. Aaltonen [et al.]{} (CDF Collaboration), CDF Conference Note 10805 (2012). V. M. Abazov [et al.]{} (D0 Collaboration), Phys. Rev. Lett. [104]{}, 071801 (2010);\ V. M. Abazov [et al.]{} (D0 Collaboration), Phys. Rev. Lett. [105]{}, 251801 (2010);\ V. M. Abazov [et al.]{} (D0 Collaboration), Phys. Lett. B [698]{}, 6 (2011);\ T. Aaltonen [et al.]{} (CDF Collaboration), Phys. Rev. Lett. [105]{}, 251802 (2010);\ T. Aaltonen [et al.]{} (CDF Collaboration), Phys. Rev. Lett. [103]{}, 101802 (2009);\ T. Aaltonen [et al.]{} (CDF Collaboration), Phys. Rev. Lett. [104]{}, 141801 (2010). T. Aaltonen [et al.]{} (CDF Collaboration), CDF Conference Note 10796 (2012). T. Aaltonen [et al.]{} (CDF Collaboration), CDF Conference Note 10798 (2012). T. Aaltonen [et al.]{} (CDF Collaboration), CDF Conference Note 10799 (2012). V. M. Abazov [et al.]{} (D0 Collaboration), Note 6220-CONF (2011). V. M. Abazov [et al.]{} (D0 Collaboration), Note 6223-CONF (2011). V. M. Abazov [et al.]{} (D0 Collaboration), Note 6256-CONF (2011). D. Acosta [et al.]{} (CDF Collaboration), Phys. Rev. D [71]{}, 032001 (2005). V. M. Abazov [et al.]{} (D0 Collaboration), Nucl. Instrum. Methods Phys. Res. A [565]{}, 463 (2006);\ M. Abolins [et al.]{}, Nucl. Instrum. Methods Phys. Res. A [584]{}, 75 (2008);\ R. Angstadt [et al.]{}, Nucl. Instrum. Methods Phys. Res. A [622]{}, 298 (2010). V. M. Abazov [et al.]{} (D0 Collaboration), Nucl. Instrum. Methods Phys. Res. A [620]{}, 490 (2010). T. Aaltonen [et al.]{} (CDF Collaboration), CDF Conference Note 10803 (2012). D. Acosta [et al.]{} (CDF Collaboration), Phys. Rev. D [71]{}, 052003 (2005). A. Abulencia [et al.]{} (CDF and CDF - Run II Collaborations), Phys. Rev. D [74]{}, 072006 (2006). M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, J. High Energy Phys. [07]{}, 001 (2003). CompHEP, E. Boos [et al.]{}, Nucl. Instrum. Methods Phys. Res. A [534]{}, 250 (2004);\ E. Boos, V. Bunichev, L. Dudko, V. Savrin, and A. Sherstnev, Phys. Atom. Nucl. [69]{}, 1317 (2006). T. Sjöstrand, L. Lonnblad and S. Mrenna, arXiv:hep-ph/0108264. J. M. Campbell and R. K. Ellis, http://mcfm.fnal.gov/.\ J. M. Campbell, R. K. Ellis, Nucl. Phys. Proc. Suppl. [205-206]{}, 10 (2010). U. Langenfeld, S. Moch and P. Uwer, Phys. Rev. D [80]{}, 054009 (2009). N. Kidonakis, arXiv:1005.3330 \[hep-ph\] (2010);\ N. Kidonakis, Phys. Rev. D [81]{}, 054028 (2010). N. Kidonakis, Phys. Rev. D [74]{}, 114012 (2006). R. Brun, R. Hagelberg, M. Hansroul, and J. C. Lasalle, [GEANT: Simulation Program for Particle Physics Experiments. User Guide and Reference Manual]{}, CERN-DD-78-2-REV;\ S. Agostinelli [et al.]{}, Nucl. Instrum. Methods A [506]{}, 250 (2003). T. Andeen [et al.]{}, FERMILAB-TM-2365 (2007). R. Hamberg, W.L. van Neerven and W.B. Kilgore, Nucl Phys. B [359]{}, 343 (1991) \[Erratum-ibid. B [644]{}, 403 (2002)\]. T. Junk, Nucl. Instrum. Methods Phys. Res. A [434]{}, 435 (1999);\ A. L. Read, J. Phys. G [28]{}, 2693 (2002). W. Fisher, FERMILAB-TM-2386-E (2006). Additional Material =================== Source --------------------- ---------- ---------- ---------- -- Luminosity $\times$ $\times$ Normalization Jet Energy Scale $\times$ $\times$ $\times$ Jet ID $\times$ $\times$ $\times$ Electron ID/Trigger $\times$ $\times$ $\times$ Muon ID/Trigger $\times$ $\times$ $\times$ $b$-Jet Tagging $\times$ $\times$ $\times$ Background $\sigma$ $\times$ $\times$ $\times$ Background Modeling Multijet Background Signal $\sigma$ $\times$ $\times$ $\times$ : \[tab:corr\]The correlation matrix for the D0 analysis channels. Uncertainties marked with an $\times$ are considered 100% correlated across the affected channels. Otherwise the uncertainties are not considered correlated, or do not apply to the specific channel. The systematic uncertainties on the background cross section ($\sigma$) and the normalization are each subdivided according to the different background processes in each analysis. [^1]: The Tevatron New-Phenomena and Higgs Working Group can be contacted at TEVNPHWG@fnal.gov. More information can be found at http://tevnphwg.fnal.gov/.
--- abstract: 'The channeling of the ion recoiling after a collision with a WIMP in direct dark matter crystalline detectors produces a larger scintillation or ionization signal than otherwise expected. Channeling is a directional effect which depends on the velocity distribution of WIMPs in the dark halo of our Galaxy and could lead to a daily modulation of the signal. Here we compute upper bounds to the expected amplitude of daily modulation due to channeling using channeling fractions that we obtained with analytic models in prior work. After developing the general formalism, we examine the possibility of finding a daily modulation due to channeling in the data already collected by the DAMA/NaI and DAMA/LIBRA experiments. We find that even the largest daily modulation amplitudes (of the order of 10% in some instances) would not be observable for WIMPs in the standard halo in the 13 years of data taken by the DAMA collaboration. For these to be observable the DAMA total rate should be 1/40 of what it is or the total DAMA exposure should be 40 times larger. The daily modulation due to channeling will be difficult to measure in future experiments. We find it could be observed for light WIMPs in solid Ne, assuming no background.' author: - Nassim Bozorgnia - 'Graciela B. Gelmini' - Paolo Gondolo title: Daily modulation due to channeling in direct dark matter crystalline detectors --- Introduction ============ The channeling effect in crystals refers to the orientation dependence of charged ion penetration in crystals. Channeling occurs when ions propagating in a crystal along symmetry axes and planes suffer a series of small-angle scatterings that maintain them in the open “channels" in between the rows or planes of lattice atoms and thus penetrate much further into the crystal than in other directions and loose all their energy into electrons. In dark matter crystalline detectors, a channeled ion recoiling after a collision with a WIMP (Weakly Interacting Massive Particle) would give all its energy to electrons, thus the quenching factor is $Q\simeq 1$ instead of the usual $Q< 1$ for a non-channeled ion. Thus channeling increases the ionization or scintillation signal expected from a WIMP. The potential importance of the channeling effect for direct dark matter detection was first pointed out for NaI (Tl) by Drobyshevski [@Drobyshevski:2007zj] and subsequently by the DAMA collaboration [@Bernabei:2007hw] in 2007. In 2008, Avignone, Creswick, and Nussinov [@Avignone:2008cw] suggested that a daily modulation due to channeling could occur in NaI crystals, which would be a background free dark matter signature. Such a modulation of the rate due to channeling is expected to occur at some level because the “WIMP wind" arrives to Earth on average from a particular direction fixed to the Galaxy. Assuming that the dark matter halo is on average at rest with respect to the Galaxy, this is the direction towards which the Earth moves with respect to the Galaxy. Earth’s daily rotation naturally changes the direction of the “WIMP wind” with respect to the crystal axes, thus changing the amount of recoiling ions that are channeled vs non-channeled. This amounts to a daily modulation of the dark matter signal detectable via scintillation or ionization. Using analytic models of channeling which started to be developed in the 1960’s, shortly after the effect was discovered, mostly by Lindhard [@Lindhard:1965] and collaborators, we recently computed channeling probabilities as function of the recoil energy $E_R$ and initial direction $\hat{\bf q}$ of a recoiling ion in different materials [@BGGI; @BGG]. We used a recursion of the addition rule in probability theory (see Eq. 5.13 in Ref. [@BGGI]) to find the probability $\chi(E_R,\hat{\bf q})$ that a recoiling ion enters into any channel in terms of the channeling fractions for single channels $\chi_i(E_R,\hat{\bf q})$ that we computed (where the index i runs over all channels, both axial and planar). The channeling fractions for axial and planar channels are given in Eqs. 5.2 and 5.4 of Ref. [@BGGI], respectively. In our previous papers [@BGGI; @BGG], we also obtained the “geometric" channeling fraction $P_{\rm geometric}(E_R)$ in the crystals we studied, by averaging the channeling probability $\chi(E_R,\hat{\bf q})$ over the initial recoil directions $\hat{\bf q}$ (assuming an isotropic distribution in $\hat{\bf q}$) $$P_{\rm geometric}(E_R)=\frac{1}{4\pi}\int{\chi(E_R, \hat{\bf q})d\Omega_q}. \label{geometric-fraction}$$ This integral was computed using the Hierarchical Equal Area iso-Latitude Pixelization (HEALPix) [@HEALPix:2005] of the recoil direction sphere (see Appendix B of Ref. [@BGGI]). Here “geometric” refers to assuming that the distribution of recoil directions is isotropic. In reality, in a dark matter direct detection experiment, the distribution of recoil directions depends on the momentum distribution of the incoming WIMPs (see Section II). Fig. \[Fraction-Na\].a and \[Fraction-Na\].b reproduced from Ref. [@BGGI], show respectively upper bounds to some channeling fractions for single channels $\chi_i(E_R,\hat{\bf q})$ for Na recoils (with $c=1$) and geometric channeling fraction of Na and I recoiling ions in a NaI crystal at room temperature for $1~{\rm keV}<E_R<20$ keV. The parameter $c$ mentioned in the figures is a number that we expect to be between 1 and 2, which regulates the importance of temperature corrections (for details see Ref. [@BGGI]). The channeling fractions are typically smaller for larger values of $c$ thus setting $c=0$, which is an unrealistic value, we get the largest upper bound to the channeling fractions that our calculations provide. In the figures we used $c=0$ and $c=1$. Notice also that the results in the figures do not take into account dechanneling effects which should also decrease the channeling fractions (we do not know how to properly take into account these effects with our analytic methods). ![(Color online) Upper bounds to the (a) channeling fractions for single channels $\chi_i(E,\hat{\bf q})$ of Na recoils for axial (black lines) and planar (green/gray lines) channels with $c=1$, and (b) geometric channeling fraction $P_{\rm geometric}(E)$ of Na (solid lines) and I recoils (dashed lines) as a function of the recoil energy $E$ for $T=293$ K in with $c=0$ (green/gray) and $c=1$ (black), always without including dechanneling.[]{data-label="Fraction-Na"}](OneChannelFrac.eps "fig:"){height="137pt"} ![(Color online) Upper bounds to the (a) channeling fractions for single channels $\chi_i(E,\hat{\bf q})$ of Na recoils for axial (black lines) and planar (green/gray lines) channels with $c=1$, and (b) geometric channeling fraction $P_{\rm geometric}(E)$ of Na (solid lines) and I recoils (dashed lines) as a function of the recoil energy $E$ for $T=293$ K in with $c=0$ (green/gray) and $c=1$ (black), always without including dechanneling.[]{data-label="Fraction-Na"}](FractionNaI-RoomT.eps "fig:"){height="144pt"}\ In this paper, we use the (upper bounds to the) channeling probability $\chi(E_R,\hat{\bf q})$ and the actual differential recoil spectrum to compute the event rate, taking into account channeled and non-channeled recoils (see Section III, in particular Eqs. \[prob\] and \[def-Rate\] and compare them with Eq. \[geometric-fraction\]). We then use this rate to compute upper bounds to the amplitude of the daily modulation due to channeling expected in NaI crystals. In Section IV, we examine the possibility that such a daily modulation might be observable in the data accumulated by the DAMA collaboration. Angular distribution of recoil directions due to WIMPs ====================================================== Consider the WIMP-nucleus elastic collision for a WIMP of mass $m$ and a nucleus of mass $M$. The 3-dimensional “Radon transform" of the WIMP velocity distribution can be used to define the differential recoil spectrum as function of the recoil momentum $\vec{\bf q}$ [@Gondolo:2002] $$\frac{dR}{ dE_R~d\Omega_q}= \frac{\rho \sigma_0 S(q)}{4\pi m \mu^2} \hat{f}_{\rm lab}\!\left( \frac{q}{2\mu}, \hat{\bf q} \right) , \label{eq: rate}$$ where $E_R$ is the recoil energy, $d\Omega_q=d\phi d\cos\theta$ denotes an infinitesimal solid angle around the recoil direction $\hat{\bf q}= \vec{\bf q}/q$, $q=\|\vec{\bf q}\|$ is the magnitude of the recoil momentum, $\mu=mM/(m+M)$ is the reduced WIMP-nucleus mass, $q/2\mu= v_q$ is the minimum velocity a WIMP must have to impart a recoil momentum $q$ to the nucleus, or equivalently to deposit a recoil energy $E_R = q^2 /2M$, $ \rho$ is the dark matter density in the solar neighborhood, $\sigma_0$ is the total scattering cross section of the WIMP with a (fictitious) point-like nucleus, and $S(q)$ is the nuclear form factor normalized to 1. We concentrate here on WIMPs with spin-independent interactions, for which $\sigma_0$ is usually written in terms of the WIMP-proton cross section $\sigma_p$ [@Alenazi-Gondolo:2008] $$\sigma_0=\frac{\mu^2}{\mu_p^2}A^2\sigma_p, \label{sigma0}$$ where $\mu_p=m m_p/(m+m_p)$ is the WIMP-proton reduced mass and $A$ is the atomic number of the nucleus. We use the Helm form factor [@Helm:1956] $$S(q)=\|F_{SI}(q)\|^2=\left(\frac{3j_1(qR_1)}{qR_1}\right)^2 e^{-q^2 s^2},$$ where $$j_1(x)=\frac{\sin x}{x^2}-\frac{\cos x}{x}$$ is the first kind spherical Bessel function, $R_1$ is an effective nuclear radius, and $s$ is the nuclear skin thickness. Following Duda, Kemper, and Gondolo [@Duda:2007] we set $$R_1=\sqrt{c^2 + \frac{7}{3} \pi^2 a^2 - 5 s^2},$$ and take $s\simeq0.9$ fm, $a\simeq0.52$ fm, and $c\simeq (1.23 A^{1/3} - 0.6)$ fm. These parameters have been chosen to match the numerical integration of the Two-Parameter Fermi model of nuclear density [@Duda:2007]. The Maxwellian WIMP velocity distribution with respect to the Galaxy, with dispersion $\sigma_v$ and truncated at the escape speed $v_{\rm esc}$ is given by [@Gondolo:2002] $$f_{\rm WIMP}({\bf v})=\frac{1}{N_{\rm esc} (2\pi \sigma_v^2)^{3/2}}\exp{\left[ -\frac{({\bf v}+{\bf V}_{\rm lab})^2}{2 \sigma_v^2} \right]}, \label{VelDist}$$ for $\|{\bf v}+{\bf V}_{\rm lab}\|<v_{\rm esc}$, and zero otherwise, where $$N_{esc}=\mathop{\rm erf}\left(\frac{v_{\rm esc}}{\sqrt{2}\sigma_v}\right)-\sqrt{\frac{2}{\pi}}\frac{v_{\rm esc}}{\sigma_v}\exp{\left[-\frac{v_{\rm esc}^2}{2\sigma_v^2} \right]}. \label{Radon-transform}$$ Here we are assuming the detector has a velocity $\textbf{V}_{\rm lab}$ with respect to the Galaxy (thus $- \textbf{V}_{\rm lab}$ is the average velocity of the WIMPs with respect to the detector). ${\bf V}_{\rm lab}$ is defined in terms of the galactic rotation velocity ${\bf V}_{\rm {Gal Rot}}$ at the position of the Sun (or Local Standard of Rest (LSR) velocity), Sun’s peculiar velocity ${\bf V}_{\rm {Solar}}$ in the LSR, Earth’s translational velocity ${\bf V}_{\rm {Earth Rev}}$ with respect to the Sun, and the velocity of Earth’s rotation around itself ${\bf V}_{\rm {Earth Rot}}$ (see Appendix B), $${\bf V}_{\rm {lab}}={\bf V}_{\rm {Gal Rot}}+{\bf V}_{\rm {Solar}}+{\bf V}_{\rm {Earth Rev}}+{\bf V}_{\rm {Earth Rot}}. \label{Vlab}$$ In this paper we take $V_{\rm GalRot}$ either 220 km/s or 280 km/s, as reasonable low and high values (as done in Ref [@Green-2010]), which correspond to $V_{\rm lab}$ either 228.4 km/s or 288.3 km/s, respectively (see Appendix B for details). Ref. [@Kuhlen] gives 100 km/s as the smallest estimate for the 1D velocity dispersion, which corresponds to a 3D dispersion $\sqrt{3}$ times larger, i.e. $\sigma_v=173$ km/s. Thus here we take $\sigma_v$ either 173 km/s or 300 km/s [@Gondolo:2002]. In order to visualize the arrival directions of WIMPs, we will plot $f_{\rm WIMP}(\hat{\bf v},v_q)$, the number of WIMPs per solid angle in the direction $\hat{\bf v}$ in several figures. If we limit ourselves to the WIMPs with speed higher than ${v_q}$, then $$f_{\rm WIMP}(\hat{\bf v},v_q)=\int_{v_q}^{v_{\rm max}(\hat{\bf v})}{f_{\rm WIMP}({\bf v}) v^2 dv}. \label{fWIMP}$$ The upper limit of the integral in Eq. \[fWIMP\] is such that $\|{\bf v}+{\bf V}_{\rm lab}\|=v_{\rm esc}$ and depends on the direction $\hat{\bf v}$, since $({\bf v}+{\bf V}_{\rm lab})^2=v^2+2v~ \hat{\bf v}.{\bf V}_{\rm lab} +V_{\rm lab}^2$, $$v_{\rm max}(\hat{\bf v})=-\hat{\bf v}.{\bf V}_{\rm lab}+\sqrt{(\hat{\bf v}.{\bf V}_{\rm lab})^2-{\bf V}_{\rm lab}^2+v_{\rm esc}^2}~,$$ and $$f_{\rm WIMP}(\hat{\bf v},v_q)=\frac{\exp{\left(-\frac{V_{\rm lab}^2}{2 \sigma_v^2}\right)}}{N_{\rm esc} (2\pi \sigma_v^2)^{3/2}} \int_{v_q}^{v_{\rm max}(\hat{\bf v})}\exp{\left(\frac{-v^2}{2 \sigma_v^2}\right)} \exp{\left(\frac{-2v~\hat{\bf v}.{\bf V}_{\rm lab}}{2 \sigma_v^2}\right)} v^2 dv.$$ This integral can be solved analytically and the result is in terms of error functions, $$\begin{aligned} f_{\rm WIMP}(\hat{\bf v},v_q)&=\frac{\exp{\left(-\frac{V_{\rm lab}^2}{2 \sigma_v^2}\right)}}{N_{\rm esc} (2\pi \sigma_v^2)^{3/2}}\left(\frac{\sigma_v}{2}\right) \bigg\{\sqrt{2\pi}\left[(\hat{\bf v}.{\bf V}_{\rm lab})^2 + \sigma_v^2\right] \exp{\left(\frac{(\hat{\bf v}.{\bf V}_{\rm lab})^2}{2\sigma_v^2}\right)}\nonumber\\ &\left[\textrm{erf}\left(\frac{\hat{\bf v}.{\bf V}_{\rm lab}+v_{\rm max}(\hat{\bf v})}{\sqrt{2}\sigma_v}\right)-\textrm{erf}\left(\frac{\hat{\bf v}.{\bf V}_{\rm lab}+v_q}{\sqrt{2}\sigma_v}\right)\right]\nonumber\\ &+(2 \sigma_v)\bigg[\left(\hat{\bf v}.{\bf V}_{\rm lab}-v_{\rm max}(\hat{\bf v})\right) \exp{\left(-\frac{v_{\rm max}(\hat{\bf v})(2\hat{\bf v}.{\bf V}_{\rm lab} +v_{\rm max}(\hat{\bf v}))}{2\sigma_v^2}\right)}\nonumber\\ &+ \left(-\hat{\bf v}.{\bf V}_{\rm lab}+v_q\right) \exp{\left(-\frac{v_q(2\hat{\bf v}.{\bf V}_{\rm lab} +v_q)}{2\sigma_v^2}\right)} \bigg]\bigg\}. \label{fTM}\end{aligned}$$ The maximum of $f_{\rm WIMP}(\hat{\bf v},v_q)$ happens when $\hat{\bf v}.{\bf V}_{\rm lab}=-V_{\rm lab}$, i.e. in the direction of the “WIMP wind” average velocity $-{\bf V}_{\rm lab}$. Dividing $f_{\rm WIMP}(\hat{\bf v},v_q)$ by this maximum we obtain a re-scaled distribution, a dimensionless number between 0 and 1, which we plot in Fig. \[Healpix-vq0\] (see the color scale/grayscale in the figure) on the sphere of velocity directions $\hat{\bf v}$ using the HEALPix pixelization [@HEALPix:2005] (see also Appendix B of Ref. [@BGGI]) for all WIMPs, which amounts to taking $v_q=0$. We took $V_{\rm lab}=288.3$ km/s, and $\sigma_v=300$ km/s or $\sigma_v=173$ km/s for Fig. \[Healpix-vq0\].a or b respectively. ![(Color online) WIMPs number density per solid angle $f_{\rm WIMP}(\hat{\bf v},v_q)$ (in Eq. \[fTM\]) for all WIMPs (namely $v_q=0$) re-scaled to be a number between 0 (black) and 1 (white) plotted on the sphere of velocity directions $\hat{\bf v}$ using the HEALPix pixelization for $V_{\rm {lab}}=288.3$ km/s and (a) $\sigma_v=300$ km/s and (b) $\sigma_v=173$ km/s. The arrow shows the direction of the average velocity of the WIMP wind, $-{\bf V}_{\rm lab}$. The North and South celestial poles are also indicated. The color scale/grayscale shown in the horizontal bar between black and white corresponds to values between 0 and 1 in increments of 0.05.[]{data-label="Healpix-vq0"}](fI-vq0-sigma300-V280.eps "fig:"){height="200pt"} ![(Color online) WIMPs number density per solid angle $f_{\rm WIMP}(\hat{\bf v},v_q)$ (in Eq. \[fTM\]) for all WIMPs (namely $v_q=0$) re-scaled to be a number between 0 (black) and 1 (white) plotted on the sphere of velocity directions $\hat{\bf v}$ using the HEALPix pixelization for $V_{\rm {lab}}=288.3$ km/s and (a) $\sigma_v=300$ km/s and (b) $\sigma_v=173$ km/s. The arrow shows the direction of the average velocity of the WIMP wind, $-{\bf V}_{\rm lab}$. The North and South celestial poles are also indicated. The color scale/grayscale shown in the horizontal bar between black and white corresponds to values between 0 and 1 in increments of 0.05.[]{data-label="Healpix-vq0"}](fI-vq0-sigma173-V280.eps "fig:"){height="200pt"} ![(Color online) WIMPs number density per solid angle $f_{\rm WIMP}(\hat{\bf v},v_q)$ (in Eq. \[fTM\]) for all WIMPs (namely $v_q=0$) re-scaled to be a number between 0 (black) and 1 (white) plotted on the sphere of velocity directions $\hat{\bf v}$ using the HEALPix pixelization for $V_{\rm {lab}}=288.3$ km/s and (a) $\sigma_v=300$ km/s and (b) $\sigma_v=173$ km/s. The arrow shows the direction of the average velocity of the WIMP wind, $-{\bf V}_{\rm lab}$. The North and South celestial poles are also indicated. The color scale/grayscale shown in the horizontal bar between black and white corresponds to values between 0 and 1 in increments of 0.05.[]{data-label="Healpix-vq0"}](ColorBar.eps "fig:"){height="30pt"}\ For a truncated Maxwellian WIMP velocity distribution with respect to the Galaxy, truncated at the escape speed $v_{\rm esc}$, the Radon-transform is [@Gondolo:2002] $$\hat{f}_{\rm lab}\!\left( \frac{q}{2\mu}, \hat{\bf q} \right)=\frac{1}{{N_{\rm esc}(2\pi \sigma_v^2)^{1/2}}}~{\left\{\exp{\left[-\frac{\left[ (q/2\mu) + \hat{\bf q} . {\bf V}_{\rm lab}\right]^2}{2\sigma_v^2}\right]}-\exp{\left[\frac{-v_{\rm esc}^2}{2\sigma_v^2}\right]}\right\}}, \label{fhatTM}$$ if $(q/2\mu) + \hat{\bf q} . {\bf V}_{\rm lab} < v_{\rm esc}$, and zero otherwise. The presence of $\hat{\bf q} . {\bf V}_{\rm lab}$ means that in order to compute the differential rate we need to orient the nuclear recoil direction $\hat{\textbf{q}}$ with respect to ${\bf V}_{\rm lab}$. The maximum of $\hat{ f}_{\rm lab} (\frac{q}{2 \mu},\hat{\textbf{q}})$ in Eq. \[fhatTM\] happens when $\hat{\bf q} . {\bf V}_{\rm lab}=-q/2\mu$, if $v_q=q/2 \mu < V_{\rm lab}$ (or in the direction of $-{\bf V}_{\rm lab}$ otherwise). Thus, we can re-scale $\hat{f}_{\rm lab}$ to obtain a dimensionless number between 0 and 1, $$\hat{f}^{\rm re-scaled}_{\rm lab} ={\left\{\exp{\left[-\frac{\left[ (q/2\mu) + \hat{\bf q} . {\bf V}_{\rm lab}\right]^2}{2\sigma_v^2}\right]}-\exp{\left[\frac{-v_{\rm esc}^2}{2\sigma_v^2}\right]}\right\}}\bigg/\bigg(1-\exp{\left[\frac{-v_{\rm esc}^2}{2\sigma_v^2}\right]}\bigg). \label{fhatTM-rescaled}$$ ![(Color online) (a) $f_{\rm WIMP}(\hat{\bf v},v_q)$ (in Eq. \[fTM\]) re-scaled to be between 0 and 1 plotted on the sphere of velocity directions $\hat{\bf v}$ and (b) $\hat{f}_{\rm lab}$ (re-scaled as in Eq. \[fhatTM-rescaled\]) plotted on the sphere of recoil directions using the HEALPix pixelization for I recoils with $E_R=10$ keV, $m=30$ GeV (thus $v_q=304.6$ km/s), $V_{\rm {lab}}=288.3$ km/s and $\sigma_v=300$ km/s. The arrow shows the direction of the average velocity of the WIMP wind, $-{\bf V}_{\rm lab}$. The North and South celestial poles are also indicated. The color scale/grayscale shown in the horizontal bar corresponds to values between 0 (black) and 1 (white) in intervals of 0.05. []{data-label="Healpix-1"}](fI-e10m30-sigma300-V280.eps "fig:"){height="200pt"} ![(Color online) (a) $f_{\rm WIMP}(\hat{\bf v},v_q)$ (in Eq. \[fTM\]) re-scaled to be between 0 and 1 plotted on the sphere of velocity directions $\hat{\bf v}$ and (b) $\hat{f}_{\rm lab}$ (re-scaled as in Eq. \[fhatTM-rescaled\]) plotted on the sphere of recoil directions using the HEALPix pixelization for I recoils with $E_R=10$ keV, $m=30$ GeV (thus $v_q=304.6$ km/s), $V_{\rm {lab}}=288.3$ km/s and $\sigma_v=300$ km/s. The arrow shows the direction of the average velocity of the WIMP wind, $-{\bf V}_{\rm lab}$. The North and South celestial poles are also indicated. The color scale/grayscale shown in the horizontal bar corresponds to values between 0 (black) and 1 (white) in intervals of 0.05. []{data-label="Healpix-1"}](fhatI-e10m30-sigma300-V280.eps "fig:"){height="200pt"} ![(Color online) (a) $f_{\rm WIMP}(\hat{\bf v},v_q)$ (in Eq. \[fTM\]) re-scaled to be between 0 and 1 plotted on the sphere of velocity directions $\hat{\bf v}$ and (b) $\hat{f}_{\rm lab}$ (re-scaled as in Eq. \[fhatTM-rescaled\]) plotted on the sphere of recoil directions using the HEALPix pixelization for I recoils with $E_R=10$ keV, $m=30$ GeV (thus $v_q=304.6$ km/s), $V_{\rm {lab}}=288.3$ km/s and $\sigma_v=300$ km/s. The arrow shows the direction of the average velocity of the WIMP wind, $-{\bf V}_{\rm lab}$. The North and South celestial poles are also indicated. The color scale/grayscale shown in the horizontal bar corresponds to values between 0 (black) and 1 (white) in intervals of 0.05. []{data-label="Healpix-1"}](ColorBar.eps "fig:"){height="30pt"}\ ![(Color online) Same as Fig. \[Healpix-1\] but for Na recoils and assuming $m=60$ GeV (so $v_q=196.7$ km/s) and $\sigma_v=173$ km/s (and all other parameters the same).[]{data-label="Healpix-2"}](fNa-e10m60-sigma173-V280.eps "fig:"){height="200pt"} ![(Color online) Same as Fig. \[Healpix-1\] but for Na recoils and assuming $m=60$ GeV (so $v_q=196.7$ km/s) and $\sigma_v=173$ km/s (and all other parameters the same).[]{data-label="Healpix-2"}](fhatNa-e10m60-sigma173-V280.eps "fig:"){height="200pt"} ![(Color online) Same as Fig. \[Healpix-1\] but for Na recoils and assuming $m=60$ GeV (so $v_q=196.7$ km/s) and $\sigma_v=173$ km/s (and all other parameters the same).[]{data-label="Healpix-2"}](ColorBar.eps "fig:"){height="30pt"}\ In Figs. \[Healpix-1\] and \[Healpix-2\] we present side by side the WIMPs velocity distribution, for WIMPs which can generate a signal of a certain energy $E$, namely with speed above $v_q$ (left panels) and the Radon transform (right panels) of the recoils of energy $E$ that WIMP collisions produce. In Fig. \[Healpix-1\].a and b we respectively plot $f_{\rm WIMP}(\hat{\bf v},v_q)$ on the sphere of WIMP velocity directions $\hat{\bf v}$ and $\hat{f}_{\rm lab}$ on the sphere of recoil directions (both re-scaled to be a number between 0 and 1) using the HEALPix pixelization [@HEALPix:2005] for I recoils assuming $V_{\rm lab}=288.3$ km/s, $E_R=10$ keV, $\sigma_v=300$ km/s and $m=30$ GeV. Fig. \[Healpix-2\].a and b show the same two distributions but for Na recoils and assuming $\sigma_v=173$ km/s and $m=60$ GeV (other parameters are the same). The color scale/grayscale plotted on the spheres indicate different values of the rescaled distributions: between 0 (black) and 1 (white) in intervals of 0.05. In Fig. \[Healpix-1\] the minimum WIMP speed required is $v_q=304.6$ km/s (I recoils), and since $v_q >V_{\rm lab}$, the maximum value of $\hat{f}^{\rm re-scaled}_{\rm lab}$, i.e. the maximum recoil rate, is in the direction of the “WIMP wind” average velocity, $-V_{\rm lab}$, which is shown with an arrow. In Fig. \[Healpix-2\] instead, $v_q=196.7$ km/s (Na recoils) and the maximum value of $\hat{f}^{\rm re-scaled}_{\rm lab}$ occurs when $-\hat{\bf q} . {\bf V}_{\rm lab}=v_q$, i.e. when $\hat{\bf q}$ is at an angle of $47^\circ$ of $-V_{\rm lab}$. Differential energy spectrum ============================ Let $p(E,E_R,\hat{\textbf{q}})dE$ be the probability that an energy $E$ is measured when a nucleus recoils in the direction $\hat{\textbf{q}}$ with initial energy $E_R$, normalized so that $$\int{p(E,E_R,\hat{\textbf{q}})dE}=1.$$ With our analytic approach we cannot estimate the importance of dechanneling mechanisms, such as the presence of lattice imperfections, impurities or dopants. Thus we disregard dechanneling, and assume that a recoiling nucleus can only either be channeled, in which case the measured energy is the whole initial recoil energy $E=E_R$ (first term in the following equation) or not channeled, in which case the measured energy is $E= Q E_R$ (second term), $$p(E,E_R,\hat{\textbf{q}})=\chi(E_R, \hat{\textbf{q}})\delta(E-E_R)+[1-\chi(E_R, \hat{\textbf{q}})]\delta(E-QE_R). \label{prob}$$ The first term accounts for the channeled (unquenched) events and the second term for the unchanneled (quenched) events, and $Q$ is the quenching factor. Using Eq. \[prob\] the differential energy spectrum, $$\frac{dR}{dE}=\int{\frac{dR}{dE_R d\Omega_q}p(E,E_R,\hat{\textbf{q}})d\Omega_q dE_R}, \label{def-Rate}$$ can be written as $$\begin{aligned} \frac{dR}{dE}&=&\int{\left[\chi(E,\hat{\textbf{q}})\, \frac{dR}{dE_R d\Omega_q}\bigg\|_{E_R=E} + [1-\chi(E/Q,\hat{\textbf{q}})]\,\frac{1}{Q} \, \frac{dR}{dE_R d\Omega_q}\bigg\|_{E_R=E/Q} \right]d\Omega_q}\nonumber\\ &=&\frac{dR}{dE}\bigg\|_{\rm U} +\;\int{\left[\chi(E,\hat{\textbf{q}})\, \frac{dR}{dE_R d\Omega_q}\bigg\|_{E_R=E} - \chi(E/Q,\hat{\textbf{q}})\frac{1}{Q} \, \frac{dR}{dE_R d\Omega_q}\bigg\|_{E_R=E/Q} \right]d\Omega_q},\end{aligned}$$ where the differential recoil spectrum with subindex “U", which stands for “Usual" (i.e. when channeling is not taken into account) is $$\frac{dR}{dE}\bigg\|_{\rm U}=\int \frac{1}{Q}\frac{dR}{dE_Rd\Omega_q}\bigg\|_{E_R=E/Q} d\Omega_q = \frac{1}{Q}\frac{dR}{dE_R}\bigg\|_{E_R=E/Q}.$$ Defining $\tilde{q} \equiv \sqrt{2EM}$ and using Eq. \[eq: rate\], the measured differential rate becomes, $$\begin{aligned} \frac{dR}{dE}&=&\frac{dR}{dE}\bigg\|_{\rm U}\;+\;\frac{\rho \sigma_0}{4\pi m \mu^2} \, \bigg[ S(\tilde{q}) \int{\chi(E,\hat{\textbf{q}}) \hat{f}_{\rm lab}\!\left( \frac{\tilde{q}}{2\mu}, \hat{\bf q} \right) d\Omega_q}\nonumber\\ &&- \frac{S(\tilde{q}/\sqrt{Q})}{Q} \int{\chi(E/Q,\hat{\textbf{q}}) \hat{f}_{\rm lab}\!\left( \frac{\tilde{q}}{2\mu \sqrt{Q}}, \hat{\bf q} \right) d\Omega_q } \bigg].\end{aligned}$$ Inserting $\sigma_0$ from Eq. \[sigma0\] in the above equation with the usual value for the mean local halo density $\rho=0.3~{\rm GeV/cm}^3$, we can write the spin-independent detection rate of WIMPs in general for a crystal that may contain more than one element $$\begin{aligned} \frac{dR}{dE}&=&\frac{dR}{dE}\bigg\|_{\rm U}\;+\;1.306\times10^{-3}\frac{\text{events}}{\text{kg-day-keV}}\times \frac{\sigma_{44}}{4\pi m \mu_p^2} \sum_n C_n \, A_n^2 \, \bigg[ S(\tilde{q}) \int{\chi_n(E,\hat{\textbf{q}}) \hat{f}_{\rm lab}\!\left( \frac{\tilde{q}}{2\mu_n}, \hat{\bf q} \right) d\Omega_q}\nonumber\\ &&- \frac{S(\tilde{q}/\sqrt{Q_n})}{Q_n} \int{\chi_n(E/Q_n,\hat{\textbf{q}}) \hat{f}_{\rm lab}\!\left( \frac{\tilde{q}}{2\mu_n\sqrt{Q_n}}, \hat{\bf q} \right) d\Omega_q } \bigg], \label{Rate}\end{aligned}$$ where $\sigma_{44}$ is the WIMP-proton cross section in units of $10^{-44}\;{\text{cm}}^2$, $\mu_p$ and $m$ are in GeV and $\int{\hat{ f}_{\rm lab}d\Omega_q}$ is in ${(\text{km/s})}^{-1}$. The sum is over the nuclear species $n$ in a crystal, and $C_n$, $\chi_n$, $Q_n$ and $\mu_n$ are the mass fraction, the channeling probability, the quenching factor and the reduced WIMP-nucleus mass for the element $n$, respectively. For example, for NaI crystals, as used in the DAMA experiment, we have $C_{\rm Na}=M_{\rm Na}/(M_{\rm Na} + M_{\rm I})$ and $C_{\rm I}={M_{\rm I}}/({M_{\rm Na} + M_{\rm I}})$, where $M_{\rm Na}$ and $M_{\rm I}$ are the atomic masses of Sodium and Iodine respectively. The integrals in Eq. \[Rate\] cannot be computed analytically. We integrate numerically by performing a Riemann sum once the sphere of directions has been divided using HEALPix [@HEALPix:2005] (see also Appendix B of Ref. [@BGGI]). HEALPix provides a convenient way of dividing the surface of a sphere into equal area sectors, and in our papers [@BGGI; @BGG] we use it for the first time to compute integrals over directions. With the same notation, the usual rate is $$\frac{dR}{dE}\bigg\|_{\rm U} = 1.306\times10^{-3}\frac{\text{events}}{\text{kg-day-keV}}\times \frac{\sigma_{44}}{4\pi m \mu_p^2} \sum_n C_n \, A_n^2 \, \, \bigg[\frac{S(\tilde{q}/\sqrt{Q_n})}{Q_n} \int{\hat{f}_{\rm lab}\!\left( \frac{\tilde{q}}{2\mu_n\sqrt{Q_n}}, \hat{\bf q} \right) d\Omega_q } \bigg].$$ Daily Modulation in NaI Crystals ================================ We present here the daily modulation amplitude due to channeling expected in NaI crystals for several WIMP masses and Na or I recoil energies. We assume that WIMPs have a truncated Maxwellian velocity distribution as in Eq. \[VelDist\] with $v_{\rm esc}=650$ km/s. We use the upper bounds to channeling fractions for single channels $\chi_i(E_R,\hat{\bf q})$ given in Ref. [@BGGI]. We take $T=293$ K, the temperature of the DAMA experiment. The spin-independent detection rate of WIMPs given in Eq. \[Rate\] has a time dependence through the Radon transform $\hat{ f}_{\rm lab}$. Notice that $\hat{ f}_{\rm lab}$ (see Eq. \[fhatTM\]) changes during a day through the $(\hat{\bf q} . {\bf V}_{\rm lab})$ factor appearing in the exponent and the dependence of ${\bf V}_{\rm lab}$ on ${\bf V}_{\rm {EarthRot}}$ (see Eq. \[Vlab\]). The expression showing the time dependence of $\hat{\bf q} . {\bf V}_{\rm lab}$ is given in Eq. \[qdotVlab\] (in Appendix B). During a day, ${\bf V}_{\rm {EarthRev}}$ which is responsible for the annual modulation changes too. Thus the rate does not return to exactly the same value after one day. For the cases we present in this paper, this difference is less than 10% of the total modulation amplitude in a day, and we did not correct for this effect. Relative Modulation Amplitudes ------------------------------ Here we show the signal rate as function of time during a particular arbitrary Solar day (September 25, 2010). We define the relative signal modulation amplitude $A_s$ (taking into account the signal only) in terms of the maximum and minimum daily signal rate $R_s$ as $$A_s= \frac{R_{s {\rm -max}}-R_{s{\rm -min}}}{R_{s{\rm-max}}+R_{s{\rm-min}}}.$$ The total relative modulation amplitude $A_T$ is defined in terms of the maximum $R_{T{\rm-max}}$ and minimum $R_{T{\rm -min}}$ total daily rates as $$A_T= \frac{R_{T {\rm -max}}-R_{T{\rm -min}}}{R_{T{\rm-max}}+R_{T{\rm-min}}}.$$ The total rate consists of signal plus background, $R_T=R_s+R_b$. Assuming that there is no daily modulation in the background, $R_{T{\rm -max}}-R_{T{\rm -min}}=$ $R_{s{\rm-max}}-R_{s{\rm -min}}$, and $A_T$ is related to $A_s$ as $$A_T=A_s(R_s/R_T), \label{AT}$$ where the average total rate due to signal and background is $R_T= (R_{T{\rm-max}}+R_{T{\rm-min}})/2$ and the average rate due to the signal alone is $R_s= (R_{s{\rm-max}}+R_{s{\rm-min}})/2$. Exploring the parameter space of WIMP mass and WIMP-proton cross section for different recoil energies we find that the relative modulation amplitudes $A_s$ can be large, even more than 10% for some combination of parameters. We explored the range of WIMP masses from a few GeV to hundreds of GeV for recoil energies between 2 keV and a few MeV. We show some examples in Fig. \[FigureTable\], where we plot the signal rate (in events/kg/day/keVee) as function of the Universal Time (UT) during 24 hours. We find that the largest $A_s$ happen when the signal is only due to channeling. This happens when there are no WIMPs in the galactic halo with large enough kinetic energy to provide the observed energy if the recoil is not channeled. The observed energies for which the rate is only due to channeling depend on the quenching factors $Q$, which are not well known. The smaller values of $Q$ make channeling more important so we take $Q_{\rm Na}=0.2$ [@Hooper-Collar] for Na and the usual $Q_{\rm I}=0.09$ for I. ![Signal rate (in events/kg-day-keVee) as function of the Universal Time (UT) during 24 hours for $m=10$ GeV, 12 GeV and 15 GeV for different energies. The parameters used are $\sigma_v=300$ km/s, $Q_{\rm Na}=0.2$, $Q_{\rm I}=0.09$, $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$, $c=1$ for temperature effects, a crystal temperature of $T=293$ K and $V_{\rm {lab}}=228.4$ km/s (top row) or 288.3 km/s (bottom row).[]{data-label="FigureTable"}](FigureTable.eps){height="190pt"} Statistical Significance ------------------------ The detectability of a particular amplitude of daily modulation depends on the exposure and background of a particular experiment. The former DAMA/NaI and the DAMA/LIBRA experiments (which we refer collectively as the DAMA experiment) have a very large cumulative exposure, 1.17 ton $\times$ year. However even with this large exposure, we find that the daily modulations we predict are not observable. To observe the daily modulation, the total number of events $N_T$ ($N_s$ signal plus $N_b$ background events) over the duration of the experiment should be divided into two bins, the “high-rate” bin with $N_{T {\rm -max}}$ events and the “low-rate” bin with $N_{T {\rm -min}}$ events, so that $N_T=N_{T {\rm -max}}+N_{T {\rm -min}}$. For the daily modulation to be observable at, say, the 3$\sigma$ level one should have $$N_{T {\rm -max}}-N_{T {\rm -min}}= A_T N_T > 3\sigma \simeq 3 \sqrt{N_T/2}, \label{RateCond}$$ where $\sigma^2 \simeq N_T/2$ because, with a small modulation, on average $N_{T {\rm -max}} \simeq N_{T {\rm -min}} \simeq N_T/2$. In principle there are other errors associated with identifying the “high-rate” and “low-rate” bins which we do not include here. Thus we are underestimating the errors. If the detector exposure is $M T$ in kg-day and we take bins of width $\Delta E$ in keVee, then $N_{T {\rm -max}} = R_{T {\rm -max}} M T \Delta E /2$, $N_{T {\rm -min}} = R_{T {\rm -min}} M T \Delta E /2$, $N_T =R_T M T~ \Delta E$ and $N_s =R_s M T \Delta E$, where the rates are in events/kg-day-keVee. Thus $(N_s/N_T)=(R_s/R_T)$ and using Eq. \[AT\], $A_T=A_s(N_s/N_T)$. Thus the condition in Eq. \[RateCond\] becomes $A_s N_s > 3 \sqrt{N_T/2}$ which implies $$N_s^2/N_T > 9 / (2 A_s^2),$$ or $$R_s^2/R_T > 9 /(2 A_s^2 M T~ \Delta E). \label{RateCond2}$$ The total rate of the DAMA experiment at low energies $4~{\rm keVee}<E<10$ keVee is $R_T \simeq 1 ~{\rm events/kg/day/keVee}$ [@DAMA-bckg]. This rate is much larger than the signal rates we predict and is, therefore, dominated by background. With this value of $R_T$, Eq. \[RateCond2\] becomes $$R_s^2 A_s^2 > {\frac{9}{2 M T ~\Delta E~{\rm kg~day~keVee}}}. \label{RateCond3}$$ We choose here a bin $\Delta E \simeq 1$ keVee, narrow enough to assume the signal rate to be constant in it and compatible with the energy resolution of DAMA. The energy resolution of DAMA is $\sigma_E(E)=(0.448~{\rm keVee})\sqrt{E/{\rm keVee}}+(0.0091) E \simeq 1$ keVee at low energies [@Bernabei:2008]. We consider the significance of the highest signal-to-noise energy bin that we found through inspection. With the cumulative exposure of DAMA, the condition in Eq. \[RateCond3\] for relative daily modulation amplitudes $A_s$ observable at 3$\sigma$ is $$R_s~A_s>3.2 \times 10^{-3}~{\rm events/kg/day/keVee},$$ or $$R_{s {\rm -max}}-R_{s {\rm -min}} > 6.4 \times 10^{-3}~{\rm events/kg/day/keVee}. \label{RateCond4}$$ For observability at the $n \sigma$ level we should multiply the right-hand side of Eq. \[RateCond4\] by $(n/3)$. Even the largest relative daily modulations we find, shown in Fig. \[FigureTable\], are not observable in the DAMA data according to Eq. \[RateCond4\]. The examples which we show here are for small WIMP masses and recoil energies. For large masses the value of $\sigma_p$ must be chosen in the region of the cross section and mass plane where XENON10/100 and CDMS impose $\sigma_p$ to be smaller by four orders of magnitude than for light WIMPs. This amounts to corresponding smaller signal rates and ($R_{s {\rm -max}}-R_{s {\rm -min}}$) differences. For small WIMP masses and large energies, $v_q$ is large and there are no WIMPs with the speed required for Na or I recoils. Thus, only small WIMP masses and recoil energies result in high modulation amplitudes. Fig. \[FigureTable\] shows the signal rate during 24 hours for three different WIMP masses $m=10$ GeV, 12 GeV and 15 GeV and different energies $E$ between 2 and 15 keVee. The other relevant parameters are $\sigma_v=300$ km/s, $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$ (close to the DAMA and CoGeNT regions [@Hooper-Collar; @Savage:2010; @CDMS:2011]), $c=1$, $T=293$ K and two values of $V_{\rm lab}$, 228.4 km/s (top row) and 288.3 km/s (bottom row). Recent bounds, e.g. those from XENON100 [@Xenon100:2011], impose smaller values of $\sigma_p$. In any event, changes in $\sigma_p$ are easy to take into account because $A_s$ is independent of $\sigma_p$ and the rate is just proportional to it, $R_s \sim \sigma_p$. We found the relative amplitude $A_s$ to be as large as 12% in the examples shown in Fig. \[FigureTable\], but even those large values are not observable according to Eq. \[RateCond4\] (even at the 1$\sigma$ level). With the choice of $V_{\rm lab}=228.4$ km/s (top row of Fig. \[FigureTable\]) we get a signal rate difference $R_{s {\rm -max}}-R_{s {\rm -min}}$ of $0.56 \times 10^{-3}$ events/kg/day/keVee for $m=10$ GeV and $E=10$ keVee (in this case $v_q=454.8$ km/s and 790.5 km/s for channeled Na and I recoils, respectively), $3.17 \times 10^{-4}$ events/kg/day/keVee for $m=12$ GeV and $E=12$ keVee (which corresponds to $v_q=441.6$ km/s and 732.9 km/s for Na and I channeled recoils, respectively), and $4.25 \times 10^{-4}$ events/kg/day/keVee for $m=15$ GeV and $E=15$ keVee (for which $v_q=430.6$ km/s and 670.6 km/s for Na and I channeled recoils, respectively). With the choice of $V_{\rm lab}=288.3$ km/s (bottom row of Fig. \[FigureTable\]), $R_{s {\rm -max}}-R_{s {\rm -min}}$ is $0.77 \times 10^{-3}$ events/kg/day/keVee for $m=10$ GeV and $E=10$ keVee (one of the energies shown), $2.95 \times 10^{-4}$ events/kg/day/keVee for $m=12$ GeV and $E=12$ keVee, and $0.58 \times 10^{-5}$ events/kg/day/keVee for $m=15$ GeV and $E=15$ keVee. Because the minimum WIMP speeds $v_q$ are large in these examples, a smaller velocity dispersion of the WIMP distribution leads to smaller rates (since a smaller amount of WIMPs have velocities larger than $v_q$). So the signal rate difference $R_{s {\rm -max}}-R_{s {\rm -min}}$ is even smaller for smaller values of $\sigma_v$. The left-bottom panel of Fig. \[FigureTable\] shows the signal rate as function of UT for $m=10$ GeV and $V_{\rm lab}=288.3$ km/s for several energies between 2 keVee and 12 keVee. The rate decreases but $A_s$ increases with increasing energy and the best conditions for observability happen at some energy where neither the rate nor $A_s$ are very small. The rates for low energies between 2 keVee and 6 keVee are dominated by the usual (i.e. non-channeled) rate and the daily modulation is due purely to the change in WIMP kinetic energy in the lab frame as the Earth rotates around itself. The rates for energies above 8 keVee (green/gray lines) are purely due to channeling, i.e. the usual rate is zero. For intermediate energies, 6 keVee to 8 keVee, the usual and channeled rates both contribute and thus the daily modulation is due to both the channeling effect and the daily change in the usual rate. For $E=2$, 4, 6, 8, 10 and 12 keVee, the values of $R_{s {\rm -max}}-R_{s {\rm -min}}$ given in events/kg/day/keVee are respectively $4.3 \times 10^{-4}$, $0.5 \times 10^{-3}$, $0.92 \times 10^{-3}$, $2.8 \times 10^{-4}$, $0.77 \times 10^{-3}$ and $0.52 \times 10^{-3}$. Notice that for all the energies shown the difference in rate is similar, but the largest $A_s$ values happen at energies above 8 keVee, for which the rate is only due to channeling. The channeling daily modulation amplitude increases as the ratio of the velocity dispersion to the average speed of the WIMPs that contribute to the signal (i.e. with $v>v_q$) decreases. This ratio is small and thus $A_s$ large for large values of $v_q$. Notice that the phase of the modulation due to channeling depends on the orientation of the crystal with respect to the Galaxy and the phase of the modulation in the usual rate does not, which would allow to distinguish both effects, if they were observable. The case of $m=10$ GeV and $E=6$ keVee has the largest rate difference, but is not observable at 3$\sigma$ according to Eq. \[RateCond4\] (not even at the 1$\sigma$ level). Choosing $\sigma_p=4 \times 10^{-40} \textrm{cm}^2$ (still within the DAMA allowed region but not compatible with the recent XENON100 result) results in a rate difference of $1.84 \times 10^{-3}$ events/kg/day/keVee for this case which would not be observable even at the 1$\sigma$ level. Finally, we would like to compare our results with those obtained in Ref. [@Avignone:2008cw] by Creswick [et al]{}. They found a relative daily modulation amplitude $A_s=$0.85% (their definition of amplitude differs by a factor of 2 from ours, so they quote 1.7%) for 5 GeV WIMP mass and 3.8 keVee measured energy (in which case $v_q=471.2$ km/s and 936.6 km/s for channeled Na and I recoils, respectively. There are no WIMPs with the speed required for I recoils, thus only Na recoils are possible). In order to compare our calculation with theirs, we compute the signal event rate as function of time for $c=1$, $T=293$ K (temperature corrections are not included in the calculation of Creswick [et al.]{}) and choosing all the other parameters very close to those used in Ref. [@Avignone:2008cw], i.e. $V_{\rm {lab}}=228.4$ km/s and $\sigma_v=300$ km/s. A WIMP mass of 5 GeV is outside the region of parameter space compatible with the annual modulation reported by DAMA [@Savage:2010]. Since $A_s$ does not depend on $\sigma_p$, we choose an arbitrary value of $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$ to plot the signal rate as a function of UT (the upper bound given by TEXONO and CoGeNT [@TEXONO:2009] is five times larger, $\sigma_p < 1 \times 10^{-39} \textrm{cm}^2$). Our result is shown in Fig. \[T293-e3m5\].a. We find $A_s=$0.16% ($R_{s {\rm -max}}-R_{s {\rm -min}}=4.4 \times 10^{-6} ~{\rm events/kg/day/keVee}$). Even when we consider the extreme choice of $c=0$ to compute temperature effects (an unrealistic value for which the channeling fractions are larger) with the same parameters, we get $A_s=0.14\%$. This case is shown in Fig. \[T293-e3m5\].b. ![Signal rate as function of UT during 24 hours for $E=3.8$ keVee and $m=5$ GeV, with $V_{\rm {lab}}=228.4$ km/s, $\sigma_v=300$ km/s, $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$, and $Q_{\rm Na}=0.2$, $Q_{\rm I}=0.09$ for (a) $c=1$ and (b) $c=0$. The daily modulation is not observable in both cases.[]{data-label="T293-e3m5"}](NaI-c1-T293K-e3.8m5.eps "fig:"){height="144pt"} ![Signal rate as function of UT during 24 hours for $E=3.8$ keVee and $m=5$ GeV, with $V_{\rm {lab}}=228.4$ km/s, $\sigma_v=300$ km/s, $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$, and $Q_{\rm Na}=0.2$, $Q_{\rm I}=0.09$ for (a) $c=1$ and (b) $c=0$. The daily modulation is not observable in both cases.[]{data-label="T293-e3m5"}](NaI-c0-T293K-e3.8m5.eps "fig:"){height="144pt"}\ Future Prospects for DAMA and other Experiments ----------------------------------------------- The daily modulation might be detectable in other experiments with smaller background or WIMP halo components with a smaller dispersion such as streams or a thick disk. The amplitude of the daily modulation increases as the WIMP velocity distribution is narrower i.e. for larger values of the average velocity and smaller values of the velocity dispersion of the detectable WIMPs (which is not $\sigma_v$), i.e. those with velocity larger than $v_q$. This is easy to understand since as the dispersion increases more channels are available for channeling of the recoiling ions. In the limit in which the velocity distribution would be isotropic with respect to the detector, the daily rotation would not introduce any difference in the rate due to channeling. Having a large relative signal modulation amplitude $A_s$ is not sufficient for observability. In Eq. \[RateCond4\] what is important is $(A_s~ R_s)=(R_{s {\rm -max}}-R_{s {\rm -min}})/2$. However, the condition in Eq. \[RateCond4\] was derived considering the total rate in the DAMA experiment, which is dominated by background. For an experiment where the background is negligible, i.e. $R_T=R_s+R_b \simeq R_s$, we can derive a different observability condition (at the 3$\sigma$ level) from Eq. \[RateCond2\], $$R_s~A_s^2= A_s~(R_{s {\rm -max}}-R_{s {\rm -min}})/2> 9 /(2 M T~ \Delta E). \label{RateCond-NoBckg}$$ This condition might be easier to satisfy in future experiments. One could ask which is the maximum level of total rate with the current DAMA exposure that would be needed to make the signal daily modulation observable. Inserting the current exposure of DAMA (1.17 ton year) in Eq. \[RateCond2\], we have $${\left(A_s~R_s\right)} ^2/R_T > 1.05 \times 10^{-5}~{\rm events/kg/day/keVee},$$ which using $A_s R_s=(R_{s {\rm -max}}-R_{s {\rm -min}})/2$, becomes $$R_T < \frac{\left(R_{s {\rm -max}}-R_{s {\rm -min}}\right)^2}{4.2 \times 10^{-5} ~{\rm events/kg/day/keVee}}.$$ Even in the case with the highest rate difference we found, i.e. $R_{s {\rm -max}}-R_{s {\rm -min}}=0.98 \times 10^{-3}$ events/kg/day/keVee (the $m=10$ GeV, $E=6$ keVee, $V_{\rm lab}=288.3$ km/s example shown in the bottom-left panel of Fig. \[FigureTable\]) observability would require $$R_T < 0.023 ~{\rm events/kg/day/keVee}, \label{maxR_T}$$ roughly $1/40$ of what is now. We could ask instead what exposure would be needed with the current total rate in the DAMA experiment to make the daily modulation observable. Setting $R_T \simeq 1$ events/kg/day/keVee in Eq. \[RateCond2\], we obtain $$\frac{M T \Delta E}{{\rm (events/kg/day/keVee)}} > \frac{9}{2 \left(A_s~R_s \right)^2}= \frac{18}{\left(R_{s {\rm -max}}-R_{s {\rm -min}}\right)^2} .$$ Again, for the case with the highest rate difference we found ($m=10$ GeV, $E=6$ keVee and $V_{\rm lab}=288.3$ km/s) and with $\Delta E \simeq$ 1 keVee we would require an exposure 40 times larger, $$M T > 51.3 ~{\rm ton ~year}.$$ We have computed the daily modulation due to channeling in other material such as Ge, solid Xe and solid Ne, and we find that it will be very difficult to observe. For light WIMPs the cross section can be larger than for heavier ones without violating experimental bounds, $\sigma_p=10^{-39} \textrm{cm}^2$ [@TEXONO:2009] and this favors the detection of the daily modulation. We find that for a WIMP mass $m=5$ GeV the daily modulation due to channeling may be observable in solid Ne if the signal would be above threshold and assuming no background. The geometric channeling fraction reaches a maximum at around 10 keV for solid Ne [@BGG], thus the largest modulation amplitude happens at that energy. For example for a solid Ne detector operating at 23 K at Gran Sasso, for $E=10$ keV, assuming $Q_{\rm Ne} =0.25$ [@Tretyak], $c=1$ and with velocity distribution parameters $\sigma_v=300$ km/s and $V_{\rm {lab}}=228.4$ km/s we find $R_s A_s^2=3.68 \times 10^{-5}$ events/kg/day/keVee. Using Eq. \[RateCond-NoBckg\] we find that the exposure needed to observe this modulation at 3$\sigma$ is $MT=0.33$ ton year. For the same parameters but for $m=7$ GeV and $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$ (parameters compatible with the possible dark matter signal found by CoGeNT and with DAMA according to Ref. [@Hooper]), we find $R_s A_s^2=7.2 \times 10^{-7}$ events/kg/day/keVee, and the exposure needed is $MT=17.1$ ton year. The usual rate is zero in both cases, and the modulation is just due to channeling. The signal rate during 24 hours and the required exposures for the two cases are shown in Fig. \[NeRate\] and Table \[table:Ne\], respectively. ![Signal rate as function of UT during 24 hours for a solid Ne detector operating at $T=23$ K at Gran Sasso for $E=10$ keVee, $Q=0.25$, $c=1$, $\sigma_v=300$ km/s, $V_{\rm {lab}}=228.4$ km/s and for (a) $m=5$ GeV and $\sigma_p=10^{-39} \textrm{cm}^2$, and (b) $m=7$ GeV and $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$.[]{data-label="NeRate"}](Ne-e10m5.eps "fig:"){height="144pt"} ![Signal rate as function of UT during 24 hours for a solid Ne detector operating at $T=23$ K at Gran Sasso for $E=10$ keVee, $Q=0.25$, $c=1$, $\sigma_v=300$ km/s, $V_{\rm {lab}}=228.4$ km/s and for (a) $m=5$ GeV and $\sigma_p=10^{-39} \textrm{cm}^2$, and (b) $m=7$ GeV and $\sigma_p=2 \times 10^{-40} \textrm{cm}^2$.[]{data-label="NeRate"}](Ne-e10m7.eps "fig:"){height="144pt"}\ --------------------- ------------------------------ ----------------- Case $\sigma_p$ ($\textrm{cm}^2$) $MT$ (ton year) \[0.5ex\] $m=5$ GeV $10^{-39}$ 0.33 $m=7$ GeV $2 \times 10^{-40}$ 17.1 \[1ex\] --------------------- ------------------------------ ----------------- : Observability in solid Ne detector \[table:Ne\] We intend to further explore the observability of a daily modulation in future experiments for different halo models in future work. G.G. and N.B. were supported in part by the US Department of Energy Grant DE-FG03-91ER40662, Task C. P.G. was supported in part by the NFS grant PHY-0756962 at the University of Utah. Crystal Orientation =================== We need to orient the crystal with respect to the laboratory. We define a reference frame fixed with the laboratory and orient its axes so that the $xy$ plane is horizontal, the $x$-axis points North, the $y$-axis points West, and the $z$-axis points to the zenith. We denote its unit coordinate vectors as $\hat{\cal N}$, $\hat{\cal W}$ and $\hat{\cal Z}$, respectively. We also define the crystal frame with $X,Y,Z$ cartesian axes fixed with the crystal. The unit coordinate vectors of the crystal frame are $\hat{\mathbf{X}}$, $\hat{\mathbf{Y}}$ and $\hat{\mathbf{Z}}$. We now want to connect the laboratory frame to the crystal frame. Let the standard orientation correspond to the configuration in which $\hat{\mathbf{X}}=\hat{\cal N}$, $\hat{\mathbf{Y}}=\hat{\cal W}$, and $\hat{\mathbf{Z}}=\hat{\cal Z}$. We start with the crystal in the standard orientation, and we turn it into any other orientation $\hat{\mathbf{X}}$, $\hat{\mathbf{Y}}$, $\hat{\mathbf{Z}}$. In this new orientation, each of the unit coordinate vectors of the crystal frame can be written in terms of unit coordinate vectors of the lab frame, $$\begin{aligned} \hat{\mathbf{X}}&=&\alpha_X~\hat{\mathbf{\mathcal{N}}}+\beta_X~\hat{\mathbf{\mathcal{W}}}+\gamma_X~\hat{\mathbf{\mathcal{Z}}},\nonumber\\ \hat{\mathbf{Y}}&=&\alpha_Y~\hat{\mathbf{\mathcal{N}}}+\beta_Y~\hat{\mathbf{\mathcal{W}}}+\gamma_Y~\hat{\mathbf{\mathcal{Z}}},\nonumber\\ \hat{\mathbf{Z}}&=&\alpha_Z~\hat{\mathbf{\mathcal{N}}}+\beta_Z~\hat{\mathbf{\mathcal{W}}}+\gamma_Z~\hat{\mathbf{\mathcal{Z}}}, \label{Crystal-Lab}\end{aligned}$$ where $\alpha_i$, $\beta_i$ and $\gamma_i$ are the “direction cosines” between the two sets of cartesian coordinates of the lab and crystal frames, for $i=X,Y,Z$. For example, the coordinate vector $\hat{\mathbf{X}}$ of the crystal has a particular angle with each of the lab frame coordinate vectors $\hat{\cal N}$, $\hat{\cal W}$, $\hat{\cal Z}$. Let $a_X$ be the angle between $\hat{\mathbf{X}}$ and $\hat{\cal N}$, $b_X$ the angle between $\hat{\mathbf{X}}$ and $\hat{\cal W}$, and $c_X$ the angle between $\hat{\mathbf{X}}$ and $\hat{\cal Z}$. The direction cosines of the unit vector $\hat{\mathbf{X}}$ are given by, $$\begin{aligned} \alpha_X &\equiv& \cos a_X =\hat{\mathbf{X}}\cdot\hat{\cal N},\nonumber\\ \beta_X &\equiv& \cos b_X =\hat{\mathbf{X}}\cdot\hat{\cal W},\nonumber\\ \gamma_X &\equiv& \cos c_X =\hat{\mathbf{X}}\cdot\hat{\cal Z}.\end{aligned}$$ We can find the direction cosines for $\hat{\mathbf{Y}}$ and $\hat{\mathbf{Z}}$ unit vectors in a similar way. From these definitions it follows that $\alpha_i ~\alpha_j+\beta_i~\beta_j+\gamma_i~\gamma_j=\delta_{ij}$ where $i,j=X,Y,Z$. We prefer using direction cosines over Euler angles because the direction cosines can easily be measured for any known orientation of a crystal in a laboratory, whereas it may be difficult to specify the Euler angles. Eq. \[Crystal-Lab\] gives the transformation from the lab frame to the crystal frame. We can also find the lab coordinate vectors in terms of the crystal coordinate vectors, $$\begin{aligned} \hat{{\bf {\cal N}}}&=&\alpha_X~\hat{{\bf X}}+\alpha_Y~\hat{{\bf Y}}+\alpha_Z~\hat{{\bf Z}},\nonumber\\ \hat{{\bf {\cal W}}}&=&\beta_X~\hat{{\bf X}}+\beta_Y~\hat{{\bf Y}}+\beta_Z~\hat{{\bf Z}},\nonumber\\ \hat{{\bf {\cal Z}}}&=&\gamma_X~\hat{{\bf X}}+\gamma_Y~\hat{{\bf Y}}+\gamma_Z~\hat{{\bf Z}}. \label{Lab-Crystal}\end{aligned}$$ In the results we show in this paper, we took $\alpha_X=\beta_Y=\gamma_Z=1$ and all the other $\alpha_i$, $\beta_i$ and $\gamma_i$ equal to zero. Choosing a different orientation for the crystal does not change the average rate, but $A_s$ may change by a factor of 2 for NaI depending on the orientation of the crystal. The observability condition is still not satisfied. Lab to equatorial transformation -------------------------------- To connect the laboratory frame to the equatorial coordinate frame, we recall the definition of the geocentric equatorial inertial (GEI) frame: its origin is at the center of the Earth, its $x_e$-axis points in the direction of the vernal equinox, its $y_e$-axis points to the point on the celestial equator with right ascension 90$^\circ$ (so that the cartesian frame is right-handed), and its $z_e$-axis points to the north celestial pole. We denote its unit coordinate vectors as $\hat{\bf x}_e$, $\hat{\bf y}_e$, and $\hat{\bf z}_e$. We want to find the transformation formulas from the laboratory frame to the GEI frame. This transformation can be achieved by two successive rotations. The first rotation is by an angle of $(90^\circ-\lambda_{\rm lab})$ counterclockwise about the laboratory $y$-axis to align the new $x' y'$ plane with the plane of the celestial equator. Here $\lambda_{\rm lab}$ is the latitude of the laboratory in degrees, with northern latitudes taken as positive and southern latitudes taken as negative. With this rotation, the new $z'$-axis points to the north celestial pole. The second rotation is by an angle $(15t_{\rm lab}+180)$ degrees clockwise about the new $z'$-axis to bring the $x'$-axis in the direction of the vernal equinox. Here $t_{\rm lab}$ is the laboratory Local Apparent Sidereal Time (LAST) in hours (the LAST is the hour angle of the vernal equinox at the location of the laboratory). One has $$t_{\rm lab} = t_{\rm GAST} + l_{\rm lab}/15,$$ where $t_{\rm GAST}$ is the Greenwich Apparent Sidereal Time (GAST) in hours and $l_{\rm lab}$ is the longitude in degrees measured positive in the eastward direction (e.g. $l_{\rm lab}=+110^\circ$ for 110$^\circ$ E and $l_{\rm lab}=-110^\circ$ for 110$^\circ$ W). The current local apparent sidereal time for any specified longitude $l_{\rm lab}$ can be computed online, for example on the website of the US Naval Observatory at http://tycho.usno.navy.mil/ sidereal.html (accessed Sept 19, 2010). As an alternative, one can use the following formula [@Hapgood; @USNO89] for the Greenwich mean sidereal time (which differs from the Greenwich apparent sidereal time by less than 1.2 seconds, completely negligible for our purposes), $$\begin{aligned} t_{\rm GAST} = (101.0308 + 36000.770 \, T_0 + 15.04107 \, {\rm UT})/15, \label{tGAST}\end{aligned}$$ where $$\begin{aligned} T_0 = \frac{ \lfloor {\rm MJD} \rfloor - 55197.5}{36525.0}.\end{aligned}$$ Here ${\rm UT}$ is the Universal Time in hours, $\lfloor {\rm MJD} \rfloor $ is the integer part of the modified Julian date (MJD), which is the time measured in days from 00:00 UT on 17 November 1858 (Julian date 2400000.5). Note that $T_0$ is the time in Julian centuries (36525 days) from 12:00 UT on 1 January 2010 to the previous midnight. At 12:00 UT on 1 January 2010, the Julian date is 2455198, and the MJD is 55197.5. Also the the $15.04107/15$ in Eq. \[tGAST\] corrects from solar time (UT) to sidereal time. Sidereal day is shorter than Solar day by 3.9 minutes. In this paper, all our results are computed for the particular arbitrary day of 25 September 2010, for which $T_0=0.00729637$. Note also that UT is different from coordinated Universal Time (UTC) which is the time scale usually used for data recording. UTC is atomic time adjusted by an integral number of seconds to keep it within 0.6 s of UT. For our purposes the difference between UT and UTC is negligible. Taking into account the two rotations explained above, one can find the transformation equations of the unit vectors, $$\begin{aligned} \hat{\bf x}_e & = -\cos(t^\circ_{\rm lab}) \left[ \sin(\lambda_{\rm lab}) \hat{\cal N} -\cos(\lambda_{\rm lab}) \hat{\cal Z} \right] + \sin(t^\circ_{\rm lab}) \hat{\cal W}, \nonumber\\ \hat{\bf y}_e & = -\sin(t^\circ_{\rm lab}) \left[ \sin(\lambda_{\rm lab}) \hat{\cal N} -\cos(\lambda_{\rm lab}) \hat{\cal Z}\right] - \cos(t^\circ_{\rm lab}) \hat{\cal W}, \nonumber\\ \hat{\bf z}_e & = \cos(\lambda_{\rm lab}) \hat{\cal N} + \sin(\lambda_{\rm lab}) \hat{\cal Z}, \label{eq:labeq3}\end{aligned}$$ where $t^\circ_{\rm lab}=15 t_{\rm lab}$ is the laboratory LAST converted to degrees. As a check, for a laboratory on the equator at local sidereal time 0, i.e. $\lambda_{\rm lab}=0^\circ$ and $t^\circ_{\rm lab}=0^\circ$, one has $\hat{\bf x}_e = \hat{\cal Z}$, $\hat{\bf y}_e = -\hat{\cal W}$, and $\hat{\bf z}_e = \hat{\cal N}$; six sidereal hours later at the same laboratory, i.e. $\lambda_{\rm lab}=0^\circ$ and $t^\circ_{\rm lab}=90^\circ$, one has $\hat{\bf x}_e = \hat{\cal W}$, $\hat{\bf y}_e = \hat{\cal Z}$, and $\hat{\bf z}_e = \hat{\cal N}$; for a laboratory at the South Pole ($\lambda_{\rm lab}=-90^\circ$), using the direction of the Greenwich meridian in place of the “North” axis $\hat{\cal N}$ so that the local sidereal time at the South Pole by convention coincides with the Greenwich sidereal time, one has $\hat{\bf x}_e = \hat{\cal N}$, $\hat{\bf y}_e = -\hat{\cal W}$, and $\hat{\bf z}_e = -\hat{\cal Z}$ at $t^\circ_{\rm lab}=0^\circ$ and $\hat{\bf x}_e = \hat{\cal W}$, $\hat{\bf y}_e = \hat{\cal N}$, and $\hat{\bf z}_e = -\hat{\cal Z}$ at $t^\circ_{\rm lab}=90^\circ$ . All of these are correctly given by Eq. \[eq:labeq3\]. The formulas in Eq. \[eq:labeq3\] can be inverted, and the transformation from the equatorial frame to the lab frame is achieved: $$\begin{aligned} \hat{\cal N}&=-\sin(\lambda_{\rm lab})\left[\cos (t^\circ_{\rm lab}) \hat{\bf x}_e +\sin (t^\circ_{\rm lab})\hat{\bf y}_e\right]+\cos(\lambda_{\rm lab})\hat{\bf z}_e, \nonumber\\ \hat{\cal W}&=\sin (t^\circ_{\rm lab})\hat{\bf x}_e-\cos (t^\circ_{\rm lab})~\hat{\bf y}_e, \nonumber\\ \hat{\cal Z}&=\cos(\lambda_{\rm lab}) \left[\cos (t^\circ_{\rm lab}) \hat{\bf x}_e +\sin (t^\circ_{\rm lab}) \hat{\bf y}_e\right]+\sin(\lambda_{\rm lab})\hat{\bf z}_e. \label{Equit-Lab}\end{aligned}$$ The latitude and longitude of Gran Sasso are $\lambda_{\rm lab}=42.45^\circ$ and $l_{\rm lab}=13.7^\circ$, respectively. Fig. \[Earth\] shows the laboratory frame ($\hat{\cal N}$, $\hat{\cal W}$, $\hat{\cal Z}$) and the equatorial coordinate frame ($\hat{\bf x}_e$,$\hat{\bf y}_e$,$\hat{\bf z}_e$) plotted on the Earth’s sphere at $UT=0$ using Eq. \[Equit-Lab\]. ![(Color online) Earth’s sphere in the equatorial frame ($\hat{\bf x}_e$,$\hat{\bf y}_e$,$\hat{\bf z}_e$) specified with black arrows. The laboratory frame (N,W,Z) specified with blue/dark gray arrows is also shown.[]{data-label="Earth"}](Earth.eps){height="247pt"} Equatorial to galactic transformation ------------------------------------- To connect the equatorial frame to the galactic coordinate frame, we recall the definition of the galactic coordinate system: its origin is at the position of the Sun, its $x_g$-axis points towards the galactic center, its $y_g$-axis points in the direction of the galactic rotation, and its $z_g$-axis points to the north galactic pole. For the epoch of January 1950.0 the transformation from the equatorial frame ($\hat{\mathbf{x}}_e, \hat{\mathbf{y}}_e, \hat{\mathbf{z}}_e$) to the galactic frame ($\hat{\mathbf{x}}_g, \hat{\mathbf{y}}_g, \hat{\mathbf{z}}_g$) is given by [@gal-equat]: $$\begin{aligned} \hat{{\bf x}}_g&=\hat{{\bf x}}_e~(-0.06699)+\hat{{\bf y}}_e~(-0.8728)+\hat{{\bf z}}_e~(-0.4835),\nonumber\\ \hat{{\bf y}}_g&=\hat{{\bf x}}_e~(0.4927)+\hat{{\bf y}}_e~(-0.4503)+\hat{{\bf z}}_e~(0.7446),\nonumber\\ \hat{{\bf z}}_g&=\hat{{\bf x}}_e~(-0.8676)+\hat{{\bf y}}_e~(-0.1883)+\hat{{\bf z}}_e~(0.4602). \label{Equit-Gal}\end{aligned}$$ The transformation from the galactic frame to the equatorial frame is given by $$\begin{aligned} \hat{\mathbf{x}}_e&=&\hat{\mathbf{x}}_g~(-0.06699)+\hat{\mathbf{y}}_g~(0.4927)+\hat{\mathbf{z}}_g~(-0.8676),\nonumber\\ \hat{\mathbf{y}}_e&=&\hat{\mathbf{x}}_g~(-0.8728)+\hat{\mathbf{y}}_g~(-0.4503)+\hat{\mathbf{z}}_g~(-0.1884),\nonumber\\ \hat{\mathbf{z}}_e&=&\hat{\mathbf{x}}_g~(-0.4835)+\hat{\mathbf{y}}_g~(0.7446)+\hat{\mathbf{z}}_g~(0.4602). \label{Gal-Equit}\end{aligned}$$ The change of Eqs. \[Equit-Gal\] and \[Gal-Equit\] from the epoch of January 1950.0 to 25 September 2010 is small and would not affect the final results in this paper. Laboratory motion ================= The velocity of the lab with respect to the center of the Galaxy can be divided into four components (as in Eq. \[Vlab\]): ${\bf V}_{\rm {Gal Rot}}$, ${\bf V}_{\rm {Solar}}$, ${\bf V}_{\rm {Earth Rev}}$ and ${\bf V}_{\rm {Earth Rot}}$. We take $V_{\rm {Gal Rot}}=220$ km/s or 280 km/s [@Green-2010], $V_{\rm Solar}=18$ km/s [@Schoenrich-2010], $V_{\rm {Earth Rev}}=29.8$ km/s and $V_{\rm {Earth Rot}}=(0.465102 ~{\rm km/s}) \cos \lambda_{\rm lab}$, where $\lambda_{\rm lab}$ is the latitude of the lab. Values of $V_{\rm {Gal Rot}}=220$ km/s or 280 km/s results in $V_{\rm lab}=228.4$ km/s or 288.3 km/s, respectively (see Appendix B.5 for the equation of ${\bf V}_{\rm {lab}}$). Thus, ${\bf V}_{\rm {lab}}$ is dominated by the galactic rotation velocity. We need to compute $\hat{\mathbf{q}} \cdot{\bf V}_{{\rm lab}}$, where $\hat{\mathbf{q}}$ is given in the crystal reference frame ($\hat{\mathbf{q}}=q_X ~\hat{{\bf X}} +q_Y ~\hat{{\bf Y}} +q_Z ~\hat{{\bf Z}}$). Therefore, we need to also write ${\bf V}_{\rm {lab}}$ in the crystal frame. We have, $$\hat{\bf q} \cdot{\bf V}_{\rm {lab}}=\hat{\bf q} \cdot{\bf V}_{\rm {Gal Rot}}+\hat{\bf q} \cdot{\bf V}_{\rm {Solar}}+\hat{\bf q} \cdot{\bf V}_{\rm {Earth Rev}}+\hat{\bf q} \cdot{\bf V}_{\rm {Earth Rot}}. \label{qdotV}$$ We will compute each term on the right-hand side of Eq. \[qdotV\] individually. Galactic rotation ----------------- The velocity of the galactic rotation ${\bf V}_{\rm {Gal Rot}}$ is defined in the galactic reference frame, $${\bf V}_{\rm {Gal Rot}}=V_{\rm {Gal Rot}} \hat{{\bf y}}_g, \label{GalacticRot}$$ where $V_{\rm {Gal Rot}}$ is the galactic rotation speed (i.e. the local circular speed), and $\hat{{\bf y}}_g$ is in the direction of the galactic rotation. Following Ref. [@Green-2010] , we take $V_{\rm {Gal Rot}}=220$ km/s or 280 km/s. Using the conversions in Eq. \[Equit-Gal\], we can write $\hat{{\bf y}}_g$ in the equatorial reference frame in terms of ($\hat{{\bf x}}_e$,$\hat{{\bf y}}_e$,$\hat{{\bf z}}_e$). Then, we use Eq. \[eq:labeq3\] to transform from the equatorial frame to the lab frame ($\hat{\cal N},\hat{\cal W},\hat{\cal Z}$), and finally we use Eq. \[Lab-Crystal\] to transform from the lab frame to the crystal frame ($\hat{{\bf X}}, \hat{{\bf Y}},\hat{{\bf Z}}$). Thus, we can use Eq. \[Lab-Crystal\] to write ${\bf V}_{\rm {Gal Rot}}$ in terms of the crystal frame coordinates, and compute $\hat{\mathbf{q}} \cdot{\bf V}_{\rm {Gal Rot}}$, $$\begin{aligned} \hat{\mathbf{q}} \cdot{\bf V}_{\rm {Gal Rot}}&=q_X V_{\rm {Gal Rot},X}+q_Y V_{\rm {Gal Rot},Y}+q_Z V_{\rm {Gal Rot},Z}.\end{aligned}$$ We have $$\begin{aligned} \hat{\mathbf{q}} \cdot{\bf V}_{\rm {Gal Rot}}&=V_{\rm {Gal Rot}}\bigg\{\bigg(\left[-0.4927\cos(t^\circ_{\rm lab}) +0.4503\sin(t^\circ_{\rm lab}) \right]\sin(\lambda_{\rm lab})+0.7446\cos(\lambda_{\rm lab})\bigg)\nonumber\\ &\left(\alpha_X q_X+\alpha_Y q_Y+\alpha_Z q_Z\right)+\bigg(0.4927\sin(t^\circ_{\rm lab})+0.4503\cos(t^\circ_{\rm lab}) \bigg)\left(\beta_X q_X+\beta_Y q_Y+\beta_Z q_Z\right)\nonumber\\ &+\bigg(\left[0.4927\cos(t^\circ_{\rm lab}) -0.4503\sin(t^\circ_{\rm lab}) \right]\cos(\lambda_{\rm lab})+0.7446\sin(\lambda_{\rm lab})\bigg)\nonumber\\ &\left(\gamma_X q_X+\gamma_Y q_Y+\gamma_Z q_Z \right)\bigg\}. \label{qdotGalacticRot}\end{aligned}$$ Eq. \[qdotGalacticRot\] has a time dependence through $t^\circ_{\rm lab}$ and would be responsible for any daily modulation in the rate. Solar motion ------------ The velocity of the Sun’s motion in the galactic rest frame is, $${\bf V}_{\rm {Solar}}=U \hat{{\bf x}}_g + V \hat{{\bf y}}_g +W \hat{{\bf z}}_g, \label{Solar}$$ where $(U,V,W)_\odot=(11.1, 12.2, 7.3)$ km/s [@Schoenrich-2010]. Using Eq. \[Equit-Gal\], we can transform from the galactic frame to the equatorial frame, and using Eq. \[eq:labeq3\] we can transform from the equatorial frame to the lab frame. Then we can use Eq. \[Lab-Crystal\] to write ${\bf V}_{\rm {Solar}}$ in terms of the crystal frame coordinates. Thus, we can compute $\hat{\mathbf{q}} \cdot{\bf V}_{\rm {Solar}}$ as $$\begin{aligned} \hat{\bf q} \cdot{\bf V}_{\rm {Solar}}&=\bigg(\big[(1.066~\textrm{km/s})\cos(t^\circ_{\rm lab})+(16.56~\textrm{km/s})\sin(t^\circ_{\rm lab})\big]\sin(\lambda_{\rm lab})+(7.077 ~\textrm{km/s})\cos(\lambda_{\rm lab})\bigg)\nonumber\\ &(\alpha_X q_X+\alpha_Y q_Y+\alpha_Z q_Z)+\bigg(-(1.066~\textrm{km/s})\sin(t^\circ_{\rm lab})+(16.56~\textrm{km/s})\cos(t^\circ_{\rm lab})\bigg)\nonumber\\ &(\beta_X q_X+\beta_Y q_Y+\beta_Z q_Z)+\bigg(-\big[(1.066~\textrm{km/s})\cos(t^\circ_{\rm lab})+(16.56~\textrm{km/s})\sin(t^\circ_{\rm lab})\big]\cos(\lambda_{\rm lab})\nonumber\\ &+(7.077 ~\textrm{km/s})\sin(\lambda_{\rm lab})\bigg)(\gamma_X q_X+\gamma_Y q_Y+\gamma_Z q_Z). \label{qdotSolar}\end{aligned}$$ Clearly, Eq. \[qdotSolar\] has a time dependence through $t^\circ_{\rm lab}$ and would be responsible of any daily modulation in the rate. Earth’s revolution ------------------ The velocity of the Earth’s revolution around the sun is given in terms of the Sun ecliptic longitude $\lambda(t)$ as [@Green] $$\begin{aligned} {\bf V}_{\rm {Earth Rev}}&=V_{\oplus}(\lambda(t)) [\cos\beta(x) \sin (\lambda(t)-\lambda_x) \hat{{\bf x}}_g \nonumber\\ &+ \cos\beta(y) \sin (\lambda(t)-\lambda_y) \hat{{\bf y}}_g + \cos\beta(z) \sin (\lambda(t)-\lambda_z) \hat{{\bf z}}_g], \label{EarthRev}\end{aligned}$$ where $V_{\oplus}=29.8$ km/s is the orbital speed of the Earth, $V_{\oplus}(\lambda(t))=V_{\oplus}[1-e \sin(\lambda(t)-\lambda_0)]$, $e=0.016722$, and $\lambda_0=13^\circ+1^\circ$ are the ellipticity of the Earth’s orbit and the ecliptic longitude of the orbit’s minor axis, respectively, and $\beta_i=(-5^\circ.5303, 59^\circ.575, 29^\circ.812)$ and $\lambda_i=(266^\circ.141, -13^\circ.3485, 179^\circ.3212)$ are the ecliptic latitudes and longitudes of the ($\hat{{\bf x}}_g$,$\hat{{\bf y}}_g$,$\hat{{\bf z}}_g$) axes, respectively. The Sun’s ecliptic longitude $\lambda(t)$ can be expressed as (p. 77 of Ref. [@Lang] and Ref. [@Green]), $$\lambda(t)=L + (1^\circ .915 - 0^\circ.0048 T_0) \sin g+ 0^\circ .020 \sin 2g,$$ where $L=281^\circ .0298 + 36000^\circ .77 T_0 + 0^\circ .04107 UT$ is the mean longitude of the Sun corrected for aberration, $g=357^\circ .9258 + 35999^\circ .05 T_0 + 0^\circ .04107 UT$ is the mean anomaly (polar angle of orbit). Using Eq. \[Equit-Gal\], we can transform from the galactic frame to the equatorial frame, and using Eq. \[eq:labeq3\] we can transform from the equatorial frame to the lab frame ($\hat{\cal N},\hat{\cal W},\hat{\cal Z}$). Then we can use Eq. \[Lab-Crystal\] to write ${\bf V}_{\rm {Solar}}$ in terms of the crystal frame coordinates. Thus, we can compute $\hat{\mathbf{q}} \cdot{\bf V}_{\rm {Earth Rev}}$ as $$\begin{aligned} \hat{\mathbf{q}} \cdot{\bf V}_{\rm {Earth Rev}}&=V_{\oplus}(\lambda(t)) \bigg\{\big[-\cos(t^\circ_{\rm lab}) \sin(\lambda_{\rm lab}) {\cal A}(t) -\sin(t^\circ_{\rm lab}) \sin(\lambda_{\rm lab}) {\cal B}(t) + \cos(\lambda_{\rm lab}) {\cal C}(t) \big]\nonumber\\ &\left(\alpha_X q_X+\alpha_Y q_Y+\alpha_Z q_Z\right) + \big[\sin(t^\circ_{\rm lab}) {\cal A}(t) -\cos(t^\circ_{\rm lab}) {\cal B}(t) \big]\left(\beta_X q_X+\beta_Y q_Y+\beta_Z q_Z\right)\nonumber\\ &+\big[\cos(t^\circ_{\rm lab}) \cos(\lambda_{\rm lab}) {\cal A}(t) + \sin(t^\circ_{\rm lab}) \cos(\lambda_{\rm lab}) {\cal B}(t) +\sin(\lambda_{\rm lab}) {\cal C}(t) \big] \left(\gamma_X q_X+\gamma_Y q_Y+\gamma_Z q_Z\right)\bigg\}, \label{qdotEarthRev}\end{aligned}$$ where $$\begin{aligned} {\cal A}(t)&=(-0.06699) \cos\beta(x) \sin (\lambda(t)-\lambda_x) + (0.4927)\cos\beta(y) \sin (\lambda(t)-\lambda_y)\nonumber\\ &+ (-0.8676)\cos\beta(z) \sin (\lambda(t)-\lambda_z),\nonumber\\ {\cal B}(t)&= (-0.8728)\cos\beta(x) \sin (\lambda(t)-\lambda_x) + (-0.4503) \cos\beta(y) \sin (\lambda(t)-\lambda_y)\nonumber\\ &+ (-0.1883) \cos\beta(z) \sin (\lambda(t)-\lambda_z),\nonumber\\ {\cal C}(t)&= (-0.4835) \cos\beta(x) \sin (\lambda(t)-\lambda_x) + (0.7446) \cos\beta(y) \sin (\lambda(t)-\lambda_y)\nonumber\\ &+ (0.4602) \cos\beta(z) \sin (\lambda(t)-\lambda_z).\end{aligned}$$ Eq. \[qdotEarthRev\] has a time dependence through $t^\circ_{\rm lab}$ and $\lambda(t)$ and would be responsible for any daily modulation in the rate. Earth’s rotation ---------------- Finally, we want to compute ${\bf V}_{\rm {Earth Rot}}$, the velocity of Earth’s rotation around itself. We have $${\bf V}_{\rm {Earth Rot}}=-V_{\rm {RotEq}} \cos \lambda_{\rm lab} \hat{\cal W},$$ where $V_{\rm {RotEq}}$ is the Earth’s rotation speed at the equator, and is defined as $V_{\rm {RotEq}}=2 \pi R_{\oplus}/({\rm {1~ sidereal~ day}})$. The Earth’s equatorial radius is $R_{\oplus}=6378.137$ km, and one sidereal day is 23.9344696 hr$=86164$ s. therefore $V_{\rm {RotEq}}=0.465102$ km/s. Using Eq. \[Lab-Crystal\] to write $ \hat{\cal W}$ in terms of the crystal frame coordinates, we can easily find $\hat{\mathbf{q}} \cdot {\bf V}_{\rm {Earth Rot}}$ as $$\hat{\mathbf{q}} \cdot{\bf V}_{\rm {Earth Rot}}=-V_{\rm {RotEq}} \cos \lambda_{\rm lab} \left(\beta_X q_X+\beta_Y q_Y+\beta_Z q_Z\right). \label{qdotEarthRot}$$ There is no time dependence in Eq. \[qdotEarthRot\], because it is written in the crystal frame, and both the lab and the crystal are rotating with the Earth. Total Velocity -------------- Now we can insert Eqs. \[qdotGalacticRot\], \[qdotSolar\], \[qdotEarthRev\] and \[qdotEarthRot\] into Eq. \[qdotV\] to compute $\hat{\mathbf{q}} \cdot{\bf V}_{{\rm lab}}$. Inserting the values of $V_{\oplus}=29.8$ km/s, $\epsilon=23.439 ^\circ$ and $V_{\rm {RotEq}}=0.465$ km/s, we have (in km/s): $$\begin{aligned} \hat{\mathbf{q}} \cdot{\bf V}_{{\rm lab}}&=\bigg\{\bigg[-\cos(t^\circ_{\rm lab})~A(t)+\sin(t^\circ_{\rm lab})~B(t)\bigg]\sin\lambda_{\rm lab} +C(t)~\cos\lambda_{\rm lab}\bigg\}\left(\alpha_X q_X+\alpha_Y q_Y+\alpha_Z q_Z\right)\nonumber\\ &+\bigg\{\sin(t^\circ_{\rm lab})~A(t)+\cos(t^\circ_{\rm lab})~B(t)-0.465 \cos \lambda_{\rm lab}\bigg\}\left(\beta_X q_X+\beta_Y q_Y+\beta_Z q_Z\right)\nonumber\\ &+\bigg\{\bigg[\cos(t^\circ_{\rm lab})~A(t)-\sin(t^\circ_{\rm lab})~B(t)\bigg]\cos\lambda_{\rm lab}+C(t)~\sin\lambda_{\rm lab}\bigg\}\left(\gamma_X q_X+\gamma_Y q_Y+\gamma_Z q_Z\right), \label{qdotVlab}\end{aligned}$$ where $$\begin{aligned} A(t)&=0.4927~ V_{\rm {Gal Rot}} - 1.066~\textrm{km/s} + (V_{\oplus}(\lambda(t)) {\cal A}(t),\nonumber\\ B(t)&=0.4503~ V_{\rm {Gal Rot}} + 16.56~\textrm{km/s} - (V_{\oplus}(\lambda(t)) {\cal B}(t),\nonumber\\ C(t)&=0.7445~ V_{\rm {Gal Rot}} + 7.077~\textrm{km/s} + (V_{\oplus}(\lambda(t)) {\cal C}(t).\end{aligned}$$ [99]{} E. M. Drobyshevski, Mod. Phys. Lett. A [23]{} (2008) 3077 \[arXiv:0706.3095 \[physics.ins-det\]\]. R. Bernabei [et al.]{}, Eur. Phys. J. C [53]{}, 205 (2008) \[arXiv:0710.0288 \[astro-ph\]\]. F. T. Avignone, R. J. Creswick and S. Nussinov, arXiv:0807.3758 \[hep-ph\]; R. J. Creswick, S. Nussinov and F. T. Avignone, arXiv:1007.0214v2 \[astro-ph.IM\]. J. Lindhard, Kongel. Dan. Vidensk. Selsk., Mat.-Fys. Medd. [34]{} No. 14 (1965). N. Bozorgnia, G. B. Gelmini and P. Gondolo, JCAP [11]{}, 019 (2010) \[arXiv:1006.3110 \[astro-ph.CO\]\]; N. Bozorgnia, G. B. Gelmini and P. Gondolo, JCAP [11]{}, 028 (2010) \[arXiv:1008.3676 \[astro-ph.CO\]\]; N. Bozorgnia, G. B. Gelmini and P. Gondolo, JCAP [11]{}, 029 (2010) \[arXiv:1009.3325 \[astro-ph.CO\]\]; N. Bozorgnia, G. B. Gelmini and P. Gondolo, arXiv:1011.6006 \[astro-ph.CO\]. K. M. Górski [et al.]{}, ApJ [622]{}, 759 (2005). P. Gondolo, Phys. Rev. D [66]{}, 103513 (2002). M.S. Alenazi and P. Gondolo, Phys. Rev. D [77]{}, 043532 (2008). R. Helm, Phys. Rev. [104]{} 1466 (1956). G. Duda, A. Kemper, and P. Gondolo, JCAP 0704 012, [12]{} (2007). A. M. Green, JCAP [10]{} 034 (2010) \[arXiv:1009.0916v2 \[astro-ph.CO\]\]. M. Kuhlen [et al.]{}, JCAP [02]{} 030 (2010) \[arXiv:0912.2358v1 \[astro-ph.GA\]\]. R. Bernabei [et al.]{}, Eur. Phys. J. C [56]{} (2008) 333 \[arXiv:0804.2741v1 \[astro-ph\]\]. R. Bernabei [et al.]{} \[DAMA Collaboration\], Nucl. Instrum. Meth. A [592]{}, 297 (2008) \[arXiv:0804.2738 \[astro-ph\]\]. D. Hooper, J.I. Collar, J. Hall, D. McKinsey and C.M. Kelso, Phys. Rev. D [82]{} 123509 (2010) \[arXiv:1007.1005v3 \[hep-ph\]\]. C. Savage, G. Gelmini, P. Gondolo and K. Freese, Phys. Rev. D [83]{} 055002 (2011) \[arXiv:1006.0972v2 \[astro-ph.CO\]\]. Z. Ahmed [et al.]{} \[CDMS Collaboration\], Phys. Rev. Lett [106]{} 131302 (2011) \[arXiv:1011.2482 \[astro-ph.CO\]\]. E. Aprile [et al.]{} \[XENON100 Collaboration\], arXiv:1104.2549v2 \[astro-ph.CO\]. S. T. Lin [et al.]{} \[TEXONO Collaboration\], Phys. Rev. D [79]{} 061101 (2009) \[arXiv:0712.1645 \[hep-ex\]\]; C. E. Aalseth [et al.]{} \[CoGeNT Collaboration\], Phys. Rev. Lett. [101]{}, 251301 (2008) \[Erratum-ibid. [102]{}, 109903 (2009)\] \[arXiv:0807.0879 \[astro-ph\]\]. V. I. Tretyak, Astropart. Phys [33]{} 40 (2010) \[arXiv:0911.3041v1 \[nucl-ex\]\]. D. Hooper, J. I. Collar, J. Hall, D. McKinsey and C. Kelso, Phys. Rev. D [82]{} 123509 (2010) \[arXiv:1007.1005v3 \[hep-ph\]\]. M. A. Hapgood, Planet. Space Sci. 40, 711 (1992). U.S. Naval Observatory, [Almanac for Computers]{} 1990. Nautical Almanac Office, U.S. Naval Observatory, Washington, D.C. Peter Duffett-Smith, “Practical Astronomy with Your Calculator”, 3rd ed. Cambridge, England: Cambridge University Press (1992). R. Schoenrich, J. Binney and W. Dehnen, Mon. Not. Roy. Astron. Soc. [403]{} 1829 (2010) \[arXiv:0912.3693\]. K. R. Lang, Astrophysical Formulae, Springer-Verlag, New York (1999). A. M. Green, Phys. Rev D [68]{} 023004 (2003).
--- abstract: 'In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane, by computing the Cartesian equation of the bifurcation curve in terms of the positions of the receivers. From that equation, we can compute its real asymptotic lines. The present manuscript completes the analysis of [@nostro]. Our result is useful to check if a source belongs or is closed to the bifurcation curve, where the localization in a noisy scenario is ambiguous.' address: 'Dipartimento di Matematica, Politecnico di Milano, I-20133 Milano, Italia' author: - 'Marco Compagnoni, Roberto Notari' bibliography: - 'biblio.bib' title: \| TDOA–based localization in two dimensions:\ the bifurcation curve --- Introduction ============ The problem of localizing an object in space is a classical topic in the literature of space–time signal processing. The first studies on the subject date back to World War II, motivating the creation of the two-dimensional LOng RAnge Navigation (LORAN) radio positioning system. LORAN was based on the measurements of the time differences of arrival (TDOA) of synchronized radio signals originated from three distinct known emitters and it required the use of hyperbolic charts to determine the position of the receiver [@Getting1993]. Nowadays, LORAN has been replaced primarily by Global Positioning System (GPS), but the mathematical models underlying LORAN and (three–dimensional) GPS localization are essentially the same [@Siouris1993]. TDOA–based localization of unknown point sources is very widespread and popular also in acoustic signal processing, because it is characterized by a reduced computational cost with respect to other solutions and robustness against noise [@Huang2004]. The main contributions to the study of TDOA–based localization come from the engineering literature, where the authors usually focus on the development of algorithms for locating the source starting from empirical TDOA data, affected by (mainly, Gaussian distributed) noise. Relevant examples are [@Smith1987; @Smith1987a; @Abel1987; @Schau1987; @Huang2000; @Huang2001; @Huang2004; @Huang2004a; @Gillette2008a; @Beck2008]. A classification of the different methods can be done according to the proposed solution: maximum likelihood principle versus least–squares estimators, linear approximation versus numerical optimization, and finally iterative versus closed forms–algorithms (for a resume of the most significant of these methods see [@Bestagini2013]). However, all of them are based on the model of geometric propagation of the signal in an isotropic and homogeneous medium. This means that, given a TDOA measurement between two receivers placed on the Euclidean plane at positions $\mathbf{m}_i$ and $\mathbf{m}_j$, the locus of source locations that are compatible with that measurement is one branch of a hyperbola of foci $\mathbf{m}_i$ and $\mathbf{m}_j$ and whose aperture depends on the range difference (TDOA times the speed of propagation of the signal). Therefore, if we consider multiple measurements, one can readily find the location of the source through the intersection of branches of hyperbole. The three dimensional localization is very similar, being equivalent to the intersection of sheets of hyperboloids. The mathematical properties of the TDOA–based localization have been investigated in several manuscripts. In particular, the closed–form solutions to the intersection problems have been provided for both configurations of three receivers in a plane and four receivers in the space [@Schmidt1972; @Bancroft1985; @Kraus1987; @Abel1991; @Chauffe1994; @Leva1995; @Hoshen1996; @Grafarend2002; @Awange2002; @Coll2009; @Spencer2010; @Coll2012; @nostro]. However, due to the nonlinearity, it is well known that in these minimal configurations of receivers there does not exist a unique admissible position of the source for any given set of TDOA measurements. In fact, there are open regions in the physical space where the intersection set is the union of two points and so the source location is intrinsically ambiguous. This fact is known in literature as the bifurcation problem and we will name bifurcation set the border between the two domains where the localization problem has respectively one or two solutions. In [@Schmidt1972] it is shown that the bifurcation set is a curve in two dimensions and a surface in three dimensions. In [@Hoshen1996], by relating TDOA–based localization and the ancient Problem of Apollonius [@Boyer1989] of drawing a circle touching three other circles or two circles and a point, Hoshen was able to analytically describe the bifurcation sets in two and three dimension in terms of polar and spherical coordinates. More recently, an analysis of the bifurcation problem in a 2–D scenario has been completed through the use of numerical simulations [@Spencer2007]. Source location error analysis in 2–D and 3–D with noisy TDOA data has also recently been considered for both closed form and numerical solutions [@Spencer2010]. A deeper description of the geometric properties of the bifurcation curve in the 2–D case was given lately by the authors in [@nostro]. Although the statistical model describing TDOA–based localization is not defined by polynomial functions, in [@nostro] it has been shown that all the relevant objects are real (semi)–algebraic varieties. In particular, the bifurcation curve is an algebraic curve and more precisely a rational quintic curve, whose explicit rational parametrization was provided. In [@nostro], we were not able to find a Cartesian polynomial equation. In this manuscript, we fill this gap by providing the explicit Cartesian equation of the bifurcation curve in terms of the positions of the receivers. This is an important step towards the study of the statistical model of localization based on TDOAs. Indeed, as observed in [@nostro], locating a source placed around the bifurcation curve is an ill-posed problem and in these situations any localization algorithm has a very low accuracy. On the other hand, checking if a point belongs to, or is close to, a curve is quite a hard problem if one starts from the parametric equation of the curve itself. On the contrary, the two above–mentioned problems can be easily solved from the Cartesian equation of the curve. The paper is organized as follows. In Section 2, we describe the deterministic model of the physical problem, and we recall the main results proved in [@nostro]. Section 3 is devoted to the computation of the Cartesian equation of the bifurcation curve and of its real asymptotic lines in terms of the positions of the receivers. In the last Section, we summarize the results, and we outline a research program for extending our study to both the real scenario, and the cases not yet covered, such as the 3–D case, or the planar case with more than $ 3 $ receivers. State of the art ================ In this section we briefly resume the main results of [@nostro] concerning the 2–D localization with three receivers in a noiseless scenario. In this setting, the physical world can be identified with the Euclidean plane and, after choosing an orthogonal Cartesian coordinate system, with ${\mathbb{R}}^2$. On the plane, we have three receivers ${{\mathbf}{m_{i}}}=(x_{i},y_{i}),\ i=0,1,2$ at known positions and a source ${\mathbf}{x} = (x,y)$. The corresponding displacement vectors are $${\mathbf}{d_i}({\mathbf}{x})={\mathbf}{x}-{{\mathbf}{m_{i}}},\qquad {\mathbf}{d_{ji}}={{\mathbf}{m_{j}}}-{{\mathbf}{m_{i}}},\qquad i,j=0,1,2,$$ whose norms are $d_i({\mathbf}{x})$ and $ d_{ji}$, respectively. Without loss of generality, we assume the speed of propagation of the signal in the medium to be equal to $1$. In the noiseless scenario we adopt, the TDOA between each pair of different microphones is equal to the difference of the ranges (the so–called pseudorange): $$\label{TDOA} \tau_{ji}({\mathbf}{x})=d_{j}({\mathbf}{x})-d_{i}({\mathbf}{x}),\quad i,j=0,1,2.$$ The three TDOAs are not independent. In fact, the linear relation $ \tau_{12}({{\mathbf}{x}}) = \tau_{10}({{\mathbf}{x}}) - \tau_{20}({{\mathbf}{x}}) $ holds for each $ {\mathbf}{x} \in {\mathbb{R}}^2.$ Hence, we are allowed to choose a microphone as the reference one, say $ {{\mathbf}{m_{0}}},$ and so we consider only the TDOAs involving $ {{\mathbf}{m_{0}}},$ without loss of information. In [@nostro] we collected $ \tau_{10}({{\mathbf}{x}})$ and $\tau_{20}({{\mathbf}{x}})$ by defining the TDOA map: $$\begin{array}{cccc} {\boldsymbol}{\tau_2}: & {\mathbb{R}}^2 & \longrightarrow & {\mathbb{R}}^2\\ & {\mathbf}{x} & \longrightarrow & \quad (\tau_{10}({\mathbf}{x}),\tau_{20}({\mathbf}{x})) \end{array}.$$ The study of the TDOA map is the heart of the mathematical characterization of the localization problem and it is the subject of [@nostro]. In fact, the main problems concerning the localization in the deterministic set–up can be formulated in terms of $ {\boldsymbol{\tau}}_2 $: given $ {\boldsymbol{\tau}}:=(\tau_1,\tau_2)\in{\mathbb{R}}^2,$ there exists a source ${\mathbf}{x}$ such that $ {\boldsymbol{\tau}}_2({\mathbf}{x}) = {\boldsymbol{\tau}}$ if, and only if, $ {\boldsymbol{\tau}}\in \mbox{Im}({\boldsymbol{\tau}}_2).$ Moreover, the uniqueness of such a source ${\mathbf}{x}$ can be equivalently written as $ \vert {\boldsymbol{\tau}}_2^{-1}({\boldsymbol{\tau}}) \vert = 1$. In this paper we assume that ${{\mathbf}{m_{0}}},{{\mathbf}{m_{1}}},{{\mathbf}{m_{2}}}$ are not collinear. The interested reader can find the complete analysis of the aligned configuration in [@nostro]. In Figure \[fig:tauimage\] we draw the image of ${\boldsymbol{\tau}}_2$, with receivers ${{\mathbf}{m_{0}}}=(0,0),\ {{\mathbf}{m_{1}}}=(2,0)$ and ${{\mathbf}{m_{2}}}=(2,2)$. Im$({\boldsymbol}{\tau_2})$ is a subset of a convex polytope $P_2$, the hexagon defined by: $$\left\{ \begin{array}{l} -d_{10} \leq \tau_1 \leq d_{10} \\ -d_{20} \leq \tau_2 \leq d_{20} \\ -d_{21} \leq \tau_2 - \tau_1 \leq d_{21} \end{array} \right. .$$ There exists a unique ellipse $E$ tangent to each facet of $P_2$ (at the six points $T_i^\pm,\ i=0,1,2$), the one defined by $$\label{eq:a} a({\boldsymbol}{\tau}) = \Vert \tau_2 {\mathbf}{d_{10}} - \tau_1 {\mathbf}{d_{20}} \Vert^2 - \Vert {\mathbf}{d_{10}} \wedge {\mathbf}{d_{20}} \Vert^2=0.$$ We name $E^-$ and $E^+$ the interior and the exterior region of the ellipse, respectively, while $ U_0, U_1, U_2$ are the three disjoint connected components of $\mathring{P_2} \setminus (E^-\cup E)$. Using this notation, we have $$\mbox{Im}({\boldsymbol}{\tau_2}) = E^- \cup \bar{U}_0 \cup \bar{U}_1 \cup \bar{U}_2 \setminus \{ T_0^\pm, T_1^\pm, T_2^\pm \}$$ and in particular $$\vert{\boldsymbol}{\tau_2}^{-1}({\boldsymbol{\tau}})\vert= \begin{cases} 2 & \text{if }\ {\boldsymbol{\tau}}\in U_0 \cup U_1 \cup U_2,\\ 1 & \text{if }\ {\boldsymbol{\tau}}\in \mbox{Im}({\boldsymbol}{\tau_2}) \setminus U_0 \cup U_1 \cup U_2. \end{cases}$$ Furthermore, it holds: (a) ${\boldsymbol{\tau}}\in E$ if, and only if, the hyperbola branches $$\label{hyp-br} A_i({\boldsymbol{\tau}}) = \{ {{\mathbf}{x}}\ \vert \ d_i({{\mathbf}{x}}) - d_0({{\mathbf}{x}}) = \tau_i \}, i = 1, 2,$$ have one of the two asymptotic lines parallel each other. This means that there could exist a source at a great distance from the microphones, in comparison to $d_{10}$ and $d_{20}$. If $ {\boldsymbol}{\tau} \in E\cap\mbox{Im}({\boldsymbol}{\tau_2})$ the hyperbola branches meet at a point at finite distance, too, which corresponds to another admissible source position. (b) ${\boldsymbol{\tau}}\in E^-$ if, and only if, the hyperbola branches $ A-1({\boldsymbol{\tau}}) $ and $ A_2({\boldsymbol{\tau}}) $ meet at one simple point and so, for a given ${\boldsymbol{\tau}}$, there exists a unique source position ${{\mathbf}{x}}$. In this case the localization is still possible even in a noisy scenario, but we lose in precision and stability as $ {\boldsymbol{\tau}}$ gets close to $E$. (c) ${\boldsymbol{\tau}}\in U_0 \cup U_1 \cup U_2$ if, and only if, the previous hyperbola branches intersect at two distinct points, which means that for a given ${\boldsymbol{\tau}}$ there are two admissible source positions. The two solutions overlap if ${\boldsymbol{\tau}}\in\partial P_2$, which corresponds to the tangential intersection of the branches. Each preimage ${{\mathbf}{x}}$ of a given $ {\boldsymbol}{\tau} \in \mbox{Im}({\boldsymbol}{\tau_2})$, i.e. the admissible source locations, is given in a very compact form through the formalism of exterior algebra over $ 3$–dimensional Minkowski space–time as: $$\label{eq:inv-image} {\mathbf}{x}({\boldsymbol}{\tau}) = {\mathbf}{L_0}({\boldsymbol}{\tau}) + \lambda({\boldsymbol{\tau}}) \ast((\tau_2 {\mathbf}{d_{10}} - \tau_1 {\mathbf}{d_{20}})\wedge {\mathbf}{e_3}),$$ where $ \lambda $ is one of the real negative solutions of a certain quadratic equation (see [@nostro] for the full details). Roughly speaking, in the physical plane there are two different regions: the preimage of the interior of the ellipse $E^-$, where the TDOA map is $1$–to–$1$ and the source localization is possible, and the preimage of the three disjoint regions $U_i,\ i=0,1,2$, where the map is $2$–to–$1$ and there is no way to uniquely locate the source. By definition, the region of transition is exactly the bifurcation curve $\tilde{E}$, that can be characterized as the inverse image of the ellipse $E$. In the remaining part of this section we recall the main results on the behavior of ${\boldsymbol}{\tau_2}$ in the physical plane. In Figure \[fig:x-plane\] we give two examples of bifurcation curve and the relative sets. (a) As we said above, $\tilde{E} ={\boldsymbol}{\tau_2}^{-1}(E)$. If ${\boldsymbol{\tau}}\in E$, $\lambda({\boldsymbol{\tau}})$ is a rational function and becomes a rational parametrization of $\tilde{E}$. In [@nostro] it has been proved that $\tilde{E}$ is a rational algebraic curve of degree $5$, singular on the complex plane but smooth on the real one. (b) The real part of $ \tilde{E} $ consists of three disjoint and unbounded arcs, one for each arc of $ E $ contained in $ \mbox{Im}({\boldsymbol}{\tau_2}).$ In particular, when ${\boldsymbol{\tau}}$ gets close to one point among the $T_i^\pm$’s in $ E \cap C $, the point ${{\mathbf}{x}}$ on $ \tilde{E} $ goes to infinity. This fact and the invariance of under the reflection ${\boldsymbol{\tau}}\rightarrow -{\boldsymbol{\tau}}$ imply that the quintic $\tilde{E}$ has three real ideal points (the ones of the lines $ r_0, r_1, r_2,$), while the remaining two ones are the (complex) preimages of the (complex) ideal points of $ E.$ Finally, the points $ {{\mathbf}{m_{0}}}, {{\mathbf}{m_{1}}}, {{\mathbf}{m_{2}}} $ do not belong to $\tilde{E},$ because their images via $ {\boldsymbol}{\tau_2} $ are not on $ E.$ (c) Let $ \tilde{U}_i $ be the inverse image of $U_i $ via $ {\boldsymbol}{\tau_2},$ for $ i=0,1,2,$ and $ \tilde{E}^- $ be the inverse image of $ E^-.$ Then, $\tilde{E}^-, \tilde{U}_0, \tilde{U}_1, \tilde{U}_2 $ are open subsets of the $ x$–plane, which are separated by the three arcs of $ \tilde{E} $. On $\tilde{E}^-$ the TDOA map is $1$–to–$1$, while it is $2$–to–$1$ on each $\tilde{U}_i, \ i=0,1,2$. Moreover, the dashed half–lines in Figure \[fig:x-plane\] outgoing from the receivers divide each $ \tilde{U}_i $ into two connected components and $ {\boldsymbol}{\tau_2} $ is $ 1$–to–$1 $ on each of them. (d) The source localization is possible if ${\boldsymbol{\tau}}\in E^-$ and consequently ${{\mathbf}{x}}\in \tilde E^-$. Otherwise, assume $ {\boldsymbol}{\tau} \in U_i$, then there are two admissible sources in the two disjoint components of $\tilde{U}_i$. As ${\boldsymbol}{\tau}$ gets close to $E$, then one of its inverse images gets close to a point on $\tilde{E}$, while the other one goes to infinity. Conversely, if $ {\boldsymbol}{\tau} $ gets close to $\partial P_2$, the inverse images of ${\boldsymbol}{\tau}$ come close to each other and get close to a point on one of the dashed half–lines. As we said before, in a realistic noisy scenario, we have poor localization in the region close to $\tilde{E}$. The algebraic equation of $\tilde{E}$ and its asymptotic lines ============================================================== In this section, we use the formalism of exterior algebra over the Euclidean plane, and we refer to Appendix A of [@nostro]) for a brief introduction and summary on the subject and for the notation. However, from the general results, it follows that $$\ast (\bold{u} \wedge \bold{v} ) = \det \left( \begin{array}{cc} u_1 & v_1 \\ u_2 & v_2 \end{array} \right)$$ where $ \bold{u} = u_1 \bold{e}_1 + u_2 \bold{e}_2, \bold{v} = v_1 \bold{e}_1 + v_2 \bold{e}_2 $ and $ (\bold{e}_1, \bold{e}_2 ) $ is an orthonormal basis of $ {\mathbb{R}}^2.$ \[def:1\] Let us define: (a) $ {\mathbf}{D_0}({{\mathbf}{x}})=d_0({{\mathbf}{x}}) {\mathbf}{d_{12}},\quad {\mathbf}{D_1}({{\mathbf}{x}})=d_1({{\mathbf}{x}}) {\mathbf}{d_{20}},\quad {\mathbf}{D_2}({{\mathbf}{x}})=d_2({{\mathbf}{x}}) {\mathbf}{d_{01}}; $ (b) $D_i({{\mathbf}{x}})=\Vert{\mathbf}{D_i({{\mathbf}{x}})}\Vert\quad and\quad \displaystyle{p_i({{\mathbf}{x}})=\frac{{\mathbf}{D_i}({{\mathbf}{x}})\cdot {\mathbf}{d_0}({{\mathbf}{x}})}{d_i({{\mathbf}{x}})}},\ i=0,1,2;$ (c) $W=\ast({\mathbf}{d_{10}} \wedge {\mathbf}{d_{20}})$; (d) $ Q({{\mathbf}{x}})=D_0({{\mathbf}{x}})^2+D_1({{\mathbf}{x}})^2+D_2({{\mathbf}{x}})^2-W^2; $ (e) $ P_{ij}({{\mathbf}{x}})={\mathbf}{D_i}({{\mathbf}{x}})\!\cdot\!{\mathbf}{D_j}({{\mathbf}{x}})\quad and\quad \displaystyle{p_{ij}=\frac{P_{ij}({{\mathbf}{x}})}{d_i({{\mathbf}{x}})d_j({{\mathbf}{x}})}},\quad i,j=0,1,2. $ An algebraic equation for the quintic curve $\tilde{E}$ is $F({{\mathbf}{x}})=0$, where: $$\label{eq:CartBif7} \begin{array}{l} F({{\mathbf}{x}})=Q({{\mathbf}{x}})^4-8\,Q({{\mathbf}{x}})^2(P_{01}({{\mathbf}{x}})^2+P_{12}({{\mathbf}{x}})^2+P_{20}({{\mathbf}{x}})^2)\,+\\ 64\,Q({{\mathbf}{x}})P_{01}({{\mathbf}{x}})P_{12}({{\mathbf}{x}})P_{20}({{\mathbf}{x}})+16\,(P_{01}({{\mathbf}{x}})^4+P_{12}({{\mathbf}{x}})^4+P_{20}({{\mathbf}{x}})^4)\\ -32\,(P_{01}({{\mathbf}{x}})^2P_{12}({{\mathbf}{x}})^2+P_{12}({{\mathbf}{x}})^2P_{20}({{\mathbf}{x}})^2+P_{20}({{\mathbf}{x}})^2P_{01}({{\mathbf}{x}})^2). \end{array}$$ The polynomial $F({{\mathbf}{x}})$ is invariant under permutation of the points ${{\mathbf}{m_{0}}},$ ${{\mathbf}{m_{1}}},{{\mathbf}{m_{2}}}$. Expanding with respect to $d_0({{\mathbf}{x}})$, we have: $$\label{eq:CartBif8} \begin{array}{ll} F({{\mathbf}{x}})\!=\!\!\!\!&((W^2+2(d_{20}^2\,p_2({{\mathbf}{x}})-d_{01}^2\,p_1({{\mathbf}{x}})))^2+ 16\,p_{12}^2\,p_1({{\mathbf}{x}})\,p_2({{\mathbf}{x}}))^2+\\ &8\,d_0({{\mathbf}{x}})^2(-W^4\,p_{12}\,(-8\, d_{01}^2 d_{20}^2+p_{01}\,p_{20}+2W^2)+\\ &2W^2((W^2+2\,p_{12}^2)(W^2+3\,p_{12}^2)(p_2({{\mathbf}{x}})-p_1({{\mathbf}{x}}))+\\ &3\,p_{12}\,(2\,d_{01}^2\,d_{20}^2+W^2)(d_{20}^2\,p_2({{\mathbf}{x}})-d_{01}^2\,p_1({{\mathbf}{x}}))+\\ &2\,p_{12}^2\,(p_{01}\,d_{20}^2\,p_2({{\mathbf}{x}})-p_{20}\,d_{01}^2\,p_1({{\mathbf}{x}})))+\\ &4(d_{20}^2\,p_2({{\mathbf}{x}})-d_{01}^2\,p_1({{\mathbf}{x}}))^2 (2(d_{01}^2+d_{20}^2)W^2+7\,p_{01}\,p_{12}\,p_{20})-\\ &8(d_{20}^2\,p_2({{\mathbf}{x}})-d_{01}^2\,p_1({{\mathbf}{x}}))(W^2p_{12}^2\,(p_2({{\mathbf}{x}})-p_1({{\mathbf}{x}}))+\\ &(W^2+4p_{12}^2)(p_{01}^2\,p_2({{\mathbf}{x}})-p_{20}^2\,p_1({{\mathbf}{x}})))-\\ &16\,p_{12}\,p_1({{\mathbf}{x}})p_2({{\mathbf}{x}})(2W^2d_{01}^2d_{20}^2+p_{12}^2\,p_{20}\,p_{01})+\\ &2\,d_0({{\mathbf}{x}})^2(p_{12}(d_{01}^2+d_{20}^2)(4\,d_{01}^2d_{20}^2W^2+\\ &p_{12}^2(2\,p_{20}\,p_{01}-p_{12}(d_{01}^2+d_{20}^2))) +d_{01}^2d_{20}^2(4d_{01}^4d_{20}^4-7W^4))-\\ &8Wd_{01}^2d_{12}^2d_{20}^2\!\ast\!\!({\mathbf}{d_{01}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})) \!\ast\!\!({\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})) \!\ast\!\!({\mathbf}{d_{20}}\wedge{\mathbf}{d_0}({{\mathbf}{x}}))). \end{array}$$ The bifurcation curve $\tilde{E}$ is the preimage of the ellipse $E$, therefore we obtain a Cartesian equation of $\tilde{E}$ by substituting $(\tau_1,\tau_2)=(\tau_{10}({{\mathbf}{x}}),\tau_{20}({{\mathbf}{x}}))$ in : $$\label{eq:CartBif} \Vert (d_2({{\mathbf}{x}})-d_0({{\mathbf}{x}})) {\mathbf}{d_{10}} - (d_1({{\mathbf}{x}})-d_0({{\mathbf}{x}})) {\mathbf}{d_{20}} \Vert^2 = (\ast ({\mathbf}{d_{10}} \wedge {\mathbf}{d_{20}}) )^2.$$ Using Definitions \[def:1\], we have the more symmetric form $$\label{eq:CartBif2} \Vert {\mathbf}{D_0}({{\mathbf}{x}}) + {\mathbf}{D_1}({{\mathbf}{x}}) + {\mathbf}{D_2}({{\mathbf}{x}}) \Vert^2 = W^2$$ and, after expanding the left hand side, $$\label{eq:CartBif3} \begin{array}{l} D_0({{\mathbf}{x}})^2+D_1({{\mathbf}{x}})^2+D_2({{\mathbf}{x}})^2+\\ 2\,({\mathbf}{D_0}({{\mathbf}{x}})\!\cdot\!{\mathbf}{D_1}({{\mathbf}{x}})+ {\mathbf}{D_1}({{\mathbf}{x}})\!\cdot\!{\mathbf}{D_2}({{\mathbf}{x}})+{\mathbf}{D_2}({{\mathbf}{x}})\!\cdot\!{\mathbf}{D_0}({{\mathbf}{x}}))=W^2. \end{array}$$ Of course, this is not an algebraic equation with respect to $x,y$. In order to obtain one, we use again Definitions \[def:1\] and we rewrite equation as $$\label{eq:CartBif4} Q({{\mathbf}{x}})+2P_{12}({{\mathbf}{x}})=-2\,(P_{01}({{\mathbf}{x}})+P_{20}({{\mathbf}{x}})).$$ By squaring both sides and reordering, we obtain: $$\label{eq:CartBif5} \begin{array}{l} Q({{\mathbf}{x}})^2-4(P_{01}({{\mathbf}{x}})^2-P_{12}({{\mathbf}{x}})^2+P_{20}({{\mathbf}{x}})^2)=\\ -4\,Q({{\mathbf}{x}})\,P_{12}({{\mathbf}{x}})+8\,P_{01}({{\mathbf}{x}})\,P_{20}({{\mathbf}{x}}) \end{array}$$ Again, the right side of the last equation is not a polynomial, but squaring once we get the algebraic equation: $$\label{eq:CartBif6} \begin{array}{l} (Q({{\mathbf}{x}})^2-4(P_{01}({{\mathbf}{x}})^2-P_{12}({{\mathbf}{x}})^2+P_{20}({{\mathbf}{x}})^2))^2= 16\,Q({{\mathbf}{x}})^2\,P_{12}({{\mathbf}{x}})^2\\ -64\,Q({{\mathbf}{x}})\,P_{01}({{\mathbf}{x}})\,P_{12}({{\mathbf}{x}})\,P_{20}({{\mathbf}{x}})+64\,P_{01}({{\mathbf}{x}})^2\,P_{20}({{\mathbf}{x}})^2, \end{array}$$ that coincides with equation $F({{\mathbf}{x}})=0$. It is straightforward to verify that is invariant with respect to permutations of the points ${{\mathbf}{m_{0}}},{{\mathbf}{m_{1}}},{{\mathbf}{m_{2}}}$. The degree of polynomial with respect to $(x,y)$ is 8 at the most. By calculating the Taylor expansion of centered at the point ${{\mathbf}{m_{0}}}$, we show that $F({{\mathbf}{x}})$ has degree $5$ and this completes the proof (the verification of expansion is a simple matter of computation). If the receivers are not collinear, we have that $F({{\mathbf}{m_{0}}})=F({{\mathbf}{m_{1}}})=F({{\mathbf}{m_{2}}})=W^8>0$. Therefore ${{\mathbf}{x}}\in \tilde{E}^-$ if, and only if, $F({{\mathbf}{x}})<0$ and ${{\mathbf}{x}}\in \tilde{E}^+$ if, and only if, $F({{\mathbf}{x}})>0$. As an example, we provide the Cartesian equations of the two bifurcation curves of Fig. \[fig:x-plane\]. The bifurcation curve on the left has equation $$\begin{split} \tilde{E}: & -4x^4y+4x^3y^2-4x^2y^3+4xy^4+2x^4+20x^3y-16x^2y^2+4xy^3 + \\ & -6y^4-10x^3-38x^2y+30xy^2+2y^3+18x^2+28xy-22y^2+ \\ & - 12x-4y+1=0 \end{split}$$ while the one of the right has equation $$\begin{split} \tilde{E}: & -20x^4y-60x^3y^2-60x^2y^3-60xy^4-40y^5+10x^4+68x^3y+ \\ & +80x^2y^2+84xy^3+82y^4-34x^3-58x^2y-10xy^2-50y^3+30x^2 + \\ & +4xy+22y^2-4x-4y+1=0. \end{split}$$ Using polynomial we can compute an algebraic expression for the real asymptotic lines of $\tilde{E}$. We refer to Appendix B of [@nostro] for an introduction to projective geometry. As a preliminary, we prove the following Lemma. \[Lemma\] $\displaystyle{W=\frac{\ast({\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}}))+ \ast({\mathbf}{d_{20}}\wedge{\mathbf}{d_1}({{\mathbf}{x}}))+ \ast({\mathbf}{d_{01}}\wedge{\mathbf}{d_2}({{\mathbf}{x}}))}{2}}$. We use the following identities $${\mathbf}{d_{01}}+{\mathbf}{d_{12}}+{\mathbf}{d_{20}}=0,\qquad {\mathbf}{d_{ij}}={\mathbf}{d_j}({{\mathbf}{x}})-{\mathbf}{d_i}({{\mathbf}{x}}),\qquad i,j=0,1,2.$$ We have $$\begin{array}{ll} {\mathbf}{d_{10}}\wedge{\mathbf}{d_{20}} &=({\mathbf}{d_{0}}({{\mathbf}{x}})-{\mathbf}{d_{1}}({{\mathbf}{x}}))\wedge{\mathbf}{d_{20}}=\\ &={\mathbf}{d_0}({{\mathbf}{x}})\wedge({\mathbf}{d_{21}}+{\mathbf}{d_{10}})-{\mathbf}{d_1}({{\mathbf}{x}})\wedge{\mathbf}{d_{20}}=\\ &={\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})+{\mathbf}{d_{20}}\wedge{\mathbf}{d_1}({{\mathbf}{x}}) +({\mathbf}{d_2}({{\mathbf}{x}})+{\mathbf}{d_{20}})\wedge{\mathbf}{d_{10}}=\\ &={\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})+{\mathbf}{d_{20}}\wedge{\mathbf}{d_1}({{\mathbf}{x}}) +{\mathbf}{d_{01}}\wedge{\mathbf}{d_2}({{\mathbf}{x}})-{\mathbf}{d_{10}}\wedge{\mathbf}{d_{20}}. \end{array}$$ The Lemma follows from Definition \[def:1\] and the last identity. Let $ {\mathbb{A}}^2 $ be the affine plane, and let $ {\mathbb{P}}^2 $ be the projective plane obtained by joining the ideal line $ \ell $ to $ {\mathbb{A}}^2.$ Let $C$ be an algebraic curve in the affine plane ${\mathbb{A}}^2$. The ideal points of $C$ are the intersection points of $C$ and the ideal line $ \ell.$ An asymptotic line of $C$ is a line in $ {\mathbb{A}}^2 $ tangent to $C$ at one of its smooth ideal points. Let $ f(x,y) = 0 $ be the Cartesian equation of a degree $ d $ algebraic curve $ C $ in $ {\mathbb{A}}^2.$ We can write it as $$f(x,y) = f_d(x,y) + f_{d-1}(x,y) + \dots + f_1(x,y) + f_0$$ where $ f_i(x,y) $ is homogeneous of degree $ i.$ We embed $ {\mathbb{A}}^2 $ into $ {\mathbb{P}}^2 $ by setting $ x = X/U, y = Y/U,$ and $ \ell: U=0 $ is the ideal line. The curve $ C \subset {\mathbb{P}}^2 $ is then defined by $$F(X,Y,U)= f_d(X,Y) + f_{d-1}(X,Y) U + \dots + f_1(X,Y) U^{d-1} + f_0 U^d.$$ The ideal points are the solutions, in the sense of projective geometry, of $ U = f_d(X,Y) = 0.$ Let $ [a:b:0] $ be a smooth ideal point of $ C.$ So, the line $ r: -bx+ay+c=0 $ is an asymptotic line for $ C $ if $ [a:b:0] $ is a solution of $ F(X,Y,U) = -bX+aY+cU = 0 $ of multiplicity at least two. The bifurcation curve $\tilde{E}$ has three real ideal points $[x_1-x_2:y_1-y_2:0],[x_2-x_0:y_2-y_0:0],[x_0-x_1:y_0-y_1:0]$ and two complex ones $[1:i:0],[1:-i:0].$\ The three real asymptotic lines of $\tilde{E}$ have Cartesian equation $$\label{eq:asintoti}\begin{array}{l} L_0:\ 4\ast({\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}}))-3W=0,\\ L_1:\ 4\ast({\mathbf}{d_{20}}\wedge{\mathbf}{d_1}({{\mathbf}{x}}))-3W=0,\\ L_2:\ 4\ast({\mathbf}{d_{01}}\wedge{\mathbf}{d_2}({{\mathbf}{x}}))-3W=0. \end{array}$$ Finally, we have $L_0\cap L_1\cap L_2=\emptyset.$ The homogeneous degree–$5$ part of polynomial is $$\label{eq:CartBif85} -64Wd_{01}^2d_{12}^2d_{20}^2\,d_0({{\mathbf}{x}})^2 \ast({\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})) \ast({\mathbf}{d_{20}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})) \ast({\mathbf}{d_{01}}\wedge{\mathbf}{d_0}({{\mathbf}{x}}))$$ and so it is straightforward to check that the five ideal points of the statement are its roots, and they all are smooth for $ \tilde{E}.$ By using a package for algebraic computations, it is easy to prove that $ L_i $ meets $ C $ at the corresponding ideal point with multiplicity $ 2,$ and this proves that the line $ L_i $ is asymptotic to $ C.$ The last statement follows from Lemma \[Lemma\]. In fact, if we sum the three polynomials defining the asymptotic lines, we obtain $$\begin{array}{l} 4\ast({\mathbf}{d_{12}}\wedge{\mathbf}{d_0}({{\mathbf}{x}})+{\mathbf}{d_{20}}\wedge{\mathbf}{d_1}({{\mathbf}{x}})+{\mathbf}{d_{01}}\wedge{\mathbf}{d_2}({{\mathbf}{x}}))-9W=-W. \end{array}$$ Thus, the three lines do not have a common intersection point: in fact, if there exists a common point $ {{\mathbf}{x}}_0,$ its coordinates satisfy also the sum of the three equations of the asymptotic lines. Such a sum is $ W = 0,$ and so $ {{\mathbf}{x}}_0 $ does not exist. As $ \tilde{E} $ is a quintic curve, each line $ L_i $ intersects $ \tilde{E} $ at either $ 1 $ or $ 3 $ real points. Hence, also if it is not evident from Fig. \[fig:x-plane\], the unbounded arcs of $ \tilde{E} $ definitely belong to different half–planes with respect to its asymptotic lines. Conclusion ========== In this paper, we recall the state of the art on the localization of sources in a plane from the TDOAs for the case of $ 3 $ receivers in the same plane. Then, we focus on a problem still open in the literature: the computation of the Cartesian algebraic equation of the bifurcation curve $ \tilde{E},$ that is to say, of the curve in the plane of source and receivers whose points are sources for which the hyperbola branches (\[hyp-br\]) have an asymptotic line parallel each other. The knowledge of such an equation allows us to easily solve the problem of finding points in the plane which are close or belong to such a curve $ \tilde{E}.$ The importance of computing the equation of $ \tilde{E} $ stems from the fact that at its points, every localization algorithm has a poor accuracy. The computation of the Cartesian equation of the bifurcation curve rests on two steps: first, by squaring a non–polynomial equation for $ \tilde{E},$ we compute a polynomial; then, by using a Taylor expansion, we get the explicit degree five equation. From such equation, it is possible to compute the real asymptotic lines of $ \tilde{E}.$ Notice that it is not possible to compute such lines from the parametric equation of $ \tilde{E}.$ In [@nostro] as well as in the present paper, we completed the geometric study of the noiseless model of localization, encoded in the TDOA map $ {\boldsymbol{\tau}}_2,$ for the case of $ 3 $ receivers in a plane. In a manuscript in preparation, we will conduct a similar study for a real scenario. Other then the deterministic case, the needed techniques come from information geometry [@Amari2000] and statistics with algebraic tools [@Draisma2013; @Kobayashi2013], and from numerical analysis. Moreover, still in preparation, we are studying the geometry of the noiseless model in the case of $ 4 $ or more receivers in a plane.

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.